5040
Ind. Eng. Chem. Res. 1997, 36, 5040-5044
Comments on “Equipartition of Forces: A New Principle for Process Design and Optimization” Jianguo Xu Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195
Sir: Sir, Recently, Ratkje and co-workers published their work on the optimal design of processes based on minimal production of entropy (Sauar et al., 1996a,b). Assuming that the rates of entropy production are proportional to the squares of the driving forces, they concluded that process designs should be optimized by equally distributing the driving forces throughout the unit process, hence “equipartition of forces”. They stated that the approach of equipartition of forces was a “new principle for process design”. I agree that analyzing and understanding optimal allocation of driving forces are very useful in process analysis and synthesis and am pleased to see publications which address this important aspect of process design. I would like to point out, however, that their basic assumption that entropy production rates are proportional to the square of driving forces is not valid for many important chemical processes. More importantly, their objective function appears to be incorrect which implies their conclusion that equipartition of forces should be the “principle” for optimal process design is not valid. The basis for this conclusion is presented below and follows the sequence of their derivation. 1. Entropy Production Is Not Proportional to the Square of the Forces in Many Cases In obtaining their conclusion, Sauar, Ratkje, and Lien (1996a,b) assumed that process design systems have such a property that (1) the flux of path n, Jn, is proportional to the force, Xn
Jn ) LnXn
(1)
where Ln is a coefficient which is independent of Xn and (2) the entropy production rate for path n, ∆S˙ in, is the product of the flux and the force:
∆S˙ in ) d∆Sin/dt ) JnXn ) LnXn2
(2)
Notice the difference between the rate of entropy change and the rate of entropy production. The latter can be understood as the rate of exergy consumption divided by T0, the ambient temperature: ∆S˙ in ) ∆E˙ /T0. Such a property is valid for many processes at conditions very close to thermodynamic equilibrium, as was pointed out by Prigogine (1961). It is, however, typically not valid when the process is far from equilibrium, as are many chemical processes. The following three examples demonstrate that for chemical reactions, turbulent flow, and gas permeation through a medium, such a relationship does not hold. (i) Chemical Reactions. Consider a reversible reaction of ideal gases A and B: A ) B. The rate of entropy production of the reaction is
d∆S˙ i/dV ) -r∆Gm/T
(3)
in which ∆Gm is the change of molar Gibbs free energy, r the reaction rate, V the reactor volume, and T the S0888-5885(97)00835-X CCC: $14.00
reaction temperature. It is also known from thermodynamics that ∆Gm ) -RT ln[K/(pB/pA)], in which K is the reaction equilibrium constant, pA and pB are the partial pressures of components A and B, and R is the universal gas constant. The reaction rate, r, is given by
r)b r (1 - pB/KpA) ) b r {1 - exp[∆Gm/RT]}
(4)
where b r is the rate of the forward reaction. Therefore, the rate of entropy production is
b{1 - exp[∆Gm/RT]}∆Gm/T d∆S˙ i/dV ) -r
(5)
When the reaction is not very close to equilibrium, this relationship is far from the quadratic form that Sauar et al. (1996a,b) assumed. The same conclusion can be drawn for the integration of d∆S˙ i/dV in the whole reactor. (ii) Turbulent Flow. For turbulent flow of gases, when the Reynolds number is large enough, the pressure drop is proportional to the square of the velocity of the flow. We have the relationship of flux and pressure drop dp and pressure as follows:
J/Ω ) fx(dp/dl)p
(6)
in which f is a constant independent of the pressure drop, Ω is the flow area in the pipe, and dp/dl is the pressure drop in unit length of the pipe. The rate of entropy production in that section of the pipe with length of dl, however, is equal to
d∆S˙ i/dt ) JR ln[(p + dp)/p]
(7)
Therefore, the rate of entropy creation is
d∆S˙ i ) Rfx(dp/dl)p ln[(p + dp)/p]Ω ≈ ΩRfx(dp/dl)p[(dp/dl)/p]dl
(8)
or rather
d∆S˙ i ≈ ΩRfx(dp/dl)/p[(dp/dl)] dl
(8′)
Since the derivative of the rate of entropy production with respect to the pipe length is not proportional to the square of the driving force, be it (dp/dl) or (dp/dl)/ p, the rate of entropy production, which is the integration of eq 8′ along the length of the pipe, will not be proportional to the square of the driving force. The same conclusion holds for turbulent flow of liquids except that the relation between the production rate of entropy and driving force is somewhat different. (iii) Permeation of an Ideal Gas Mixture through a Medium. When J moles of ideal gas permeate from the higher pressure side to the lower pressure side per unit of time, there is a loss of pressure, equivalent to © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 5041
pressure exergy loss rate of
∆E˙ ) JRT0 ln(ph/pl)
(9)
in which ph and pl are the pressures on the higher pressure and on the lower pressure side, respectively, and T0 is the ambient temperature. On the other hand, there is a change in composition chemical potential on the two sides of the medium. From Xu and Agrawal (1996) we know that the rate of change in composition chemical potential is M
∆F˙ ) RT
[Ji ln(xil/xih)] ∑ i)1
(10)
for an M component system. xil and xih are the mole fraction of component i on the low-pressure side and that on the higher pressure side of the membrane, respectively. Therefore, the rate of entropy production is M
[Ji(pih/pil)] ∑ i)1
∆S˙ i ) ∆E˙ /T0 ) -∆F˙ /T ) R
(11)
where M is the number of components in the gas mixture. If the permeation process follows Fick’s law, we have dJi ) Di(pih - pil) dA, in which A is the area for permeation and Di is a constant related to the diffusion coefficient of component i through the medium and the thickness of the permeation medium. Therefore, we have
d∆S˙ i dA
unrelated to energy are frequently not affected by the design policy, minimizing the sum of operating cost and equipment cost is equivalent to minimizing the sum of energy cost and equipment cost in these cases:
Min(energy cost + equipment cost) S Min(operating cost + equipment cost) (14) The objective for optimal design then becomes
Min(energy cost + equipment cost)
(15)
From thermodynamics, we know that energy is really not consumed. It is merely converted to a different form in a process. It is exergy that is consumed in the process. Therefore, energy cost is really the cost of exergy consumption. The unit costs of different forms of exergy are different. For example, electricity is typically more expensive than thermal exergy, since electricity is typically converted from thermal exergy. There are inefficiencies in energy conversion and transmission processes, in addition to the equipment cost for the conversion and transmission (including maintaining them) which utility customers have to pay. For the sake of convenience, we will include the cost of equipment for energy conversion and transmission (such as compressors, pumps, expanders, furnaces, transformers, motors, generators, etc., and their accessories) in “energy cost”, or rather “cost of exergy”, not in the “equipment cost”. Doing this allows for analysis of individual unit processes. When several forms of exergy are involved, the unit costs of different forms of exergy are different. In that case, the objective function for optimal design becomes
M
[Di(pih - pil) ln(pih/pil)] ∑ i)1
)R
∑ j)1
Min[
Integration of eq 12 in area gives the total entropy production of the permeation process. Again, eq 12 does not follow the quadratic relationship in eq 2, no matter what is considered as the driving force between pih pil and pih/pil, nor will the integration of d∆S˙ i/dA in the whole area. The list can go on and on. However, it really does not matter whether this assumption is correct or not, since the objective function for their derivation seems to be incorrect, as will be shown in the text below. 2. The Objective Function for Minimization Is the Production Cost The purpose of investment in a process plant, aside from the social and environmental aspects, is to seek return on the investment. Therefore, the objective of the optimal design should be to maximize that return. When production of the plant is fixed, it becomes minimization of the production cost. Production cost is the sum of equipment depreciation cost and operating cost when safety, flexibility, and controllability issues do not affect the design policy. It can also be expressed as the total capital cost of the equipment and the overall operating cost during the life time of the plant (with gross return on investment including taxes, interest rates, and net return on investment, etc., factored in):
Min(operating cost + equipment cost)
M
(12)
(13)
Since the operating cost is composed of energy cost and other costs unrelated to energy use and since the costs
N
Bi(Ci)] ∑ j)1
Aj(∆Exj) +
(16)
in which M is the number of forms of exergy consumed in the process and N the number of equipment items excluding those for energy conversion and transmission; Aj and Bi are the cost functions of exergy form j and equipment item i. ∆Exj is consumption of exergy form j which is equal to the negative value of the change in that type of exergy. Ci is the size of equipment item i. Exergy consumption is related to the rate at which exergy is consumed. If the on-stream life of the plant is known and the gross return on investment is given, eq 16 can be rewritten as follows: M
∑ j)1
Min[
N
Ej(∆E˙ xj) +
Bi(Ci)] ∑ j)1
(16′)
Equipment cost includes all the costs related with the equipment. In the chemical process industry, the relationship between equipment cost and equipment size can often be expressed by eq 17
Bi ≈ aiCiRi
(17)
in which ai is a constant that is related with the material and labor intensity of equipment item i but unrelated to the size of the equipment. Ri is a constant smaller than unity, typically in the range of 0.6-1. Equipment size is related to the rate, r, and the capacity. Examples include chemical reactors, heat exchangers (with respect to the rate of heat transfer per unit area of heat exchanger area), etc. In these cases
5042 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997
the size of the equipment can be written as
Ci )
∫0G
i
dgi ri
∂ a ∂∆(1/T)
Q dq ∂ + eT0 ∆(1/T) dq ) 0 ∫0QL∆(1/T) ∫ 0 ∂∆(1/T)
(23)
(18) a)
in which Gi is the capacity of process unit i (the flux). The optimal design problem becomes
[
Min
M
N
( )]
Ej(∆E˙ xj) + ∑ Bi ∫0 ∑ j)1 i)1
dgi
Gi
ri
(19)
Notice that this objective function is very different from that in Sauar et al. What they did was to minimize the entropy production with the constraint that the total flux is constant. The cost of equipment did not enter their objective function for optimization. 3. Optimal Driving Force Allocation in Heat Exchanger Design As was mentioned in section 1, many processes do not follow the phenomenological relations in the process industry. One exception is heat transfer when the pressure drop exergy losses of the heat-exchanging streams are negligible relative to the exergy losses resulting from heat transfer. Keep in mind that such a restriction may be a very significant one. For example, Agrawal and Woodward (1991) showed that, in the main heat exchanger of a cryogenic nitrogen generator, the exergy loss due to pressure drops was actually greater than that due to heat transfer. Relatively small exergy losses due to ∆p will occur if gaseous heat-transfer streams are at relatively high pressures or if the heat-transfer streams are liquid or undergo a phase change. We will discuss such a case for the sake of simplicity and to show that the results thus obtained are very different from that of equipartition of forces even in such applications. Let us assume that the Fourier heat-transfer equation takes the form
ri ≈ Li∆(1/Ti)
(20)
following Sauar et al. ri is the rate of heat transfer at the ith location. Equation 20 holds when Ti . ∆Ti. Li in the equation is the heat-transfer coefficient at that location. It is approximately equal to uiT2, in which ui is the heat-transfer coefficient in the conventional heattransfer equation. The heat-transfer area of the heat exchanger is thus
C)
dq ∫0QL∆(1/T)
(21)
in which Q is the duty of the heat exchanger and L independent of the driving force. Temperature T is a function of q when exchange of sensible heat is involved. The rate of exergy loss follows as
∆E˙ x ) T0
∫0Q∆(T1 ) dq
(22)
in which T0 is the ambient temperature. In order to optimize the driving force, we will have to set the derivative of the cost function with respect to the set of driving forces ∆(1/T) to zero. Notice that ∆(1/T) is the set of driving forces ∆(1/T) at any location i. Therefore, eq 19 becomes
dB(C) dC
e)
dE(∆E˙ x) d∆E˙ x
(23′)
a and e can be considered to be the unit costs of the heat-transfer area and thermal exergy, respectively. Very often e can be expressed as the product of the derivative of the cost of exergy per unit time (e.g., in years) with respect to exergy consumption at the beginning of the project, dE(∆Ex)|t)0/d∆Ex and the sum of the present value of a dollar in the lifetime (in years) of the plant:
e)
dE˙ (∆E˙ x)|t)0 d∆E˙ x
∫0τf(t) dt
Notice that τ is the life of the plant (in years), while f(t) ) 1/(1 + I)t in which I is the gross (annual) return on investment, and t is time (in years). Notice that a and e are not necessarily equal to the total cost of the heat exchanger divided by the area and the total cost of energy divided by the total exergy consumption. Since the policy at one point in the heat exchanger does not affect the objective function at a different position in this particular problem, eq 23 is equivalent to eq 24 for all the locations in the heat exchanger:
-dq a + eT0 dq ) 0 Li∆(1/T)i2
(24)
For eq 24 to hold, we should have
eT0 -
a )0 Li∆(1/T)i2
(25)
that is
∆(1/T)i )
x
a eT0Li
(26)
In the conventional form, we have
∆Ti ≈ Ti
x
a eT0ui
(27)
Eq 27 is exactly what Steinmeyer (1984, 1992) suggested (without proof) for determining the optimal ∆T in reboilers. Equation 26 or eq 27 gives the optimal driving force in a heat exchanger if this driving force can be manipulated in any way one likes to. Notice that Li or ui does not have to be a constant. When there are several heat exchangers involved, eqs 26 and 27 can be derived for the different heat exchangers, and the terms have the same meanings. This means that the optimal driving force for heat transfer ∆(1/T) is proportional to the square root of the unit cost of the heat-transfer area and inversely proportional to the square root of the unit cost of thermal exergy, e, defined by eq 23′ and to the square root of the heat-transfer coefficient L, which is a function of temperature. When the driving force is expressed in the form of heat-transfer temperature difference, it is also proportional to the absolute temperature. In a heat recovery system, when the unit cost of the heat exchanger changes or when the heat-transfer coefficient changes, the optimal heat-transfer driving
Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 5043
force should be different. This makes intuitive sense since we typically use a smaller heat-transfer temperature difference in a heat exchanger in which a liquid stream boils on one side and a vapor stream condenses on the other side. A liquid/liquid heat exchanger typically also has a smaller-heat transfer temperature difference than a heat exchanger with gas streams of relatively low pressures on the two sides because the former has a significantly greater heat-transfer coefficient at typical heat-transfer conditions. Other factors, such as the greater exergy loss due to pressure drops in the gas/gas heat exchanger, further increase such a trend. In a heat recovery system that raises superheated steam of, say, over 10 MPa, the average heattransfer driving force in the superheater is typically significantly greater than those in the boiler or boiler feed water heater. That is, because the temperature in the superheater is greater, the heat-transfer coefficient is typically smaller than those in the boiler and the water heater, while more expensive materials have to be used in building the superheater than the boiler and the water heater, so that the unit area cost of the superheater is greater. When the unit cost of thermal exergy, e, is not known, eqs 26 and 27 are still useful. When the value of e can be considered the same for all heat exchangers in consideration, then the ratios of the driving forces in different heat exchangers are related by
∆(1/T)1
x
x
L1 ) ∆(1/T)2 a1
L2 a2
( )x ( )x u1 ∆T2 ) a1 T2
u2 a2
Applying the Lagrange method leads to
e
[
∂ dq ∫0Q∆(1/T) dq] + λ ∂∆(1/T) ∫0QL∆(1/T)
∂ [T ∂∆(1/T) 0
]
)
0 (32)
in which e is defined by eq 23′. Again, since the optimal policy at one point is assumed to have no effect on a different point, eq 32 is equivalent to
e
[
]
∂ dq ∂ [T ∆(1/T)i dq] + λ )0 ∂∆(1/T)i 0 ∂∆(1/T)i Li∆(1/T)i (33)
at any location i. Solving eq 33 gives the optimal driving force
∆(1/T)i )
x
λ eT0Li
(34)
Substituting eq 34 into eq 31-2 gives
λ)
c) (29)
Min[E(∆E˙ x)]
(30-1)
area ) A
(30-2)
in which A is a constant. Substituting the expressions for exergy consumption and area into eqs 30-1 and 30-2 gives
∫0Q∆(1/T) dq)]
(31-2)
eT0 2 c A2
(35)
in which
In other words, it is the value ∆(1/T)(L/a)1/2 or rather (∆T/T)(u/a)1/2 that should be compared and made as constant as possible for a constant valuation of the thermal exergy. Therefore, such terms should be used rather than the T-H diagram or the “heat availability diagram” of Umeda et al. (1979a,b) in the optimal allocation of driving forces in heat exchangers. The T-H diagram can still be used in combination with the (∆T/T)(u/a)1/2-H diagram to relate temperature with the term to be made constant. The heat availability diagram of Umeda et al. (1979a,b) can also be used to indicate the loss of exergy as enthalpy changes (and temperature changes if the T-H diagram is not used). However, pinch analysis should be performed using the (∆T/T)(u/a)1/2-H diagram, not the other two diagrams. Assume for whatever reason the heat exchanger area is already fixed and T . ∆T in the heat exchanger. In that case, the problem of minimizing production cost is that of minimizing cost of exergy consumption subject to the constraint that the total area is a constant.
Min[E(T0
dq )A ∫0QL∆(1/T)
(28)
or rather
∆T1 T1
with constraint
(31-1)
∫0Q dq ≈ ∫0Q xLi
dq
Txui
(36)
which is independent of location i. Substituting eq 35 into eq 34 gives
∆(1/T)i )
1 c A L x i
(37)
Notice that the driving force is inversely proportional to the square root of Li. Using the conventional driving force and heat-transfer coefficient gives
Ti c A u x i
(38)
∆Ti c u ≈ Ti x i A
(39)
∆Ti ≈ or rather
This shows that when the heat-exchanger area is fixed, the optimal heat transfer ∆T should be proportional to the absolute temperature and inversely proportional to the square root of the heat-transfer coefficient ui. That is to say, when the heat-transfer coefficient changes, the optimal driving force changes also. If a heat exchanger has a boiling/condensing zone and a vapor superheating/ gas cooling zone, the optimal heat transfer ∆T should be much smaller in the boiling/condensing zone than in the superheating zone. Following Sauar et al., the driving force ∆(1/T) would be the same (i.e., the heat
5044 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997
irrespect of the change in the heat-transfer coefficient. transfer ∆T should be proportional to T2 instead of T) 4. Summary In short, equipartition of forces does not seem to be a good approach for process design. Its basic assumptions are invalid for many processes in the process industry. Its objective function appears to be incorrect as well. The conclusion thus obtained that the driving force should be equally distributed in the unit process (i.e., equipartition of forces) is different from my derivation using the production cost as the objective function for minimization even for a case in which the phenomenological relationship holds, which states that in a heat exchanger in which the pressure drop exergy losses are negligibly small, the heat transfer ∆T should be proportional to the absolute temperature and the square root of the unit cost of the heat-transfer area and inversely proportional to the square root of the heattransfer coefficient. Following the theory of equipartition of forces of Sauar et al., the optimal heat transfer ∆T would be proportional to the square of the absolute temperature and independent of the heat-transfer coefficient. The functions for equipment costs and operating costs are usually rather complex. I would not consider some rules based on certain approximations and assumptions as principles for process design even when they are correct, no matter how valuable such rules are in process synthesis. To me, a principle is something more fundamental. The ultimate objective of the optimal design of a process plant is to maximize the return on investment. The same objective should be the starting point for any rules or design policies as well. Therefore, maximizing return on investment is an objective closest to something called a principle for process design. Acknowledgment The author is grateful to Dr. Keith Wilson for his valuable comments and suggestions. Notation A ) reactant of chemical reaction A ) B A ) heat-transfer area; area for permeation Aj ) cost of exergy form j a ) unit cost of heat-transfer area defined in eq 23′ B ) product of chemical reaction A ) B B ) heat-transfer area cost Bi ) cost of equipment item i Ci ) size of equipment item i c ) number defined by eq 36 Di ) diffusion coefficient of component i E ) cost of exergy as a function of rate of exergy consumption E˙ ) rate of cost of exergy consumption ∆Exj ) consumption of exergy form j, the negative value of change of exergy form j ∆E˙ ) rate of consumption of exergy form j e ) unit cost of exergy, defined in eq 23′ F˙ ) rate of change in composition chemical potential f ) coefficient relating the molecular flow rate with pressure gradient in a pipe
Gi ) capacity of equipment item i ∆Gm ) change of molar Gibbs free energy J ) flux l ) pipe length K ) equilibrium constant of a chemical reaction L ) heat-transfer coefficient defined by eq 20; L ≈ uT2 (T . ∆T) M ) number of exergy forms; number of components in the gas mixture for permeation through a medium N ) number of equipment items p ) pressure ph, pl ) pressures on the higher pressure side and on the lower pressure side of a permeation medium, respectively pih, pil ) partial pressures of component i on the higher pressure side and on the lower pressure side of a permeation medium, respectively Q ) heat-transfer duty q ) heat transferred R ) universal gas constant r ) rate (of heat transfer, reaction, etc.) b r ) rate of the forward reaction ∆Si ) entropy production, defined by ∆Si ) ∆Ex/T0 ∆S˙ i ) rate of entropy production T ) absolute temperature Ti ) absolute temperature at location i T0 ) ambient temperature t ) time u ) heat-transfer coefficient V ) reactor volume xih, xil ) mole fraction of component i on the higher pressure side and lower pressure side of a permeation medium Greek Letters Ri ) constant defined in eq 17 λ ) Lagrange multiplier τ ) on-stream life of the plant Ω ) flow area
Literature Cited Agrawal, R.; Woodward, D. Efficient cryogenic nitrogen generators: an exergy analysis. Gas Sep. Purif. 1991, 5, 139. Prigogine, I. Thermodynamics of irreversible processes, 2nd ed.; Interscience Publishers: New York 1961. Sauar, E.; Ratkje, S. K.; Lien, K. M. Equipartition of forces: a new principle for process design and optimization. Ind. Eng. Chem. Res. 1996a, 35, 4147. Sauar, E.; Ratkje, S. K.; Lien, K. M. Equipartition of forcessa new principle for process design. AIChE Annual Meeting, Chicago, 1996b. Steinmeyer, D. E. Process energy conservation. Encyclopedia of Chemical Technology, supplement volume; Wiley: New York, 1984. Steinmeyer, D. E. Optimum ∆P and ∆T in heat exchange. Hydrocarbon Process. 1992, April, 53. Umeda, T.; Niida, K.; Shiroko, K. A thermodynamic approach to heat integration in distillation systems. AIChE J. 1979a, 25, (3), 423. Umeda, T.; Harada, T.; Shiroko, K. A thermodynamic approach to the synthesis of heat integration systems in chemical processes. Comput. Chem. Eng. 1979b, 25 (3), 273. Xu, J.; Agrawal, R. Membrane separation process analysis and design strategies based on thermodynamic efficiency analysis. Chem. Eng. Sci. 1996, 51 (3), 365.
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