Symbolic mathematicsbased simulation of cylinder spaces for regenerative gas cycles
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Abstract
Regenerative gas cycles, including the Stirling engine, are sealed machines using pistons within cylinders to transfer energy from a heat source to a colder reservoir using a gas as working substance. For the optimal design of these cycles, we need a detailed description of gas dynamic behavior. This contribution deals with the simulation of cylinder spaces without internal combustion (as we find for regenerative gas cycles). For the simulation, we suggested a symbolic mathematicsbased strategy to describe the dynamic system behavior based on partial nonlinear differential equations for the conserved quantity. The renunciation of numerical approximation gives the advantage that the underlying physical mechanisms are characterized by exact expressions and parameters. Using some assumptions, the dynamic behavior of the gas within the cylinder is already described by ordinary nonlinear differential equations. Depending on the selected boundary conditions analytical solutions can be obtained for some cases. Finally, the developed solution is based on it and will be received as a series expansion. Additionally, for the simulationbased optimization of the processes it is possible to calculate directly the periodicalsteady state of the system with the help of the symbolic solution. The simulation is suitable for fundamental theoretical investigations, as well as for the implementation in simulation software for different regenerative gas cycles.
Keywords
Modelling simulation Cylinder space Regenerative gas cycle Stirling engineList of symbols
 \( A_{\text{q}} \)
Heat transfer surface area
 \( c_{ 0} \)
Integration constant
 \( c_{\text{p}} \)
Spec. isobaric heat capacity
 \( c_{\text{v}} \)
Spec. isochoric heat capacity
 \( d \)
Cylinder diameter
 \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{D} } \)
Diagonal matrix
 \( \dot{H} \)
Enthalpy flow rate
 \( i \)
Loop variable
 \( j \)
Loop variable
 \( k \)
Overall heat transfer coefficient
 n
Count
 \( Nu \)
Nusselt number
 \( m \), \( \dot{m} \)
Mass, mass flow rate
 p
Pressure
 P
Power
 \( Q \), \( \dot{Q} \)
Heat, rate of heat flow
 \( R \)
Spec. gas constant
 \( Re \)
Reynolds number
 \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{S} } \), \( s \)
Structure matrix, components of the matrix
 \( T \)
Absolute temperature
 t
Time
 \( U \)
Internal energy
 V
Eigenbasis
 \( V \), \( \dot{V} \)
Volume, volumetric flow rate
 \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{X} } \)
Square matrix
 \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{y} \), \( y \)
Substitution vector, components of the vector
 \( z \)
Transformation variable
 \( \delta \)
Volume parameter
 \( \varepsilon \)
Pressure parameter
 \( \kappa \)
Isentropic exponent
 λ
Eigenvalue
 \( \mu \)
Heat transfer parameter
 ϱ
Density
 φ
Phase
 \( \omega \)
Angular velocity
Indices
 \( 0 \)
Initial value
 \( {\text{ges}} \)
Total/whole process
 K
Component
 U
Surrounding
 x
Coordinate
 in
Inflow
Introduction

cylinder space (displacement or working cylinder),

heat exchanger (cooler or heater for external heat transfer),

regenerator (internal heat transfer).
For the optimal design of these components different methods for modeling and simulation are used (cf. [4]). The differential equations for the conserved quantities: mass, energy, and momentum are usually not directly solvable, i.e., do not have closed form solutions. Instead, solutions must be approximated using numerical methods. For purposes of simplified calculations, one may assume that the process takes place from initiation to completion without a change in a special state quantity of the system. In this case also analytic solutions exist, such as the Schmidt model for Stirling engines [11]. One more model that has occasionally been used is the model for the regenerator due to Hausen [2].
The process pressure sets again a boundary condition for the single components. Furthermore the components are coupled by the energy flow of the involved masses as well as without mass via the surrounding.
Simulation models for the component of the cylinder space exist for regenerative gas cycles and also for reciprocating compressors or internalcombustion engines. Analytic solutions for the cylinder space may be obtained by assumption of an adiabatic, polytropic, or isothermal process (cf. [1]), as with the Schmidt model [11]. In contrast, the studies of multidimensional effects like heat transfer [13] or the pressure loss are solely based on numerical methods. Other models are also using numerical methods simulating the cylinder spaces for piston engines based on ordinary differential equations, like [5] for Vuilleumier engines, [8] for diesel engines, or [10] for Stirling engines.
The analytical or semianalytical methods are only partially suitable for aims of simulationbased optimization because gross simplifications for the real situation must be superinduced. The numerical approximations do not deal with this problem but with an analytical model the solution is held in form of an equation. It will then be easier to handle and one can clearly recognize the characteristics of the physical process. Additionally, for the simulation of regenerative gas cycles it is possible to calculate directly the periodicalsteady state of the system. This paper aims at exploiting the benefits of an analytical calculation method without using gross simplifications for the modeling.

the determination of the influence parameters of the cylinder space,

their effects and especially the influence of individual parameters change,

the simplifications that result from fixing certain parameters for the simulation, and

the conditions for the isothermal process, which is widely accepted in the field.
Finally, for simulation purposes analytic solutions are generally considered superior to numerical approximations.
Energy equation

\( T_{\text{x}} = T_{\text{in}} (\varphi ) \), a transient function for the gas taking up cylinder

\( T_{\text{x}} = T \), the temperature inside the cylinder space for the gas delivering cylinder
The working gas within the cylinder is considered as ideal mixed. The reason is that the gas streams alternately incoming and leaving the cylinder provide in each case an intense turbulence.
Furthermore, no basic differentiation of the energy balance is made between a displacement and working piston because that has only a constructive meaning. The appearing leaky mass flow about the piston seals (which can be predicted mathematically only inexactly) is extremely low at a displacement compressor due to the lower pressure differences, and the effects for the process can be neglected.
To calculate the rate of heat transfer the temperature at the wall surface is accepted as steady, but in [5] a transient consideration was also implemented. Besides in the periodicalsteady state, the gas temperature in the cylinder is characterized by a sine approximation that leads to the propagation of the penetration depth of the periodical temperature variation in the wall. The divergence of the heat transfer compared to a model with a steady inside temperature by metallic materials amounts to merely a few percent. Hence this approach is suggested, if necessary, in exclusively isolated cylinder rooms.

\( \delta = f\left( \varphi \right) \) in case of the gas taking up cylinder space, and

\( \delta = 0 \) in case of the gas delivering cylinder space.
Solution of cylinder space temperature
Boundary conditions  General solution  

\( \mu = 0 \)  \( T = p^{{\frac{\kappa  1}{\kappa }}} V^{  1} \left( {c_{0} + \mathop \smallint \nolimits p^{{\frac{1  \kappa }{\kappa }}} T_{\text{in}} {\text{d}}V} \right) \)  (14) 
\( \mu = 0 \), \( \varepsilon = 0 \)  \( T = V^{  1} \left( {c_{0} + \mathop \smallint \nolimits T_{\text{in}} {\text{d}}V} \right) \)  (15) 
\( \mu = 0 \), \( \delta = 0 \)  \( T = T_{0} \left( {\frac{p}{{p_{0} }}} \right)^{{\frac{\kappa  1}{\kappa }}} \)  (16) 
The difficulty still exists in the consideration of the usually transient pressure progression, because the functions to be integrated always contain exponents of nonnatural numbers. With this consideration an analytical solution for (14) is limited, and whether it gives a suitable solution generally depends naively on the function of the pressure. Therefore, simplifications deliver only solutions for special boundary conditions, but not for the general case. In particular, the pressure progression is given during the simulation as a rule not as a function, and will be received first by the whole energy balance (1). Therefore, another procedure must be applied to get the solution.
The Matrix \( \underline{\underline{S}} \) in (19) is not independent of \( \varphi \) and so (22) does not give the whole solution to solve the matrix initial value problem. If one insists on having an exponential representation, the exponent needs to be corrected with the rest of the Magnus series.
Apart from that the solution of the matrix exponential (29) coincides with the gas taking up cylinder space.
In this case the initial temperature \( T_{0} (\varphi = 0) \) is not discretionary anymore; it is a function of the cylinder space temperature at maximum gas capacity. This is received again from the solution of the Riccati equation with \( \varphi = \pi \) for the gas taking up cylinder.
Plot of Solution

the heat transfer parameter \( \mu \),

the pressure parameter \( \varepsilon \),

the volume parameter \( \delta \),

the temperature of the surrounding area \( T_{\text{U}} \), and

the temperature of the incoming gas \( T_{\text{in}} \).
List of boundary conditions for the temperature profile calculation of the cylinder spaces
Type  Value 

Cylinder diameter  \( d = 0.096 \;{\text{m}} \) 
Angular velocity  \( \omega = 52,\!36\;{\text{rad}}/{\text{s}} \) 
Temperature of surrounding area  \( T_{\text{U}} = 1000\;{\text{K}} \) 
Test function: Pressure  \( p = \left( {17  \frac{11}{\pi }\varphi + \frac{7}{{\pi^{2} }} \varphi^{2} } \right){\text{bar}} \) 
Test function: Gas taking up temperature  \( T_{\text{in}} = \left( {750  \frac{100}{\pi }\varphi } \right)\text{K} \) 
Incoming volumetric flow  Volume coefficient  
\( \dot{V} = {\text{const}} . \)  \( \delta = \varphi^{  1} \)  (30) 
\( \dot{V} = \dot{V}_{ \hbox{max} } \sin \left( \varphi \right) \)  \( \delta = \tan \left( {\frac{\varphi }{2}} \right)^{  1} \)  (31) 
The Eqs. (30) and (31) are a formal statement of the wellknown fact that at \( \varphi = 0 \), there is no gas inside the cylinder (\( \delta = \infty \)). While the cylinder volume increases, the influence of the temperature of the incoming gas is reduced by decrease of the volume coefficient. Finally, the cylinder reaches the maximum gas capacity and the movement of the piston stops (\( \delta = 0 \)). In the following time interval the parameter for the volume stays at zero; the cylinder delivers the gas to the connected component.
Conclusion
The working gas inside a machine of a regenerative gas cycle is pushed through the heat exchangers and the regenerator by cylinder piston assemblies. For the optimal design of these components one requires the information of the gas mass flow emitting and receiving by the cylinder spaces. In that regard, account must be taken of the modeling of the mass change within any cylinder space, based on differential equations for the conserved quantities mass, energy, and momentum, which are theoretically as well as numerically difficult to treat.
The proposed model of the cylinder space describes the dynamic behavior already by an ordinary nonlinear differential equation, in particular due to the assumption that the alternating incoming and outgoing gas mass flow ensures an intense turbulence. Depending on the selected boundary conditions analytical solutions can be obtained for some instances. In addition, by using a special mathematical transformation, the energy equation of the cylinder space could be obtained as a system of linear differential equations, where the general solution is based on the matrix exponential function. Of course, the energy equations could also be treated by any of the numerical algorithms to solve ordinary differential equations. The important point to be emphasized here is that the main reason for the usefulness of the exponential algorithms lies in the fact that it preserves structural intrinsic properties of the exact solution.
 1.
the pulsating process pressure,
 2.
the heat transfer to the surrounding,
 3.
the temperature of the incoming gas flow, and
 4.
usually oscillatory volumetric flow from the gear in question.
For the temperature within the cylinder space the theoretical investigations showed an isentropic change in state for the influence of the process pressure depending on the isentropic exponent. To retain the isothermal state a heat transfer to the surrounding must be carried out, whereby the influence of the overall heat transfer coefficient would have to be very pronounced to compensate the pressure change. In addition, the temperature change by the pressure is pulsating. However, the heat flow has only one direction: cooling down or heating up. Finally, the temperature of the gas taking up cylinder depends on the temperature of the incoming volumetric flow, which is a function of the bordering component. Some people assume that the process takes place without an increase or decrease in a state quantity of the system like the temperature. The impact of the investigated parameters indicates this assumption is not always correct; the general solution appears indispensable for realistic simulation.
The information of the parameters can be provided to the featured cylinder space simulation continuously or discreetly, as this is present as a finite number of discrete time steps. The division of the time steps depending on required simulation accuracy is in this case virtually arbitrary since there is an analytical solution for the matrix exponential function. Thus an efficient symbolic solution, in comparison to the numerical approximation, of the underlying differential equation is obtained.
The presented model of the cylinder space can be used for design simulating software of regenerative gas cycles. By the modular construction of the simulation program the external boundary conditions of the cylinder space are provided via the coupling of the other modules. Number and arrangement of the individual modules of a regenerative gas cycle is determined by the process configuration and therefore the cylinder space module is suitable for simulating different machines. Furthermore, additional physical models such as for combustion or pressure loss may be integrated. Thus, the basic approach used for the symbolic mathematic based simulation of cylinder spaces opens up many fields of application like different regenerative gas cycles, reciprocating compressors or internalcombustion engines.
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