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Advances in Information Science and Applications - Volume I A method for optimization of plate heat exchanger Václav Dvořák a method for configuration optimization. Fábio et al. in work [3] presented an algorithm for the optimization of heat exchange area of plate heat exchangers. The algorithm was based on the screening method. For each kind of plate, subject to certain constraints, optimal configurations were found which presented the smallest area. Each of these found configurations had local optima characteristics. Similarly Arsenyeva et al. [4] discussed the developments in design theory of plate heat exchangers, as a tool to increase heat recovery and efficiency of energy usage. The optimal design of a multi-pass plate-and-frame heat exchanger with mixed grouping of plates was considered. The optimizing variables included the number of passes for both streams, the numbers of plates with different corrugation geometries in each pass, and the plate type and size. The mathematical model of a plate heat exchanger was developed to estimate the value of the objective function in a space of optimizing variables. To account for the multi-pass arrangement, the heat exchanger was presented as a number of plate packs with coand counter-current directions of streams, for which the system of algebraic equations in matrix form was readily obtainable. The exponents and coefficients in formulas to calculate the heat transfer coefficients and friction factors were used as model parameters to account for the thermal and hydraulic performance of channels between plates with different geometrical forms of corrugations. These parameters were reported for a number of industrially manufactured plates. The described approach was implemented in software for plate heat exchangers calculation. In another work Gut et al. [5] presented a screening method for selecting optimal configurations for plate heat exchangers. The optimization problem was formulated as the minimization of the heat transfer area, subject to constraints on the number of channels, pressure drops, flow velocities and thermal effectiveness, as well as the exchanger thermal and hydraulic models. The configuration was defined by six parameters, which are as follows: number of channels, numbers of passes on each side, fluid locations, feed relative location and type of channel flow. The proposed method relied on a structured search procedure where the constraints were successively applied to eliminate infeasible and sub-optimal solutions. The method can be also used for enumerating the feasible region of the problem; thus any objective function can be used. Examples showed that the screening method was able to Abstract— Research of devices for heat recovery are currently focused on increasing the temperature and heat efficiency of plate heat exchangers. The goal of optimization is not only to increase the heat transfer or even moisture but also reduce the pressure loss and possibly material costs. During the optimization of plate heat exchangers using CFD, we are struggling with the problem of how to create a quality computational mesh inside complex and irregular channels. These channels are formed by combining individual plates or blades that are shaped by molding, vacuum forming, or similar technology. Creating computational mesh from the bottom up manually is time consuming and does not help later optimization. The paper presents a method of creating meshes based on dynamic mesh method provided by software Fluent. Creating of mesh by pulling is similar to the own production process, i.e. it is perpendicular to the plates. The advantages of this method are: The ability to change quickly the whole geometry of the plate, possibility to use optimization algorithms, ability to control the size of the wall adjacent cells and similarity of meshes even in completely different geometries. The paper discusses the problems with very narrow gaps and distortions of the mesh. Using this method, a row of cases with waves and ridges were created. The resulting dependence of efficiency and pressure loss on the ridges count are replaced by mathematical relationships. An objective function is suggested and verified to optimize heat transfer surface of the exchanger. Keywords—Dynamic mesh, heat exchanger, optimization. T I. INTRODUCTION he development of recuperative heat exchangers in recent years focused on increasing efficiency. Another challenge is the development of so-called enthalpy exchangers for simultaneous heat and moisture transport, i.e. transport of both sensible and latent heat, as presented by Vít et al. in work [1]. A lot of others researchers dealt with design and optimization of plate heat exchangers. For example Gut et al. [2] developed a mathematical model in algorithmic form for the steady simulation of plate heat exchangers with generalized configurations. The configuration is defined by the number of channels, number of passes at each side, fluid locations, feed connection locations and type of channel-flow. The main purposes of this model were to study the configuration influence on the exchanger performance and to further develop Author gratefully acknowledges financial support by Czech Technological Agency under the project TACR TA01020313. V. Dvořák is with the Technical university of Liberec, Faculty of mechanical engineering, Department of Power Engineering Equipment. Address: Studentska 2, 46117, Liberec. Phone: +420 485 353 479; e-mail: [email protected]. ISBN: 978-1-61804-236-1 193 Advances in Information Science and Applications - Volume I successfully determine the set of optimal configurations with a reduced number of exchanger evaluations. However the optimal design of plates itself is not under investigation in these mentioned studies, while several researchers tried to optimize even the shape of heat exchanger area. E.g. multi-objective optimization of a cross-flow plate fin heat exchanger (PFHE) by means of an entropy generation minimization technique was described by Babaelahi et al. in study [6]. Entropy generation in the PFHE was separated into thermal and pressure entropy generation as two objective functions to be minimized simultaneously. The Pareto optimal frontier was obtained and a final optimal solution was selected. By implementing a decision-making method, here the LINMAP method, the best trade-off was achieved between thermal efficiency and pumping cost. This approach led to a configuration of the PFHE with lower magnitude of entropy generation, reduced pressure drop and pumping power, and lower operating and total cost in comparison to singleobjective optimization approaches. Kanaris et al. [7] suggested a general method for the optimal design of a plate heat exchanger (PHE) with undulated surfaces that complies with the principles of sustainability. They employed previously validated CFD code to predict the heat transfer rate and pressure drop in this type of equipment. The computational model was a three-dimensional narrow channel with angled triangular undulations in a herringbone pattern, whose blockage ratio, channel aspect ratio, corrugation aspect ratio, angle of attack and Reynolds number are used as design variables. An objective function that linearly combines heat transfer augmentation with friction losses, using a weighting factor that accounts for the cost of energy, was employed for the optimization procedure. New correlations were provided for predicting Nusselt number and friction factor in such PHEs. The authors stated that the results were in very good agreement with published data. Han et al. in article [8] numerically investigated the thermalhydrodynamic characteristics of turbulent flow in chevron-type plate heat exchangers with sinusoidal-shaped corrugations. The computational domain contained a corrugation channel, and the simulations adopted the shear-stress transport κ-ω model as the turbulence model. The numerical simulation results in terms of Nusselt number and friction factor were compared with limited experimental data and existing correlations in order to verify the accuracy of the numerical model. The corrugation depth, corrugation angle, corrugation pitch, and fluid inlet velocity were identified as design variables, and 200 samples were selected using the maximum entropy design method to build the metamodel for obtaining the heat transfer coefficient as well as the pressure drop per unit length. A multi-objective genetic algorithm was utilized as the optimizer. The optimization results were presented in the form of Pareto solution set, which clearly showed its dominance over the entire design space and the tradeoff between the two optimization objectives: maximizing heat ISBN: 978-1-61804-236-1 transfer coefficient and minimizing drop per unit length. Also, the Pareto optimal designs were validated against the values directly obtained from numerical simulations. The approximation-assisted optimization shows that all optimal designs have largest enlargement factor values inside the design space, and the optimal corrugation angle increases with the increase of maximum heat transfer coefficient. This work is motivated by the need to optimize heat transfer area of plate heat recovery heat exchangers. Heat exchangers are assembled from plates. Plates are made of metal, paper or plastic and are shaped by press molding. Ridges and grooves are supposed to increase the heat (and mass) transfer and also determine the plate pitch and carry the heat exchanger. To optimize a heat exchanger, we have to create a model and a computational mesh and use computational fluid dynamic (CFD) software. By assembling the heat exchanger, complicated and irregular narrow channels are created. Disadvantages of repeated generation of computational meshes are: It is slow, meshes made in different models are not similar and parameterization of the model is problematic. Further, even a small change of geometry requires to go through the whole process of model creation and mesh generation again. As a result, there is high probability of creation errors of model and low quality of mesh cells. It is necessary to setup the solver, boundary conditions and all models for all computed variants. Furthermore meshes are not similar, i.e. the size, shape, height of wall adjacent cells are not the same for different topologies. E.g. Novosád in work [9], investigated the influence of oblique waves on the heat transfer surface. The biggest problem in this work was the creation of custom geometry. Each option had to be modeled separately and a meshed. Each model had to be loaded into the solver, set the boundary conditions and subsequently evaluated by calculation. Therefore, a new method for generation computational variants was developed and is presented in this work. II. METHODS A. Method for rapid generation of computational model In this paper, we discuss the case of a counter flow heat exchanger, which has symmetrical heat transfer area. Processes in such heat exchanger can be investigated by modeling the flow around only one plate using symmetrical boundary conditions. How such a model appears can be seen from Fig. 1. The heat transfer surface is divided into two parts. Input and output portions (reported as wall) is fixed, and serve to develop the velocity profiles before the central portion (main wall) which will be deformed. Input boundary conditions are specified by mass flow rates (mass flow inlets), the output boundary conditions are specified as pressure outlets with static pressure 0 Pa. In this study, we used turbulence model SST κ-ω, medium was air considered as ideal gas. As a results, we obtained pressure, velocity, turbulence and temperature fields inside the computational domain for average inlet velocity of air 3 m/s. 194 Advances in Information Science and Applications - Volume I method for model generation, we calculated and evaluated two successions of geometries, each with a pitch plates of δ = 3 (mm). In the first case, the geometry of corrugation was defined as waves by function y −y π ⋅ n , z = cos 0 2 ⋅ y 0 (1) where is relative vertical coordinate of the wall, (m) is width of the model, (m) lateral coordinate and number of waves. In the second case, the geometry of corrugation was defined as ridges by function Fig. 1 Model of heat exchange surface of counter flow recuperative heat exchanger. y −y z = arcsin − cos 0 π ⋅ n y 2 ⋅ 0 The new method is based on dynamic mesh method provided by software Fluent. First a simple model with straight wall, see Fig. 1 is created. This model is fully functional, i.e. read into Fluent, boundary conditions are set and it is possible to calculate the flow and heat transfer. In a subsequent step, the computational mesh is deformed using the so-called UDF (User defined functions). Actually, individual nodes of computational mesh are manipulated with. The result is a transformation of the mesh, see Fig. 2 The figure also shows that during the deformation, it is possible to change the spacing of the plates, to create dents or corrugations and also define the size of the cells adjacent to the wall. (2 ⋅ π ) , (2) were created on the tops while tabs of the width of the ridges, as it is obvious from Fig. 3. B. Theory of counter flow heat exchangers Most of the recuperative heat exchangers in air conditioning works in the isobaric mode, where mass flow rates of warm and cold air are equal, i.e. m c = m h . Assuming equality of specific heat capacities, c p c = c p h , we can write the coefficient of efficiency as η= th i − th o t h i − tc i ⋅ 100 (%) , (3) Where t h i (℃) is the inlet temperature of hot air. Furthermore index c denotes a cold stream, index i inlet into the heat exchanger and index o the outlet of the heat exchanger. For the pressure drop assessment, it is used local loss coefficient ξ . It is the ratio of total pressure los between the Fig. 2 Computational mesh after deforming. It is obvious that, as for commonly created mesh, the refinement of the mesh defines minimal size of geometrical details which can be described by the mesh. It can be seen from Fig. 3 that shapes of peaks and valleys of each ridge vary according the grid and ridge spacing. inlet and outlet ∆p and dynamic pressure pd , i.e. ξ= ∆p p0 i − po = pd pd (4) where ∆p is the pressure difference between average total pressure in the pressure inlets a) pressure at the pressure outlets p0 i (Pa) and average static po (Pa). The dependence between the heat balance and efficiency η is expressed as b) Fig. 3 Cut of the computational mesh, a) – initial mesh, b) – mesh after deforming and ridges creating. ⋅ cp ⋅ k⋅A=m For this initial study, no direct optimization method were used to control the deforming of the surface. Using mentioned ISBN: 978-1-61804-236-1 η 1 −η where Pkg/s) is the mass flow rate, 195 (5) c p (J/(kg·K)) is isobaric Advances in Information Science and Applications - Volume I specific heat capacity, k (W/(m2·K)) heat transfer coefficient and A (m2) is the area of heat transfer surface. III. RESULTS The results of computations of both cases are plotted in Fig. 4. It is clearly evident from the figure that with the increasing number of waves in the model the efficiency of the heat exchanger increases. The course for both designs of the heat exchanger are similar, but it is evident for smaller count of waves that wall with ridges has higher efficiency than wall with waves. The course of the pressure loss in contrast, is such that for the first few waves, the pressure loss decreases, reaches a minimum for n = 3 ÷ 3.5 and then begins to grow. For comparison, the diagram also shows the results plotted for the wave number n = 0, i.e. for the flat plate. There is clearly apparent advantages of such an arrangement. Variant with waves or ridges needs at least n = 15 to achieve the same efficiency, but the pressure loss is higher comparing to the flat plates. Fig. 5 Surface heat transfer coefficient – pressure loss coefficient for all calculated variants. Pressure loss coefficient ξ1 similarly counted the pressure loss only in the deformed part of the heat exchanger surface and is calculated from the relationship ξ1 = ξ − ξ 0 L0 , L0 + L1 (7) Where ξ (1) is the total pressure loss coefficient of the case and ξ 0 (1) is the total loss coefficient for the case of flat plates. Fig. 4 Diagram efficiency – pressure loss for all variants. The same results are also plotted in a diagram in Fig. 5, where the heat transfer is represented by heat transfer coefficient k1 (W/(m2·K)), which takes into account the heat transfer only in the middle deformed part of the exchanger model and is calculated from the relationship ⋅ cp ⋅ B ⋅ (k1 ⋅ L1 + k 0 ⋅ L0 ) = m η , 1 −η (6) Fig. 6 Dependency of heat exchanger efficiency on count of ridges. where k0 (W/(m2·K)) is the heat transfer coefficient of a flat plate, B (m) is the width of the model, L0 and L1 (m) are Generally, as we can see from Fig. 4 and Fig. 5, the higher efficiency is obtained for cases with higher pressure loss. However, in practice, pressure loss and efficiency are required when optimizing the heat exchanger. Usually, the maximal pressure loss is specified and for that given pressure restrain, a solution with maximal efficiency is sought. We used obtained dependence of efficiency on count of ridges, which is lengths of flat and deformed parts of the wall respectively, see Fig. 1. Let us mention yet that the heat transfer coefficient is evaluated without respect for the increase in heat transfer area heat caused by deforming. This approach is fully in line with the practice, for which the area of used material is critical. ISBN: 978-1-61804-236-1 196 Advances in Information Science and Applications - Volume I presented in Fig. 6, while the dependency of pressure loss is in Fig. 7. function for required values of pressure loss of the heat exchanger is in Fig. 8. Fig. 7 Dependency of pressure loss of the heat exchanger on count of ridges. Fig. 9 Course of objective function during optimization for required pressure loss ∆p R . The courses of objective function during optimization for given required pressure loss, while a simple gradient method was used, are in Fig. 9. IV. CONCLUSIONS Method has been developed for the rapid generation of computational models. The method simulates the forming process, wherein the heat exchanger plates are made, and allows arbitrarily shaped heat exchanging surface. Among its advantages are the ability to change the pitch of heat exchanger plates and control the size of the cell adjacent to the wall. Another advantage is the similarity of calculating variant and the ability to use the method in the optimization. It is also advantageous to use obtained calculated data for initialization of the next computed variant. The method was tested on calculation of dependencies of efficiency and pressure drop for two variants of the corrugation of the heat exchange surfaces of heat exchanger. The flow and heat transfer in the heat exchanger with symmetrical heat exchange surface were studied. For forming walls of the heat exchanger, waves and ridges, which proved to be more advantageous, were used. The obtained dependencies were replaced by polynomial functions and an objective function was designed for possible optimization. The function is based on the practice that maximum efficiency of the heat exchanger is required for a given pressure drop constraints. Suitable constants of the function were found and the functions were tested by optimization using simple gradient methods. Only one parameter – number of ridges – was optimized. The method for rapid generation of computational models and suggested objective function will be used in further work to optimize the shape of the heat transfer surface dependent on more than one parameter. Fig. 8 Course of objective function for various required pressure loss ∆p R . Because in practice we optimize for maximum efficiency for a given pressure drop, was designed a target function as ∆p − ∆p R F = η − cη cE ⋅ ∆p R N , (8) where ∆p R (Pa) is the reference or required pressure loss, cE (1) is maximal relative deviation of the required pressure drop ∆p R , N is exponent and cη is penalization of the objective function if the actual pressure ∆p loss differs from required. The shape of the objective function was optimized, and it was found that the optimum parameters for the fastest convergence are N = 1, cη = 2% and cE = 0.01 . The course of the target ISBN: 978-1-61804-236-1 197 Advances in Information Science and Applications - Volume I REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] T. Vít., P. Novotný, Nguyen Vu, V. Dvořák, “Testing method of materials for enthalpy wheels,” Recent Advances in Energy, Environment, Economics and Technological Innovation, Paris, France, 29th – 31st October, 2013. Jorge A. W. Gut, José M. Pinto, “Modeling of plate heat exchangers with generalized configurations,” International Journal of Heat and Mass Transfer 46, no. 14, 2003, pp. 2571 - 2585. Fábio A. S. Mota, Mauro A. S. S. Ravagnani, E. P. Carvalho, “Optimal design of plate heat exchangers,” Applied Thermal Engineering, Volume 63, Issue 1, 5 February 2014, pp. 33–39. Olga P. Arsenyeva, Leonid L. Tovazhnyansky, Petro O. Kapustenko, Gennadiy L. Khavin, “Optimal design of plate-and-frame heat exchangers for efficient heat recovery in process industries,” Energy Volume 36, Issue 8, August 2011, pp. 4588–4598 Jorge A. W. Gut, José M. Pinto, “Optimal configuration design for plate heat exchangers,” International Journal of Heat and Mass Transfer 47, no. 22, 2004, pp. 4833 - 4848. M. Babaelahi, S. Sadri, H. Sayyaadi, “Multi-Objective Optimization of a Cross-Flow Plate Heat Exchanger Using Entropy Generation Minimization,” Chemical Engineering & Technology, Volume 37, Issue 1, January, 2014, pp. 87–94 A. G. Kanaris, A. A. Mouza, S. V. Paras, “Optimal design of a plate heat exchanger with undulated surfaces,” International Journal of Thermal Sciences 48, no. 6, 2009, pp. 1184 - 1195. W. Han, K. Saleh, V. Aute, G. Ding, Y. Hwang, R. Radermacher, “Numerical simulation and optimization of single-phase turbulent flow in chevron-type plate heat exchanger with sinusoidal corrugations,” HVAC&R Research 17, 2011, pp. 186 - 197. J. Novosád, V. Dvořák, “Investigation of effect of oblique ridges on heat transfer in plate heat exchangers, Experimental Fluid Mechanics 2013, November 19.-22., 2013, pp. 510 - 514. ISBN: 978-1-61804-236-1 198