Introduction

Micro-heat sinks (MHS) have become very popular in recent years due to their extensive cooling and mixing capabilities [1]. Tuckerman and Pease [2] were the first ones to use liquid coolants in MHS in order to improve cooling performance. More than two decades later, Koşar et al. [3] employed MPFs for this purpose. They performed experiments with MPFs with H/D ratios of 1 and 2 and achieved lower FF with the latter case. In addition, staggered arrangement and diamond cross section of MPFs resulted in higher FF compared to the in-line arrangement and circular MPFs, respectively. Furthermore, thermal and hydrodynamic boundary layers affected the heat transfer coefficient (HTC) and FF where Re and H/D were the dominant factors [4].

Koşar and Peles [4] held end-wall effects responsible for the discrepancies between their results and the ones obtained from earlier correlations for a staggered arrangement of circular MPFs (H/D = 1 and 2.43, 5 < Re < 128). With similar system and flow conditions using R-123 as the coolant, end-wall effects diminished for Re > 100 [5]. With a single circular MPF (0.5 < H/D < 5, 20 < Re < 150) [6], end-wall effects and FF decreased with H/D and Re, whereas Nu increased with Re. Renfer et al. [7] considered circular MPFs (D = 100,Footnote 1H = 100 and 200, Re < 330). For H/D = 2, the slope of pressure drop versus flow rate curve increased for Re > 200 which suggested a transition in flow patterns. In addition, heat transfer enhanced more in the case of H/D = 2 due to larger flow mixing, whereas dominant end-wall effects hindered generation of vortices and increased pressure drop for H/D = 1. Zhao et al. [8] considered staggered arrangements of circular, elliptical, diamond, square and triangular MPFs (50 < Re < 1800) and observed a general pattern where FF increased with Re due to dominating end-wall effects (Re < 100) and eddy dissipation (Re > 300).

The effect of MPF cross section has also been studied. Zhao et al. [8] mentioned that the triangular MPFs had the lowest FF in the laminar regime and the highest FF in the turbulent flow regimes compared to all other types of MPFs. A reverse pattern was reported with the elliptical ones. In an extension to this study, Guan et al. [9] considered triangular, circular and diamond MPFs (100 < Re < 900, HF < 150). At a fixed mass flux, pressure drops and FF increased with HF for all cases. However, FF curves reached a plateau beyond a certain Re. For both circular and diamond MPFs, HTCs increased with HF, whereas an increasing–decreasing trend was observed with the triangular MPFs. The values with the triangular case were always larger than the other cases. Koşar and Peles [10] considered circular, hydrofoil, cone-shaped and rectangular MPFs (14 < Re < 720). For all non-streamlined MPFs, Nu dependency on Re changed at a certain Re while for the hydrofoil case, the dependency level remained the same. The rectangular MPFs triggered more intense flow separation which increased the wake length, pressure drop and HTC. These studies suggested that the MPF cross section needs to be chosen based on the flow condition. Izci et al. [11] considered single MPF with different cross sections (20 < Re < 120, 200 < HF < 300). Rectangular MPFs had the largest HTC and Nu but also had the largest flow separations, which resulted in the largest pressure drop and FF. This highlights the other challenge where a good heat removal capacity is usually accompanied with a larger pressure drop penalty.

Beside the MPF cross section, some studies have considered the effect of MPF density. Koşar and Peles [10] reported larger HTCs and pressure drops with denser MPF configurations. Similarly, Koşar et al. [12] reported higher FF with staggered arrangements and denser configurations of circular, diamond and hydrofoil MPFs (1 < Re < 2500). The hydrofoil MPFs had lower pressure drop while diamond MPFs had significantly larger pressure drop. Hence, denser configuration are not favorable hydrodynamically but may enhance the thermal performance.

To manipulate the hydrodynamic penalty, recent studies have employed different MPF densities along the MHS. Rubio-Jimenez et al. [13] considered three of such designs with square, circular, elliptical and chamfered rectangular MPFs. The first one consisted of four sections with MPFs of different cross sections [H = 200 and length (L) = 75, 100 and 150]. The second one had four sections with rectangular MPFs but with varying heights from 100 to 300 µm. The third one had three sections with 200-µm-high MPFs. Heat dissipation rate was influenced by the ratio of heating area to the length of the sections with MPFs. The case with 100-µm-long rectangular MPFs had the best thermal performance. Pressure drop was significantly affected by the MPF shape and increased with the length of sections with MPFs. Sections with no MPFs were not recommended in this design. Gonzalez-hernandez et al. [14] studied another design with increasing MPF density along the channel. Two heights of 100 and 200 µm were considered for MHS, and for each of them, four cases with different MPF heights (20–50% of the channel height) were evaluated for flow rates between 72.765 and 145.53 mL min−1 [HF = 153.18 (from both top and bottom surfaces)]. For the highest flow rate, the 200 µm MHS with 60-µm-high MPFs had the best temperature uniformity with only 2.24 k difference in different regions. For the lowest flow rate, the 200 µm MHS with 80-µm-high MPFs had the best temperature uniformity with only 2.91 k temperature difference. This shows the importance of proper geometry selection based on the flow condition. In addition, for both flow rates and for both MHS heights, pressure drop increased with H which shows the same challenge between heat transfer gain and pressure drop penalty.

To study the effect of gap size, Liu et al. [15] considered in-line and staggered arrangements of circular MPFs (250 < H < 1000) in a 1-mm-high MHS (25 < Re < 800) where FF increased with the gap size. In-line arrangements had lower FF as a result of fewer interactions among generated vortices. Roth et al. [16] considered circular MPFs (65 < D < 100, 251.6 < H < 264.4) with a fixed gap size of 50 µm (9 < Re < 238.4). At lower Re, heat transfer increased with D. In addition, Nu increased sharply with Re and reached a plateau.

To study the effect of pitch ratios, John et al. [17] considered in-line arrangements of circular and square MPFs (50 < Re < 500). Horizontal and vertical distances between MPFs (HPD and VPD) were varied between 350 and 650 µm and between 150 and 300 µm, respectively. For the circular MPFs, pressure drop and thermal resistance were significantly affected by HPD. Liu et al. [18] considered pitch ratios between 0.9 and 1.1 in different in-line and staggered arrangements of circular MPFs. Nu decreased and FF slightly changed with pitch ratios. Sluggish flow regions were highlighted as the reason for overprediction of earlier correlations for FF and Nu at low Re.

To study the effect of nanoparticles, Bahiraei et al. [19] considered circular MPFs (H = 200, 400, 600 and 800, and D = 600, 900 and 1200) in a 1-mm-high MHS using graphene nano-platelets. The bottom surface temperature and thermal resistance decreased, and the pumping power and temperature distribution uniformity increased with particle concentration, H and D. Although a significantly better thermal performance was reported when employing nanoparticles, pressure drop didn’t change much. Bahiraei et al. [20] also considered a step microchannel design which had 20 parallel channels (L = 29.23 mm and W = 1.2 mm) using Al2O3–water as the coolant. The parallel channels had a uniform flow which ensured a uniform temperature distribution (400 < Re < 1000). The surface temperature decreased and the power consumption increased with either Re and particle concentration.

Although many studies have focused on the effect of geometrical specifications of MPFs and the flow conditions, the results are not easily comparable due to high number of variables. In an effort to do a systematic study, Mohammadi and Koşar [21, 22] did a two-part study considering ten in-line [21] and ten staggered [22] arrangements of circular MPFs (D = 50, 100 and 200). Two horizontal and vertical pitch ratios (HPR and VPR) of 1.5 and 3 were considered [20 < Re < 160, HF = 30 (through the bottom surface)]. For the in-line arrangements, Nu decreased and pressure drops increased with both VPR and H/D. Besides, pressure drops increased and thermal performance indices (TPIs) decreased with HPR. Distinct decreasing/increasing trends were observed for TPIs with both VPRs. For the staggered arrangements, both Nu and pressure drops were significantly affected by the wake length behind MPFs which was intensified with both Re and HPR. The velocity gradients became milder, pressure drops decreased and Nu increased with VPR. For H/D = 1 and 2, TPIs had an increasing–decreasing trend with Re with the turning point somewhere between Re = 40 and 80.

Although the amount of heat flux through the bottom surface of all arrangements in these two studies [21, 22] was the same, the generated heat through MPFs was different. Hence, to ensure a fair comparison, the heat flux through all heating surfaces are set to be 30 W cm−2 in this study. In addition, to further simplify the comparison, only the arrangements with H/D = 1 are considered which leaves the arrangement type (in-line or staggered), HPR (1.5 or 3) and VPR (1.5 or 3) as the variables. Hence, four in-line and four staggered arrangements of circular MPFs are considered. Assuming a single-phase laminar flow, the simulations were run at five Re equal and below 150 which results in a total of 40 study cases.

Geometrical modeling and methodology

Four in-line and four staggered arrangements of circular MPFs in a rectangular microchannel with the dimensions of 5 × 1.5 × 0.1 mm3 are considered. The diameter and height of MPFs in all cases are 0.1 mm. Two values (0.15 and 0.3 mm) are considered for both the horizontal (SL) and vertical (ST) distances between MPFs. This results in HPR (SL/D) and VPR (SH/D) of 1.5 and 3, respectively. These eight configurations are labeled in Table 1 for an easier reference. I and S on the first letter stand for in-line and staggered arrangements, respectively. V and H on the second and forth letters stand for vertical and horizontal directions, respectively. Finally, d and s on either the third or fifth letters stand for dense (0.15 mm) and sparse (0.3 mm) spacing. For instance, IVdHd refers to an in-line arrangement with a dense vertical and a dense horizontal spacing. For all cases, a constant number of 30 MPFs is considered for a better comprehensive comparison. Hence, in cases with ST/D of 1.5, three columns of MPFs exist, while in cases with ST/D of 3, six MPF columns exist. For the IVdHd and IVdHs configurations, an additional three MPFs are needed behind the last column of MPFs to have a total of 30 MPFs, as shown in Fig. 1. The total heating area in each case is 8.2 × 10−6 m2, which consists of a 7.265 × 10−6 m2 microchannel-free base area and a 9.425 × 10−7 m2 MPF surface area.

Table 1 Geometrical specifications of all configurations
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Fig. 1
figure 1

Three-dimensional model of the IVdHd configuration

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For all cases, the initial section is MPF-free to let the flow become hydrodynamically fully developed before encountering the first column of MPFs. This length is calculated as [23]:

$$L_{\text{h}} = 0.05D_{\text{h}}\, Re$$
(1)

where Dh is the hydraulic diameter which depends on the channel inlet cross-sectional area, Ac, and its perimeter, p:

$$D_{\text{h}} = \frac{{4A_{\text{c}} }}{p}$$
(2)

Reynolds number is calculated as:

$$Re = \frac{{\rho U_{\text{in}} D_{\text{h}} }}{\mu }$$
(3)

where ρ and μ are the fluid (water) density and dynamic viscosity at T = 300 K, respectively, and Uin is the fluid inlet velocity. A distance of 10D is considered behind the last column of MPFs to ensure the vortices are significantly damped.

Numerical modeling

For each configuration, the simulations are run at five Re values (20, 40, 80, 120 and 150). The fluid inlet temperature is 300 K while a constant HF = 30 W cm−2 is applied through the bottom section of the microchannel and the liquid interacting surfaces of the MPFs. The top and side surfaces of the microchannel are thermally isolated. The outlet boundary opens to the atmosphere. Thermophysical properties (density, thermal conductivity, specific heat and dynamic viscosity) of the fluid are set to vary with temperature for higher accuracy [24]. Table 2 summarizes the boundary conditions:

Table 2 Boundary conditions
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The set of steady compressible continuity, momentum and energy equations (Eqs. 4–6) were solved using the ANSYS FLUENT 14.5 software. Each of the simulations took about 20 h on average to satisfy the residual criterion of 10−5 using workstations with double-processor Intel Xenon 2.4 GHz CPU and 12 GB of RAM. The least-square cell-based interpolation and the second-order upwind methods were used to resolve the gradients of solution variables and interpolation of the field variables, respectively:

$$\frac{{\partial \left( {\rho u} \right)}}{\partial x} + \frac{{\partial \left( {\rho v} \right)}}{\partial y} + \frac{{\partial \left( {\rho w} \right)}}{\partial z} = 0$$
(4)
$$\begin{aligned} \frac{{\partial (\rho u^{2} )}}{\partial x} + \frac{{\partial \left( {\rho uv} \right)}}{\partial y} + \frac{{\partial \left( {\rho uw} \right)}}{\partial z} = - \frac{\partial P}{\partial x} + \frac{1}{Re}\left( {\frac{{\partial \tau_{\text{xx}} }}{\partial x} + \frac{{\partial \tau_{\text{xy}} }}{\partial y} + \frac{{\partial \tau_{\text{xz}} }}{\partial z}} \right) \hfill \\ \frac{{\partial \left( {\rho uv} \right)}}{\partial x} + \frac{{\partial (\rho v^{2} )}}{\partial y} + \frac{{\partial \left( {\rho vw} \right)}}{\partial z} = - \frac{\partial P}{\partial y} + \frac{1}{Re}\left( {\frac{{\partial \tau_{\text{yx}} }}{\partial x} + \frac{{\partial \tau_{\text{yy}} }}{\partial y} + \frac{{\partial \tau_{\text{yz}} }}{\partial z}} \right) \hfill \\ \frac{{\partial \left( {\rho uw} \right)}}{\partial x} + \frac{{\partial \left( {\rho vw} \right)}}{\partial y} + \frac{{\partial (\rho w^{2} )}}{\partial z} = - \frac{\partial P}{\partial z} + \frac{1}{Re}\left( {\frac{{\partial \tau_{\text{zx}} }}{\partial x} + \frac{{\partial \tau_{\text{zy}} }}{\partial y} + \frac{{\partial \tau_{\text{zz}} }}{\partial z}} \right) \hfill \\ \end{aligned}$$
(5)
$$\frac{{\partial \left( {uE} \right)}}{\partial x} + \frac{{\partial \left( {vE} \right)}}{\partial y} + \frac{{\partial \left( {wE} \right)}}{\partial z} = - \frac{{\partial \left( {uP} \right)}}{\partial x} - \frac{{\partial \left( {vP} \right)}}{\partial y} - \frac{{\partial \left( {wP} \right)}}{\partial z} - \frac{1}{Re \times Pr}\left( {\frac{{\partial q_{\text{x}} }}{\partial x} + \frac{{\partial q_{\text{y}} }}{\partial y} + \frac{{\partial q_{\text{z}} }}{\partial z}} \right) - \frac{1}{Re}\left( {\frac{\partial }{\partial x}\left( {u\tau_{\text{xx}} + v\tau_{\text{xy}} + w\tau_{\text{xz}} } \right) + \frac{\partial }{\partial y}\left( {u\tau_{\text{yx}} + v\tau_{\text{yy}} + w\tau_{\text{yz}} } \right) + \frac{\partial }{\partial z}\left( {u\tau_{\text{zx}} + v\tau_{\text{zy}} + w\tau_{\text{zz}} } \right)} \right).$$
(6)

Grid sensitivity analysis

The microchannel length is divided into three sections: (1) before MPFs, (2) MPF section and (3) behind MPFs, as presented in Table 3 for different configurations. In the first and third sections, structured rectangular grids and in the second section, unstructured tri/rectangular grids are used. In addition, to capture the velocity and temperature gradients close to the heating sections, inflation layers are used to refine the grid size in these regions.

Table 3 Length of different sections
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Table 4 presents the details of the three grid networks used to test sensitivity of the results to grid size for the IVdHd configuration. The maximum obtained static temperature in the whole domain is used to compare these cases. Since the results obtained by the second and third grid networks are close, the second network is used for this case. Similar networks are used for all the other configurations.

Table 4 Grid sensitivity analysis
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Validation of numerical results

To validate the numerical model, experimental data of Koşar and Peles [10] (2CLD case which includes 28 columns and 12 rows of circular MPFs in a staggered arrangement with ST = 150 μm and SL = 300 μm) are used (by applying the same operating conditions). Figure 2a, b compares the experimental data and the results obtained from simulation. For the pressure data, the average and maximum differences are 13.9% and 23.4%, respectively. For the temperature difference between the inlet and outlet sections, the average and maximum differences are 5.1% and 10.2%, respectively, which provide sufficiently acceptable accuracy.

Fig. 2
figure 2

Comparison of the simulation data with the results of Koşar and Peles [10]—ΔP and ΔT versus Re

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Post-processing of the results

The hydrodynamic performance of MHS is assessed with two parameters: (1) pressure drop and (2) friction factor. These are expressed in Eqs. 8 and 9, respectively [12]:

$$P_{\text{in}} = \frac{{\int_{\text{in}} {P\,{\text{d}}A} }}{{A_{\text{in}} }}$$
(7a)
$$P_{\text{out}} = \frac{{\int_{\text{out}} {P\,{\text{d}}A} }}{{A_{\text{out}} }}$$
(7b)
$$\Delta P = P_{\text{in}} - P_{\text{out}}$$
(8)
$$f_{\exp } = \frac{2\rho \Delta P}{{N_{\text{col}} G^{2} }}$$
(9)

where ρ is the volumetric averaged density of the fluid, Ncol is the number of MPF columns and G is the mass flux, which is expressed as [12]:

$$G = \frac{\rho Q}{{A_{\hbox{min} } }}$$
(10)

where Amin for the in-line and staggered arrangements is obtained with Eqs. 11a and 11b, respectively [12]:

$$A_{\hbox{min} } = \left( {w^{\prime } - ND} \right)h^{\prime }$$
(11a)
$$A_{ \hbox{min} } = \frac{{S_{\text{T}} - D}}{{S_{\text{T}} }}w^{\prime } h^{\prime }$$
(11b)

The thermal performance of the systems is assessed using Nusselt number. For this purpose, the inlet, outlet, average and wall surface temperatures are calculated using Eqs. 12a–12d, respectively:

$$T_{\text{in}} = \frac{{\int_{\text{in}} {T\,{\text{d}}A} }}{{A_{\text{in}} }}$$
(12a)
$$T_{\text{out}} = \frac{{\int_{\text{out}} {T\,{\text{d}}A} }}{{A_{\text{out}} }}$$
(12b)
$$T_{\text{m}} = \frac{{T_{\text{in}} + T_{\text{out}} }}{{A_{{{\text{h}} . {\text{s}} .}} }}$$
(12c)
$$T_{{{\text{h}} . {\text{s}} .}} = \frac{{\int_{{{\text{h}} . {\text{s}} .}} {T\,{\text{d}}A} }}{{A_{{{\text{h}} . {\text{s}} .}} }}$$
(12d)

which leads to obtaining heat transfer coefficient and Nusselt numbers:

$$h = \frac{q}{{A_{{{\text{h}} . {\text{s}} .}} \left( {T_{{{\text{h}} . {\text{s}} .}} - T_{\text{m}} } \right)}}$$
(13)
$$Nu = \frac{{hD_{\text{h}} }}{k}$$
(14)

where q is the heat input and k is the volumetric averaged thermal conductivity of the fluid.

Finally, the thermal–hydraulic performance index (TPI) is used to take both thermal and hydrodynamic performances into account [25]. It represents the ratio of the normalized Nusselt number over the third root of the normalized friction factor. The normalization is done using the values obtained for the IVdHd configuration at a constant pumping power:

$${\text{TPI}} = \frac{{Nu /Nu_{{{\rm IV}_{\rm d} {\text{H}}_{\rm d} }} }}{{\left( {f /f_{{{\rm IV}_{\rm d} {\text{H}}_{\rm d} }} } \right)^{1/3} }}.$$
(15)

Results and discussion

Hydrodynamic performance

As the fluid passes over the sections with MPFs, the streamlines get disturbed. At a sufficiently high Re, they separate from the surface of the obstacle which creates a negative pressure gradient behind MPFs. In addition to the MPF cross section, the number of MPFs, arrangement type and Re affect the pressure drop and friction factors. In the following sections, the effects of Re, SL/D, ST/D and the arrangement type are analyzed in detail.

Pressure drop

Figure 3 shows the pressure drop values as a function of Re, and Table 5 presents the same data for an easier reference. For all cases, pressure drop increases with Re which is mainly attributed to larger wake length behind MPFs, Fig. 4.

Fig. 3
figure 3

Pressure drop versus Re

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Table 5 Pressure drops for different configurations at different Re values
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Fig. 4
figure 4

Velocity profiles for the SVsHd configuration at the mid-span height at Re = 40, 80 and 120

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Figure 5 depicts the velocity profiles of the IVdHd and IVdHs configurations at the mid-span height at Re = 80. Table 6 presents the wake length behind each MPF column in these two configurations (the values nondimensionalized by D) as well as the ratio of the wake length of these two cases behind each MPF column. As can be seen in Fig. 5, the wake behind the first two columns of the IVdHd configuration is restricted with the MPF of the next column and cannot extend much whereas in the IVdHs configuration, such restriction does not exist. This can be noticed in the wake length ratios which varies between 0.43 (Re = 20) to 3.71 (Re = 150) and is mostly above 1.8 for the first two columns whereas for the third column, the ratio varies between 0.38 (Re = 20) to 0.86 (Re = 120). A similar comparison can be made between the IVsHd and IVsHs configurations which has the same arrangement type and VPR.

Fig. 5
figure 5

Velocity profiles of the IVdHd and IVdHs configurations at the mid-span height (Re = 80)

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Table 6 Nondimensionalized wake lengths for the IVdHd and IVdHs configurations and their ratio
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For staggered arrangements with the same VPR, distinct patterns are seen with VPR = 1.5 and 3. For VPR = 1.5, pressure drop values are very close for the SVdHd and SVdHs configurations. The former has higher values at Re = 20, 40 and 80, whereas the latter has higher values at Re = 120 and 150. For VPR = 3, the SVsHd configuration has higher pressure drops compared to the SVsHs configuration for all Re which shows an opposite trend to the in-line arrangements.

For configurations with the same arrangement type and HPR, such as the IVdHd and IVsHd configurations, the one with VPR = 1.5 has significantly higher pressure drop which is attributed to larger velocity gradients as a result of less available cross section, Fig. 6. A similar comparison can be made for all the other similar pairs.

Fig. 6
figure 6

Velocity profiles of the IVdHd and IVsHd configurations at the mid-span height (Re = 80)

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Finally, for configurations with the same VPR and HPR but different arrangement types, such as the IVdHd and SVdHd configurations, the one with the staggered arrangement has higher pressure drops. The difference is high enough to result in a larger pressure drop even when comparing a staggered arrangement with HPR = 1.5 (e.g., SVdHs), to an in-line arrangement with VPR = 3 (e.g., IVdHd). This is attributed to more numerous regions with significant velocity gradients with the staggered configuration (Fig. 7).

Fig. 7
figure 7

Velocity profiles of the IVsHd and SVsHd configurations at the mid-span height (Re = 80)

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Friction factor

Friction factor directly depends on the density and pressure drop and is inversely dependent on the number of MPF columns and square root of mass flux. The number of MPF columns for the configurations with VPR = 1.5 is 3 (neglecting the additional MPFs in the last column of in-line arrangements), whereas Ncol is 6 for the configurations with VPR = 3. Figure 8 illustrates FF variation with configuration type at different Re values, and Table 7 presents the same data for a better comparison. For each configuration, FF decreases with Re. This reduction, despite the increase in ΔP with Re, suggests that mass flux has a dominant effect.

Fig. 8
figure 8

Friction factor versus configuration type at different Re values

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Table 7 Friction factor for different configurations at different Re values
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The comparison between configuration pairs with the same arrangement type and VPR resembles the related comparison of ΔP in the previous section. The number of Ncol is not important in this comparison as it is the same for each pair leg. For the in-line arrangements, the configurations with HPR = 3 have larger FF as a result of larger ΔP. For staggered arrangements, two patterns exist. For VPR = 1.5, the FF values are very close to each other such that the SVdHd configuration has higher values at Re = 20, 40 and 80 and the SVdHs configuration has larger FF at the other two Re. For VPR = 3, the SVsHd configuration has larger FF.

The comparison between the configurations with the same arrangement type and HPR is more delicate as Ncol in the second leg of the pair is twice the first one. For the in-line arrangements with either the dense or sparse HPR, the configuration with VPR = 3 has larger FF at Re = 20 and 40 and the configuration with VPR = 1.5 has larger FF at higher Re. The additional three MPFs added to the IVdHd and IVdHs configurations make a more detailed comparison difficult. For the staggered arrangement with either the dense or sparse HPR, the configuration with VPR = 3 has a higher FF at all Re.

The comparison of the configuration pairs with the same HPR and VPR but different arrangement types suffers from the additional MPFs as well. Therefore, only the IVsHd and SVsHd, and the IVsHs and SVsHs configurations are compared where the ones with the staggered arrangement have larger FF in each pair.

Thermal performance

Nusselt number is used as the parameter to compare the thermal performance. It depends on the convective HTC, Dh and k. Since Dh is the same for all cases and k does not change significantly, Nu comparison mainly depends on h, which itself is a function of q, Ah.s. and (Th.s. − Tm). Since q and Ah.s. are equal for all configurations, the comparison of h or Nu mainly depends on ΔT.

Figure 9 and Table 8 present the Nu for different configurations. For each configuration, Nu increases with Re. The comparison between configuration pairs with the same arrangement type and VPR shows that at Re = 80, 120 and 150, the configurations with HPR = 3 have higher Nu which is due to a smaller difference between the fluid and heating surface temperatures. At Re = 20 and 40, the pattern is not unanimous as the temperature difference between cases with HPR = 1.5 and 3 are close and surpass each other with no regular pattern.

Fig. 9
figure 9

Nusselt number versus configuration type at different Re values

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Table 8 Nusselt number for different configurations at different Re values
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The comparison of the configuration pairs with the same arrangement type and HPR shows that the configurations with the VPR = 1.5 have higher Nu (with no exception) which is also due to a smaller temperature difference between the fluid and heating surfaces (as a result of less space between MPFs and hotter regions of fluid). Finally, the comparison of the configuration pairs with the same HPR and VPR shows that the staggered arrangements have higher Nu which can be attributed to higher mixing and thus less temperature difference between the heating surfaces and the liquid. Figure 10 shows the temperature profiles at the 20% and 50% height of the SVsHs configuration at Re = 80. It is seen that the flow washes away the heat generated through the MPF surfaces toward the end of the channel.

Fig. 10
figure 10

Temperature profiles of the SVsHs configuration at 20% and 50% heights (Re = 80)

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Thermal–hydraulic performance

Figure 11 illustrates the thermal–hydraulic performance index (TPI) as a function of Re, and Table 9 presents the same data. Since FF and Nu of the IVdHd configuration are used as the normalizing factors, the TPI for this case remains 1 for all Re. The TPI of five configurations is below 1. For the in-line configurations, the patterns are mostly increasing which indicate that as Re increases, the heat removal gain of the systems are better compared to the pressure drop penalty (and FF). The only exception is the IVdHs configuration between Re = 20 and 40. Comparing the configuration pairs with the same VPR, the configuration with HPR = 1.5 has a larger TPI which is mainly due to their higher FF. The only exception is between the IVsHd and IVsHs configurations at Re = 20.

Fig. 11
figure 11

Thermal performance index (TPI) of different configurations versus Re

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Table 9 TPI values for different configurations at different Re values
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For the staggered configurations, two patterns are seen with VPR. For VPR = 1.5, TPI mostly drops with Re which indicates a larger effect of the pressure drop (and FF) compared to the gain of heat transfer. The only exception is the SVdHs configuration at Re = 150. Despite this decreasing trend, TPIs of the SVdHs and SVdHs configurations are the only ones above 1 for all Re which is a result of relatively high Nu and low FF. For VPR = 3, the patterns for both configurations are decreasing at Re = 20, 40 and 80 and increasing at Re = 120 and 150. Comparing the configurations pairs with the same VPR, the ones with HPR = 3 mostly have the higher TPI. The exception is the SVdHd and SVdHs configurations at Re = 20 and 40.

Summary and conclusions

In this study, the hydrodynamic and thermal performances of eight MHS with different in-line and staggered arrangements of MPFs were compared, where the focus was on the effect of arrangement type and pitch ratios. These parameters changed the available cross-sectional area at the MPF locations and affected the wake length behind MPFs which both affect the evaluative parameters. The findings of this study can be summarized as follows:

  • A larger VPR, which provides a larger available cross section, generally results in lower pressure drops and higher Nu.

  • A larger HPR allows the wake to extend more which generally results in higher pressure drops, FF and Nu.

  • With the same HPR and VPR, staggered arrangements generally result in higher pressure drops, FF and Nu.

  • The best overall TPI was achieved with the staggered arrangement with VPR = HPR = 1.5. This configuration has 9.3% (at Re = 20), 7% (Re = 40) and about 6% (Re = 80, 120 and 150) higher TPI compared to the reference case.