Schmidt Analysis

Content of this page :

1. Introduction and Definitions
2. Pressure
   2.1 Pressure as Function of Crank Angle
   2.2 Minimum and maximum Pressure
   2.3 Average Pressure
3. Work
   3.1 Work done by Expansion Space
   3.2 Work done by Compression Space
   3.3 Net Work done by Engine
4. Thermodynamic Efficiency
5. Optimization
   5.1 Some general Remarks
   5.2 Influence of V_clc, V_k, V_r, V_h, and V_cle
   5.3 Optimal volume phase lag (vpl)
   5.4 Mathematical Properties of Eq. (22)
   5.5 Optimization for Compression and Expansion Space
6. Optimization Program
   6.1 What it needs and does
   6.2 Entry to Optimization Program < < <Entry to Program
7. Heat Transfer Calculations
   7.1 Introduction
   7.2 Basic Equations used on all subspaces
   7.3 Heat transfer inside compression space
   7.4 Heat transfer inside expansion space
   7.5 Heat transfer inside kooler space
   7.6 Heat transfer inside heater space
   7.7 Heat transfer inside the regenerator
   7.8 Summary

1. Introduction and Definitions

The Schmidt Analysis ( after Gustav Schmidt, who published this anlysis in 1871 ) takes the isothermal analysis of Stirling engines one step futher by assuming that the volume of the expansion and the compression space vary sinusoidally.

We use here very much the same notation as used by Urieli.

The following assumptions are basis of the subsequent analysis :

• 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)    p V = m R T

• The pressure, p , is the same everywhere inside the engine and varies only with time.
  
  
• The heat transfer conditions are sufficient to keep the gas inside the compression space, volume V_c, and inside the kooler space, volume V_k, at constant temperature T_c at all times. Therefore, the masses of gas inside the compression and the kooler space, respectively, are given by :

  (2)    m_c = p V_c / ( R T_c )

  (3)    m_k = p V_k / ( R T_c )

• The heat transfer conditions are sufficient to keep the gas inside the expansion space, volume V_e, and heater space, volume V_h, at constant temperature T_h at all times. Therefore, the masses of gas inside the expansion and the heater space, respectively, are given by :

  (4)    m_e = p V_e / ( R T_h )

  (5)    m_h = p V_h / ( R T_h )

• 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. Therefore the mass of gas inside the regenerator space is given by :

  (6)    m_r = p V_r ln(T_h/T_c) / ( R ( T_h - T_c ) )

• The volume of the compression space varies in sinusoidal fashion :

  (7)    V_c = V_clc + 0.5 V_swc( 1 + cos(Θ) )

  Θ = crank angle, proportional to time
  V_clc = clearance volume = minimum of V_c
  V_swc = swept volume , max{V_c} = V_clc + V_swc

• The volume of the expansion space varies in sinusoidal fashion :

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

  Θ = crank angle, proportional to time
  V_cle = clearance volume = minimum of V_e
  V_swe = swept volume , max{V_e} = V_cle + V_swe
  δ = volume phase lag angle. If positive, V_c is lagging behind V_e in time.

• The total mass of gas inside the engine, m_tot, does not vary in time.

  (9)    m_tot = m_c + m_k + m_r + m_h + m_e

  
  
  

2. Pressure

2.1 Pressure as Function of Crank Angle

An equation for pressure p as function of angle Θ can now be found by substituting Eq.(2) to (8) into Eq.(9) and solving for p. To simplify the resulting equation a bit we use an arbitrarily chosen reference volume, V_ref and define 3 dimensionless variables e, c, and d :

(10)    e = 0.5 V_swe / V_ref

(11)    c = (0.5 V_swc/V_ref) ( T_h / T_c )

(12)   

With that :

Eq. (13) allows us to determine the pressure p at any angle Θ for given volumes and temperatures T_h and T_c. In order to determine mean pressure and work output for which we need to perform integration, it is of advantage to bring Eq.(13) into a different form using the geometric identity :

         cos(Θ+δ)   =   cos(Θ) cos(δ) - sin(Θ) sin(δ)

where :

B = e + c + d

C = √[ e² + 2 e c cos(δ) + c² ]

and β can be obtained from :

sin(β) = e sin(δ) / C

cos(β) = (c+e cos(δ)) / C

2.2 Minimum and maximum Pressure

According to Eq.(14) the minimum pressure is reached when cos(Θ+β) = +1 and the maximum is reached when this cosine-term is equal to -1 :

It takes a just little algebra to show that

B > C

under all circumstances and of course physics requires the same ( that is p_max > 0).

2.3 Average Pressure

The average, over angle Θ, of the pressure can be calculated from

         p_ave   =   1/(2π) p dΘ  =  …… (1/π) ( B + C cos(Θ + β) )^-1 dΘ

The solution to the integral is explained in Integrals in support of Schmidt Analysis (look for integral I₁) and we arrive at :

It is interesting to note that the average pressure is exactly equal to the geometric mean of the minimum and maximum pressure.

3.1 Work done by Expansion Space

By definition :

         W_e = ∫ p dV_e

in which the integration has to be performed over a complete cycle. Differentiating Eq. (8) with respect to Θ :

         dV_e = - 0.5 V_swe sin( Θ+δ ) dΘ

and using Eq. (14) for the pressure we obtain :

         W_e = - m_tot R T_h e sin(θ+δ) / ( B + C cos(Θ+β) ) dΘ

The solution to the integral is explained in Integrals in support of Schmidt Analysis (look for integral I₃) with which we get :

         W_e = - m_tot R T_h e 2π/C [ B/√(B²-C²) - 1 ] sin(β-δ)

Using the definition for β from Eq.(14) and some trigonometry we finally get :

(18)   W_e = m_tot R T_h (2π e c /C²) [ B/√(B²-C²) - 1 ] sin(δ)

3.2 Work done by Compression Space

By definition :

         W_c = ∫ p dV_c

in which the integration has to be performed over a complete cycle. Differentiating Eq. (7) with respect to Θ :

         dV_c = - 0.5 V_swc sin( Θ ) dΘ

and using Eq. (14) for the pressure we obtain :

         W_c = - m_tot R T_h (0.5 V_swc/V_ref) sin(θ) / ( B + C cos(Θ+β) ) dΘ

The solution to the integral is explained in Integrals in support of Schmidt Analysis (look for integral I₂) with which we get after some algebra and trigonometry :

(19)   W_c = - m_tot R T_c (2π e c /C²) [ B/√(B²-C²) - 1 ] sin(δ)

The minus sign indicates that work has to be done on this space to perform the required piston motion.

3.3 Net Work done by Engine

         W_net = W_e + W_c

With the results from Eq. (17) and (18) we get :

(20)   W_net = m_tot R ( T_h - T_c ) (2π e c /C²) [ B/√(B²-C²) - 1 ] sin(δ)

An important note is in order concerning the expressions for the works concerning the term m_tot R in which R was the specific gas constant. This term gives the correct impression that the work output of a Stirling engine increases as the mass of the gas inside the machine is increased. But it also gives the impression that exchanging a gas with another one with a higher value of its gas constant is advantageous. Unfortunately, that is not correct. To demonstrate :

Let's say you have a space of volume V which you fill with an ideal gas at pressure p and at temperature T then, by virtue of the ideal gas law :

m_tot R = p V / T

which says that the product of mass m_tot and specific gas constant R has the same value ( namely = pV/T ) irrespective of the specific gas you use.

4. Thermodynamic Efficiency

         η = W_net/(Q_h + Q_e)

To determine Q_h and Q_e we look at the law of conservation of energy for each of the subspaces involved and obtain :

Q_c = W_c
Q_k = W_k = 0
Q_r = W_r = 0
Q_h = W_h = 0
Q_e = W_e

It is the assumption of isothermal behaviour of all subspaces which simplifies the law of conservation of mass so drastically because the gas masses flowing in and out of each subspace have at all times a temperature identical to that of the subspace itself and therefore the net energy transport per cycle of these mass flows is zero for each subspace. In addition, the works for the kooler, regenerator, and heater are zero because the volumes of these spaces do not change in time ( dV = 0 in ∫ p dV ).

With that :

(21)   η = W_net/W_e = 1 - T_c/T_h

using Eq. (18) and (19) for W_e and W_net, respectively. (Those familiar with the Carnot efficiency and its underlying physical principle shouldn't be surprised at all).

5. Optimization

5.1 Some general Remarks

Within the confines of the Schmidt analysis we have available for modification the temperatures T_c and T_h, the volumes of the individual subspaces, the total mass m_tot, the type of gas, and the volume phase lag δ.

There is no need to discuss the temperatures T_c and T_h much, their ratio directly affects the efficiency in an obvious way and their difference the net work, per cycle, W_net in an equally obvious way. Hence, in the following we assume T_c and T_h to be given.

Because for given temperatures T_c and T_h the efficiency is identical for all Schmidt engines we need to optimize W_net. Eq.(20) for W_net is not quite suitable because it contains m_tot as a factor which will change as we change the volume of one or several subspaces or the phase lag δ unless we want to compare different engines operating at different pressures.

Instead of keeping m_tot constant we compare engines ( in order to find optimal configurations ) which have the same average pressure , p_ave

To do that, we solve Eq. (17) for m_tot and substitute into Eq. (20) :

(22)  

In Eq. (22) we now consider P_ave, V_ref, and the temperatures T_h and T_c as fixed constants and study first the influence of some of the subspaces.

5.2 Influence of V_clc, V_k, V_r, V_h, and V_cle

It is generally accepted in the Stirling community that the volumes of these sub spaces are to be kept at a minimum as much as heat transfer and manufacturing considerations allow. It is interesting to note that the Schmidt analysis points in the same direction.

The only influence the volumes V_clc, V_k, V_r, V_h, and V_cle have on Eq. (22) is through the term B which increases as any of these volumes increases. But an increase in B leads to a decrease in W_net according to Eq. (22) !!! ( Take the derivative of W_net with respect to B, it'll be negative.

From Eq. (12) and (14) we also see that B increases more dramatically with an increase in (V_clc + V_k) than with an increase in (V_h+V_cle), namely by a factor of (T_h/T_c). The affect of the regenerator volume V_r is similary enhanced by the factor T_h/(T_h -T_c) ln(T_h/T_c), the value of which lies somewhere between 1 and T_h/T_c.

5.3 Optimal volume phase lag (vpl)

The optimum for the volume phase lag angle δ is achieved for example by keeping all parameters in Equation (22) constant but varying the volume phase angle δ systematically until the largest value for the network Wnet is achieved. δ influences Wnet mostly through the sin(δ)-term at the right end of Eq.(22) but also through the term C² ( see Eq.(14) ).

Mathematically speaking, we have to take the derivative of Eq.(22) with respect to δ and setting it to zero. After some algebra an equation for the optimum volume phase lag can be obtained :

(23)  

(24)  

For the definitions of B and C² see Eq.(14). Eq.(23) can not be solved explicitly for δ because δ enters also the term C², but it is an excellent iterative equation. This means that you can take an estimate for δ ( for example 90° ) to determine the term C², then determine μ from Eq. (24) and finally find an improved value for δ via Eq. (23) which you can use as a starting value for another go-around. Usually, after 2 iterations you obtain changes for angle δ only of a thousands of a degree or less. For cases I have investigated the optimum value for δ is always close to 90° which is an indication that the behavior of Wnet as given by Eq.(22) is dominated by the sine-term. That in turns tells us that the volume phase lag is not an extremely critical parameter, for example sin(60°)=sin(120°)=0.866 which does not compare badly with sin(90°) = 1.

5.4 Mathematical Properties of Eq. (22)

Eq.(22) displays a remarkable symmetry property related to the variables e and c as defined in Eq. (10) and (11), respectively. The values of c and e can be interchanged without affecting W_net.

As an example, let's assume a certain engine has

0.5 V_swe/V_ref = 0.4
0.5 V_swc/V_ref = 0.2
Th/T_c = 3

By definition, that makes : e = 0.4      c = 0.6

Then we can built a second engine which will have the same W_net as the first but the following characteristics :

0.5 V_swe/V_ref = 0.6
0.5 V_swc/V_ref = 0.133
Th/T_c = 3

That makes : e = 0.6      c = 0.4

We could not prove mathematically that W_net is a strictly increasing function of the dimensionless expansion space volume, e, and dimensionless compression space volume, c. Instead, we verified numerically that the derivative of W_net with respect to "e" ( and with the symmetry property that means w.r.t. "c" also ) is always positive at millions of combinations for e,c, and volume phase lag angle δ. In addition, we could prove mathematically that the derivative of W_net with respect to "e" is positive as e→0 and tends to go to +0 as e→∞, again supporting the claim that W_net in Eq. (22) is a strictly increasing function.

In practice, this means that increasing the expansion or the compression space or both will result in additional work generated by a Stirling engine according to the Schmidt analysis, but with diminishing return as ( because of derivative going to +0 as e→∞ ) these two spaces become large.

5.5 Optimization for Compression and Expansion Space

Based on the previous findings that W_net continuously increases as the compression and/or expansion space are increased, we must employ an additional restriction on these volumes.

For example, we could demand that the sum of these two volumes is a constant and then ask for how best to divide up the available space.

Because we could not find an analytical solution which optimizes the division of a given volume into compression and expansion space a computer program is provided.

6. Optimization Program

6.1 What it needs and does

A sample case demonstrates the capabilities of this program.

As input to the program the user provides his/her engine specifications, for example :

The program in turn provides three sets of responses :

One pertaining exactly to the user's configuration, a second which keeps all of user's input data except optimizes the volume phase lag, and thirdly a set of results which optimizes volume phase lag and V_swc/V_swe keeping the value of V_swc+V_swe at its original value.
The program also produces a contour plot of the network produced as function of volume phase lag and swept-volume ratio. Typically, such graphs show that the optimum is fairly flat, meaning that even larger changes deviations from the optimal values for VPL or V_swc/V_swe will not reduce the network per cycle dramatically.

6.2 Entry to Optimization Program

7. Heat Transfer Calculations

7.1 Introduction

The derivation of equation and associated discussions in this Section are not directly linked to optimization of the geometric layout of a Stirling engine but are related to the discussion of what working gases are preferrable.

It is commonly argued that the choice of working gas of a Stirling engine has an influence on the amount of net work produced due to different properties of gases. In particular, it is thought that flow friction losses and heat transfer rates can be positively influenced. Helium and Hydrogen are commonly mentioned as preferable over air.

In addition to these arguments, we propose that the amount of heat which has to be transferred in the different subspaces of a Stirling engine ( compression space , kooler , regenerator , heater , expansion space ) is influenced quite dramatically by the value of the ratio of specific heats, κ=c_p/c_v, of the gas. Roughly speaking the value of κ is 1.667 for monatomic, 1.4 for diatomic and less than 1.3 for molecules with a higher number of atoms ( A short note on ideal gases). This influence was demonstrated during a more in depth analysis of The ideal Stirling Cycle and Heat Load on the Regenerator which showed that the heat load per cycle is proportional to the value of 1/(1-κ). Hence, the amount of heat to be removed from/ added to the gas by the regenerator matrix increases by a factor of over 2 (two) when switching from Helium to CO₂ or methane. The basic reason for this is that the energy balance on each individual subspace is affected by the specific heat at constant volume, c_v, and - in a different way - by the specific heat at constant pressure, c_p. The program, Schmidt , now calculates various heat transfer rates based on the equations derived below.

In the equations below, the letter "d" in front of a quantity as in dV_e denotes a (infinitesimal) small change of the quantity itself, in this example V_e. Such equations can be obtained by differentiation

7.2 Basic Equations used on all subspaces

Change of volume of compression space versus change of crank angle Θ. Obtained by differentiating Eq. (7) :

dV_c = -0.5*V_swcsin(Θ) dΘ

Change of volume of expansion space versus change of crank angle Θ. Obtained by differentiating Eq. (8) :

dV_e = -0.5*V_swesin(Θ+δ) dΘ

Change in pressure p as function of changes in volume of compression and expansion space. Obtained by differentiating Eq. (13) :

dp = - p²/(m_tot R T_h) (dV_e + (T_h/T_c) dV_c)

Part of the assumptions underlying the Schmidt analysis is to assume the working medium to be an ideal gas with constant specific heat capacities. As a result the specific internal energy, u , of an ideal gas at temperature T is :

u = c_v ( T - T₀ ) [kJ/kg]

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

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 the linear relationship. For monatomic gases like He, Ne, Ar etc. 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.

Also know that for ideal gases the following relationship holds exactly :

c_p - c_v = R

7.3 Heat transfer inside compression space

Because the temperature is assumed to stay constant, the specific internal energy for this space is :

u = c_v ( T_c - T₀)

and the specific enthalpy of the gas leaving is :

h = c_p T_c - c_v T₀

Because the kooler - exchanging mass with the compression space - is also at temperature T_c the equation for the enthalpy is also correct when mass is flowing into the compression space.

Let dm_c be a small amount of gas entering the compression space while the crank rotates by a small angle dΘ and let dQ_c be a small amount of heat flowing into the gas. Conservation of energy then becomes :

dm_c c_v ( T_c - T₀) = dQ_c - p dV_c + dm_c ( c_p T_c - c_v T₀)

The temperature T₀ drops out and rearranging for dQ_c which is the quantity we wish to calculate :

dQ_c = p dV_c - ( c_p - c_v ) T_c dm_c = p dV_c - R T_c dm_c

The small volume change dV_c is known and dm_c can be obtained by differentiating the ideal gas law in which for the compression space the pressure p, the volume V_c, and the mas m_c change with time :

p V_c = m_c R T_c

p dV_c + V_c dp = dm_c R T_c

Using this to eliminate dm_c from the equation for dQ_c we arrive at :

dQ_c = - V_c dp

In the computer program "Schmidt" we determine dQ_c incrementing dΘ be small amounts and summing up ( basically integrating ) over a complete cycle. Because during a cycle the pressure increases (dp>0) and decreases (dp<0) heat is transferred in and out of the gas during a complete cycle with more heat removed than gained. The program keeps track separately of the sum of all the positve dQ_c and the negatives.

7.4 Heat transfer inside expansion space

The analysis is exactly identical to that of the compression space in section 7.3.

dQ_e = - V_e dp

Integrating over a complete cycle produces a positive Q_e representing the net heat flowing into the gas in the expansion space.

7.5 Heat transfer inside kooler space

Because the temperature is assumed to stay constant, the specific internal energy for this space is :

u = c_v ( T_c - T₀)

and the specific enthalpy of the gas leaving is :

h = c_p T_c - c_v T₀

Because the kooler receives gas from the compression space and the regenerator with a temperature T_c , the equation for the enthalpy is also correct when mass is flowing into the kooler space regardless as to whether it is coming from the compression space or the regenerator. Because the volume of the kooler space is constant the energy balance becomes :

dm_k c_v ( T_c - T₀) = dQ_k + dm_k ( c_p T_c - c_v T₀)

Re-arranging :

dQ_k = - ( c_p - c_v ) T_c dm_k = - R T_c dm_k

The small change of mass inside the kooler space, dm_k, can be obtained by differentiating the ideal gas law in which for the kooler space only the pressure p and the mass m_k change with time :

p V_k = m_k R T_c

V_k dp = dm_k R T_c

Using this to eliminate dm_k from the equation for dQ_k we arrive at :

dQ_k = - V_k dp

Because during a cycle the pressure in the engine arrives back at its starting value Q_k = ∫ dQ_k = 0 which is the well-known artifact of the Schmidt analysis. The heat transferred inside the kooler space into the gas during that part of the cycle while the pressure is decreasing :

Q_k = V_k ( p_max - p_min )

is exactly transferred out of the gas while the pressure is increasing.

7.6 Heat transfer inside heater space

The analysis for the heater space is exactly like that for the kooler space with the results :

dQ_h = - V_h dp

7.7 Heat transfer inside the regenerator

Things are a bit more complicated for the regenerator because on its inside the temperature varies (linear variation is the classical assumption) from T_c on the kooler side to T_h on the heater side. In addition the enthalphy of the gas exchanged between kooler and regenerator has an enthalphy corresponding to T_c while for the gas exchange between heater and regenerator T_h determines the enthalpy.

We irst look at the internal energy of the regenerator. Let "A" be its free cross-section and "L" be its length and let "dx" be a small slice thereof. In the following all integrals go from x=0 ( kooler side ) to x=L ( heater side ) and the temperature varies linearly :

(7.7.1)    T = T_c + x/L ( T_h - T_c )

The mass inside the regenerator with ρ=density becomes now :

(7.7.2)    m_r = ∫ ρ A dx

and with ρ = p / ( R T ) (from ideal gas law) we get

(7.7.3)    m_r = ∫ p / ( R T ) A dx = p A / R &int dx/T = p V_r / ( R T_r )

Here we used V_r = A L = volume of regenerator and T_r = effective temperature of regenerator defined by :

(7.7.4)    T_r = ( T_h - T_c ) / ln(T_h/T_c)

The equation to keep for later use is that for small changes of m_r :

(7.7.5)    dm_r = V_r / ( R T_r ) dp

In similar fashion we can derive an equation for the internal energy of the gas inside the regenerator.

(7.7.6)    U = ∫ c_v ( T - T₀ ) ρ A dx = c_v ∫ T ρ A dx - c_v T₀ ∫ ρ A dx = c_v ∫ p/R A dx - c_v T₀ m_r

Finally :

(7.7.7)    U = V_r (c_v/R) p - c_v T₀ m_r

(7.7.8)    dU = V_r (c_v/R) dp - c_v T₀ dm_r

We are now in the position to set up the energy balance for the regenerator but still need "dm₁" and "dm₂" the masses leaving the regenerator to the kooler and the heater space, respectively.

(7.7.9)    dU = V_r (c_v/R) dp - c_v T₀ dm_r = dQ_r - dm₁ ( c_p T_c - c_v T₀ ) - dm₂ ( c_p T_h - c_v T₀ )

Because of conservation of mass : dm_r = - (dm₁+dm₂) and all terms containing T₀ drop out again to give :

(7.7.10)    dQ_r = V_r (c_v/R) dp + c_p T_c dm₁ + c_p T_h dm₂

We have already an expression for "dp" but don't know dm₁ and dm₂ yet. But by virtue of conservation of mass and previous expressions for dm_k and dm_c :

(7.7.11)    dm₁ = dm_k + dm_c = V_k/(R T_c) dp + V_c/(R T_c) dp + p/(R T_c) dV_c

In the same fashion :

(7.7.12)    dm₂ = dm_h + dm_e = V_h/(R T_h) dp + V_e/(R T_h) dp + p/(R T_h) dV_e

Substituting back :

(7.7.13)    dQ_r = [V_r c_v/R + (V_c+V_k+V_h+V_e) c_p/R) ] dp + p c_p/R ( dV_e + dV_c )

With &kappa = c_p/c_v we get :

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

Again, program "Schmidt" sums up separately all positive dQ_r ( heat transferred from the matrix into the gas ) and all negatives and verifies that the net heat transfer during a complete cycle is zero as indicated by above equation.

7.8 Summary

The program and above equations have been triple checked.

We performed calculations on what is referred to as the ross90 data which are presented in Section 6.1 of this webpage. The Input-webpage for the program "Schmidt" offers an easy option to input these data into the program with the additional option to choose the working gas ( air , helium , hydrogen , carbon dioxide , methane ).

As predicted theoretically, a change in working gas has no affect on the efficiency, net work and heat transfer in the compression, kooler, heater, and expansion space. The mass of working gas is proportional to the inverse of the specific gas constant. The heat transferred inside the regenerator into the gas while it is being moved from the heater to the kooler (Qr1 in the table below) is strongly affected by κ = ratio of specific heats. ( the same amount of heat is transferred out of the gas when the gas streams in the reverse direction). The table below shows the results for the ross90 geometry when volume phase lag and swept-volume ratio are not optimized.

When comparing - for example - the two gases Helium and Carbon Dioxide the analysis of the ideal Stirling cycle predicted that the heat transfer rates would change by a factor of (1.667-1)/(1.289-1) = 2.308 while the Schmidt analysis produces a ratio of 41.0438/22.1790 = 1.851 for the ross90 geometry. Although not quite as bad as predicted, the increase in heat transfer load on the regenerator with decreasing value of ratio of specific heats, κ, is quite significant. Note , that there is no significant difference between air and hydrogen.

For the above tabulated cases the net work per cycle was a constant 3.6136 [J] which is significantly lower than the heat load on the regenerator. This emphasizes once more the importance of the regenerator.