Ind. Eng. Chem. Res. 2000, 39, 4431-4433
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CORRESPONDENCE Comments on “Equipartition of Forces: A New Principle for Process Design and Optimization” Tore Haug-Warberg Department of Technology, Høgskolen i Telemark, N-3914 Porsgrunn, Norway
Sir: In a paper devoted to process design and optimization, Sauar et al.1 discuss a principle called the equipartition of forces for minimizing the entropy production in heat- and mass-transfer process equipment. The principle is claimed to be new and generally valid for processes with parallel transport paths and does not assume that the phenomenological transport coefficients are constant over the space variables. The validity of their findings is seriously questioned by Xu,2 who gives several examples on practical situations where the principle obviously leads to wrong predictions. The rebuttal to Xu’s comments (Sauar et al.3) does not resolve any of the disagreements, and I find it appropriate to write another comment to the work of Sauar et al.1 In this letter I will show that the minimum entropy production in a two-section heat exchanger with fixed hot side temperatures, with the sections being composed of different materials, does not correspond to the principle of equipartition of forces. The state of optimum is, in fact, superior to the equipartition of forces principle (see Figure 1). My results are restricted to one-dimensional heat transfer only, and the interested reader is guided to the work of Schechter4 for a full discussion of the general entropy production problem. The outcome of Schechter’s elegant analysis supports the simpler results achieved in this letter, however, and on p 145 Schechter writes, “Thus, the principle of minimum entropy production cannot be valid unless the phenomenological coefficients are constants.” That is to say, the principle of minium entropy production does not yield the classical condition ∇J ) 0 for steady-state heat conduction, except when the phenomenological heat-transfer coefficient λ is constant throughout the entire control volume. It should be mentioned that the heat flux versus temperature gradient relationship is then taken to be J ) λ∇(T-1) and not Fourier’s law J ) κ∇T. Assuming Schechter’s interpretation is correct, this is in itself sufficient to disprove the validity of the equipartition of forces principle. However, it is likely that the problem description is quite oblique to the casual reader, and I will therefore do a specific analysis of a simple heat exchanger to show that there are boundary conditions yielding lower entropy productions than those predicted by the equipartition of forces. For one-dimensional heat conduction in the z direction, the entropy production θ in a macroscopic volume can be written (see, e.g., Schechter,4 p 145):
Figure 1. Optimal entropy production (lower curve) and entropy production according to the equipartition of forces principle (dashed line).
θ)A
∫0sJ‚∇(T-1) dz ) A∫0sJz ∂z∂ (T-1) dz
(1)
At a constant heat flux component Jz, this integral has a particular simple solution:
(
θ ) AJz
)
1 1 ) AJz∆(T-1) T(s) T(0)
(2)
Equation 2 shall next be applied to a two-section heat exchanger with parallel transport paths. The heat transfer in section one follows Fourier’s law (Jz,1 ) κ∇zT), while the heat transfer in section two is written according to the theory of linear irreversible thermodynamics (Jz,2 ) λ∇zT-1). To keep things simple, both κ and λ are assumed to be temperature-independent parameters. The close connection between Fourier’s law and linear irreversible thermodynamics is made visible by rewriting the last relation to Jz,2 ) -λT-2∇zT ) -λ′(T) ∇zT. In particular, it should be stressed that the alternative forms of Jz have a heat-transfer coefficient that may depend on the intensive thermodynamic state variables but not on the driving force. This observation is true regardless of whether ∇zT or ∇z(T-1) is chosen as the driving force. However, the force-flux relation is a local property, and linearity in this respect is not the same as to claim that the global entropy production is a linear function of the boundary conditions, i.e. ∆T or ∆(T-1). In my opinion, this is a weak point in the work of Sauar et al.1 However, to serve the purpose of a simple illustration, the flux equations can be given the following integral forms:
10.1021/ie001094o CCC: $19.00 © 2000 American Chemical Society Published on Web 09/29/2000
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Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000
A1Jz,1 ) A1κs1-1(T1(0) - T1(s1)) ) A1κs1-1∆T1 (3) A2Jz,2 ) A2λs2-1
(
)
1 1 ) A2λs2-1∆(T2-1) (4) T2(s2) T2(0)
) A1Jz,1
(
)
(
)
1 1 1 1 + A2Jz,2 T1(s1) T1(0) T2(s2) T2(0)
(
)
1 1 + T1(s1) T1(0) 1 1 A2λs2-1 T2(s2) T2(0)
) A1κs1-1(T1(0) - T1(s1))
(
)
2
(
)
σopt ) σ(Ropt,βopt)
(5)
The heat transfers are assumed parallel and the total heat flux reads
J ) A1Jz,1 + A2Jz,2
1 1 c1xRopt3 + 1 - c1x + c2(1 - x) Ropt2 - c2(1 - x) ) 0 2 2 (16) To calculate σopt for given area x and heat-transfer parameters c1 and c2, the temperature ratios Ropt and βopt are first solved from eqs 15 and 16 and then inserted into eq 14 to yield
The total entropy production Θ in the body is
Θ ) θ1 + θ2
cubic polynomial in the temperature ratio Ropt:
(6)
(17)
Numerically, it is found that only the largest real root of Ropt has a physical meaning. The next task is to compare the minimum entropy production from eq 17 to that predicted by the equipartition of forces principle. With common hot end temperatures in the two sections, the equipartition of forces requires that the cold end temperatures are common as well, regardless of whether the “driving force” is ∆T or ∆(T-1). The cold side boundary condition is therefore T2(s2) ) T1(s1), or alternatively
Combined with eqs 3 and 4, the last equation can be written as
βEoF ) REoF
T1(s1) κT1(0) T2(0) λ 1+ A2 -1 s1 T1(0) s2T2(0) T2(s2)
The temperature ratio REoF is determined by solving eq 13 substituted for eq 18:
J ) A1
(
)
(
)
(7)
Things can be made a little easier by introducing the definitions
c1 )
κT1(0) (A1 + A2) Js1
c2 )
λ(A1 + A2)
x)
Js2T2(0) A1 A1 + A2 T1(s1)
R)
β)
T1(0) T2(s2) T2(0)
(10)
(11)
(12)
(13)
ΘT(0) ) c1x(1 - R)(R-1 - 1) + J c2(1 - x)(β-1 - 1)2 (14)
The optimum value of β is
βopt ) 2R2/(1 + R2)
(19)
σ ) c1x(1 - R)(R-1 - 1) + (1 - xc1(1 - R))(β-1 - 1) (20)
(9)
At constant hot side temperature T2(0) ) T1(0) ) T(0), the total entropy production in eq 5 can also be put on a dimensionless form
σ)
c1xREoF2 + (1 - c1x + c2(1 - x))REoF - c2(1 - x) ) 0 Combining eqs 13 and 14 yields
(8)
and then substituting into eq 7:
1 ) xc1(1 - R) + c2(1 - x)(β-1 - 1)
(18)
(15)
Compare definitions (11) and (12) with eq 31 in the Appendix. Inserted for βopt, eq 13 can be written as a
and if eq 18 is inserted into eq 20, a particular simple relation is found to hold for the entropy production according to the principle of equipartition of forces:
σEoF ) REoF-1 - 1
(21)
One calculated result is shown in Figure 1. Note that the parameters c1 ) 1.5 and c2 ) 0.5 have been chosen such that the entropy production according to the equipartition of forces principle is independent of the area ratio x. This deliberate choice was made to illustrate the curvature of the optimal entropy production, which for random choices of c1 and c2 would be masked by a large linear contribution. The figure shows that there are indeed states being more optimal than those predicted by the equipartition of forces. At the global minium point occurring at x = 0.7, the temperature ratios are as follows:
Ropt ) 0.4071 βopt ) 0.2844 REoF ) 0.3333 If, for example, the hot side temperature is 1000 K, the optimal cold side temperatures are 407.1 and 284.4 K for sections 1 and 2, respectively. The corresponding heat fluxes are 0.6225 and 0.3775, measured as fractions of the total heat flux. The equipartition of forces, on the other hand, predicts a cold side temperature of 333.3
Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4433
K for both sections with heat flux contributions 0.7000 and 0.3000, respectively. The depicted situation is rather extreme and, for example, for c1 ) 10 and c2 ) 9, the temperature ratios are as follows (this time at x ) 0.5):
(
)
A1κ 1 ∂Θ 1 )s1 T1(s1) T1(0) ∂T1(s1) A1κ
(T1(0) - T1(s1)) (23) s1(T1(s1))2
(
Ropt ) 0.9025
2A2λ ∂Θ 1 1 )∂T2(s2) s2(T2(s2))2 T2(s2) T2(0)
βopt ) 0.8977
A1κ ∂J )s1 ∂T1(s1)
)
(24)
(25)
The necessary conditions for an optimum are
REoF ) 0.9000 The difference in entropy production between the optimal state and the equipartition of forces state is now calculated to be less than 0.07%, and for all practical purposes, the state determined by the equipartition of forces can be claimed to be nearly optimal. However, this result is a mere consequence of the phenomenological transport coefficient λ being nearly constant in both sections of the heat exchanger and only demonstrates the limitation of the principle. My conclusion is, therefore, that the equipartition of forces is an insignificant principle in the current context. Furthermore, for those cases in which it is found to yield correct results, it is probably included in the work of Schechter.4 Appendix
(27)
∂Θ ∂J +π )0 ∂T2(s2) ∂T2(s2)
(28)
When eqs 23-28 are combined and π is eliminated, the following condition emerges
(
)
1 1 1 + (T1(0) - T1(s1)) ) T1(s) T1(0) (T1(s1))2 1 1 2 (29) T2(s2) T2(0)
(
)
which on a rearranged form reads
(
)(
) (
)
1 1 1 T1(0) 1 1 ) (30) +1 2 T1(s1) T1(s1) T1(0) T2(s2) T2(0) or
The total entropy production Θ in a two-segment heat exchanger with parallel heat-transfer paths is to be minimized for given heat-transfer areas A1 and A2, a given total heat flux J, and a given temperature T2(0) ) T1(0) on the hot side. The necessary conditions for the minimum correspond to finding a stationary point of the Lagrange function
∂J ∂Θ +π )0 ∂T1(s1) ∂T1(s1)
(( ) ) (
1 T2(0) T1(0) 2 T1(0) T1(s1)
2
-1 )
T2(0)
T2(s2)
)
-1
(31)
According to eq 31, there is no equipartition of forces determining the cold side temperatures T2(s1) and T2(s2). In fact, neither ∆T nor ∆(T-1) is constant in the heat exchanger. Literature Cited
L ) Θ + π(J - J°)
(22)
with respect to the temperatures T1(s1) and T2(s2) at the cold side. Here π is the Lagrange multiplier and J° is the specified heat flux. To identify the stationary state of L, the following partial derivatives are needed:
A2λ ∂J )∂T2(s2) s2(T2(s2))2
(26)
(1) Sauar, E.; Ratkje, S. K.; Lien, K. M. Equipartition of Forces: A New Principle for Process Design and Optimization. Ind. Eng. Chem. Res. 1996, 35, 4147-4153. (2) Xu, J. Comments on “Equipartition of Forces: A New Principle for Process Design and Optimization”. Ind. Eng. Chem. Res. 1997, 36, 5040-5044. (3) Sauar, E.; Ratkje, S. K.; Lien, K. M. Rebuttal to Comments on “Equipartition of Forces: A New Principle for Process Design and Optimization”. Ind. Eng. Chem. Res. 1997, 36, 50455046. (4) Schechter, R. S. The Variational Method in Engineering; McGraw-Hill: New York, 1967; Chapter 4.
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