A Decision-Analytic Model for the Bi-Objective Sizing of Heat Exchangers Under Uncertainty

2.1. Heat Exchanger Design Under Uncertainty

There are several studies assessing the behavior of heat exchangers under uncertainty. Badar et al. [3], Clarke et al. [4] and Azarkish and Rashki [5] apply Monte Carlo simulation to determine the reliability of the thermal designs under different probability distributions of several design parameters like tube diameter, thickness, and conductivity, and the convective thermal coefficients. Sensitivity analysis procedures are shown in Taylor et al. [6] and Tan et al. [7], which study the effect of thermal properties on the performance of the heat exchangers used in cryogenic CO₂ processes, while Lambert and Gosselin [8] study the impact of the cost and transport coefficients correlation uncertainties on the equipment field performance. Finally, Caputo et al. [9] compare the performance of five “rules of thumb” used in heat exchanger oversizing against parametric uncertainty.

Heat exchanger design methods that consider uncertainty are shown in Bernardo et al. [10] who use a stochastic programming formulation to optimize the design of an exchanger and a reactor, Giugno et al. [11] and Khan et al. [12] who address the robust design of plate and fin exchangers, the latter through neural networks and Jafari-Asl et al. [13] which propose a hybrid metaheuristic, based on k-clustering and the whale optimization algorithm, to minimize total costs while maintaining the equipment performance level.

Recently, several researchers have studied the effect of uncertainty on the design of ground heat exchangers, which use the soil or ground layers at shallow depths as thermal sources. Proposals for the determination of the relevant conductivities using, respectively, regression and Kalman filters are found in Zhang et al. [14] and Shoji et al. [15], while Garber et al. [16] and Dusseault and Pasquier [17] evaluate the financial risk of the designs, the former with respect to uncertainty in demand and the latter using the concept of net present value at risk. Finally, a metaheuristic for the design of these devices is shown by Mikhaylova et al. [18].

2.2. Design of Heat Exchanger Networks Under Uncertainty

The synthesis of heat exchanger networks for maximum energy integration, in uncertainty about the inlet streams' flows and temperatures, is discussed in the pioneering work of Floudas and Grossmann [19]. More recently, Tellez et al. [20] and Chen and Hung [21] applied fuzzy logic concepts to the synthesis, the latter in a multi-objective mixed integer linear programming scheme. Other approaches to consider parameter uncertainty are found in Gálvez et al. [22], who use gray programming, Pintarič and Kravanja [23], who put forward a set of heuristics to be used when designing the network, and Khulaifi and Mutairi [24], whose model considers the variability in heat capacity, viscosity, and conductivity. Recently, Sudhanshu and Chaturvedi [25] used robust linear programming to consider variable heat capacities and inlet temperatures in network design, and Tian and Li [26] applied Shapley's value, a concept used in bargaining theory, and fuzzy logic to allocate network costs among plants with a common heat recovery network.

2.3. Software for Heat Exchanger Design

Several proprietary software for the design of tube and shell heat exchangers are available, as is the case with HTRI's XChanger Suite, Aspen Exchanger Design & Rating, and CHEMCAD [27]. Custom-built programs for specific cases can be found in Mohanty and Manandhar [28] for designing kettles and furnaces, Cui et al. [29] for designing and simulating vertical ground heat exchangers, Wu et al. [30] for sizing spiral heat exchangers with lateral phase change, and Bensafi et al. [31] and Jiang et al. [32] for sizing heat exchangers used in refrigeration systems. Meanwhile, Cartaxo et al. [33] developed an educational software for the transient-state simulation of tube and shell heat exchangers.

The RESHEX program of Saboo et al. [34] is one of the earliest reports on software for designing heat exchanger networks with maximum energy recovery and minimum cost. Other programs for executing pinch energy integration analysis and designing the relevant tube and shell heat exchangers are shown in Lona et al. [35], Chin et al. [36], and Martín and Mato [37]. Finally, Bozan et al. [38] show the BatcHEN program to design low-cost heat exchanger networks for multipurpose batch plants' service.

2.4. State of the Art Summary

Decision Analysis, as introduced by Howard [2], is a hybrid of systems engineering and decision theory, and is considered a branch of operational research. By applying its procedures when analyzing a choice situation, the user is guaranteed to find the best alternative in accordance with a set of axioms of rational decision-making [39]. Previous reports on heat exchanger design under uncertainty do not take a Decision Analytic perspective. As presented here, said perspective allows highlighting the interrelationship between uncertainty and the preferences of the decision-maker in selecting the best design. Additionally, previous reported research doesn’t address the design of exchangers based on ranges of heat transfer coefficient values, which are common in engineering handbooks and may be the only information accessible to the engineer in the early steps of process design. Finally, regarding the existing reports of software for heat exchanger design, the described programs don´t have provisions to handle parameter uncertainty and are coded in software requiring a license purchase for its use, while in this work the developed user-friendly interface is coded as a Visual Basic Macro and executable in Microsoft Excel, which is widely accessible by the academic and engineering communities in several countries.

3.1. Problem Statement

A process stream at an inlet temperature T_I,P must be cooled to a temperature T_O,P by a utility stream at a temperature T_I,U. The mass flow of the process stream is W_P, while its heat capacity per unit mass and that of the service stream are C_P,P, and C_P,U, respectively (Fig. 2).

When sizing the heat exchanger, the utility stream outlet temperature T_O,U can be calculated from a recommended temperature approach [1], while an energy balance allows determining the mass flow of this stream, W_U. The exchanger is governed by two energy conservation equations (Eqs. 1-2), a heat transfer relation (Eq. 3), and two additional equations (Eqs. 4-5).

(4)

(5)

Where Q is the amount of energy exchanged, U is the overall heat transfer coefficient, A is the exchanger area, and ∆T_LMTD is the logarithmic mean of the temperature differences. U is calculated from the thickness ε of the wall separating the currents in the exchanger, the thermal conductivity k of the metal, and the internal h_I and external h_O thermal convective coefficients. These last two coefficients are strongly dependent on the fluid flow and temperature conditions, so that in the early stages of design, it is impossible to know their value precisely. However, ranges of typical values of these coefficients can be found in engineering handbooks [40]. The ranges are denoted h_I∈[h_I^MIN, h_I^MAX] and h_O∈[h_O^MIN, h_O^MAX].

3.2. Heat Exchanger Sizing as a Decision Problem

In practice, the operation of the exchanger will likely involve a control system adjusting the flow of the utility stream to keep the outlet temperature of the process stream at a desired value or set-point (T_O,P^SP), as shown in Fig. (3). This would compensate for inaccuracies in the data used in sizing the exchanger.

The sizing problem consists of setting the area of the exchanger (A) and the maximum installed capacity of the utility stream (W_U,MAX). An influence diagram of this decision is shown in Fig. (4). Circles represent uncertainties, squares stand for decisions, the circle with a double border means a deterministic calculation, and the hexagon represents the consequences. The finding of an optimal solution by using influence diagrams implies that the recommended design is chosen from a discrete set of alternatives and that the relevant probability distributions have been discretized. This is computationally convenient but can limit the quality of the solution found. More refined solution-searching approaches and simulation-based techniques for uncertainty propagation can be used to maintain the continuous nature of decisions and uncertainties, but these techniques would have increased the computational effort required to provide a solution.

The convective transfer coefficients h_I and h_O are the uncertainties of the problem, while A and W_U,MAX represent the decisions. The arrows reaching the double-bordered circle indicate that when the mentioned four elements are known, the outlet temperature of the process stream, T_O,P, is fixed. Consequences are outcomes that are important to the decision-maker, in this case, the probability that the process outlet temperature is at its set-point P(T_O,P=T_O,P^SP) and the total cost C_T, which depends on the design decisions. The decision model can be split into a factual model, based on equations 1 to 5, that calculates the probability distribution of T_O,P given the decisions, and a value model providing a preference metric for consequences P(T_O,P=T_O,P^SP) and C_T. More precisely, the value model consists of a utility function that assigns a desirability value (utility) to the performance metric and exchanger size, the latter used as a proxy of cost. The expected utility of each alternative is calculated from its performance probability distribution and size. The chosen alternative is the one that gets the maximum expected utility.

3.2.1. Factual Model

The ranges of h_I and h_O are discretized as sets of N values [h_I¹= h_I^MIN, h_I²,..., h_I^N=h_I^MAX] and [h_O¹= h_O^MIN, h_O²,..., h_O^N=h_O^MAX], with each value having the same probability 1/N. In the same way, it is assumed that the exchanger area and maximum installed utility stream capacity are to be chosen, respectively, from discrete sets of a and w alternatives A∈[A¹,A²,...,A^a] and W_U,MAX∈[W_U,MAX¹, W_U,MAX²,..., W_U,MAX^w]. For a set of values h_Iⁱ, h_O^k, A^m, and W_U,MAXⁿ the algorithm shown in Fig. (5) calculates T_O,P, and W_U. The third step of Fig. (5) determines whether the configuration can keep the process stream outlet temperature at its set-point, first by calculating the maximum possible value of the logarithmic mean of the temperature differences (corresponding to a constant utility stream temperature and the outlet temperature of the process stream at its set-point). The minimum UA value, (UA)_MIN, is determined from the heat needed to bring the process stream temperature to its set-point and the maximum value of the logarithmic mean of the temperature differences. If the specified value of UA is greater than (UA)_MIN, the required utility current flow W_U is calculated. If this W_U value is less than the maximum installed capacity W_U,MAXⁿ, the outlet temperature of the process stream is set to T_O,P^SP, and the utility stream flow to W_U. However, if the calculated value of W_U is bigger than W_U,MAXⁿ or the value of UA is below (UA)_MIN, the outlet temperatures of both streams T_O,P, and T_O,U are calculated with the flow of the utility stream at its maximum value W_U,MAXⁿ. In this case, the process stream outlet temperature cannot be kept at its set-point. So, given the parameters included in the top box of Fig. (5), the factual model allows the discretized probability distributions of the heat transfer coefficients to result in a probability distribution of the outlet stream tempera-

ture, as each combination of possibilities of the heat transfer coefficients produce an outlet temperature possibility, being the probability of said temperature the joint probability of the heat transfer coefficient possibilities. From this probability distribution, the performance metric (probability of maintaining the outlet temperature at its set point) is determined.

3.2.2. Value Model

To balance the objectives of keeping the outlet temperature of the process stream at its set-point and minimizing equipment cost, an additive value (or utility) function, Eq. (6), is assumed

UG=kSP USP(P(TO,P=TO,PSP))+k$U$(CT)

(6)

The weights k_SP and k_$ add up to one. U_SP takes a value of one for the greatest probability, among the alternative designs considered, of maintaining T_O,P at its set point, P_MAX(T_O,P=T_O,P^SP), and takes the value of zero for the corresponding smallest probability P_MIN(T_O,P=T_O,P^SP), as shown in Eq. (7).

(7)

The total cost C_T, Eq. (8), is assumed to be linear with respect to the exchanger area and the maximum installed capacity of the utility stream

CT=αA A + αW WU,MAX

(8)

Where α_A and α_W are unitary cost factors. The function U_$ is calculated based on C_T according to Eq. (9):

(9)

Since the maximum cost C_T^MAX is that of the alternative specifying the greatest exchanger area and utility stream capacity (A^MAX and W_U,MAX^MAX), and the minimum cost C_T^MIN is that of the alternative setting the minimum values of these decisions (A^MIN and W_U,MAX^MIN), Eq. 9 can be rearranged as Eq. (10):

(10)

The selected design will be the one producing the highest expected value of the overall utility U_G. Oversize percentages of area and utility stream capacity, X_A and X_WU,MAX, with respect to some base values A^B and W_U,MAX^B, are defined as in Eqs. (11 and 12).

(11)

(12)

For instance, if there is no overdesign in area, X_A=0 and the area is equal to A^B, while an overdesign of 200% (X_A=200) corresponds to an area three times the size of A^B. In terms of oversized variables, U_$ is expressed as Eq. (13):

(13)

Where X_A^MAX, X_A^MIN, X_WU,MAX^MAX, and X_WU,MAX^MIN are the maximum and minimum overdesign percentages considered in the alternatives, respectively, for area and maximum flow capacity of the utility stream, while γ_A and γ_A are defined by Eqs. (14-15):

(14)

(15)

Equation 13 is rearranged to Eq. (16), so U_$ can be calculated from the γ_A/γ_W ratio

(16)

The weights k_SP and k_$ depend on the decision maker´s trade-off between the temperature control objective and the equipment cost. To calculate these weights, a transition ∆T_SP is defined as changing the lowest probability of keeping the process stream outlet temperature at its set point (among the considered design alternatives), to the corresponding highest probability. ∆T_SP is formally stated in Eq. (17), where the arrow means “is changed into”

∆TSP: PMIN(TO,P=TO,PSP)→ PMAX(TO,P=TO,PSP)

(17)

If the decision maker views this probability improvement as equivalent to a monetary amount C^e (Eq. 18):

∆TSP ∼ Ce

(18)

Where “∼” means “is indifferent between”, and ∆C_T is the breadth of the cost range among the alternatives, as in Eq. (19),

∆CT=αA(AMAX−AMIN) +αW (WU,MAXMAX− WU.MAXMIN)

(19)

Then the weights can be derived from the indifference condition of Eq. (18), translated to

kSP(USP(PMAX(TO,P=TO,PSP))−USP(PMIN(TO,P=TO,PSP)))=k$(U$(CTMIN)− U$(CTMIN + Ce))

(20)

Eq. (20) and k_SP and k_$ adding up to unity produce Eqs. (21 and 22):

(21)

(22)

3.3. Results and Discussion

The exchanger shown in Fig. (6) is considered, where the objective is to cool a process stream of 0.4 kg/s, from 70°C to 50°C. For this purpose, a utility stream at 20°C is available.

The heat capacities of the streams are C_P,P=C_P,U=4.180 kJ/kg K. With a minimum temperature approach of 30°C, the outlet temperature of the service stream is calculated as T_O,U=40°C and its flow determined to be W_U=0.4 kg/s. The exchanger metal wall thickness and thermal conductivity are ε=0.005 m and k=80 W/m K, while the convective coefficients h_I and h_O are known to be anywhere between 600 and 12'000 W/m²K [41].

The base value of the area is calculated with the values of h_I and h_O at the middle of the range of possibilities (6300 W/m²K), which produces an area A^B=0.423 m². The base value of the maximum capacity of the utility stream is equal to the value calculated by the heat balance, W_U,MAX^B=0.4 kg/s._. By discretizing the range of h_I and h_O into eleven equally spaced points, each one with the same probability, and applying the algorithm in Fig. (5), the cumulative probability distribution of T_O,P for the base design is calculated (Fig. 7). It is observed that the probability of getting the process stream to its set-point temperature is around 36%, while there is certainty of keeping this temperature below 68°C.

The probability distribution in Fig. (8) shows that for the base design, the most likely scenario is that the temperature will be greater than 50°C and below 58°C.

Fig. (9) shows the effect of oversizing the area and maximum utility stream flow capacity on the probability of T_O,P being at its set-point. It is observed that a 300% overdesign in both decisions achieves a probability of 91% of maintaining the outlet stream process temperature at its desired value. Fig. (10) shows the recommended overdesigns given k_SP and γ_A/γ_W values. As k_SP grows (to the right in the figure), maintaining the outlet stream temperature at its set-point becomes increasingly important compared with the equipment cost. Thus, moving to the right causes bigger overdesigns to be prescribed. Moving upward in the chart, the relative cost of heat exchanger area with respect to the cost of the utility stream maximum flow capacity (γ_A/γ_W) increases, so moving in this direction makes the model recommend smaller exchangers and larger provisions for the utility stream maximum flow.

3.4. Excel Macro Interface for Heat Exchanger Sizing

The calculations described in Section 3.2 were coded in Visual Basic, to be run as a macro of Microsoft Excel. While several high-end mathematical software could have been used in developing and solving the model, a choice was made to use Excel in the current work, as it doesn’t require the purchase of a license in addition to that of MS-Office and is widely used by practicing engineers. The user opens an Excel file enabled to run macros and clicks on the “Calculate/show Interface” button (Fig. 11), which displays the main interface shown in Fig. (12). The data for the problem described in Section 3.3 appears as defaults in the relevant fields. Near the bottom left corner, the user can specify the limits of the ranges of thermal convective coefficients. By clicking “Calculate base dimensions”, the base values of exchange area and flow of the utility stream are calculated, using the middle range values of the coefficients.

A dialog box asks the user if the probability plots of the outlet process stream temperature are required. If so, the graphs of Fig. (13) are produced.

Once the base dimensions are calculated, the “Calculations for oversized design” button and the controls to specify oversizing percentages become active, as shown in Fig. (14) for a case in which a 50% overdesign is specified both in area and maximum flow of the utility stream. After clicking on the above-mentioned button, the augmented values appear in their respective boxes, as does the probability of maintaining the process stream at its set-point. A dialog box asks the user if a graph of probabilities for the augmented design is desired, in which case the graphs of Fig. (15) are produced.

Finally, the button “Generate SP probability vs. oversizing chart” button produces a chart similar to Fig. (9), while the button “Generate recommended oversizing vs. preference parameter plot” produces one similar to Fig. (10).