Modeling and Second Law Based Optimization of Plate Fin and Tube Heat Exchanger Using MOPSO

In the present study, a comprehensive thermal modeling and optimal design of plain Fin-and-Tube Heat Exchanger (FTHE) is performed. Hence, method is applied to estimate the heat exchanger pressure drop and effectiveness. The design parameters of this scientific study are selected as: longitudinal pitch, transversal pitch, fin pitch, number of tube pass, tube diameter, cold stream flow length, no-flow length and hot stream flow length. In addition, Multi Objective Particle Swarm Optimization (MOPSO) is applied to obtain the minimum number of entropy generation units and total annual cost (sum of investment and operation costs) as two objective functions, simultaneously. The results of optimal designs are a set of multiple optimum solutions, called ‘Pareto optimal solutions’. It reveals that any geometrical changes which decrease the number of entropy generation units lead to an increase in the total annual cost and vice versa. Moreover, for prediction of the optimal design of the FTHE, an equation for number of entropy generation units versus the total annual cost is derived for the Pareto front. Furthermore, the sensitivity analysis of change in optimum number of entropy generation units and total annual cost with changes in design parameters of the fin tube heat exchanger is also performed in detail. n exponent of nonlinear increase with area increase Nu nusselt number (-) NP number of tube passes (-) Nr number of tube row (-) NS number of entropy generation units Nt total number of tube (-) Nl number of plate in L1 direction (-) NTU number of transfer units (-) Pr Prandtl number (-) r interest rate (-) Re Reynolds number (-) St Stanton number (-) S rate of entropy generation (W/K) U overall heat transfer coefficient (W/M2 k) V heat exchanger volume (m3) Vt volumetric flow rate (m 3/s) Xl longitudinal pitch(m) Journal of Applied Mechanical Engineering J o u r n al o f A pp lied ical Eninee r i n g


Introduction
Compact heat exchangers are characterized by a large heat transfer surface area per unit volume of the exchanger. Fin Tube Heat Exchanger (FTHE) as shown in Figure 1, is a typical compact heat exchanger that is widely used in many industrial power generation plants, chemical, petrochemical, and petroleum industries. Fins or extended surface elements are introduced to increase the heat transfer area [1]. Some of commonly used fins are, wavy, offset strip, louver, perforated, and plain fins [2]. Recently, evolutionary algorithms have received a lot of attention as an optimization method particularly in heat exchanger design. Sanaye and Hajabdollahi applied NSGA-II and optimized the rotary regenerator, plate fin and shell and tube heat exchangers [3][4][5]. Dincer et al. obtained optimum cooling water temperature during condensation of saturated water vapor within a shell and tube condenser [6]. Hilbert et al. also, used a multi objective optimization technique to maximize the heat transfer rate and to minimize the pressure drop in a tube bank heat exchanger [7]. Xie et al. minimized the total volume as well as the total annual cost of a compact heat exchanger by considering three shape parameters as decision variables [8]. Wang et al. applied genetic algorithm to optimize primary energy saving, annual total cost saving, and carbon dioxide emission reduction [9]. Guo et al. employed genetic algorithm to optimize the field synergy number which is defined as the indicator of the synergy between the velocity field and the heat flow [10]. Sahin et al. optimized the design parameters of a heat exchanger with rectangular fins by Taguchi experimental-design method [11]. Doodman et al. minimized the total annual cost of air cooled heat exchangers using global sensitivity analysis [12]. Foli et al. estimated the optimum geometric parameters of micro channels in micro heat exchangers by maximizing the heat transfer rate and minimizing the pressure drop as two objective functions [13]. Liu and Cheng optimized a recuperator for the maximum heat transfer effectiveness as well as minimum exchanger weight and pressure loss [14]. Gholap and Khan also studied air cooled heat exchangers by minimizing the energy consumption of fans and material cost as two objective functions [15].
In this paper, it is aimed to first conduct a thermal modeling of a FTHE and later optimize this equipment by minimizing the number of entropy generation units as well as minimizing the total annual cost, simultaneously. Particle swarm optimization technique is applied to provide a set of the Pareto multiple optimum solutions. The sensitivity analysis of change in optimum values of number of entropy generation units and total annual cost with change in design parameters was performed and the results are reported. In summary, the following are the specific contributions of this paper to the area:

Thermal modeling of a plain type fin tube heat exchanger
Applying multi-objective optimization for fin and tube heat exchanger with number of entropy generation units and total annual cost as two objectives using particle swarm optimization algorithm.
Selecting the longitudinal pitch, transversal pitch, fin pitch, number of tube pass, tube diameter, cold stream flow length, no-flow length and hot stream flow length as design parameters (decision variables).
Proposing a closed form equation for the total annual cost in term of number of entropy generation units at the optimal design point.
Performing sensitivity analysis of change in objective functions when the optimum design parameters vary and find the degree of each parameter on objective functions conflicting.

Thermal modeling
Cold stream Hot stream Figure 1: Typical plain fin and tube heat exchanger with staggered tube arrangement.
The NTU ε − method is applied here for predicting the heat exchanger performance. The effectiveness of cross-flow heat exchanger with both fluids unmixed is proposed as [16]: Where is the modified Bessel function. Number of Transfer Units (NTU) and heat capacity ratio * ( ) C are defined as follows [1]: Here, o U is the overall heat transfer coefficient based on fin side, computed from: The fouling resistance and wall conduction resistance have been assumed to be negligible. Hear the , i c A is the total heat transfer area in inside of tube and , tot h A is outside total heat transfer surface area including fins and tubes as follow: , , where f A and Also s η in Equation 4 is overall surface efficiency defined as [1]: and f η is efficiency of a single fin.
The Fanning factor f and Colburn factor j (respectively representative of pressure drop and thermal performance) defined in Equations (9) and (10) by Wang et al [18] for plain flat fins on staggered tube banks as:  The Reynolds number and hydraulic diameter are defined as [18]: and G is mass flux.
The heat transfer coefficient is defined as follow [1]: where St is Stanton number defined as: Furthermore, the pressure drop was also estimated from [1]: where flow A minimum free-flow area on the tube outside.
The number of entropy generation units is defined as follows [19]: where . S is the rate of entropy generation: and: . ln ln

Influence of longitudinal heat conduction
To evaluate the influence of heat conduction in the flow direction either in the solid wall or in the fluid, the following approximation for longitudinal conduction analysis has been also performed. Fluids where t N and l N are the total tube number and the number of plate in L 1 direction. h is the heat transfer coefficient, δ is the fin thickness and L 1 , L 2 and L 3 are cold stream flow length, hot stream flow length and no-flow length, respectively ( Figure 1).
generally have a low thermal conductivity (liquid metal excepted), but the wall conductivity may be quite high. Consequently, only wallconduction effects will be considered in the following treatment. The influence of longitudinal conduction is to reduce exchanger effectiveness for a given number of transfer units, and this reduction may be quiet serious in exchangers with short flow lengths design for high effectiveness values ( 90% ε > ) [1]. Assuming the temperature difference T δ for the hot fluid, is of the same magnitude for the cold fluid (C min /C max ≈ 1), and also for the wall then the wall temperature gradient is T / L δ , with L being the flow length and the longitudinal heat transfer by conduction is of the order [1]. If the wall cross-section area for longitudinal conduction is designated k A : where A k , is the wall cross-section area for longitudinal conduction and k w , is the wall conduction coefficient.
The convection heat transfer rate is also given by energy-balance considerations as: and then: Velocity and position updating: The movement of the particle towards the optimum solution is governed by updating its position and velocity attributes. The velocity and position update equations are given as [20]: where i represents the ith particle in the population and k represents each iteration.
The term g k p specifies the best global solution obtained by the swarm and i p is the best position of each particle over time. In addition w, c 1 and c 2 are the inertia weight, self confidence factor and swarm confidence factor respectively and rand is the random number generated uniformly in the range [0-1]. Best position of each particle ( i p ): In MOPSOA the individual experience of the particle is captured, that corresponds to the best performance attained so far by it in its flight. Actually the present solution is compared with the previous solutions, and it replaces the latter only if it dominates that solution. If neither of them is dominated by the other, then one of them is selected randomly [21]. p solution is done from the archive on the basis of the diversity of the solutions. In MOPSOA the diversity measurement has been done using a concept similar to the crowding-distance measure in [24].

Best global position of swarm (
Crowding distance: The standard crowding distance proposed by Deb [24] is utilized, where the crowding distance of an individual is the perimeter of the rectangle with its nearest neighbors at diagonally opposite corners. So, if two individuals have the same rank, each one who has a larger crowding distance is better.
Historical Archive: The MOPSOA algorithm has been modified to include an archive of the historically non-dominated individuals. Archive is used to update the data at each iteration.

Objective Functions, Design Parameters and Constraints
In this study, number of entropy generation units (defined in Equation 17) and total annual cost are considered as two objective functions. Total annual cost is includes investment cost (the annualized cost of the heat transfer surface area) and operating cost of compressor/ pump to flow the fluid as follows [8]: Here C A and K el are the heat exchanger investment cost per unit surface area and the electricity unit cost respectively, n is a constant and

Particle Swarm for multi-objective optimization
Multi-objective optimization: A multi-objective problem consists of optimizing (i.e. minimizing or maximizing) several objectives simultaneously with a number of inequality or equality constraints. The problem can be formally written as follows: where x is a vector containing N param design parameters, ( ) 1,....,  [20]. Like evolutionary algorithms PSO is also a population based heuristic, where the population of the potential solutions is called a swarm and each individual solution within the swarm, is called a particle. There have been several recent attempts to use PSO for Multi-Objective Optimization (MOO) [21][22][23]. This algorithm is called Multi-Objective Particle Swarm Optimization Algorithm (MOPSOA) which is coupled with the objective functions developed in this study for optimization.

Case Study
In this part, our case study has been considered. Outlet gases from the compressor with mass flow rate 2.5 kg/s and temperature 425 K enters the fin tube pre-cooler heat exchanger. On the other hand, water with 3.2 kg/s and temperature 285 K flow in the other side. The FTHE metal is made of stainless steel with thermal conductivity about k w =25 W/m.K. Also, the ratio of inside to outside tube diameter is taken to be 0.8. Operating conditions and the cost function constant values are listed in Table 1. The thermophysical properties of air such as Prandtl number, viscosity and specific heat were considered as temperature dependent.

Longitudinal heat conduction
To quantify the effect of longitudinal heat conduction in comparison with the convection heat transfer, the numerical values of λ (equation 22) were computed for k w =25 W/m.k for all optimum design cases as is shown in Figure 2. It was found that the distribution of λ value in the whole optimal output domain show the numerical values less than 0.005 (0.5%). This shows that assuming the negligible conduction heat transfer in our analysis in flow direction is acceptable for the studied problem.

Verification of modeling and optimization results
To verify the modeling results, the simulation output were compared with the corresponding reported results given in reference [1]. The comparison of our modeling results and the corresponding values from reference [1], for the same input values listed in Table 2, are given in Table 3. The results show that the difference percentage points of two mentioned modeling output results are acceptable.

Optimization results
To minimize the number of entropy generation units and the total annual cost, eight design parameters including longitudinal pitch, transversal pitch, fin pitch, number of tube pass, tube diameter, cold stream flow length, no-flow length and hot stream flow length were selected. Design parameters (decision variables) and the range of their variations are listed in Table 4. The number of iterations for finding the global extremum in the whole searching domain was about10 25 . System was optimized for depreciation time Compressor or pump efficiency 0.65      particle swarm optimization was performed for 500 iterations, using 100 particles, inertia weight of w=0.4, self confidence factor of c 1 =1.75 and swarm confidence factor c 2 =1.8. The results for Pareto-optimal front are shown in Figure 3, which clearly reveal the conflict between two objectives, the number of entropy generation units and the total annual cost. Any geometrical change that decreases the number of entropy generation units, leads to an increase in the total annual cost and vice versa. This shows the need for multi-objective optimization techniques in optimal design of a FTHE. It is shown in Figure 3, that the minimum number of entropy generation units exists at design point A (0.001694), while the total annual cost is the biggest at this point. On the other hand the minimum total annual cost occurs at design point (193.3 $/year), with a biggest number of entropy generation units value (0.002533) at that point. Design point A is the optimal situation at which, number of entropy generation units is a single objective function while design point E is the optimum condition at which total annual cost is a single objective function.
The optimum values of two objectives for five typical points from A to E (Pareto-optimal fronts) for input values are given in table 1 are  listed in table 5.
To provide a useful tool for the optimal design of the FTHE, the following equation for number of entropy generation units versus the total annual cost was derived for the Pareto curve ( Figure 3).
which is valid in the range of 0.001694<NS<0.002533 for number of entropy generation units. The interesting point is that considering a numerical value for the number of entropy generation units in mentioned range, provides the minimum total annual cost for that optimal point along with other optimal design parameters.
The selection of a single optimum point from existing points on the Pareto front needs a process of decision-making. In fact, this process is mostly carried out based on engineering experiences and importance of each objective for decision makers. The process of final decisionmaking in Figure 3 is usually performed with the aid of a hypothetical point named as equilibrium point, that both objectives have their optimal values independent of the other objectives [25]. It is clear that it is impossible to have both objectives at their optimum point, simultaneously. The equilibrium point is not a solution located on the Pareto frontier. In this paper, LINMAP method was used to find the final optimum solution in Pareto front [25].
In the LINMAP method, each objective is non-dimensionalized using the following relation: ii. The other design parameters are almost constant for all optimum cases.
Since the optimum values of two decision variables (cold stream flow length and no flow length) have scattered distribution in their lower domains, one may predict that these two parameters have important effects on the conflict between the number of entropy generation units and total annual cost. The other six design parameters have no effect on the conflict between the two objective functions and improve both objective functions at a specific value.  Figure 5a, it is found that by increase in the cold stream flow length, number of entropy generation units is increased continuously and annual cost decreases. Therefore, this increment leads to conflicting behavior on objective functions. The relatively scattering distribution of optimal points in Figure 4a confirms this idea.

Hot stream flow length:
As it is shown in Figure 5b, increasing the hot stream flow length results in increase in the both number of entropy generation units as well as total annual cost. Therefore, both objectives improve by decreases of hot stream flow length. The numerical value of hot stream flow length located at its minimum value in Figure 4b verified this point.      Figure 5g, by increase of the tube passes, both the number of entropy generation units and annual cost decrease, simultaneously. Therefore, the higher value of this parameter is suitable until it does not violence the constraints. Hence, number 3 is selected for tube passes as shown in Figure 4g.

Number of tube pass: Based on
Tube diameter: As shown in Figure 5h by increasing the tube diameter, both annual cost and number of entropy generation units decrease (except in a small region for number of entropy generation units). Selecting the maximum value for the tube diameter in Figure 4h confirms this point.

Conclusions
In the present paper, thermal modeling of the plain fin and tube heat exchanger has been carried out. For this task, a fin tube heat exchanger is optimally designed using multi objective optimization technique. The design parameters (decision variables) are longitudinal pitch, transversal pitch, fin pitch, number of tube pass, tube diameter, cold stream flow length, no-flow length and hot stream flow length. In the presented optimization problem, the number of entropy generation units and total annual cost were two objective functions (both of them were minimized, simultaneously). A set of Pareto optimal front points were shown. Further, the distribution of each design parameters in their allowable range is shown. The results revealed the level of conflict between the two objectives. Furthermore the correlation between the optimal values of two objective functions is proposed. It is found that each point in the Pareto frontier would be a final optimized point which strongly depends on the decision maker. However, a decision making method based on LINMAP method was presented to specify the final optimum design. Finally, the cold stream flow length and no flow length are found to be important design parameters which caused a relative conflict between number of entropy generation units and the total annual cost. The results show that by decrease of hot stream flow length and by increase of fin pitch, transversal pitch, longitudinal pitch, number of tube pass and tube diameter, both the number of entropy generation units and annual cost improved, simultaneously.