^+$#>Theory  Overview

The aim of this section is to provide the user with detailed notes on some of the engine simulation input options and to describe some of the theory behind the structure and operation of the simulation program. As far as is possible the variable names used in the description of the Data Module have been maintained.

The theory notes are arranged under the following sub-headings:

"         Pipes
"         Governing Equations of Gas Flow
"         Numerical Method
"         Pipe Wall Friction
"         Pipe Wall Heat Transfer
"         Pipe Bends
"         Tapered pipes
"         Junctions
"         Cylinders and Plenums
"         Gas Properties
"         Fuel Properties
"         Fuel / Combustion Systems
"         ^ACombustion Models
"         In-Cylinder Heat Transfer
"         Cylinder Scavenging
"         Plenum Heat Transfer
"         Ports
"         Valves
"         Turbochargers
"         Superchargers
"         Charge Coolers
"         Mechanical Links
"e DynamicsTHEORY_ENG_DYNAM
"         Friction

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^+$#>Theory - Pipes

Pipes are one-dimensional elements, the properties of which vary as a function of space and time. The modelling of these elements is perhaps the most crucial aspect of ensuring simulation robustness and accuracy. It is essential for the simulation engineer to understand the limitations and assumptions of the models which are being applied therefore an extensive description of the pipe governing equations and solution technique is presented here.

The pipe theory section is split into the following sections:

"         Governing Equations of Gas Flow
"         Numerical Method
"         Pipe Wall Friction
"         Pipe Wall Heat Transfer
"         Pipe Bends
"         Tapered pipes

Each of these sections provides basic theory related to its particular topic. For more detailed information the user should consult Refs. 1-4.

A pipe is defined by specifying its diameter at various points along its length and some information about its wall properties. In this way, complex pipe shapes can be defined (see the Pipe Data section).

Note: It is important to note that the governing equations of one-dimensional flow are valid only when the fluid adheres to the walls of the duct considered. When separation occurs over extended sections of the duct the one-dimensional assumption is invalid. Separation will occur in pipes with severe increases in area in the downstream direction, or at any geometrical discontinuity. In these cases boundary models, such as sudden enlargements or contractions, should be used to mimic the flow behaviour.


References:

1. Winterbone, D.E. and Pearson, R.J., Design techniques for engine manifolds. Wave action methods for I.C. engines. Professional Engineering Publications, 1999 (ISBN 1-86058-179 X).

2. Winterbone, D.E. and Pearson, R.J., Theory of Engine Manifold Design. Wave action methods for I.C. engines. Professional Engineering Publications, 2000 (ISBN 1-86058-209 5).

3. Benson, R.S., The thermodynamics and gas dynamics of internal combustion engines (Volume 1), Clarendon Press, 1982. (ISBN 0-19-856210-1)

4. Horlock, J.H. and Winterbone, D.E., The thermodynamics and gas dynamics of internal combustion engines (Volume 2), Clarendon Press, 19862. (ISBN 0-19-856212-8)

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^+$#>Theory  Pipes: Governing Equations of Gas Flow

If the pressure wave phenomena which exist in engine manifolds, and have a strong influence on the engine performance, are to modelled then pipe models must include at least one spatial dimension. In fact, since waves in pipes rapidly become plane after encountering geometrical discontinuities, one-dimensional model of pipe gas dynamics, provide a good compromise between solution accuracy and computer run-time.

The conditions within pipe elements are calculated at each time step (calculation crankangle) by solving a set of conservation equations for mass, momentum and energy. The following section describes how these equations are derived. By following this process the user can gain an understanding of the limitations of the pipe model and this provides a sound foundation to the successful modelling of manifold systems as equivalent one-dimensional pipe networks. Further information on gas flows in the manifolds of internal combustion engines can be found in Refs. 1 and 2.

Conservation Laws
The fundamental equations of fluid mechanics are mathematical statements that define the conservation of mass, momentum, and energy for a control volume. A conservation law asserts that the rate of change of a conserved flow property in a fixed volume is the result of the net effect of the flux of the property across the boundary of the volume and the change in that property due to internal sources. Consider the flow of a compressible fluid through an infinitesimal section of pipe in which the area of the cross-section perpendicular to the axis of the pipe varies, as shown in Fig. 1. If the area variation is gradual the fluid properties are approximately uniform across any cross-section and can be taken as functions of x and t only - the flow is then said to be quasi-one-dimensional.

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Fig. 1. Fluid control volume in duct.

The Continuity Equation
Conservation of mass dictates that its rate of change within the control volume shown in Fig. 1 is equal to the net mass flow rate through the element. If the length of the duct element is dx and its cross-sectional area is F then the rate of change of mass within the control volume is . The term represents the gradient of the mass flux and the product of this quantity with the length dx gives the net mass flow across the element. Thus the continuity equation can be expressed as

                           .                                         (1)

The Momentum Equation
The momentum equation embodies the requirement that the sum of the pressure forces and the shear forces acting on the surface of the control volume is equal to the sum of the rate of change of momentum within the control volume and the net efflux of momentum out of the control volume. The resultant force on the control volume is caused by the pressure difference between the end faces and the component, in the x-direction, of the pressure on the sides of the volume. The difference in the pressure forces across the end faces of the control volume is given by the product of the gradient of the force with the length of the element,
                           ,                                                           (2)
and the pressure on the sides of the control volume produces a force in the x-direction of
                           .                                                           (3)
Note that the presence of the minus sign in the term (2) arises from the convention that the forces are regarded as positive in the x-direction. For flows in engine manifolds the pipe walls can be assumed to be non-distensible so that the pipe area is a function of x alone.

The shear forces on the control volume arise due to the friction between the moving fluid and the stationary duct walls and can be modelled simply as a shear stress, w , opposing the fluid motion, as shown in Fig. 1. For the infinitesimal control volume shown the surface force is given by
                           ,                                                           (4)
where D is an equivalent, or hydraulic, diameter of the duct. Expressing the shear stress in terms of the pipe wall friction coefficient, f, as
                           ,                                                          (5)
enables the surface force on the control volume to be represented as
                           .                                                  (6)
In one-dimensional models of the gas dynamic processes in engine manifolds the inclusion of this term is usually the only concession to recognizing the presence of fluid viscosity; the character of the governing equations remains essentially inviscid.

The rate of change of momentum within the control volume is given by
                                                                                       (7)
and the net efflux of momentum from the control surface is
                           .                                                          (8)

Hence the momentum equation is given by
         .             (9)


The Energy Equation
The energy equation can be derived by applying the first law of thermodynamics to the control volume shown in Fig. 1, in the form
                           ,                                        (10)
where E0 is the total stagnation internal energy of the control volume and H0 is the total stagnation enthalpy. The first term on the right-hand side of eqn (10) can be written in terms of the specific stagnation internal energy as
                           ,                                                          (11)
where e0 is defined as
                           .                                                          (12)

The second term on the right-hand side of eqn (10) represents the net efflux of stagnation enthalpy across the control surface and is given by
                           ,                                                          (13)
where h0 is the stagnation enthalpy of the gas, which is related to the stagnation internal energy via the equation
                           .                                                          (14)
Radial heat transfer from the gas to the manifold wall, or vice versa, is easily incorporated into the energy equation. If the heat transfer rate per unit mass of gas is denoted as q then the total heat transfer rate from / to the control volume is, using the convention that heat transfer is positive into the control volume,
                           .                                                          (15)

The work done by or on the system, , is zero for gas flow in a pipe element of an engine manifold.

In terms of the above quantities the energy equation takes the form
                           .                               (16)

The governing equations for the one-dimensional flow of a compressible fluid in a pipe with area variation, wall friction, and heat transfer are thus:


Summary

continuity
                           ;                                                 (17)
momentum
                          ;             (18)
energy
                           .                               (19)
These relationships are a set of non-linear hyperbolic partial differential equations.


Equations in Conservation Law Form
Expanding and re-arranging eqns (17) - (19) gives

                           ;                                        (20)

                       ;                      (21)

                           ,                      (22)
where
                           ,                                                 (23)
and the term is used to ensure that the pipe wall friction always opposes the fluid motion.

Equations (20) - (22) can be written in symbolic vector form as

                                                                    (24)
where

, , .                 (25)

When there is no pipe area variation, wall friction, or heat transfer the equations reduce to

                                                  (26)

and are known as the one-dimensional Euler equations. This presentation of the equations is referred to as the Weak Conservation Law Form since the equations can be obtained directly from the integral conservation equations of mass, momentum, and energy applied to the fixed control volume shown in Fig.1.

Equations (17) to (19) can be expressed in a manner which enforces more directly the conservation of the fluid properties when the flow in pipes with area variation is considered. By retaining the pipe cross-sectional area in the differential terms the governing equations become
                                                                    (27)
where

, .                  (28)

The continuity equation is now strictly homogeneous (since it contains no source terms) and if there is no fluid friction or heat transfer the source vector contains only a term representing the extra pressure force arising from the change of area across the control volume. This form of the governing equations is known as the Strong Conservation Law Form.

In Lotus Engine Simulation it is possible to use either the Strong Form. (equation 28) or the Weak Form (equation 25) of the governing equations. The strong form of the governing equations can give benefits in mass conservation in pipes of varying cross-sectional area when numerical methods employing flux limiter functions are used in order to achieve second-order accuracy (as in Lotus Engine Simulation). The weak form of the governing equations can increase the calculation stability in the region of changes in pipe wall gradient.

In tapered pipes in which there is initially no flow or disturbance due to pressure wave excitation it may be found that a small amount of spurious information is generated when using the strong form of the governing equations. This is due to the inclusion of the pipe cross-sectional area term in the solution vector, W, in equation (28). The flux limiter function is calculated using differences in the value of the solution vector along the pipe. For quiescent conditions the density and stagnation internal energy along the pipe do not vary, and the velocity is zero, but because the cross-sectional area, F, varies down the pipe the solution vector varies, and therefore the flux limiter will modify the solution very slightly. The amplitude of this noise is generally very small compared with that of, say, the pressure waves once they reach the pipes concerned but if the user wishes to eliminate it the weak form of the governing equations should be used.

The Source Term Splitting option provides an approach where the flux limiter function is applied only to the homogeneous form of the governing equations (weak conservation law form without source terms) so that new extrema introduced by the source term are not directly limited.

It is important to note that the governing equations of one-dimensional flow are valid only when the fluid adheres to the walls of the duct considered. When separation occurs over extended sections of the duct the one-dimensional assumption is invalid. Separation will occur in pipes with severe increases in area in the downstream direction, or at any geometrical discontinuity. In these cases boundary models, such as sudden enlargements or contractions, should be used to mimic the flow behaviour.

References:

1. Winterbone, D.E. and Pearson, R.J., Design techniques for engine manifolds. Wave action methods for I.C. engines. Professional Engineering Publications, 1999 (ISBN 1-86058-179 X).

2. Winterbone, D.E. and Pearson, R.J., Theory of Engine Manifold Design. Wave action methods for I.C. engines. Professional Engineering Publications, 2000 (ISBN 1-86058-209 5).

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^+$#>Theory  Pipes: Numerical Method

A shock-capturing finite volume scheme is used to solve the governing equations of gas flow in pipes. A significant amount of background theory is required to present a numerical method in a proper context. Only a brief description of the underlying theory is given here. The interested user should refer to References 1 and 2, at the end of this section, for a full account of numerical methods for gas dynamics in engine manifolds.

The numerical method used in the Lotus Engine Simulation program is based on the two-step Lax-Wendroff scheme, used in conjunction with a symmetric non-linear flux limiter, giving second-order spatial and temporal accuracy. This scheme is a member of the class of shock-capturing finite difference schemes which are capable of handling shock waves and super-sonic flows that can occur in the manifolds of high-performance engines. The flux limiter, which is based on the total variation diminishing (TVD) criterion (TVD) (see later), helps to prevent the occurrence of spurious oscillations in the solution when shock waves and contact discontinuities are encountered.

The Two-Step Lax-Wendroff (Richtmyer) Method
The two-step Lax-Wendroff method is a space-centred scheme based on the computational stencil shown below in Fig. 1.

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Fig. 1. Computational stencil for two-step Lax-Wendroff scheme.

The set of equations used to characterise the flow in engine manifolds can be expressed in symbolic vector notation as (see section on Governing Equations of Gas Flow)

                           .                                        (1)

The first step of the scheme uses a space-centred differences about the points [(i+1/2)x,nt] and [(i-1/2)x,nt] whilst the second step is a calculation which uses a time difference centred about the point (ix, (n+1/2)t). Thus the scheme can be expressed in the form

         ;    (2)
              (3)
and

         .    (4)

The Godunov Theorem (see Ref. 2) states that all second-order schemes having constant coefficients will generate spurious oscillations at discontinuities such as shock waves and contact surfaces. This obstacle to the development of numerical methods for hyperbolic equations can be circumvented by the construction of non-linear difference schemes in which the coefficients of the scheme are functions of the solution itself. One approach to constructing non-linear difference schemes is based on the total variation diminishing (TVD) criterion which is a measure of the variation of the solution at any given time step, given by
                           .                               (5)
In order to prevent the occurrence of spurious oscillations the total variation of the solution must satisfy the condition
                           .                                        (6)
This criterion can be utilised in a numerical scheme in the form of a smoothness monitor which tests the sign of consecutive gradients of the solution between pipe meshes.

The two-step Lax-Wendroff scheme can be modified to fulfil the TVD criterion by appending the term

         (7)

after the second-step, where

                                                           (8)
and
     .                 (9)
This approach to producing a symmetric TVD scheme was proposed by Davis (see Refs. 3 and 4).

The local Courant number is defined as

                                                                    (10)

where is given by

                           .                               (11)

The flux limiter can be defined the flux limiter as

                                                               (12)

This limiter constrains the Courant number of the scheme to 0.7.

The interface between the intra-pipe gas dynamic calculations and the boundary conditions is dealt with by using the Mesh Method of Characteristics  this well-established technique is covered comprehensively in Refs. 2 and 5.

Mesh Length and the Courant-Friedrichs-Lewy Stability Condition
In setting up the computational domain for any problem the value of the mesh size, x, is determined by the user, or the programmer (when automatic mesh generation is requested) by establishing criteria which fixes the compromise between accuracy and computational speed. The upper limit for the mesh length is dictated by the size of the smallest pipe element in the system: this gives the model with the lowest possible spatial accuracy for a given numerical method for non-linear waves. The value of the time step, t, however is subject to constraints imposed through stability considerations which arise from the well known criterion of Courant, Friedrichs, and Lewy (CFL) (see Ref. 6). This criterion requires that information (in the form of disturbances, or waves) cannot travel more than one mesh length in one calculation time increment, and this is expressed through the equation

                                                                            (13)
where
                           ,                                                          (14)

and represents the largest wave speed present is the entire solution domain at time level n. The parameter CCFL is known as the Courant, or CFL, number and clearly the time marching procedure will be most efficient when the value of this parameter is close to 1.

The method of characteristics is based on a transformation of the governing equations which enables the paths of disturbances to be tracked explicitly as they propagate through the flow field. For this technique it is clear that the physical interpretation of the case CCFL=1 corresponds to a situation where, in at least one computational cell, a wave starts from [(i-1)x, nt] or [(i+1)x, nt] and reaches [ix,(n+1)t]. It is only strictly safe to use a Courant number of 1 if the wave maximum wave speed, , does not increase as the wave travels across the cell. When the flow field is non-homentropic, however, the wave speed will not be constant over the cell and a more cautious (i.e. lower) value of CCFL should be used. The TVD scheme used in the Lotus Engine Simulation code dictates a Courant number of 0.7.

For non-linear waves can be estimated using the relationship

                                    .                               (15)

The number of meshes used in the pipes will determine the accuracy of the pipe flow calculations. It is difficult however to generalise on the mesh requirements. The stability requirement imposed on the calculations is that a wave cannot traverse a mesh in one time step. Mesh lengths of between 15-20 mm for inlet pipes and 25-30 mm for exhaust pipes is usually sufficient. It should be noted that the speed of simulation will slow dramatically with increasing mesh density. In general simulation run times increase with mesh density to the power of 1.5. The Pipe Auto-Mesh facility (activated from the Data menu on the Toolbar) gives a good compromise between accuracy and computer run-time.

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Fig. 2. Pipe graphical display

The meshes within a pipe can be visualised together with the pipe geometry by clicking the Pipe Graphical Display icon in the Pipe Property Sheet. Fig. 2 shows a complex pipe geometry which forms part of the exhaust system of a two-stroke motorcycle. The numbers in black (along the top of the pipe) and the black circles indicate the sections at which the user has specified the pipe equivalent diameter. The red circles indicate the position of the mesh points within the pipe. For accurate definition of the pipe geometry there should be more mesh points than sections at which the pipe geometry is defined.

Additional pipe meshes may improve computational stability, especially in pipes containing severe Tapers. The Lotus Engine Simulation features Automatic Mesh Refinement.

References:

1. Winterbone, D.E. and Pearson, R.J., Design techniques for engine manifolds. Wave action methods for I.C. engines. Professional Engineering Publications, 1999 (ISBN 1-86058-179 X).

2. Winterbone, D.E. and Pearson, R.J., Theory of Engine Manifold Design. Wave action methods for I.C. engines. Professional Engineering Publications, 2000 (ISBN 1-86058-209 5).

3. Davis, S.F. TVD finite difference schemes and artificial viscosity. NASA CR 172373, 1984.

4. Davis, S.F. A simplified TVD finite difference scheme via artificial viscosity. SIAM J. Sci. Stat. Comput., 8, 1, 1-18, 1987.

5. Benson, R.S., The thermodynamics and gas dynamics of internal combustion engines (Volume 1), Clarendon Press, 1982. (ISBN 0-19-856210-1)

6. Courant, R., Isaacson, E., and Rees, M. On the solution of non-linear hyperbolic differential equations by finite differences. Commun. Pure Appl. Math. 5, 243-249, 1952.

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^+$#>Theory  Pipes: Automatic Mesh Refinement

The size of the pipe meshes used in the simulation represent a compromise between the accuracy and speed of the calculation  See the Numerical Method section. In regions of the model where the wave speed is not constant over the solution domain, the solution may become unstable. This is especially true in strong tapers, where the source terms relating to the area variation can have a destabilising effect on the calculation. The Lotus Engine Simulation features an automated mesh refinement routine which can yield significant benefits in model robustness, whilst not necessarily causing the computational penalty of defining finer meshes for the base model.

Refinement Criteria
When the Automatic Mesh Refinement option is enabled, the simulation checks the spatial and temporal variation in pressure and density. The variation in density and pressure between each pipe mesh point and the adjacent mesh point is checked for the current time level. Additionally, the variation in density and pressure between each pipe mesh point at the current time level, with those at the previous time level are checked. If the variation in density or pressure is found to be above the refinement limit, the number of meshes in that particular pipe is doubled. The current time-step is re-evaluated for all of the pipes in the model. The user can limit the how many times the number of pipe mesh points is doubled. If the variation in the parameters for all of the meshes in a given pipe are below the de-refinement limits and that pipe is currently at a higher state of refinement than the base model, then the number of meshes in that pipe will be halved.

The number of times a given pipe can be refined in any time-step is only limited by the user definable refinement level limit, or by the maximum allowable number of meshes in a pipe. Pipes are only allowed to de-refine once per calculation time-step.

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Automatic Mesh Refinement Parameter Variation Gradient User Area Valve Properties Menu

The refinement criteria are all normalised by dividing the variation in the parameter by the previous time-step value for the mesh point under consideration, such that the test for variation in pressure becomes:
Pressure variation for mesh point x

The pressure variation value obtained from the relationship above is tested against the user specified pressure refinement and de-refinement parameters. If the pressure variation at any mesh point is found to be greater than the refinement parameter, the pipe is refined. This is repeated for the density variation parameter, which is evaluated in the same way as the pressure variation parameter. Once the pipe has been refined, the current time-step calculation is repeated. If the pressure and density variations for all of the mesh points in the pipe is below the de-refinement limit then the pipe may be de-refined.

Refining a single pipe still has a significant impact on the calculation run time, as the calculation time-step is based on the shortest time that information (in the form of disturbances, or waves) can travel one mesh length in one calculation time increment  See the Numerical Method section.

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^+$#>Theory  Pipes: Wall Friction

The pipe wall friction factor, f, is defined as (English  not US - definition)
                           .                                        (1)
It is common practice, in wave-action simulations, to use a constant value of f in the region of 0.004-0.01; in pipes containing bends higher values are often used. In fact the curve on the Moody diagram for a smooth pipe (surface roughness k 2.5 m) gives values in the range 0.0035-0.008 for Reynolds numbers in the range 1104-5105.

As described in the Pipe Data Variables section there are three ways to define the pipe wall friction factor in the Lotus Engine Simulation code. The first of these methods is to specify the wall friction factor directly. This requires some experience on the part of the user, and some knowledge of the cycle-averaged Reynolds numbers in the manifold pipes. The other two options set the pipe wall friction factor indirectly, either based on a value of the pipe wall surface roughness specified by the user, or by using a default value for the pipe wall surface roughness based on the material type of the pipe wall which has been specified by the user. In the latter case the default values for the material surface roughness are given in Table 1.

Table 1. Surface roughness values for different pipe materials in program.
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For Reynolds numbers in the range 3.5103Re108, and relative roughness values in the range 10-6(k/D)10-2, the program uses the equation           
         ,                               (2)

(see Ref. 1) to evaluate the pipe wall friction factor, where D is the pipe diameter. Reynolds number in this equation is given by
                                                                                      (3)
The gas viscosity, , is a function of its temperature and is evaluated by the code. For Reynolds numbers less than 3500 the flow is assumed to be laminar and the pipe wall friction factor is given by the expression

                                    .                                                 (4)

Equations (2) and (4) can be applied to give either a value for f at every mesh point and time step of the calculation, or to give an average value for each pipe section comprising the manifold. In the interests of maintaining reasonable computer run times the latter course is followed in the Lotus Engine Simulation code.

Values for the friction factor calculated for each pipe over each engine cycle are printed in the .MRS file.


References

1. Swamee, P.K. and Jain, A.K., Explicit equations for pipe-flow problems. J. Hydraulic Div. Proc. ASCE, pp. 657-664, May 1976

2. Winterbone, D.E. and Pearson, R.J., Theory of Engine Manifold Design. Wave action methods for I.C. engines. Professional Engineering Publications, 2000 (ISBN 1-86058-209 5).

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^+$#>Theory  Pipes: Wall Heat Transfer

The heat transfer term, q, in the energy equation presented in the section on the Governing Equations of Gas Flow) is used to represent simple convective heat transfer in the radial direction from the gas to the pipe.

An approximate treatment for convective heat transfer, due to Benson (see Refs. 1 and 2), is adopted in the Lotus Engine Simulation code. The approach is based on the assumption that the analogy between heat and momentum transfer in steady flow can be extended to non-steady flow. This assumption is not strictly true. In addition to the fact that the Reynolds analogy oversimplifies the mechanism of turbulent heat transfer, it also ignores the existence of any laminar sub-layer. The approach, however, is reasonable as a first approximation and is described below. The heat transfer rate per unit mass is
                                         (1)
where h is the convective heat transfer coefficient and Tw and Tg are the temperatures of the pipe inner wall and gas, respectively. Reynolds' analogy gives the convective heat transfer coefficient as
,                                                 (2)
where f is the pipe wall friction factor (which can be set independently of the value used in the wall friction term in the momentum equation). Equation (1) then becomes
,                                        (3)
and, for an ideal gas,
.                               (4)

At the end of each cycle the total heat transferred to the walls at all the meshes in the pipe is summed and used to perform a simple one-dimensional heat transfer calculation to determine the pipe inner wall temperature that should be used for the next cycle. Thus it is necessary to specify the pipe wall thickness, material type and method of cooling in the Pipe Data Variables.

The assigned wall material properties for the default option are given in Table 1.

Table 1. Pipe material properties.
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Note - the density and specific heat are only used in a thermal transient simulation  this facility is not available in the current version of the Lotus Engine Simulation code.

The air gap pipe data was drawn from a finite element and CFD model of an exhaust pipe of 35 mm ID 50 mm OD and employing a 3mm air gap (Ref. 3).

The default external cooling properties are given in Table 2.

Table 2. Pipe coolant data.
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References:

1. Benson, R.S., The thermodynamics and gas dynamics of internal combustion engines (Volume 1), Clarendon Press, 1982. (ISBN 0-19-856210-1)

2. Winterbone, D.E. and Pearson, R.J., Theory of Engine Manifold Design. Wave action methods for I.C. engines. Professional Engineering Publications, 2000 (ISBN 1-86058-209 5).

3. Sandford, M.H., and Jones, R.D., Powerplant systems and the role of CAE - Part 1 Exhaust Systems. SAE paper no. 920396, 1992.

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^+$#>Theory  Pipe Bends

Pipe bends are handled in the model in essentially the same way as a conventional pipe. The pipe bend angle and radius are specified as properties of the pipe (see Pipe Data Variables) and empirical data, based on Ref. 1, is used in order to infer an equivalent pipe-wall friction factor that mimics the pressure-loss effects of the bend on the gas flow.

The length of the pipe, , which forms the bend is required  if this value is less than the product of the bend radius and angle , the value of the bend radius is reduced to give the correct length. If the specified pipe length exceeds the product of the bend radius and angle the bend is placed in the centre of the pipe and the inlet and outlet pipe length surrounding the bend are set equal to half the difference .

The pressure-loss due to the secondary flows and separated regions within the bend, and the redevelopment of the flow downstream of it, can be expressed as (see ref. 1)
                           ,                                        (1)
where Kb is the bend loss coefficient. Considering the shear stress developed over a length of pipe x enables the pipe wall friction factor, defined as (see theory on Pipe Wall Friction)
                          ,                                        (2)
to be expressed in the form
                           ,                                                 (3)
and combining this with eqn (1) gives
                           .                                                 (4)

Miller (Ref. 1) gives data for the basic loss coefficient, Kb*, as a function of bend angle and the radius-to-pipe diameter (r/D) ratio at a Reynolds number of . This basic loss coefficient is then modified to give the corrected loss coefficient as
                                                                    (5)
where the C values are correction factors which account for variations in Reynolds number (CRe), outlet pipe length (Co), and surface roughness (Cf). In this way the pipe friction factor may be increased by a factor of 3 or 4 in pipe bends.

Fig. 1 below shows the variation of the basic loss factor, Kb*, with pipe bend angle and r/D ratio for Re =. The surface roughness correction factor, Cf, for bends of and is given by

                                                                             (6)

where fsmooth is the friction factor for a hydraulically smooth pipe and frough is the friction factor obtained using the assumed pipe and bend roughness. For and the value of Cf is obtained from eqn. (6) using .

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Fig. 1. Variation of Kb* with bend angle and bend radius / diameter ratio for Re = 106.

The outlet pipe length correction factors used in conjunction with the basic loss-coefficient Kb* are shown in Fig. 2. For precise details of the way in which this, and the other correction factors are used see Ref. 1.

{
Fig. 2. Variation of outlet correction factor with outlet length / diameter.

Values for the equivalent pipe-wall friction factor calculated for each pipe over each engine cycle are printed in the .MRS file.

Reference

1. Miller, D.S., Internal flow systems. Second Edition. BHR Group Ltd., 1990.

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^+$#>Theory  Tapered Pipes

Tapered pipes are handled in the model in essentially the same way as the pipe bends. The pipe diameter at various distances along the pipe can be specified as properties of the pipe (see Pipe Data Variables). The equations presented in the Numerical methods section assume that the gas flow uniformly fills the entire pipe. Thus, secondary flow losses caused by flow separation in steep diffuser sections, are not accounted for. Empirical data, based on Ref. 1, has been used to develop a relationship which is used to infer an equivalent pipe-wall friction factor that mimics the pressure-loss effects of the diffuser sections on the gas flow.

{

Fig. 1. Schematic of diffuser.

The pressure-loss due to the secondary flows and separated regions within the diffuser, and the redevelopment of the flow downstream of it, can be expressed as (see ref. 1)
                           ,                                        (1)
where Kd is the diffuser loss coefficient. Considering the shear stress developed over a length of pipe x enables the pipe wall friction factor, defined as (see theory on Pipe Wall Friction)
,                                        (2)
to be expressed in the form
                           ,                                                 (3)
and combining this with eqn (1) gives
                           .                                                 (4)

The diffuser loss can be expressed as
.                                        (5)
The function c(q) can be approximated as
.                               (6)

A Reynolds number correction factor, cRe, can be applied to the diffuser loss. This can be approximated by
                           .                                        (7)

The Lotus Engine Simulation applies the diffuser loss on a mesh-wise basis. The length of the diffuser, , is taken as the pipe mesh length. Areas A1 and A2 are the area at the upstream and downstream nodes respectively. The diffuser angle, q, is simply a geometric function of A1, A2 and l.

Values for the equivalent pipe-wall friction factor calculated for each pipe over each engine cycle are printed in the .MRS file.

Reference

1. Miller, D.S., Internal flow systems. Second Edition. BHR Group Ltd., 1990.

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^+$#>Theory  Pipe Junctions
When pipes are connected together the program checks the junction type. If only two pipe ends are connected and the diameters of both pipes at the junction are the same then as smooth transition (equal area junction) is modelled and there is no pressure discontinuity at the junction. If the two pipes have different diameters then a step change in area is modelled and a pressure discontinuity will be produced by the requirement for mass and momentum continuity. In this case the sudden enlargement and sudden contraction models detailed in Ref. 1 are used.

If more than two pipe ends are connected at the junction then a constant pressure junction is modelled. A constant pressure junction can be transformed into a pressure-loss junction by dropping the pressure-loss junction icon on the pipe junction concerned.

The propagation of pressure waves through junctions in engine manifolds is an intrinsically multi-dimensional phenomenon. The modelling of such junctions within a one-dimensional simulation presents a major challenge, since the geometry of the junction cannot be fully represented and can have a significant influence on the pressure waves that propagate through them.

Variations of two boundary models have been most widely used in wave action engine simulations for dealing with multi-pipe junctions: the constant pressure junction, and the pressure-loss junction approaches. With both of these junction models it is necessary to assume that the flows entering and leaving the junction are one-dimensional, and that the physical dimensions of the junction are negligible compared with the overall dimensions of the pipe network.

In turbocharged diesel engines the assumption of constant-pressure junctions is often acceptable because the flow velocities are quite low. The situation is changed if pulse converter junctions (see Fig. 1) are fitted because these are basically junctions which have been designed to give high pressure losses in preferential directions. In petrol (or gasoline) engines the gas flow velocities are substantially higher and then it becomes increasingly important to take account of these flow losses, since they can have an affect on the volumetric efficiency of the engine.

{

Figure 1  Schematic of a pulse converted exhaust system.

References

1. Benson, R.S. The thermodynamics and gas dynamics of engine manifolds. Vol. 1, Eds. J.H. Hor-lock and D.E. Winterbone, Oxford University Press. 1982.

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^+$#>Theory  Pipe Junctions: Two-Pipe Equal Area Junctions

When two pipes if the same cross-sectional area are joined together they form an equal area junction which produces no pressure-loss effects on the gas flow through it. The boundary equations for this type of junction can be calculated using either the Method of Characteristics or the two-step Lax-Wendroff method  these options can be selected from the Data menu. In the former case identical algorithms to those described for a pipe interior mesh calculation in references 1 and 2 are used. In the latter case a special computational stencil for the two-step Lax-Wendroff method is constructed in order to cope with different mesh sizes in the pipes which are joined. Figure 1 shows how a virtual mesh point is introduced in the largest mesh adjacent to the boundary. The gas properties at this mesh point are interpolated from the values at the surrounding mesh points and then used in the finite difference calculation which is carried out over equally-spaced meshes.

The finite difference option will handle transonic and supersonic flow conditions more robustly than the method of characteristics option.

{

Figure 1  Creation of virtual mesh point at equal area junction.

References

1. Benson, R.S. The thermodynamics and gas dynamics of engine manifolds. Vol. 1, Eds. J.H. Hor-lock and D.E. Winterbone, Oxford University Press. 1982.

2. Winterbone, D.E. and Pearson, R.J., Theory of Engine Manifold Design. Wave action methods for I.C. engines. Professional Engineering Publications, 2000 (ISBN 1-86058-209 5).

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^+$#>Theory  Pipe Junctions: Constant Pressure Junctions

The simplest method of dealing with a multi-pipe junction is to assume that the static pressure at all of the pipe ends comprising the junction is uniform, so that

                          ,                                                       (1)

^+$#>Theory  Pipe Bundles

The pipe bundle is a simple mechanism for representing a group of similar pipes by a single pipe. It is useful for the modelling of exhaust catalyst bricks or charge-cooler passages.

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^+$#>Theory  Pipe Junctions: Pressure Loss Junctions

All junctions result in pressure losses, these losses are more significant in some junctions than in others. In order to allow inclusion of geometry induced effects, a new pressure loss junction model was developed. This requires data, particular to the junction, relating the pressure drop across the junction for various flow configurations to the pressure ratio, mass flow ratio, and mass flow rate. This data is usually expressed in terms of steady-flow pressure loss coefficients. Ref. 1 provides a source of these for many three-pipe junction configurations.

The use of steady-flow pressure loss coefficients in wave action simulations is based on the assumption that the pressure drop between any two branches of a junction, when experiencing an unsteady flow field, is instantaneously equivalent to the pressure drop between the branches when subjected to a steady flow. This forms the basis of the quasi-steady assumption which is normally used in the boundary models of engine simulation codes, for a more detailed discussion see Ref. 2.

In the Lotus Engine Simulation, a generalised technique is used for evaluating the instantaneous pressure loss between the branches of the junction. A detailed description of this model can be found in Ref. 3. This generalised technique is based on consideration of the fluid momentum and has the advantage that it allows junctions formed by any number of branches to be considered. The only additional data required by this model, over the constant pressure junction model, is the angular relationship between the various branches which form the junction. See the pressure-loss junction data page for details.

Definition of the Pressure Loss Coefficient
The pressure loss junction model requires data, particular to the junction, relating the pressure drop across the junction for various flow configurations to the pressure ratio, mass flow ratio, and mass flow rate. This data is expressed in terms of a steady-flow pressure loss coefficients, which can be expressed in terms of the stagnation pressure drop, as
.                    (1)
Here Ki represents any loss coefficient, and the subscripts up and down are used to denote the upstream and downstream branches between which the loss coefficient applies and com denotes the branch which carries the entire flow passing through the junction.
Twelve separate coefficients are required to characterise even a simple three-pipe junction. The loss coefficients must either be obtained from steady flow tests (eg. Miller (1)), or from empirical (see Ito and Imai (4)) or simplified analytical formulations (eg Bassett et al. (5)). A description of the procedure for measuring the steady-flow pressure-loss coefficients of a junction is given in Ref. 2. These loss coefficients simply express the stagnation pressure drop caused as the flow passes between two branches of the junction and can be incorporated into the junction boundary equations, which are then solved using an iterative process, which is described fully in Ref. 2.
It may be anticipated that the pressure-loss junction model would perform poorly in unsteady flow situations, as it employs steady-flow pressure-loss data. However, Bassett et al. (6) have demonstrated that it can perform well, even in flows which contain shock waves.
Estimation of the Loss Coefficient
Expressions relating the pressure drop between the various branches of the junction are required. It may be argued that when more than one pipe contains flow towards the junction, that the pressures at the ends of each of those pipe ends must be equal. In the pressure-loss model built into the Lotus Engine Simulation code, the junction branch which contains the largest mass-flow rate towards the junction at any given time-step is identified and set as the datum branch for the time-step under consideration. Thus, only a relationship for estimating the pressure loss between the branch designated as the datum branch and the branches with flows away from the junction is required.

Consider a junction formed by n-branches; at any given instant, some of these branches will contain fluid flowing towards the junction, and others will contain fluid flowing away from the junction. It can be shown (see Ref. 3) that an expression can be obtained for the pressure loss coefficient between the datum branch (denoted dat) and any other branch, j, containing flow away from the junction, as

,                             (2)


where the area ratio between the datum branch and any other branch is defined as
.                                                        (3)
and the mass flow ratio is defined as
.                                                        (4)
{

^+$#>Theory - Cylinders and Plenums

Cylinders and plenums are so-called zero dimensional elements in that they have properties of mass, pressure, temperature and volume but NOT length. The conditions within these elements are calculated at each crank angle by solving the energy equation in the form

                                        (1)

The solution procedure is summarised as follows;

Calculate heat release due to combustion
Calculate enthalpy change due to gas flows
Calculate heat transfer using old cylinder temperature

Estimate change in cylinder pressure due to energy and volume changes

where:
mcyl     =       cylinder mass
cv       =       specific heat at constant pressure
Tcyl     =       cylinder temperature at previous increment
dV       =       change in volume during increment
Vcyl     =       cylinder volume.at previous increment
    =        ratio of specific heats

Estimate displacement work

where
pcyl     = cylinder pressure at previous increment

Estimate temperature change


(1) Enter iteration loop to converge on cylinder temperature



Calculate cylinder pressure

where
pnew     = new cylinder pressure
Vnew     = new cylinder volume
Rcyl     = Universal gas constant for gases in the cylinder.

Calculate displacement work


Recalculate heat transfer based on mean gas temperature during increment.

Calculate energy change due to this gas temperature



Calculate internal energy change due to change in cylinder temperature.



Where Enew and Ecyl are the internal energies of the gas in the cylinder at this and the previous time steps respectively.

Calculate the error in temperature due to the mismatch in changes in internal energies


If dT is greater than 0.01 K repeat calculations from (1).

When converged on temperature recalculate all conditions within the cylinder.

The above methodology is the most simple approach to solving the energy equation for zero dimensional elements. Other programs use more complex predictor - corrector algorithms which can be more computationally efficient. The authors have tested these but found the above approach to be the most robust.

In order to ensure stability under all test conditions the crank angle increments are limited to ensure that the change in mass of a zero dimensional element does not exceed 25% of the current mass in a particular time step. This is performed by assuming that the rate of change in mass from the previous step will also apply to the current step. This limit is most often invoked on high compression ratio four stroke engines at TDC overlap.

Much of the zero-dimensional element theory was derived and adapted from the following publications.

References:

1. The Thermodynamics and Gas Dynamics of Internal Combustion Engines (Volume 1) R.S.Benson (section 1.3.3 pp 36 & section 4.9 pp 182) (ISBN 0-19-856210-1)

2. The Thermodynamics and Gas Dynamics of Internal Combustion Engines (Volume 2) J.H.Horlock & D.E.Winterbone (section 10 pp 583 & section 18 pp 1016) (ISBN 0-19-856212-8)

3. Internal Combustion Engines (Volume 2) R.S.Benson & N.D.Whitehouse (chapter 8 pp 271 & chapter 9 pp 303) (ISBN 0-08-022720-1)

4. Turbocharging the Internal Combustion Engine. N.Watson & M.S.Janota (section 15.5 pp 528) (ISBN 0-333-24290-4)

^+$#>Theory - Gas Properties

Gas is transferred to all elements as a mixture of 11 gases plus fuel. The properties of the individual gases are calculated as functions of temperature with these properties being averaged as molar fractions to give the overall properties of the mixture. The main benefit of this approach is that a wide range of fuels and air fuel ratios can be accurately simulated. With the effects of gas composition on parameters such as the speed of sound in exhaust systems being correctly calculated.

The properties calculated for each gas mixture are;

Enthalpy H (J)
Internal Energy U (J)
Heat Capacity @ const p (J/K)
Heat Capacity @ const V (J/K)
Specific Enthalpy h (J/kg or J/kmole)
Specific Internal Energy u (J/kg or J/kmole)
Specific Heat Capacity @ const p Cp (J/kg.K or J/kmole.K)
Specific Heat Capacity @ const V Cv (J/kg.K or J/kmole.K)
Gamma

The gas species considered are;

CO2
CO
N2
H2O
O2
H2
C8H18
C12H26
CH4
H
N
NO
O
OH

The gas property model is based on polynomial curve fits to thermodynamic data for each species.

For each species i at temperature T the enthalpy and specific enthalpy are given by;





The internal energy and specific internal energy are given by;




The heat capacity and specific heat capacity at constant pressure (cp & scp) are given by;





The heat capacity and specific heat capacity at constant volume (cv & scv) are given by;





The ratio of specific heats



The constants for the polynomials are;

{

^+$#>Theory - Fuel Properties

Default values for calorific value, relative density, hydrogen carbon ratio and molecular weight for each fuel option are provided. These are;

Fuel	Calorific Value kJ/kg	Relative Density	H/C Ratio	O/C Ratio	Molecular Weight	Mal- Distribution
1: Gasoline	43000	0.75	1.8	0.0	114.23	1.0
2: Diesel	42700	0.84	1.9	0.0	170.0	1.0
3: Methane	46280	0.7373E-3	3.87	0.0	17.423	0.0
4: Methanol	20000	0.79	4.0	1.0	32.04	1.0
The user is free to specify any of the fuel properties. The simulation will automatically adjust the combustion chemistry and heat release rates as appropriate. The combustion chemistry employed is described in detail in reference 1 with corrections for CO and O2 as decried below. The maldistribution factor The use of full chemical kinetics combustion models is not appropriate for the majority of simulation work as the models are computationally expensive on computer resources. An alternative method of catering for the dissociation effects on effective heat release has therefore been developed through the use of a so called maldistribution factor. The maldistribution factor is incorporated to allow for a reduction in effective calorific value of the fuel due to poor charge mixing and dissociation. A factor of 0.0 implies almost perfect mixing and a high effective calorific value for the fuel. If a factor of 1.0 is used, a reduction in %CO2 and increase in %CO and %O2 is used to re-calculate the effective calorific value of the fuel. The effective calorific value is defined as the calorific value minus the effects of combustion to CO (rather than CO2) and to H2 (rather than H2O). The assumed energy release rates are; "         C to CO2        32760 kJ/kg "         C to CO         9100 kJ/kg "         H2 to H20        120000 kJ/kg A typical gasoline engine would have a maldistribution factor of between 1.0 and 3.0. Values less than 1.0 imply better combustion and may be appropriate for gas fuelled engines. The effective calorific value calculated by the program is provided in the .MRS output file. The relative proportions of CO2, CO and O2 produced by different maldistribution factors are shown on the Eltinge chart (see below) References 1. Internal Combustion Engine Fundamentals J.B.Heywood (section 4.2 pp 130) (ISBN 0-07-028637-X) 2. Fuel-Air Ratio and Distribution from Exhaust Gas Composition L.Eltinge SAE 680114 { Eltinge Chart (Fuel H/C ratio = 1.8, Water Constant = 3.5)]

^+$#>Theory - Fuel / CombustionSystem

Four combustion systems are catered for in the Lotus Engine Simulation code  these are:
"         Carburetted
"         Port Injected
"         Direct Injected
"         Indirect Injected
The combustion system option performs two functions. Firstly it controls the method by which fuel is introduced to the model and secondly it sets the defaults for the combustion models and heat transfer options

If the carburettor option is selected then fuel is mixed with the air prior to introduction to the model at the inlet boundary. It is recognised that this is restrictive and this option will be improved in future versions of the program.

The port-injected model introduces fuel to the cylinder as air flows through the inlet valve. The model assumes that the fuel is fully evaporated and it thus displaces fresh charge that might other wise flow into the cylinder.

The direct and indirect-injection options are identical with respect to the way by which fuel is introduced to the cylinder. Fuel is assumed to be introduced to the cylinder at the same rate as it is combusted. The only effect of specifying and indirect injection combustion system is to change some of the default combustion and heat transfer settings.

Fuelling
Three fuelling options are provided. However not all are available with each combustion system type. The fuelling options are;

"         User specified trapped air fuel ratio (DI & IDI)
"         User specified equivalence ratio (CARB & PI)
"         User specified fuelling (DI & IDI)

The trapped air fuel ratio option uses the amount of oxygen in the cylinder at the beginning of compression to calculate the quantity of fuel that is to be injected. This option is particularly useful when the predicted performance at a limiting air fuel ratio is required. Users should however be aware that it can sometimes cause instabilities in the simulation when small variations in fuelling cause significant changes in airflow. For example on turbocharged engines the modulation in fuelling from one cycle to the next can cause a similar modulation in turbocharger speed and hence airflow.

The equivalence ratio option is used to specify the quantify of fuel that should be mixed with fresh charge when it flows through an inlet boundary or inlet port. The flow routines have been developed to ensure that over-fuelling does not occur with reverse flow.

Users should note the difference in definition between equivalence ratio and lambda.



The fixed fuelling option simply injects the same quantity of fuel into the cylinders irrespective of air flow. If the air fuel ratio become too rich a warning will be issued.

^+$#>Theory - Combustion Models

The program employs a single zone heat release model. This means that during combustion the heat released is used to heat the whole of the combustion space. The main implication of this assumption is that the bulk gas temperature is generally lower than the core combusted gas temperature behind the flame front. This may have an effect on detailed in-cylinder heat transfer, however since the semi-empirical heat transfer models make gross assumptions regarding heat transfer coefficient and wall temperature the effects of this assumption are small. The program will be extended in future versions to allow the use of a two zone combustion model.

The heat release rate can be defined either using one of two empirical heat release functions or to be defined explicitly by the user in the form of an angle verses heat release rate curve. The empirical heat release functions are derived from the Wiebe equation (reference 1) and adapted to diesel combustion characteristics by the addition of a pre-mixed combustion phase by Watson & Pilley (reference 2)

Wiebe Function

The Wiebe function define the mass fraction burned as

,

where

A        =        A coefficient in Wiebe equation

M        =        M coefficient in Wiebe equation

    =        actual burn angle (after start of combustion)

b   =       total burn angle (0-100% burn duration)


Two Part Wiebe Function

The two part Wiebe function defines the mass fraction burned in the premixed combustion period as



The mass fraction burned during the diffusion combustion period is defines as




,

where,

A        =        A coefficient in Wiebe equation

M        =        M coefficient in Wiebe equation

C1       =        cp1 coefficient in Watson & Pilley equation

C2       =        cp2 coefficient in Watson & Pilley equation

    =        fraction of premixed combustion to total combustion

    =        delay angle between premixed and diffusion combustion

    =        actual burn angle (after start of combustion)

b   =        total burn angle (0-100% burn duration)


The main advantage of the Wiebe functions are that they are normalised by the combustion duration. Thus the user may quickly change the total combustion duration and be confident of achieving a realistic heat release rate.

Wiebe Function Defaults

The single part Wiebe function is used by default for all combustion systems. The model coefficients are set as a function of the fuel type. The default coefficients are;








{

Fuel	A	M
1  Gasoline	10.0	2.0
2  Diesel	6.9	0.5
3  Methane	5.0	2.2
4  Methanol	10.0	2.0
At present there are no defaults for the two part heat release equation. Typical values for the constants for a turbocharged DI diesel engine are however;          A = 10.0 M = 0.4          C1= 2.0          C2 = 5500          = 0.05          = 0.0 Combustion Duration The definition of the combustion duration is a function of the type of fuel being used. It is notoriously difficult to reliably measure both the start and end of combustion in spark ignited gasoline and methanol fuelled engines. An approach has therefore been adopted by which the combustion duration of these engines is defined as the number of crank degrees between 10% and 90% mass fraction burnt. For diesel (and some gas) engines however the start and end of combustion are more easily obtained. Thus for all other engines the combustion duration is defined as the number of crank degrees between 0 and 100% mass fraction burned. Default combustion durations are available for several fuel / combustion system combinations. These are mainly intended to allow the user to quickly develop a new model and should not be relied upon for accurate modelling of each combustion system / fuel type combination. The available defaults are; Fuel	Carburettor	Port Injected	Direct Injection	Indirect Injection
Gasoline	Eqn.1	Eqn.1	Eqn.1	Eqn.1
Diesel			Eqn 2	Eqn 3
Methane
Methanol	Eqn.1	Eqn.1	Eqn.1	Eqn.1
11       With the default combustion durations defined by; Eqn.1 Eqn.2 Eqn.3 Combustion Phasing The definition of the combustion phasing is a function of the type of fuel being used. It is notoriously difficult to reliably measure both the start of combustion in spark ignited gasoline and methanol fuelled engines. An approach has therefore been adopted by which the combustion phasing of these engines is defined as the number of crank degrees after TDC firing at which 50% of the fuel has been burnt. (Note a negative combustion phasing value for these engines implies an angle of 50% burn before TDC). For diesel (and some gas) engines however the start and end of combustion are more easily obtained. Thus for all other engines the combustion phasing is defined as the number of crank degrees before TDC at which combustion starts. (Note a negative combustion phasing value for these engines implies a start of combustion timing after TDC). 0        Default combustion phasings are available for several fuel / combustion system combinations. These are mainly intended to allow the user to quickly develop a new model and should not be relied upon for accurate prediction of performance or maximum cylinder pressure. The available combustion phasing defaults are; Fuel	Carburettor.	Port Injected	Direct Injection	Indirect Injection
Gasoline	A50%-10.atdc	A50%-10.atdc	A50%-10.atdc	A50%-10.atdc
Diesel			SOC- 5        5.btdc	SOC- 6        0.btdc
Methane
Methanol	A50%-10.atdc	A50%-10.atdc	A50%-10.atdc	A50%-10.atdc
Definition of Heat Release Angles Maximum Cylinder Pressure Targets (IHRPHO,TPMAX) Some simulation studies require that performance is predicted at a specified maximum cylinder pressure or that cylinder pressures are limited so as not to exceed a specified limits. Both of these options are provided through the IHRPHO and TPMAX input data. The two options are described as; "         Target PMAX "         PMAX retard With target PMAX the simulation program will automatically adjust the heat release phase in order that the predicted maximum cylinder pressure matches that specified by the user. The heat release phase will be either advanced or retarded as required. An algorithm by which the simulation rapidly converges on the required phasing is used, however there are no explicit convergence checks that prevent the program from stopping if the maximum cylinder pressure does not match that required by the user. The PMAX retard option is similar to the target PMAX option but in this case the heat release phase may only be retarded. This means that if the maximum cylinder pressure does not achieve the target then the heat release phasing remains unchanged. If the maximum cylinder pressure is found to exceed the target then the heat release phase is retarded until the target maximum cylinder pressure is achieved. This option is particularly useful when trying to mimic the effects of knock in gasoline engines. An assumption is usually made is that at a given engine speed, knock will always occur at the same maximum cylinder pressure. Thus in a simulation study, if the changes in engine specification produce an increase in volumetric efficiency, then the increase in predicted performance is limited by the imposition of heat release retard. Test results have shown that the combustion duration increases with ignition retard. Thus if the PMAX retard option is used in conjunction with any of the wiebe functions, then the 10-90% burn duration is automatically increased by 3.75 degrees per degree of retard. The above model provides an approximation to the effects of knock. It is however only an approximation and the detailed response of every engine to knock and ignition retard will differ. In fact on some gasoline engines the maximum cylinder pressure achieved with lower volumetric efficiency can be higher that that achieved with high volumetric efficiency.          User Specified Combustion This option enables the user to specify the burn curve with a series of angle, mass fraction burnt ordinate pairs. The first angle burn angle and the first mass fraction burnt must equal 0.0. The last angle entered is taken as the burn angle (0-100%) and should be accompanied by a mass fraction burnt figure of 1.0. The user must check that the mass fraction burn curve is monotonically increasing (i.e. there are no negative rates). This check is NOT performed by the program. When deriving this type of data from measured cylinder pressure data it is strongly recommended that the cylinder pressure data is first smoothed. This will help to ensure a smooth mass fraction burn curve. References 1. Habempirische Formel fur die Verbrennungsgeschrwindigkeit Verlag der Akademie der Wissenschaften der VdSSR I.Wiebe Moscow (1956) 2. A Combustion Correlation for Diesel Engine Simulation. N.Watson, A.D.Pilley & M.Marzouk. SAE 800029.

^+$#>Theory - Cylinder Heat Transfer

Heat transfer to and from the cylinder gases are calculated at every crank angle increment. These calculations require a knowledge of the wall area, wall temperatures and surface heat transfer coefficient.

Cylinder Wall Area

The cylinder surface areas are calculated via the default or user specified surface area to bore area ratios. Where;



The default surface to bore area ratios are a function of the combustion system, as given below:

Combustion System	Head/Bore Area Ratio	Piston/Bore Area Ratio
Carburetted	1.2	1.1
Port Injected	1.2	1.1
Direct Injected	1.0	1.4
Indirect Injected	2.0	1.0
The liner area is calculated at each increment by summing the piston displacement from TDC with the bump clearance. The default bump clearance is calculated from the compression ratio and assuming the cylinder employs a disk combustion chamber. Cylinder Wall Temperatures The cylinder wall temperatures are either specified explicitly by the user or calculated via a simple one dimensional heat transfer calculation for the cylinder head and liner walls. The cylinder walls are assumed to have a wall thickness that is a directly proportional to the bore diameter; "         Head flame face thickness = 0.13 x Bore "         Liner thickness = 0.07 x Bore The thermal conductivity of the walls is specified by the material index. The assigned wall material properties are; Material	Conductivity(W/m/K)
Cast Iron	45
Aluminium	150
Steel	48
Zirconium	4.1
The coolant temperature is assumed to be 100 oC and the coolant connective heat transfer coefficients are assumed to be 10000 W/m2/K, for the cylinder head and 8000 W/m2/K for the liner. Thus from a knowledge of the heat transfer rate the gas side wall temperature may be calculated. The heat transfer rate for the first cycle is estimated from the fuel flow rate. On subsequent cycles it is obtained from the previous cycles heat transfer results. 0        The heat transfer rate for the liner wall temperature calculation is assumed to be 44% of the heat transfer rate to the cylinder head. This approach is adopted because of the changing liner surface area and the subsequent difficulty in deriving a meaningful heat transfer rate per unit area. The mean surface temperature of a cylinder head on most modern four stroke engines is heavily dominated by the valve temperatures. Valve head temperatures are calculated for both inlet and exhaust valves as a function of fuel type and air fuel ratio. For gasoline and methanol engines; AFR < 11.5          Inlet Valve (oC)                  = 5.8.AFR + 367.6          Exhaust Valve (oC)       = 25.7.AFR + 418.5 11.5 < AFR < 18.0          Inlet Valve (oC)                  = -0.5 .AFR3 + 19.1.AFR2 - 236.5 AFR + 1389.8          Exhaust Valve (oC)       = -0.89.AFR3 + 31.6.AFR2 - 344.1.AFR + 1860.1 18.0 < AFR < 26.0          Inlet Valve (oC)                  = -38.25.AFR + 1094.5          Exhaust Valve (oC)       = -69.50 AFR + 1907.0 26.0 < AFR          Inlet Valve (oC)                  = 100.0          Inlet Valve (oC)                  = 100.0 For diesel engines; AFR < 25.0          Inlet Valve (oC)                  = -4.1 AFR + 504.2          Exhaust Valve (oC)       = -4.2 AFR + 663.0 25.0 < AFR < 80.0          Inlet Valve (oC)                  = -4.1 AFR + 504.2          Exhaust Valve (oC)       = -0.003.AFR3+0.611.AFR2-41.92.AFR + 1260.1 80.0 < AFR < 200.0          Inlet Valve (oC)                  = -0.635.AFR + 227          Exhaust Valve (oC)       = -1.667 AFR + 433.6 200.0 < AFR          Inlet Valve (oC)                  = 100.0          Exhaust Valve (oC)       = 100.0 For gas engines; Equivalence Ratio (EQV) > 1.27 Inlet Valve (oC)                  = 84.7 / EQV + 367.6          Exhaust Valve (oC)       = 375.2 / EQV + 418.5 1.27 < EQV < 0.81          Inlet Valve (oC)                  = -1556/EQV3 + 4071/EQV2 - 3453/ EQV + 1389.8          Exhaust Valve (oC)       = -2770/EQV3 + 6736/EQV2 - 5023/.EQV + 1860.1 0.81 < EQV < 0.56          Inlet Valve (oC)                  = -558.5/EQV + 1094.5          Exhaust Valve (oC)       = -1015/EQV + 1907.0 0.56 < EQV          Inlet Valve (oC)                  = 100.0          Exhaust Valve (oC)       = 100.0 The cylinder head temperature is calculated as the area average of the wall temperature and the valve temperature. The piston temperature is assumed to equal to the area averaged cylinder head temperature. This is a gross assumption, however, it is the only one that can reasonably be made given the wide variety of piston geometrys and materials. Users who do not wish to use the above valve temperature and piston assumptions but do wish to employ the simple one dimensional model may specify the conductance for the head, piston and liner. Where The one dimensional calculation is performed individually for the head, piston and liner thus giving a greater flexibility to the wall temperature model. Cylinder Heat Transfer Models The heat transfer models proposed by Annand (reference 1), Woschni (reference 2) and Eichleberg (references 3 & 4) are provided in the program. All these models have been derived from a basic Nusselt Number / Reynolds Number correlation for flow in pipes. Each model employs coefficients that have been developed to best reproduce the heat transfer results obtained by experiment. The coefficients used by the program may either be the default values or may be tuned by the user to best suit the engine being studied. Annand The connective heat transfer model proposed by Annand is defined as;          where h        =        heat transfer coefficient A        =        Annand open or closed cycle A coefficient B        =        Annand open or closed cycle B coefficient k        =       thermal conductivity of gas in the cylinder Dcyl     =       cylinder bore Re       =       Reynolds number based upon mean piston speed and the engine bore.The density is that calculated for the cylinder contents at each crank angle. Thus the heat transfer per unit area of cylinder wall is defined as; where; dQ/F     =       heat transfer per unit area C        =       Annand closed cycle C coefficient. The first part of the heat transfer equation is the connective heat transfer and the second part the radiative heat transfer. Radiative heat transfer is only modelled during combustion. Thus C is only required for the closed cycle model. Default coefficients are provided for the Annand model. The choice of coefficients being a function of the combustion system type. Open Cycle Coefficients are; Combustion System	A	B
Carburetted or Port Injected	0.2	0.8
Direct or Indirect Injected	1.1	0.7
Closed Cycle Coefficients are; Combustion System	A	B	C
Carburetted Port Injected	0.12	0.8	4.29E-9
Direct or Indirect Injected	0.45	0.7	3.271E-8
The radiative heat transfer term should more correctly be a function of the fuel type with a higher number being used for diesel fuel and the lower for the other fuel types. Often it is necessary to tune the coefficients of the in-cylinder heat transfer model to achieve good correlation both for volumetric efficiency and heat transfer. It is recommended that only the A coefficient is tuned with the B coefficient being set at 0.8. Typical values for A range between 0.1 and 0.3. (see Ref. 5). Woschni The connective heat transfer model proposed by Woschni is defined as; , where h        =        heat transfer coefficient A        =        Woschni open or closed cycle A coefficient B        =        Woschni open or closed cycle B coefficient C        =        Woschni open or closed cycle C coefficient D        =        Woschni closed cycle D coefficient p        =       Cylinder pressure T        =        Cylinder temperature          V        =        Cylinder volume Dcyl     =       Cylinder bore     =        Mean piston speed     =       Mean swirl velocity Tsoc     =       Cylinder gas temperature at start of combustion psoc     =       Cylinder gas pressure at start of combustion Vsoc     =       Cylinder volume at start of combustion pmotor   =        Motoring cylinder pressure The mean swirl velocity is given by;              (i.e. half periphery gas speed) Srat     =        Woschni open or closed cycle swirl ratio 0        N        =        Engine speed [rev/min] The motoring cylinder pressure is given by; where G        =       Woschni ratio of specific heats. The last term (factored by D) in the Woschni model is a so called combustion term and is thus used only during the closed cycle. The heat transfer per unit area of cylinder wall is defined as; 0        Default coefficients are provided for the Woshni model. The choice of coefficients being a function of the combustion system type. Open Cycle Coefficients are; Combustion System	A	B	C	Srat
Carburetted or Port Injected	3.26	9.12	0.834	0.0
Direct or Indirect Injected	3.26	6.18	0.417	0.0
Closed Cycle Coefficients are; Combustion System	A	B	C	D	G	Srat
Carburetted or Port Injected	3.26	4.56	0.616	0.00324	1.33	0.0
Direct or Indirect Injected	3.26	2.28	0.308	0.00324	1.33	0.0
Note the default coefficients provided for the direct and indirect injection engines are the same as those reproduced by Heywood (reference 6), with the same units being employed by the equations as shown in that text. The coefficients used for the carburetted and port injected engines are those which have been found to best match the measured performance and heat transfer results from test engines at Lotus. Often it is necessary to tune the coefficients of the in-cylinder heat transfer model to achieve good correlation both for volumetric efficiency and heat transfer. It is recommended that the B and C coefficients are tuned. An inexperienced user may find it more convenient to tune the swirl ratio term only. Eichelberg The convective heat transfer model proposed by Eichelberg is defined as; where h        =        heat transfer coefficient A        =        Eichelberg open or closed cycle A coefficient B        =        Eichelberg open or closed cycle B coefficient     =        mean piston speed p        =       Cylinder pressure T        =        Cylinder temperature The heat transfer per unit area of cylinder wall is defined as; Default coefficients are provided for the Eichelberg model. Open Cycle Coefficients are; Combustion System	A	B
All Combustion Systems Types	2.43	0.5
Closed Cycle Coefficients are; Combustion System	A	B
All Combustion Systems Types	2.43	0.5
This was the first and most simple of the published heat transfer correlations. The user is recommended to tune the A coefficient as required. References 1. Heat Transfer in the Cylinder of Reciprocating Internal Combustion Engines. W.J.D.Annand (Proc.I.Mech.E 177.973 (1963)) 2. Experimental Investigation of Instantaneous Heat Transfer in the Cylinder of a High Speed Diesel Engine. K.Sihling & G.Woshni. SAE 790833 3. Investigation of Internal Combustion Engine Problems. G.Eichelberg Engineering Oct 1939 Vol 148, 463 & 547 4. Unsteady Heat Transfer in Engines. V.D.Overbye et al. SAE Transactions NY 1961 461 5. The Thermodynamics and Gas Dynamics of Internal Combustion Engines (Volume 2) J.H.Horlock & D.E.Winterbone (section 12.4.3 pp 767) (ISBN 0-19-856212-8) 6. Internal Combustion Engine Fundamentals J.B.Heywood (section 12.4.3 pp 678) (ISBN 0-07-028637-X)

^+$#>Theory - Cylinder Scavenging

The in-cylinder scavenging model controls the way by which charge gas is mixed with the gas that is currently in the cylinder prior to the cylinder gas being exhausted. There are four scavenging models available. These are described as;

"         Perfect Mixing
"         Perfect Displacement
"         Brandham Benson Displacement Mixing Model
"         Blair Stripping Scavenging Model

It is important to note that all the published scavenge models assume an isobaric, isothermal, constant volume flow process. This is very different to the conditions found in the internal combustion engine. The scavenge models have been implemented in such a way that when the simulation model is constructed to simulate an isobaric, isothermal constant volume flow process then the classical scavenging response is obtained. In more conventional simulations the instantaneous scavenging response at each crank angle increment is assumed to be that defined by the scavenge model under isobaric, isothermal constant volume conditions. The overall scavenging response however is often very different to that produced by the classic models.

The definitions of the scavenging terms used in the program are as follows;

0        Scavenging efficiency
                                                          (1)

0        Scavenging Ratio

                                                          (2)

0        Charging Efficiency

                                                          (3)

0        Trapping Efficiency

                                                          (4)

0        In the above equations the terms are defined as:

             =        mass of air trapped in the cylinder;
             =        mass of residual gas trapped in the cylinder;
                =        mass of air supplied to the cylinder;
             =        mass of air in cylinder at bdc and reference conditions.

Note the use of the reference mass as the denominator in the scavenge ratio equation in a cycle simulation program produces an incorrect scavenging response due to significant changes in both cylinder pressure and volume.

Perfect Mixing Model

With the perfect mixing model the assumption is made that any charge gas entering the cylinder is instantaneously, homogeneously mixed with the gas currently in the cylinder. Thus the subsequent transfer of gas to the exhaust will cause some of the charge gas to be removed from the cylinder. This is the default scavenging model for all cylinders and results in the most pessimistic performance results.

Perfect Displacement Model

With the perfect displacement model the assumption is made that any charge gas entering the cylinder is NOT mixed with the gas currently in the cylinder. The subsequent transfer of gas to the exhaust will cause only residual exhaust gas to be removed from the cylinder. Under prolonged scavenging a point comes at which all the residual gas has been exhausted. Following this fresh charge air must be exhausted. This scavenging model produces the most optimistic results as the least amount of residual gas remains in the cylinder.

The perfect displacement scavenging model is used for all non-cylinder simulation elements. This ensures that if a reverse flow of cylinder gas to an inlet plenum is produced then that reverse flowed gas is first returned to the cylinder before any fresh charge air is flowed.

Displacement Mixing Model

Benson and Brandham (reference 2) suggested a hybrid scavenging model by which the initial part of the scavenging process exhibited perfect displacement scavenging up to a defined scavenge ratio (SCRA) at which the gas in the cylinder is assumed to instantaneously homogeneously mix. Subsequent scavenging produces perfect mixing results. Despite the period of displacement scavenging this model still tends to under predict the scavenge efficiency of ported cylinders at high scavenge ratios.

Stripping Model

Blair (reference 3) adapted and refined the displacement mixing model by proposing what this author has called the gas stripping model. The basic principal of the model is that the gas in the cylinder is contained in two discrete volumes, a so called perfect displacement volume and a so called mixing volume. As a packet of air enters the cylinder a proportion M is stripped off and homogeneously mixed with the residuals in the mixing volume. The remaining portion 1.0-M is placed in the displacement volume. The transfer of gas to the exhaust system draws gas from the mixing volume only, up to the point where there is no mixing charge remaining in the cylinder.

The key to this model is that the proportion of mixed air M is a continuous function of scavenge ratio.

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^+$#>Theory - Plenum Heat Transfer

Heat transfer in plenums is calculated using the connective heat transfer coefficient supplied by the user. For the majority of simulations the heat transfer coefficient may be set to 0.0. The following notes however provide a guide as to how a heat transfer coefficient of the correct order of magnitude may be calculated by the user.

The Nusselt number/Prandtl number/Reynolds number correlation usually applied to turbulent flow in pipes is;

where



and
h        =        heat transfer coefficient                 (W/m2/K)

k        =        gas conductivity                  (W/m/K)

cp       =        specific heat capacity            (kJ/kg/K)

    =        gas density                        (kg/m3)

v        =        gas velocity                       (m/s)

    =        dynamic viscosity                 (kg./ s.m)

d        =        characteristic length             (m)

Re arranging the above equation and assuming that the Prandtl number remains constant at around 0.7 yields


The following table provides typical air properties over a range of temperatures;

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^+$#>Theory - Ports

0        Modelling of Intake and Exhaust Ports
1        In modelling the intake and exhaust ports of engines the geometry of the port should be included in the pipe model, as the port element contains no length, it merely contains data relating to the valve flow coefficient at various valve lifts.

2        Modelling the Flow Through a Valve
3        When gas flows through a valve the development of separation and recirculation regions gives rise to a vena-contracta where the actual cross-sectional area of the gas stream (effective area) is less than the geometric area of the orifice. This phenomenon cannot be simulated directly using a one-dimensional model and has to be characterised using empirical data. Data giving measured effective valve areas, or flow coefficients (), are required as input values to Lotus Engine Simulation. There are several other boundary features which require similar information or data giving the variation of pressure drop with mass flow rate across the device (for example Throttles).

5        The effective area of a valve is a hypothetical concept which enables the mass flow through the valve to be evaluated for a given pressure difference across it. A mathematical model of the flow through the valve is developed, from which the effective area of the valve throat can be derived from the measured values of pressure across the valve and the mass flow rates. The value of effective area obtained is dependent on the particular mathematical model (Woods and Khan [1]) and therefore if the data is to supplied to a wave-action simulation program it is imperative that the model used to analyse the steady-flow data matches that employed in the boundary model of the computer program. In this way the use of effective flow area measured using a steady-flow rig enables the mass flow rate obtained in the experiments, for a particular valve lift and pressure difference across it, to be reproduced by Lotus Engine Simulation. The Port Flow Tool Section describes the measurement procedure in detail.

6        Calculation of the Effective Area,
8        The purpose of this section is to outline general principles and not to review the details of various models for predicting the flow of gas through a valve. The specific example of subsonic flow through an exhaust valve will be used to develop an expression for the effective flow area of the valve. The form of the final expression giving the mass flow rate as a function of pressure ratio and effective flow area is identical for subsonic flow through an inlet valve into a cylinder. In the latter case, the static pressure in the intake pipe is used as the upstream pressure and the stagnation pressure in the cylinder, , is the downstream pressure value.

10       Consider a single-cylinder engine, as shown below, where the flow exits from the cylinder through a single valve into an exhaust port/pipe of constant cross-sectional area.

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^+$#>Theory - Valves

0        Valves may be specified by one of five options;

"         Poppet valve
"         Self acting reed valve
"         Disc valve
"         Piston port
"         User specified angle area curve

         Poppet Valves

The valve lift profiles may be specified by one of four options;

"         Default fast lift polynomial
"         Default slow lift polynomial
"         User specified polynomial
"         User specified angle/lift ordinate data

With each of the options the valve lift duration is specified by the number of crank degrees between valve opening (AVO) and valve closing (AVC). When the user specified angle/lift ordinate data option is used the lift profile data are linearly scaled so that the lift duration matches that specified with AVO and AVC. The advantage of this scaling is that the user may specify one generic valve lift profile and perform valve timing sensitivity studies by changing only one or two numbers (ie AVO and AVC) in the input data file.

With each of the lift profile options the maximum valve lift is specified by the maximum valve lift AVLM. When the user specified angle/lift ordinate data option is used the lift profile is linearly scaled so that the maximum valve lift matches that specified with AVLM. Users who wish to perform valve timing sensitivity studies should be aware that the maximum achievable valve lift reduces with reducing lift duration. Thus in order to generate realistic valve timing trade-offs the maximum lift should be adjusted with the valve lift duration.

         Polynomial Lift Curves

The default lift curves employ a polynomial consisting of four coefficients and four exponents. The nature of the polynomial is such that the sum of the coefficients is -1.
The coefficients of the default lift curves are;

Fast Lift	Slow Lift
Coefficient	Exponent	Coefficient	Exponent
-1.2423	2	-1.507928	2
0.2553	12	0.541945	7
-0.1148	68	-0.048289	30
0.1019	70	0.014273	40
These are shown below The default and user specified polynomial lift options allow the user to input a maximum lift dwell angle. This is the number of degrees at which the valve remains at maximum lift after the opening before starting to close. The dwell angle should not be a negative number. { Polynomial Valve Lift Curves User Specified Angle/Lift Ordinates The user specified angle/lift ordinate data option allows the user to provide the actual cam design data as input to the simulation. This data is specified in crank angle / valve lift ordinate pairs. The first crank angle should be 0.0 and the last the lift opening duration (although the duration may be subsequently scaled as described above). The first and last valve lift ordinates should be 0.0. The figure above compares the a Lotus designed direct acting 235 cam valve lift ordinate curve with those generated by the default slow and fast lift curves generated for the same to of ramp duration and maximum lift. The most significant difference between the cam design curve and the fast lift polynomial is during the ramp at the beginning and end of lift. It is recommended that not all of the ramps are included in the angle/lift ordinate data. Experience has shown that best simulation results are achieved when angle/lift ordinate data are included for approximately 10 crank degrees before the top of the opening ramp and after the top of the closing ramp. The most appropriate extensions to the lift curve will change from engine to engine depending on the tappet clearance and flexibility of the valve train. The strategy employed to convert cam design data into valve lift ordinate data for input to the simulation is summarised as follows; { Cam Profile Corrections Self Acting Reed Valves A relatively simple self acting reed valve model is employed in the program. The model shown schematically below, employs a spring mass representation of the valve/reed that is forced to move between the valve seat and the lift stop by the pressure on either side of the valve and the area over which this pressure acts. { Self Acting Reed Valve Model The force on the valve is given by                                         (1) where              =        area of petal;              =        the stiffness of the valve;              =        valve lift;              =        mass of the valve;              =        acceleration of the valve. The valve velocity is integrated as                                                                    (2) and the valve displacement is then calculated from the equation          .                                                         (3) The valve lift can then be integrated using the equation          .                                                         (4) Finally, the flow area is evaluated as          where              =        the geometric flow area available;              =        the discharge coefficient of the passage. The model assumes that there is no valve bounce on either the valve stop or the valve seat. This implies that the self-acting valve is well matched to the application. Disc valve The disc valve model calculates the flow area of a port which is covered and uncovered by a disc which rotates at crankshaft speed. The model is shown diagramatically below,. { Disc Valve Model The flow area is calculated from the area of the port that is uncovered by the disc valve and the disc valve discharge coefficient. The discharge coefficient is assumed to reduce with increasing area from 1.0 to the value for the fully uncovered port provided by the user. Piston Ported Valve The piston ported valve model calculates the flow area of a port which is covered and uncovered by moving piston. The model is shown diagramatically below. { Piston Ported Valve Model The flow area is calculated from the area of the port that is uncovered by the piston and the port discharge coefficient. The discharge coefficient is assumed to reduce with increasing area from 1.0 to the value for the fully uncovered port provided by the user.

^+$#>Theory - Throttles

The throttle option specifies the characteristics of constant area flow devices that are used to connect one element to another. Note that only one pipe, plenum, or other element can be connected to each side of a throttle.

Essentially two items of data are required by the throttle element: geometric flow area and flow coefficient (). The product of the geometric flow area and the value then gives the effective flow area of the throttle.

Geometric Area
Throttles may be specified as one of the following types:
"         Simple Area
"         Butterfly
"         Slide Plate
"         Slide Valve
"         Barrel Valve

Lotus Engine Simulation calculates the geometric area at a given throttle position, normal to the direction of flow, for each of these throttle types.

The throttle flow coefficient can be supplied to Lotus Engine Simulation directly, however, it is important to ensure that the throttle area option selected is consistent with the way in which the throttle data has been processed, (i.e. so that the reference area, , is consistent - See Ports).

Effective Flow Area
When gas flows through a throttle valve the development of separation and recirculation regions gives rise to a vena-contracta where the actual cross-sectional area of the gas stream (effective area,) is less than the geometric area of the orifice. This phenomenon cannot be simulated directly using a one-dimensional model and has to be characterised using empirical data. Data giving measured effective throttle valve areas, or flow coefficients (), are required as input values to Lotus Engine Simulation. There are several other boundary features which require similar information or data giving the variation of pressure drop with mass flow rate across the device (for example Ports).

The effective area of the throttle is a hypothetical concept which enables the mass flow through the throttle to be evaluated for a given pressure difference across it. A mathematical model of the flow through the throttle is developed, from which the effective flow area of the throttle can be derived from the measured values of pressure across the throttle and the mass flow rate through it. In this way the use of effective flow area measured using a steady-flow rig enables the mass flow rate obtained in the experiments, for a particular throttle opening and pressure difference across it, to be reproduced by Lotus Engine Simulation. The measurement procedure for throttles is described briefly below. The Port Flow Tool Section describes the measurement procedure in detail for poppet vavles.

Steady Flow Test
If an experiment is performed in which the mass flow rate through, and pressure drop across, a device are measured, then the effective flow area, , of the section of the device lying between the upstream and downstream pressure tappings can be evaluated.

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 rig it can be assumed that, provided that a reasonable entry bellmouth is used and there are no significant pressure losses between the entry and the throttle, the upstream stagnation pressure is equal to the reservoir, or ambient, pressure,

1      , is used. For a

 rig it is assumed that the upstream stagnation pressure, 2         , is equal to the ambient pressure, 3       . For a

 rig it is assumed that the downstream static pressure, 4    , is equal to the ambient pressure, 5       . Thus:

^+$#>Theory - Turbochargers

Turbochargers are modelled as compressors and turbines on a common free spinning (or compounded) shaft. The general approach is the same as that published in references 1, 2 and 3, with the instantaneous compressor and turbine performance being derived from non dimensionalised characteristic maps.

The input data structure has been designed to be as similar as possible to that published in the SAE J1826 turbocharger gas stand test recommended practice. The provision of mass flow, pressure ratio, speed and efficiency scaling factors to allow the user to scale a base map to fine tune a particular compressor / turbine characteristic to a given engine application. The compressor and turbine routines are designed to be very robust. Thus smoothing and extrapolation of test data is not essential prior to input to program. The extrapolation assumptions made within these routines are described in the following sections.

The accurate simulation of free spinning turbochargers demands that the simulation converge on a shaft speed that provides an exact work balance between compressors and turbines. Convergence is judged to have been achieved when the turbine work is within 2% of the compressor work. At the end of each cycle the simulation examines the shaft work balance and automatically increases or decreases the shaft speed as appropriate. Within each cycle the shaft speed is permitted to fluctuate in response dynamic imbalance between compressor and turbine work. The amplitude of this imbalance is controlled by the compressor and turbine inertias.

Compressors

Compressor maps must be defined as a series of constant speed lines defining mass flow, pressure ratio and efficiency. The speed lines must each employ the same number of mass flow points per curve and must be monotonically increasing in order. The input data order is summarised by the following diagram.

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Compressor Map Data Entry Order

Extrapolated Constant Speed Line

Turbine Map Data Entry Sequence

Typical Turbine Map

Extrapolated Constant Speed Line

^+$#>Theory - Superchargers

The Model

The objective of the supercharger model in Lotus Engine Simulation is to calculate the pressure and temperature rise of the gas, and its mass flow rate as it passes through the device. Fig. 1 shows the variation in the state values, upstream and downstream of the compressor, on a T-s diagram.

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Fig. 1. Temperature  entropy diagram for flow through compressor

 this is achieved using the equation
                  .                                                                 (7)

It is important to note that the isentropic and adiabatic efficiencies are used for calculating different quantities.

Defining the Input Data

^+$#>Theory - Expanders

The Model

The objective of the expander model in Lotus Engine Simulation is to calculate the pressure and temperature drop of the gas, and its mass flow rate as it passes through the device. Fig. 1 shows the variation in the state values, upstream and downstream of the expander, on a T-s diagram.

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Fig. 1. Temperature  entropy diagram for flow through expander

 this is achieved using the equation
                                                                           (7)

It is important to note that the isentropic and adiabatic efficiencies are used for calculating different quantities.

Defining the Input Data

^+$#>Theory - Charge Coolers

Charge coolers provide a means by which heat is subtracted from (or supplied to) the gas in the engine simulation model. The characteristics of the charge cooler are supplied in the form of pressure loss, coolant temperature and effectiveness verses mass flow rate ordinate data. At each instant the simulation program calculates the mass flow across the charge cooler flow device from the instantaneous pressure drop. The accompanying coolant temperature and effectiveness data similarly derived.

The provision of mass flow information should more correctly be in the form of a volume flow rate ordinate data from which the mass flow rate could be calculated from the inlet conditions. However mass flow data appears to be more readily available and is therefore used. If required alternative forms of charge cooler input data could be provided.

The principal assumption of the charge cooler model is that the effectiveness and coolant temperature characteristics within the engine cycle are quasi static. This implies that the charge cooler has no thermal inertia.

The charge cooler effectiveness, , is defined as



where
=Charge cooler gas inlet temperature
=Charge cooler gas outlet temperature
= Charge cooler coolant temperature

^+$#>Theory - Mechanical Links

Compressor and turbines are linked to shafts via a specified gearing and mechanical efficiency. The mechanical efficiency is that efficiency by which work is transmitted to or absorbed from the shaft. This may be used to model the bearing losses in a turbocharger.

The inertias specified are for the compressor of turbine wheel only. The inertia referred to the shafts by the gearing is automatically calculated within the program.

Both compressors and turbines may be linked to the crankshaft. This causes the flow devices to operate at one speed only. The power absorbed by or transmitted from these flow devices is added to the TOTAL engine performance and economy results printed in the .MRS file.

References:

1. The Thermodynamics and Gas Dynamics of Internal Combustion Engines (Volume 1) R.S.Benson (section 9 pp 479) (ISBN 0-19-856210-1)

2. Internal Combustion Engines (Volume 2) R.S.Benson & N.D.Whitehouse (chapter 10 pp 339) (ISBN 0-08-022720-1)

3. Turbocharging the Internal Combustion Engine. N.Watson & M.S.Janota (section 15 pp 517) (ISBN 0-333-24290-4)

^+$#>Theory  Engine Dynamics

Simulation of engine transient performance requires the calculation of the engine mechanism dynamics. The engine dynamics can be calculated in LES based on cycle-averaged values of brake torque and engine inertia so that the engine speed is updated once per cycle. Alternatively the intra-cycle variation of torque and inertia can be considered so that the engine speed is updated at every calculation time step.

The basic equation for calculating the engine acceleration is

                           ,                              (1)

where is the engine speed in rev/sec, is the brake torque, is the load torque, is the total engine inertia referred to the crankshaft, is the load inertia, and is the engine speed. The brake torque is given by

                               ,                                   (2)

where is the torque generated by the gas pressure forces, is the resisting torque generated by the engine friction, and is torque generated by the engine inertia forces at any particular crank position and is given by

                                                                                     (3)

The engine inertia referred to the crankshaft varies as a function of crank angle and is given by

,   (4)

where is the rotational inertia of the crankshaft about its centreline, is the rotational inertia of the valvetrain system, is the rotational inertia of the engine accessaries, is the need of the valvetrain system relative to the crankshaft, and is the speed of the accessary drive relative to the crankshaft.

In equations (3) and (4) the quantity represents the reciprocating mass of each piston / cylinder assembly (including a contribution from the connecting rod mass  see below). The parameter represents the rotating component of the connecting rod mass, is the distance from the crankshaft centreline to the piston-pin centre, is the crank throw, is the connecting rod length, is the crank angle with respect to TDC, is the angle between the connecting rod and the cylinder / crank axis, is the inclination of the cylinder / crank axis from the vertical, is the acceleration due to gravity.

The term represents the residual inertia of the two-mass representation of the connecting rod. Using this approach simplifies the engine dynamics calculations by lumping the mass of the connecting rod at its extremities (large- and small-end centres). The two lumped masses obey the relationships

                                                                            (4)

and

                  ;                     .                     (5)

In these equations and are the distances of the rotating and reciprocating mass components from the centre of gravity (C of G) of the connecting rod, shown in Fig. 1. This two-mass representation of the connecting rod requires the addition of a residual inertia component in order to generate the actual inertia of the connecting rod using the two lumped masses from the equation

                                                          (6)

where is the actual inertia of the connecting rod about its centre of gravity in an axis parallel to the axis of the crankshaft centreline. The residual inertia can also be expressed as a function of the radius of gyration of the connecting rod in the form

                                                                   (7)

The input data variables related to the quantities in the above equations are described in the Input Data Section of this Help File.

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.                              (8)

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^+$#>Theory - Friction

The mechanical friction of the engine may be either, calculated using one of four simple empirical correlations or specified explicitly by the user in the form of a friction mean effective pressure or a mechanical efficiency. Note that the input data is the mechanical friction only and should not include pumping losses as these are calculated by the program. Thus unadjusted motoring loss data cannot be used as input data.

The available friction models are summarised as

Modified Barnes-Moss

(reference 1)

,

where   

N        =       engine speed [rev/min]

    =       mean piston speed [m/s].


Modified Millington & Hartles DI

(reference 2)



where

CR       =        Compression ratio

Chen & Flynn Model for Large Engines

(reference 3)

,

where
    =        Maximum cylinder pressure [bar]

The recommended procedure is to calculate the engine friction using a modified version of the Patton & Heywood model (reference 4) that requires crankshaft and camshaft configuration and bearing dimensions. This model is coded into the Lotus Friction Tool and data can be written directly from this code into the input data of a Lotus Engine Simulation model.

References

1. A Designers Viewpoint. H.W.Barnes-Moss. I.MECH.E C343/73

2. Frictional Losses in Diesel Engines. B.W.Millington & E.R.Hartles.
SAE 680590 (1968)

3. Development of a single cylinder compression ignition research engine.
S.K.Chen & P.F.Flynn. SAE 650733

4. Development and Evaluation of a Friction Model for Spark Ignition Engines. K.J.Patton, R.G.Nitschke & J.B.Heywood SAE 890836

^+$#>Theory  Silencer Modelling and Noise Prediction

The approach to modelling silencer elements in the Lotus Engine Simulation code follows the theoretical approach described by Onorati in Ref. 1. Two types of silencer element can be modelled using the built in Silencer Super-Elements within Lotus Engine Simulation, these are simple reactive silencer elements and perforate/resistive elements.

Modelling Simple Reactive Silencers
Reactive silencers achieve the abatement of sound pressure levels by reflecting the acoustic power, carried by pressure waves, back to the noise source. They exploit the mechanism of reflection and transmission of sound waves at geometrical discontinuities (abrupt area changes, junctions of pipes, etc.) to control the acoustic power generated by the source and transmitted downstream along the pipe-system, through the interaction between the engine noise source and the silencing device. These silencers are distinct from absorptive, or dissipative, devices, which make use of sound-absorptive material to dissipate the acoustic energy as heat when pressure waves travel along the ducts of the acoustic element.

The simplest reactive silencer configurations that may be introduced in intake and exhaust pipe systems are expansion chambers, and Helmholtz and column side (quarter-wave) resonators. The correct calculation of the acoustic characteristics of these elements is very important, since they represent the fundamental elements used to build up more complex silencers through acoustically equivalent schemes.

Expansion chambers, shown generically in Fig. 1a, have a broad-band attenuation, which approaches zero for the resonant, or transparency, frequencies of the system: these frequencies are given by

                                                                            (1)

where n = 1, 2, 3,..& Resonators have a narrow-band action, with intense attenuation only at the resonant frequencies. The frequency of a Helmholtz resonator, shown in Fig. 1b, is

                  ,                                                (2)

where F and ln are the cross sectional area and the length of the neck respectively, V is the volume of the chamber. The resonant frequency of a column (or quarter wave) resonator, shown in Fig. 1c, is

                 ,                                                                (3)

where lr is the column resonator length and n=1, 3, 5, ...)

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as
                                                          (3)
where the harmonics f

n (for a four-stroke engine) are given by:
                                        (4)
and f

n of the spectral components of the pressure, so that the corresponding r.m.s. values are and the sound pressure level spectrum is given by:

^+$#>Combustion Analysis Tool  Overview

The Combustion Analysis Tool is essentially a stand-alone combustion analysis program. However, it also allows the user to quickly create user-defined heat release phase and period combustion data for use in the test conditions section of the Lotus Engine Simulation code models.

The program uses a simple heat release approach to analyse cylinder pressure / crank angle data and calculates the burn duration, phase, and mass fraction burned, which can be used as input data for the Lotus Engine Simulation code combustion model. (See Data module  Heat release phase or period)

The program also calculates the rate of pressure rise and heat release and enables graphical display of these quantities.

The Combustion Analysis Tool can also be used in conjunction with a database.

^+$#>Combustion Analysis Tool  Opening the Combustion Analysis Tool

There are three methods of opening the Combustion Analysis Tool:

Firstly, after loading the Lotus Engine Simulation code, if the Start Wizard is active, then the user is able to select the Combustion Analysis Tool option directly from the wizard.

However, if the start wizard had been disabled or the user is already working within the Lotus Engine Simulation code, they must select either Tools / Combustion Analysis Tool from the main menubar or click on the Combustion Analysis Icon near the top of the window.

^+$#>Combustion Analysis Tool  Closing the Combustion Analysis Tool

In order to close the Combustion Analysis Tool, either click on the Close Icon at the top right of the window or select File / Close from the combustion analysis tool menubar.

On the Combustion Analysis File menu, there is another close option named Close (make current), as shown below. This also closes the Combustion Analysis program but at the same time, also copies the calculated data into the relevant sections of the current Lotus Engine Simulation code model.

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^+$#>Combustion Analysis Tool  Solving

Once all required data has been entered, it can be solved by selecting File / Solve Update from the Combustion Analysis Tool menubar, as shown below. This will produce results, which can be viewed through the Text Results and Graphical Results sections.

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^+$#>Combustion Analysis Tool - Viewing Graphical Results

Graphical results can be viewed by left-clicking on the Graphical Results button, as shown below. This will display the graphical results window which contains a graph on the left hand portion of the window and a display control section on the right hand side of the display.

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^+$#>Combustion Analysis Tool - Listing Graph Values

If the user wishes to accurately read off particular values from the displayed graph, then they should firstly select Graphical Results / List from the Combustion Analysis Tool menubar. When this has been done, cross-hairs will appears as the user moves the mouse pointer over the graph area. To list a graph value, click on the graphical display at the point of interest. X axis (Engine RPM) and Y axis (from whichever graph is selected) values will be displayed at the bottom of the graph area, as shown below. The colour of the text indicates which graph values are being displayed. The value displayed will relate to the point at which the vertical cross-hair crosses the line which is closest to the cross point of the cross-hairs. Click with the cross-hair cross point as close as possible to the point of interest. To remove the cross hairs when finished listing values, click the right mouse button.

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^+$#>Combustion Analysis Tool - Listing Database Entries

When there is data stored in the database scratch file (see Database Structure) then it is possible to list the stored database entries. This is done by selecting Database / List Entries from the Combustion Analysis Tool menubar, as shown below. After performing this task, a window will appear with a spreadsheet-style layout of the database data. Particular entries can be highlighted by clicking on them with the left mouse button.

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^+$#>Combustion Analysis Tool - Reverting to Original Database Order

In order to return the database order back to its original order, when the database listing has been displayed, press the right mouse button whilst the mouse pointer is positioned anywhere on the database listing and select Revert to Original Order from the pop-up menu, as shown below.

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^+$#>Combustion Analysis Tool - Clipping Columns

An alternative method of hiding certain database entries is to clip columns. This allows the user to hide the entries above, below or on either side of specific column values. In order to do this, position the mouse pointer over the column heading of interest and then press the left mouse button to select the column. Then press the right mouse button to bring up the pop-up menu. From the listing, select either High Clip Selected Column (To hide entries with column values above a certain value), Low Clip Selected Column (To hide entries with column values below a certain value) or Pass Clip Selected Column (To hide entries above and below certain values). After selecting the type of clip, a dialogue box will appear, requesting the relevant column value(s). Enter the value(s) to complete the procedure, as depicted below.

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^#Combustion Analysis Tool Icon

^+$#>Concept Builder  Overview

What is the Lotus Engine Simulation code Concept Builder?

The Lotus Engine Simulation code Concept Builder is a powerful tool allowing the user to quickly gain an appreciation of the parameters associated with a particular engine configuration. It considers the dimensions of and gas flow through, the intake system, cylinders and the exhaust system. The Lotus Engine Simulation code Concept Builder can be used in isolation from the simulation program as a stand-alone analysis tool, or it can be used to quickly generate the basis of a pipe network model for a simulation. It uses established theory combined with Lotus Engineering engine knowledge to produce an engine model that can provide a starting point for the engine development process.

^+$#>Concept Builder  Starting the Concept Builder

The Concept Builder can be accessed in three ways. The Concept Builder can be entered from the wizard which appears when the simulation code is started up. It can be activated by clicking on the appropriate icon at the top of the Network Builder screen. Alternatively, the Concept Builder can be accessed by clicking Concept Tool within Network Builders Tools menu.

^+$#>Concept Builder  Layout of the Concept Builder

The Concept Builder tool consists of a single interface window. Fundamental engine parameters including the number of cylinders, total swept volume and the maximum power speed are located at the top of the window. Directly below are the pressures and temperatures at the inlet and the exhaust respectively. Default values are used within Concept Builder for these, but user defined values can be entered.

The central regions of the window display the boxes associated with all the basic dimensions of the intake system and the exhaust system. An outline diagram of the engine system indicates each dimension graphically.

The final section of Concept Builder is highlighted in blue and includes all non-dimensional parameters calculated by the Concept Builder code. These include valve and valve timing details, tuning speeds, gas flow parameters and the mean piston speed.

The parameters which have notepad symbols next to them in the Concept Builder interface window allow the user to define how the values of the parameter are calculated from a given list of other parameters. In this way the user is able to over-ride the Lotus devised criteria for the determination of many of the Concept Builder parameters.

The Extended Data section enables the user to modify some of the criteria which are used to specify the intake and exhaust options.

^+$#>Concept Builder  Running the Concept Builder

The simplest way to run the Concept Builder is by entering only the No. of Cylinders, Swept Volume and the Maximum Power Speed for the desired engine. The Concept Builder will run automatically when enter is pressed. It should be noted that these three values must always be entered in order to run the Concept Builder.

Each of the parameter boxes has a padlock icon next to it. By activating this icon the user will fix any value that has been entered into the box. This feature allows the user to fix parameters relating to the engine. The Concept Builder will then calculate all the other parameters based these values. Fixed values are highlighted in purple.

Lotus Engineers have specified working ranges for the values within Concept Builder. The ranges recommended include all reasonable parameter values according to Lotus knowledge. If a value is entered that is outside of this range, or if a value is calculated that is outside of this range, the relevant box or boxes will be highlighted in red, as shown in the screen-shot below.

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^+$#>Concept Builder  System Dimensions

System Dimensions

Dimensional data relating to the intake and exhaust systems within the Concept Builder engine is calculated according to limits and ratios dictated by Lotus Engineering. These limits and ratios have been found to provide the most desirable engine performance characteristics.

Calculations Performed

"         Bore (1) is initially calculated by assuming a Bore/Stroke ratio of 1:1.

"         Maximum stroke is calculated using the maximum engine speed and assuming a maximum permitted piston speed of 20ms-1. For the purposes of this calculation, Bore(1) is used.

"         Actual stroke is calculated for the engine to achieve the specified swept volume. If this stroke exceeds the maximum stroke calculated, its value will be re-set to the limited value.

"         Actual Bore is calculated from the Actual Stroke. This value replaces the Bore(1) value calculated previously.

"         Inlet throat diameter is calculated from the bore diameter. The throat area is taken as 23% of the bore area.

"         Inlet throat gas velocity is calculated using the continuity equation. This considers the expanding volume of the cylinder as the piston moves down and calculates the corresponding velocity of the gas through the throat, assuming that the gas is an incompressible fluid. If the gas flow speed exceeds 80ms-1, Concept Builder changes the throat diameter to that which will result in a maximum gas velocity of 80ms-1.

"         Exhaust throat diameter is calculated as a proportion of the inlet diameter. Lotus Engineering guidelines dictate that the exhaust throat area is 70% of the inlet throat area.

"         Inlet port diameters are calculated according to a throat/port area ratio of 1: 0.8

"         Inlet port gas velocity is calculated using the standard continuity equation. If the calculated gas velocity exceeds 110ms-1, the port diameter is set to a value that will limit the gas flow to this maximum value.

"         Exhaust port diameters are calculated according to a throat/bore area ratio of 1: 0.9

^+$#>Concept Builder  Tuning Equations & Theory

It is critical that within an engine both the intake system and the exhaust system are tuned so that the desired torque and power curves can be realised. Engine tuning considers the propagation of pressure waves through the system and their reflection. Pressure waves can be used to enhance the volumetric efficiency of the intake system and to aid the removal of residual exhaust gas in the exhaust system. Concept Builder uses two basic tuning equations.

The intake system is tuned using the Helmholtz resonator equation. Concept Builder sets the Helmholtz tuning speed as Max Power Speed  1500 rev/min, although a user defined value can be set. The Helmholtz speed is the point of maximum volumetric efficiency and is therefore the point of maximum torque - See Concept Builder Theory  Helmholtz resonator method.

The exhaust system is tuned using a simple wave propagation equation based on the wave propagation speed in the gas. The calculation is performed on the basis of the blow-down pulse being reflected as a rarefraction wave at the end of the exhaust primary pipe and this wave returning to the exhaust valve within a crank angle period of 120. This aims to ensure the reflection of the peak of the blow down pulse during the valve overlap period to assist scavenging.


Concept Builder Theory - Exhaust Tuning.


Concept Builder Theory


Helmholtz Resonator Method

The Concept Builder uses the Helmholtz Resonator method to calculate the intake pipe dimensions required to achieve a desired tuning speed. The tuning speed of an engine is the speed at which the induction process matches the natural frequency of the combined pipe and cylinder system.

The Helmholtz Resonator method considers the gas within the intake pipe as a finite incompressible mass. The volume of gas is considered as a spring with no inertia. Deceleration of the gas plug causes a peak in pressure at BDC.

Concept Builder can be used in a number of ways in relation to the tuning speed. It can either be used to calculate the Helmholtz speed directly, or it can be used to calculate the exhaust length required to provide tuning at a user defined engine speed. The user can set a specified engine speed by entering a value in the Helmoltz engine speed box and activating the locking device indicated by the padlock icon.


Helmoltz Resonator Equation




where    N = engine speed (rev/min)
Fp = Pipe cross sectional area
Lp = Pipe length (m)
Vc = Mean cylinder volume = 0.5 * cylinder swept volume + clearance volume.

a = speed of sound =

where    = Ratio of specific heats
        R = Gas constant
        T = Temperature (K)

N.B - The values of Fp and Lp are modified to take into account the tapering of the intake pipe.


Tuned Exhaust Speed


Concept Builder uses a standard wave propagation equation to calculate the exhaust length for a user specified maximum power speed. Alternatively Concept Builder can be used to determine the tuning speed for a specified exhaust length.

Concept Builder calculates the necessary exhaust pipe length by first calculating the speed of sound for the fluid. It then determines the theoretical distance travelled by the wave during a 120 period. By multiplying this time duration by the speed of sound, the wave propagation distance may be calculated. The exhaust length is half the total distance calculated.

Calculation of engine speed from a user defined exhaust length is calculated using a rearrangement of the same equation.


Exhaust Tuning Equation

Time for blow down and pulse return
where    N = engine speed (rev/min)


Exhaust Length = Time for blow down and pulse return * speed of sound * 0.5


Gulp Factor

The volumetric efficiency is a ratio of the mass of air trapped in a cylinder to the mass of air that could be trapped within the swept volume if the air was at inlet manifold density. This efficiency must be high in order to maximise the performance of an engine. If all other parameters remain constant, the mean effective pressure is directly proportional to it.

The Concept Builder calculates the so-called Gulp Factor of the system to indicate the limitation of the breathing system. Firstly Concept Builder takes a default Lotus valve lift profile. This profile can be scaled to suit a user specified valve lift duration if required. Next it pescribes a flow coefficient curve corresponding to the bore/stroke ratio for the engine in question. From this curve the flow coefficient at each individual valve lift point can be determined.

A mean effective valve area for the duration of the valve open period is determined by integration of the valve lift curve. Finally the Gulp Factor is calculated.

The Gulp Factor is in effect the Mach Index for the fluid, although the Mach Index does not take into account the flow coefficient. The Mach Index is the average Mach Number over the entire valve open period and it is proportional to the ratio of the bore area to the mean inlet valve area. Increasing Mach number beyond a threshold value corresponds to decreasing volumetric efficiency. This trend is a consequence of the flow within the inlet valve approaching sonic speeds and thus choking.


Inlet Gas Velocity/ (Mean Flow Coefficient * Mach Number) = Gulp Factor



= Mach Index

= Bore Area
= Piston Speed
= Mean Inlet Valve Area
= Speed of sound in gas =

^+$#>Concept Builder  Editing Equations and Functions

The func tions and empirical relationships used within the concept tool are based on a combination of Lotus Engineering Experience and fundamental gas dynamics. All of these relationships are expressed in terms of Fortran syntax that the user can edit and replace the default settings with their own. These changes are saved to the users ini file.

To identify whether a variables relationship can be changed look for the edit icon next to the data field. The example sectional screen shot below shows several data fields with the function editor adjacent to them, (ringed).

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Example Screen shot showing Edit Icon

Function Editor

^+$#>Concept Builder  Extended Data Setting

Some of the model network geometry values can be changed via the Extended Data tab on the concept tool. This employs the same user Fortran function method as with the main data fields.

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Extended Data Setting

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^#>Concept Builder Edit Icon

^+$#>Data Checking Tool - Overview
Overview

The data checking wizard provides a tool that allows the user to check the validity and quality of the current data. A large number of checks are performed and a list is given for each data section, of the number of Errors, Warnings and Comments found in the current data. A message is given for each item in the list that identifies the particular data variable at fault.

The data checking wizard is run in one of two modes, either directly as a interactive window, or indirectly as a summary message dialogue.

The data checking wizard is run directly through the menu item Tools / Data-check Wizard.
This displays a window that shows the list of messages in a scrollable text region adjacent to the appropriate data section icon.

The data checking wizard is run indirectly every time a calculation is performed, the data values are checked and if any discrepancies identified a simple summary of the number of errors, warnings and comments is displayed.

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^+$#>Data Checking Tool  Data Checking Fail Types

Data Checking Fail Types

Three types of message are displayed by the data checker, these are Error, Warning and Comment. Due to the complexity of the data requirements and the inter dependency it is not always clear cut as to the appropriateness of a particular value or flag setting. Some solution types will use different data values and thus adds further vagaries to their validity.

The first category of Error is used when a data value(s) or type is felt to be in error in all possible scenarios. Typical examples of this are failure to enter a value for a compulsory variable, or incorrectly entered, negative or out of range numbers.

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^+$#>Data Checking Wizard  Opening the Data Checking Wizard

Opening the Data Checking Wizard

To open the data checking wizard select the menu Tools / Data-check Wizard from the main window menubar. Alternatively the Data Checking Icon can be selected. Whilst the wizard is open the icon remains indented and the pull down menu item is ticked.

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^+$#>Data Checking Wizard  Updating the Data Checking Wizard Display

Updating the Data Checking Wizard Display

If the data checking wizard window has been left open whilst changes have been made to data, its display will potentially no longer reflect the true No. of errors, warnings and comments. To update the display select Functions / Update from the wizard menubar. The current data will then be checked and the wizard display updated.

^+$#>Data Checking Wizard  Closing the Data Checking Wizard

Closing the Data Checking Wizard

To close the data checking wizard select either the close icon at the top right corner of the wizard window, the wizard window menu at the top left, the menu item Functions / Close from the wizard menubar, or alternatively the Data Checking Icon can be un-selected.

^+$#>Input Data - Overview

The Lotus Engine Simulation interface allows the user to enter data, read in, save models, create new models, and adjust data in existing models. Data entered via the interface is written to an input data file which has the extension .sim. This file is read by the program Solve Module when the calculation begins.

Icons representing the various model elements are associated with property sheets which allow the user view and edit the data for that element. Graphical features allow the user to view the result of changes to some of the specific data-sets and adjust data. The network builder interface gives a visual representation of the engine model. More detailed descriptions of the models used by the program can be found in the Theory section of this help file.


Data Sub-Components

The sub-components of the engine model are:

"         Base Engine Data
"         Fuel and Fuel System Data
"         Combustion and Heat Transfer Data
"         Scavenge Model Data
"         Ports and Valves Data
"         Pipes and Plenums Data
"         Throttle Data
"         Turbocharger and Compressor Data
"         Inlet Data
"         Exit Data
"         Intake/Exhaust Super Elements
"         Test Conditions Data

When an element is placed on the builder is selected the relevant property sheet is displayed which enables the user to edit the properties associated with the element. Some property sheets spawn subsidiary windows in which the user can enter more detailed information related to a particular sub-model. Property sheets may also contain spreadsheets. Certain functions can be performed on elements from the Right Mouse Button menu, e.g. pipes can be automatically split at a nominated point.

0        Note that when editing property sheets variables are held in memory after editing when the user selects another elements or submits the data to be run.


0        Model Structure

Simulation models of the engine system are created through defining elements. Six element types are provided:

"         Cylinders (zero-dimensional elements with combustion, work and heat transfer);
"         Plenums (zero dimensional elements with work (optional) and heat transfer);
"         Pipes (one-dimensional elements with wall friction and heat transfer);
"         Inlets (infinite source of inlet gas at specified pressure and temperature);
"         Exits (exhaust boundary specified pressure);
"         Closed end (special element used for pipes end connections).

These elements are connected by so called flow devices which regulate the flow of gas between the elements. The currently available flow devices are;

"         Valves (both cam operated valves, piston-ported valves, reed-valves and disc valves);
"         Throttles (defines a flow area and discharge coefficient);
"         Compressors (full turbocharger compressor map model);
"         Turbines (full turbocharger turbine map model);
"         Charge Coolers (flow device with pressure loss and heat transfer);

Each element may be connected to another via any flow device with the exception of a multi-pipe junction. Two pipe junction models are available:

"         the Constant Pressure model  produced by simply connecting together pipe ends;
"         the Pressure-Loss model  produced by placing an icon over an existing constant pressure junction an supplying junction pipe branch angles.

The pressure-loss model is particularly suited to modelling junctions in high-speed engines and those with pulse-converter manifolds.

^+$#>Input Data  The Sim File

The Lotus Engine Simulation model data is stored in an ASCII text file, with a key word based structure that allows individual data sections to be identified by the applications file reader. Historically the structure of this file was relatively simple and fully documented in the help file, such that experienced users were able to edit the file using the viewing/editing tools provided, to perform model data changes.

With the introduction of a fully functioning drag and drop style interface the use for direct editing as a user technique has become restricted and is no longer recommended. The sim file format is no longer documented in the help file.

With future updates it is anticipated that the inclusion of protected data sections in the file and indeed whole file encryption will remove direct editing of the sim file as an end-user function.

The current release include two tools for viewing and editing the sim files, but the support for these as end-user features will be withdrawn at a future release.

The sim file viewer can be opened from the File / File View menu item, whilst the sim file editor can be opened from the File /File Edit menu item. These two text viewers are identical in function with the exception that the user cannot edit the text in the viewer. The most useful commands with these tools are the File / Get Current and File /Make Current options.

^+$#>Input Data - Parameter Limits

The Lotus Engine Simulation code employs parameterisation of practically every array within the program. This means that the limits can be easily changed upon request. For example the maximum number of cylinders is parameterised as 20. If a user wished to model 24 cylinders then a one number change in an include file and a recompilation would facilitate this.

The parameter limits can be found in the Element Summary.

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^+$#>Input Data - How to Load a Model

To load a previously created model or one of the supplied examples, select the file open icon from the main window, or File/Open from the menu-bar. This brings up the standard windows file-browser.

As an alternative to the standard file browser the File/Open (preview) main menu item can be used to scan through folders to locate and graphically preview any located model files without the need to load them into the interface.

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^+$#>Input Data - How to Extract a Model from an *.mrs File

Models can be extracted from previously created *.Mrs results files. Select File/Extract Model from .mrs File from the menu-bar, as shown below. This brings up the standard windows file-browser.

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^+$#>Input Data - How to Save a Model
To save a model, select the file save icon from the main window tool-bar or the menu-bar option File/Save, as shown below. If no change has been made to the model, this automatically brings up the browser to add a new file-name. Otherwise the file is overwritten.

To save the current model unchanged or otherwise, select File/Save As from the menu-bar or the file save as from the main window tool-bar. This will automatically bring up the browser and prompt the user to enter a new filename. If the same or another used filename is entered the user is prompted to accept overwriting of that file.

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^+$#>Input Data - How to Change an Option
To change an option, for instance the type of fuel burnt in the engine (Data/Fuel and Fuel System/Fuel Type), use the mouse to select the arrow at the right of the display box. This presents the available options and allows selection from the list.

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^+$#>Input Data - How to Use Spreadsheets
To manipulate data in a spreadsheet, for instance the valve lift data, first ensure that a map is available. If not enter a value for the number of points, as shown below, and then press <RETURN>

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Copying data from a Spreadsheet

^+$#>Input Data  Data Import Tool
Selecting Data/Manage Data Import from the main window tool-bar, as shown below, opens the Data Import Tool.
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Data Import Tool Window

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^+$#>Input Data - Base Engine Data - General

The base engine data that is required by Lotus Engine Simulation can be broken down into the following categories:

Cycle Type

This data is entered using the Data/Cycle Type menu on the tool bar. This enables the user to specify the cycle type of the engine.

Engine Geometry

Data such as bore, stroke and connecting rod length are entered via the property sheet associated with each Cylinder element in the builder.

Engine Inertia

For Transient Calculations data on the mass and inertia of various components needs to be specified. This is again done from the property sheet associated with each Cylinder element in the builder.

Cylinder and Valve Event Phasing

The timing of each cylinder with respect to TDC of cylinder 1 needs to be specified. Again this is done from the property sheet associated with each Cylinder element in the builder. The timing of the valves is specified via the property sheet associated with each Valve.

The Cylinder Timing Display can be used to view the relative phasing of the cylinder motion and valve events.

^+$#>Input Data - Base Engine Data Variables

Bore: Cylinder bore [mm]

Stroke: Cylinder stroke [mm]

Cyl Swept Volume: Displays the swept volume of the current cylinder based on the cylinder dimensions (bore and stroke) entered. This field is provided for information only and is not a user definable property.

Total Swept Volume: Displays the total swept volume of all of the cylinders in the current model. This field is provided for information only and is not a user definable property.

Con-rod length: Length of connecting rod from centre of little-end to centre of big-end [mm].

Pin Off-Set: Piston pin off-set [mm]. Positive towards ant-thrust side of piston.

Compression Ratio: Compression ratio  must be greater than 1.0. (Clearance vol.+swept vol.)/(clearance vol.).

Combustion and Heat Transfer: The Combustion and Heat Transfer data section is concerned with defining the types of models to be used for representing the combustion and heat transfer processes and the surface areas and temperatures of various components within the cylinder.

Phase: Phasing of cylinder firing with respect to TDC firing of cylinder 1 [deg.].

Note that the Cylinder Phase Display button can be used to visualise the firing the firing-order (and Valve Lift Profiles Valve) of the cylinders which have been included in the model, as shown below:

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^+$#>Input Data - Cylinder Timing Display

The cylinder timing display is used to show cylinder position and associated valve lifts, through the engine cycle. It can be used purely from the Data Entry Module as a visual data checking tool, or it can also be opened from the Results Module as a post processing tool.

The display opened from the data entry module will be similar to the display shown below. This has the timing rose diagram in the bottom left of the screen, the scaled 2d view of the cylinder in the centre and the relevant data to the right of the display. The example shown is for a single intake valve element and a single exhaust valve element. (This can be identified either from the fact that the rose diagram has only one line for inlet and one for exhaust, or from the data widgets the second column of which is greyed out).

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Cylinder Element and Transient Property Sheet Section

^{+$ #}Input Data - Fuel and Fuel System Data - General

This data is accessed using the Fuel and Fuel System element on the builder interface.

The Fuel and Fuel System data section is concerned with defining the method of introducing the fuel in to the engine (i.e. the combustion system) and specifying the type of fuel to be burnt.

Gasoline, Diesel, Methane, and Methanol fuels can be simulated. The manner by which fuel is introduced to the model is closely linked to the specified combustion system type. For all direct injection / indirect injection engines, fuel is introduced to the cylinder at the same rate as it is combusted. For other combustion system types the fuel is either port injected, where fuel is mixed with the fresh charge flowing through the inlet valves, or added via a carburettor, were fuel is pre-mixed with charge air before being introduced via an inlet.

The properties of each fuel type are displayed but are only editable if the type of fuel selected is User Defined. The fuel types are limited to those composed of C,H, and O atoms only.

^+$#>Input Data - Fuel and Fuel System Data Variables

Fuel System: Combustion / fuel delivery system type:

"         Carburettor
"         Port-Injection
"         Direct Injection
"         Indirect Injection

Fuel Type: Type of fuel to be burnt in the engine:

"         Gasoline
"         Diesel
"         Methane
"         Methanol
"         User defined

If the fuel type is User Defined the following data needs to be supplied:

Calorific Value: Calorific value (specific heating value) of fuel [kJ/kg]

Relative Density: Relative density of fuel

Hydrogen / Carbon Ratio of Fuel: Ratio of number of hydrogen atoms (moles) to number of carbon atoms (moles) in fuel.

Oxygen / Carbon Ratio of Fuel: Ratio of number of oxygen atoms (moles) to number of carbon atoms (moles) in fuel.

Fuel Molecular Mass: Mass per kilo-mole of fuel.

Maldistribution Factor: This factor is used to allow for a reduction in the effective calorific value of the fuel due to running rich, dissociation effects, and poor charge mixing. Suggested values for this parameter are: 1.0 for gasoline, diesel, or methanol, and 0.0 for methane. Further information can be obtained in the Theory section.

^+$#>Input Data - Combustion and Heat Transfer Data - General

Combustion and Heat Transfer data is accessed through the cylinder property sheet.

The Combustion and Heat Transfer data section is concerned with defining the types of models to be used for representing the combustion and heat transfer processes. Wiebe functions are used to define the heat release rates.

The first window to appear allows direct editing of the Combustion Data. The Heat Transfer data is accessed from the bottom half of this window and is sub-divided into the categories of

"         Open cycle
"         Closed cycle
"         Component Surface Areas
"         Component Surface Temperatures

These options are selected using buttons from the lower portion of the main window.

^+$#>Input Data - Combustion and Heat Transfer Data - Combustion Model

The option of single or two-part Wiebe functions is available; the two-part Wiebe function should only be used for simulating diesel combustion systems as the second part of the function models the diffusion burning process. For more information on the combustion models used in the Lotus Engine Simulation code see the Theory section.

Data Variables

Type: Type of model for heat release rate:

"         Single Wiebe function
"         Two-part Wiebe function  for diesel combustion systems only

Single Wiebe
Wiebe A: Coefficient A in Wiebe equation (see Theory section)

Wiebe M: Coefficient M in Wiebe equation (see Theory section)

Two-Part Wiebe
Wiebe A: Coefficient A in Wiebe equation (see Theory section)

Wiebe M: Coefficient M in Wiebe equation (see Theory section)

CP1: Coefficient CP1 in Wiebe equation (see Theory section)

CP2: Coefficient CP2 in Wiebe equation (see Theory section)

Fract: Fraction of premixed combustion (between 0 and 1) (see Theory section)

Delay: Delay angle between first and second parts of Wiebe function [deg.] (see Theory section)

Default and User Defined options are available for both single and two-part Wiebe models. The option not selected is greyed out.

^+$#>Input Data - Combustion and Heat Transfer Data - Heat Transfer Model

Heat transfer data is accessed via the menu options Open cycle HT and Closed cycle HT which appear in the cylinder property sheet on the righthand-side of the builder interface when a cylinder element is clicked on.

A choice of the Annand, Woschni, or Eichelberg models is available for in-cylinder heat transfer in both the open and closed periods. All three models generate values for the convective heat transfer coefficient in the cylinder; the closed period Annand model includes a term for radiative heat transfer.

The same model is used for all cylinders.

For further information on the heat transfer models used in the Lotus Engine Simulation code see the Theory section.

Data Variables

The variables below are entered for both the open and closed period parts of the cycle, unless indicated otherwise.

Annand Model
A: Annand A coefficient (see Theory section)
B: Annand B coefficient - exponent of Reynolds number (see Theory section)
C: Annand C coefficient - for radiation term in closed period only (see Theory section)

Woschni Model
A: Woschni A coefficient (see Theory section)
B: Woschni B coefficient - mean piston speed factor (see Theory section)
C: Woschni C coefficient  Swirl speed factor (see Theory section)
D: Woschni D coefficient - factor for closed period pressure differential (see Theory section)
G: compression / expansion index  closed period (see Theory section)
SR: swirl ratio

Eichelberg
A: Eichelberg A coefficient (see Theory)
B: Eichelberg B coefficient  exponent of product of cylinder pressure and temperature (see Theory)

Default values for all the above coefficients are provided by the interface but it is often necessary to tune these values to achieve a good correlation for both volumetric efficiency and heat transfer. For the Annand model it is recommended that only the A coefficient is tuned. For the Woschni model it is recommended that the B and C coefficients are tuned, but an experienced user may wish to adjust only the swirl ratio term. For the Eichelberg model it is suggested that the A coefficient should be adjusted.

An essential element in modelling the heat transfer in an engine is the specification of the component surface areas and temperatures. The data windows for this information are described below.

^+$#>Input Data - Combustion and Heat Transfer Data - Component Surface Areas

Component surface area data is accessed through the cylinder property sheet.

Because the detailed combustion chamber geometry is not entered as data it is necessary to provide other means of defining the relevant areas for heat transfer calculations. This is done simply by defining the cylinder head and piston surface areas as factors of the cylinder bore area.

Default values can be selected for surface area ratios or can be defined by the user. If the user defined option is selected the required data is entered into a spreadsheet. This data can be assigned as being common to all the cylinders or can be defined for each individual cylinder.

Note that in order to enter data into the spreadsheet the correct number of cylinders must first be set in the Base Engine data window.

Data Variables

Head / Bore: Ratio of cylinder head area to cylinder bore cross-sectional area

Piston / Bore: Ratio of piston surface area to cylinder bore cross-sectional area

Exp. Liner: Length of liner exposed by piston at TDC [mm].

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^+$#>Input Data - Combustion and Heat Transfer Data - Component Surface Temperatures

This data is accessed through the cylinder property sheet.

The heat transfer calculation requires values for the gas-side surface temperatures of the combustion chamber. This can be achieved by entering the component temperatures directly or by specifying the material type and coolant properties or the thermal resistance of the cylinder head / piston crown / cylinder liner wall. The data required by the various options is described below. Information on the thermal network model itself is given in the Theory section. The simulation calculates the gas-side wall temperature using a one-dimensional heat flux calculation.

Define material and coolant properties
This option enables the user to select a material type from a drop-down menu. If the material type desired does not feature on the list a user defined material type may be created by entering the thermal conductivity in the appropriate value box. The data can be defined as common to all cylinders, or can be defined on an individual basis. Arrow buttons are used to toggle through the number of cylinders.

As part of the thermal network calculation it is necessary to specify a coolant temperature, a wall / coolant heat transfer coefficient, and a component wall thickness. Default values (given in the Theory section) are available or user may specify these values directly. The data can be defined as common to all cylinders, or can be defined on an individual basis. Arrow buttons are used to toggle through the number of cylinders.

Define overall thermal resistance and coolant temperatures
It is possible to define the overall thermal resistance for the transfer of heat from the cylinder to the coolant using this option. The respective coolant temperatures are also required  default or user defined values may be specified.

Define inner wall temperatures for components
This option enables the user to enter the component surface temperature directly.

Note: Some values for the piston are difficult to define and in the cases where data for the piston is not requested the heat transfer rate though this component is calculated as a ratio of that through the cylinder head.

Data Variables

Cylinder head / Piston / Liner: material properties (thermal conductivity [W/m/K]) / coolant properties (temp. [oC]; wall / coolant heat transfer coefficient [W/m2/K]; wall thickness [mm]), or thermal resistance [mm2/K/W] and coolant temperature [oC].

^+$#>Input Data - Scavenge Model Data - General

This data is accessed via the cylinder property sheet.

The Scavenge Model Data is used to define the way in which the in-coming charge to the cylinder is mixed with the cylinder contents. These models affect the values of volumetric efficiency predicted by the code.

The simplest model is the Perfect Mixing model and this assumes that any gas entering the cylinder is instantaneously and homogeneously mixed with the gas in the cylinder. This is the default scavenging model for all cylinders and results in the most pessimistic performance predictions because it releases some intake charge to the exhaust in the gas exchange process as soon as any inflow to the cylinder has occurred.

In the Perfect Displacement model assumes that any charge gas entering the cylinder does not mix with the gas currently held within it. This ensures that any gas flowing out of the exhaust valve during the valve overlap period is composed entirely of combustion products until all the residual gas has been removed.

The Benson and Brandham model is a hybrid of the perfect mixing and perfect displacement models in which a pre-defined portion of the scavenging process is characterised by the perfect displacement model, after which the perfect mixing model takes over.

For further details of these and the Blair model see the Theory.

No additional variables are required by the Perfect Mixing and the Perfect Displacement models. The Blair model requires additional data that is empirically derived.

^+$#>Input Data - Scavenge Model Data Variables

Constant A: Scavenge ratio up to which displacement scavenging is used in Benson / Brandham model, or Constant A in Blair model (see Theory).

Constant B: Constant B in Blair model (see Theory section).

Constant C: Constant C in Blair model (see Theory section).

^+$#>Input Data - Port Data - General

This data is accessed via property sheets associated with the port element in the builder interface.

The Port Data property sheet allows the user to specify the port flow characteristics obtained from steady flow rig tests. This data is then used in conjunction with valve lift data in order to calculate the effective flow area of port and valve assemblies at any crank angle in the engine cycle. Note that a port data should only be used in conjunction with a poppet valve element in the builder.

Ideally the user should be in possession of flow rig data measured for the port / valve assembly concerned (User Cf curve ..). If this data is not available Default Good and Default Poor port data can be selected which are derived from curve fits of the Lotus port flow data base. The default characteristics differ for intake and exhaust ports.

The option also exists for the user to specify the port flow coefficient at 0.3 L/D (User Cf at 0.3 L/D). With this option the program interpolates between (and extrapolates beyond) the default good and poor flow curves in order to generate a flow characteristic that achieves the required flow coefficient at 0.3 L/D.

For further information on the heat transfer models used in the Lotus Engine Simulation code see the Theory section.

^+$#>Input Data - Port Data Variables

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^+$#>Input Data - Valve Data - General

This data is accessed via property sheets associated with the port element in the builder interface

The user may select any one of the following valve element options:

"         Poppet valve;
"         Self-acting reed valves;
"         Disc valves;
"         Piston port;
"         User specified angle area curve.

These options are selected from the element list menus on the left of the builder interface. Any combination of valve types can be used on an engine. Note that port data should only be used in conjunction with the poppet valve option.

A description of the data variables required by each valve type can be seen by clicking on the links above.

More detailed descriptions of the models can be found in the Theorysection

^+$#>Input Data  Valve Data - Poppet Valve Lift Options

The valve lift profiles may be specified by one of four options:
"         Default fast lift polynomial;
"         Default slow lift polynomial;
"         User specified polynomial;
"         User specified angle / lift data

With each of the options the valve lift duration is specified by the number of crank degrees between valve opening (AVO) and valve closing (AVC). Valve timings can be modified either directly by changing the opening and closing timings or by changing the MOP (maximum opening point) value  this enables the user to advance or retard the cam timing and maintain the period by adjusting only a single number. Note that if the opening and closing timings are being edited the MOP value box is greyed-out and is modified automatically. Conversely, if the MOP value is edited the opening and closing value boxes are greyed out and are modified automatically.

When the user specified angle/lift ordinate data option is used the lift profile data are linearly scaled so that the lift duration matches that specified with AVO and AVC. The advantage of this scaling is that the user may specify one generic valve lift profile and perform valve timing sensitivity studies by changing only one or two numbers (ie AVO and AVC) in the input data file.

With each of the lift profile options the maximum valve lift is specified by the maximum valve lift AVLM. When the user specified angle/lift ordinate data option is used the lift profile is linearly scaled so that the maximum valve lift matches that specified with AVLM. Users who wish to perform valve timing sensitivity studies should be aware that the maximum achievable valve lift reduces with reducing lift duration. Thus in order to generate realistic valve timing trade-offs the maximum lift should be adjusted with the valve lift duration.

It should be noted that the Lotus Concept Valvetrain tool can be used to generate actual cam profiles which can be downloaded directly into the poppet valve lift data using the Close Make Current option.

Polynomial Lift Curves
The default lift curves employ a polynomial consisting of four coefficients and four exponents. The nature of the polynomial is such that the sum of the coefficients is -1.
The coefficients of the default lift curves are given in the Theory section.

The default and user specified polynomial lift options allow the user to input a maximum lift dwell angle. This is the number of degrees at which the valve remains at maximum lift after the opening before starting to close. The dwell angle should not be a negative number.

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Importing Valve Angle/Lift Ordinate Data

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^+$#>Input Data  Valve Data - Self Actuating Reed Valves

A relatively simple self acting reed valve model is employed in the Lotus Engine Simulation code. The model employs a spring mass representation of the valve/reed that is forced to move between the valve seat and the lift stop by the pressure on either side of the valve and the area over which this pressure acts. It is assumed that there is no valve bounce on either the valve stop or the valve seat. This implies that the self-acting valve is well matched to the application. For details of the reed valve model see the Theory section.

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^+$#>Input Data  Valve Data - Disc Valves

The disc-valve model calculates the flow area of a port which is covered and uncovered by a disc which rotates at crankshaft speed. The flow area is calculated from the area of the port that is uncovered by the disc-valve and the disc-valve discharge coefficient. The discharge coefficient is assumed to reduce with increasing area from 1.0 to the value for the fully uncovered port provided by the user. For further details of the disc-valve model see the Theory section.

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