CHEMICAL ENGINEERING TRANSACTIONS VOL. 81, 2020 A publication of The Italian Association of Chemical Engineering Online at www.cetjournal.it Guest Editors: Petar S. Varbanov, Qiuwang Wang, Min Zeng, Panos Seferlis, Ting Ma, Jiří J. Klemeš Copyright © 2020, AIDIC Servizi S.r.l. ISBN 978-88-95608-79-2; ISSN 2283-9216 The Sizing of Plate-Fin Exchangers to Fixed Dimensions Within a Volume Design Region Jorge L. García-Castillo, Martín Picón-Núñez* Department of Chemical Engineering, University of Guanajuato, Noria Alta S/N, Guanajuato, Gto., Mexico picon@ugto.mx This paper shows the development of a design approach for plate and fin heat exchangers to meet fixed dimensions. This approach adopts the concept of volume design region that establishes the limits within which the physical dimensions (length, width and height) of a specific design problem can be set. The design region is determined by minimum and maximum dimensions. The heat exchanger volume is dictated by the problem specifications and the type of secondary surface used on each of the fluids. High density surfaces tend to produce small volumes, while the opposite applies to low density surfaces. In principle, if the heat transfer and friction factor correlations for secondary surfaces are expressed as a function of the geometrical parameters that define the fin density, then it possible to fix the surface density boundaries that give the smallest and largest exchanger volume. The design methodology presented in this work enables to include exchanger dimensions as a design objective along with the heat load and the pressure drop. To achieve these objectives, surface design is a central strategy. In this work, triangular, louvered, rectangular and offset surfaces are used to demonstrate the methodology. 1. Introduction Plate and fin heat exchangers were originally developed for gas to gas applications. However, the new manufacturing techniques have made it possible to construct them in almost any kind of material and geometry (Hathaway et al., 2018), making them suitable for application with liquids and at higher temperatures and pressures (Mortean et al., 2016). In the plate and fin technology, fluids flow through channels separated by metal walls. Between these plates, secondary surfaces are placed to provide structural strength, to increase the heat transfer area, and for heat transfer enhancement. The thermal performance of a plate and fin heat exchanger depends mainly on the thermohydraulic characteristics of the heat transfer surface. A key issue in design, is their appropriate specification. Secondary surfaces tend to produce high heat transfer coefficients and pressure drop at low Reynolds numbers, and for the purposes of design, is of paramount importance the know the way the heat transfer coefficient and the friction factor behave as a function of Reynolds and the fin geometry. To date, a large amount of experimental data for compact surfaces has been published since the first largest collection reported by Kays and London (1984). Since then, studies have extended the availability of experimental data and semiempirical correlations (Rui et al., 2017). Further studies have demonstrated that higher fin densities improve the heat transfer performance of these exchangers (Yang et al., 2017). The design of plate and fin exchangers has been taken the route of optimisation studies seeking to find the design solutions for minimum heat exchanger volume by surface selection (Kunpeng et al., 2015) and by multi-objective optimisation (Khan and Li, 2017). Other authors have recognized the importance of surface selection when space is a limitation (Tao et al., 2017). Recent research developments in plate and fin heat exchangers have centred around the innovative production of new heat transfer surfaces aiming at improving heat transfer and friction performance. Such investigations demonstrate the important role that secondary surfaces play in the performance, size and cost of these devises. The present paper introduces a design approach for plate and fin heat exchangers where block dimensions become an additional design objective. The design approach is based on the engineering of secondary surfaces to meet a specific thermo-hydraulic performance. The work is organized as follows: The design principles for plate and fin exchangers is revised; then the thermohydraulic DOI: 10.3303/CET2081094 Paper Received: 03/04/2020; Revised: 23/05/2020; Accepted: 24/05/2020 Please cite this article as: García-Castillo J.L., Picón-Núñez M., 2020, The Sizing of Plate-Fin Exchangers to Fixed Dimensions Within a Volume Design Region, Chemical Engineering Transactions, 81, 559-564 DOI:10.3303/CET2081094 559 aspects for the design of triangular, rectangular, offset and louvred surfaces are presented. Finally, the design approach is demonstrated on a case study. 2. Design equation The geometry of a plate and fin heat exchanger requires the definition of the exchanger height, width, and length. For each stream, the type of secondary surface employed, and the number of passages must also be specified. The type of surface is a design element that must be fixed at the outset of a design approach. The general expression for the design of a heat exchanger is: 𝑄 = 𝑈 𝐴 𝐹 ∆𝑇𝐿𝑀 (1) Where U is the overall heat transfer coefficient (W/m2K), A is the total surface area (m2), F is the correction factor of the logarithmic mean temperature difference and ΔTLM is the logarithmic mean temperature difference (K). For plate and fin heat exchangers, the total heat transfer area per unit volume is greater compared with other technologies; this feature is referred as area density β (m2/m3); for this reason, is common to express their dimensions as a function of the total exchanger volume, VT (m 3). Similarly, the total surface area for the hot and cold sides may vary significantly with the type of secondary surface used. One way of dealing whit this is by linking the total surface area for each side to the total volume of the heat exchanger (Picón-Núñez et al., 1999). This is represented by the term α, and is calculated as follows: 𝛼𝑖 = 𝐴𝑖 𝑉𝑇 ; 𝑖 = 1,2 (2) Where α (m2/m3) is the ratio to the total surface area of one side of the exchanger to the total exchanger volume (VT). The term i denotes the hot and cold side; for each side, α is calculated from the geometrical characteristics of the type of surface employed as: 𝛼𝑖 = 𝛽𝑖 ( 𝛿1 𝛿1+𝛿2+2𝐹𝑡ℎ ) ; 𝑖 = 1,2 (3) The term β is the area density and relates the surface area on one side of the heat exchanger to the volume on that side, δ is the plate spacing (m) and Fth is the plate thickness (m). The total surface temperature effectiveness of the fin can be determined from (Kays and London, 1984): 𝜂𝑜 = 1 + 𝑓𝑠 { 𝑡𝑎𝑛ℎ[(2ℎ/𝑘𝐹𝑡ℎ ) 1/2(𝛿/2)] [(2ℎ/𝑘𝐹𝑡ℎ ) 1/2(𝛿/2)] − 1} (4) The term fs is the ratio of the secondary surface area to that of the total surface area, for triangular surfaces is expressed as: 𝑓𝑠 = (𝛿 − 𝐹𝑡ℎ )/𝑐𝑜𝑠𝜃 [(𝑎𝑡 − 𝐹𝑡ℎ ) + (𝛿 − 𝐹𝑡ℎ )/𝑐𝑜𝑠𝜃] (5) Where at is half of the base of the triangular fin (m) and θ is the characteristic angle (°). Introducing α into heat transfer expression Eq(1), for a counter current arrangement (F = 1) and free of fouling, the resulting expression is: 𝑉𝑇 = 𝑄 Δ𝑇𝐿𝑀 [ 1 (𝜂𝑜 ℎ𝐴)1 + 1 (𝜂𝑜 ℎ𝐴)2 + 𝑅𝑤 ] (6) Where Rw is the resistance to heat transfer due to the thermal conduction through the metal wall (K/kW). For the design of a plate and fin exchanger, volume is a more precise variable to account for the size of the unit. 3. Surface engineering For a heat exchanger to transmit the required heat load within the limitations imposed by the pressure drop and within a set of desired dimensions, surface geometry becomes a degree of freedom that can be manipulated to simultaneously achieve the three design objectives. Surface engineering is the procedure whereby the surface geometry that meets a specific thermal performance is found. The thermal performance depends on three terms: The heat transfer coefficient (h), the total surface area (A) and the total surface temperature effectiveness (ηo). The assumptions in the development of the approach for surface engineering are: steady state operation, single phase heat transfer process, constant fluid properties, adiabatic operation, negligible longitudinal conduction effects, uniform heat transfer coefficients and uniform flow distribution. Figure 1 shows the main geometrical 560 dimensions that determine the thermo-hydraulic performance of secondaries surfaces such as triangular, rectangular, offset, and louvred. Figure 1: Geometry of secondary surfaces: a) Triangular, b) Rectangular, c) Louvered, d) Offset strip-fin The heat transfer and friction performance of the secondary surfaces are determined from the expressions presented by several authors. The pressure drop due to friction across the core of the heat exchanger is expressed by: Δ𝑃 = 2𝑓𝐿 𝑚2 𝜌𝑑ℎ 𝐴𝑐 2 (7) Where f is the friction factor, ρ is the fluid density (kg/m3), L the flow length (m), m is the mass flow rate (kg/s) and Ac is the free flow area (m 2). For the complete specification of fin surface, dh is the hydraulic diameter (m) and is calculated from the surface parameters as a function of the fin height and fin pitch. 4. Volume design approach The volume design region represents the volume space where a feasible heat exchanger exits. A volume region has minimum and maximum boundaries. These are determined, when the highest surface area density (with the largest number of fins per inch) is used and when the lowest surface area density is used. For a two-stream heat exchanger the total volume is calculated using Eq(6). The type of secondary surface employed in design determines the shape and dimensions of the exchanger. For instance, a high-density surface results in a heat exchanger with a large frontal area and short flow length. With a low-density surface, the resulting exchanger exhibits low frontal area and long flow length. Table 1 shows heat transfer and friction correlation for different types of surfaces. As mentioned above, the relation between heat exchanger dimensions and fin geometry depends on the number of fins that can be accommodated per unit length in the flow direction. A high-density fin is designed when the values of the variables at,ar,ao, and al (Figure 1) take the smallest possible values. This is when the variables approximate the fin thickness: at = al = Fth and ar = ao = 2Fth. A low-density surface is obtained when the number of fins per inch equals 1. In a pure countercurrent arrangement, only one of the streams can fully absorb the pressure drop allocated for design (Picon-Núñez et al., 1999). In this case, the stream chosen to maximise its pressure drop is referred to as the critical stream. In other arrangements such as the crossflow, both streams can fully absorb their pressure drop. For a given surface geometry, the pressure drop of the critical stream will fix the flow length and free flow area. To demonstrate the concept of volume design region, the maximum and minimum volumes are calculated using the same surface type and same surface density on a two-stream problem. The volume design region is calculated for triangular, rectangular, offset strip-fin and louvered surfaces. Figure 2 depicts the volume design region. The exchanger width is a degree of freedom that can be used to produce a design with a specific aspect ratio. 561 Table 1: Correlations for several heat transfer surfaces Expression Range of validity Std dev Notes Rectangular surfaces 𝑗 = 0.233𝑅𝑒 −0.48 ( 𝐹𝑝𝑖𝑡𝑐ℎ 𝛿 ) 0.192 [ 𝐹𝑡ℎ 𝛿 ] −0.208 (8) 2,700 < 𝑅𝑒 < 10,000 ±5.3% (Diani et al., 2012) 𝑓 = 0.029𝑅𝑒−0.09 ( 𝐹𝑝𝑖𝑡𝑐ℎ 𝛿 ) 0.034 [ 𝐹𝑡ℎ 𝛿 ] −0.169 (9) 2,700 < 𝑅𝑒 < 10,000 ±3.4% (Diani et al., 2012) 𝑗 = ℎ𝐴𝑐 𝑚𝐶𝑝 𝑃𝑟 2/3 (10) Triangular surfaces 𝑗 = 0.718𝑅𝑒 −0.625[𝛿 𝐹𝑝𝑖𝑡𝑐ℎ⁄ ] 0.765 [𝐹𝑡ℎ 𝐹𝑝𝑖𝑡𝑐ℎ⁄ ] 0.765 (11) 100 < 𝑅𝑒 < 1,000 ±12% (Chennu, 2018) 𝑗 = 0.789𝑅𝑒 −1.1218[𝛿 𝐹𝑝𝑖𝑡𝑐ℎ⁄ ] 1.235 [𝐹𝑡ℎ 𝐹𝑝𝑖𝑡𝑐ℎ⁄ ] −0.764 (12) 1,000 < 𝑅𝑒 < 10,000 ±12% (Chennu, 2018) 𝑓 = 3.12𝑅𝑒−0.852[𝛿 𝐹𝑝𝑖𝑡𝑐ℎ⁄ ] 0.156 [𝐹𝑡ℎ 𝐹𝑝𝑖𝑡𝑐ℎ⁄ ] −0.184 (13) 100 < 𝑅𝑒 < 1,000 ±11% (Chennu, 2018) 𝑓 = 2.69𝑅𝑒−0.918[𝛿 𝐹𝑝𝑖𝑡𝑐ℎ⁄ ] 0.355 [𝐹𝑡ℎ 𝐹𝑝𝑖𝑡𝑐ℎ⁄ ] −0.175 (14) 1,000 < 𝑅𝑒 < 10,000 ±11% (Chennu, 2018) Offset strip-fin surfaces 𝑗 = 0.6522𝑅𝑒−0.5403 𝜉−0.1541 𝛿0.1499𝜂−0.0678(1 + 5.269𝑥10−5𝑅𝑒1.34𝜉0.504𝛿0.456𝜂−1.055)0.1 (15) 300 < 𝑅𝑒 < 3,500 Rui et al., 2017) 𝜉 = 𝑎𝑜 𝑏𝑜 (16), 𝛿 = 𝐹𝑡ℎ 𝐿𝑓𝑜 (17) 𝜂 = 𝐹𝑡ℎ 𝑎𝑜 (18) 𝑓 = 9.6243𝑅𝑒−0.7422 𝜉−0.1856 𝛿0.3053 𝜂−0.2659(1 + 1.7669𝑥10−8𝑅𝑒4.429𝜉0.92𝛿3.767𝜂0.236)0.1 (19) 300 < 𝑅𝑒 < 3,500 (Rui et al., 2017) 𝑑ℎ = 4𝑎𝑜𝑏𝑜𝐿𝑓𝑜 2(𝑎𝑜𝐿𝑓𝑜+𝑏𝑜𝐿𝑓𝑜+𝐹𝑡ℎ𝑏𝑜)+𝐹𝑡ℎ𝑎𝑜 (20) ln(𝑗) = −0.0264136(𝑙𝑛𝑅𝑒)3 + 0.555843(𝑙𝑛𝑅𝑒)2 − 4.09241𝑙𝑛𝑅𝑒 + 6.21681 ln(𝑓) = 0.132856(𝑙𝑛𝑅𝑒)2 − 2.28042𝑙𝑛𝑅𝑒 + 6.79634 (21) 300 < 𝑅𝑒 < 3,500 300 < 𝑅𝑒 < 3,500 (Rui et al., 2017) 𝑑ℎ = 2𝑎𝑜𝑏𝑜 𝑎𝑜+𝑏𝑜 (23) 𝜉 = 𝑎𝑜 𝑏𝑜 (24) (22) Louvered surfaces. 𝑗 = 𝑅𝑒 [−0.484− 1.887 𝑙𝑛𝑅𝑒 ] [ 𝐹𝑑 𝐿𝑝 ] 0.157 [2.24 − 0.588𝑙𝑛 ( 𝐹𝑝𝑖𝑡𝑐ℎsin𝐿𝛼 𝐿𝑝 )] 𝑓 = 𝑅𝑒−0.433 [ 𝐹𝑑 𝐿𝑝 ] 0.185 (1.10 + 4.31 ( 𝐿𝛼 90 ) 2 + 0.836 ln ( 𝐹𝑝𝑖𝑡𝑐ℎ 𝐿𝑝 ) ( 𝐹𝑝𝑖𝑡𝑐ℎ 𝐿𝑝 ) 2 ) (25) 100 < 𝑅𝑒 < 3,000 (Erbay et al., 2017) 𝑅𝑒 = �̇�𝐿𝑝 𝜇𝐴𝑐 (27) (26) 100 < 𝑅𝑒 < 3,000 (Erbay et al., 2017) Figure 2: Pictorial representation of the volume design region: a) maximum volume, b) minimum volume 5. Case study The case study refers to the design of a two-stream heat exchanger using plate and fin technology. Table 2 presents the operational data and physical properties of a problem taken from the literature (Smith, 1994). The design approach will provide the volume design region where feasible solutions exist. The fin and plate thickness used for the problem are 0.0003 m and 0.002 m, and the plate spacing (δ) is 0.0065 m. 562 Table 2: Operating data and physical properties for case study Flow stream parameters Hot gas Cold air Mass flowrate (kg/s) 24.68 24.32 Pressure drop (Pa) 2,659.6 3,562.9 Inlet Temperature (K) 702.6 448.2 Outlet Temperature (K) 521.3 637.9 Physical properties mean values Prandtl number 0.670 0.670 Cp(J/kg K) 1,084.80 1,051.90 Viscosity (Pa·s) 0.000030 0.000028 Thermal conductivity (W/m K) 0.0488 0.0447 Density (kg/m3) 0.577 5.827 Heat capacity mass flow rate CP (kW/K) 26.78 25.58 Using the operating information in Table 2 as input parameters, an iterative approach is implemented, and the design results are shown in Table 3. For each type of surface, the two columns represent the design using the lowest and highest fin density. As can be seen, for the case of the highest fin density, the louvered fin gives the lowest volume VT = 0.39 m 3 when compared with other geometries; however, it is not the case for the lowest fin density. In this circumstances, the offset strip-fin surface exhibits lower volume. This situation comes about as a result of the louvered fin having similar shape to the triangular surface, and, for a low fin density, the offset strip fin has a larger heat surface area. From the results in Table 3, it can be seen that the triangular surface gives the higher volumes for both conditions, VT,max = 21.81 m 3 and VT,min = 0.61 m 3 . The flow length that corresponds to the minimum volume is L = 0.057 m and the free flow area is 3.37 m2. The flow length for the maximum volume is L = 1.77 m and the free flow area 2.97 m2. It is important to mention that the volume design region is case sensitive. Now, these results can be expressed in a different way: any flow length between L = 0.057 m and 1.77 m and free flow area between 3.37 m2 and 2.97 m2, can be achieved if the fin density is modified accordingly. The fixing of the plate width (W) fixes the plate height (H), the shape of the frontal area (aspect ratio) can be accommodated to desired relative dimensions. For a near 1 aspect ratio, Figure 3 depicts the upper and lower limits for the volume design region of the feasible solutions. Table 3: Volume and block dimensions for the case study. Dimension Triangular Rectangular Offset strip-fin Louvered Fin (fins/in) Fin (fins/in) Fin (fins/in) Fin (fins/in) Fin =1 Fin =28.2 Fin =1 Fin =28.2 Fin =1 Fin =28.2 Fin =1 Fin =28.2 Volume (m3) 21.81 0.61 7.03 0.35 6.24 0.51 9.88 0.39 Width (m) 3.5 3.25 3.40 1.04 2.70 2.8 2.38 3.24 Height (m) 3.52 3.26 3.41 1.05 2.67 2.7 2.37 3.25 Length (m) 1.77 0.06 0.07 0.3151 0.86 0.07 1.75 0.06 Surface area (m2) 3,802 1,319 1,387.7 388.6 1,148 503.4 1,721 854.1 Pressure drop (Pa) 2,660 161.10 2,660 161.10 2,660 127.20 2,660 127.20 2,660 217.70 2,660 156.6 2,660 778.0 2,660 778.0 Figure 3: Volume design region using triangular surfaces: a) 𝐹𝑖𝑛 = 1, b) 𝐹𝑖𝑛 = 28.22, c)Volume design region 563 6. Conclusions This paper has introduced a design methodology for plate and fin heat exchangers where block dimensions are viewed as a new design objective. The main conclusions of this work are: • Secondary heat transfer surfaces are a degree of freedom that can be used to achieve specific dimensions as a design objective. • Surface engineering is a design strategy based on the selection of fin density through which, heat duty, pressure drop, and block dimensions can simultaneously be achieved. • For a given design problem, the volume design region defines the minimum and maximum volume achievable and is case dependent. • Any heat exchanger can be designed within the limits imposed by the volume design region. • One of the limitations of this approach is the range of validity of the generalized expressions for heat transfer and friction factor. This is the case of very low viscosity fluids, that tend to exhibit large Reynolds numbers which go beyond the range of validity of the expressions. Acknowledgements The support of the National Council for Science and Technology of Mexico (CONACYT) for the development of this project is gratefully acknowledged. References Chennu R., 2018, Numerical analysis of compact plate-fin heat exchangers for aerospace applications, International Journal of Numerical Methods for Heat & Fluid Flow, 24, 395-412. Diani A., Mancin S., Rossetto L., 2012, Experimental and numerical analyses of different extended surfaces, Journal of Physics Conference Series, 395, 012045. Erbay L.B., Doğan B., Öztürk M., 2017, Comprehensive study of heat exchangers with louvered fins, In Heat Exchangers Advances Features and Applications, Ed. Mushed S. and Matos-Lopes M., InTechOpen, Rijeka, Croatia. Hathaway B. J., Garde K., Mantell S. C., Davidson J. H., 2018, Design and characterization of an additive manufactured hydraulic oil cooler, International Journal of Heat and Mass Transfer, 117, 188–200. Kays W.M. and London A. L., 1984, Compact Heat Exchangers, Mac Graw Hill, New York, USA. Khan T.A., Li W., 2017, Optimal design of plate-fin heat exchanger by combining multi-objective algorithms, International Journal of Heat and Mass Transfer, 108, 1560–1572. Kunpeng G., Shang N., Smith R., 2015, Optimization of fin selection and thermal design of counter-current plate- fin heat exchangers, Applied Thermal Engineering, 78, 491-499. Mortean M.V.V., Cisterna LHR., Paiva KV., Mantelli MBH., 2016, Development of diffusion welded compact heat exchanger technology, Applied Thermal Engineering, 93, 995–1005. Picón-Núñez M., Polley G.T., Torres-Reyes E., Gallegos-Muñoz A.,1999, Surface selection and design of plate fin heat exchangers, Applied Thermal Engineering, 19, 917-931. Rui S., Mengmeng C., Jianjun L., 2017, A correlation for heat transfer and flow friction characteristics of the offset strip fin heat exchanger, International Journal of Heat and Mass transfer, 115, 695–705. Smith E.M.,1994, Direct thermal sizing of plate and fin heat exchangers, Proceedings of the 10th International Heat Transfer Conference, Brighton, UK, 1994, 55-66. Tao C., Jie W., Wei P., 2017, Flow and heat transfer analyses of a plate-fin heat exchanger in an HTGR, Annals of Nuclear Energy, 108, 316–328. Yang Y., Li Y., Si B., Zheng J., 2017, Heat transfer performances of cryogenic fluids in offset strip fin-channels considering the effect of fin efficiency, International Journal of Heat and Mass Transfer, 114, 1114–1125. 564