Relative comparison of goodness factor for various channel shapes for counter flow microchannel heat exchanger

Abstract

As microelectronic devices have become increasingly complex and miniaturized, the demand for effective cooling solutions has grown substantially. The increasing heat fluxes in integrated circuit (IC) chips, driven by the rapid increase in transistors per square millimeter, have challenged the efficacy of traditional rectangular microchannels. To address these challenges, it is imperative to propose optimized structures or alternative methods for enhanced heat transfer, thereby improving the heat dissipation in these confined spaces and extending the lifespan of components or ICs. Therefore, a three-dimensional numerical analysis was conducted to establish a validated and robust methodology for evaluating novel microchannel shapes that facilitate more efficient heat transfer. The computational 3D model was developed using experimentally tested microchannel heat exchangers (MCHE) and validated with experimental data, showing a variation of 3–5% for hot fluid and 6–12% for cold fluid across the entire design of experiments (DoEs) for the baseline design. This study aims to identify novel channel shapes for MCHE through numerical analysis, which can achieve a higher j-factor and lower f-factor under the same operating conditions as the baseline design. To examine and identify novel channel configurations, segmented channel microchannel heat exchangers (MCHEs) with varying channel cross sections were assessed. The findings were compared to a baseline design of straight MCHEs with equivalent volumes and areas under identical operating conditions, focusing on the goodness factor (j/ƒ) and effectiveness. Upon comparing the goodness factor across all 24 configurations, it was determined that the trend of the superior design mirrored that observed in the hot fluid temperature and balanced flow design of experiments (DoEs), albeit with differing percentage changes. It was further concluded that a 40% step height at the bottom of a perforated straight channel (with holes) exhibited the best performance, followed by a 40% step height at the bottom of a non-perforated straight channel (without holes). The performance of a 20% step height at both the top and bottom surfaces was also comparable to the 40% step height at the bottom. The change in the goodness factor increased by approximately 20%-30% across the entire unbalanced flow DoE, balanced flow DoE, and hot fluid temperature DoE for the aforementioned designs compared to the baseline design. This study aims to guide researchers and industry professionals for developing and modifying their baseline design, when seeking enhanced heat-transfer efficiency with the same volume and area installation.

Introduction

As the name implies, a microchannel heat exchanger (MCHE) is characterized by tubes or channels that are considerably smaller than those found in conventional heat exchangers. Specifically, the hydraulic diameter (D_h) of these tubes’ ranges from 200 μm to greater than 10 μm. Over the past two decades, advancements in thermodynamic systems such as cooling systems and heat pumps have been realized owing to the enhanced understanding derived from prior studies on these systems. Some research also suggests that channels with hydraulic diameters (D_h) between 500 μm and greater than 5 μm can be classified as microchannels. This accumulated knowledge enables designers and process engineers to refine their designs and optimize their functionality. The utilization of microchannel heat exchangers (MCHEs) is expanding in response to the increasing demand for faster and more efficient heat-transfer rates. The primary advantages of MCHEs over traditional heat exchangers are:

• The compact design of the MCHE significantly reduces spatial requirements, thereby allowing the installation of multiple heat exchangers within a limited area. This characteristic enhances the ease of shipping and transportation compared to traditional heat exchangers.

• A microchannel heat exchanger (MCHE) is characterized by a reduced internal volume owing to the presence of tubes with a minimal hydraulic diameter. The incorporation of a thin stationary buffer layer within the MCHE facilitated enhanced heat transfer by reducing the thickness of the boundary layer.

• The microchannel heat exchanger (MCHE) demonstrated an accelerated rate of heat transfer owing to its enhanced surface area relative to its volume.

• The MCHE experienced a substantial reduction in pressure, which enhanced the durability of the system and increased its robustness.

Microelectromechanical systems (MEMS) are highly miniaturized machines enabled by recent advancements in microfabrication and assembly of various small systems, including microchannel heat exchangers, microreactors, and microsensors. These devices collectively fall under the umbrella term microelectromechanical systems¹. Microchannels are characterized by high surface-to-volume ratios, elevated heat-transfer coefficients, and minimal thermal resistance. These systems are typically comprised of electromechanical devices with dimensions ranging from 1 mm to 1 μm, including microchannel heat exchangers, microreactors, and microsensors². Microchannel heat exchangers are particularly advantageous because of their capacity to dissipate substantial amounts of heat from confined spaces. This attribute renders them ideal for applications requiring compact heat energy removal, such as biomedical processes, metrology, telecommunications, cooling high-flux microelectronics, automotive systems, nuclear reactor barriers, fuel processing, aerospace, and chemical industries³. The performance advantages of microchannel heat exchangers, including their application in integrated chemical processing, are well-documented in the literature^4–6. Sommers et al.⁷ provided a comprehensive review of the advantages of ceramic heat exchangers in various applications. Ceramic materials offer several potential benefits over their metal counterparts, notably, their ability to function at elevated temperatures. In addition, ceramics exhibit greater resistance to corrosive chemical environments than metals. The intermittent introduction of oxygen to burn off deposits may effectively remove fouling in oxide ceramics, which are capable of withstanding strongly oxidizing conditions. To optimize the performance of counter-flow heat exchangers, materials with low thermal conductivities were employed to minimize axial wall conduction. Furthermore, heat exchangers (HX) can be categorized based on the surface area density (β), as proposed by Shah and Sekulic (2002)⁸.

• Macro-type HX: β < 700 m²/m³.

• Compact-type HX: When β is 700 m²/m³ < β < 3000 m²/m³.

• Meso type HX: When β is 3000 m²/m³ < β < 15,000 m²/m³.

• Micro-type HX: When β was greater than 15,000 m²/m³.

Ceramics exhibit lower thermal conductivities than other metals. Researchers have conducted experiments to establish correlations between the flow and heat transfer characteristics of heat exchangers. Most authors have developed j and f models that utilize geometrical parameter ratios with minor modifications. However, the development of a unique model is challenging owing to the availability of diverse geometric variables in the literature^2,9,10. Wieting¹¹ formulated empirical correlations for heat transfer and flow friction in rectangular offset fin plate-fin heat exchangers by employing the Reynolds number and structural parameters as independent variables. Joshi and Webb¹² proposed a correlation between the heat transfer factor, or Colburn factor (j), and the fanning friction factor (f) across various Reynolds numbers, based on experimental data from 21 serrated fin groups. Mochizuki et al.¹³ investigated 18 test cores with varying fin geometries in a wind tunnel and derived f and j correlation equations for offset strip fin cores. Shah and Bhatti¹⁴ provided a comprehensive summary of the friction factors and Nusselt numbers of various duct geometries. Kurosaki et al.¹⁵ conducted experiments on heat transfer within louvre arrays. Rachkovskij et al.¹⁶ explored heat transfer in cross-flow micro heat exchangers, focusing on variations associated with reduced tube size and length. Albakhit et al.¹⁷ performed a numerical analysis of heat transfer and flow in parallel-flow microchannel heat exchangers. They employed a hybrid approach to solve the nonlinear momentum equations for one or two channels using computational fluid dynamics (CFD) codes. The user-developed code utilizes a velocity field to solve the energy equation, examining the heat transfer for thermally developing laminar flow in two parallel rectangular channels representing heat exchangers. These findings suggest that the development of velocity profiles in the entrance region enhances the heat-transfer coefficients. Brandner et al.¹⁸ compared the thermal performance of micro-structured cross-flow heat exchangers. Researchers have determined that reducing the hydraulic diameter of microchannels enhances heat transfer in micro heat exchangers. Foli et al.¹⁹ proposed two methods for optimizing microchannel design parameters in MCHEs to maximize the heat transfer rate within specified constraints. The first method integrates CFD with analytical solutions for momentum and heat transfer to optimize microchannel dimensions with a predetermined geometry. Neagu et al.²⁰ employed the finite element method across various channel geometries and identified the optimal channel shape based on MEMS heat exchangers.

In the power generation sector, there is a growing demand for small-scale power plants, such as microscale power plants, owing to the identification of a new working fluid, carbon dioxide (CO₂), which can function in a supercritical regime. These plants are referred to as supercritical CO₂ Brayton-cycle micropower plants. CO₂ is typically selected for feasibility studies of supercritical micro-power plants because of its stability, relative inertness, moderate critical pressure, well-documented thermodynamic properties, non-toxicity, abundance, and low cost. Supercritical CO₂-based Brayton cycle power plants exhibit a higher efficiency than conventional steam power plants²¹. The spatial footprint of these micro power plants is significantly smaller for the same power output when compared to combined cycle, gas, or steam power plants, because CO₂ possesses a higher density than traditional fluids. The efficiency of these plants can be further enhanced by employing a MCHE to extract additional heat from hot flue gases before they are released into the environment, transferring this heat to CO₂ prior to its entry into the generator. This process reduces the power requirements of the boiler, evaporator, or generator²². Furthermore, because of their increased effectiveness and compactness, closed Brayton supercritical CO₂ power cycles present a viable solution for waste heat recovery in bottoming cycles compared with Rankine steam bottoming cycles²³. Honeywell is engaged in the development of a microchannel heat exchanger featuring advanced heat transfer surfaces aimed at enhancing thermal management systems for a fuel-to-air heat exchanger. The utilization of fuel as a heat sink is prevalent in military aircrafts. Typically, fuel-to-liquid heat exchangers are employed within heat transport loops to facilitate the cooling of various components, including engines and hydraulic oil, as well as electronic devices. Honeywell also conducted design trade studies on microchannel heat exchangers, which underscores the potential advantages over conventionally used heat exchangers in aircraft. Specifically, metallic microchannel-based heat exchangers offer a potential reduction in volume and/or weight of 20–30% compared to the current compact plate-fin design²⁴. The substantial rise in copper prices in recent years has prompted engineers and researchers to seek alternative materials that are lighter and possess equal or superior thermal conductivities. MCHE present a viable solution by reducing the equipment weight and enhancing the compactness of the devices, thereby decreasing the overall footprint and weight of the system. Additionally, refrigeration and air-conditioning industries are increasingly incorporating smart and electronic features into their product designs. This trend necessitates the development of more efficient heat transfer methods that do not contribute to the increased product bulk. For example, small-scale and miniature vapor compression refrigeration systems can be utilized to cool individuals in extreme environments through portable devices or cooling electronic devices²⁵. Rapid advancements in the Internet of Things (IoT), artificial intelligence (AI), and machine learning (ML) are expected to generate enormous volumes of data, thereby necessitating the development of data centers with substantial capacity. A significant challenge associated with these datacenters is their cooling requirement. The International Technology Roadmap for Semiconductors (ITRS 2011) has projected that the power dissipation from a microprocessor chip will exceed 800 W by 2026, a substantial increase from the estimated 270 W in 2015²⁵. For instance, the ORACLE SPARC T4 processor used in high-performance computing exhibited a heat dissipation value of 240 W in 2011²⁵. This processor, with a typical die size of approximately 403 mm², resulted in a heat flux value of approximately 0.6 MW/m²²⁵. Current cooling techniques are inadequate for managing high heat flux values observed in such scenarios. Zhu et al.²⁶ performed a numerical study to evaluate the heat transfer effectivenss of wavy microchannel heat sinks with left-right and up-down configurations. The analysis incorporated variables such as wavy amplitudes, wavelengths, channel aspect ratios, and the width ratios of the channel to pitch, while ensuring constant pumping power and channel cross-sectional area. The findings indicated that the up-down wavy configuration consistently outperformed the left-right design when small wavelengths were utilized, regardless of the wavy amplitude, channel aspect ratio, or width ratio of the channel to pitch. Conversely, at larger wavelengths, the performances of both the designs were nearly identical. Jet impingement cold plates (CPs) serve as cooling modules in electric vehicles; however, the existing designs have not been optimized for metal additive manufacturing. Kaood et al.²⁷ evaluated four novel CPs using ANSYS Fluent. CP1–CP3 are dual-chamber designs, with CP2 having jet extensions and CP3 featuring a curved baffle. CP4 was a triple-chamber CP with fluid extraction. Testing at flow rates between 0.212 and 2.121 LPM showed CP4 has the lowest pressure drop (37.36 kPa at 2.121 LPM), while CP3 demonstrated best thermal performance with lowest surface temperatures (326.5 K at 2.121 LPM), lowest peak temperature (336.4 K), highest uniformity (0.9915), and lowest thermal resistance (0.042 K/W). The results indicated an improved cooling potential through changes in the flow architecture.

Hasan et al.²⁸ evaluated internal finning of copper tubes numerically. Finned RCPs were assessed for cooling capacity, temperature uniformity, and hydraulic losses. Triangular, rectangular, and circular fin profiles were studied using varying parameters. The results showed cooling improvements of 4.64 and 45.2% using eight triangular fins (area fraction of 0.3, aspect ratio of 2.5). The cooling rate increased by 27.1% with 12 rectangular fins (area fraction = 0.3, aspect ratio = 2.0). Using a weighted performance index with equal criteria, an optimal design (12 triangular fins, area fraction 0.2, aspect ratio 2.0) improved the cooling capacity and rate by 1.54 and 17.7%, respectively, while increasing the temperature nonuniformity by 3.7% and pressure losses by 3.4 times. Pandey et al.²⁹ examined a microjet impingement microchannel heat sink with porous media through numerical analysis. The performance was compared with that of non-porous media at Reynolds number of 200–500 for fully filled porous channels, bottom porous slabs, side porous slabs, and slotted porous configurations. The fully filled and slotted porous designs showed lower temperatures but higher pressure drops. The slotted porous slabs reduced the pressure drop with a moderate temperature increase. The heat sink with 0.15 mm thick base porous slabs performed better at high Reynolds numbers, while the 0.5 mm slotted porous slab improved at low Reynolds numbers. Nusselt number correlations were developed to optimize the design.

Energy conservation and recovery represent critical areas of focus, driven by the substantial demand and environmental concerns associated with energy consumption. In recent decades, advancements in semiconductor technology have facilitated the production of numerous micro- and nano-scale devices. These technological advancements have enabled the integration of billions of transistors within a compact area, thereby increasing transistor density per unit area. This heightened transistor density results in significant heat generation during device operation, potentially leading to a notable increase in heat flux owing to die shrinkage and other complex design factors. Consequently, localized hotspots may form as a result of the highly nonuniform heat distribution. As microelectrical devices continue to evolve in complexity and miniaturization, the development of effective cooling solutions has become increasingly essential. The escalating heat fluxes in integrated circuit (IC) chips, driven by the exponential increase in transistors per square millimeter, have pushed traditional rectangular microchannels to their limits to address these challenges. It is imperative to propose optimized structures or alternative enhanced heat transfer methods to improve the heat dissipation within these confined spaces and extend the lifespan of components or ICs. MEMS-based MCHEs offer advantages such as superior heat exchange properties, compact design for industrial throughput, and reduced internal volume. Its popularity stems from its versatility and ability to meet the diverse needs of various industries. MCHSs represent a viable solution for cooling microelectrical devices, and researchers are actively investigating innovative designs and optimization techniques to ensure that these cooling solutions meet the evolving demands of modern electronics.

Computational model of existing MCHE and validation

Model description

Developing an accurate numerical simulation approach is essential for virtually evaluating different designs through an analysis. To confirm the simulation outcomes and associated methodology, a simulation model was constructed using the details of the experimental setup and geometry, as published in a previous study³⁰. Comprehensive studies on fluid dynamics and heat transfer were performed using the commercial software CFD-ACE v2021.0 to augment and refine existing research on ceramic-compact counterflow microchannel heat exchangers (CFMCHEs).

Numerical methodology validation

Figure 1 illustrates a MCHE with a flow layer measuring 50 mm × 100 mm, constructed from ceramic materials. The flow layer contains ten microchannels, each with a height of 550 μm and width of 2.8 mm. A plate with a thickness of 600 μm separated the two streams originating from the hot and cold channels. The spacing between each microchannel column and row was 4 mm and 2.83 mm, respectively. The experimental MCHE design and construction incorporated four alumina flow layers, with two layers subjected to heating and two subjected to cooling. The computational model investigated the configuration of four plates, comprising two hot and two cold plates, by altering their positions. All internal flow layers were identical, except for a 180° rotation between adjacent layers, to accommodate the counter-flow arrangement for better heat transfer. Cold fluid entered the system from the right and exited to the left. In a counterflow arrangement, the hot fluid enters from the left and exits to the right, with the objective of optimizing heat transfer. The implementation of parallel layers is driven by the objective of minimizing production costs. The computational domain is characterized by the presence of inlet and outlet pipes, where hot and cold fluids enter through a central aperture located at the bottom and exit through two outlet apertures at the top, as illustrated in Fig. 1. The computational model was developed using CFD-GEOM v2021.0 and analyzed using CFD-ACE v2021.0, as referenced in the study. The assembly features one inlet and two outlet manifolds for each fluid. The microchannel heat exchanger (MCHE) was constructed using alumina material, and air served as the heat-transfer medium in the reference study.

Fig. 1.

The reference geometry details: (a) Plate drawing with dimensions; (b): Computational domain.

The 3D computational model was validated against 21 experimental data points (hot fluid temperature variation (6), balanced flow (6), and unbalanced flow (9)) scenarios, as shown in Tables 1, 2 and 3, with a variation of 3%-5% for hot fluid and 6%-12% for cold fluid across the entire design of experiments (DoEs) with experimental data³¹. The reference geometry and computational domain are shown in Fig. 1. CFD-ACE v2021.0 is utilized to ascertain the flow velocity, pressure, and temperature within the channels of a CFMCHE. The finite volume method (FVM) was employed to convert the governing equations into algebraic equations, accomplished through an “upwind” scheme. The SIMPLE algorithm was implemented to ensure mass conservation and to derive the pressure field. The segregated solver was applied to resolve the governing integral equations for the conservation of mass, momentum, and energy.

Table 1.

Hot fluid Inlet temperature variation study.

S.No.	Mass flow rate		Inlet temperature
	Cold fluid	Hot fluid	Cold fluid	Hot fluid
	kg/s	kg/s	°C	°C
1	0.001970	0.001970	40	500
2	0.001970	0.001970	40	550
3	0.001970	0.001970	40	600
4	0.001970	0.001970	40	650
5	0.001970	0.001970	40	700
6	0.001970	0.001970	40	750

Table 2.

Balanced flow study.

S.No.	Mass flow rate		Inlet temperature
	Cold fluid	Hot fluid	Cold fluid	Hot fluid
	kg/s	kg/s	°C	°C
1	0.000493	0.000493	40	500
2	0.000986	0.000986	40	500
3	0.001479	0.001479	40	500
4	0.001972	0.001972	40	500
5	0.002465	0.002465	40	500
6	0.002958	0.002958	40	500

Table 3.

Unbalanced flow study.

S.No.	Mass flow rate		Inlet temperature
	Cold fluid	Hot fluid	Cold fluid	Hot fluid
	kg/s	kg/s	°C	°C
1	0.001180	0.001158	50	700
2	0.001279	0.001158	50	700
3	0.001378	0.001158	50	700
4	0.001476	0.001158	50	700
5	0.001575	0.001158	50	700
6	0.001674	0.001158	50	700
7	0.001773	0.001158	50	700
8	0.001871	0.001158	50	700
9	0.001970	0.001158	50	700

The results of the computational model and corresponding experimental data of the reference geometry are listed in Table 4.

Table 4.

Outlet temperature during validation results (Experimental vs Numerical).

S.No.	Mass flow rate		Inlet temperature		Experimental Outlet		Numerical Outlet
	Cold air	Hot air	Cold air	Hot air	Cold air	Hot air	Cold air	Hot air
	kg/s	kg/s	°C	°C	°C	°C	°C	°C
1	0.001180	0.001158	50	700	466	390	473	379
2	0.001279	0.001158	50	700	451	381	451	370
3	0.001378	0.001158	50	700	436	370	431	360
4	0.001476	0.001158	50	700	424	361	415	352
5	0.001575	0.001158	50	700	410	351	398	344
6	0.001674	0.001158	50	700	396	344	383	338
7	0.001773	0.001158	50	700	385	335	367	330
8	0.001871	0.001158	50	700	376	329	353	325
9	0.001970	0.001158	50	700	369	325	341	319

The analysis in Table 4 indicates that the prediction accuracy for the hot-fluid outlet temperature consistently surpassed that of the cold fluid. Notably, the geometrical configurations of the hot and cold fluid sections, including all piping, were identical. Given that an identical mesh was employed for both the hot and cold plate sections, including the piping, the accuracy of the cold fluid outlet temperature prediction was inferior to that of the hot fluid. A plausible explanation for this discrepancy is the potential loss of calibration of the thermocouple associated with the cold fluid outlet during testing at the outlet location.

A performance comparison was conducted between the straight, wavy, and venturi channel MCHE under identical conditions. The geometrical details, operating conditions, and results of the velocity, temperature change, and pressure drop can be found in^31,32.

Channel section exploration

The enhanced thermo-hydraulic performance of MCHEs can be determined if any one of the following three criteria is met for identical boundary conditions compared to the baseline design (straight-channel MCHEs, as shown in Fig. 1).

• Attain equivalent effectiveness with reduced pressure loss.

• Enhance the effectiveness of the MCHE while maintaining the same level of pressure drop.

• Enhances the effectiveness of the MCHE while reducing the pressure drop.

Different channel configurations or forms have been investigated to improve thermal effectiveness, as shown in Fig. 2. To evaluate how the transverse shape of the channel affects the confirmed MCHE, the design modification assessment has the following limitations.

Fig. 2.

Various Channel shape Exploration for MCHE^31,32.

• The total number of channels was maintained at ten.

• Each channel measured 14 mm and had the same length.

• The longitudinal space between adjacent channels was maintained at 4 mm.

• The locations and dimensions of the inlet and two outlets remained unchanged for each circuit.

• The microchannels were spaced 4 mm between columns and 2.83 mm between rows.

• The flow remained counterflow, with air as the working fluid and Alumina as MCHE material.

• The partition plate thickness, channel height, and total stack height persisted at 600, 550, and 5650 micrometers, respectively.

Compared to straight MCHE, wavy MCHE showed 12–25% improved effectiveness but 60–80% higher pressure drops under the same conditions, requiring a performance trade-off evaluation³¹. The Venturi CFMCHE showed a similar thermal performance with increased pumping power owing to its throat Sect.³². To enhance the heat transfer performance, longitudinal ribs were added in the middle of each MCHE channel, increasing the heat transfer surface area by 2%, while reducing the flow area. A secondary fluid channel was incorporated into the channel partition with the longitudinal ribs, as shown in Fig. 3. The configurations of the nonperforated and perforated channels are shown in Fig. 3. The hole in the perforated channel partitions was located 0.275 mm from the base at the channel height center. The holes were positioned in the flow direction for all designs (Straight, Wavy and Venturi). Slanted coplanar holes were used in the perforated straight and venturi channels, whereas perpendicular holes were used in the perforated wavy channels.

Fig. 3.

Various Channel shape Exploration for MCHE.

In three out of the four shapes, steps were added with heights of either 20% or 40% of the channel’s height. The terms “BTM” and “TOP_BTM” were assigned to indicate the step located at the bottom and on both the top and bottom surfaces of the flow channels, respectively, as shown in Fig. 3. The designations “20pc” and “40pc” were utilized to indicate step heights that correspond to 20% and 40% of the channel height, respectively. All three basic channel sections (without steps or longitudinal ribs), with and without perforation, are shown in Fig. 4.

Fig. 4.

Perforated and non-perforated for each major channel shape for MCHE.

Success criteria for the two best channel shapes

The baseline design was a straight channel without any steps. Each design’s goodness factor and effectiveness are compared with the baseline to identify two optimal MCHE channel shapes that provide a high goodness factor and effectiveness under identical conditions. These criteria ranked the best shapes from the 24 analyzed models, comparing the outlet temperature, pressure drop, Colburn j factor, fanning friction factor (f), and goodness factor (j/f). The top two shapes are presented when comparing all designs under the operating scenarios from Tables 1, 2 and 3.

Heat exchangers were compared based on their effectiveness, pressure loss, cost, space requirements, and weight. The channel shape selection depends on the mechanical design and thermal performance factors, assessed using the friction factor (f) and Colburn j factor. Conjugate heat transfer has been studied numerically, and temperature-dependent fluid and solid properties have been used in computational fluid dynamics (CFD)³¹. Fluid properties such as density, viscosity, specific heat, thermal conductivity, velocity, and pressure drop were determined using area-weighted cell-centered values during the CFD-VIEW v2021.0 post-processing.

The following two dimensionless parameters were utilized to evaluate the performance of all the channel shapes of the CFMCHE when operating under identical operating conditions to identify the most efficient channel section under the constraints specified in Sect. "Numerical methodology validation".

1. The j factor or Colburn j factor or the Colburn-Chilton j factor³¹.

The Colburn-Chilton factor (j factor) can be expressed as.

1

where.

Nu = Nusselt number.

Re = Reynolds number.

Pr = Prandtl number.

2

• b)Darcy friction factor () or Fanning friction factor ().

The correlation between Darcy and Fanning friction factor can be written as³²

3

To calculate the Darcy friction factor, many formulae and correlations have been listed in the literature. There is a considerable variation in the density of the cold fluid between the inlet and outlet ports, which are separated by a small length of ~ 67 mm. Therefore, the Mach (M) number based on the Darcy friction factor equation is used³³.

4

where.

= Average Darcy friction factor along the duct,

L_max = Maximum attainable duct length with M at the inlet, ft (or m),

D_h = Hydraulic diameter, ft (or m),

γ = Ratio of specific heat of flowing gas.

M₁ = Mach number of gas flow at duct inlet.

M₂ = Mach number of gas flow at duct outlet.

• c)Goodness factor calculationn.

The Goodness factor, defined as the quotient of the j factor and the fanning friction factor, evaluates the heat exchanger effectiveness under identical conditions. A higher goodness factor is preferable, and the shapes with the highest values are selected. This factor can be improved by increasing the j factor with constant friction, reducing friction with a constant j factor, or optimizing both factors simultaneously. An analysis was conducted on all shapes, as shown in Fig. 3, to determine the optimal channel shape for enhanced heat transfer under the same conditions³¹.

5

Governing equations

Numerical studies were performed under steady-state conditions during this stage of research. Therefore, all time-dependent terms in the set of governing equations become zero. The following assumptions were made in the numerical analysis:

a) The fluid flow was laminar, incompressible and continuum.

b) The heat transfer was steady.

c) Radiation heat transfer was negligible.

d) The MCHE was designed and fabricated without distortion in reference study.

e) The internal channel surfaces of the MCHE are smooth.

The commercial package CFD-ACE v2021.0 was used in this study, where geometry was created, analyzed, and post-processed in CFD-GEOM v2021.0, CFD-ACE v2021.0, and CFD-VIEW v2021.0. As researchers, it is essential to know what is happening in the commercial code background while solving the problem. The proper velocity, pressure, and temperature profiles in the computational domain were obtained by solving the continuity, momentum, and energy equations. The simplified set of 3D Cartesian equations for continuity, momentum, and energy conservation after applying the aforementioned assumptions is as follows:

Continuity equation³¹

6

Momentum equation³¹

7

8

9

For steady state, = 0, = 0, = 0, = 0.

= dynamic viscosity, = density, u = velocity in x direction, v = velocity in y direction, w = velocity in z direction, p = pressure.

Energy equation³¹

10

For steady state, = 0.

T = temperature, = specific heat at constant pressure, = thermal conductivity.

Grid generation and grid independence test

The geometries of the hot- and cold-fluid sections were identical, which facilitated the use of a linear extrusion method to construct the model from the bottom plane to the top, thereby ensuring a consistent grid across all sections. This technique enhances control over both the quantity and quality of the grid. By employing this strategy, an adequate number of elements was incorporated into the microchannel, along with a clearly defined boundary layer element. An appropriate grid strategy and appropriate physics applications are essential for successful numerical validation, assuming that uncertainties in the experimental data are disregarded. The full 3D graded domain is shown along with an enlarged view of the meshed surface in Fig. 5.

Fig. 5.

Grid independence test (GIT).

The performance of a computational model is contingent upon the extent of problem discretization and the incorporation of the underlying physics. An appropriate grid must be established to capture the trends accurately. The size of the grid elements and the number of elements within a grid are critical factors in the design. If the grid size is excessively small, it increases the number of grid points, leading to longer computation times and delayed results without additional benefits. Conversely, if the number of grid points or elements is insufficient, the accuracy of the results is questionable, potentially leading to incorrect conclusions. Therefore, it is imperative to perform grid optimization or a grid independence test (GIT) before analyzing the model for all required operating conditions to ensure accurate results. An extensive evaluation of computational models with varying grid numbers, where grid counts were varied from 1x to 3 × (4.2 M to 12.6 M) during the grid independence test to obtain confidence in the results, was undertaken to confirm the consistency of results and trends across the entire DoE.

During the grid independence test, an extensive DoE was executed by varying the hot fluid temperature from 500 to 750 °C in steps of 50 °C, while all other conditions were kept constant. For each scenario, the computational model yielded a temperature drop and rise in the hot and cold fluids, respectively, as the output. The outlet temperature variations in the hot and cold fluids are plotted against the inlet hot-fluid temperature for the computational models in Fig. 6 for the operating conditions listed in Table 1. The analysis reveals that increasing the grid count beyond the optimal grid number has a negligible effect (less than 1% variation) on the outlet temperature of hot and cold fluids with respect to the output from the optimum grid model. The grid was optimized in a similar fashion using a GIT to ensure that the results of the computational domain were unaffected by the type and grid count across all 24 geometries. The comprehensive process and methodology of the GIT have been discussed^31,32.

Fig. 6.

(a) Hot fluid exit temperature during Grid independence test (GIT). (b) Cold fluid exit temperature during Grid independence test (GIT).

Boundary conditions

The three fundamental channel designs, each with eight subvariations, resulted in a total of 24 design variations. These 24 designs are assessed through a comprehensive Design of Experiments (DoE) approach, employing the same methodology utilized during the validation phase for straight channel MCHEs using the same material and boundary conditions, as detailed in Tables 1, 2 and 3.

Results and discussion

The computational models of various MCHE configurations with identical boundary conditions showed that the hot fluid lost heat flowing from the inlet to the outlet, while the cold fluid absorbed heat and gained temperature. Therefore, the densities of the hot and cold fluids increased and decreased, respectively, along the flow direction. The density change in the cold fluid was more significant than that in the hot fluid during the heat exchange. The following conclusions are drawn when the performances of all configurations discussed in Fig. 3 are compared with the existing straight channel without step design in terms of the goodness factor (j/f) and effectiveness (ε) for the same and identical conditions. The Mach number was calculated across the entire domain for all DoE conditions and design configurations. The maximum Mach number observed within the domain was approximately 0.2. Since this maximum Mach number was below 0.3, the flow was deemed incompressible. The nomenclature, “"Wi” is used for with scenarios respectively in legends in this section. Similarly, the nomenclature, “BTM” and “TOP_BTM” are used for step placed at bottom surface and top and bottom surface of flow channels respectively in legends.

Hot fluid Inlet temperature effect

Comparing the results for the goodness factor and effectiveness across configurations (Figs. 7 and 8), the wavy channel shows superior effectiveness owing to its torturous path and longer residence time compared to the straight and venturi configurations, despite causing a higher pressure drop. Steps in perforated channels perform better than non-perforated channels in terms of goodness factors.

Fig. 7.

(a) Goodness factor for without step configuration. (b) Goodness factor for 20% BTM steps configuration. (c) Goodness factor for 20% TOP_BTM steps configurations. (d) Goodness factor for 40% BTM steps configuration.

Fig. 8.

(a) Effectiveness for without steps configuration. (b) Effectiveness for 20% BTM steps configuration. (c) Effectiveness for 20% TOP_BTM steps configuration. (d) Effectiveness for 40% BTM steps configuration.

The j factor in the Venturi channel was lower than that in the straight channel owing to the higher Reynolds number at the throat. Given that the throat length constitutes 30% of the channel length, this effect of the high Reynolds number was not observed in the remaining 70% of the channels. Consequently, the Venturi channel results in lower temperature variations in both cold and hot fluids than straight channels.

The friction factor was determined using the Mach number to account for the changes in the density. An analysis of the goodness factor (j/f) indicates that a 40% step height at the bottom of perforated straight channels yields the best performance, followed by non-perforated straight channels. The Goodness factor increased by approximately 20%-25% compared to the baseline straight channel, as illustrated in Fig. 7(d). Similar changes were observed at a step height of 20% %in the top-bottom configuration, as shown in Fig. 7(c). Venturi designs with 20% top-bottom and 40% bottom steps surpassed the baseline, but underperformed relative to the other configurations.

The effectiveness of the MCHE configurations was calculated and compared, as shown in Fig. 8. Under identical conditions, the top four designs with enhanced goodness factors demonstrated approximately 10% greater effectiveness than the baseline. The wavy channels, both perforated and non-perforated, with a 40% bottom-channel configuration, exhibited an approximately 25% increase in effectiveness relative to the baseline. Comparable changes in effectiveness were observed for configurations with a step height of 20% %in both top and bottom arrangements. Further investigation is still needed to check whether the same observations are observed in balanced and unbalanced flow, when all these designs are further evaluated for identical balanced and unbalanced flow conditions.

Balanced flow effect

The balanced flow scenario was studied using a computational model of all 24 MCHE configurations, and the goodness factor (j/f) was calculated. A comparison of the goodness factor across configurations showed the same trend as the hot fluid temperature DoE, though with different percentages. The design points corresponding to identical subsets in the goodness factor graph overlap at the lowest flow rate. As the cold and hot fluid flow rates increased, the goodness factor points spread with varying slopes, as shown in Fig. 9.

Fig. 9.

(a) Goodness factor for without steps configuration. (b) Goodness factor for 20% BTM steps configuration. (c) Goodness factor for 20% TOP_BTM steps configuration. (d): Goodness factor for 40% BTM steps configuration.

The Goodness factor at 0.000493 kg/s flow is similar across straight, wavy, and venturi designs for identical subsets, but varies with an increased balanced flow rate. At the lowest flow rate, conduction-dominated heat transfer compared to forced convection became dominant as the balanced flow increased. This explains why the values start similarly and diverge with an increased balanced flow across all designs.

The step height of 40% at the bottom in the perforated straight channel performed the best, followed by 40% step height in the nonperforated straight channel. The goodness factor increases by ~ 25%-30% from the baseline at the highest balanced flow rate, while at the lowest flow rate, all designs perform similarly owing to conduction dominance, as shown in Fig. 9(d). Similar changes in the goodness factor were observed at a step height of 20% for the top and bottom configurations in the perforated and non-perforated straight channels from the baseline, as shown in Fig. 9(c). Venturi designs with perforated and non-perforated channels performed better for 20% top-bottom and 40% bottom steps than the baseline but lower than the first two designs.

The effectiveness of the MCHE configurations is calculated and compared in Fig. 10 for a balanced flow. The top four designs with high goodness factors were 2%-10% more effective than the baseline design under identical conditions, as shown in Fig. 10(d). The perforated and non-perforated wavy channels with a 40% bottom channel provide ~ 10%-30% higher effectiveness than the baseline design across flow rates. Similar effectiveness changes were observed at a step height of 20% in the top and bottom configurations compared to the baseline, as shown in Fig. 10(c).

Fig. 10.

(a): Effectiveness for without steps configuration. (b) Effectiveness for 20% BTM steps configuration. (c) Effectiveness for 20% TOP_BTM steps configuration. (d) Effectiveness for 40% TOP_BTM steps configuration.

Unbalanced flow effect

When analyzing unbalanced flow through computational modeling of 24 MCHE configurations with identical boundary conditions, the goodness factor (j/f) results follow trends similar to those of the balanced flow DoE in Fig. 11. The outperforming design trend matches that of the hot fluid temperature and balanced-flow DoEs. The 40% step height at the bottom of the perforated straight channel performs the best, followed by the 40% step height at the bottom in the non-perforated straight channel. The goodness factor increased by ~ 22%-25% from the baseline design for the unbalanced flow rate DoE range, as shown in Fig. 11(d).

Fig. 11.

(a) Goodness factor for without steps configuration. (b) Goodness factor for 20% BTM steps configuration. (c) Goodness factor for 20% TOP_BTM step configuration. (d) Goodness factor for 40% BTM steps configuration.

Similar goodness factor changes were observed at 2 step height of 20% for the top and bottom configurations in the perforated and nonperforated straight channels from the baseline, as shown in Fig. 11(c). Venturi designs with perforated and non-perforated channels performed better for the 20% top-bottom and 40% bottom-step configurations than the baseline, but lower than the first two designs. The effectiveness of the MCHE configurations is calculated and compared in Fig. 12 for unbalanced flow. The top four designs with high goodness factors were 5% more effective than the baseline design under identical conditions across the DoE flow range, as shown in Fig. 12(d). The perforated and non-perforated wavy channels with a 40% bottom channel show 20%-25% higher effectiveness than the baseline design for low to high flow rates. Future studies can optimize the hole location, size, and number to minimize the friction factor and increase the Coburn factor.

Fig. 12.

(a) Effectiveness for without steps configuration. (b) Effectiveness for 20% BTM steps configuration. (c) Effectiveness for 20% TOP_BTM steps configuration. (d) Effectiveness for 40% BTM steps configuration.

Goodness factor correlations

A total of 504 cases were analyzed by evaluating 24 designs (Fig. 3) for 21 data points (Tables 1, 2 and 3). Twelve data points per design built the Colburn j factor correlation in terms of Reynolds and Prandtl numbers. The Coefficients and equations are presented in Tables 5, 6, 7, 8, 9 and 10.

Table 5.

Straight channel with perforation configurations.

		Perforation section geometry
No.	Config.	Avrg. & RMS	Section	Constant	j factor
1	Straight Channel with No Step	Avrg.= 4.9915E-03		C	0.033219
		RMS = 1.3375E-04		n	-0.18749
				m	1.2961
2	Straight Channel with 20pc BTM Step	Avrg.= 5.0236E-03		C	0.038043
		RMS = 4.1499E-05		n	-0.20416
				m	1.3325
3	Straight Channel with 20pc TOP BTM Step	Avrg.= 5.0883E-03		C	0.038621
		RMS = 3.9265E-05		n	-0.20544
				m	1.3360
4	Straight Channel with 40pc BTM Step	Avrg.= 5.1604E-03		C	0.035620
		RMS = 3.6838E-05		n	-0.19592
				m	1.2958

Table 6.

Straight channel with non-perforation configurations.

		Non-perforation section geometry
No.	Config.	Avrg. & RMS	Section	Constant	j factor
1	Straight Channel with No Step	Avrg.= 4.7914E-03		C	0.02297
		RMS = 1.2356E-04		n	-0.2
				m	0.063285
2	Straight Channel with 20pc BTM Step	Avrg.= 5.0167E-03		C	0.037922
		RMS = 4.0949E-05		n	-0.20367
				m	1.32980
3	Straight Channel with 20pc TOP BTM Step	Avrg.= 5.0368E-03		C	0.038564
		RMS = 3.8534E-05		n	-0.20526
				m	1.33450
4	Straight Channel with 40pc BTM Step	Avrg.= 5.2046E-03		C	0.038626
		RMS = 3.4846E-05		n	-0.20512
				m	1.3351

Table 7.

Wavy channel with perforation configurations.

		Perforation section geometry
No.	Config.	Avrg. & RMS	Section	Constant	j factor
1	Wavy Channel with No Step	Avrg.= 4.9485E-03		C	0.036039
		RMS = 3.8573E-05		n	-0.19709
				m	1.3141
2	Wavy Channel with 20pc BTM Step	Avrg.= 5.0193E-03		C	0.036231
		RMS = 3.7405E-05		n	-0.19757
				m	1.3155
3	Wavy Channel with 20pc TOP BTM Step	Avrg.= 5.0204E-03		C	0.036249
		RMS = 3.5391E-05		n	-0.19741
				m	1.3155
4	Wavy Channel with 40pc BTM Step	Avrg.= 5.0397E-03		C	0.037118
		RMS = 3.3167E-05		n	-0.20019
				m	1.3225

Table 8.

Wavy channel with non-perforation configurations.

		Non-perforation section geometry
No.	Config.	Avrg. & RMS	Section	Constant	j factor
1	Wavy Channel with No Step	Avrg.= 4.9775E-03		C	0.036238
		RMS = 3.9235E-05		n	-0.19670
				m	1.3135
2	Wavy Channel with 20pc BTM Step	Avrg.= 5.0236E-03		C	0.036398
		RMS = 3.6927E-05		n	-0.19809
				m	1.3166
3	Wavy Channel with 20pc TOP BTM Step	Avrg.= 5.0483E-03		C	0.036404
		RMS = 3.6523E-05		n	-0.19804
				m	1.3162
4	Wavy Channel with 40pc BTM Step	Avrg.= 5.0519E-03		C	0.036913
		RMS = 3.1954E-05		n	-0.19944
				m	1.3203

Table 9.

Venturi channel with perforation configurations.

		Perforation section geometry
No.	Config.	Avrg. & RMS	Section	Constant	j factor
1	Venturi Channel with No Step	Avrg.= 4.7668E-03		C	0.035586
		RMS = 3.2737E-05		n	-0.19998
				m	1.2323
2	Venturi Channel with 20pc BTM Step	Avrg.= 4.7939E-03		C	0.03647
		RMS = 1.2381E-04		n	-0.20147
				m	1.2845
3	Venturi Channel with 20pc TOP BTM Step	Avrg.= 4.8621E-03		C	0.040806
		RMS = 1.2282E-04		n	-0.21276
				m	1.3420
4	Venturi Channel with 40pc BTM Step	Avrg.= 4.9059E-03		C	0.03717
		RMS = 9.7309E-05		n	-0.20076
				m	1.311131

Table 10.

Venturi channel with non-perforation configurations.

		Non-Perforation section geometry
No.	Config.	Avrg. & RMS	Section	Constant	j factor
1	Venturi Channel with No Step	Avrg.= 4.8143E-03		C	0.036687
		RMS= 3.6123E-05		n	-0.19952
				m	1.3225
2	Venturi Channel with 20pc BTM Step	Avrg.= 4.8483E-03		C	0.037377
		RMS= 3.5985E-05		n	-0.20162
				m	1.3280
3	Venturi Channel with 20pc TOP BTM Step	Avrg.= 4.9320E-03		C	0.037129
		RMS= 4.2828E-05		n	-0.20087
				m	1.3222
4	Venturi Channel with 40pc BTM Step	Avrg.= 4.9842E-03		C	0.037676
		RMS= 3.3748E-05		n	-0.20190
				m	1.3294

Conclusions

A detailed simulation study was performed to validate the computational model using experimental data published in reference. set-up discussed. The results were validated with experimental data with 3% -5% variation for hot fluid and 6%–12% for cold fluid for the entire design of experiments (DoEs). Further study was also carried out by applying a similar methodology for segmented straight, wavy, and venturi channel shapes of MCHE with perforated and non-perforated channels with different step heights for identical conditions, and the results were compared with the same volume and area straight channel without step MCHE baseline design for various identical operating conditions. During the study and identification of novel channel shapes, segmented channel MCHE with different channel cross sections were evaluated, and the results were compared with the same volume and area straight MCHE baseline design for various identical operating conditions in terms of the goodness factor (j/f) and effectiveness. When the Goodness factor is compared for all 24 configurations, it is concluded that trend of outperforming design is identical as observed in hot fluid temperature and balanced flow DoEs though the % change is differed. It is also concluded that 40% step height at bottom in perforated straight channel (With holes) still performs best followed by 40% step height at bottom in non-perforated straight channel (Without hole). The change in Goodness factor is increased ~ 20%-30% for entire unbalanced flow DoE, balanced flow DoE and hot fluid temperature DoE for aforementioned designs from baseline design. The ranking of all evaluated channel cross-sections is given in descending order of thermohydraulic performance as follows:

Straight_40pc_BTM_Wi_Holes > Straight_40pc_BTM > Straight_20pc_TOP_BTM_Wi_Holes > Straight_20pc_TOP_BTM > Venturi_4mm_40pc_BTM_Wi_Holes > Venturi_4mm_40pc_BTM > Venturi_4mm_20pc_TOP_BTM_Wi_Holes > Venturi_4mm_20pc_TOP_BTM > Straight_20pc_BTM_Wi_Holes > Straight_20pc_BTM > Venturi_4mm_20pc_BTM_Wi_Holes > Venturi_4mm_20pc_BTM > Straight_Wi_Holes > Straight > Venturi_4mm_Wi_Holes > Venturi_4mm > Wavy_40pc_BTM_Wi_Holes > Wavy_40pc_BTM > Wavy_20pc_TOP_BTM_Wi_Holes > Wavy_20pc_TOP_BTM > Wavy_20pc_BTM_Wi_Holes > Wavy_20pc_BTM > Wavy_Wi_Holes > Wavy.

The recommended design has been tested experimentally in the laboratory after building through additive manufacturing and numerically varied³⁴.

List of symbols

hydraulic diameter, mm

A_a

wave amplitude

P_f

fin pitch

Hp

plate space

H_f

fin height

T

temperature

C_p

specific heat

K_t

thermal conductivity

u

velocity in x direction

v

velocity in y direction

w

velocity in z direction

p

pressure

f

friction factor, dimensionless

j

Colburn j factor

distance between two inlet ports of hot and cold fluids, m

Nu

Nusselt number, dimensionless

Pr

Prandtl number, dimensionless

Re

Reynolds number, dimensionless

St

Stanton number, dimensionless

Average Darcy friction factor along the duct

L_max

Maximum attainable duct length with M at the inlet, ft (or m)

D

Duct diameter, ft (or m)

γ

Ratio of specific heats of flowing gas

M

Mach number of the gas flow at the duct inlet

Greek Symbols

effectiveness

δ_f

fin thickness

έ

ineffectiveness

µ

viscosity, kg/m-s

λ

fin length of unit offset strip and, which is also wavelength

pressure drop, Pa

density, kg/m³

Subscripts

i

inner

o

outer

Abbreviations

CFD

Computational fluid dynamics

DoE

Design of experiment

HTC

Heat transfer coefficient

MCHE

Microchannel heat exchangers

MEMS

Microelectromechanical systems

Avrg.

Average value

RMS

Root Mean Square value

Author contributions

Vishwas Kumar Pandey: Investigation, Data curation, Writing-original draft, Methodology, and ValidationVinay Pratap Singh Negi: Writing-original draft, Review, and EditingChennu Ranganayakulu: Resources, Editing, Supervision, and Final Approval.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.