A simple heat exchanger network (HEN), illustrated in Fig. 1.5, consists of one cold feed, one hot effluent stream and a heat exchanger (HE), a heater and a cooler. The cold feed stream enters at 30C (= TC,in) and it has to be heated to 125C (= TC,out) while the hot effluent stream enters at 115C (= TH,in) and it has to be cooled to 40C (= TH,out). The mass flow rate and heat capacity of the feed stream are respectively 16 kg/s and 4 kJ/(kg.K), and the mass flow rate and heat capacity of the effluent stream are respectively 20 kg/s and 3.8 kJ/(kg.K).
Fig. 1.5 Schematic of a heat exchanger network (HEN) for heat recovery and reuse number
HE in Fig. 1.5 helps to recover and reuse thermal energy, thus reducing both steam required in the heater and cooling water required in the cooler. This reduces the operating cost. However, inclusion of HE is likely to increase the required investment (also known as capital cost).
Thus, there will be a trade-off between operating cost and investment, which is the case in many engineering applications. The design question is whether HE should be included and, if so, what should be the size of HE. This can be formulated as an optimization problem with one or two objectives, and DE can be used to solve it. This section describes the formulated optimization problem with two objectives, and then presents results obtained using IMODE program (described in Chapter 5), which is based on multi-objective DE and is implemented in MS Excel environment. Note that optimization problem formulation requires background and knowledge in the relevant discipline. For example, HEN
Heat Exchanger
(HE) Heater (QH)
Steam Feed
T2
TC,out = 125oC TC,in = 30oC
Cooler (QC) T1
TH,in = 115oC TH,out = 40oC
Cooling Water Effluent
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problem formulation (described below) is simple and easy to understand by chemical and mechanical engineers.
HE, cooler and heater in Fig. 1.5 are all assumed to be counter-current type, which is the most common. Temperatures of the effluent and feed streams after HE are denoted as T1 and T2 respectively. If there is no HE, then T1 = TH,in and T2 = TC,in. HEN under consideration is at the design stage, and so the area of the heat exchanger, heater and cooler (denoted by AHE, AH and AC respectively) can be varied for optimization. The governing (model) equations and the optimization problem for the HEN can be developed based on heat transfer principles and suitable cost correlations/data for the investment and operating cost. One of the governing equations is the energy balance, which, for HE, is given by:
16 × 4 × T − T , = 20 × 3.8 × (T , − T ) (1.3) The above equation can be re-arranged to find T as follows:
T = T , + × .
× × T , − T (1.4)
The two objective functions are:
Minimize Investment ($) = IH + IHE + IC (1.5a) Minimize Operating Cost ($/year) = CSteam + CCW (1.5b) Here, IH, IHE and IC are respectively the investment for the heater, HE and cooler. Each of them is given by:
Ii = 38000 + 520Ai0.9 for i = Heater, HE and Cooler (1.6) Here, Ai is the heat transfer area of equipment i, which is given by the following equations obtained from the heat transfer rate equation involving log mean temperature difference.
A = × ×( , )×
,
, ( ) (1.7a)
A = × . ×( , )×
, ,
, , (1.7b)
A = × . ×(( ), )× ,
, (1.7c)
In the above equations, overall heat transfer coefficients for heater, HE and cooler are assumed to be UH (= 0.78 kW/m2-K), UHE (= 0.50 kW/
m2-K) and UC (= 0.50 kW/m2-K), respectively. Further, saturated steam
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Introduction 19
Differential Evolution in Chemical Engineering 9in x 6in b2817-ch01
(heating medium) enters the heater at 180oC and leaves as condensate at the same temperature, thus giving the latent heat of condensation for heating purpose. Cooling water enters and exits the cooler at 30oC and 40oC, respectively, due to heat transferred from the effluent.
In the operating cost equation (1.5b), steam cost (CSteam) and cooling water cost (CCW) are given by:
CSteam = 400 × 16 × 4 × (T , − T ) (1.8a)
CCW = 25 × 20 × 3.8 × (T − T , ) (1.8b) Here, utility costs are taken to be 400 US$/(kW.y) for steam and 25 US$/(kW.y) for cooling water.
In equations 1.3 to 1.8, TH,in = 115oC, TH,out = 40oC, TC,in = 30oC and TC,out = 125oC from the problem statement. Further, knowing T1, all other unknown quantities in these equations can be calculated (i.e., T2 from equation 1.4, Ai from equation 1.7, investment from equation 1.6 and operating cost from equation 1.8). Hence, HEN optimization in Fig. 1.5 has only one decision (independent) variable, namely, T1. Lower bound for this variable is TH,out = 40oC, which still ensures a minimum approach temperature of 5oC, and upper bound is 115oC (i.e., TH,in in the absence of HE). In addition, the calculated/dependent variable, T2 is also constrained between 30oC (in the absence of HE) and 110oC (= 115 - 5 to ensure a minimum approach temperature of 5oC).
In summary, the optimization problem is simultaneous minimization of two objectives in equation 1.5 with respect to T1 between 40 and 115oC subject to two inequality constraints to keep T2 between 30 and 110oC. Note that equations 1.4, 1.6, 1.7 and 1.8 are essentially equality constraints but they can be solved easily one by one. The latter approach is better since stochastic optimizers are not effective in handling equality constraints (Sharma and Rangaiah, 2013b).
HEN optimization problem is entered into the Objectives &
Constraints worksheet of IMODE program (Fig. 1.6a), and it is linked with the Main Program Interface worksheet (Fig. 1.6b). Algorithm parameters are the default values along with population size of 40 and maximum number of generations of 100. HEN design problem is solved by clicking ‘Run IMODE’ icon. Optimal values of objectives (Invest-ment and Operating Cost), decision variable (T1) and two inequality constraints on T2 are plotted in Fig. 1.7. This figure also contains plots showing the variation of Operating Cost with area of heater, heat exchanger and cooler, respectively.
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Fig. 1.6a HEN optimization problem entered into the Objectives & Constraints worksheet of the IMODE program
Fig. 1.6b Main program interface of the IMODE program for the HEN optimization problem
A
Worksheet for Providing the Decision Variables, Functions and Constraints (Scroll down for instructions on calculation and linking of objectives and constraints.)
HEN example
Decision Variable and its Bounds Description
Objectives & Constraints MOOSetup Intermediate Results Results at ChiTC Results at SSTC Results after MNG Value
IMODE (Improved Multi-Objective Differential Evolution) for problems with continuous/integer variables and inequality constraints Algorithm:
Objectives & Constraints MOOSetup Intermediate Results Results at ChiTC Results at SSTC Results after MNG 20
Generation Interval: Saving Intermediate Result
(This should be less than the maximum number of generations)
H I J K L M N
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Introduction 21
Differential Evolution in Chemical Engineering 9in x 6in b2817-ch01
As expected and as can be seen in Fig. 1.7, Investment and Operating Cost objectives are conflicting. More importantly, MOO provides quantitative trade-off between Investment and Operating Cost, and also many optimal solutions along with the optimal values of decision variable, constraints and dependent variables (namely, areas of heater, heat exchanger and cooler). All these give a deeper insight into the process on hand and for selecting one of the optimal solutions based on the preferences of the decision maker and other considerations.
Before using MOO results, it is desirable to analyze and explain qualitatively the trends of objectives, decision variables and other quantities, in order to ensure their validity. For example, increasing T1
leads to increased energy recovery in the heat exchanger, thus resulting in the decreased requirement of steam (in the heater) and cooling water (in the cooler), which reduces Operating Cost. In terms of equipment size, heat exchanger area increases, and areas of the heater and cooler decrease as T1 increases. These variations are consistent with the expectations based on the knowledge in the heat transfer field. The overall outcome of area changes is the increased investment cost.
Application described in this section, although realistic, is relatively simple. More complex and realistic applications are described and discussed in many chapters of this book. For example, IMODE program and its application are presented in Chapter 5, retrofitting of large HENs is covered in Chapter 6, and Chapter 10 describes optimization of bioethanol separation by the hybrid process of distillation and vapor permeation. The next section outlines scope and contributions of all chapters in this book.