Ideal Adiabatic Simulation of Stirling Engines

Content of this and related pages :

1. Introduction, Assumptions, and Definitions
2. Derivation of relevant Differential Equations
2.1 Conservation of Mass and Energy, general
2.2 Compression Space
2.3 Expansion Space
2.4 Kooler Space
2.5 Heater Space
2.6 Regenerator
2.7 Putting it all together
3. Sample Calculations and Optimization
3.1 Some Observation on an Example Calculation
3.2 Influence of Type of Gas
3.3 Optimization for Volume Phase Lag and Swept Volume Ratio
4. Entry Points to web-based Programs
5. Case Studies, optimizing volume phase lag and swept volumes :
5.1 Case Study I, ross90.dat
5.2 Case Study II, SESUSA 2004 Engine

1. Introduction, Assumptions, and Definitions

The Stirling engines to be simulated consist of five subspaces connected in linear fashion as shown in the Figure below. The cooler, regenerator, and heater space have constant volumes V_k, V_r, and V_h, respectively. Both, the compression and the expansion space , are each divided into a fixed clearance volume, V_clc and V_cle respectively, and a time-dependent volume which varies between 0 and their respective maxima, V_swc and V_swe. The precise variation of V_swc and V_swe during a complete cycle depends on the mechanical drives employed. Typical arrangements of the five subspace are shown below. For the alpha-type everything can be stacked into a single cyclinder. In the beta-type both the power piston and displacer are housed in a single cyclinder while in the gamma-type they occupy separate cyclinders.

Most commonly used arrangements

In principle, the analysis brought in this paper is not restricted as to the precise mechanism employed to change the volumes of the compression and expansion space although the programs built upon it offer only a few options.

In this paper we use very much the same notation as employed by Urieli. In particular we use the subscripts c, k, r, h, and e to refer to the compression, kooler, regenerator, heater, and expansion space, respectively. As it is common practice in engineering thermodynamics, we use m [kg] for mass, p [N/m²] for pressure, T [°K] for temperature, V [m³] for volume, u [kJ/kg] for specific internal energy, h [kJ/kg] for enthalpy, Q [J] for heat, and W [J] for work.

We frequently use the letter "d" in front of a quantity to denote a (infinitesimal) small change of the quantity as a result of a small change of the crank angle. i.e dm_c is the change in mass of the compression space as the crank angle changes by the amount dΘ. The letter "d" is not used for any other purposes.

The following assumptions are basis of the subsequent analysis :

1. The engine turns at constant speed. Therefore, time and crank angle Θ are proportional to each other.
   
   
2. Any pressure drops due to flow resistances and pressure differentials needed to accelerate the working gas are neglected. Hence, the pressure, p , has at a given instant the same value everywhere inside the engine and varies only with time. Correspondingly, kinetic energies of the working gas are neglected in the law of conservation of energy.
   
   
3. Leakage of gas to the outside of the engine, including crank case, is assumed to be zero.
   
   
4. Both, the compression and expansion space, volumes V_c and V_e, are adiabatic. This means that no heat gets exchanged in these two spaces between the gas and the surrounding walls nor with the piston/displacer surfaces. The temperature in these spaces is uniform but varies during a cycle due to changes in pressure, volume and gas coming from/leaving to the adjacent kooler and heater space, respectively.
   
   
   
5. The heat transfer conditions for the kooler space are sufficiently good to keep the gas inside the kooler space, volume V_k, at constant uniform temperature T_c at all times. The same holds true for the heater space, volume V_h and uniform temperature T_h.
   
   
6. The heat transfer conditions are sufficient to keep the temperature distribution inside the regenerator, volume V_r , linear, varying from T_c where the regenerator is connected to the kooler to T_h at the heater side.
   
   
   
7. An ideal gas is used as working fluid and the ideal gas law connecting pressure p, volume V, mass m, and temperature T to each other via the specific gas constant R :

   (1.1)    p V = m R T

   Also, the specific heat capacities and constant volume, c_v, and constant pressure, c_p, are connected to the specific gas constant :

   (1.2)    c_p - c_v = R

   We furthermore assume that the working gas has temperature independent values for the specific heat capacities, c_v and c_p, and their ratio κ is known :

   (1.3)    κ = c_p/c_v    (i.e 1.4 for air and hydrogen)

   As a result the specific internal energy, u , of an ideal gas at temperature T is :

   (1.4)    u = c_v ( T - T₀ ) [kJ/kg]

   And then by definition (h=u+p*v) the specific enthalpy becomes :

   (1.5)    h = c_p T - c_v T₀ [kJ/kg]

   The temperature T₀ is inserted to provide the best fit between the real behavior of u and constant-c_v relationship. For monatomic gases like He, Ne, Ar etc. Eq. (1.4) holds true with T₀=0 over a temperature range of a few thousand degrees. We will later see that T₀ itself will drop out of the important equations to follow. See also ( A short note on ideal gases)

   
   
   

2. Derivation of relevant differential equations

2.1 Conservation of mass and energy

In addition to the above restrictions and assumptions the laws of conservation of mass and energy form the basis of the development of the relevant differential equations.

For mass we have :

(2.1.1)    dm_c + dm_k + dm_r + dm_h + dm_e = 0

which states that the gain of mass in one subspace (dm>0) has to be compensated by losses of mass in other subspaces ( dm<0).

Conservation of energy for a single subspace with index "s" which is connected to two adjacent subspaces "s1" and "s2" :

(2.1.2)    d(m_s u_s ) = dQ_s - p dV_s + h_s1 dm_s1 + h_s2 dm_s2

The term h_s1 dm_s1 denotes the energy brought into the subspace from its adjacent subspace "s1" with the gas. The enthalpy h_s1 corresponds to the enthalpy of the case inside subspace "s1" if dm_s1>0 (mass coming from "s1") and corresponds to enthalpy h_s for dm_s1<0 (mass leaving "s"). Because of conservation of mass :

(2.1.3)    dm_s = dm_s1 + dm_s2

Special forms of Eq. (2.1.2) arise for the different subspaces. For the compression and expansion space we have by assumption zero heat transfer, hence dQ_s = 0 , and only a single connection with an adjacent space. For the kooler , regenerator, and heater we have constant volume. Hence, dV_s = 0 for these subspaces.

In the following sections we will investigate the individual subspaces. One objective is to express for each subspace the change in mass in terms of the overall pressure change "dp" and volume change dV (where applicable) and substitute these equations into Eq. (2.1.1) to obtain a single differential equation for the pressure p driven by the volume changes in the compression and expansion space. Equally important is the development of equation for the heat fluxes in each of the subspaces and mass flow rates. Finally, in Section 2.7, we pull everything together and summarize the resulting equations.

2.2 Compression Space

The ideal gas law states :

(2.2.1)    p V_c = m_c R T_c

As the crank angle varies, so do pressure p, mass m_c and temperature T_c, and volume V_c. Differentiation gives :

(2.2.2)    R T_c dm_c = V_c dp + p dV_c - m_c R dT_c

In order to facilitate sustitution of dm_c in Eq.(2.1.1) we need to eliminate dT_c from this equation. We do that using conservation of energy. Because this space is adiabatic and has only one adjacent subspace ( the kooler) the general energy equation, Eq. (2.1.2) simplifies somewhat.

(2.2.3)    d(m_c u_c ) = - p dV_c + h_ck dm_c

We substitute now u_c = c_v (T_c - T₀) and h_ck = c_p T_ck - c_v T₀ in accordance with Eq. (1.4) and (1.5) :

m_c c_v dT_c + c_v ( T_c - T₀ ) dm_c = - p dV_c + (c_p T_ck - c_v T₀) dm_c

Note that the temperature T₀ cancels out and we can rearrange :

(2.2.4)    m_c c_v dT_c = - p dV_c + (c_p T_ck - c_v T_c) dm_c

The newly introduced temperature T_ck reflects the different energy content of the gas being exchanged between compression and kooler space :

dT_c of Eq. (2.2.4) can now be substituted into Eq. (2.2.2) and after some algebra :

(2.2.6)    dm_c = ( κ p dV_c + V_c dp ) / ( κ R T_ck )

which we later substitute into the total mass balance, Eq. (2.1.1) to eliminate from dm_c in favor of dp and the known change in compression space volume dV_c.

2.3 Expansion Space

The expansion space is treated in exactly the same fashion as the compression space. The relevant equation to be used later can be simply gained from those of compression space by subscript substitution :

(2.3.6)    dm_e = ( κ p dV_e + V_e dp ) / ( κ R T_eh )

2.4 Kooler Space

The ideal gas law states :

(2.4.1)    p V_k = m_k R T_k

Because volume and temperature are constant for this space, differentiation of Eq. (2.2.1) gives :

(2.4.2)    dm_k = (V_k/(R T_k)) dp

which is all what we need for substitution into the total mass balance, Eq. (2.1.1) to eliminate dm_e in favor of dp.
The heat flow rate for the kooler space is determined from the energy balance, Eq. (2.1.2). Again we take into account that volume and temperature are constant and express internal energy and enthalpies in terms of specific heat capacities. We also note that the temperature of the gas exchanged between kooler and regenerator is always T_k.

dQ_k = c_v ( T_k - T₀ ) dm_k + (c_p T_ck - c_v T₀ ) dm_c - (c_p T_k - c_v T₀ ) ( dm_k + dm_c )

Here the term (dm_k + dm_c) reflects the mass loss of kooler and compression space combined to the regenerator taking with it energy. Again the reference temperature T₀ cancels out and terms can be regrouped :

(2.4.3)    dQ_k = c_p ( T_ck - T_k ) dm_c - R T_k dm_k

2.5 Heater Space

The analysis of this subspace is completely identical to that of the kooler space. The relevant equation can be taken over, only subscripts need to be adjusted.

(2.5.2)    dm_h = (V_h/(R T_h)) dp

(2.5.3)    dQ_h = c_p ( T_eh - T_h ) dm_e - R T_h dm_h

in which the temperature T_eh is the same as that defined by Eq. (2.3.5).

2.6 Regenerator

The analysis for the regenerator is identical for the ideal adiabatic simulation and the Schmidt analysis. For details we refer the reader to that paper (Heat transfer inside the regenerator). The only adjustment needed is that for the Schmidt analysis the temperature in the adjacent kooler space was referred to as T_c while it is now T_k. We therefore list only the important equations.

The effective temperature of the regenerator is :

(2.6.1)    T_r = ( T_h - T_k ) / ln(T_h/T_k)

It is constant in time and can be used to calculate the mass inside the regenerator using the ideal gas law and taking into account a linear temperature distribution. With the volume, V_r, being constant, a change in mass is simply related to a change in pressure :

(2.6.2)    dm_r = V_r / ( R T_r ) dp

For the energy balance we have to take into account the heat exchange with the walls, the change in internal energy (due to change in mass) and the energies transported by the gas flowing between the regenerator and its adjacent subspaces (kooler and heater). Based on the derivation for the Schmidt analysis we obtain :

(2.6.3)    dQ_r = { [ V_r + κ(V_c+V_k+V_h+V_e) ] dp + κ p ( dV_e + dV_c ) } / { κ-1 }

2.7 Putting it all together

We substitute now the Equations (2.2.6), (2.3.6), (2.4.2), (2.5.2), and (2.6.2) into Eq. (2.1.1) and solve for dp. The results is :

(2.7.1)   

V_c and V_e and their derivatives are determined by the drive geometry. For example for a sinusoidal drive ( see Schmidt Analysis for definitions of the terms ) :

V_c = V_clc + 0.5 V_swc( 1 + cos(Θ) )

V_e = V_cle + 0.5 V_swe( 1 + cos(Θ + δ) )

dV_c = -0.5*V_swcsin(Θ) dΘ

dV_e = -0.5*V_swesin(Θ+δ) dΘ

Here δ is the swept volume phase lag (VPL).

When integrating Eq. (2.7.1) as function of crank angle care must be taken of the temperatures T_ck and T_eh as they change values as the crank angle Θ advances :

Our current strategy is to integrate dm_c and dm_e as function of the crank angle as well using :

(2.2.6)    dm_c = ( κ p dV_c + V_c dp ) / ( κ R T_ck )

and

(2.3.6)    dm_e = ( κ p dV_e + V_e dp ) / ( κ R T_eh )

and calculate the temperatures in the compression and heater space using the ideal gas law for either space :

(2.2.1)    p V_c = m_c R T_c

and

p V_e = m_e R T_e

3. Sample Calculations and Optimization

3.1 Some Observation on an Example Calculation

In this section we discuss the results obtained by our simulation program, IdealAdiabatic, using the following input ( known also as ross90.dat) :

Basically, the program integrates the differential equations from Section 2. using a standard 4^th Order Runge-Kutta Method over a number of cycles until a "steady state" is achieved with an average pressure equal to that demanded by the input ( 200 kPa in the above sample input ). The term "steady state" refers to a state in which all properties like pressure and temperature and mass in each space are equal at the beginning ( crank angle &Theta = 0 ° ) and the end ( crank angle Θ=360°) of a cycle to within some specified limit ( at present a relative error limit of 10^-5 is employed). After achieving "steady state", the simulator program provides the following summary output :

These summary data are followed by a listing of the values of 39 variables as function of crank angle Θ. These data are formatted in such a way that they can easily exported (swipe and paste) into a file on any computer. This file can be read by either a spread sheet program ( space separated columns , merge delimiters ) with the objective of producing desired graphs or any other plotting software available on your computer.

Presently, the following quantities make up this list :

Theta[deg] = crank angle
p [bar] = pressure
Tc [K] = temperature in compression space
Te [K] = temperature in expansion space
Vc [cc] = volume of compression space
Ve [cc] = volume of expansion space
Vtot [cc] = total volume
Wc [J] = work done in compression space
We [J] = work done in expansion space
Wtot [J] = Wc + We
Qk [J] = heat in kooler
Qr [J] = heat in regenerator
Qh [J] = heat in heater
dWc [W] = instantaneous power output, compression space
dWe [W] = instantaneous power output, expansion space
dWtot [W] = dWc + dWe
dQk [W] = heat flow rate in kooler space
dQr [W] = heat flow rate in regenerator space
dQh [W] = heat flow rate in heater space
mc [g] = mass in compression space
mc [g] = mass in kooler space
mr [g] = mass in regenerator space
mh [g] = mass in heater space
me [g] = mass in expansion space
ck [g/s] = mass flow rate from compression into kooler space
kr [g/s] = mass flow rate from kooler into regenerator space
rh [g/s] = mass flow rate from regenerator into heater space
he [g/s] = mass flow rate from heater into expansion space
CP [cc] = location of compression piston with lowest position at 0
EP [cc] = location of expansion piston
0.1 [cc] = location of gas particle, tagged 0.1
0.2 [cc] = location of gas particle, tagged 0.2
0.3 [cc] = location of gas particle, tagged 0.3
0.4 [cc] = location of gas particle, tagged 0.4
0.5 [cc] = location of gas particle, tagged 0.5
0.6 [cc] = location of gas particle, tagged 0.6
0.7 [cc] = location of gas particle, tagged 0.7
0.8 [cc] = location of gas particle, tagged 0.8
0.9 [cc] = location of gas particle, tagged 0.9

Theta[deg] p[bar]  Tc[K]  Te[K]      Vc[cc]      Ve[cc]    Vtot[cc]       Wc[J]       We[J]     Wtot[J]       Qk[J]       Qr[J]       Qh[J]      dWc[W]      dWe[W]    dWtot[W]      dQk[W]      dQr[W]      dQh[W]       mc[g]       mk[g]       mr[g]       mh[g]       me[g]     mtot[g]     ck[g/s]     kr[g/s]     rh[g/s]     he[g/s]      CP[cc]      EP[cc]     0.1[cc]     0.2[cc]     0.3[cc]     0.4[cc]     0.5[cc]     0.6[cc]     0.7[cc]     0.8[cc]     0.9[cc]
000.0  1.4604e+00 282.15 783.87  6.9045e+01  3.7560e+01  2.0121e+02  0.0000e+00  0.0000e+00  0.0000e+00  0.0000e+00  0.0000e+00  0.0000e+00  4.8879e-01 -1.1150e+03 -1.1145e+03 -1.1184e+02 -1.5020e+03  5.3082e+02  1.2452e-01  5.2937e-02  3.2025e-02  1.5717e-02  2.4382e-02  2.4958e-01 -2.0583e+00 -3.3573e+00 -4.1431e+00 -4.5288e+00  0.0000e+00  2.0121e+02  1.3839e+01  2.7678e+01  4.1517e+01  5.5356e+01  6.9204e+01  8.3919e+01  9.8633e+01  1.2008e+02  1.6261e+02
005.0  1.4743e+00 283.00 785.99  6.8929e+01  3.4924e+01  1.9846e+02 -1.7067e-02 -3.8678e-01 -4.0385e-01 -4.3421e-02 -4.8908e-01  1.7639e-01 -9.8073e+01 -1.1118e+03 -1.2099e+03 -1.3804e+02 -1.3164e+03  4.8492e+02  1.2512e-01  5.3442e-02  3.2330e-02  1.5867e-02  2.2825e-02  2.4958e-01 -1.3902e+00 -2.9935e+00 -3.9634e+00 -4.4394e+00  1.1615e-01  1.9858e+02  1.3866e+01  2.7615e+01  4.1365e+01  5.5115e+01  6.8864e+01  8.3429e+01  9.8005e+01  1.1844e+02  1.5982e+02
010.0  1.4912e+00 283.96 788.55  6.8581e+01  3.2330e+01  1.9552e+02 -6.8624e-02 -7.7130e-01 -8.3992e-01 -9.5984e-02 -9.1309e-01  3.3609e-01 -1.9813e+02 -1.1020e+03 -1.3001e+03 -1.6450e+02 -1.1272e+03  4.3477e+02  1.2548e-01  5.4052e-02  3.2700e-02  1.6048e-02  2.1302e-02  2.4958e-01 -6.9769e-01 -2.6082e+00 -3.7640e+00 -4.3313e+00  4.6371e-01  1.9598e+02  1.4105e+01  2.7745e+01  4.1386e+01  5.5027e+01  6.8668e+01  8.3058e+01  9.7469e+01  1.1701e+02  1.5716e+02
015.0  1.5110e+00 285.05 791.53  6.8005e+01  2.9799e+01  1.9241e+02 -1.5516e-01 -1.1512e+00 -1.3064e+00 -1.5778e-01 -1.2707e+00  4.7765e-01 -2.9948e+02 -1.0853e+03 -1.3848e+03 -1.9094e+02 -9.3307e+02  3.8043e+02  1.2560e-01  5.4770e-02  3.3134e-02  1.6261e-02  1.9820e-02  2.4958e-01  2.1786e-02 -2.1996e+00 -3.5435e+00 -4.2031e+00  1.0400e+00  1.9345e+02  1.4554e+01  2.8067e+01  4.1581e+01  5.5094e+01  6.8608e+01  8.2807e+01  9.7030e+01  1.1578e+02  1.5464e+02
020.0  1.5338e+00 286.27 794.93  6.7204e+01  2.7349e+01  1.8916e+02 -2.7707e-01 -1.5241e+00 -1.8011e+00 -2.2706e-01 -1.5570e+00  5.9945e-01 -4.0190e+02 -1.0615e+03 -1.4634e+03 -2.0825e+02 -7.1788e+02  3.2111e+02  1.2546e-01  5.5598e-02  3.3635e-02  1.6507e-02  1.8386e-02  2.4958e-01  7.8914e-01 -1.7560e+00 -3.2957e+00 -4.0514e+00  1.8407e+00  1.9100e+02  1.5210e+01  2.8580e+01  4.1949e+01  5.5319e+01  6.8688e+01  8.2682e+01  9.6692e+01  1.1474e+02  1.5230e+02
025.0  1.5597e+00 287.65 798.75  6.6185e+01  2.4999e+01  1.8579e+02 -4.3471e-01 -1.8874e+00 -2.3221e+00 -3.0269e-01 -1.7681e+00  7.0002e-01 -5.0515e+02 -1.0304e+03 -1.5355e+03 -2.2756e+02 -4.9967e+02  2.5824e+02  1.2505e-01  5.6539e-02  3.4204e-02  1.6786e-02  1.7010e-02  2.4958e-01  1.5828e+00 -1.2883e+00 -3.0252e+00 -3.8776e+00  2.8597e+00  1.8865e+02  1.6070e+01  2.9280e+01  4.2490e+01  5.5701e+01  6.8911e+01  8.2682e+01  9.6460e+01  1.1389e+02  1.5015e+02
030.0  1.5888e+00 289.17 802.97  6.4956e+01  2.2768e+01  1.8233e+02 -6.2829e-01 -2.2386e+00 -2.8669e+00 -3.8543e-01 -1.9028e+00  7.7819e-01 -6.0889e+02 -9.9161e+02 -1.6005e+03 -2.4924e+02 -2.7800e+02  1.9220e+02  1.2435e-01  5.7593e-02  3.4842e-02  1.7099e-02  1.5697e-02  2.4958e-01  2.4032e+00 -7.9534e-01 -2.7303e+00 -3.6800e+00  4.0892e+00  1.8642e+02  1.7126e+01  3.0163e+01  4.3200e+01  5.6237e+01  6.9283e+01  8.2808e+01  9.6333e+01  1.1321e+02  1.4821e+02
035.0  1.6211e+00 290.84 807.59  6.3525e+01  2.0672e+01  1.7880e+02 -8.5792e-01 -2.5750e+00 -3.4329e+00 -4.7618e-01 -1.9599e+00  8.3294e-01 -7.1275e+02 -9.4487e+02 -1.6576e+03 -2.7368e+02 -5.2343e+01  1.2345e+02  1.2337e-01  5.8761e-02  3.5548e-02  1.7446e-02  1.4458e-02  2.4958e-01  3.2500e+00 -2.7635e-01 -2.4097e+00 -3.4566e+00  5.5199e+00  1.8432e+02  1.8371e+01  3.1223e+01  4.4074e+01  5.6925e+01  6.9800e+01  8.3056e+01  9.6312e+01  1.1270e+02  1.4649e+02
040.0  1.6564e+00 292.64 812.59  6.1904e+01  1.8726e+01  1.7524e+02 -1.1236e+00 -2.8936e+00 -4.0172e+00 -5.7594e-01 -1.9379e+00  8.6343e-01 -8.1625e+02 -8.8982e+02 -1.7061e+03 -3.0121e+02  1.7784e+02  5.2582e+01  1.2209e-01  6.0043e-02  3.6324e-02  1.7827e-02  1.3301e-02  2.4958e-01  4.1221e+00  2.6919e-01 -2.0617e+00 -3.2056e+00  7.1409e+00  1.8238e+02  1.9796e+01  3.2451e+01  4.5105e+01  5.7760e+01  7.0449e+01  8.3423e+01  9.6396e+01  1.1236e+02  1.4501e+02
045.0  1.6949e+00 294.56 817.94  6.0105e+01  1.6947e+01  1.7166e+02 -1.4250e+00 -3.1917e+00 -4.6167e+00 -6.8584e-01 -1.8350e+00  8.6904e-01 -9.1885e+02 -8.2614e+02 -1.7450e+03 -3.3211e+02  4.1315e+02 -1.9682e+01  1.2050e-01  6.1438e-02  3.7167e-02  1.8241e-02  1.2236e-02  2.4958e-01  5.0175e+00  8.4134e-01 -1.6851e+00 -2.9250e+00  8.9398e+00  1.8060e+02  2.1389e+01  3.3838e+01  4.6287e+01  5.8736e+01  7.1224e+01  8.3903e+01  9.6582e+01  1.1217e+02  1.4377e+02
050.0  1.7364e+00 296.61 823.62  5.8142e+01  1.5347e+01  1.6810e+02 -1.7618e+00 -3.4661e+00 -5.2279e+00 -8.0709e-01 -1.6495e+00  8.4945e-01 -1.0199e+03 -7.5353e+02 -1.7734e+03 -3.6659e+02  6.5416e+02 -9.2480e+01  1.1860e-01  6.2944e-02  3.8078e-02  1.8688e-02  1.1274e-02  2.4958e-01  5.9328e+00  1.4395e+00 -1.2787e+00 -2.6128e+00  1.0903e+01  1.7900e+02  2.3138e+01  3.5374e+01  4.7609e+01  5.9845e+01  7.2115e+01  8.4491e+01  9.6866e+01  1.1214e+02  1.4278e+02
055.0  1.7810e+00 298.76 829.60  5.6029e+01  1.3939e+01  1.6458e+02 -2.1334e+00 -3.7136e+00 -5.8470e+00 -9.4097e-01 -1.3791e+00  8.0462e-01 -1.1186e+03 -6.7175e+02 -1.7904e+03 -4.0474e+02  9.0145e+02 -1.6480e+02  1.1638e-01  6.4558e-02  3.9055e-02  1.9167e-02  1.0426e-02  2.4958e-01  6.8635e+00  2.0625e+00 -8.4191e-01 -2.2673e+00  1.3016e+01  1.7759e+02  2.5032e+01  3.7048e+01  4.9064e+01  6.1080e+01  7.3113e+01  8.5179e+01  9.7245e+01  1.1226e+02  1.4204e+02
060.0  1.8284e+00 301.01 835.85  5.3784e+01  1.2733e+01  1.6112e+02 -2.5386e+00 -3.9312e+00 -6.4698e+00 -1.0887e+00 -1.0218e+00  7.3495e-01 -1.2142e+03 -5.8058e+02 -1.7948e+03 -4.4654e+02  1.1555e+03 -2.3547e+02  1.1383e-01  6.6277e-02  4.0095e-02  1.9678e-02  9.7048e-03  2.4958e-01  7.8032e+00  2.7080e+00 -3.7436e-01 -1.8871e+00  1.5261e+01  1.7639e+02  2.7054e+01  3.8847e+01  5.0639e+01  6.2432e+01  7.4207e+01  8.5960e+01  9.7713e+01  1.1252e+02  1.4155e+02
065.0  1.8786e+00 303.35 842.34  5.1422e+01  1.1738e+01  1.5777e+02 -2.9764e+00 -4.1154e+00 -7.0918e+00 -1.2516e+00 -5.7502e-01  6.4121e-01 -1.3056e+03 -4.7992e+02 -1.7855e+03 -4.9177e+02  1.4166e+03 -3.0319e+02  1.1096e-01  6.8095e-02  4.1195e-02  2.0217e-02  9.1215e-03  2.4958e-01  8.7440e+00  3.3729e+00  1.2356e-01 -1.4711e+00  1.7623e+01  1.7539e+02  2.9190e+01  4.0757e+01  5.2324e+01  6.3891e+01  7.5387e+01  8.6826e+01  9.8265e+01  1.1294e+02  1.4132e+02
070.0  1.9312e+00 305.76 849.02  4.8962e+01  1.0963e+01  1.5453e+02 -3.4450e+00 -4.2630e+00 -7.7080e+00 -1.4307e+00 -3.6302e-02  5.2468e-01 -1.3917e+03 -3.6977e+02 -1.7615e+03 -5.4000e+02  1.6849e+03 -3.6652e+02  1.0776e-01  7.0005e-02  4.2350e-02  2.0785e-02  8.6889e-03  2.4958e-01  9.6759e+00  4.0526e+00  6.5071e-01 -1.0189e+00  2.0083e+01  1.7462e+02  3.1424e+01  4.2764e+01  5.4105e+01  6.5446e+01  7.6640e+01  8.7768e+01  9.8895e+01  1.1350e+02  1.4134e+02
075.0  1.9862e+00 308.22 855.86  4.6422e+01  1.0413e+01  1.5144e+02 -3.9424e+00 -4.3707e+00 -8.3131e+00 -1.6271e+00  5.9683e-01  3.8715e-01 -1.4714e+03 -2.5028e+02 -1.7216e+03 -5.9057e+02  1.9603e+03 -4.2395e+02  1.0424e-01  7.1998e-02  4.3556e-02  2.1376e-02  8.4197e-03  2.4958e-01  1.0587e+01  4.7414e+00  1.2050e+00 -5.3053e-01  2.2623e+01  1.7407e+02  3.3738e+01  4.4854e+01  5.5969e+01  6.7085e+01  7.7956e+01  8.8775e+01  9.9595e+01  1.1421e+02  1.4160e+02
080.0  2.0431e+00 310.72 862.79  4.3823e+01  1.0091e+01  1.4852e+02 -4.4662e+00 -4.4353e+00 -8.9015e+00 -1.8412e+00  1.3268e+00  2.3094e-01 -1.5432e+03 -1.2174e+02 -1.6649e+03 -6.4254e+02  2.2422e+03 -4.7391e+02  1.0040e-01  7.4061e-02  4.4804e-02  2.1989e-02  8.3263e-03  2.4958e-01  1.1463e+01  5.4321e+00  1.7834e+00 -7.2698e-03  2.5222e+01  1.7374e+02  3.6116e+01  4.7009e+01  5.7903e+01  6.8796e+01  7.9322e+01  8.9840e+01  1.0036e+02  1.1507e+02  1.4209e+02
085.0  2.1017e+00 313.23 870.37  4.1183e+01  1.0002e+01  1.4579e+02 -5.0132e+00 -4.4538e+00 -9.4670e+00 -2.0736e+00  2.1581e+00  6.4022e-02 -1.6058e+03  1.5316e+01 -1.5905e+03 -6.9552e+02  2.5440e+03 -4.8637e+02  9.6279e-02  7.6183e-02  4.6088e-02  2.2619e-02  8.4149e-03  2.4958e-01  1.2279e+01  6.0950e+00  2.3539e+00  5.1789e-01  2.7862e+01  1.7365e+02  3.8538e+01  4.9214e+01  5.9890e+01  7.0501e+01  8.0726e+01  9.0951e+01  1.0120e+02  1.1607e+02  1.4280e+02
090.0  2.1615e+00 315.75 878.93  3.8523e+01  1.0144e+01  1.4327e+02 -5.5803e+00 -4.4233e+00 -1.0004e+01 -2.3242e+00  3.0946e+00 -1.0636e-01 -1.6578e+03  1.6018e+02 -1.4977e+03 -7.4705e+02  2.8469e+03 -4.9421e+02  9.1882e-02  7.8350e-02  4.7399e-02  2.3262e-02  8.6920e-03  2.4958e-01  1.3029e+01  6.7458e+00  2.9444e+00  1.0788e+00  3.0522e+01  1.7380e+02  4.0987e+01  5.1451e+01  6.1915e+01  7.2213e+01  8.2155e+01  9.2097e+01  1.0214e+02  1.1722e+02  1.4372e+02
095.0  2.2219e+00 318.25 887.99  3.5862e+01  1.0518e+01  1.4099e+02 -6.1633e+00 -4.3412e+00 -1.0505e+01 -2.5922e+00  4.1359e+00 -2.7864e-01 -1.6978e+03  3.1186e+02 -1.3859e+03 -7.9524e+02  3.1477e+03 -4.9722e+02  8.7238e-02  8.0540e-02  4.8724e-02  2.3912e-02  9.1697e-03  2.4958e-01  1.3695e+01  7.3734e+00  3.5488e+00  1.6718e+00  3.3183e+01  1.7417e+02  4.3443e+01  5.3703e+01  6.3963e+01  7.3926e+01  8.3598e+01  9.3269e+01  1.0317e+02  1.1851e+02  1.4483e+02
100.0  2.2823e+00 320.70 897.03  3.3222e+01  1.1120e+01  1.3895e+02 -6.7578e+00 -4.2055e+00 -1.0963e+01 -2.8760e+00  5.2809e+00 -4.5102e-01 -1.7241e+03  4.6913e+02 -1.2550e+03 -8.3796e+02  3.4431e+03 -4.9480e+02  8.2381e-02  8.2732e-02  5.0050e-02  2.4563e-02  9.8582e-03  2.4958e-01  1.4256e+01  7.9650e+00  4.1591e+00  2.2912e+00  3.5823e+01  1.7477e+02  4.5888e+01  5.5953e+01  6.6018e+01  7.5629e+01  8.5044e+01  9.4460e+01  1.0429e+02  1.1995e+02  1.4613e+02
105.0  2.3422e+00 323.08 905.59  3.0623e+01  1.1946e+01  1.3718e+02 -7.3589e+00 -4.0144e+00 -1.1373e+01 -3.1733e+00  6.5269e+00 -6.2153e-01 -1.7355e+03  6.3044e+02 -1.1050e+03 -8.7289e+02  3.7287e+03 -4.8638e+02  7.7351e-02  8.4900e-02  5.1361e-02  2.5207e-02  1.0765e-02  2.4958e-01  1.4691e+01  8.5065e+00  4.7653e+00  2.9292e+00  3.8422e+01  1.7560e+02  4.8303e+01  5.8184e+01  6.8065e+01  7.7310e+01  8.6485e+01  9.5660e+01  1.0552e+02  1.2155e+02  1.4760e+02
110.0  2.4005e+00 325.36 913.33  2.8083e+01  1.2989e+01  1.3568e+02 -7.9611e+00 -3.7669e+00 -1.1728e+01 -3.4811e+00  7.8696e+00 -7.8800e-01 -1.7305e+03  7.9398e+02 -9.3653e+02 -8.9760e+02  3.9995e+03 -4.7151e+02  7.2195e-02  8.7017e-02  5.2642e-02  2.5835e-02  1.1896e-02  2.4958e-01  1.4978e+01  8.9830e+00  5.3562e+00  3.5763e+00  4.0962e+01  1.7664e+02  5.0670e+01  6.0379e+01  7.0006e+01  7.8958e+01  8.7910e+01  9.6862e+01  1.0684e+02  1.2330e+02  1.4924e+02
115.0  2.4567e+00 327.52 920.05  2.5623e+01  1.4242e+01  1.3447e+02 -8.5586e+00 -3.4625e+00 -1.2021e+01 -3.7953e+00  9.3028e+00 -9.4812e-01 -1.7082e+03  9.5765e+02 -7.5052e+02 -9.0970e+02  4.2496e+03 -4.4986e+02  6.6968e-02  8.9053e-02  5.3873e-02  2.6440e-02  1.3251e-02  2.4958e-01  1.5099e+01  9.3792e+00  5.9190e+00  4.2208e+00  4.3422e+01  1.7790e+02  5.2971e+01  6.2521e+01  7.1816e+01  8.0563e+01  8.9311e+01  9.8058e+01  1.0826e+02  1.2521e+02  1.5103e+02
120.0  2.5098e+00 329.53 925.69  2.3261e+01  1.5695e+01  1.3356e+02 -9.1451e+00 -3.1016e+00 -1.2247e+01 -4.1111e+00  1.0818e+01 -1.0995e+00 -1.6676e+03  1.1191e+03 -5.4848e+02 -9.0697e+02  4.4723e+03 -4.2125e+02  6.1731e-02  9.0977e-02  5.5038e-02  2.7011e-02  1.4827e-02  2.4958e-01  1.5036e+01  9.6802e+00  6.4400e+00  4.8497e+00  4.5784e+01  1.7935e+02  5.5189e+01  6.4593e+01  7.3554e+01  8.2116e+01  9.0678e+01  9.9240e+01  1.0978e+02  1.2728e+02  1.5296e+02
125.0  2.5590e+00 331.36 930.26  2.1016e+01  1.7337e+01  1.3296e+02 -9.7143e+00 -2.6853e+00 -1.2400e+01 -4.4231e+00  1.2405e+01 -1.2397e+00 -1.6084e+03  1.2759e+03 -3.3244e+02 -8.8758e+02  4.6607e+03 -3.8572e+02  5.6550e-02  9.2760e-02  5.6116e-02  2.7541e-02  1.6617e-02  2.4958e-01  1.4776e+01  9.8718e+00  6.9049e+00  5.4488e+00  4.8029e+01  1.8099e+02  5.7305e+01  6.6580e+01  7.5211e+01  8.3608e+01  9.2006e+01  1.0040e+02  1.1139e+02  1.2951e+02  1.5502e+02
130.0  2.6034e+00 332.99 933.82  1.8903e+01  1.9155e+01  1.3267e+02 -1.0260e+01 -2.2158e+00 -1.2475e+01 -4.7253e+00  1.4051e+01 -1.3665e+00 -1.5303e+03  1.4254e+03 -1.0490e+02 -8.5021e+02  4.8074e+03 -3.4352e+02  5.1495e-02  9.4372e-02  5.7091e-02  2.8019e-02  1.8608e-02  2.4958e-01  1.4310e+01  9.9422e+00  7.2999e+00  6.0031e+00  5.0142e+01  1.8281e+02  5.9304e+01  6.8466e+01  7.6777e+01  8.5031e+01  9.3286e+01  1.0159e+02  1.1310e+02  1.3190e+02  1.5719e+02
135.0  2.6424e+00 334.41 936.43  1.6940e+01  2.1136e+01  1.3268e+02 -1.0775e+01 -1.6961e+00 -1.2471e+01 -5.0112e+00  1.5739e+01 -1.4774e+00 -1.4338e+03  1.5650e+03  1.3114e+02 -7.9423e+02  4.9052e+03 -2.9513e+02  4.6638e-02  9.5782e-02  5.7945e-02  2.8438e-02  2.0781e-02  2.4958e-01  1.3634e+01  9.8816e+00  7.6116e+00  6.4975e+00  5.2105e+01  1.8479e+02  6.1170e+01  7.0113e+01  7.8246e+01  8.6378e+01  9.4511e+01  1.0282e+02  1.1488e+02  1.3445e+02  1.5946e+02
140.0  2.6751e+00 335.58 938.20  1.5141e+01  2.3265e+01  1.3301e+02 -1.1253e+01 -1.1300e+00 -1.2383e+01 -5.2745e+00  1.7451e+01 -1.5706e+00 -1.3197e+03  1.6920e+03  3.7231e+02 -7.1982e+02  4.9472e+03 -2.4124e+02  4.2053e-02  9.6967e-02  5.8661e-02  2.8790e-02  2.3113e-02  2.4958e-01  1.2751e+01  9.6840e+00  7.8284e+00  6.9178e+00  5.3904e+01  1.8692e+02  6.2890e+01  7.1576e+01  7.9609e+01  8.7642e+01  9.5676e+01  1.0409e+02  1.1674e+02  1.3711e+02  1.6183e+02
145.0  2.7009e+00 336.51 939.18  1.3520e+01  2.5525e+01  1.3365e+02 -1.1689e+01 -5.2240e-01 -1.2211e+01 -5.5088e+00  1.9167e+01 -1.6442e+00 -1.1891e+03  1.8040e+03  6.1492e+02 -6.2803e+02  4.9274e+03 -1.8277e+02  3.7810e-02  9.7903e-02  5.9228e-02  2.9068e-02  2.5576e-02  2.4958e-01  1.1671e+01  9.3469e+00  7.9411e+00  7.2511e+00  5.5525e+01  1.8918e+02  6.4450e+01  7.2905e+01  8.0861e+01  8.8817e+01  9.6774e+01  1.0538e+02  1.1867e+02  1.3978e+02  1.6427e+02
150.0  2.7194e+00 337.16 939.45  1.2089e+01  2.7899e+01  1.3460e+02 -1.2077e+01  1.2114e-01 -1.1955e+01 -5.7085e+00  2.0865e+01 -1.6969e+00 -1.0438e+03  1.8989e+03  8.5506e+02 -5.2079e+02  4.8412e+03 -1.2082e+02  3.3974e-02  9.8573e-02  5.9633e-02  2.9266e-02  2.8138e-02  2.4958e-01  1.0408e+01  8.8722e+00  7.9429e+00  7.4868e+00  5.6956e+01  1.9155e+02  6.5837e+01  7.4093e+01  8.1995e+01  8.9898e+01  9.7800e+01  1.0669e+02  1.2064e+02  1.4244e+02  1.6681e+02
155.0  2.7301e+00 337.54 939.05  1.0860e+01  3.0369e+01  1.3584e+02 -1.2412e+01  7.9434e-01 -1.1617e+01 -5.8686e+00  2.2520e+01 -1.7276e+00 -8.8592e+02  1.9747e+03  1.0888e+03 -4.0085e+02  4.6855e+03 -5.6606e+01  3.0605e-02  9.8964e-02  5.9869e-02  2.9382e-02  3.0764e-02  2.4958e-01  8.9859e+00  8.2662e+00  7.8308e+00  7.6171e+00  5.8185e+01  1.9402e+02  6.7041e+01  7.5135e+01  8.3006e+01  9.0878e+01  9.8749e+01  1.0799e+02  1.2266e+02  1.4507e+02  1.6938e+02
160.0  2.7330e+00 337.65 938.05  9.8407e+00  3.2916e+01  1.3736e+02 -1.2690e+01  1.4903e+00 -1.1200e+01 -5.9853e+00  2.4110e+01 -1.7359e+00 -7.1792e+02  2.0302e+03  1.3123e+03 -2.7162e+02  4.4590e+03  8.5581e+00  2.7754e-02  9.9069e-02  5.9933e-02  2.9414e-02  3.3415e-02  2.4958e-01  7.4303e+00  7.5391e+00  7.6050e+00  7.6373e+00  5.9204e+01  1.9657e+02  6.8054e+01  7.6027e+01  8.3890e+01  9.1753e+01  9.9615e+01  1.0928e+02  1.2469e+02  1.4766e+02  1.7198e+02
165.0  2.7280e+00 337.47 936.47  9.0400e+00  3.5521e+01  1.3917e+02 -1.2909e+01  2.2019e+00 -1.0707e+01 -6.0561e+00  2.5608e+01 -1.7215e+00 -5.4252e+02  2.0643e+03  1.5218e+03 -1.3700e+02  4.1630e+03  7.3337e+01  2.5462e-02  9.8886e-02  5.9822e-02  2.9359e-02  3.6054e-02  2.4958e-01  5.7730e+00  6.7054e+00  7.2695e+00  7.5464e+00  6.0005e+01  1.9917e+02  6.8866e+01  7.6763e+01  8.4641e+01  9.2518e+01  1.0040e+02  1.1053e+02  1.2673e+02  1.5019e+02  1.7458e+02
170.0  2.7152e+00 337.01 934.34  8.4637e+00  3.8165e+01  1.4124e+02 -1.3066e+01  2.9214e+00 -1.0144e+01 -6.0799e+00  2.6992e+01 -1.6849e+00 -3.6259e+02  2.0766e+03  1.7140e+03 -1.1627e+00  3.8007e+03  1.3643e+02  2.3759e-02  9.8421e-02  5.9541e-02  2.9221e-02  3.8643e-02  2.4958e-01  4.0479e+00  5.7825e+00  6.8319e+00  7.3469e+00  6.0581e+01  2.0182e+02  6.9425e+01  7.7340e+01  8.5254e+01  9.3169e+01  1.0110e+02  1.1172e+02  1.2874e+02  1.5265e+02  1.7717e+02
175.0  2.6948e+00 336.29 931.70  8.1161e+00  4.0826e+01  1.4355e+02 -1.3160e+01  3.6414e+00 -9.5182e+00 -6.0568e+00  2.8239e+01 -1.6269e+00 -1.8108e+02  2.0672e+03  1.8861e+03  1.3173e+02  3.3775e+03  1.9661e+02  2.2661e-02  9.7683e-02  5.9094e-02  2.9002e-02  4.1144e-02  2.4958e-01  2.2904e+00  4.7903e+00  6.3026e+00  7.0448e+00  6.0929e+01  2.0448e+02  6.9779e+01  7.7753e+01  8.5728e+01  9.3702e+01  1.0174e+02  1.1284e+02  1.3070e+02  1.5503e+02  1.7971e+02
180.0  2.6674e+00 335.31 928.58  8.0000e+00  4.3485e+01  1.4609e+02 -1.3191e+01  4.3545e+00 -8.8364e+00 -5.9887e+00  2.9330e+01 -1.5487e+00 -8.9288e-01  2.0367e+03  2.0358e+03  2.5775e+02  2.9009e+03  2.5281e+02  2.2174e-02  9.6688e-02  5.8493e-02  2.8707e-02  4.3523e-02  2.4958e-01  5.3582e-01  3.7502e+00  5.6948e+00  6.6491e+00  6.1045e+01  2.0714e+02  6.9944e+01  7.8000e+01  8.6057e+01  9.4113e+01  1.0228e+02  1.1387e+02  1.3258e+02  1.5730e+02  1.8220e+02
185.0  2.6332e+00 333.83 924.97  8.1161e+00  4.6121e+01  1.4885e+02 -1.3160e+01  5.0533e+00 -8.1068e+00 -5.8830e+00  3.0238e+01 -1.4512e+00  1.7518e+02  1.9860e+03  2.1612e+03  3.3765e+02  2.3060e+03  3.0843e+02  2.2306e-02  9.5449e-02  5.7743e-02  2.8339e-02  4.5748e-02  2.4958e-01 -1.3063e+00  2.6153e+00  4.9877e+00  6.1520e+00  6.0929e+01  2.0977e+02  6.9912e+01  7.8073e+01  8.6234e+01  9.4395e+01  1.0272e+02  1.1477e+02  1.3436e+02  1.5946e+02  1.8461e+02
190.0  2.5924e+00 331.24 920.85  8.4637e+00  4.8715e+01  1.5179e+02 -1.3069e+01  5.7312e+00 -7.3382e+00 -5.7558e+00  3.0924e+01 -1.3349e+00  3.4450e+02  1.9161e+03  2.2606e+03  3.9356e+02  1.6536e+03  3.5950e+02  2.3080e-02  9.3971e-02  5.6849e-02  2.7900e-02  4.7785e-02  2.4958e-01 -3.1160e+00  1.4549e+00  4.2201e+00  5.5773e+00  6.0581e+01  2.1237e+02  6.9669e+01  7.7958e+01  8.6248e+01  9.4537e+01  1.0303e+02  1.1552e+02  1.3597e+02  1.6147e+02  1.8692e+02
195.0  2.5458e+00 327.82 916.26  9.0400e+00  5.1246e+01  1.5489e+02 -1.2921e+01  6.3817e+00 -6.5397e+00 -5.6105e+00  3.1383e+01 -1.2022e+00  5.0469e+02  1.8289e+03  2.3336e+03  4.4146e+02  9.9364e+02  4.0325e+02  2.4462e-02  9.2283e-02  5.5828e-02  2.7399e-02  4.9613e-02  2.4958e-01 -4.8012e+00  3.2602e-01  3.4278e+00  4.9501e+00  6.0005e+01  2.1490e+02  6.9213e+01  7.7654e+01  8.6095e+01  9.4536e+01  1.0321e+02  1.1610e+02  1.3737e+02  1.6334e+02  1.8912e+02
200.0  2.4944e+00 323.95 911.23  9.8407e+00  5.3696e+01  1.5814e+02 -1.2720e+01  6.9992e+00 -5.7205e+00 -5.4500e+00  3.1613e+01 -1.0557e+00  6.5375e+02  1.7267e+03  2.3804e+03  4.8092e+02  3.4085e+02  4.3930e+02  2.6402e-02  9.0420e-02  5.4700e-02  2.6846e-02  5.1216e-02  2.4958e-01 -6.3355e+00 -7.4987e-01  2.6292e+00  4.2876e+00  5.9204e+01  2.1735e+02  6.8507e+01  7.7161e+01  8.5776e+01  9.4391e+01  1.0324e+02  1.1650e+02  1.3853e+02  1.6502e+02  1.9118e+02
205.0  2.4391e+00 319.94 905.78  1.0860e+01  5.6046e+01  1.6151e+02 -1.2468e+01  7.5789e+00 -4.8895e+00 -5.2774e+00  3.1620e+01 -8.9794e-01  7.9010e+02  1.6116e+03  2.4017e+03  5.1183e+02 -2.9146e+02  4.6754e+02  2.8847e-02  8.8414e-02  5.3487e-02  2.6250e-02  5.2586e-02  2.4958e-01 -7.6999e+00 -1.7552e+00  1.8411e+00  3.6060e+00  5.8185e+01  2.1970e+02  6.7581e+01  7.6483e+01  8.5293e+01  9.4103e+01  1.0314e+02  1.1670e+02  1.3945e+02  1.6650e+02  1.9310e+02
210.0  2.3808e+00 316.03 899.97  1.2089e+01  5.8277e+01  1.6497e+02 -1.2172e+01  8.1168e+00 -4.0554e+00 -5.0955e+00  3.1412e+01 -7.3177e-01  9.1261e+02  1.4862e+03  2.3988e+03  5.3437e+02 -8.9188e+02  4.8813e+02  3.1733e-02  8.6302e-02  5.2209e-02  2.5623e-02  5.3717e-02  2.4958e-01 -8.8831e+00 -2.6768e+00  1.0779e+00  2.9205e+00  5.6956e+01  2.2193e+02  6.6464e+01  7.5621e+01  8.4647e+01  9.3673e+01  1.0289e+02  1.1671e+02  1.4011e+02  1.6778e+02  1.9485e+02
215.0  2.3205e+00 312.34 893.82  1.3520e+01  6.0373e+01  1.6850e+02 -1.1836e+01  8.6096e+00 -3.2263e+00 -4.9071e+00  3.1003e+01 -5.5975e-01  1.0205e+03  1.3529e+03  2.3734e+03  5.4893e+02 -1.4512e+03  5.0143e+02  3.4998e-02  8.4114e-02  5.0886e-02  2.4974e-02  5.4612e-02  2.4958e-01 -9.8810e+00 -3.5055e+00  3.5142e-01  2.2443e+00  5.5525e+01  2.2403e+02  6.5167e+01  7.4581e+01  8.3841e+01  9.3102e+01  1.0250e+02  1.1651e+02  1.4052e+02  1.6884e+02  1.9644e+02
220.0  2.2589e+00 308.92 887.37  1.5141e+01  6.2319e+01  1.7207e+02 -1.1465e+01  9.0551e+00 -2.4097e+00 -4.7151e+00  3.0408e+01 -3.8431e-01  1.1134e+03  1.2138e+03  2.3272e+03  5.5610e+02 -1.9625e+03  5.0797e+02  3.8577e-02  8.1884e-02  4.9537e-02  2.4311e-02  5.5276e-02  2.4958e-01 -1.0695e+01 -4.2364e+00 -3.2914e-01  1.5885e+00  5.3904e+01  2.2597e+02  6.3700e+01  7.3367e+01  8.2880e+01  9.2393e+01  1.0199e+02  1.1611e+02  1.4067e+02  1.6969e+02  1.9783e+02
225.0  2.1970e+00 305.79 880.65  1.6940e+01  6.4098e+01  1.7565e+02 -1.1064e+01  9.4516e+00 -1.6125e+00 -4.5217e+00  2.9645e+01 -2.0769e-01  1.1914e+03  1.0712e+03  2.2625e+03  5.5657e+02 -2.4209e+03  5.0841e+02  4.2407e-02  7.9638e-02  4.8178e-02  2.3645e-02  5.5717e-02  2.4958e-01 -1.1332e+01 -4.8680e+00 -9.5741e-01  9.6183e-01  5.2105e+01  2.2775e+02  6.2075e+01  7.1988e+01  8.1769e+01  9.1551e+01  1.0137e+02  1.1552e+02  1.4055e+02  1.7033e+02  1.9904e+02
230.0  2.1353e+00 302.92 873.70  1.8903e+01  6.5698e+01  1.7921e+02 -1.0639e+01  9.7983e+00 -8.4069e-01 -4.3293e+00  2.8732e+01 -3.1881e-02  1.2545e+03  9.2688e+02  2.1814e+03  5.5113e+02 -2.8240e+03  5.0344e+02  4.6429e-02  7.7403e-02  4.6826e-02  2.2981e-02  5.5946e-02  2.4958e-01 -1.1803e+01 -5.4018e+00 -1.5294e+00  3.7109e-01  5.0142e+01  2.2935e+02  6.0304e+01  7.0451e+01  8.0515e+01  9.0579e+01  1.0065e+02  1.1475e+02  1.4017e+02  1.7073e+02  2.0004e+02
235.0  2.0746e+00 300.29 866.57  2.1016e+01  6.7106e+01  1.8273e+02 -1.0194e+01  1.0095e+01 -9.9590e-02 -4.1396e+00  2.7689e+01  1.4197e-01  1.3034e+03  7.8270e+02  2.0861e+03  5.4024e+02 -3.1642e+03  5.0418e+02  5.0587e-02  7.5200e-02  4.5493e-02  2.2327e-02  5.5977e-02  2.4958e-01 -1.2122e+01 -5.8475e+00 -2.0516e+00 -1.8870e-01  4.8029e+01  2.3076e+02  5.8398e+01  6.8767e+01  7.9125e+01  8.9484e+01  9.9842e+01  1.1382e+02  1.3952e+02  1.7092e+02  2.0084e+02
240.0  2.0153e+00 297.89 859.42  2.3261e+01  6.8312e+01  1.8618e+02 -9.7353e+00  1.0342e+01  6.0626e-01 -3.9547e+00  2.6542e+01  3.2070e-01  1.3387e+03  6.4011e+02  1.9788e+03  5.2428e+02 -3.4334e+03  5.2535e+02  5.4833e-02  7.3053e-02  4.4194e-02  2.1689e-02  5.5815e-02  2.4958e-01 -1.2308e+01 -6.2192e+00 -2.5355e+00 -7.2764e-01  4.5784e+01  2.3197e+02  5.6372e+01  6.6960e+01  7.7608e+01  8.8271e+01  9.8934e+01  1.1275e+02  1.3860e+02  1.7087e+02  2.0142e+02
245.0  1.9581e+00 295.69 852.37  2.5623e+01  6.9307e+01  1.8954e+02 -9.2661e+00  1.0539e+01  1.2731e+00 -3.7760e+00  2.5310e+01  5.0704e-01  1.3612e+03  5.0037e+02  1.8616e+03  5.0468e+02 -3.6463e+03  5.4798e+02  5.9121e-02  7.0977e-02  4.2939e-02  2.1073e-02  5.5474e-02  2.4958e-01 -1.2374e+01 -6.5129e+00 -2.9669e+00 -1.2266e+00  4.3422e+01  2.3296e+02  5.4239e+01  6.5056e+01  7.5972e+01  8.6947e+01  9.7922e+01  1.1155e+02  1.3741e+02  1.7060e+02  2.0178e+02
250.0  1.9032e+00 293.67 845.48  2.8083e+01  7.0082e+01  1.9277e+02 -8.7913e+00  1.0689e+01  1.8977e+00 -3.6047e+00  2.4015e+01  7.0144e-01  1.3719e+03  3.6450e+02  1.7364e+03  4.8218e+02 -3.8067e+03  5.7160e+02  6.3414e-02  6.8987e-02  4.1734e-02  2.0482e-02  5.4967e-02  2.4958e-01 -1.2336e+01 -6.7355e+00 -3.3475e+00 -1.6848e+00  4.0962e+01  2.3373e+02  5.2015e+01  6.3068e+01  7.4230e+01  8.5522e+01  9.6813e+01  1.1026e+02  1.3594e+02  1.7009e+02  2.0191e+02
255.0  1.8509e+00 291.81 838.77  3.0623e+01  7.0632e+01  1.9586e+02 -8.3147e+00  1.0792e+01  2.4777e+00 -3.4415e+00  2.2672e+01  9.0412e-01  1.3715e+03  2.3329e+02  1.6048e+03  4.5745e+02 -3.9188e+03  5.9568e+02  6.7677e-02  6.7092e-02  4.0588e-02  1.9920e-02  5.4307e-02  2.4958e-01 -1.2207e+01 -6.8941e+00 -3.6800e+00 -2.1025e+00  3.8422e+01  2.3429e+02  4.9716e+01  6.1009e+01  7.2394e+01  8.4004e+01  9.5614e+01  1.0889e+02  1.3422e+02  1.6936e+02  2.0182e+02
260.0  1.8015e+00 290.11 832.31  3.3222e+01  7.0954e+01  1.9878e+02 -7.8400e+00  1.0851e+01  3.0111e+00 -3.2873e+00  2.1298e+01  1.1152e+00  1.3610e+03  1.0737e+02  1.4684e+03  4.3105e+02 -3.9873e+03  6.1965e+02  7.1882e-02  6.5301e-02  3.9504e-02  1.9388e-02  5.3510e-02  2.4958e-01 -1.2002e+01 -6.9958e+00 -3.9671e+00 -2.4807e+00  3.5823e+01  2.3461e+02  4.7358e+01  5.8893e+01  7.0476e+01  8.2405e+01  9.4333e+01  1.0748e+02  1.3232e+02  1.6842e+02  2.0151e+02
265.0  1.7551e+00 288.54 826.13  3.5862e+01  7.1043e+01  2.0151e+02 -7.3707e+00  1.0867e+01  3.4965e+00 -3.1424e+00  1.9907e+01  1.3344e+00  1.3413e+03 -1.2794e+01  1.3285e+03  4.0350e+02 -4.0169e+03  6.4298e+02  7.6004e-02  6.3618e-02  3.8487e-02  1.8888e-02  5.2588e-02  2.4958e-01 -1.1734e+01 -7.0473e+00 -4.2123e+00 -2.8209e+00  3.3183e+01  2.3470e+02  4.4959e+01  5.6736e+01  6.8513e+01  8.0736e+01  9.2980e+01  1.0604e+02  1.3026e+02  1.6726e+02  2.0098e+02
270.0  1.7117e+00 287.12 820.25  3.8523e+01  7.0901e+01  2.0403e+02 -6.9096e+00  1.0843e+01  3.9329e+00 -3.0072e+00  1.8513e+01  1.5616e+00  1.3133e+03 -1.2688e+02  1.1864e+03  3.7518e+02 -4.0122e+03  6.6515e+02  8.0023e-02  6.2048e-02  3.7537e-02  1.8422e-02  5.1554e-02  2.4958e-01 -1.1412e+01 -7.0549e+00 -4.4187e+00 -3.1250e+00  3.0522e+01  2.3455e+02  4.2537e+01  5.4552e+01  6.6567e+01  7.9010e+01  9.1564e+01  1.0460e+02  1.2809e+02  1.6590e+02  2.0023e+02
275.0  1.6716e+00 285.82 814.71  4.1183e+01  7.0527e+01  2.0632e+02 -6.4597e+00  1.0779e+01  4.3198e+00 -2.8820e+00  1.7125e+01  1.7962e+00  1.2776e+03 -2.3469e+02  1.0429e+03  3.4644e+02 -3.9777e+03  6.8569e+02  8.3923e-02  6.0594e-02  3.6657e-02  1.7990e-02  5.0420e-02  2.4958e-01 -1.1048e+01 -7.0241e+00 -4.5899e+00 -3.3952e+00  2.7862e+01  2.3418e+02  4.0110e+01  5.2358e+01  6.4605e+01  7.7240e+01  9.0096e+01  1.0318e+02  1.2585e+02  1.6436e+02  1.9927e+02
280.0  1.6347e+00 284.65 809.53  4.3823e+01  6.9925e+01  2.0836e+02 -6.0233e+00  1.0680e+01  4.6567e+00 -2.7668e+00  1.5753e+01  2.0376e+00  1.2352e+03 -3.3609e+02  8.9909e+02  3.1754e+02 -3.9175e+03  7.0416e+02  8.7689e-02  5.9256e-02  3.5847e-02  1.7593e-02  4.9199e-02  2.4958e-01 -1.0648e+01 -6.9600e+00 -4.7288e+00 -3.6338e+00  2.5222e+01  2.3358e+02  3.7695e+01  5.0168e+01  6.2641e+01  7.5441e+01  8.8587e+01  1.0180e+02  1.2357e+02  1.6249e+02  1.9811e+02
285.0  1.6010e+00 283.60 804.73  4.6422e+01  6.9099e+01  2.1013e+02 -5.6028e+00  1.0546e+01  4.9437e+00 -2.6616e+00  1.4407e+01  2.2850e+00  1.1866e+03 -4.3104e+02  7.5551e+02  2.8869e+02 -3.8355e+03  7.2020e+02  9.1312e-02  5.8034e-02  3.5108e-02  1.7230e-02  4.7900e-02  2.4958e-01 -1.0220e+01 -6.8669e+00 -4.8385e+00 -3.8431e+00  2.2623e+01  2.3275e+02  3.5311e+01  4.8000e+01  6.0689e+01  7.3628e+01  8.7050e+01  1.0047e+02  1.2129e+02  1.6032e+02  1.9675e+02
290.0  1.5705e+00 282.67 800.31  4.8962e+01  6.8056e+01  2.1163e+02 -5.2001e+00  1.0381e+01  5.1809e+00 -2.5664e+00  1.3092e+01  2.5375e+00  1.1324e+03 -5.1955e+02  6.1283e+02  2.6002e+02 -3.7351e+03  7.3348e+02  9.4782e-02  5.6928e-02  3.4439e-02  1.6902e-02  4.6533e-02  2.4958e-01 -9.7687e+00 -6.7487e+00 -4.9218e+00 -4.0252e+00  2.0083e+01  2.3171e+02  3.2976e+01  4.5869e+01  5.8762e+01  7.1814e+01  8.5497e+01  9.9181e+01  1.1904e+02  1.5794e+02  1.9521e+02
295.0  1.5432e+00 281.86 796.31  5.1422e+01  6.6803e+01  2.1283e+02 -4.8172e+00  1.0186e+01  5.3689e+00 -2.4811e+00  1.1815e+01  2.7941e+00  1.0732e+03 -6.0165e+02  4.7152e+02  2.3164e+02 -3.6193e+03  7.4374e+02  9.8092e-02  5.5937e-02  3.3840e-02  1.6608e-02  4.5107e-02  2.4958e-01 -9.2989e+00 -6.6085e+00 -4.9810e+00 -4.1822e+00  1.7623e+01  2.3046e+02  3.0707e+01  4.3790e+01  5.6874e+01  7.0017e+01  8.3942e+01  9.7868e+01  1.1684e+02  1.5540e+02  1.9349e+02
300.0  1.5190e+00 281.18 792.72  5.3784e+01  6.5350e+01  2.1374e+02 -4.4556e+00  9.9637e+00  5.5081e+00 -2.4056e+00  1.0581e+01  3.0536e+00  1.0094e+03 -6.7743e+02  3.3201e+02  2.0361e+02 -3.4908e+03  7.5074e+02  1.0124e-01  5.5060e-02  3.3309e-02  1.6347e-02  4.3630e-02  2.4958e-01 -8.8137e+00 -6.4489e+00 -5.0183e+00 -4.3162e+00  1.5261e+01  2.2900e+02  2.8521e+01  4.1780e+01  5.5040e+01  6.8300e+01  8.2397e+01  9.6544e+01  1.1472e+02  1.5269e+02  1.9162e+02
305.0  1.4979e+00 280.61 789.56  5.6029e+01  6.3708e+01  2.1435e+02 -4.1169e+00  9.7161e+00  5.5992e+00 -2.3397e+00  9.3929e+00  3.3150e+00  9.4159e+02 -7.4697e+02  1.9462e+02  1.7597e+02 -3.3519e+03  7.5429e+02  1.0421e-01  5.4296e-02  3.2847e-02  1.6120e-02  4.2111e-02  2.4958e-01 -8.3157e+00 -6.2719e+00 -5.0355e+00 -4.4287e+00  1.3016e+01  2.2736e+02  2.6435e+01  3.9854e+01  5.3273e+01  6.6692e+01  8.0876e+01  9.5223e+01  1.1270e+02  1.4985e+02  1.8960e+02
310.0  1.4798e+00 280.16 786.83  5.8142e+01  6.1890e+01  2.1464e+02 -3.8024e+00  9.4455e+00  5.6431e+00 -2.2834e+00  8.2546e+00  3.5770e+00  8.6998e+02 -8.1034e+02  5.9637e+01  1.4872e+02 -3.2047e+03  7.5421e+02  1.0701e-01  5.3642e-02  3.2451e-02  1.5926e-02  4.0557e-02  2.4958e-01 -7.8063e+00 -6.0790e+00 -5.0341e+00 -4.5213e+00  1.0903e+01  2.2554e+02  2.4464e+01  3.8025e+01  5.1586e+01  6.5147e+01  7.9392e+01  9.3914e+01  1.1079e+02  1.4690e+02  1.8746e+02
315.0  1.4648e+00 279.82 784.54  6.0105e+01  5.9909e+01  2.1462e+02 -3.5134e+00  9.1539e+00  5.6405e+00 -2.2365e+00  7.1687e+00  3.8384e+00  7.9493e+02 -8.6764e+02 -7.2707e+01  1.2183e+02 -3.0506e+03  7.5037e+02  1.0963e-01  5.3097e-02  3.2122e-02  1.5764e-02  3.8973e-02  2.4958e-01 -7.2865e+00 -5.8715e+00 -5.0155e+00 -4.5953e+00  8.9398e+00  2.2356e+02  2.2624e+01  3.6308e+01  4.9991e+01  6.3675e+01  7.7958e+01  9.2629e+01  1.0901e+02  1.4386e+02  1.8520e+02
320.0  1.4527e+00 279.61 782.69  6.1904e+01  5.7780e+01  2.1429e+02 -3.2510e+00  8.8434e+00  5.5924e+00 -2.1989e+00  6.1373e+00  4.0977e+00  7.1673e+02 -9.1894e+02 -2.0221e+02  9.5287e+01 -2.8911e+03  7.4264e+02  1.1207e-01  5.2660e-02  3.1857e-02  1.5635e-02  3.7367e-02  2.4958e-01 -6.7567e+00 -5.6500e+00 -4.9805e+00 -4.6519e+00  7.1409e+00  2.2143e+02  2.0928e+01  3.4715e+01  4.8501e+01  6.2288e+01  7.6588e+01  9.1380e+01  1.0735e+02  1.4077e+02  1.8284e+02
325.0  1.4436e+00 279.51 781.28  6.3525e+01  5.5520e+01  2.1365e+02 -3.0163e+00  8.5162e+00  5.5000e+00 -2.1704e+00  5.1621e+00  4.3536e+00  6.3560e+02 -9.6429e+02 -3.2869e+02  6.9032e+01 -2.7273e+03  7.3092e+02  1.1432e-01  5.2329e-02  3.1657e-02  1.5537e-02  3.5745e-02  2.4958e-01 -6.2166e+00 -5.4148e+00 -4.9298e+00 -4.6917e+00  5.5199e+00  2.1917e+02  1.9389e+01  3.3258e+01  4.7128e+01  6.0997e+01  7.5293e+01  9.0178e+01  1.0582e+02  1.3765e+02  1.8041e+02
330.0  1.4374e+00 279.54 780.32  6.4956e+01  5.3146e+01  2.1271e+02 -2.8102e+00  8.1743e+00  5.3641e+00 -2.1510e+00  4.2443e+00  4.6047e+00  5.5175e+02 -1.0037e+03 -4.5199e+02  4.3012e+01 -2.5599e+03  7.1512e+02  1.1638e-01  5.2104e-02  3.1521e-02  1.5470e-02  3.4111e-02  2.4958e-01 -5.6657e+00 -5.1661e+00 -4.8639e+00 -4.7156e+00  4.0892e+00  2.1680e+02  1.8020e+01  3.1950e+01  4.5880e+01  5.9810e+01  7.4084e+01  8.9034e+01  1.0443e+02  1.3453e+02  1.7791e+02
335.0  1.4341e+00 279.68 779.80  6.6185e+01  5.0676e+01  2.1147e+02 -2.6337e+00  7.8197e+00  5.1860e+00 -2.1406e+00  3.3853e+00  4.8496e+00  4.6536e+02 -1.0373e+03 -5.7193e+02  1.7164e+01 -2.3896e+03  6.9514e+02  1.1825e-01  5.1983e-02  3.1448e-02  1.5434e-02  3.2472e-02  2.4958e-01 -5.1029e+00 -4.9035e+00 -4.7829e+00 -4.7238e+00  2.8597e+00  2.1433e+02  1.6829e+01  3.0799e+01  4.4769e+01  5.8738e+01  7.2974e+01  8.7959e+01  1.0317e+02  1.3156e+02  1.7538e+02
340.0  1.4336e+00 279.93 779.73  6.7204e+01  4.8129e+01  2.0994e+02 -2.4876e+00  7.4546e+00  4.9669e+00 -2.1392e+00  2.5858e+00  5.0869e+00  3.7658e+02 -1.0649e+03 -6.8833e+02 -8.5781e+00 -2.2168e+03  6.7093e+02  1.1992e-01  5.1966e-02  3.1437e-02  1.5429e-02  3.0832e-02  2.4958e-01 -4.5270e+00 -4.6266e+00 -4.6869e+00 -4.7165e+00  1.8407e+00  2.1178e+02  1.5828e+01  2.9815e+01  4.3802e+01  5.7789e+01  7.1971e+01  8.6961e+01  1.0204e+02  1.2881e+02  1.7282e+02
345.0  1.4360e+00 280.31 780.10  6.8005e+01  4.5524e+01  2.0814e+02 -2.3728e+00  7.0808e+00  4.7081e+00 -2.1466e+00  1.8467e+00  5.3150e+00  2.8557e+02 -1.0866e+03 -8.0103e+02 -3.4283e+01 -2.0417e+03  6.4243e+02  1.2139e-01  5.2053e-02  3.1490e-02  1.5455e-02  2.9198e-02  2.4958e-01 -3.9364e+00 -4.3346e+00 -4.5755e+00 -4.6937e+00  1.0400e+00  2.0918e+02  1.5022e+01  2.9005e+01  4.2987e+01  5.6970e+01  7.1086e+01  8.6051e+01  1.0103e+02  1.2629e+02  1.7026e+02
350.0  1.4413e+00 280.80 780.92  6.8581e+01  4.2880e+01  2.0607e+02 -2.2899e+00  6.7006e+00  4.4107e+00 -2.1631e+00  1.1688e+00  5.5324e+00  1.9246e+02 -1.1022e+03 -9.0979e+02 -6.0020e+01 -1.8643e+03  6.0959e+02  1.2265e-01  5.2244e-02  3.1605e-02  1.5511e-02  2.7575e-02  2.4958e-01 -3.3294e+00 -4.0265e+00 -4.4482e+00 -4.6552e+00  4.6371e-01  2.0653e+02  1.4420e+01  2.8376e+01  4.2331e+01  5.6287e+01  7.0325e+01  8.5235e+01  1.0015e+02  1.2400e+02  1.6772e+02
355.0  1.4494e+00 281.42 782.17  6.8929e+01  4.0219e+01  2.0376e+02 -2.2397e+00  6.3160e+00  4.0764e+00 -2.1884e+00  5.5297e-01  5.7377e+00  9.7384e+01 -1.1118e+03 -1.0144e+03 -8.5852e+01 -1.6845e+03  5.7238e+02  1.2370e-01  5.2538e-02  3.1784e-02  1.5599e-02  2.5968e-02  2.4958e-01 -2.7040e+00 -3.7011e+00 -4.3044e+00 -4.6004e+00  1.1615e-01  2.0387e+02  1.4024e+01  2.7932e+01  4.1840e+01  5.5748e+01  6.9696e+01  8.4523e+01  9.9349e+01  1.2193e+02  1.6522e+02
360.0  1.4604e+00 282.15 783.87  6.9045e+01  3.7560e+01  2.0121e+02 -2.2228e+00  5.9293e+00  3.7065e+00 -2.2228e+00 -4.9077e-08  5.9293e+00  4.8879e-01 -1.1150e+03 -1.1145e+03 -1.1184e+02 -1.5020e+03  5.3082e+02  1.2452e-01  5.2937e-02  3.2025e-02  1.5717e-02  2.4382e-02  2.4958e-01 -2.0583e+00 -3.3573e+00 -4.1431e+00 -4.5288e+00  0.0000e+00  2.0121e+02  1.3839e+01  2.7678e+01  4.1517e+01  5.5356e+01  6.9204e+01  8.3919e+01  9.8633e+01  1.2008e+02  1.6261e+02

The following graphs pertain to the above listed input parameters.

Pressure as function of crank angle Θ

As expected the pressure fluctuates between a minimum ( 1.433 bar at Θ = 340 °) and a maximum ( 2.733 bar at Θ = 160 ° ). The precise location is of course determined by the drive employed ( sinusoidal here) and the size of the kooler, regenerator, and heater space.

Figure 3.1a : Pressure as function of crank angle

Total volume as function of crank angle Θ

This merely is a reflection of the drive type employed and the size of the kooler, regenerator, and heater space. It can be expected that the ratio of maximum to minimum volume strongly affects the ratio of maximum to minimum pressure.

Figure 3.1b : Total volume as function of crank angle

p-V Diagram

The shown curve is typical for almost all internal and external combustion engines. The area enclosed by the loop equals the net work per cycle ( 3.706480 J ) produced by the engine while the area between the Vtot-axis and the upper-most part of the loop corresponds to the heat comsumed per cycle. The ratio between the two equals the actual efficiency of the engine, here 0.625116. This value is of course less than the Carnot-efficiency ( 0.674973 ) based on the kooler and heater temperature. The thermodynamic reason for this decline in efficiency is the mixing of gas leaving the compression and expansion space, respectively, at temperatures different from that of the gas in the kooler and heater space, respectively.

Figure 3.1c : Pressure-Volume Diagram

Temperatures in Compression and Expansion Space

Due to the lack of heat transfer in these spaces the temperatures of the gas oscillates between a minimum and maximum during the course of a complete cycle, although the gases entering these spaces from the kooler and heater space, respectively, enter at fixed temperatures (T_k and T_h, respectively). The reason for oscillations in temperatures are the pressure changes.

Figure 3.1d :Temperature in compression and expansion space

Heat Flows in the Kooler, Regenerator, and Heater Spaces

The shown curves represent the heat which flows into theses spaces ( positive when heat flows from the walls into the gas , negative otherwise ) during the time which has elapsed between the crank being at position Θ=0° and the Θ given on the horizontal axis.
For the kooler this means that some 6.08 Joules of heat are removed from the gas during the first 170° (the values for the angles are only approximate ) of crank rotation (until the minimum in Q_k is reached). During the remaining 190° for a complete cycle some 3.86 Joules are added back into the gas for a net loss of heat of 2.222786 Joules. Hence, to some extent the kooler space acts as a recuperator adding heat to the gas during some part of a cycle and removing heat during other parts. Important is to note that the total amount of heat exchanged in the kooler space ( 6.08 + 3.86 Joules ) is a factor of 4.5 higher than the net heat of 2.22 Joules alone would have indicated adding to the problem of designing the kooler space.
The fate of the heater space is similar although not quite as dramatic. During the first 40° of crank rotation heat is added (0.86 Joules), during the next 130° heat is removed (2.60 Joules) while between Θ=170° and 360° some 7.67 Joules are added back into the gas for a net heat transfer of 5.929266 Joules into the gas. Hence, the total amount of heat transferred, 11.3 = 0.86 + 2.60 + 7.67 Joule, is only about twice as large as the net heat would indicate.
For the regenerator we see that heat is removed from the gas during the first 30° (until the minimum is reached). Then heat is transferred into the gas from 30° to 200°, which is then removed between 200 and 360° with a net heat transfer of zero. As indicated in the summary, the heat added during one part of the cycle and then removed during the remaining part is 33.605684 Joule which is about a factor of 9 (nine) higher than the net work produced by the engine.

Figure 3.1e : Heat flow rates

Mass Flow Rates between Spaces

The mass flow rates [ mass per unit time ] depend but are proportional to the rotational speed of the engine. As such, the curves in the graph below scale linearly with rotational speed but retain their shape and size with respect to each other. The mass flow rate from the compression to the kooler space is counted positive when the gas actually flows from the compression into the kooler space and negative when the gas flows the other way. The same agreement is underlying the other mass flow rates shown.
Most obvious are the two "singularity" points where all 4 curves intersect, that is when all 4 mass flow rates are equal in value. At Θ=160° the gas is flowing everywhere in the direction from compression to expansion space with the pressure being at its maximum. At Θ=340° the flowrate is again equal everywhere but in opposite direction with the pressure at its minimum. One can show that these two "singularity" points exists in principle for the Ideal Adiabatic Simulation of Stirling engines and that they always coincide with the location of minimum and maximum pressure.
Noticable in the graph below is also the feature that the regenerator space experiences gas entering from both ends (heater and kooler) simultaneously (between 40° where the purple curve is intersecting the horizontal axis and 60° where the green lines has its intersect). Similarily, there is an interval (200 to 220°) when the regenerator space is bleeding on both ends. Equivalent statements can be made about the kooler and the heater space.

Figure 3.1f : Mass flow rates

In Figure 3.1g below, we trace 9 tagged gas particles (grey lines) as they move among the five different subspaces during a complete engine cycle. The subspaces are separated by the solid black, horizontal lines. As an example for such a motion, the particle tagged 0.6 starts at crank angle Θ=0° in the middle of the cooler section, volume V_k, moves up into the regenerator, volume V_r, and after reaching its "top"-most location at Θ≈205° returns back to its starting position (Θ=360°). The numbers used as tags indicate the fraction of total mass inside the engine which lies between the power piston and the thus tagged particle. As we look at particles closer to either piston ( 0.1 at the compression piston, 0.9 at the expansion piston ) the gas particles follow closely the movement of the respective pistons. Note also, that the spacing between the grey lines is much larger near the expansion piston because of the higher temperatures in this region.

Figure 3.1g : Movement of tagged gas particles

3.2 Influence of Type of Gas

Two properties of the working gas enter the Ideal Adiabatic Analysis, namely the specific gas constant, R, and the ratio of the specific heat capacities, κ. The values for these properties are known for all gases of interest, see for example A short note on ideal gases.

In the equations governing the ideal adiabatic analysis the specific gas constant is always multiplied with a mass. As a result, changing the value of the specific gas constant for a given engine will modify only those quantities which are proportional to mass, i.e. the total mass , the masses inside the various subspaces of the engine and the mass flow rates between these spaces. Efficiency, work output, and heat flow rates as well as pressure and temperature history changes throughout a cycle, for example, are independent of the value of the specific gas constant. This was used successfully as a test for our program. One can compare for example air and hydrogen which have practically identical κ ( 1.400 vs. 1.405 )

The influence of the value of κ is more insidious with its value changing from 1.667 for Helium, to 1.4 for air, down to 1.299 for methane and even lower for propane and octane. Our analysis of The ideal Stirling Cycle and Heat Load on the Regenerator already indicated that the heat load on the regenerator is proportional to 1/(κ-1) favoring strongly Helium as working gas. This was in principal confirmed by the subsequently conducted Schmidt Analysis.

The following table reflects the results of simulations of completely identical engines with several gases which had identical specific gas constant but different values for the ratio of specific heats, κ. (Of course not all of these gases exists in reality)

The row labelled "Heat load [J], regenerator" reflects the amount of heat transfer into the working gas per cycle, most of it during the time the gas is streaming from the kooler towards the heater. The same amount is of course removed from the gas during the remaining part of the cycle. The numbers show that this heat load is dramatically affected ( by a factor of nearly 2.5 ) when we go from a gas with high to a gas with low κ. All other listed quantities change as well as κ is changed ( the total work output the least, surprisingly ) but to a much lesser extent. In practical terms, the drastic change in heat load of the regenerator means that if one builds an engine with air (κ=1.4) as working gas but under-sizes the regenerator, a change to Helium ( κ=1.667 ) gives an improved engine performance because of the lower demand on the regenerator. According to the ideal adiabatic analysis a change from air (κ=1.400) to hydrogen (κ=1.405) will result in insignificant changes in engine performance.

3.3 Optimization for Volume Phase Lag and Swept Volume Ratio

Because the concept of the ideal adiabatic simulation assumes ideal heat transfer conditions and zero flow losses one cannot expect to obtain the size and other design details of kooler, regenerator, and heater spaces. Still, the ideal adiabatic simulation can provide useful information. One result - known to many in the Stirling engine business - is for example, that an decrease in the volume of any one or several of these spaces results in an increase in the power output and efficiency. How small these spaces can be made is a question of heat transfer and flow losses. The Schmidt Analysis, section 5.2 suggested a further refinement of this statement, namely that changes in the volume of the kooler space influence the power output by a factor of T_h/T_k more than a change in volume of the heater space, with the regenerator lying roughly halfway in between.

The following tabulated values pertain to variations in volume with the base equal to the sample input as provided in Section 3.1 on this web page with T_h=923 K and T_k=300 K and all three spaces of similar volume as shown in the 2^nd column of the following table. In columns 3, 4, and 5 we reduced each volume by 3 cm³, respectively, which is about 10 %, keeping the other two volumes at their original value.

The numerical results obtained under the assumption of Ideal Adiabatic Simulation confirm the prediction that reductions of the kooler space have a far greater impact than reductions of the heater space and indeed, using the numbers for the Work/cycle from Table 3.3a we see that ratio of the change in work/cycle for a reduction in kooler space to that of the reduction in heater space is :

( 3.813474 - 3.706480 ) / ( 3.740601 - 3.706480 ) = 3.136

which is close to the ratio of the absolute temperatures :

923 / 300 = 3.08

The influence of the volumes of the crompression and expansion space is more complex. In general an increase in either or both of these spaces leads to an increase in the work per cycle.

Figure 3.3.a : Work per Cycle and Efficiency as Function of the swept Volume of Expansion and Compression Space

Figure 3.3.a shows lines of constant work-per-cycle ( the lines with 20 , 15 , 10 , and 5 [J] ) and lines of constant efficiency ( curves represented as symbols with efficiency ranging from 0.64 to 0.55 ) with the swept volumes of the compression (Vswc) and the expansion space (Vswe) as independent variables for a Stirling Engine with the following input parameters :

Surprisingly, the efficiency of the engine does not vary dramatically as one or both of these volumes are changed. For example ( using best estimates off Figure 3.3.a ) , if we wanted to build an engine with 10 [J] we could choose Vswc = 45 [cc], Vswe = 350 [cc] at an efficiency of η = 0.59 or choose Vswc = 105 [cc], Vswe = 125 [cc] at η = 0.60. From the point of view of efficiency the best choice would be Vswc = 69 [cc], Vswe = 185 [cc] with η = 0.61. From the point of view of minimizing the total volume ( Vswc+Vswe ) and with that the size of the engine one would choose Vswc = 93 [cc], Vswe = 137 [cc] with an efficiency of η = 0.605 for 10 [J] engine. Hence, upon first glance it seems that engine efficiency does not provide a clear guide line as to how to choose Vswc and Vswe. At a minimum, we need to look at how the heat loads in the kooler, regenerator, and heater vary upon changes of the swept volumes for the compression and expansion space. We pick up this discussion as part of 2 case studies, caseI.html and caseII.html, which utilize the web-based programs of Section 4.

4. Entry Points to web-based Programs

The equations derived in Section 2. of this paper comprise a set of ordinary first order differential equations with time ( or crank angle ) as independent variable. We chose a standard Runge-Kutta 4th order method to solve these equations by starting the engine with arbitrarily choosen initial conditions. After some 8 to 10 revolutions of the crank shaft the solutions reach steady state, that is the conditions at the end of a cycle everywhere inside the engine are - to within presribed accuracy (10^-5 relative) - identical to those at the beginning of the cycle. Once steady state has been achieved, the pressure inside the engine is adjusted to the requested average pressure (averaged over a complete cycle).

Presently, there are three entry points available to slightly different versions of the same programs evaluating a Stirling engine based on the ideal adiabatic simulation as outlined in Section 2. The first program

Program IdealAdiabatic v.1.x

requires the user to enter data as outlined in Section 3. into individual boxes on its web page. It therefore can only perform the analysis of a single engine at a time but provides all the output mentioned in Section 3 after a few seconds of excution time. Presently, only a sinusoidal variation of the compression and expansion space is implemented.

Program IdealAdiabatic v.2.x

requests from the user to specify the working gas to be used in one box and to enter the remaining engine data into another box in arbitrary order. This allows the user to save all engine data (except for gas type ) into a file on the user's computer and re-enter them at some later point conveniently by simply swipe-and-paste. Optionally, the user can specify for the swept volume of the compression space or of the expansion space or both a range of values in the form : minimum , maximum , increment to compute net work per cycle and efficiency for many different engine designs in one swoop. This facility was used to generate Figure 3.3.a . Presently, only a sinusoidal variation of the compression and expansion space is implemented.

Program IdealAdiabatic v.3.x

This is an optimization program using again the ideal adiabatic simulation as its physical/mathematical background as described on this page. The user specifies the gas to be used, the volume of kooler, regenerator, and heater space plus clearance spaces in the compression and expansion space. Also to be specified are the average pressure, the desired work-output per cycle, the RPMs and the phase-lag between the sinusoidal variation of the compression and expansion space. In return, the program varies the swept volumes of the compression and expansion space systematically such that the point Vswc=Vswe and the point of most efficient engine are included for a total of some 100 different configuration for which all parameters specified by the user are identical, including mean pressure and work per cycle. For each such configuration the values of 20 different characteristics are produced in a format suitable to be copied into a file on the user computer for plotting purposes. Presently, only a sinusoidal variation of the compression and expansion space is implemented.