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1. Introduction and Definitions 1. Introduction and DefinitionsGustav Schmidt (Sep 16,1826 - Jan 27, 1883) published the isothermal analysis of the Stirling cycle in 1871 and provided a closed form solution for the case of sinusoidally varying volumes of the expansion and the compression space. We use here very much the same notation as used by Urieli. The following assumptions are basis of the subsequent analysis :
2. Pressure2.1 Pressure as Function of Crank AngleAn 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. It is of advantage to operate as much as possible with dimensionless variables which can be achieved by simple algebraic rearrangement. appropriate reference variables. As a results units do not matter anymore (but temperature must always measured in absolute degrees) because they naturally cancel out. We define 3 dimensionless variables e, c, and d : (10) e = 0.5 (11) f = 0.5
(12) d =
With that : (13) p =
Eq.(13) allows us to determine the pressure p for any value of the crank angle Θ provided we know all volumes, temperatures, the phase angle, and the total mass of working gas. The latter can be determined by knowing the average pressure inside the engine. To this end we need to integrate Eq.(13). 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)
(14a) p =
where : (14b) B = e + f + d (14c) C = √ e² + 2 e f cos(δ) + f² and β can be obtained from : (14d) sin(β) = f sin(δ) / C cos(β) = (c+f cos(δ)) / C 2.2 Average PressureThe average, over angle Θ from zero to 2π, of the pressure can be calculated from
p_{ave} =
The solution to the integral is explained in Integrals in support of Schmidt Analysis (look for integral I_{1}) and we arrive at :
(15) p_{ave} =
It is interesting to note that the average pressure is exactly equal to the geometric mean of the minimum and maximum pressure. 2.3 Minimum and maximum PressureAccording to Eq.(14a) the minimum pressure is reached when cos(Θ+β) = +1 and the maximum is reached when this term is equal to -1 :
(16) p_{min} =
(17) (17) p_{max} =
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). 3.1 Work done by Expansion SpaceBy 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.(14a) for the pressure we obtain : W_{e} = - m_{tot} R T_{h} e ∫0 2π
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}
Using the definition for 'e', sin(β) and cos(β) from Eq.(14d) and some trigonometry we finally get : (18a) W_{e} = m_{tot} R T_{h}
Or alternatively, (18b) W_{e} = p_{ave} V_{swe}
3.2 Work done by Compression SpaceBy 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} = - e V_{swc} sin( Θ ) dΘ and using Eq.(14a) for the pressure we obtain : W_{c} = - m_{tot} R T_{h} e (V_{swc}/V_{swe}) ∫0 2π
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 : (19a) W_{c} = - m_{tot} R T_{c}
Or alternatively : (19b) W_{c} = - p_{ave} V_{swe}
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 EngineW_{net} = W_{e} + W_{c} With the results from Eq.(17) and (18) we get : (20a) W_{net} = m_{tot} R ( T_{h} - T_{c} )
Or alternatively : (20b) W_{net} = p_{ave} V_{swe} ( 1 -
An important insight from Eq.(20b) is that the net work is independent of the type of working gas being used. The gas constant does not appear in any of the appearing coefficients. The same holds true for the work of the compression space, W_{c} Eq.(19b), and the work of the expansion space, W_{e} Eq.(18b). 4. Thermodynamic EfficiencyThe thermodynamic efficiency for heat engines is generally defined as the ratio of benefit ( W_{net} ) to costs ( Q_{h} + Q_{e} = heat supplied per cycle ).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} 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 =
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. Optimization5.1 Some general RemarksWithin 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 can only optimize W_{net} as given in Eq.(20b). The influence of p_{ave} and V_{swe} is quite obvious. 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.(20b) is through the term B which decreases as any of these volumes decreases. Any decrease in B leads to an increase in W_{net} according to Eq.(20b) !!! ( Take the derivative of W_{net} with respect to B, it'll be negative.) From Eq.(12) and (14b) we also see that B decreases more dramatically with a decrease in (V_{clc} + V_{k}) than with a decrease 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, δThe optimum for the volume phase lag angle δ is achieved for example by keeping all parameters in Eq.(20b) constant but varying the volume phase angle δ systematically until the largest value for the network W_{net} is achieved. δ influences Wnet mostly through the sin(δ)-term at the right end of Eq.(20b) but also through the term C², see Eq.(14c). This suggest that the optimum for the volume phase angle is close to 90°, a value commonly suggested. Mathematically speaking, we have to take the derivative of Eq.(20b) with respect to δ and setting it to zero. After some algebra an equation containing the optimum volume phase lag can be obtained : (22) cos(δ) = √1 + μ² - μ (23) μ =
For the definitions of B and C² see Eq.(14b) and (14c). Eq.(22) can not be solved
explicitly for δ because δ appears to the left of the equal sign
and on the right through the term C² and there doesn't seem to be a
way to solve for δ or its cosine or sine explicitly. But it renders
itself nicely to an iterative solution. 5.4 Optimization for Compression and Expansion SpaceAlthough it seem to be impossible to provide an exxact proof experience with the computer based determination of W_{net} indicates that the net work per cycle will increase as either the swept volume of the expansion space, V_{swe}, or that of the compression space, V_{swc} is increased, albeit with deminishig return. On the other hand, increasing these volumes increases the size and construction cost of an engine. 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 Program6.1 What it needs and doesA 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. 6.2 Entry to Optimization Program7. Heat Transfer Calculations7.1 IntroductionThe 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 7.2 Basic Equations used on all subspacesChange 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 7.3 Heat transfer inside compression spaceBecause 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. 7.4 Heat transfer inside expansion spaceThe 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 spaceBecause 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. 7.6 Heat transfer inside heater spaceThe analysis for the heater space is exactly like that for the kooler space with the results : dQ_{h} = - V_{h} dp and the same conclusion are to be drawn.7.7 Heat transfer inside the regeneratorThings 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. 7.8 SummaryFor each value of the crank angle Θ from 0 to 2π in steps of some small increment dΘ the above equations are evaluated. First V_{c} and V_{e} and their increments dV_{c} and dV_{e} and from that the change in pressure dp. With that all heat flow increments dQ_{c} etc. can be evaluated and appropriately summed to the contributions from the previous Θ's.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. |