4550
Ind. Eng. Chem. Res. 1996, 35, 4550-4558
A Simplified System Model for Rapid Evaluation of Disturbance Propagation through a Heat Exchanger Network Y. H. Yang, J. P. Gong, and Y. L. Huang* Department of Chemical Engineering and Materials Science, Wayne State University, Detroit, Michigan 48202
This paper describes a simplified system model for rapidly evaluating disturbance propagation through a heat exchanger network. The model depicts cause-effect relationships between a set of stream output variables (target temperature fluctuations) and a set of input variables (disturbances of source temperatures and heat capacity flowrates). The model, which consists of a set of linear equations of disturbances, can precisely evaluate the propagation caused by even severe temperature disturbances and/or by moderate heat capacity flowrate fluctuations. It is easy to use, computationally efficient, and particularly helpful in analyzing integrated process systems where disturbance propagation is always a major concern. Case studies demonstrate the applicability of the model in process analysis and improvement. Introduction A chemical process plant is usually operated under the presence of various disturbances and fluctuations. These disturbances propagate throughout the plant and affect the stability of process operations (Linnhoff and Kotjabasakis, 1986). If disturbance propagation is very severe, the process will be extremely difficult to operate or even uncontrollable regardless of advances of control techniques. It is very important, therefore, that disturbance propagation be detected early and evaluated completely. During the last 2 decades, great progress has been made in identifying disturbance propagation and rejecting disturbances during control system design (McLane and Davison, 1970; Commault et al., 1984, 1991; Dion et al., 1994). The models are applicable only to linear, but not to nonlinear, systems. A premise of using their models is that the process structure is fixed and not to be changed. These models are not applicable to the enhancement of process controllability through process synthesis. Probably, the earliest work on disturbance propagation analysis for process synthesis was pioneered by Kotjabasakis and Linnhoff (1986). They introduced a procedure for considering flexibility right at the design stage and established a trade-off between flexibility and total cost of a heat exchanger network (HEN). Their procedure makes extensive use of what they termed sensitivity tables; thus, it is inefficient to the analysis of a complex HEN. The procedure was later improved by Ratnam and Patwardhan (1991), who adopted Kern’s (1950) well-known linear heat-exchanger model. The basic idea of their model is that, for any new operating conditions considered, all temperatures in the network should be computed. That is, the system matrices in their model must be recalculated for each new process operating condition, even if there is a single change of an operating variable. Thus, the model is still computationally inefficient. Moreover, the model does not provide an explicit relationship between a set of disturbances and a set of target temperatures. It is very difficult, therefore, to trace any piece of propagation of a specific disturbance. This makes the analysis and * Author to whom all correspondence should be addressed. Telephone: 313-577-3771. Fax: 313-577-3810. E-mail:
[email protected].
S0888-5885(96)00321-1 CCC: $12.00
improvement of a process very difficult. Furthermore, their model is for flexibility rather than controllability. More recently, Li et al. (1994) developed linear equations for modeling the disturbance propagation in a HEN. Their model is also for process flexibility analysis. Li and co-workers introduced an average temperature difference to approximate a logarithmic mean temperature difference and derived approximate linear relationships between source and target temperature disturbances. Unfortunately, their model completely ignores the cross effect of the disturbances of temperatures and those of heat capacity flowrates. This leads to considerable computational errors when both types of disturbances exist. In addition, their model relies on the simulation of a given HEN structure when flowrate variations are considered. Through simulation, piecewise linearization for each temperature concerned can be obtained. This is very cumbersome when a number of process alternatives are to be analyzed. Different from the aforementioned models for process flexibility analysis, the approach developed by Huang and Fan (1992) is for process controllability analysis during the process synthesis phase. The approach can be used to identify and quantify disturbance propagation in a process by means of artificial intelligence techniques. This methodology has been used to design highly controllable HEN’s, mass-exchanger networks, and work-exchanger networks (Huang and Fan, 1994, 1996; Huang and Edgar, 1995). The quantification of disturbance propagation in the methodology is based on approximate reasoning. They classified disturbances into three degrees of severity, control requirements into three levels of precision, and propagation into four patterns. These classifications may introduce noticable errors for some cases, and the solutions may not be preferable when more precise process evaluation is needed (Sabharwal et al., 1995; Yang and Huang, 1996). This methodology needs to be improved. In the present work, a simplified system model is introduced for evaluating quickly and precisely disturbance propagation in a HEN. The model is developed for the analysis of controllability, rather than flexibility, during the process design stage. It is compared with rigorous models to conclude its applicability. The efficacy of using the model is demonstrated by solving practical industrial problems. © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4551
Figure 1. Flowsheet of an integrated plant with various disturbance propagation.
Figure 2. Representation of the HEN in the integrated plant: (a) grid diagram; (b) disturbance propagation table.
Disturbance Propagation In an integrated process plant, HEN’s are always adopted to recover energy. The introduction of the HEN’s to the plant usually introduces various interconnections among process streams. Improper interconnections, allowing various undesirable disturbance propagation, always cause process structural problems. Industrial Example. An integrated process is illustrated in Figure 1. This process contains many disturbance propagation paths. As an example, feed stream A (stream H1) and the bottom stream of distillation column D2 (stream H2) experience severe disturbances. These disturbances propagate through heat exchangers E1 and E2 to disturb the temperature of stream C1 which enters reactor R1. The disturbance propagation paths can be readily identified if the HEN is depicted in a grid diagram as shown in Figure 2a. All types of disturbance propagations in it are summarized in a disturbance propagation (DP) table (Huang and Fan, 1992). In Figure 2b, the digits of 0-3 represent the number of heat exchangers involved in the propagation. For instance, the disturbance at the inlet of stream H1 propagates through heat exchangers E1, E2, and E3 in sequence to the outlet of stream H2. Thus, a digit of 3 is assigned to the entry [(δThs 1, δMcPh1), δTht 2] in the DP table. This disturbance affects, in turn, the output temperatures of streams H1, C1, and H2 and the intermediate temperature of stream
C2. In this network, the temperature fluctuations at locations A, B, C, D, E, and F should be examined in order to identify the stability of the output temperatures of all streams and, further, to reduce or eliminate operational problems during process design. To precisely evaluate disturbance propagation in a process, a simple and predictable disturbance propagation model is needed. Unfortunately, no such model is available. Here, we avoid the development of a rigorous system model due to the following reasons: (i) Since numerous stream segments exist in an integrated process plant and a variety of process variables should be evaluated, a simple model is highly desirable for quick estimation of disturbance propagation. (ii) In process design, detailed disturbance information is always unavailable; this information is accessible only after a designed process is operated. However, rough information on disturbances can always be evaluated prior to process design. This includes all possibly largest process fluctuation ranges and basic control requirements. The information is, in reality, enough for process design where only the worst situation should be taken into account; all other disturbances (moderate or light) can be easily dealt with during process control design. To facilitate process analysis, the system model should maintain explicit physical meaning in terms of model variables and parameters. This requires that the first principles be the basis in model development. Estimation of Disturbance Propagation. For a heat exchanger operated in a given normal operating condition, the severity of disturbance propagation from one stream to another is largely dominated by the fluctuation of source temperature and that of heat capacity flowrate of each stream (δTsh, δTsc, δMcPh, and δMcPc). Note that these fluctuation data refer to the largest fluctuation ranges possibly occurring during process operation; they can be either positive or negative. A major concern of disturbance propagation through the heat exchanger is the stability of the target temperature of each stream which is characterized by the largest deviations from the normal operating point; they are designated as δTth and δTtc. For any heat exchanger, the following energy balance and heattransfer equations hold.
Q ) McPh∆Th ) McPc∆Tc ts ∆Tst hc - ∆Thc Q ) UA ∆Tst hc ln ts ∆Thc
(1)
(2)
where
∆Th ) Tsh - Tth
(3)
∆Tc ) Ttc - Tsc
(4)
s t ∆Tst hc ) Th - Tc
(5)
t s ∆Tts hc ) Th - Tc
(6)
Here, ∆Th and ∆Tc are, respectively, the temperature change of a hot stream and that of a cold stream; ts ∆Tst hc and ∆Thc are the temperature difference at the
4552 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
hot end and that at the cold end of a heat exchanger, respectively. Equations 1 and 2 can be extended when disturbances are taken into account.
Q + δQ ) (McPh + δMcPh)[∆Th + (δTsh - δTth)] ) (McPc + δMcPc)[∆Tc + (δTtc - δTsc)]
δTt ) (δTth
δTtc)T
(12)
δTs ) (δTsh
δTsc)T
(13)
(
δMcPc)T
δMcP ) (δMcPh (7)
Q + δQ ) s t ts t s [∆Tst hc + (δTh - δTc)] - [∆Thc + (δTh - δTc)] UA (8) s t ∆Tst hc + (δTh - δTc) ln t s ∆Tts hc + (δTh - δTc) The target temperature disturbances, δTth and δTtc, can be evaluated by solving these two equations. Due to the existence of a logarithmic term, it is nearly impossible to obtain analytical expressions of δTth and δTtc in terms of δTsh, δTsc, δMcPh, and δMcPc. If the models are used to evaluate disturbance propagation through a process network, especially the one with bypasses and loops, it becomes very cumbersome. If the models are to be embedded into a process synthesis procedure, it becomes extremely complicated. These models should be greatly simplified.
(
1-
)
a A ) a11 21
∆Th
∆Th
∆Tss hc
∆Tss hc ∆Tc 1∆Tss hc
a12 a22 ) ∆Tc
∆Tss hc
(
(
)
b B ) b11 21
b12 b22 ) ∆Th ∆Th 22McPh ∆Tss
(
hc
∆Th∆Tc
)
)
∆Tc∆Th -
2McPc∆Tss hc
(
∆Tc ∆Tc 22McPc ∆Tss
2McPh∆Tss hc
hc
s s ∆Tss hc ) Th - Tc
)
)
(14)
(15)
(16)
(17)
Since both ∆Th and ∆Tc are smaller than ∆Tss hc, all elements in A must be positive and less than 1; they are correlated in the following way:
Unit-Based Disturbance Propagation Model The model simplification can be initiated with the replacement of a logarithmic mean term by an arithmetic mean term under such a reasonable assumption that a process is always operated in the region not too far away from its normal operating point. This type of replacement has been adopted in many cases (Alsop, 1987; Johnson and Palsson, 1991; Ratnam and Patwardhan, 1991). With this replacement, eqs 2 and 8 become respectively
Q ) UA
ts ∆Tst hc + ∆Thc 2
(9)
Q + δQ ) s t ts t s (∆Tst hc + δTh - δTc) + (∆Thc + δTh - δTc) UA (10) 2 Simplified Model. A manipulation of eqs 1, 7, 9, and 10 can result in a formula relating target temperature disturbances to the disturbances of source temperatures and those of heat capacity flowrates. In derivation, we adopt the following commonly used additional assumptions: (i) no phase change occurs in any heat exchanger; (ii) changes in stream pressure drops are not serious; (iii) the second-order terms, such as the product of disturbances (δ‚δ), are neglected. The first two assumptions imply constant heat capacity for each process stream and constant heat-transfer coefficient for each heat exchanger. The third assumption is mainly used for the linearization of the unitbased model; this is safe when temperature and heat capacity flowrate disturbances are not significant. A more detailed derivation can be found in the appendix. The manipulation generates
a11 ) 1 - a12
(18)
a22 ) 1 - a21
(19)
and
Alternatively, matrix A can be written as:
(
1 - a12 A) a 21
s
δT ) AδT + BδMcP where
(11)
)
(20)
Likewise, an alternative matrix B can be derived below:
B)
(
Rh(2 - a12) Rha21
-Rca12 -Rc(2 - a21)
)
(21)
where
Rh )
∆Th 2McPh
(22)
Rc )
∆Tc 2McPc
(23)
For convenience, matrix B can be expressed as the product of matrices C and D below:
B ) CD
(24)
where
C)
(
(2 - a12) a21 D)
t
a12 1 - a21
(
Rh 0
-a12 -(2 - a21) 0 Rc
)
)
(25)
(26)
Since each element of A is less than 1, any single disturbance of source temperature will be decayed
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4553
during propagation through a heat exchanger. That is, δTth caused by either δTsh or δTsc will be smaller than either of them. The same conclusion can be made for δTtc. In the disturbance propagation model, the elements in matrices A and B are determined by only four parameters, a12, a21, Rh, and Rc, which are correlated as:
Rh a12 McPc ) Rc a21 McPh
(27)
Interestingly, eqs 18 and 19 show that the sum of the elements in each row of matrix A is equal to unity; the subtraction of the second element from the first one in each row of matrix C is equal to 2. These facilitate the determination of the element values of the matrices. System Disturbance Propagation Model Any heat exchanger performs as a disturbance distributor, which is quantified by eq 11. In a HEN, a single disturbance may propagate through a series of heat exchangers and affect the stability of many other streams. The development of a general system model can help calculate the fluctuation of the output temperature of each stream affected by any known disturbance. The unit-based model in eq 11 can form the basis in developing such a system model. System Model. A system model for characterizing all disturbance propagation through a HEN can be developed in the following form.
δTt ) AδTs + BδMcP
(28)
δT )
(δTht 1
δTht 2
‚‚‚
δTht N
h
δTs ) (δThs 1 δThs 2 ‚‚‚ δThs N
h
δTct 1
δTct 2
δTct N )T
‚‚‚
(29)
c
δTcs1 δTcs2 ‚‚‚ δTcsN )T (30) c
1
Ts (°C)
McP δTs(+) δTs(-) δMcP(+) δMcP(-) δTt (kW/°C) (°C) (°C) (kW/°C) (kW/°C) (°C)
Tt (°C)
204.4 65.6 248.9 121.1 93.3 93.3 65.6 65.6 37.8 37.8
H1 H2 C1 C2 C3
δMcPh
2
‚‚‚ δMcPh
Nh
δMcPc
1
δMcPc
2
‚‚‚ δMcPc )T Nc
(31)
System matrices A and B contain temperaturerelated and heat capacity flowrate-related system disturbance propagation information, respectively. They can be obtained through the following procedure. Determination of System Matrices A and B. Assume that Nh hot streams and Nc cold streams are connected by Ne heat exchangers in a HEN. Application of eq 11 to each heat exchanger generates the following equation:
δTEout ) AEiδTEini + BEiδMcPEi i
δT*out ) A*δT*in + B*δMcP*
(33)
2 4 1 2 1
-2 -3 -0.6 -2.5 -3
0.4 0.1 0.05 0.1 0.3
-0.4 -0.2 -0.1 -0.05 -0.4
δT*in ) ((δTEin1)T (δTEin2)T ‚‚‚ (δTEinN )T)T
(5.5 (3 (6 (7 (1
(34)
e
)T (δTEout )T ‚‚‚ (δTEout )T)T (35) δT*out ) ((δTEout 1 2 N e
T (δMcPE )T ‚‚‚ (δMcPE )T)T δMc* P ) ((δMcPE ) 1 2 Ne (36)
A* ) diag(AE1, AE2, ‚‚‚, AENe)
(37)
B* ) diag(BE1, BE2, ‚‚‚, BENe)
(38)
The dimensions of δT*in, δT*out, and δMc*P are all 2Ne × 1, and the dimensions of A* and B* are all 2Ne × 2Ne. Note that δT*in and δT*out contain (2Ne - Nh Nc) intermediate temperatures; δMc* P contains (2Ne Nh - Nc) redundant heat capacity flowrates. Correspondingly, A* and B* contain extra information which is not our interest. To obtain eq 28 from eq 33, vectors δT*in and δT*out should be permuted to δTin and δTout, respectively, in the following manner:
h
δTcs1 ‚‚‚ δTcsN
c
m δTm ‚‚‚ δT2N )T 1 e-Nh-Nc
) ((δTs)T (δTm)T)T δTout ) (δTht 1 ‚‚‚ δTht N
(39)
δTct 1 ‚‚‚ δTct N
c
m δTm ‚‚‚ δT2N )T 1 e-Nh-Nc
) ((δTt)T (δTm)T)T
(40)
In δMc*P, all redundant heat capacity flow rates should be eliminated. This reduces δMc*P to δMcP. With these definitions, we can obtain
( ) (
A δTt ) A11 m 21 δT
)( ) ( )
A12 δTs B1 A22 δTm + B2 δMcP
(41)
where δTs, δTt, and δMcP are all of the dimension of (Nh + Nc) × 1; δTm is of the dimension of (2Ne - Nh Nc) × 1. The two equations can be written separately from the above equation:
i ) 1, 2, ..., Ne (32)
where δTEini and δTEout are the vectors of stream temi perature disturbances at the inlet and outlet of a heat exchanger Ei, respectively; δMcPEi is the vector of fluctuated heat capacity flowrates of the streams through the heat exchanger Ei; AEi and BEi are of the same structures as A and B, respectively, in eq 11. The above Ne equations can be lumped to generate the following system equation:
13.29 16.62 13.03 12.92 11.40
where
h
δMcP ) (δMcPh
stream no.
δTin ) (δThs 1 ‚‚‚ δThs N
where t
Table 1. Stream Data for the H5SP1R Synthesis Problem
δTt ) A11δTs + A12δTm + B1δMcP
(42)
δTm ) A21δTs + A22δTm + B2δMcP
(43)
From eq 43, we can have
δTm ) (I - A22)-1A21δTs + (I - A22)-1B2δMcP (44) Substituting eq 44 into eq 42 gives the system model:
δTt ) AδTs + BδMcP where
(28)
4554 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
Figure 3. Grid diagrams of the five solutions of the HEN problem H5SP1R. Table 2. Comparison of the Deviations of Stream Target Temperatures in Five Solution Alternatives for the H5SP1R Synthesis Problem target temp (°C)
control requirement
sol. A
sol. B
sol. C
sol. D
sol. E
δTh1t δTh2t δTc1t δTc2t δTc3t
5.5 3 6 7 1
4.35/-5.60 2.78/-0.20 3.88/-0.05 3.04/-3.25 0/0
4.11/-5.22 3.14/-3.74 3.58/-2.82 3.78/-3.91 0/0
3.50/-4.71 3.93/-4.22 3.88/-2.96 3.70/-4.04 0/0
3.50/-4.71 3.28/-3.24 0/0 3.70/-4.04 7.14/-6.11
3.48/-3.94 5.08/-3.63 0/0 8.24/-4.4 6.17/-3.99
A ) A11 + A12(I - A22)-1A21
(45)
B ) B1 + A12(I - A22)-1B2
(46)
Applications The system propagation model has been successfully used to evaluate process alternatives and to modify existing process systems. Two case studies are illustrated below. Evaluation of Process Alternatives. The HEN design problem, namely, H5SP1R, consists of two hot streams and three cold streams. The design data including disturbance and control information are listed in Table 1 (Huang and Fan, 1992). The pinch point of
this synthesis problem is located at 43.4 °C. The minimum energy requirement is 884.6 kW of steam. The system needs five heat-transfer units as the minimum. The design problem requires precise control of the target temperatures of streams H2 ((3 °C) and C1 ((1 °C), with relatively large tolerable deviation ranges for the other three streams. With those process restrictions, there only exist five solution alternatives which are identified by the distributed strategy (Huang and Fan, 1992); they are as shown in Figure 3. The system model in eq 28 is then used to evaluate all five alternatives. The computational results of the deviations of stream target temperatures of each solution are summarized in Table 2. Note that, in solutions A, B, and C, the target temper-
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4555
ature of streams C3 can be easily controlled with no fluctuation due to the placement of a heater on this stream. For the target temperature of stream H2, solution A is more desirable since the temperature deviation is within the tolerable range ((3 °C); solutions B and C have larger temperature deviations. Solutions D and E are entirely undesirable due to severe deviations of the target temperatures of streams H2 and C3. Overall, solution A should be the first choice. For this solution, if we introduce the disturbance vectors and control variable vector
δTt ) (δTht 1 δTht 2 δTct 1 δTct 2 δTct 3)T
(47)
δTs ) (δThs 1 δThs 2 δTcs1 δTcs2 δTcs3)T
(48)
δMcP ) (δMcPh
1
δMcPh
2
δMcPc
1
δMcPc
2
δMcPc )T 3
(49) then the system matrices are
(
A) 0.0739 0 0 0.7431 0
(
0.0463 0.2539 0.7140 0.0616 0
0.0590 0.3237 0.2860 0.0785 0
0.0878 0.4225 0 0.1168 0
)
(50)
B) 4.4388 0 0 1.7472 0
0.7329 0 0 0 0
0.5260 4.1079 1.8731 0.7002 0
-0.2517 -1.3798 -5.4825 -0.3347 0
-1.0498 -0.8567 0 -3.8809 0
)
-2.8611 0 0 0 0 (51)
Elements aij in matrix A and bij in matrix B represent the severity of the propagation from the inlet of stream i to the outlet of stream j. The larger the absolute value of an element, the more severe the disturbance propagation. For instance, two zero elements in the second row of matrix A mean that there exist no disturbance propagation paths from streams H1 to H2 and from streams C3 to H2. Structurally, the main contribution of disturbance propagation to the outlet of stream H2 is from stream C2, followed by that from streams C1 and H2. It also shows that the disturbance at the inlet of stream H1 is decayed by 92.6% when reaching its own outlet since the value of element a11 is very small. The last rows of matrices A and B contain only zero elements. This is due to a commonly adopted assumption that a heater can absorb all disturbance by adjusting steam flow rate through it. This means that the target temperature of stream C3 can be controlled very precisely. As a comparison, Huang and Fan (1992) used the AIbased distributed strategy to evaluate the structural controllability of all five alternatives. It turns out that solution A has the largest index value (0.958). By contrast, solutions B-E are of the index values of 0.917, 0.750, 0.083, and 0.083. This indicates that solution A is clearly the best. Solution B is much better than solution C. Solutions D and E are not acceptable. While the evaluations by two different approaches demonstrate a good agreement, in general, the simpli-
Figure 4. Flowsheet of a heat-integrated reactor-separation process.
fied model can provide reliable and quantitative information of disturbance propagation through a network which makes designers much more confident in process analysis. The model-based evaluation also suggests that solutions A, B, and C are all acceptable, although solution A is more preferable. Process designers should evaluate all three solutions when other design criteria are considered. Improvement of a Heat-Integrated ReactionSeparation System. The flowsheet of an industrial process is shown in Figure 4. A recycle stream from other process is heated from 98.9 to 123.9 °C and fed to distillation column D1. Feed A is preheated in a series of heat exchangers and then enters reactor R1 where a slight endothermic reaction takes place. The product stream of reactor R1 and the column overhead stream, after cooling, are mixed with feed B. This mixture stream is fed to reactor R2 in which catalyst is very sensitive to its operating temperature; its performance will be seriously degraded if the temperature is unstable and above 110 °C. This requires that the feed stream temperature of reactor R2 be strictly controlled between 96.1 and 98.9 °C. However, this process faces many perturbations including (a) 10% of mass flowrate change in the recycle stream, (b) 8.3% of mass flowrate change in the column overhead stream, (c) column overhead temperature from 132.2 to 165.6 °C, (d) feed A temperature between 37.8 and 65.6 °C, and (d) feed B temperature between 32.2 and 48.9 °C. These make the process inoperable, especially for reactor R2. Through an analysis by the extended distributed strategy, the operational problems are found to be caused by improper design of the HEN inherited (Yang and Huang, 1996). The grid diagram of the HEN is shown in Figure 5. This process has the pinch point at 37.8 °C. The minimum energy requirement is 2022 kW, with a minimum of six heat transfer units. For this system, if we define the following disturbance vectors and control variable vector
δTt ) (δTht 1 δTht 2 δTct 2 δTct 3)T
(52)
δTs ) (δThs 1 δThs 2 δTcs2 δTcs3)T
(53)
δMcP ) (δMcPh
1
δMcPh
2
δMcPc
2
δMcPc )T (54) 3
then the system matrices are of the following values:
4556 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
Figure 5. Grid diagram of the HEN in the heat-integrated reactor-separation process. Figure 7. Flowsheet of a modified heat-integrated reactorseparation process.
temperature disturbances and introduces one more unknown variable, the derived form of the system model in eq 28 remains unchanged. A complete process flowsheet embedding this HEN is depicted in Figure 7, with its system matrices A and B listed below.
Figure 6. Grid diagram of a modified HEN for the heat-integrated reactor-separation process.
(
0.3042 0.2909 A) 0.3573 0
(
1.3562 0.6117 B) 0.1099 0
0 0.3633 0 0
0 1.4618 0 0
0.2288 0.2189 0.6427 0
-0.0488 -0.0467 -0.3506 0
0.4670 0.1279 0 0
-0.9820 -2.4967 0 0
)
)
(55)
(56)
Note that, in matrices A and B, the last rows contain only zero elements, since stream C3 is eventually heated by a heater which is capable of absorbing all disturbances. Also note that stream C1 is not included in all vectors and matrices, since this stream is not involved in heat exchange with any stream through a heat exchanger. Instead, this stream is mixed with streams H1 and H2; the mixed stream enters reactor R2 with the temperature fluctuation from 91.4 to 111.7 °C. This fluctuation is far beyond the tolerable range (96.1-98.9 °C). Thus, the system cannot be operated regardless of control techniques used. Figure 6 shows a modified HEN obtained by the extended distributed strategy (Yang and Huang, 1996). The derivation of this solution is out of the scope of this paper. The modified HEN contains a bypass and a loop design for improving system performance. In fact, when a bypass is considered, the system model only needs to include an energy balance equation describing the relationship of the temperature disturbances before and after mixing. Since the equation gives linearity in
(
(
0.4 0 A) 0.5 0
0.7738 0 B) 0.2763 0
0 0.1995 0 0
0 3.3824 0 0
0.6 0 0.5 0
-0.2303 0 -0.5759 0
0 0.8005 0 0
)
0 -2.9450 0 0
(57)
)
(58)
In this modified system, the stream mixing H1, H2, and C1 before entering a heater is between 83.8 and 98.2 °C. Through the heater, the stream entering reactor R2 can be easily controlled at 98.9 °C. Also note that the target temperature of stream C2 fluctuates less severely than the original design; the temperature of stream C3 entering a heater can be operated between 164.6 and 167 °C which shows a smaller range than that in the original design. With these changes, the original operational problems are eliminated structurally. Moreover, the modified system requires one heat exchanger less than the original system. Hence, the capital cost through process improvement can be reduced while energy cost still remains the same. Discussion. While the model applications in the above cases have shown excellent agreement with the results by rigorous models based on eqs 7 and 8, this may not be true for some other applications. This is due to the replacement of a logarithmic mean term by an arithmetic mean term in model development. The resultant unit-based model in eq 11 shows that the fluctuations of the target temperatures are quantified in two parts: (a) The fluctuations due to source temperature disturbances quantified as AδTs. This quantification is proven to be rigorous and provides no errors regardless of the severity of any single source temperature disturbance or any combination of source temperature changes over any range. (b) The fluctuations caused by heat capacity flowrate disturbances quantified as BδMcP. The prediction
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4557
based on this quantification may be erroneous when δMcPh and δMcPc are significant (g20% of their nominal values). To obtain a precise evaluation, we suggest replacing BδMcP by CcBDcδMcP, where Cc and Dc are of the following forms:
Cc )
Dc )
(
(
f1(a12, a21) 0
g1(McPh, δMcPh) 0
0 f2(a12, a21)
)
0 g2(McPc, δMcPc)
)
δQ ) maximum deviation of the heat duty of a heat exchanger U ) overall heat-transfer coefficient Superscripts s ) source t ) target
(59)
Subscripts
(60)
h ) hot stream c ) cold stream e ) heat exchanger
The two matrices may contain nonlinear forms, depending on the precision expected. With these corrections, prediction errors by the system model can be reduced to the range of 10%. Certainly, this error range is acceptable for conceptual design. (c) The developed model is for the analysis on disturbance propagation during the first and conceptual phase of process design, i.e., process synthesis. Thus, more practical issues, such as design parameters of any piece of equipment and energy storage effects, are not considered in this work. These issues, however, must be addressed if the model is used during detailed process design and system optimization stages. Concluding Remarks An evaluation of disturbance propagation through a process system is always an important step in process analysis, modification, and operational improvement. This is especially true for an integrated process plant where numerous disturbances propagate to many streams whose target variables should be controlled precisely. In the present work, a simplified system model is proposed to precisely evaluate the propagation caused by even severe temperature disturbances and/ or by moderate heat capacity flowrate fluctuations in a HEN. It is easy to use and computationally very efficient. For any HEN problems tackled, the computations can be accomplished nearly spontaneously. It is particularly helpful in analyzing integrated process systems where disturbance propagation is always a major concern. The system model can be easily converted to those for evaluating concentration disturbance propagation through a mass exchanger network. We are exploring the application opportunities of this model for conceptual design in order to design highly structural controllable process plants. Acknowledgment Financial support from the National Science Foundation (CTS-9414494) is gratefully acknowledged. Nomenclature A ) heat-transfer area of a heat exchanger A ) temperature disturbance propagation matrix B ) heat capacity flowrate disturbance propagation matrix C ) matrix related to steady-state temperatures D ) matrix related to steady-state temperatures and heat capacity flowrates I ) identity matrix McP ) heat capacity flowrate δMcP ) maximum disturbance of heat capacity flowrate T ) stream temperature δT ) maximum temperature deviation from nominal setting ∆T ) temperature difference Q ) heat duty of a heat exchanger
Appendix. Derivation of the Unit-Based Disturbance Propagation Model The derivation of the unit-based disturbance propagation model in eq 11 can be initiated at rewriting eqs 7 and 10. This gives rise to:
Q + δQ ) McPh∆Th + McPh(δTsh - δTth) + δMcPh∆Th + δMcPh(δTsh - δTth) ) McPc∆Tc + McPc(δTtc - δTsc) + δMcPc∆Tc + δMcPc(δTtc - δTsc)
ts ∆Tst hc + ∆Thc + 2 (δTsh - δTtc) + (δTth - δTsc) UA (A-1) 2
) UA
By applying eq 9 and neglecting the second-order term, the following relationship holds:
UA [(δTsh - δTtc) + (δTth - δTsc)] ) 2 δMcPh∆Th + McPh(δTsh - δTth) (A-2) UA [(δTsh - δTtc) + (δTth - δTsc)] ) 2 δMcPc∆Tc + McPc(δTtc - δTsc) (A-3) Substituting eqs 1, 2, and A-3 into eq A-2 gives
∆Th ∆Tst hc
+ ∆Tts hc
(δTsh - δTtc + δTth - δTsc) ) ∆Th δMcPh + (δTsh - δTth) (A-4) McPh
Similarly, the following relationship can be obtained:
∆Tc ∆Tst hc
+ ∆Tts hc
(δTsh - δTtc + δTth - δTsc) ) ∆Tc δMcPc + (δTtc - δTsc) (A-5) McPc
Dividing eq A-5 by eq A-4 gives
∆Th δMcP + (δTsh - δTth) McPh
∆Th ) ∆Tc ∆Tc δMcPc + (δTtc - δTsc) McPc
(A-6)
or
δTth ) δTsh +
∆Th s ∆Th t ∆Th δTc δTc + δMcPh ∆Tc ∆Tc McPh ∆Th δMcPc (A-7) McPc
4558 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
Substituting eq A-7 into eq A-4 and neglecting the second-order term gives
∆Tc
δTtc )
∆Tss hc
δTsh +
∆Tst hc δTsc ss ∆Thc
∆Th∆Tc +
2∆Tss hcMcPh
δMcPh -
st ∆Tc(∆Tss hc + ∆Thc) δMcPc (A-8) 2∆Tss hcMcPc
Similarly, substituting eq A-8 into eq A-5 gives
δTth )
∆Tts ∆Th s hc δTsh + δTc + ss ∆Thc ∆Tss hc
ts ∆Th(∆Tss ∆Tc∆Th hc + ∆Thc) δMcPh δMcPc (A-9) ss 2∆ThcMcPh 2∆Tss hcMcPc
These two equations can be expressed into the following matrix form.
( ) ( ) ( ) δMcPh δTth δTsh ) A t s + B δMc δTc δTc Pc
( )(
(A-10)
where
A)
∆Tts hc
∆Th
∆Tss hc
∆Tss hc ∆Tst hc ∆Tss hc
∆Tc
(
∆Tss hc
1-
∆Th
∆Th
∆Tss hc
∆Tss hc ∆Tc 1∆Tss hc
) ∆T c ∆Tss hc
ts ∆Th(∆Tss hc + ∆Thc)
B)
2∆Tss hcMcPh
(
-
∆Th ∆Th 22McPh ∆Tss ∆Th∆Tc
2McPh∆Tss hc
2∆Tss hcMcPc st ∆Tc(∆Tss hc + ∆Thc)
∆Th∆Tc
2∆Tss hcMcPh
)
∆Th∆Tc -
(
)
hc
)
2∆Tss hcMcPc ∆Tc∆Th
-
2McPc∆Tss hc
(
∆Tc ∆Tc 22McPc ∆Tss
hc
)
)
(A-11)
) (A-12)
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Received for review June 10, 1996 Revised manuscript received October 1, 1996 Accepted October 1, 1996X IE960321C X Abstract published in Advance ACS Abstracts, November 15, 1996.
