Effect of Pump Performance Curves and Geometric Characteristics of Offset Fins on Heat Exchanger Design Optimization

Abstract

This study analyzes several design cases to identify the optimum geometric structure of the offset fin, determined by two design factors: the number of fins in the flow direction and the number of fins in the vertical direction. Increasing the number of fins in the vertical direction has relatively minor effects on the heat transfer rate and surface area. In contrast, adding more fins in the flow direction results in enhancement of thermal performance. Correlations for the Colburn j factor and the Fanning f factor, incorporating N_fin,v and N_fin,f, are established. The resistance curve of each case is yield based on the f factor correlation, and the heat transfer rate at the actual coolant flow rate is obtained the j factor correlation. A difference in the pressure drop resulted in a different coolant flow rate despite using the same circulation pump, showing a difference of 1.67 times between the minimum and maximum values. Although a different coolant circulates in each case, there was no reversal in the heat transfer rate compared to the situation in which a constant amount flows. The impact of the pump’s performance curve on the heat transfer rate becomes relatively pronounced with an increased pressure drop. When N_fin,f was 30, the ratio of maximum to minimum heat transfer rate was 8.73% with a constant coolant flow rate, but this ratio increased to 13.08% when considering the actual coolant flow rate facilitated by a pump.

1. Introduction

The advent of electric vehicles (EVs) in public transportation marks a significant shift in the transportation sector. The International Energy Agency reports that over 26 million EVs were on the road in 2022, reflecting a sales increase of approximately 60% compared to 2021 [1]. As sales escalate, there is a growing market concern regarding the thermal management systems (TMSs) of EVs [2]. Essential components of a TMS in battery EVs include the air conditioning system for the cabin, the battery TMS (BTMS), and the driving motor TMS [3]. BTMSs are typically categorized into air-cooled and liquid-cooled types. Many commercial EV manufacturers opt for air-cooled BTMSs due to their simple structure, low initial costs, lightweight characteristics, and ease of maintenance [4]. Liquid-cooled systems, despite their higher complexity, initial cost, and weight, offer compact structures and superior cooling capabilities [5]. The liquid-cooled type operates by circulating a coolant around the batteries to absorb heat generated during their charge and discharge cycles, subsequently transferring this heat through a heat exchanger to the vapor compression cycle. Higher charge or discharge rates correspond to increased heat generation [6], necessitating advancements in BTMSs as EV performance improves. Although higher coolant flow rates enhance heat transfer, they also induce greater pressure drops [7,8].

Plate heat exchangers, such as chevron, wavy, dimple, and offset-fin types, are utilized in EV BTMSs. Offset-fin and dimple types, in particular, are prevalent due to their efficiency [8]. Offset fins, known for significantly enhancing heat transfer by periodically interrupting boundary layers and creating oscillating velocity in the fin wakes [9], simultaneously cause considerable pressure drops. This necessitates a balanced analysis of thermo-hydraulic performance in heat exchangers. To explain, the larger pressure drop results in reduced coolant flow rate if the same circulation pump is used. Since the lower coolant flow rate causes a reduction in the heat transfer rate, comparing various heat exchangers at the same coolant flow rate is not reasonable. Dimple configurations are recognized for achieving high heat transfer enhancement with relatively low pressure losses [10]. The remainder of this section summarizes previous research on the thermal and hydraulic performance of heat exchangers.

Fernandez-Sera et al. [11] conducted an experimental analysis on single-phase heat transfer between water and ethylene glycol solutions (10 to 30 wt.%) using an offset-fin heat exchanger. They developed an empirical correlation, utilizing the Wilson plot method, to determine the convection heat transfer coefficient as a function of the Reynolds number. This method is prevalent in various studies [12]. Yang et al. [13] introduced a dimensionless parameter representing the ratio of the effective fin surface area to the square of fin thickness for offset fins, suggesting that this parameter influences the advantages and disadvantages of reducing fin length. Park et al. [14] established correlations for the Colburn j and friction factors from experiments with offset-fin heat exchangers, also detailing an optimization technique involving six independent variables to enhance the thermo-hydraulic performance of offset-strip heat exchangers. Mada et al. [15] proposed a correlation for heat transfer between the gaseous phases of R-134a and R-1234yf based on computational fluid dynamics (CFD) simulations. They highlighted the effectiveness of CFD analysis in deriving correlations by comparing their j factor with existing literature data. Ismail et al. [16], Piper et al. [17], and Mocnuk et al. [18] also confirmed the utility of CFD in heat transfer analysis. Research extends beyond single-phase heat transfer, encompassing various correlations for evaporation heat transfer found in the literature. Jige et al. [19] studied the evaporation heat transfer of R-1234ze(E) and R-32 in a plate-fin heat exchanger, noting that mass flux directly impacts the heat transfer coefficient more than heat flux. Similar findings were reported by Prabakaran et al. [20,21], although some studies [22,23] indicate that heat flux and saturation temperature significantly influence evaporation heat transfer. Kim et al. [24] examined the thermo-flow characteristics of offset-fin heat exchangers by a numerical method in laminar regimes. They provided multiple correlations that are applicable in each specific range of beta, which is a parameter calculated from total and vacant area. At present, there are many studies on optimization of heat exchangers according to several design parameters based on the genetic algorithm. Guan et al. [25] derived correlations of the j and f factors by numerical analysis, then carried out optimization of PFHE. Cao et al. [26] also provided j and f factor correlations. Here, the correlations for both laminar and turbulent flow are distinguished, and a standard for the evaluation of the Reynolds number is provided. The genetic algorithm is also used to optimize fin design. Jamil et al. [27] conducted not only thermal and hydraulic analysis but also economic and exergy analysis according to three fin geometries using numerical codes, and Moon et al. [28] utilized the genetic algorithm for PFHE thermal and economic optimization.

The discharge process in EV batteries generates multiple forms of heat, including ohmic, reaction, polarization, and secondary reaction heat, all of which contribute to temperature rise [29]. The heat generated by batteries varies and is influenced by factors such as charging or discharging current, battery temperature, and voltage [30]. Thus, EVs frequently require enhanced cooling capacity to maintain battery efficiency. The recommended operating temperature range for lithium-ion battery packs is between 20 and 50 °C [3]. Generally, increasing the coolant’s mass flow by increasing the pump’s RPM can improve the heat transfer rate. The mass flow rate is determined by the pump’s performance curve and the heat exchanger’s resistance curve, which indicates pressure loss.

As highlighted previously, optimizing heat exchangers requires careful consideration of both heat transfer and pressure loss, particularly for offset fins, due to their significant impact on pressure drop. Increased pressure loss across a heat exchanger reduces the mass flow rate [31], which, in turn, affects the heat transfer rate [11]. It is crucial to assess the thermal performance of various heat exchanger designs based on actual mass flow rates derived from the performance and resistance curves. Although studies on optimization of offset fin structure and development of correlations are very common, it is hard to find analysis combining the thermo-hydraulic performance of heat exchangers with pump performance curves.

Consequently, this research aims to (1) provide comprehensive design data on the geometric effects of offset fins. This analysis provides dimensionless numbers such as j and f factors for various fin design cases to understand the thermo-hydraulic characteristics. The next aim is to (2) establish correlations for single-phase heat transfer based on CFD analysis. The correlation for the f factor enables generation of resistance curves by inputting Re and the number of fins. Since the point where the resistance curve intersects with the performance curve means the actual coolant flow rate, the heat transfer rate in this condition can be calculated based on the correlation for the j factor. The last aim is to (3) offer a practical comparison of heat exchanger models reflecting the performance curve of the pump and the resistance curve of the heat exchanger. Since the convective heat transfer coefficient of each case at the actual coolant flow rate is obtained from correlations, the expected heat transfer rate in this condition is calculated using the effectiveness–NTU method.

2. Geometric Model and Simulation Conditions

2.1. Offset-Fin Heat Exchanger

Figure 1 presents a schematic of the offset-fin heat exchanger utilized in this study. Similar to traditional plate heat exchangers, offset heat exchangers comprise several fin layers, as shown in Figure 1, with the refrigerant and coolant flows circulating through each layer in succession. The analysis was confined to the central rectangular area of a single layer. Hence, comparisons drawn between various cases are intended to highlight relative differences in thermal and hydraulic performance.

The configuration of the offset fins is determined by two design parameters: the number of fins aligned with the flow direction (N_fin,f) and the number of fins in the vertical direction (N_fin,v). Consequently, the geometric dimensions of each fin are determined by these parameters, given the constant rectangular installation surface area of 73 × 65 mm². Following

Table 1. Specific fin dimensions and hydraulic diameter in each case.

Nfin,f [EA]	Nfin,v [EA]	s [mm]	h [mm]	t [mm]	l [mm]	Dh [mm]
18	34	1.633	2.1	0.2	1.897	1.698
18	40	1.634	2.1	0.2	1.613	1.675
18	46	1.634	2.1	0.2	1.402	1.653
18	52	1.634	2.1	0.2	1.24	1.632
18	56	1.634	2.1	0.2	1.112	1.611
24	34	1.135	2.1	0.2	1.894	1.356
24	40	1.135	2.1	0.2	1.613	1.337
24	46	1.135	2.1	0.2	1.402	1.318
24	52	1.135	2.1	0.2	1.24	1.301
24	56	1.135	2.1	0.2	1.112	1.283
30	34	0.835	2.1	0.2	1.897	1.096
30	40	0.835	2.1	0.2	1.613	1.08
30	46	0.835	2.1	0.2	1.402	1.065
30	52	0.835	2.1	0.2	1.24	1.05
30	56	0.835	2.1	0.2	1.112	1.035

summarizes the specific geometric dimensions of the fins according to the number of fins in each direction. It is evident that length of the fins reduces as the number of fins increases in the vertical direction, since more fins will be installed in the same space. For the same reason, the spacing of the fins becomes smaller as the number of fins in the flow direction increases. The hydraulic diameter of the offset fins is determined by Equation (1) [14]. The range for N_fin,v was 34 to 58, analyzing three specific cases for N_fin,v: 18, 24, and 30. The fin height remained constant across all cases.

$$D_h={\displaystyle \frac{4shl}{2\left(sl+hl+th\right)+ts}}$$

(1)

Figure 2 presents a comparison of the heat transfer surface area for each case. An increase in the number of fins, both in the flow direction and vertically, leads to a larger surface area. However, increasing N_fin,v yields relatively modest increases in surface area compared to increasing N_fin,f. Specifically, an N_fin,v of 58 results in only a 5.19% larger surface area compared to an N_fin,v of 34. In contrast, increasing N_fin,f significantly enlarges the area; adding 6 and 12 fins in the flow direction enhances the surface area by 15.9% and 32.8%, respectively.

2.2. Simulation Conditions and Properties

The aim was to derive a correlation for single-phase convective heat transfer with the coolant, necessitating varying the coolant mass flow rate. This range was established based on the temperature difference between the coolant’s inlet and outlet sides. A standard mass flow rate of 0.0225 kg/s per layer was identified, where the temperature difference between the inlet and outlet was approximately 5 °C for the standard design with N_fin,f of 18 and N_fin,v of 34. Analyses included several rates lower and higher than this standard flow rate.

Notably, this research did not simulate the evaporation heat transfer of a refrigerant. Instead, a fixed value for the evaporation heat transfer coefficient and the refrigerant stream’s saturation temperature were used as boundary conditions. Typically, the evaporation heat transfer coefficient is determined experimentally. However, for the purposes of comparing cases with different geometric designs of coolant side fins, assuming a reasonable value for the evaporation heat transfer coefficient is considered acceptable.

A mixture of water (50%) and ethylene glycol (50% each) was used as the coolant. The simulation accounted for changes in its viscosity, thermal conductivity, and specific heat capacity corresponding to temperature variations. Below, Equations (2)–(4) present the calculation formulas for those temperature-dependent values.

$$\mu \left(T\right)=\left(1.216\times {10}^{-6}\right) {T_{\mathrm{c}}}^2-\left(8.328\times {10}^{-4}\right) T_{\mathrm{c}}+\left(1.439\times {10}^{-1}\right)$$

(2)

$$k\left(T\right)=\left(6.290\times {10}^{-4}\right)T_{\mathrm{c}}+\left(2.072\times {10}^{-1}\right)$$

(3)

$$C_{\mathrm{p}}\left(T\right)=\left(-1.600\times {10}^0\right){T_{\mathrm{c}}}^2+\left(9.882\times {10}^2\right)T_{\mathrm{c}}-\left(1.487\times {10}^5\right)$$

(4)

Table 2. Information about model-related parameters.

Software	ANSYS Fluent 2023 R2
Node number	997,119
Element number	5,073,776
Maximum skewness	0.847
Viscous model	Laminar
Solver type	Density-based steady state
Fluid/solid type	Water 50% ethylene glycol 50%/aluminum
Boundary condition: fluid inlet/outlet	Mass flow inlet/pressure outlet
Boundary condition: solid	Wall convection
Convergence criteria	1 × 10−06 (coolant outlet temperature)

, below, shows information about model-related parameters including boundary conditions, meshing information, and convergence criteria. The iteration was repeated until the coolant outlet temperature met the absolute criterion of 1 × 10⁻⁰⁶. Additionally, it was double-checked that the net heat flux and mass flux became zero, which normally means the model has been converged.

3. Results and Discussion

3.1. Thermal and Hydraulic Performance at Standard Flow Rate

Figure 3 presents a comparison of heat transfer rates at the standard coolant flow rate. Increasing the number of fins in the flow direction significantly enhances the heat transfer rate. In contrast, increasing the number of fins in the vertical direction has a relatively minor impact on thermal performance. The heat transfer rate is influenced by both the heat transfer area and the convective heat transfer coefficient. Figure 4 shows the relative ratio of the heat transfer rate and surface area. The term “relative ratio” refers to the comparative degree of heat transfer rate or surface area relative to the standard configuration, where N_fin,f and N_fin,v are 18 and 34, respectively. The diminishing relative difference in heat transfer rates for configurations with N_fin,f values of 18, 24, and 30, compared to their surface areas, indicates a decrease in the convective heat transfer coefficient with an increasing number of fins in the flow direction. However, variations in the number of fins in the vertical direction do not significantly affect the convective heat transfer coefficient.

Figure 5 details the convective heat transfer coefficient as a function of the numbers of fins in both the flow and vertical directions. Consistent with previous mentions, a lower convective heat transfer coefficient is observed with a higher N_fin,f, and adding more fins in the vertical direction does not substantially improve the convective heat transfer coefficient, except when N_fin,f is 30. The reasons for those can be explained by Figure 6, which reveals velocity vectors of coolant flow where both the color and the size of the vector represent the degree of velocity. Firstly, it is evident that coolant flow circulates along every surface of the fin closely with reasonable velocity, which contributes to heat transfer performance positively. However, the vectors present lower velocity as more fins are installed in the flow direction. This is because the coolant flow is disturbed as the gap between fins becomes narrow to install more fins, which negatively affects thermal performance. This issue is partially mitigated by adding more fins in the vertical direction, which provides additional pathways for coolant flow.

The relationship between coolant flow rate through different fin configurations and pressure loss is also explored. Increased flow resistance, corresponding to reduced velocity vector distribution, results in higher pressure loss, as shown in Figure 7. Complex fin structures in the flow direction increase pressure loss, whereas additional fins in the vertical direction can reduce it. This is because more fins in the vertical direction create additional paths for the flow, since more fins will be installed in the same space. The detailed dimensions of the fins according to the number of fins can be found in Table 1. Thus, while enhancing vertical fin count does not significantly improve heat transfer area or rate at a constant coolant flow rate, it can effectively decrease pressure loss, facilitating increased coolant flow through the fins. Therefore, assessing thermal performance across various fin configurations without considering the actual coolant flow rate may be inadequate. This research seeks to incorporate the pump’s performance curve and the heat exchanger’s resistance curve for a more accurate comparison.

3.2. Performance According to Reynolds Number

3.2.1. Nusselt Number and Colburn j Factor

Utilizing the Colburn j factor as a dimensionless number facilitates an intuitive understanding of the thermal performance of heat exchangers. The Colburn j factor is calculated using the Nusselt, Reynolds, and Prandtl numbers, as shown in Equation (5). Establishing correlations is an efficient method to determine the heat transfer coefficient for a given coolant flow rate. Equation (6) introduces the correlation for the Nusselt number, which indicates the relative ratio between thermal convection and conduction. The Nusselt number is derived from the convective heat transfer coefficient, calculated using Equations (7)–(9) based on CFD results. Specifically, the heat transfer rate (Q) can be computed using the coolant outlet temperature obtained from CFD analysis, which is then used to calculate UA given the logarithmic mean temperature difference (LMTD) as shown in Equation (8). By considering UA to be the inverse of the sum of thermal resistances, the convective heat transfer coefficient (h_i) is ultimately determined. In this study, the evaporation heat transfer coefficient (h_o) is assumed to be a constant value of 1500 W/m² K, and thermal resistance due to conduction is disregarded due to its negligible magnitude. The evaporation heat transfer coefficient is affected by the geometric characteristics of fins, the heat flux, the mass flux, and the saturation temperature. It is inevitable that assuming the constant value of h_o may bring out errors in absolute value; however, it is believed that overall methodologies of this research are applicable to analyze the effects of considering the pump performance curve and heat exchanger resistance curve. Further experimental or CFD research regarding evaporation according to the number of fins and operating conditions is needed for more accurate analysis.

$$j={\displaystyle \frac{Nu}{ReP{r}^{\frac{1}{3}}}}$$

(5)

$$Nu={\displaystyle \frac{h_{\mathrm{i}}{D}_{\mathrm{h}}}{k}}$$

(6)

$$Q=m_{\mathrm{c}}{C}_{\mathrm{p},\mathrm{c}}\left(T_{\mathrm{c},\mathrm{in}}-T_{\mathrm{c},\mathrm{out}}\right)=UA∆T_{LMTD}$$

(7)

$$UA={\displaystyle \frac{Q}{∆T_{LMTD}}}={\left({\displaystyle \frac{1}{h_{\mathrm{i}}A}}+{\displaystyle \frac{1}{h_{\mathrm{o}}A}}\right)}^{-1}$$

(8)

$$h_{\mathrm{i}}={\displaystyle \frac{1}{A}}{\left({\left({\displaystyle \frac{Q}{∆T_{\mathrm{LMTD}}}}\right)}^{-1}-{\displaystyle \frac{1}{h_{\mathrm{o}}A}}\right)}^{-1}$$

(9)

The Reynolds number for offset-fin heat exchangers is calculated using Equation (10), with values ranging from 34 to 274. This suggests that the small hydraulic diameter of the offset fins results in laminar flow.

$$Re={\displaystyle \frac{\rho V_{\mathrm{ave}}{D}_{\mathrm{h}}}{\mu }}$$

(10)

Figure 8 presents the Nusselt number and Colburn j factor as functions of the Reynolds number. No significant variance in the heat transfer coefficient is evident according to N_fin,v; thus, Nusselt numbers are similar for identical N_fin,f values. However, an increase in fins in the flow direction significantly influences the Nusselt number. The Colburn j factor, providing an intuitive estimation of thermal performance, yields values marginally higher than those found in the literature (

Table 3. Summary of Colburn j factor values and correlations from previous research.

Ref.	Type	Re	j Factor	Correlation
[14]	Offset	50–200	0.12–0.2	$j=0.6833R{e_{\mathrm{c}}}^{-0.3199}{\left({\displaystyle \frac{{\mu }_{\mathrm{c}}}{{\mu }_{\mathrm{wall}}}}\right)}^{0.14}$
[13]	Offset	200–2500	0.003–0.03	$j=C_0{\left({\displaystyle \frac{l}{D_{\mathrm{h}}}}\right)}^{C_1}{\varphi }^{C_2}{\alpha }^{C_3}{\gamma }^{C_4}{\delta }^{C_5}R{e}^{*}P{r}^{C_7}{{\eta }_{\mathrm{f}}}^{C_8}$
[32]	Wavy	1000–10,000	0.007–0.018	$j=0.527R{e}^{-0.512}P{r}^{-3.187}{\left({\displaystyle \frac{D}{L}}\right)}^{0.423}{\left({\displaystyle \frac{L}{H}}\right)}^{-0.256}{\left({\displaystyle \frac{A}{W}}\right)}^{0.161}$
[33]	Offset	500–6000	0.011–0.025	$j=X_1+\left[X_2+c_{0}R{e}^{X_3}\right]e^{X_4}$
[33]	Wavy	500–6000	0.008–0.02	$j=X_1+\left[X_2+c_{0}R{e}^{X_3}\right]e^{X_4}$
[34]	Offset	100–10,000	0.007–0.02	$j=A_0{\left({\displaystyle \frac{1}{D_{\mathrm{e}}}}\right)}^{A_1}{\phi }^{A_2}{\alpha }^{A_3}{\gamma }^{A_4}{\delta }^{A_5}R{e}^{*}$
[35]	Offset	300–4000	0.006–0.03	$j=0.6522R{e}^{-0.5403}{\alpha }^{-0.1541}{\delta }^{0.1499}{\gamma }^{-0.0678}{\left[1+5.269·{10}^{-5}R{e}^{1.340}{\alpha }^{0.504}{\sigma }^{0.456}{\gamma }^{-1.055}\right]}^{0.1}$
[36]	Offset	100–1500	0.003–0.04	$j=0.489R{e}^{-0.445}$
[37]	Offset	100~550	0.02–0.05	$j=0.357R{e}^{-0.436}$
[38]	Wavy	400–8000	0.005–0.02	$j=0.657R{e}^{-0.3338}{\left({\displaystyle \frac{H_{\mathrm{p}}}{D_{\mathrm{h}}}}\right)}^{0.571}{\left({\displaystyle \frac{P_{\mathrm{f}}}{D_{\mathrm{h}}}}\right)}^{0.283}{\left({\displaystyle \frac{2A_{\mathrm{a}}}{D_{\mathrm{h}}}}\right)}^{0.0569}{\left({\displaystyle \frac{L_{\mathrm{d}}}{D_{\mathrm{h}}}}\right)}^{-1.011}{\left({\displaystyle \frac{\lambda }{D_{\mathrm{h}}}}\right)}^{0.229}{\left({\displaystyle \frac{\delta }{D_{\mathrm{h}}}}\right)}^{-0.047}$
[39]	Offset	2000	N/A	$j=1.37{\left({\displaystyle \frac{l}{D_{\mathrm{h}}}}\right)}^{-0.25}{\alpha }^{-0.184}R{e}^{-0.67}$
[40]	Offset	1500	N/A	$j=0.53R{e}^{-0.5}{\left({\displaystyle \frac{l}{D_{\mathrm{h}}}}\right)}^{-0.15}{\left({\displaystyle \frac{s}{h-t}}\right)}^{-0.14}$

), attributed to the direction of coolant flow relative to the fin structure. Lee et al. [14] highlighted similar findings, highlighting the impact of flow direction on performance characteristics.

The Colburn j factor correlation, incorporating Re, Pr, N_fin,f, N_fin,v, and various constants, is expressed in Equation (11).

$$j=CR{e}^a{N_{\mathrm{fin},\mathrm{v}}}^b{N_{\mathrm{fin},\mathrm{f}}}^d$$

(11)

Table 4. Constants for correlation of j factor in Equation (11).

Constant	C	a	b	d
Values	21.4111	−0.8354	−0.9081	0.0305

lists the fitted constants, and Figure 9 compares the j factors derived from CFD simulation and the correlation of Equation (11), showing a reasonable agreement with errors within ±10%.

3.2.2. Pressure Drop and Fanning f Factor

The relationship between the pressure drop and the Reynolds number is depicted in Figure 10. An increase in N_fin,f and a decrease in N_fin,v resulted in a greater pressure loss, with the maximum pressure drop reaching approximately 88 kPa at a Reynolds number of 184.3. The pressure drop exhibits an exponential increase as the Reynolds number increases. In contrast to the Nusselt number’s behavior with respect to the Reynolds number, increasing N_fin,v significantly influences the result, especially when N_fin,f is 30. This is attributed to the reduction in flow channel area with more fins installed in the flow direction, complicating coolant flow. Additionally, more fins in the vertical direction provide more channels for the coolant’s exit, explaining why the highest pressure drop can be observed when N_fin,f is at its maximum and N_fin,v at its minimum.

The Fanning f factor is calculated using Equation (12) [33]. A notable distinction between the f factor and the j factor is that N_fin,v influences the f factor results but not the j factor.

$$f={\displaystyle \frac{D_{\mathrm{h}}}{2L}}{\displaystyle \frac{∆P}{\rho {V_{\mathrm{ave}}}^2}}$$

(12)

Similar to the j factor, vertical coolant flow results in significantly larger f factor values than other heat exchanger configurations, as summarized in

Table 5. Summary of Fanning f factor values and correlations from previous research.

Ref	Type	Re	f Factor	Correlation
[14]	Offset	50–200	3–4	$f=8.2488R{e_{\mathrm{c}}}^{-0.1968}$
[32]	Wavy	1000–10,000	0.03–0.09	$f=25.534R{e}^{-0.347}P{r}^{2.954}{\left({\displaystyle \frac{D}{L}}\right)}^{0.832}{\left({\displaystyle \frac{L}{H}}\right)}^{-0.321}{\left({\displaystyle \frac{A}{W}}\right)}^{0.296}$
[33]	Offset	500–6000	0.071–0.142	$f=Y_1+\left[Y_2+b_{0}R{e}^{Y_3}\right]e^{Y_4}$
[33]	Wavy	500–6000	0.025–0.12	$f=Y_1+\left[Y_2+b_{0}R{e}^{Y_3}\right]e^{Y_4}$
[34]	Offset	100–10,000	0.048–0.09	$f=B_0{\left({\displaystyle \frac{1}{D_{\mathrm{e}}}}\right)}^{B_1}{\phi }^{B_2}{\left({\displaystyle \frac{1}{1-2\alpha }}\right)}^{B_3}{\left({\displaystyle \frac{1}{1-2\gamma }}\right)}^{B_4}{\delta }^{B_5}R{e}^{*}$
[35]	Offset	300–4000	0.03–0.02	$f=9.6243R{e}^{-0.7422}{\alpha }^{-0.1856}{\delta }^{0.3053}{\gamma }^{-0.2659}{\left[1+7.669·{10}^{-8}R{e}^{4.429}{\alpha }^{0.920}{\sigma }^{3.767}{\gamma }^{-0.236}\right]}^{0.1}$
[42]	Offset	10–200	0.2–4	$f=20.362R{e}^{-0.7661}$
[36]	Offset	10–950	0.03–0.2	$f=10R{e}^{-0.68}$
[38]	Wavy	400–8000	0.04–0.11	$f=3.869R{e}^{-0.3569}{\left(\frac{H_{\text{p}}}{D_{\text{h}}}\right)}^{-0.519}{\left(\frac{P_{\text{f}}}{D_{\text{h}}}\right)}^{0.050}{\left(\frac{2A_{\text{a}}}{D_{\text{h}}}\right)}^{0.7825}{\left(\frac{L_{\text{d}}}{D_{\text{h}}}\right)}^{0.0867}{\left(\frac{\lambda }{D_{\text{h}}}\right)}^{-0.417}{\left(\frac{\delta }{D_{\text{h}}}\right)}^{0.0577}$
[39]	Offset	2000	N/A	$f=5.55{\left({\displaystyle \frac{l}{D_{\mathrm{h}}}}\right)}^{-0.32}{\alpha }^{-0.092}R{e}^{-0.67}$
[40]	Offset	1500	N/A	$f=8.12R{e}^{-0.74}{\left({\displaystyle \frac{l}{D_{\mathrm{h}}}}\right)}^{-0.41}{\left({\displaystyle \frac{s}{h-t}}\right)}^{-0.02}$

. Additionally, Table 5 lists several correlations describing the f factor from previous research. This study incorporated variables representing the number of fins into the correlation presented by Beale [41], as shown in Equation (13).

$$f=a{(b+{\displaystyle \frac{1}{Re}})}^c{N_{fin,f}}^d{N_{fin,v}}^e$$

(13)

Table 6. Constants for correlation of f factor in Equation (13).

Constant	C	a	b	d	e
Values	83.5675	0.0056	0.4684	0.2401	−0.7588

presents the values of the fitted constants, and Figure 11 compares the f factor obtained from CFD analysis with the calculated values from the correlation. This comparison also demonstrates reasonable agreement, with errors within ±10%.

3.2.3. Area Goodness Factor

The concept of the flow area goodness factor, represented by the ratio of the Colburn j factor to the Fanning f factor versus the Reynolds number, serves as a common approach to jointly assess thermal and hydraulic performance. Figure 12 illustrates this ratio according to the Reynolds number. A higher flow area goodness factor indicates that a smaller frontal area for the heat exchanger is required [43], which also implies that the heat transfer rate per unit of frontal area is large [44]. As a result, the area goodness factor was found to be larger for higher N_fin,v and lower N_fin,f values, signifying an improvement in pressure loss and the f factor. In contrast, the Colburn j factor was less influenced by the geometric characteristics of the fins.

Compared to previous research, vertical offset fins demonstrated a relatively low flow area goodness factor [10,44,45]. The diminished area goodness factor for vertical heat exchangers is primarily due to a high Fanning f factor compared to other types of heat exchangers, with the relative ratio of the Colburn j factor among them being smaller. Although this dimensionless parameter facilitates a relative comparison of the thermo-hydraulic performance of various heat exchangers, it does not intuitively indicate which design is more advantageous. Hence, the importance of considering a pump characteristic curve and a resistance curve is underscored.

3.3. Performance Comparison at Actual Flow Rate

3.3.1. Characteristic Curve

The thermal performance of heat exchangers is significantly influenced by the coolant flow rate, which is determined by the performance curve of the pump and the resistance curve of the heat exchanger. To assess the impact of pump characteristics on thermal performance, two hypothetical pumps with different performance curve shapes are considered. As illustrated in Figure 13, Pump A is capable of circulating a higher flow rate at the same pump head than Pump B, which exhibits steeper performance curves. The difference in flow rates between the two pumps at a given head is particularly notable in the flow rate range exceeding 20 L per minute (LPM). The flow rates discussed here pertain to the total coolant flow rate rather than the flow rate through a single layer. That is, it is assumed that an identical flow rate uniformly circulates through 15 layers by reflecting heat generation amount of electric vehicle batteries. Wang et al. [46] revealed that average rate of heat generation of a lithium ion battery module ranges from 12,416.25 to 104,790 W/m³. Additionally, Berjoza and Jurgena [47] reported that the weight of batteries for EVs ranges from 385 to 544 kg, and the equivalent density of 18,650 battery cells is about 2018 kg/m³ [48]. Considering those values from related research, the number of layers is assumed to be 15, whose heat transfer rate becomes around 7 kW.

Additionally, it is assumed that the RPM of both pumps remains constant. In an EV’s TMS, it should be possible to control the coolant flow rate according to the heat load. Therefore, changes in performance curves according to RPM need to be considered for more detailed and practical application in the future research.

3.3.2. Resistance Curve

The resistance curve is derived from pressure drop data related to the flow rate through a single layer. For the purpose of calculating the total coolant volume flow rate, it is assumed that an identical flow rate uniformly circulates through 15 layers. By integrating the performance and resistance curves, the actual flow rate circulating through the heat exchanger can be determined. The intersection of these two curves indicates the flow rate at which the pump can supply enough force to overcome the heat exchanger’s pressure loss. Figure 14 shows the coolant volume flow rate for each case relative to the resistance and performance curves.

Table 7. Summary of coolant volume flow rates determined by pumps (in LPM).

Pump A	Nfin,f (↓)/Nfin,v (→)	34	40	46	52	58
	18	30.25	32.16	34.18	35.14	36.00
	24	26.64	28.26	29.84	31.15	32.01
	30	21.49	23.05	24.54	25.96	26.97
Pump B	Nfin,f (↓)/Nfin,v (→)	34	40	46	52	58
	18	27.28	28.64	29.58	30.14	30.61
	24	24.48	25.54	26.63	27.55	28.23
	30	20.25	21.41	22.70	23.80	24.75

lists the coolant flow rates determined in each case for Pump A and Pump B, respectively.

An observed difference of 6.1–17.6% in flow rates through the same geometry highlights the impact of pump type on heat transfer performance, indicating that the achievable heat transfer rate with the same heat exchanger varies depending on the pump used. A comparison of flow rates across all cases reveals significant discrepancies; the geometry with the minimum flow rate has a flow rate value that is 40.3% and 33.8% lower than the maximum for Pump A and Pump B, respectively. This highlights the impracticality of comparing the thermal performance of heat exchangers with varying geometries under a uniform coolant flow rate without considering the actual flow rate achievable based on pump characteristic curves and heat exchanger resistance curves.

3.3.3. Heat Transfer Rate at Actual Flow Rate

The heat transfer rate at the coolant flow rate achievable using two different pumps can be determined by using geometric information and correlations previously introduced. The process is as follows: With the obtainable flow rate established, this flow rate is converted into the Reynolds number, as indicated by Equation (10). In this equation, viscosity is obtained from Equation (2), and hydraulic diameter is calculated using Equation (1). The coolant’s density is calculated according to assumed to be constant at 1080 kg/m³. Given that the average coolant velocity for each case at various mass flow rates is computed from CFD analysis, a first-order linear fitting equation between coolant flow rate and average velocity is generated. After computing the Reynolds number at a specific coolant flow rate, the Nusselt number for this Reynolds number is calculated using the j factor correlation from Equation (11). Accordingly, the convective heat transfer can be calculated using Equation (6).

The UA value is obtained via Equation (8), with all necessary values provided, allowing for the calculation of each geometric case’s effectiveness as shown in Equation (14), utilizing the mass flow rate and specific heat capacity of the coolant.

$$\epsilon =1-\exp \left({\displaystyle \frac{-UA}{\dot{m_c}{C}_{\mathrm{p},\mathrm{c}}}}\right)$$

(14)

The effectiveness–NTU method is subsequently employed to calculate the actual heat transfer rate as outlined in Equation (15).

$$Q_{actual}=\epsilon Q_{ideal}=\epsilon h_{\mathrm{i}}A\left(T_{\mathrm{c},\mathrm{in}}-T_{\mathrm{s}}\right)$$

(15)

Figure 15 presents a comparison of the heat transfer rate at the actual coolant flow rate, aiming to identify differences from the constant coolant flow rate scenario depicted in Figure 3. No reversals in results between geometric cases are identified; the configurations with the largest N_fin,v and N_fin,f exhibit the highest heat transfer rates. However, the disparity between cases, when considering a consistent coolant flow rate and the actual flow rate, is notably distinct. Specifically, results tend to show a more pronounced increase with an increase in N_fin,v. This is attributable to the decrease in pressure loss with higher N_fin,v, allowing a greater coolant flow rate and, thus, enhanced heat transfer. Moreover, as the installation of more fins in the flow direction reduces the coolant flow rate, the performance gap attributable to N_fin,v narrows when the actual coolant flow rate is taken into account. Interestingly, the type of pump utilized does not significantly impact the heat transfer rate. The maximum difference in heat transfer rate between cases using Pump A and Pump B was only 1.15%, indicating that distinct pump performance curves do not markedly affect outcomes.

On the other hand, the difference in heat transfer rate between the cases of constant coolant flow rate and actual flow rate is quite distinguishable. Figure 16 depicts the comparison of relative ratio at each case. The term “relative ratio” refers to the comparative degree of heat transfer rate relative to the standard configuration, where N_fin,f and N_fin,v are 18 and 34, respectively. The impact of the pump’s performance curve on the heat transfer rate becomes more pronounced with increased pressure drop. For instance, when N_fin,f is 30, the ratio of the maximum to the minimum heat transfer rate was 8.73% with a constant coolant flow rate, but this ratio increased to 13.08% when considering the actual coolant flow rate facilitated by a pump. Considering both the performance and resistance curves facilitates a more precise comparison between geometric configurations.

4. Conclusions

This research aimed to provide a comprehensive analysis focusing on (1) the impact of offset fin geometry on design, (2) correlations for single-phase heat transfer derived from CFD, and (3) a practical comparison of heat exchanger models that incorporates both the pump’s performance curve and the heat exchanger’s resistance curve. The key findings are as follows:

The heat transfer rate tends to increase with more fins in both directions. However, the increase in the heat transfer rate according to N_fin,v was significantly different depending on N_fin,f. The difference in heat transfer rate between N_fin,v values of 34 and 58 was about 4.48% when N_fin,f was 18, while it was 8.74% for N_fin,f of 30. The increase in the heat transfer rate can be attributed to two factors: an increase in surface area and an enhanced convective heat transfer coefficient. In the case of N_fin,f of 18, the increase in surface area between N_fin,v values of 34 and 58 was 4.94%, which is only a 0.45% difference compared to the increase in heat transfer rate, indicating that the heat transfer rate was primarily enhanced due to the enlarged surface area. On the other hand, in the case of N_fin,f of 30, the increase in surface area between N_fin,v values of 34 and 58 was 5.34%, suggesting that the remaining increase was due to the enhanced convective heat transfer coefficient.

The convective heat transfer coefficient and pressure drop according to N_fin,v presented inverse symmetry characteristics. That is, more fins in the vertical direction were beneficial not only in reducing pressure loss but also in enhancing the heat transfer coefficient. Similarly, the increase or decrease in these parameters varied according to N_fin,f. Increasing N_fin,v was advantageous in both thermal and hydraulic aspects, whereas increasing N_fin,f had a negative impact by causing greater pressure loss and a reduced convective heat transfer coefficient. However, the increase in surface area compensated for the reduced convective heat transfer coefficient.

Further CFD analysis, incorporating various coolant flow rates, facilitated the establishment of relationships between different parameters (such as the Nusselt number, Colburn j factor, and Fanning f factor) in relation to the Reynolds number. The correlations defining the j and f factors with respect to the Reynolds number and the number of fins in each direction have been successfully derived, demonstrating errors of less than ±10%.

The study assumed two hypothetical pumps with differing performance curves to calculate the actual coolant flow rate for each pump. The actual flow rate circulating through each geometric case was determined by identifying the intersection of the performance curve and the resistance curve of each case. Differences in pressure drop resulted in different coolant flow rates despite using the same circulation pump, showing a difference of 1.67 times between the minimum and maximum values. Although different coolant flow rates circulated in each case, there was no reversal in the heat transfer rate compared to the situation with a constant amount flowing. The impact of the pump’s performance curve on the heat transfer rate becomes relatively pronounced with increased pressure drop. When N_fin,f was 30, the ratio of maximum to minimum heat transfer rates was 8.73% with a constant coolant flow rate, but this ratio increased to 13.08% when considering the actual coolant flow rate facilitated by a pump.

Lastly, the correlations and methodology of this research can be utilized in future research analyzing the energy consumption of EV TMSs according to battery heat generation. In a TMS, the cooling capacity can be controlled by adjusting the coolant flow rate and/or changing the evaporation temperature of the refrigerant. The comparison of energy consumption according to control strategies in response to varying battery loads will be possible based on this research.