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Ind. Eng. Chem. Process Des Dev. 1982, 21, 690-695
Energy and Exergy Analysis of a Chemical Process System with Distributed Parameters Based on the Enthalpy-Direction Factor Diagram Masaru Ishida and Katsuhlto Kawamura Research laboratory of Resources Utilization, Tokyo Institute of Technology, Nagatsuta, Yokohama, Japan 227
The enthalpy-direction factor diagram is proposed for energy and exergy analysis of chemical process systems. Focusing on the exchange of energy between the energy-donating processes and the energyaccepting ones, this diagram makes it possible to analyze systematically the effect of the distributed parameters of the process. Various kinds of exergy annihilation caused by heat exchange, reaction, and so forth are obtained as the area on it. It is compared to the structured energy-exergy flow diagram (SPEED). The latter focusses on the transformation of exergy among the processes and is developed to analyze hierarchical structure of the process system on the lumped-parameterbasis. It is shown that the letter diagram is approprhte for exergy integration tasks to compose the skeleton of the hierarchical system structure and the former is suitable for energy integration tasks to polish it up.
Introduction Nowadays much attention has been paid to efficient usage of energy in a chemical process system, and the concept of the exergy or available energy is proposed to compare various kinds of energies in a unified manner (Denbigh, 1956; Gaggioli, 1961,1962; Rieckert, 1973). One method to represent exergy flow in a complex chemical process system is to represent the magnitude of the exergy flow of each stream as the width of the arrow similar to the Sancky diagram for the energy flow analysis. Although it can compactly show the overall exergy flow, its disadvantage is seen in the features that it is not suited for computer-aided treatment, especially for large sytems, and that we must specify the reference state for the definition of exergy definitely. Recently Oaki et al. (1981) proposed a structured process energy-exergy flow diagram (SPEED) which revealed the hierarchical structure of chemical process systems and facilitated computer-aided energy and exergy calculations. In their method, attention was paid on the changes in energy (enthalpy) and exergy in the process rather than t h w for the streams. The parameters for the process such as temperature, pressure, and concentrations were lumped and their distribution in the process was not taken into consideration. On the other hand, the distributed parameter problem has been analyzed in the field of heat exchangers. Nishida et al. (1971) proposed the heat content diagram in which the quantity of heat transferred is represented as the area on their diagram. Hohmann (1971) made a heat-temperature diagram and discussed the minimum approach temperature. Umeda et al. (1978) used ( T - To)/T instead of T to make the area represent the exergy annihilation for heat exchange operations. However, the discussions in those works were limited only to heat exchangers. The purpose of this paper is to propose the enthalpydirection factor diagram by introducing the concept of the direction factor. This diagram makes it possible to clearly represent exergy annihilation caused by not only heat exchange but also chemical reactions and so forth. The advantages of this diagram are compared to those of SPEED with illustrative examples. 0196-4305/82/1121-0690$01.25/0
Thermodynamic Approach A Process with Lumped Parameters. Since both enthalpy and entropy are state variables, we may obtain their changes in a process, AH and AS, as the difference in their values at the outlets and the inlets. In this paper, such a process is symbolized by a circle with arrows denoting inputs and outputs, as shown in Figure la. The resultant values of AH and AS may be represented by a vector on the (AH, TOAS)plane (Oaki and Ishida, 1981; Ishida and Oaki, 1981) shown in Figure lb. This diagram may be called the thermodynamic compass, because the direction of the vector implies the procedure to fulfill the process. The exergy change Ac (=AH- TOAS) for the process is obtained by decomposing this vector into the component of the work Won the abscissa and that of the dissipated heat Q at Toon the oblique line marked with AH = TOAS. When n processes constitute a process system, which is represented by enclosing the constituent processes by a dashed line as shown in Figure 2a, the following two criteria hold. n
C AHi = 0 (first law of thermodynamics)
i=l
(1)
R
CASi 1 0 (second law of thermodynamics)
(2)
i=l
These criteria correspond to the fact that the summation of the vectors for the processes in a process system yields a vertical vector whose length corresponds to the exergy annihilation as shown in Figure 2b, since we have the following equation between the exergy annihilation and the overall entropy increase
-CAE,= -C(AHi - TOASJ = ToCASi 2 0 (3) Each process in a process system may be divided into two types: the energy-acceptingprocess with A H > 0 and the energy-donating process with AH < 0. Let us first consider a binary process system consisting of only two processes. Then we obtain the following equation.
0 1982 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 4, 1982 891
AH (0)
(b) VECTOR ON
SYMBOL
(AH,
TOAS) PLANE
Figure 1. Representation of a lumped-parameter process. Exergy TOAS onni h i 1 at ion
(a) To generate
Exerqy anniha lot i o n
/
work -W
(b) To be driven by work -W
Figure 3. Definition of the direction factor D based on an ideal process system consisting of P, -W, and -8.
I \
--'
(0)
SYMBOL
(b)
(AH, TOAS)
( c ) (AH,
0)
Figure 2. Representation of a process system.
The subscripta ha and hd denote energy-accepting and energy-donating, respectively, and D is the following direction factor indicating the slope of the vector in Figure lb.
D = ToAS/AH
(5)
Consequently, the exergy annihilation of a process system is also obtained as the shaded area in the enthalpy-direction factor diagram shown in Figure 2c. From eq 5 we have
TOAS = AH-D ALE= AH*(l- D )
(6)
(7) Therefore TOASand Ac, respectively, are represented by the area between the line for D and the abscissa in Figure 2c and between the line for D and the line for D = 1. For a general process system consisting of m energyaccepting processes and n energy-donating ones, we have m+n
TO
a=1
m a
i
n
= C mhajDha,i i-1
j=l
ImhdjlDhdj
(8)
where we have the following relation from eq 1. m
n
n
This indicates that the magnitude ICAHil for energy-accepting processes on the abscissa in Figure 2c is always equal to that for the energy-donating processes. Let us examine some characteristics of the direction factor, D . Evidently it corresponds to the slope of the vector on the thermodynamic compaas shown in Figure lb. Since the vector P (AH, TOAS)denoting a process may be decomposed to a vector of the work Wand a vector of the heat Q a t To,the three vectors, P, -W, and -Q shown in Figure 3a constitute an ideal process system with ToCASi = 0. Hence we may say that when the process P generates the work -W, the heat of the quantity -TOAS a t To is dissipated. Then the ratio of the dissipated heat to the total energy -AH (= -ALE- TOAS)becomes equal to the direction factor D. However, when we consider a reverse process system in which a process is performed with the help of the work -W as shown in Figure 3b, the heat of the quantity -TOAS may be absorbed from the heat reservoir a t To. Consequently, we may say that the process which has a great value for D may dissipate a lot of energy to generate work but may absorb a lot of heat from the heat reservoir at To to be driven by work. Hence the
(a) System o f distributedparameter processes
(b)
System o f multistaged lumpedparameter prooeeees
Figure 4. Decomposition of distributed-parameter process into multistaged lumped-parameterprocesses.
exergy annihilation is not related to the direction factor itself but is proportional to its difference, Dha - Dhd,as given in eq 4. As already shown elsewhere (Ishida and Oaki, 1981),the direction factor D is given as the reciprocal of various dimensionless temperatures such as
D = To/Th D = To/[(c,/c,)Th] D = To/T,
(for thermal processes)
(10)
(for polytropic processes) (11)
(for chemical reaction, absorption, mixing, separation, etc.) (12)
In deriving eq 10 for thermal processes to heat or cool process fluids, constant heat capacity is assumed, and Th denotes the logarithmic mean of the inlet and outlet temperatures. Polytropic processes satisfy the relation, P P = constant with the polytropic exponent m,and c, is the polytropic heat capacity defined by c , = c,(m - y ) / ( m1). Teqin eq 12 is the equilibrium temperature at which the Gibbs free energy AG (= AH - TAS) becomes zero. In deriving eq 12, the fact that AH and AS for reactions scarcely depend on the temperature is applied. Although Tw has the dimension of temperature, it may become negative in the case when AH and A S are of different signs, say in the case of regular mixing or separation. So far, the direction factor D is taken as the ordinate of Figure 2c. When we prefer the exergy change ALEto TOAS,the availability factor A (= A e / A H = 1 - D ) may also be used as the ordinate. Then the exergy annihilation can be represented by the area between Ahdand Ahain a similar manner. In this study, however, we mainly use the direction factor D because it corresponds to the slope of the vector in Figure lb. A Process with Distributed Parameters. The direction factor D varies with position in a process with distributed parameters. Even such a process can be treated as a lumped-parameter process by using the overall direction factor calculated from the overall values of AH and AS for the energy-accepting and energy-donating processes, as shown in Figure 4a. When we take its distribution into account, however, more information about the process can be derived. A simple method to analyze
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Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 4, 1982
emperature fluid
Dhd (n)
Dhd for high-temperature fluid
__--
0 AHha (l) AHha (2)
0 '
AHha (3)
AHho
Figure 5. Enthalpy-direction factor diagram.
Figure 6. Enthalpy-direction factor diagram for a cocurrent-type heat exchanger.
the effect of the distributed parameter is to decompose the process into multistage lumped-parameter processes in such a manner that each process subsystem may satisfy eq 1 and 2, as shown in Figure 4b. Then the summation of the exergy annihilation in each process subsystem gives
T0CAS,(l)+ ToCAS,(2)+ ... +ToCAS,(n)= AHha(l)(Dha(')- Dhd'')) + Hha(2)(Dha(2) - Dhd(2)) ... = To(ASh(l)+ Mhd") + Mha(0)+ ... + Ash,'")) = To(ASh, + AShd) = AHh(Dha - D M ) (13)
+
The last two terms are just equal to the exergy annihilation when each of the heat-accepting and heat-donating processes is lumped, giving rise to the overall direction factor Dha and Dhd. Accordingly, the exergy annihilation is given as the shaded area between Dha(')and DM('),as shown in Figure 5. When the number of subprocesses is increased, the width of each AHha(&) is decreased, resulting in continuous change in Dh(i) and D,(I). Since eq 4 should hold for each subsystem, the criterion for a distributed-parameter process to proceed is given as
Dha(')2 Dhd(')
(for i = 1, ..., n )
(14)
Hence, we may conclude that the process for which Db(*) < DM(')holds for some ith stage is infeasible even if the overall value Dhais greater than Dh+ The point at which Dha(L)becomes equal to Dhd(,)is called the pinch point (Umeda et al., 1978).
Unit Process Systems Let us now examine the characteristics of typical unit process systems. Heat Exchangers. For a heat exchange process system without any change of phases, the direction factor D is given as T o / Tby eq 10. By applying the relation dH/dT = cp, we have D = To/" = T0/(Tmlet+ A H / c p )
(15)
The temperature increase of the low-temperature fluid is the heat-accepting process while the temperature decrease of the high-temperature fluid is the heat-donating process. Hence the direction factor for the low-temperature fluid appears greater than that for the high-temperature one, aa shown in Figure 6, and the shaded area corresponds to the exergy annihilation caused by the heat transfer. Figure 6 for the heat exchangers is equivalent to the heat vs. ( T - T o ) / Tdiagram proposed by Umeda et al. (1978). When there is a change in the phase, each process is decomposed into three parts, T C Tb, Tb C T < T d ,and T > T d ,where Tb and Td are bubble and dew tempera-
0
0 0
0 (0)
Uniform temperature
0
(b)
Adiabatic
(c) Noniaothermal
Figure 7. Enthalpy-direction factor diagram for an exothermic reaction.
tures, respectively. For a single phase at T C Tb or T > T d ,eq 15 may be applied. On the other hand, for multiphases between Tb and Td,we may define the overall heat capacity including the enthalpy change for the evaporation as follows. cp
=
(Hd
- Hb)/(Td - Tb)
(16)
As the first-step approximation, the direction factor a t some temperature between Tb and Tdmay be obtained by substituting eq 16 into eq 15. Reactors. First consider a tubular reactor kept at a constant temperature Tfi When an exothermic reaction is taking place, the reaction itself is the energy-donating process and its direction factor D, (= To/Te,)should appear below the direction factor for the process fluid temperature Df (= T o / T f )as , shown in Figure 7a. The dotted area between D, and Df represents the quantity of the exergy annihilation caused by this exothermic reaction. In this case, AHh, the amount of the heat of reaction, is equal to that of the heat transferred through the reactor wall. Therefore, when the heat released by the reaction is removed by the cooling medium at T,, the shaded area between Df and D, (= To/Tm)in Figure 7a becomes the exergy annihilation caused by the heat transfer. It is to be noted that D, may become negative. In such a region, Tw calculated by eq 8 is negative. In spite of this fact, Nishio et al. (1980) substituted the preheated temperatures for T in ( T - To)f T for reactions. Since the preheated temperature is always positive, their derivation may give erroneous results. When an exothermic reaction proceeds under adiabatic conditions, the increase in the temperature for the process fluid, Tf,is the energy-acceptingprocess. Then, the dotted area between D,and Df in Figure 7b is the exergy annihilation caused by the reaction. A more general case is the combination of the previous two cases (a) and (b). Namely, a part of the heat of reaction is used to increase the process fluid temperature, while the rest is transferred through the reactor wall. Hence, there are two energy-accepting processes. In such a case, we may regard the generation of heat by reaction
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 4, 1982 803
direction factor diagram. As already discussed in the previous section, various kinds of exergy annihilation are obtained as the area between the curves for D for the energy-accepting process and the energy-donating one, and the prominent advantage lies in the fact that the analysis of the distributed parameter is possible. However, the disadvantage is seen in the feature that the analysis is not hierarchical. In the previous paper (Oaki et al., 1981), it was shown that the hierarchical structure of chemical process systems D=T,/ (cp/cm)TI may be revealed by taking into account the exergy exchange among the target process and coupled ones such (a) (b) as exergy donors, exergy acceptors, and accelerators. Also Figure 8. Enthalpy-direction factor diagrams for (a) a compressor by introducing the concepts of subtargets and mediators, and (b) a turbine. we proposed the structured process energy-exergy flow diagram (abbreviated as SPEED). Hence let us apply that and the change in the temperature of the process fluid as diagram to the example process systems. just one nonisothermal process, as follows. Figure 9 shows the SPEED for the synthesis of methanol fluids at the inlet condition from methane, although the system is oversimplified. The fluids at the outlet condition (17) target of this process is the production of methanol from Then the exchange of heat between the process fluid and natural gas. I t is decomposed into two subtargets; the the cooling medium may be represented as shown in Figure production of synthesis gas and that of methanol. The 7c, where AHh: and D,' are the increase in the enthalpy target and these subtargets are marked by double quotafor the nonisothermal reaction denoted by eq 17 and its tion marks. These two subtargets are further decomposed direction factor. to lower-rank subtargets, and the lower-rank subtargets For other cases, we may draw similar diagrams by disare supplied with coupled processes to make each subtarget tinguishing energy-accepting processes from energy-doand the coupled processes constitute a process subsystem nating ones. It is to be noted that the criterion given by satisfying eq 1and 2. There are three types for coupled eq 14 holds also under nonisothermal conditions. processes: the exergy donor denoted by - giving exergy Compressors and Turbines. The changes in enthalpy to the target, the exergy acceptor denoted by receiving and entropy of n moles of an ideal gas from the state at exergy from the target, the accelerator denoted by = in the Tl and P1to T2 and P2 can be obtained as case when both target and coupled process lose exergy (Oaki et al., 1981). AH = ncp(T2- T,) (18) In Figure 9, only headings of the processes are listed to A S = n h [(T2/T1)c~.(P2/P1)-R] (19) show the hierarchical system structure clearly. In actual SPEEDS, however, each process is described by one-diwhere R is the gas constant. mensional linear chemical formulas so that the computer For the polytropic compression, in which the increase reads them and prints out the values of AH, AE, and D for in the fluid temperature caused by the friction is taken into each process, as described in the previous paper (Ishida consideration (Traupel, 1966), we have and Oaki, 1981). PV" = constant (20) Figure 10 shows the corresponding enthalpy-direction (21) (m- l ) / m = [(Y - 1)/Y1/7]pc factor diagram. The number of the process in it is the same as that in SPEED in Figure 9. For the steam re-w = AH = [ m / ( m- l)qp,]nPlv,[(P,/P,)~~-'~~~ - 11 former, the temperature of the furnace is assumed to be (22) 1573 K (Process 8) and the endothermic reaction and the heating of the reactant stream are treated as a single A S = nc, In ( T 2 / T 1 ) (23) nonisothermal heat-accepting process (Process 7). The where -W and y, respectively, are the work required for exergy annihilation for the reactor is divided into two parts compression and the ratio of heat capacities c /cv. The by drawing the curve for Df for the reactor temperature polytropic compression efficiency qpc denotes t i e ratio of in the similar manner as in Figure 7c. To the compression the enthalpy change used to compression, S V dP,to the of synthesis gas (Processes3 and 15),Figure 8a is applied. applied work. Hence, (1- rlpc) corresponds to the fraction For Process 15, however, three-staged compressors are used of the heat of friction. with intermediate cooling (Process 17). The compression Then the direction factor for the polytropic compression of the recycled gas is omitted from Figures 9 and 10 just is given as eq 11. Since the value of ( m - y ) is positive, for simplicity. For exothermic methanol synthesis reaction, the polytropic heat capacity c, becomes positive, giving an adiabatic reactor with feed of cool recycled gas at the rise to the enthalpy-direction factor diagram shown in four portions is assumed (Process 20). Hence the stagewise Figure 8a. The dotted area represents the amount of the distribution for Dh and Dhd is obtained as shown in Figure exergy annihilation in this compression system. 10. Since the overall exergy annihilation is obtained from For the polytropic expansion, on the other hand, the the marked area in Figure 10,the next step is to find better value of (m- y) becomes negative and the reciprocal of heat exchange networks to reduce the exergy annihilation. polytropic expansion efficiency, l/qp, is substituted for Another example is given for the steam power generaq, in eq 21 and 22, where q, denotes the ratio of work tion, of which SPEED is shown in Figure 11. In this case, generated to the theoretical value of -1V dP. Then the the fuel is assumed to be methane and its combustion is enthalpy-direction factor diagram is given as Figure 8b. also taken into consideration. Although the system is Large Process Systems oversimplified, SPEED shows clearly the role of the steam as the energy carrier, and the processes for the change of Synthesis of methanol and steam power generation are the state of water are treated as looping mediators and are analyzed as illustrative examples based on the enthalpyI
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Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 4, 1982 , I If II II I, ,I I, I, I, , I
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