Experimental Investigation and Predictive Modeling of Two-Phase Flow Resistance in Superhydrophilic Bi-Porous Microstructures

Abstract

Superhydrophilic micro/nano-porous media have extensive applications in electronic thermal management and energy storage systems. Predicting two-phase pressure drop in complex porous structures is of great importance for system design and optimization while remaining highly challenging. This study systematically investigates the two-phase flow resistance characteristics of bi-porous microstructures with multiple particle sizes and porosities under varying boiling regimes. Experimentally, porous samples were fabricated via vacuum sintering using nickel powders and pore-forming agents (CaCl₂), which exhibit superhydrophilicity and enhanced wicking characteristics. A visualized experimental platform was constructed to investigate the impact of pore size combinations, flow velocities, and boiling states on pressure drop. The dataset obtained through multi-factor saturated boiling experiments was further used to derive a semi-empirical model for the two-phase flow pressure drop based on the classic Kozeny-Carman (K-C) and Akagi-Chisholm (A-C) correlations. Results show that the pore size combinations and boiling states have a significant impact on the resistance performance. The proposed model achieves an average prediction deviation below 15.7%, confirming its reliability and its effectiveness as a design framework for optimizing high-capillary-force porous wicks in advanced thermal management systems.

1. Introduction

The exponential surge in power density within modern electronics has rendered thermal management a critical bottleneck for system reliability. Consequently, Loop Heat Pipes (LHPs) have emerged as a premier solution, distinguished by their passive operation and superior heat transfer efficiency driven entirely by phase-change mechanisms [1,2,3,4,5,6]. The performance of an LHP is fundamentally governed by its capillary wick, a porous component characterized by composite micro/nanoporous structures that generate the necessary pumping pressure to sustain fluid circulation. To accommodate escalating heat loads, wick architectures have evolved from traditional sintered powders [7,8,9] and reticulated foams [10,11,12,13,14] to advanced porous metasurfaces [15,16] and high-performance manifold microchannels [17,18]. Currently, research on capillary wicks spans interdisciplinary fields such as thermodynamics, fluid mechanics, and materials science. However, a persistent design paradox remains: structures optimized for high capillary force often incur excessive flow resistance, potentially choking the system—a challenge that has emerged as a pivotal research direction [19]. The pressure drop across the porous structure within the LHP evaporator is a decisive parameter for evaluating the thermodynamic cycle performance. Unlike periodic microchannels where flow dynamics are readily predictable, quantifying the two-phase pressure drop in random, multi-scale porous media remains a formidable challenge due to their stochastic pore connectivity [20,21]. Understanding these complex flow resistance characteristics is not merely a theoretical gap but a decisive factor for optimizing the thermodynamic performance of next-generation cooling systems.

Existing studies on boiling pressure drop and heat transfer characterization in porous media have been focused on three aspects, including numerical simulation, experimental measurement, and analytical interpretation. Experimentally, various researchers have systematically analyzed the thermal and hydraulic properties within the aforementioned porous structures [22,23,24,25,26,27,28,29], aiming to achieve a delicate balance between these two factors by leveraging their trade-off relationship, rather than quantifying the influence of thermophysical properties of the working fluid, geometric parameters of porous media, and operating conditions on the flow resistance. Lage et al. [30] reported two types of nonlinear pressure-drop-versus-flow-rate relations. Dukhan et al. [31] assessed the pressure drops of water flow through metal foams in various flow regimes using an experimental dataset. Zhong et al. [32] proposed a novel charge method for pressure drop evaluation of air in porous sintering SUS 316L samples. The calculation showed good agreement with the testing results when combined with validated permeability and inertia coefficients, while the thermal effect of fluids was not considered. Vanderlaan and Van Sciver [33,34] studied heat-flow-induced pressure drops of superfluid helium (He II) contained in porous media. The steady-state pressure drop across the random packs of uniform-sized polyethylene spheres varies with different heating powers. Hu et al. [35,36] experimentally analyzed the effect of tube diameter on pressure drop characteristics of refrigerant-oil mixture flow boiling in metal-foam-filled tubes, which demonstrated a pressure drop decrease resulting from incomplete cells and randomly chopped ligaments near the wall. Liao et al. [37] systematically explored the relationship between pressure drop and flow rate of heated porous steel alloy samples using air as the test fluid. Interestingly, the effect of heating power on the average density of air in porous materials on flow resistance was studied using local thermal non-equilibrium conditions. For numerical modeling, Yue et al. [38] employed the Lattice Boltzmann method (LBM) to analyze horizontal gradient porous metal pool boiling, revealing that thickness gradients in porous metals separate liquid-gas channels, thereby reducing bubble rise resistance. Additionally, hydrophobic side regions enhance heat transfer under low heat flux conditions. Sivasankaran et al. [39] investigated the significant effect of copper nanofluids on improving boiling heat transfer coefficients using CFD simulations with Eulerian-Eulerian two-phase models, while the enhancement plateaus beyond a critical particle concentration. Qin et al. [40] developed a numerical boiling model for metal foams via LBM simulations, finding that increasing foam thickness and pore density degrade pool boiling performance due to enhanced bubble escape resistance. The numerical studies related to nanofluid flow in metal foam-embedded pipes conducted by Mohammed et al. [41] demonstrated that foam structures enhanced steam generation and heat transfer during elevated pressure drop. Ranjan et al. [42] optimized wick thickness for bubble growth in microscale porous media using a volume-of-fluid (VOF) multiphase fluid model, while Mondal et al. [43] used a modified LBM to show that porous media accelerate nucleation and heat flux. Dhavale et al. [44] evaluated heat exchangers with partial metal foam filling, identifying optimal configurations via Nusselt number and Colburn j-factor analyses.

Few studies have involved the analytical modeling of pressure drop during phase transitions. Bamorovat et al. [22] developed a semi-empirical correlation for pressure drop prediction of metal-foam-filled mini tubes. The new model was optimized based on the Cicchitti model (empty tube), considering the effects of pore density, mass flux, and vapor quality. Additionally, Charnay et al. [45] summarized the popular correlations for predicting pressure drops in mini-tubes. Inayat et al. [46,47] derived a universal pressure drop correlation for open-cell metal foams, combining the basic Hagen-Poiseuille equation and geometric tortuosity, which accurately characterized the influence of geometric specific surface area on pressure drop. Subsequently, Weise et al. [48] described the pressure drop of two-phase flow in metal sponges at high mass fluxes using a homogeneous model derived from the single-phase Forchheimer equation. A total of 80% of all experimental data can be accurately predicted with known geometric properties of the corresponding metal foams.

Collectively, previous endeavors have primarily focused on evaluating the pressure drop of heat exchangers with regular geometric shapes, e.g., metal foams. However, predicting the resistance characteristics of two-phase flow in bi-porous structures is rarely mentioned, which is of significance in assessing the performance of capillary wicks in electronic cooling and thermal storage applications. Such bi-porous structures feature two distinct pore scales: macropores (typically 10–100 μm), which facilitate bulk flow, and micropores (sub-μm), which enhance capillary retention. This hierarchical configuration optimizes fluid transport by striking a balance between high permeability and strong capillary forces [49,50,51,52]. Figure 1 illustrates the application scenario of a passive two-phase loop cooling system in integrated electronic devices (Figure 1a) and the boiling heat transfer process inside the porous evaporator. By comparing single- and two-phase flow within bi-porous media, it can be seen that as a large number of bubbles emerge from the nucleation sites, the frictional resistance between the vapor and liquid phases significantly impedes the mainstream flow. In contrast, single-phase flow is mainly affected by near-wall friction with the solid matrix.

The discrepancy underscores the imperative to develop a mechanistic model characterizing the coupled effects of pore size combination, flow velocity, and boiling states on pressure drop, which is the focus of the current study. However, it should be noted that the two-phase pressure drop distribution was experimentally investigated under controlled slow-flow rather than pure capillary suction process, aiming to ensure a stable two-phase state within the porous medium, and to avoid dryout with insufficient liquid replenishment, which could otherwise affect the evaluation of the mechanisms between flow resistance and multiple parameters. We developed a semi-empirical pressure drop correlation derived from the Akagi-Chisholm (A-C) model, a widely recognized approach for multi-phase flow systems that effectively captures gas-liquid interactions within porous media. Within this framework, the Kozeny-Carman (K-C) equation was also employed to predict the pressure gradient of the single-phase (liquid) flow, explicitly linking flow resistance to the medium’s porosity and permeability.

Building upon this theoretical framework, the present study specifically focuses on superhydrophilic bi-porous microstructures under saturated boiling conditions. To validate the proposed model and address the challenge of quantifying flow resistance in random media, bi-porous nickel samples with superhydrophilic properties and enhanced capillary performance were prepared and systematically investigated using a visualization-based experimental platform. Through this approach, a direct relationship between multiscale structural features and hydraulic performance is established, thereby providing theoretical support for the design and optimization of high-capillary-force wicks in loop heat pipe systems.

2. Experimentation

2.1. Sample Preparation and Characterization

This work focuses on the pressure losses experienced by two-phase fluids flowing through nickel-based bi-porous microstructures, which exhibit superhydrophilicity and the desired capillary performance. Samples were prepared by vacuum thermoforming (ZT-40-21Y, Shanghai Chenhua Technology, Shanghai, China) of chain-like nickel powder (Shanghai Maoguo Technology, Shanghai, China, purity $>99.9\%$) as the metal substrate, and CaCl₂ (Hunan Zhicheng Chemistry, Zhuzhou, China, purity $>96.0\%$) powders were used as additives to enhance pore connectivity and surface wettability. It should be noted that CaCl₂, applied as a pore-forming agent, was dissolved via post-processing ultrasonic treatment (Figure S1), which creates nanopit morphologies on the metal particle surface during the vacuum sintering, thereby significantly enhancing hydrophilicity. A total of 24 samples with varying mass fractions (f) of CaCl₂ ($2\%$, $4\%$, and $6\%$), particle sizes of CaCl₂ ($d_{{\mathrm{CaCl}}_2}$) (80 μm and 150 μm), and nickel ($d_{\mathrm{Ni}}$) (2 μm, 15 μm, 30 μm, and 50 μm) were fabricated to examine the influence of various combinations of formed and interstitial pores on the pressure drop of two-phase flow [53,54]. Besides, other four samples were prepared and measured for model validation. As listed in

Table 1. Bi-porous sample preparation used for flow resistance testing.

Sample No.	${\mathit{d}}_{\mathbf{Ni}}$ (μm)	${\mathit{d}}_{{\mathbf{CaCl}}_2}$ (μm)	${\mathit{f}}_{{\mathbf{CaCl}}_2}$ (%)	$\mathit{\epsilon }$	${\mathit{d}}_{\mathit{p}}$ (μm)
Ni-2-80(2)	2	80	2	0.602	6.4
Ni-2-80(4)	2	80	4	0.635	5.8
Ni-2-80(6)	2	80	6	0.651	3.9
Ni-2-150(2)	2	150	2	0.588	5.3
Ni-2-150(4)	2	150	4	0.637	5.9
Ni-2-150(6)	2	150	6	0.674	6.4
Ni-15-80(2)	15	80	2	0.532	12.3
Ni-15-80(4)	15	80	4	0.487	15.9
Ni-15-80(6)	15	80	6	0.518	16.7
Ni-15-150(2)	15	150	2	0.475	20.9
Ni-15-150(4)	15	150	4	0.530	25.4
Ni-15-150(6)	15	150	6	0.608	23.1
Ni-30-80(2)	30	80	2	0.345	40.9
Ni-30-80(4)	30	80	4	0.378	34.6
Ni-30-80(6)	30	80	6	0.423	37.9
Ni-30-150(2)	30	150	2	0.386	29.8
Ni-30-150(4)	30	150	4	0.451	35.4
Ni-30-150(6)	30	150	6	0.470	36.3
Ni-50-80(2)	50	80	2	0.314	19.7
Ni-50-80(4)	50	80	4	0.352	28.3
Ni-50-80(6)	50	80	6	0.362	24.4
Ni-50-150(2)	50	150	2	0.383	23.6
Ni-50-150(4)	50	150	4	0.421	13.9
Ni-50-150(6)	50	150	6	0.439	15.8
Ni-2-50(3) val_1	2	50	3	0.612	6.21
Ni-15-80(5) val_2	15	80	5	0.469	17.2
Ni-30-150(4) val_3	30	150	4	0.351	39.4
Ni-50-220(6) val_4	50	220	6	0.363	22.1

, the sample porosities (ε) were measured by a Mercury Intrusion Porosimetry (MIP) (AutoPore V, Micromeritics). The microstructure of typical samples is shown in Figure 2, demonstrating that the combination of samples with different particle sizes produces significantly different connected metal frameworks as the flow channel, which substantially affects the single- and two-phase flow characteristics within porous media. Details on sample pre-treatment, sintering, and post-treatment processes can be found in Figure S1.

The microstructures of typical samples were characterized using Scanning Electron Microscope (SEM) techniques, as shown in Figure 2a–l. It is demonstrated that the particle size (Ni and CaCl₂) combination significantly affects their pore distributions and connectivity. The addition of CaCl₂ forms larger pores through space occupation, which is particularly prominent in the Ni-2 μm samples (Figure 2a–c) and Ni-15 μm samples (Figure 2d–f). For those with large particle sizes, (Ni-30 μm and Ni-50 μm) as shown in Figure 2g–l, maintaining large sizes with irregular geometries results in a porous framework with superior connectivity, composed of interstitial and formed pores.

Furthermore, the pore size distribution (PSD) obtained from the MIP testing illustrates that all samples exhibit a considerable high pore volume fraction (>1%) in the sub-10 μm range, including the Ni-30 μm and Ni-50 μm samples, as shown in Figure 2o,p. This suggests the presence of fine pore networks across all porous structures, regardless of the intial power size. Besides, the effect of powder size on PSD is more pronounced in the large-pore-size range (10–200 μm). As revealed by the magnified PSD analysis, the Ni-2 μm samples display distinct peaks at approximately 80 μm and 150 μm (Figure 2m), which are attributed to the formed pores induced by CaCl₂. In contrast, for the Ni-30 μm and Ni-50 μm samples, besides the formed pores, large interstitial pores generated by particle stacking are observed. These interstitial pores possess larger characteristic dimensions than those in the Ni-2 μm and Ni-15 μm samples, resulting in an enhanced PSD intensity around 150 μm, as shown in Figure 2o,p. The PSD characteristics are consistent with the microstructural features observed in the SEM images. The capillary penetration behavior further reflects the influence of pore structure on transport performance. As shown in Figure 2q–s, the Ni-2-80(6) samples completely absorbs a 2-μL droplet within 25.8 milli-seconds (ms), demonstrating its superhydrophilic properties. With increasing powder size, the suction times (Figure S2) increase to 171.6 ms, 514.9 ms and 724.5 ms for the Ni-15 μm, Ni-30 μm and Ni-50 μm samples, respectively. Notably, the 2 μm samples exhibit the most rapid penetration, primarily driven by the enhanced capillary pumping effect in the finer pores. This trend is attributed to the enlargement of pore size, which weakens capillary driving force while simultaneously reducing the internal flow resistance. These observations provide a microstructural basis for understanding the subsequent pressure drop behavior of the porous media.

2.2. Experimental Setup and Measurement

The experimental setup for measuring the boiling pressure drop in superhydrophilic bi-porous microstructures is shown in Figure 3a, including an evaporation testing section, a heating module, a circulation module, a measurement module, and a visualization module. As the core of the testing system, the evaporation section is where the boiling process occurs in the porous media, driven by six nichrome electronic heaters covered by heat-resistant magnesium rods and 304 stainless steel sleeves with a diameter of 6 mm and a length of 50 mm. Cartridge heaters were installed into the oxygen-free copper (OFC) block (${\lambda }_{\mathrm{OFC}}=401\mathrm{W}·{\mathrm{m}}^{-1}·{\mathrm{K}}^{-1}$ at 20 °C) as a heating base and powered by a stabilized direct current (D.C.) power supply (APS400B, Ivy Tech, Shenzhen, China) that permits to provide up to 720 W. A circulation pump with volume flow rate monitor (DP6G14-PC-14, Suzhou Zhonglu, Suzhou, China) was applied to drive the working fluid (deionized water) through the entire setup, maintaining a controllable inlet temperature and flow rate before boiling pressure drop testing. During the experiments, the DC current was maintained at a constant level of 0.6 A. The input power was regulated by adjusting the voltage, and the heating power was calculated based on the measured voltage and current values ($P=U\times I$). A mass flowmeter (DN6, Asmik, accuracy of $0.2\%$) was installed to measure the mass flow rate of the outflow, which can be used to estimate the quality, as explained in the data reduction. The measurement system monitors the temperature and pressure of the system. The visualization system enables the observation of the boiling state within the porous structures during the testing process, which comprises a high-speed camera (FASTCAM MINI AX 100, Photron, Tokyo, Japan) and a microscope lens (2X F-MOUNT, NAVITAR, Rochester, NY, USA). Fluoroelastomer (FKM) was applied to ensure a seal between all components in the boiling testing section.

A total of six K-type thermocouples (a diameter of 1 mm) were used for temperature measurement. Four of them were embedded in the OFC heating block, arranged at specific locations to monitor the temperature gradient and calculate the heating power density, as shown in Figure S3. Two thermocouples ($T_{\mathrm{in}}$ and $T_{\mathrm{out}}$) were installed at the center of the flow channel near the fluid inlet and outlet. All temperature data were recorded by the data acquisition system (2638A, Fluke, Everett, WA, USA) at intervals of 0.5 s with an accuracy of ±0.1 °C. The measured values used for data reduction were time-averaged over 2 min under pseudo-steady conditions, defined as temperature changes of less than 0.2 °C within 1 min. As displayed in Figure 3b,c, the pressure drop when water flowed through the samples and volume flow meter was measured using two differential pressure transducers (MIK-2051, Asmik, Hangzhou, China) located at the inlet and outlet of the flow meter and evaporation section. Moreover, two absolute pressure transducers (PT131H, Shanghai YAMEN, Shanghai, China) situated on the inlet flange of the volumetric flow meter and testing section were applied for calibration.

2.3. Experimental Method

Before the pressure drop experiments, nickel samples (Figure 3d) were bonded to the OFC heating block using a high-temperature lead-free solder paste (Sn₉₉Ag_0.3Cu_0.7) to reduce contact thermal resistance. Bi-porous nickel samples shaped in strips were machined using wire electrical discharge machining (EDM), which were subsequently truncated into small blocks ($10\times 10\times 7$ mm) to maintain the original cross-sectional surface at the inflow and outlet directions to avoid the pore blockage caused by EDM processing. The inlet working fluid was kept at 70 °C ($\Delta T_{\mathrm{subcool}}$ = 30 °C), and the volume flow rates ($\dot{v}$) were set at 0.4, 0.6, 0.8, and 1.0 mL/min for the varying flow rate scenarios. The selection range of flow velocity was determined according to the capillary performance of the self-driven loop heat pipe [9]. The heating power was increased in 1 W increments from 12 W to 15 W to maintain various boiling states, while no heat flux was introduced in the single-phase flow testing. Each sample was tested five times under specific working conditions to ensure repeatability. To ensure accurate thermal boundary conditions, the test section was enclosed in a PEEK housing, and the entire flow loop was wrapped with thermal insulation cotton (∼$0.030\mathrm{W}·{\mathrm{m}}^{-1}·{\mathrm{K}}^{-1}$) to minimize heat loss to the ambient.

2.4. Uncertainty Analysis and Data Reduction

2.4.1. Uncertainty Analysis

(1)

Uncertainty in pressure measurement

The uncertainty of pressure measurement is estimated based on the accuracy of the pressure transducers. Two types of pressure sensors were employed in this experiment: an absolute pressure transducer with a full-scale range of 0–100 kPa and a differential pressure transducer with a full-scale range of 0–2 kPa. The manufacturer-specified accuracies are ±0.25 kPa and ±0.002 kPa, corresponding to 0.25% and 0.1% of full scale, respectively.

The maximum indicated error for the absolute pressure transducer:

$$\Delta p_{abs}=0.25\mathrm{kPa}$$

(1)

The maximum indicated error for the differential pressure transducer:

$$\Delta p_{diff}=0.002\mathrm{kPa}$$

(2)

Assuming a rectangular distribution, the Type-B standard uncertainty of a single pressure reading from the absolute pressure sensor and the differential pressure transducer, respectively, are:

$$u\left(p_{abs}\right)={\displaystyle \frac{\Delta p_{abs}}{\sqrt{3}}}=0.144\mathrm{kPa}$$

(3)

$$u\left(p_{diff}\right)={\displaystyle \frac{\Delta p_{diff}}{\sqrt{3}}}=0.00115\mathrm{kPa}$$

(4)

If the pressure drop is determined from two independent absolute pressure measurements, the combined standard uncertainty is

$$u(\Delta p)=\sqrt{u{\left(p_{abs}\right)}^2+u{\left(p_{diff}\right)}^2}=0.204\mathrm{kPa}$$

(5)

(2)

Uncertainty in temperature measurement

The K-type thermocouples used in this experiment have a measuring range of 200 to 800 °C with an individual measurement error of ±0.5 °C. The lowest temperature recorded in the experiment is 200 °C, and the uncertainty in temperature measurement is calculated as follows:

$$\left|U\left(T\right)\right|={\displaystyle \frac{0.5}{200}}=0.25\%$$

(6)

(3)

Uncertainty impact of wall temperature ($T_w$) on heat transfer coefficient (HTC)

Because the wall temperature $T_w$ is obtained by extrapolating from temperature sensors embedded inside the heating block, its sensitivity to the sensor location uncertainty $\Delta z$ can be estimated using a one-dimensional conduction approximation:

$$\Delta T\approx {\displaystyle \frac{q^{″}\Delta z}{{\lambda }_{OFC}}}$$

(7)

In this work, the heating block is oxygen-free copper (OFC) with ${\lambda }_{OFC}=401\mathrm{W}·{\mathrm{m}}^{-1}·{\mathrm{K}}^{-1}$. Considering machining and installation tolerances for the embedded sensors (PT100/thermocouples), a conservative location uncertainty of $\Delta z=0.1–0.2\mathrm{mm}$ is assumed. For a representative high heat flux condition in this study, $q^{″}=350{\mathrm{W}/\mathrm{cm}}^2$, the resulting extrapolation uncertainty is

$$\Delta T_w(\Delta z=0.1\mathrm{mm})\approx 0.87 \mathrm{K}$$

(8)

$$\Delta T_w(\Delta z=0.2\mathrm{mm})\approx 1.74 \mathrm{K}$$

(9)

Therefore, at the highest heat flux level of this study, the extrapolation uncertainty introduced solely by sensor location is typically within about 0.9–1.7 K. When combined with the temperature measurement accuracy (±0.1 °C) and the uncertainty associated with gradient fitting, the absolute uncertainty of $T_w$ can reasonably be considered to be on the order of about 1–2 K. The relative uncertainty of HTC can be approximated by

$${\left({\displaystyle \frac{u\left(h\right)}{h}}\right)}^2\approx {\left({\displaystyle \frac{u\left(q^{″}\right)}{q^{″}}}\right)}^2+{\left({\displaystyle \frac{u(\Delta T)}{\Delta T}}\right)}^2$$

(10)

For the typical superheat range in this work, $\Delta T\approx 20$–$60\mathrm{K}$, if $u\left(T_w\right)\approx 1$–$2\mathrm{K}$, the contribution from the temperature difference term is approximately 1.7–10%. Accordingly, the absolute uncertainty in HTC associated with the temperature difference term is on the order of several percent up to about 10%.

(4)

Uncertainty in power supply measurement

The uncertainty associated with the D.C. power supply and related components is determined using the Type B uncertainty formula, expressed as:

$$U\left(E\right)={\displaystyle \frac{O}{K}}$$

(11)

where O represents the nominal error specified in the instrument manual, given as ±1%. K is the coverage factor ($\sqrt{3}$), resulting in a power supply uncertainty of 0.577%.

(5)

Uncertainty in input heat load

The uncertainty in the input heat load is calculated using the following formula:

$$U\left(Q_{in}\right)={\displaystyle \frac{\sqrt{{\left(I·u\left(V\right)\right)}^2+{\left(V·u\left(I\right)\right)}^2}}{V·I}}$$

(12)

The calculation results indicate that the error remains within the acceptable range.

2.4.2. Data Reduction

The quality (x), defined as the vapor mass fraction of outflow, was calculated based on measured volumetric (${\dot{v}}_{total}$) and mass flow rates (${\dot{m}}_{total}$), which were used as essential input parameters in the analytical model of two-phase flow pressure drop. It can be given as:

$$x={\displaystyle \frac{{\rho }_g{\rho }_l(V{\rho }_l-1)}{{\rho }_l-{\rho }_g}}$$

(13)

$$V={\displaystyle \frac{{\dot{v}}_{total}}{{\dot{m}}_{total}}}$$

(14)

where V represents the specific volume (${\mathrm{m}}^3/\mathrm{kg}$). ${\rho }_g$ and ${\rho }_l$ ($\mathrm{kg}/{\mathrm{m}}^3$) are the densities of the liquid and gas phases.

2.5. Results Discussion

Figure 4 systematically demonstrates the experimental pressure gradients as a function of flow velocity and boiling regimes (heating power) under varying nickel particle sizes, powder sizes, and mass fractions of CaCl₂. The structural parameters summarized in Table 1 establish the framework for the controlled-variable analysis shown in Figure 4, ensuring that individual factors can be examined independently. To isolate the effect of nickel powder granularity on pressure drop gradients, samples across all scales were selected with the minimum particle size of CaCl₂ (80 μm) and $f_{{\mathrm{CaCl}}_2}$ ($2\%$), thereby minimizing confounding variables. Experimental results demonstrated a monotonic decrease in pressure drop gradient with increasing $d_{\mathrm{Ni}}$. A mechanistic explanation attributes this behavior to the microstructural characteristics of Ni-2 μm and Ni-15 μm samples, where only submicron interstitial pores and sporadically distributed isolated formed pores are observed (Figure 4a–d), precluding the formation of interconnected pore networks with uniform pore size. The constricted channel geometry in fine-powder samples exacerbates frictional losses along the flow path during the flow boiling process, with partial relief observed only in rare macroporous regions. This paradigm explains the counterintuitive observation of an elevated pressure drop in the Ni-2 μm sample, despite its higher porosity (0.588–0.674), which arises from the anomalous pore architecture disrupting the conventional inverse proportionality between porosity and pressure drop. The results highlight the dominant role of pore connectivity and structural hierarchy over porosity in determining the two-phase pressure drop of bi-porous microstructures. Besides, as the flow rate increases from 0.4 to 1.0 mL/min, the two-phase boiling intensity (vapor quality) decreased at the same heating conditions due to insufficient boiling periods. However, the results showed that the pressure drop still increased with the rise in flow velocity under the same heating power, though the magnitude of increase diminished, which indicates that the influence of flow rate on pressure drop remains more significant than that of boiling states for superhydrophilic samples in this study (Figure 4a–d).

Figure 4e–h illustrate the impact of CaCl₂ powder size (which determines the size of the formed pores) on samples with varying Ni matrix particle sizes. As revealed by testing on samples with two sizes (80 μm and 150 μm), a monotonic decrease in pressure drop gradient as the pore size increased, indicating that larger formed pores will alleviate internal flow resistance in porous structures. Notably, the Ni-2 μm samples exhibited a pronounced discrepancy between the pressure drop gradient of samples with different CaCl₂ powder sizes. A reasonable explanation is that the Ni-2 μm porous samples contain an abundance of nucleation sites, which enhance liquid boiling and cause massive vapor evolution. In this scenario, larger formed pores are crucial for effectively mitigating vapor-induced flow resistance, resulting in a significant reduction in pressure drop. Interestingly, the Ni-50 μm samples also demonstrated significant pressure drop reductions with larger CaCl₂ sizes, which may be analogous to the Ni-2 μm samples. Visualized boiling images (Figure 4m–p) clarify that for the Ni-50 μm samples, the porous framework with superior connectivity results in an enhanced evaporation effect and more frequent bubble escape compared to the Ni-30 μm samples. Additionally, the generated bubbles in the Ni-50 μm samples exhibited significantly larger escape volume than those in the Ni-2 μm and Ni-15 μm samples, due to the larger pore size. Figure 4i–l demonstrates that the pressure drop gradient decreases as the CaCl₂ ratio increases ($f_{{\mathrm{CaCl}}_2}$), indicating that both the proportion and particle size of the pore-forming agent play a similar role in influencing pressure drop: introducing larger and more formed pores creates spatial pathways for vapor escape, thereby preventing flow disturbance caused by vapor accumulation.

To explore the intrinsic relationship between single-phase and two-phase flow resistances within microstructures, the influence of porosity, equivalent pore diameter ($d_p$), and flow rate on the single-phase (liquid) flow resistance of bi-porous media was examined. As shown in Figure 4q, the single-phase flow resistance of samples across multiple scales slightly increases with increasing porosity. Whereas the trend is not pronounced, it contradicts the porosity-pressure drop trend within the homogeneous glass bead packs under the Darcy flow regime [55]. It can be attributed to the combined effect of both porosity and equivalent pore diameter of samples on pressure drop performance. For instance, the $d_p$ of Ni-2 μm samples is smaller than that of other samples, leading to higher pressure drop even with higher porosities. This finding is validated by the variation relationship between single-phase pressure drop and effective pore diameter (Figure 4r). The single-phase pressure drop gradient exhibits an opposite trend to $d_p$, which is consistent with the tendency predicted by the classic K-C correlation ($\Delta P_l\propto 1/d_p$). Moreover, as shown in Figure 4s, the pressure drop gradient increases linearly with flow velocity, matching the model predictions. Therefore, we derived analytical models for single-phase and two-phase pressure drops in bi-porous structures based on the K-C model. Additionally, it is noted that the single-phase pressure drop is significantly lower than the two-phase counterpart, demonstrating the inaccuracy of single-phase models under two-phase conditions and the necessity of boiling flow resistance modeling.

3. Modeling and Validation

3.1. Analytical Modeling

Given that the existing models reported previously are inaccurate in evaluating the pressure drop of bi-porous media at various boiling states, this study presents a parsimonious model derived from revising the classic Kozeny-Carman and Akagi-Chisholm correlations based on our original experimental dataset.

Here, we proposed a semi-empirical method modeling two-phase flow resistance, based on the Akagi-Chisholm paradigm, by coupling the inherent relationship between single-phase (liquid) and boiling pressure drops gradients with the porosity and vapor quality of samples, thus necessitating accurate single-phase modeling.

To simplify the analytical simulation, the following assumptions were made:

(1)

The interstitial and formed pores are distributed homogeneously.

(2)

The fluid velocity reaching the sample inlet is uniform.

(3)

The influence of temperature drop during liquid flow through the sample on its density and viscosity is negligible.

3.1.1. Single-Phase Modeling

The single-phase pressure drop of samples can be elaborated as variants of the K-C model [56]. Considering the influence of variable pore diameters in bi-porous microstructures, the equivalent pore diameter was considered as pore size to be substituted in calculations, which was estimated via weighted averaging of the PSD results obtained by MIP.

$${\displaystyle \frac{\Delta P_l}{L}}=a{\displaystyle \frac{\mu {(1-\epsilon )}^2}{d_p^2{\epsilon }^3}}v$$

(15)

where $\Delta P_l$ (Pa) is the single-phase pressure drop. L (m) is the length of the testing samples. $\Delta P_l/L(\mathrm{Pa}/\mathrm{m})$ represents the local pressure gradient along the testing section. $\rho$ ($\mathrm{kg}/{\mathrm{m}}^3$) and $\mu$ ($\mathrm{Pa}·\mathrm{s}$) are the density and the dynamic viscosity of the liquid. $d_p$ (m) is the equivalent pore diameter of porous media, which can be estimated based on MIP results. v represent the inlet flow velocity (m/s). a is an empirical coefficient, taken as 180 k for the classic K-C expression. k represents the Kozeny coefficient, which is highly related to the microgeometric structural complexity of porous media.

Figure 5a–d report the fitting parameters a of the single-phase pressure gradient prediction models derived from experimental data for samples with various particle sizes. For low speed flow conditions (Darcy flow), previous studies have well established that the pressure drop gradient exhibits a linear relationship with flow velocity, as defined by the concept of permeability [25]. Notably, this study confirms that bi-porous samples demonstrated a linear relationship between the independent variable $[\mu {(1-\epsilon )}^2/\left(d_p^2{\epsilon }^3\right)]$ and pressure drop gradient. Thereby, the coefficient of the governing equation can be developed by incorporating velocity-dependent pressure drop gradient and substituting the corresponding density (liquid), porosity, and equivalent pore diameters, as listed in Table 1. The $R^2$ values of all four groups of models are higher than 0.85, indicating the prediction reliability of the obtained expressions. Furthermore, the a of the Ni-2 μm and Ni-50 μm particle samples are significantly lower than the empirical values of the K-C model. Given that the Kozeny factor is closely related to the pore connectivity in porous media, the factor gradually decreases when the arrangement is regular and particles are packed loosely [57,58]. The internal pore connectivity of the porous media with good capillary properties is in high level, resulting in smaller empirical coefficients for bi-porous samples than packed beds. For the Ni-50 μm samples, the pore connectivity is higher than that of Ni-15 μm and Ni-30 μm samples confirmed by SEM images. The comparative results of pore connectivity have been verified by the tortuosity ($\tau$) measured in AMP testing, as labeled in Figure 5a–d. Physically, tortuosity is defined as the ratio of the actual tortuous flow path length ($L_e$) to the straight-line distance (L) across the medium ($\tau =L_e/L$). This parameter serves as a quantitative metric for the complexity of the porous structure; a higher $\tau$ value signifies a more convoluted pathway for fluid transport, inevitably leading to increased flow resistance and diminished effective connectivity.

3.1.2. Two-Phase Modeling

The Akagi-Chisholm model couples the Lockhart-Martinelli parameter with experimental results has been applied to resolve adiabatic two-phase flow resistance [59,60]. Based on the single-phase pressure drop of bi-porous microstructures obtained by data fitting, we considered the modified form of the A-C model to develop a semi-empirical formula for two-phase (boiling) cases. It can be seen from A-C correlation that a porosity-dependent relationship between single-phase and two-phase pressure drops was mathematically described during the flow of gas-liquid or vapor-liquid mixtures in pipes [61]. However, this model was only verified when predicting the two-phase pressure gradients in separated flow under specific gas-liquid fractions [62]. According to our experimental results of one- and two-phase flow resistance, boiling became more intense inside the same porous sample as the heating power increased, and the pressure drop increased accordingly. The phenomenon can be attributed to that the vapor generation perturbs the fluid flow, exacerbating the internal friction between fluids and the frictional effect between the solid wall and the liquid, which raises the flow resistance. Furthermore, the gas boiling state gradually weakened with increasing flow velocities under the same heating conditions. A reasonable explanation is that the mass of liquid flow through the specific heat transfer area increases due to the increase in flow velocity, resulting in insufficient heating. Therefore, the vapor quality, characterizing the boiling state (gas-liquid composition) within porous media, was introduced as another variable into the mathematical description of boiling effect on pressure drops within bi-porous microstructures, while ignoring the complex influences of parameters such as flow velocity and heating power on the boiling process to simplify the model and to ensure its physical significance.

In the two-phase flow pressure drop model, new assumptions were considered to simplify the model without having a decisive effect, including:

(1)

The heating flux at the bottom of the bi-porous microstructures is uniform.

(2)

The liquid superheat in macro- and micropores is the same, and the nucleation initiation temperatures are consistent.

(3)

Frictional pressure gradients are assumed to be separable by phase.

(4)

Higher-order effects of inter-phase slip velocity are ignored.

Total boiling pressure drop can be expressed by the summation of frictional, accelerational and gravitational sections as [63].

$$\Delta P_s=\Delta P_{s,F}+\Delta P_{s,A}+\Delta P_{s,G}$$

(16)

where Accelerational pressure gradient $\Delta P_{s,A}/L$ can be evaluated based on the previous model [62],

$${\displaystyle \frac{\Delta P_{s,A}}{L}}=-{\int }_0^L{G}^2{\displaystyle \frac{d}{dl}}\left[{\displaystyle \frac{V_g{x}^2}{\alpha }}+{\displaystyle \frac{V_l{(1-x)}^2}{(1-\alpha )}}\right]$$

(17)

where G ($\mathrm{kg}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$) represents mass velocity, $V_g$ and $V_l$ (${\mathrm{m}}^3/\mathrm{kg}$) are the specific volume of gas and liquid, respectively. $\alpha$ is the void fraction, which can be estimated by appropriate techniques based on separated flow model. Since the flow is in porous microstructures, $\Delta P_{s,G}$ is thus negligible. The frictional pressure gradient in this study was expressed by combining the A-C modeling [63].

$${\displaystyle \frac{\Delta P_{s,F}}{\Delta P_l}}=C_0{\left({\displaystyle \frac{x}{1-\epsilon }}\right)}^z(0.314\le \epsilon \le 0.673)$$

(18)

where $\Delta P_l$ represents the single-phase (liquid) of bi-porous microstructures as we modeled in Equation (15). x is the vapor quality of the outflow. $C_0$ and z are the modified Chisholm parameter and fitting exponent, respectively. The former one is typically correlated with flow regime, pipe geometry, and fluid properties, while the latter one depends on the geometric characteristics of porous media.

It is essential to acknowledge that the Equation (16) represents the total pressure drop as a superposition of frictional, accelerational, and gravitational components, while in mini- and micro-channels, surface tension forces play a dominant role. As discussed by Abiev [64], the energy dissipation associated with the formation and maintenance of the interfacial area, particularly during bubble coalescence and breakup, contributes to the total pressure loss. Although there is no explicit term in the equation that directly describes the surface energy (e.g., in terms of Weber number or Capillary number), the establishment of the model is based on experimental data which inherently encompasses these effects. Therefore, the contribution of the interfacial formation work is implicitly captured by the empirical coefficients obtained from the data regression.

3.2. Model Validation

As shown in Figure 5e–h, the corresponding parameters $C_0$ and z of four samples with different particle sizes were yielded by experiment fitting. In four particle-size cases, the pressure drop ratios ($\Delta P_{s,F}/\Delta P_l$) of samples, with porosity ranging from 0.314 to 0.673, show good fitting results with independent variables $[x/(1-\epsilon \left)\right]$, except for the Ni-50 μm sample with an $R^2$ of 0.70. It can be attributed to the uncertainty of particle fragmentation, leading to a non-uniform distribution of vaporization nuclei. Besides, the two-phase pressure drops of Ni-15 μm and Ni-30 μm samples exhibit high sensitivity to the vapor quality of two-phase flow, with z values of 2.29 and 2.03, respectively, due to their high tortuosities of the internal space [46]. As the heating power increased, boiling became more intense, and the internal friction of the two-phase flow significantly enhanced with intensifying boiling state, resulting in high flow resistances. For the experiments of Ni-2 μm and Ni-50 μm, the two groups of samples displayed more vaporization nuclei due to surface pits, which enabled higher bubble departure frequencies than those of the other two groups of samples. Therefore, a large number of dispersed small bubbles were sporadically formed in Ni-2 μm and Ni-50 μm samples rather than concentrated to grow into large bubbles. In this way, the resistance of separated small bubbles exerted on two-phase flow is low compared with that of large bubbles, thus leading to lower z values in correlations. Details on vapor quality can be found in Table S1.

Figure 6a,b shows the dependence of the experimentally determined two-phase pressure drop gradients on their calculated value according to the described model for four validated bi-porous samples, including Ni-2-50(3), Ni-15-80(5), Ni-30-150(10), and Ni-50-220(6). Percentages of predicted pressure drop with absolute error within ±15% and ±30% compared to experimental data are denoted as ${\xi }_{15}$ and ${\xi }_{30}$, respectively. It demonstrates a good qualitative agreement between the pressure drop gradient obtained by experiments and calculation with an average of 14.8% for single-phase flow and 15.7% for boiling flow within the scope of this research, indicating the reasonable applicability of the proposed modified correlations. Deviations for Ni-2 μm and Ni-15 μm at high two-phase pressure drop gradients can be attributed to the disturbance of bubbles by high flow rates. Modification of correlations is limited to laminar regimes for slow-flow inlet liquid.

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left({\displaystyle \frac{{(\Delta P_s/L)}_{\mathrm{CAL}}^i-{(\Delta P_s/L)}_{\mathrm{EXP}}^i}{{(\Delta P_s/L)}_{\mathrm{CAL}}^i}}\right)}^2}$$

(19)

where ${(\Delta P_s/L)}_{\mathrm{CAL}}$ and ${(\Delta P_s/L)}_{\mathrm{EXP}}$ are the calculated and measured two-phase pressure drop gradients. N represents the number of data points. Numerous semi-empirical correlations have been proposed for predicting saturated flow boiling pressure drop in rectangular channels based on the SFM to handle frictional pressure gradients. Herein, we compiled classical discrete-phase-based boiling pressure drop correlations in

Table 2. Summary of correlations for pressure drop of saturated flow boiling based on SFM.

Model	Equation(s)	Remarks	Ref.
Müller-Steinhagen & Heck (1986)	$\begin{array}{cc}\hfill {\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{F}}=& \left\{{\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{l}}+2\left[{\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{g}}-{\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{l}}\right]x\right\}{(1-x)}^{1/3}\hfill \\ & +{\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{g}}{x}^3\hfill \end{array}$	$D=4–392\mathrm{mm}$	[65]
Yan & Lin (1998)	${\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{F}}=-0.22R{e}^{-0.1}{\displaystyle \frac{G^2}{D_h}}[V_l+x{V}_{lg}]$	$D=2.0\mathrm{mm}$	[66]
Yu et al. (2002)	$\begin{array}{c}{\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{F}}={\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{l}}{\varphi }_l^2;\hfill \\ {\varphi }_l^2={\left[18.65{\left({\displaystyle \frac{V_l}{V_g}}\right)}^{0.5}\left({\displaystyle \frac{1-x}{x}}\right){\displaystyle \frac{R{e}_g^{0.1}}{R{e}_l^{0.5}}}\right]}^{-1.9}\hfill \end{array}$	$D=2.98\mathrm{mm}$	[67]
Sun & Mishima (2009)	$\begin{array}{c}{\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{F}}={\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{l}}{\varphi }_l^2\hfill \\ \mathrm{For}R{e}_l<2000\mathrm{and}R{e}_g<2000\hfill \\ {\varphi }_l^2=1+{\displaystyle \frac{C}{X}}+{\displaystyle \frac{1}{X^2}};\hfill \\ C=26\left(1+{\displaystyle \frac{R{e}_l}{1000}}\right)\left\{1-exp\left({\displaystyle \frac{-0.153}{0.27N_{conf}+0.8}}\right)\right\}\hfill \end{array}$	$\begin{array}{c}{D}_h=0.506–12\mathrm{mm}\hfill \\ R{e}_l=10–\mathrm{37,000}\hfill \\ R{e}_g=3–4\times {10}^5\hfill \end{array}$	[68]
Ma et al. (2024)	$\begin{array}{c}{\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{F}}={\left({\displaystyle \frac{dP}{dL}}\right)}_{\mathrm{l}}{\varphi }_f^2;\hfill \\ {\varphi }_f^2=\left[1+2690\left({\displaystyle \frac{B{d}^{0.97}}{X_{tt}^{0.43}{P}_r^{0.95}}}\right)\right]R{e}_l^{1.03}W{e}_l^{0.53}\hfill \\ X_{tt}={\left({\displaystyle \frac{1-x}{x}}\right)}^{0.9}{\left({\displaystyle \frac{{\rho }_g}{{\rho }_l}}\right)}^{0.5}{\left({\displaystyle \frac{{\mu }_l}{{\mu }_g}}\right)}^{0.1}\hfill \end{array}$	$\begin{array}{c}{D}_h=0.1–2.6\mathrm{mm}\hfill \\ G=50–3000\mathrm{kg}/{\mathrm{m}}^2\mathrm{s}\hfill \\ p=101–3970\mathrm{kPa}\hfill \end{array}$	[63]

and selected six of them with good predictive performance under microscale conditions and appropriate working fluid (water) for comparative analysis [60,63,65,66,67,68].

The evaluation results calculated using these models are presented in Figure 6c–h. Lockhart and Martinelli’s model [60] yields relatively good predictive performance, while significant deviations were observed for large particle sizes (Ni-30 μm and Ni-50 μm). Following this model, the predictive model proposed by Müller-Steinhagen and Heck [65] demonstrates the large deviation (RMSE = 237.4%), which can be attributed to the fact that the M-H model is applicable to larger channel sizes rather than the complex internal geometry of porous media. Models proposed by Yan and Lin [66] and Yu et al. [67] show low predictive accuracy, resulting from the high flow rate conditions considered during their development. Generally, Lockhart and Martinelli’s and Sun and Mishima’s [68] correlations exhibit narrow scatter of predictions with around half of the datapoints calculated within ±30%. However, there is still a certain gap compared with the model proposed in this study. Considering the foundation of the pressure drop model for bi-porous media dominated by formed pores requires taking into account the influence of various flow patterns in connective pores, which necessitates further experimental validation of Reynolds-dependent pressure drop and vapor quality.

4. Conclusions

This study systematically investigates the two-phase flow resistance characteristics of superhydrophilic bi-porous nickel microstructures with varying particle sizes and porosities under multiple boiling regimes. Porous samples were fabricated via vacuum sintering using micro nickel powders and pore-forming agents (CaCl₂), and a visualized experimental platform was established to monitor two-phase pressure drops. Based on experimental data, a semi-empirical correlation was derived by modifying the Kozeny-Carman and Akagi-Chisholm models for the evaluation of resistance characteristics in two-phase flow passive thermal management systems. Key findings can be summarized as:

(1)

Pore size combinations and boiling states significantly affect the pressure drop. Larger formed pores (induced by CaCl₂) alleviate flow resistance by facilitating vapor escape, while the confinement effect of fine channels within Ni-2 μm samples leads to high pressure drop despite high porosity, highlighting the significance of pore connectivity and structural hierarchy in flow resistance.

(2)

The single-phase flow pressure drop model of bi-porous microstructures was developed based on the K-C equation, incorporating equivalent pore diameter and porosity, with fitting coefficients ($R^2\ge 0.85$) confirming its reliability. The fitting empirical parameter (a) is closely related to the pore tortuosity of bi-porous samples.

(3)

Single-phase resistance is significantly lower than two-phase cases, highlighting the necessity of boiling pressure drop modeling. The two-phase pressure drop increases with heating power while the effect diminishes at higher flow velocities, resulting from gradually weak phase-changing processes with insufficient heating.

(4)

The two-phase model was derived using vapor quality and porosity via modified Chisholm parameters ($C_0$ and z), which demonstrates good prediction performance (RMSE = 15.7%). A strong relationship between vapor quality and two-phase pressure drop is observed for samples with few and concentrated nucleation sites (Ni-15/30 μm), which promotes the aggregation of bubbles and thus causes hydrodynamic obstruction.

It can be concluded that the presented simplified correlation used for two-phase pressure drop modeling of bi-porous microstructures with high hydrophilicity and capillary property provides insights into performance evaluation and optimization of miniature thermal management systems. Besides, it offers methodological references for pressure drop prediction in similar porous media, integrating microstructural characteristics and two-phase flow mechanisms. Future works can focus on improving the semi-empirical model with CFD simulations to characterize the effect on microscale bubble dynamics and interphase slip.