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

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}

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 :

m_{tot} R T_{h} |
1 | ||||

(13) | p | = | |||

V_{ref} |
(e+c+d) + c cos(&Theta)
+ e cos(Θ+δ) |

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)

m_{tot} R T_{h} |
1 | ||||

(14) | p | = | |||

V_{ref} |
B + C cos(Θ + β) |

where :

B = e + c + d

C = √[ e² + 2 e ccos(δ) + c² ]

and β can be obtained from :

sin(β) = esin(δ) / C

cos(β) = (c+ecos(δ)) / C

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 :

m_{tot} R T_{h} |
1 | ||||

(15) | p_{min} | = | |||

V_{ref} |
B + C |

m_{tot} R T_{h} |
1 | ||||

(16) | p_{max} | = | |||

V_{ref} |
B - C |

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).

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 *I _{1}*) and we arrive at :

m_{tot} R T_{h} |
1 | ||||

(17) | p_{ave} | = | |||

V_{ref} |
√(B²-C²) |

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

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 *I _{3}*) 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)

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 *I _{2}*) 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.

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

which says that the product of mass m_{tot}R = p V / T_{tot}and specific gas constant R has the same value ( namely = pV/T ) irrespective of the specific gas you use.

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).

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.

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}.

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)

With the coefficient μ to be calculated according to :(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.

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.

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.

A sample case demonstrates the capabilities of this program.

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

Average pressure | p_{ave} |
200.0 | kPa |

Working gas | Air | ||

Heater temperature | T_{h} |
923.0 | °K |

Kooler temperature | T_{c} |
300.0 | °K |

Volume phase lag | VPL | 95.569 | ° |

Clearance volume, compressions space | V_{clc} |
8.0 | cm³ |

Swept volume, compression space | V_{swc} |
61.045 | cm³ |

Volume of kooler space | V_{k} |
31.21 | cm³ |

Volume of regenerator | V_{r} |
34.89 | cm³ |

Volume of heater space | V_{h} |
28.51 | cm³ |

Clearance volume of expansion space | V_{cle} |
10.0 | cm³ |

Swept volume of expansion space | V_{swe} |
61.045 | cm³ |

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.

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 CO_{2} 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

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

dV_{c} = -0.5*V_{swc}sin(Θ) dΘ

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

dV_{e} = -0.5*V_{swe}sin(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 - T_{0} ) [kJ/kg]

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

h = c_{p} T - c_{v} T_{0} {kJ/kg]

The temperature T_{0} 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. T_{0}=0 over a temperature range of a few
thousand degrees. We will later see that T_{0} 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

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

u = c_{v} ( T_{c} - T_{0})

and the specific enthalpy of the gas leaving is :

h = c_{p} T_{c} - c_{v} T_{0}

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} - T_{0})
= dQ_{c} - p dV_{c} + dm_{c} (
c_{p} T_{c} - c_{v} T_{0})

The temperature T_{0} 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.

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.

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

u = c_{v} ( T_{c} - T_{0})

and the specific enthalpy of the gas leaving is :

h = c_{p} T_{c} - c_{v} T_{0}

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} - T_{0})
= dQ_{k} + dm_{k} (
c_{p} T_{c} - c_{v} T_{0})

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.

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

dQ_{h} = - V_{h} dp

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 - T_{0} ) ρ A dx
= c_{v} ∫ T ρ A dx - c_{v} T_{0}
∫ ρ A dx
= c_{v} ∫ p/R A dx - c_{v} T_{0} m_{r}

Finally :

(7.7.7)
U = V_{r} (c_{v}/R) p - c_{v} T_{0} m_{r}

(7.7.8)
dU = V_{r} (c_{v}/R) dp - c_{v} T_{0} dm_{r}

We are now in the position to set up the energy balance for the
regenerator but still need "dm_{1}" and "dm_{2}" 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} T_{0} dm_{r}
= dQ_{r}
- dm_{1} ( c_{p} T_{c} -
c_{v} T_{0} )
- dm_{2} ( c_{p} T_{h} -
c_{v} T_{0} )

Because of conservation of mass : dm_{r} = - (dm_{1}+dm_{2}) and all terms containing T_{0} drop out again to give :

(7.7.10)
dQ_{r} = V_{r} (c_{v}/R) dp
+ c_{p} T_{c} dm_{1}
+ c_{p} T_{h} dm_{2}

We have already an expression for "dp" but don't know dm_{1}
and dm_{2} yet. But by virtue of conservation of mass and
previous expressions for dm_{k} and dm_{c} :

(7.7.11)
dm_{1} = 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)
dm_{2} = 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.

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.

Rspec [J/(kg k)] | k=c_{p}/c_{v}
| m_{gas [g]} | Qr1 [J] | |
---|---|---|---|---|

Air | 287.0 | 1.400 | 0.2476 | 31.7887 |

Hydrogen | 4124.0 | 1.405 | 0.0172 | 31.4917 |

Helium | 2076.9 | 1.667 | 0.0342 | 22.1790 |

Carbon Dioxide | 188.9 | 1.289 | 0.3762 | 41.0438 |

Methane | 518.2 | 1.299 | 0.3762 | 39.9276 |

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.

Zig Herzog; hgn@psu.edu Last revised: 01/03/08