Numerical Investigations of a Single Loop Pulsating Heat Pipe for Cryogenic Applications

A pulsating heat pipe (PHP) is an energy-efficient heat transfer device to transport heat across long distances (a few meters in length), effectively utilizing the two-phase (primarily evaporation and condensation) heat transfer. Akachi [1] has proposed it first during the 1990s. A PHP is constructed as a lengthy capillary tube bent into a serpentine configuration within its heating and cooling sections. A standard PHP consists of three sections: the evaporator, condenser, and adiabatic section. The absence of any wick structure inside a PHP distinguishes it from conventional heat pipes. PHPs can generally be classified into three categories: closed-loop PHP, open-loop PHP, and closed-loop PHP with additional check valves [2]. A closed-loop PHP outperforms open-loop PHP because of its efficient fluid circulation [2]. A specific flow pattern involving liquid slugs and vapor plugs is required for a PHP to function effectively [3,4]. However, the formation of the liquid-vapor interface primarily relies on channel diameter, often quantified in terms of the dimensionless Bond number [4].

The triple point and critical points of the working fluid are the two bounds of the operating temperature limits of a PHP. This flexibility in the operating temperature range enables the application of PHPs across a spectrum from high to extremely low cryogenic temperatures [5]. In the relatively less-known cryogenic environment applications, cryocooled superconducting magnets necessitate efficient conduction cooling to maintain the magnets in their superconducting state [6]. However, connecting the cryocooler and the magnet by simple copper straps, essentially transferring heat only through solid conduction, becomes inadequate when the distance between them is significant [7]. Using a PHP, with its exceptional heat transport capabilities achieved through two-phase heat transfer, could be an appropriate alternative to circumvent this challenge.

During the past few decades, extensive research has been performed in the cryogenic field to explore the heat transfer capabilities of the PHPs at low temperatures, particularly as a thermal link between the low-temperature sink and superconducting magnets [6,8]. PHPs have been built for operation with different cryogenic fluids like nitrogen [8–12], argon [12], neon [13–15], hydrogen [16–19], and helium [20–22]. Various structural (such as diameter, length, and number of turns [13,18,20,21]) and operational parameters (like heat load, inclination angle, and fill ratio [10,11,14,19]) have been examined in numerous experimental studies. One of the primary objectives of these studies is to understand the PHP working mechanisms and enhance heat transfer performance.

It is evident from the various studies that the fill ratio is a parameter notably influencing PHP performance. The fill ratio (also called the liquid volume ratio [23]) has been defined as the fraction of the total volume in the liquid state. For example, Fonseca et al. [10] have observed the best performance in a 40-channel N₂ PHP at a 20% fill ratio, with an effective thermal conductivity of 70,000 W/mK. While studying PHPs with different working fluids like neon, nitrogen, and argon across different fill ratios in closed and open-loop setups, Barba et al. [12] have noted a critical minimum fill ratio for initiation of pulsation, ensuring minimal heat transfer and an optimal fill ratio for the peak performance.

The thermo-physical properties of the working fluid also heavily influence these fill ratios. Liang et al. [14] studied a 10-channel neon PHP operating between 15.3% and 51.5% fill ratios. It has been observed that effective thermal conductivity increases with heat input for a high fill ratio. In contrast, the effective thermal conductivity reaches a maximum when heat input increases for a low fill ratio. Sun et al. [16] have noted significant changes in the fill ratio of a cryogenic PHP relating its operating temperature. As the fill ratio increases, there is a simultaneous increase in the heat transfer limit. An experimental investigation has been conducted by Li et al. [20] on a 48-turn PHP filled with helium over a wide range of fill ratios (16.4%–94.2%). The PHP exhibits the best performance with a 48.8% fill ratio at a low heat rate (<0.6 W), while 66.1% is the fill ratio that results in the highest effective thermal conductivity at a higher heating rate (0.8–1.3 W).

Besides experimental investigations, numerical studies have also been performed to understand the complex thermo-hydrodynamics of the PHPs [24]. The complexity of the two-phase flow systems is often handled using simplified assumptions in the model [25]. Computation fluid dynamics (CFD), on the other hand, has been one of the promising methods to solve the liquid-vapor flow and capture their interface inside the PHP. Table 1 includes the numerical investigations implemented on PHPs with different geometry, working fluid, fill ratio, and boundary conditions [26–37]. A liquid nitrogen-based cryogenic PHP with different fill ratios and evaporator temperatures has been simulated by Sagar et al. [27]. They found that higher evaporator temperatures led to increased slug circulation velocity and a shift from chaotic to periodic fluid flow patterns. Vo et al. [32] have developed a 3D CFD model consisting of 8 turns and R123 as its working fluid. The wall temperatures predicted by the 3D model indicate similar flow motion and direction trends as reported in their experiment. Li et al. [33] have investigated a two-channel PHP with water as a working fluid operating under different heat loads and adiabatic lengths of PHP. The study revealed that with an increased heat load, thermal resistance decreases but increases with the adiabatic length. Kang et al. [36] have correlated the PHP performance with the separating wall thickness in their numerical investigation of a 2D single loop PHP. An increase in the wall thickness has caused a performance enhancement of 14% at a particular fill ratio of 70%. Mucci et al. [37] have performed CFD simulations to understand the effect of the number of turns in the thermo-hydrodynamics of PHP.

While the numerical studies concerning room-temperature PHPs, primarily with water as a working fluid [29–31,33–37], are in plenty, comprehensive research on cryogenic PHPs in the existing literature is rare. Since the thermo-physical properties of the working fluids significantly influence the PHP performance, while the fluid properties of cryogens are heavily temperature dependent, room-temperature PHP results may not be suitable at low temperatures. In particular, the low vapor pressure of the cryogens makes a critical difference. The literature review points out the necessity for an elaborate and comprehensive numerical model for cryogenic PHPs to understand thermo-hydrodynamics, emphasizing the mode of heat transfer when subjected to different heat loads at different fill ratios.

Thus, the primary objective of this study is to perform a two-dimensional CFD analysis using FLUENT in a single-loop PHP where nitrogen is the working fluid. This investigation intends to determine the impacts of varying fill ratios (ranging from 20% to 70%) on the thermal performance of PHPs, specifically looking at the transient behavior of velocity and temperature fields inside cryogenic PHPs to gauge how eventually, heat is getting distributed in the form of latent and sensible heat.

Starting with a physical description of the PHP used in this work, the theoretical study elaborates on the governing equations and boundary conditions, followed by the grid independence test. Finally, it ends with the validation of the model.

For nitrogen, the value of $Dcrit$ is 2.1 mm at atmospheric conditions. Accordingly, a tube of 1.5 mm internal diameter and 3.16 mm external diameter has been chosen for the PHP.

The analysis of the PHP described above begins with writing the governing equations.

The continuity, momentum, and energy equations are generally sufficient for addressing single-phase fluid dynamics issues but not adequate in two-phase situations. In a two-phase system consisting of both liquid and vapor phases, it is crucial to identify their dynamic interfaces and implement their changes in the governing equations.

Monitoring and characterizing the interface between the two phases can be accomplished through various multiphase models, such as the Eulerian model, mixture model, and volume of fluid (VOF) model [38], to name a few of the significant ones.

The VOF method has been routinely used in several liquid vapor two-phase studies [28–30]. The entire PHP control volume in the VOF method is divided into multiple computational cells, each containing either one or multiple phases. In each cell, a variable is introduced, which is the volume fraction of the cell, which accounts for tracking the liquid-vapor interface. If the volume fraction of vapor phase in a computational cell is expressed as $αg,$ following three conditions exists:

• $αg=0$⁠, the computational cell is entirely devoid of vapor phase

• $αg=1$⁠, the computational cell is completely filled with vapor phase

• $αg<1$⁠, the computational cell contains an interface of liquid and vapor phase

where $m˙fg$ represents the transfer of mass from the vapor phase to the liquid phase. The source term $Sαg$ has been considered zero in the present situation.

It is evident from Eq. (8) that the value of $βc$ depends on the saturation vapor pressure.

The CSF framework incorporates the influence of wall adhesion at the interface between fluid and solid by adjusting the surface normal in cells adjacent to the solid wall. The contact angle between the fluid and the wall is set as $300$⁠.

The properties, including density (⁠$ρ$⁠) and viscosity (⁠$μ$⁠), as defined in Eq. (9) and Eq. (10), are determined based on the volume fraction of all the phases.

The symbol $(Sh)$ represents the energy source term which results from the phase change and can be determined as: $Sh=(m˙fghfg)$ = −$( m˙gfhfg)$⁠.

In Eq. (14), $Sh,w$ symbolizes the total heat transfer across the surface between the outer and inner walls. The symbols $kw$ and $ρw$ represent the thermal conductivity and density of the tube wall, respectively.

In FLUENT, a 2D double-precision solver has been used. Unsteady time conditions have been chosen for the simulation. Moreover, a laminar viscous model with the implicit body force and gravity terms has been selected.

While a second-order upwind scheme is selected for the density and energy computations, the pressure–velocity coupling has been achieved through “PISO” (pressure-implicit with splitting of operators) within the solution model [38]. The significance of gravity in PHP operations is evident from Eq. (9), where it appears as a body force term $ρg$⁠, to dominate the momentum equation. When the body force term, such as buoyancy, governs the momentum equation, the pressure calculations are done using the “body force weighted” scheme recommended in ansysfluent [38]. A compressive scheme is adopted for the volume fraction calculations. Moreover, a user-defined function (UDF) is compiled to address the phase interactions between the liquid and vapor through Eq. (7), leading to the calculation of the energy source term $(Sh)$ in Eq. (12). Nitrogen has been chosen as the operating fluid inside the PHP, with liquid nitrogen as the primary phase. The temperature variations of the thermo-physical fluid properties have been considered in the model [42]. However, the vapor density is assumed to obey the ideal gas equation, while the saturation temperature and latent heat follow the polynomial function of pressure. The PHP tube material is stainless steel (SS), but the thermophysical properties of SS have been assumed independent of temperature and taken as constant.

Once the fluid properties are set accurately, the system is initialized with the specific fill ratio. In this study, the PHP is simulated for six different fill ratios between 20% and 70% with an increment of 10%. The PHP is initialized with a temperature of 75 K and the equivalent saturation pressure of 76,043 Pa.

where$keff$ is the effective thermal conductivity of PHP in $(W/mK)$⁠. $Leff$ is the effective length of the PHP, which can be stated as $[La+0.5(Le+Lc)]$⁠. In Eq. (15), the symbols $Te$ and $Tc$⁠, represent the evaporator and the condenser temperature, respectively, and $Q$ denotes the heat load, while $Acr$ is the cross-sectional area of the inner diameter of the capillary tube.

The PHP has been meshed using structured grids separately for the fluid and the solid domains. The fluid domain grids near the wall have been densified to capture the wall effects (see Fig. 2).

Subsequently, a mesh independence test is conducted with three grids, namely, 35,000, 57,000, and 71,000 cells. It is evident from Table 2 that the evaporator temperature changes negligibly beyond 57,000 mesh size. Accordingly, a final mesh with 57,000 cells has been chosen for the present model.

The grid invariance test is followed by the time-independent study with five different time-step sizes, namely, 0.001, 0.0005, 0.0002, 0.0001, and 0.00005 s. Table 3 shows the value of the evaporator temperature against different time-step sizes. The evaporator temperature shows insignificant change when the time-step is reduced from 0.0001 s to 0.00005 s. Hence, a 0.0001 s time-step size is opted for further investigations.

To begin with, the CFD model has been validated against the published experimental data. The experimental investigation conducted by Sagar et al. [11] has been considered the basis for validating the proposed numerical model. However, the overall dimensions of the PHP unit used in the numerical study have been modified to make it identical to the experimental counterpart [11]. Accordingly, the PHP is made of tubes with an internal and external diameter of 1.3 mm and 3.2 mm, respectively, while the evaporator, condenser, and adiabatic section lengths have been set at 50 mm, 60 mm, and 60 mm, respectively. The PHP has been initially filled to a fill ratio of 38% and tested for three heat loads, namely, 1 W, 2 W, and 4 W. Finally, the effective thermal conductivity is calculated using Eq. (15) based on the evaporator temperature, obtained from the numerical simulation.

Figure 3 showcases the experimental data and the numerically predicted results of the effective thermal conductivity. A close agreement between the numerical model and the experimental findings has been noticed. For example, the maximum relative error observed corresponding to 4 W heating is less than 8%.

The validation of the numerical model against the experimental results is followed by multiple simulations of the PHP subjected to different heat loads and fill ratios. The study examines the temperature and velocity fields (of both the liquid and vapor phases) to ultimately derive the performance parameters like effective thermal conductivity and other factors.

In the following sections, the effects of heat load and fill ratio on the different thermo-hydrodynamic parameters of a PHP is elaborated. The factors to be investigated include temperature, pressure, and fluid velocity.

The single loop PHP simulation has been carried out at five different heat loads between 0.5 and 3.0 W. The evaporator temperature varies with the heat load.

The evaporator temperature plotted at 0.5 W at different fill ratios is shown in Fig. 4.

It has been observed that initially, the evaporator temperature shows a steep rise and then gradually diminishes to reach a quasi-steady-state after some time. The time needed for the temperature to achieve its first peak is called the “startup period” or the “startup phase.” Afterward, when the evaporator temperature reaches a quasi-steady-state, it is known as the “stable period” or the “stable phase.” It is apparent from Fig. 4 that the startup time varies with the fill ratios. The higher the fill ratio, the lower the maximum temperature rise and the average temperature of the evaporator section. A higher fill ratio also helps to reach a stable phase quickly. A small fill ratio, implying a smaller liquid content, absorbs heat quickly to generate vapor bubbles and an associated increase in the evaporator temperature. It may also be noted that in the PHP operation with a small fill ratio, the condensed liquid falling from the condenser section is also proportionately less. Thereby, a higher average evaporator temperature results owing to inadequate cooling.

Figure 5 shows the evaporator temperature variations for incremental heat load conditions from 0.5 W to 3.0 W for different fill ratios between 20% and 70% with an increment of 10%. As discussed earlier, the evaporator temperature increases before reaching a steady-state of operating condition for a given heat input. When the heat load is augmented from 0.5 W to 1.0 W, the evaporator temperature shifts from a steady-state to another with an intermediate transient state. Similar behavior can also be observed for a heat load change from 1.0 W to 1.5 W, and so on. The evaporator temperature variations with the heat load are analogous to the experimental results of the other researchers [11,14,17].

It may also be noted (from Fig. 5) that the PHP operation with a higher fill ratio has a lower quasi-steady-state temperature for any heat load conditions. For example, a PHP operation with a fill ratio of 70% is always linked to the minimum temperature rise at any given heat input. This observation is consistent with our conclusions (in the previous section) that the higher fill ratio (with a large liquid mass fraction) always corresponds to a smaller rise in the evaporator temperature for a specific heat input.

A discontinuity in the evaporator temperature versus the time graph (Fig. 5) is discernible for heat input beyond 1.0 W in the case of a 20% fill ratio operation. Instability in the PHP operation for a fill ratio of 20% is elaborated further in Fig. 6, showcasing the steady rise in the evaporator temperature, failing to reach a quasi-steady-state for an enhancement in the heat input from 0.5 W to 1.0 W.

Similarly, heat addition of more than 2.0 W destabilizes the PHP operation for a fill ratio of 30%. It is evident from Fig. 5 that the PHP designated for large heat transport operations should have a high fill ratio (fill ratio); the desired effect cannot be achieved with a small liquid content or low fill ratio.

When the circulating two-phase working fluid in the confinement of the PHP loop experiences temperature variation, it is evident that there would be associated pressure fluctuations. The time versus pressure oscillations inside the PHP subjected to varying heat loads are shown in Fig. 7. At the onset, when set under operation, the PHP is charged with nitrogen corresponding to the saturation pressure at 75 K. For a given heat input, the vapor generated in the evaporator section increases system pressure, and the PHP attains the maximum pressure during the startup phase, as shown in Fig. 7. For instance, the evaporator temperature reaches the maximum value at around 9 s for a 50% fill ratio, as illustrated in Fig. 4. It is the same time (called as the startup time) during which the system pressure attains the peak value. However, it is crucial to note that the time to achieve the maximum evaporator temperature and pressure varies for different fill ratios.

Once the startup stage is over, a thermally excited oscillating motion inside the PHP begins, and the pressure inside the PHP comes to a quasi-steady-state. It is discernible from Fig. 7 that the pressure fluctuations at a particular fill ratio and given heat input are not widely dissimilar. The system equilibrates to a new pressure level when there is a change in the heat input, if not causing instability. However, one may notice an indication of destabilization from the slightly elevated operating pressure level (as is evident from Fig. 7) before the intended heat load augmentation, in the case of PHP operations with 20% and 30% fill ratios, leading to instability.

The fluctuations in the overall system pressure (in Fig. 7) indicate a possibility of pressure difference between the two ends of the PHP. Figure 8 depicts a small yet finite differential pressure existing between the two arbitrarily chosen points, E and C (shown in Fig. 8).

The change in the fluid velocity inside the PHP loop caused by the pressure gradient between the evaporator and condenser is expected to hold significant information. Accordingly, Fig. 9 shows the quantitative time variation of the fluid velocities at two arbitrarily chosen locations marked in the evaporator and condenser sections in the figure.

In 2D analysis, the velocity should ideally comprise both x and y-components. Since the x-component is negligible in the marked position, it is predominantly the y-velocity, abbreviated as velocity only. The graph corresponds to a 50% fill ratio with various heat loads.

One may note (from Fig. 9) that the velocity oscillations during the startup time are negligible compared to those during steady-state operation. The velocity oscillation begins at the end of the startup time. However, it is observed during stable operation that the velocity oscillations occur around zero with both positive and negative amplitudes (∼±0.5 m/s). It may be emphasized that the positive velocity corresponds to an upward fluid motion toward the condenser, while the negative velocity is the downward flow from the condenser. It is evident from the velocity reversal depicted in Fig. 9 that the pulsating fluid flow has been set alongside pressure and temperature oscillations within the PHP.

As it has been observed earlier that an augmentation in heat input from one level to another for a specific fill ratio may lead to system instability, a renewed, in-depth probe relating velocity variations during the process with a 20% fill ratio situation has been investigated. Figure 10 shows the velocity variations during the heat input transition from 0.5 W to 1.0 W, corresponding to the fill ratio of 20%.

While the changes in the velocity before the “startup” (around 10 s) remain insignificant, periodic fluctuations in velocity around the zero line set in later. However, a sharp fall in the velocity follows soon after (30 s to 40 s) the heat input is increased from 0.5 W to 1.0 W. The situation continues as long as the heat input remains the same. A scaled-up view (beyond 70 s, shown in the subset) indicates that the oscillation of the liquid slugs and vapor plugs has been transformed into a uniform and almost always positive velocity (toward the condenser), leading to an unstable operating condition.

While studying the thermohydraulics in the PHP, the thermal performance evaluation of a PHP is eventually essential as it is meant for heat transportation. Since the effective thermal conductivity is the key performance parameter (as stated in Eq. (15)), its variations during various heating stages and fill ratios are demonstrated in Fig. 11.

It may be noted from Fig. 11 that the effective thermal conductivity for a 20% fill ratio is assessed for a single heat input of 0.5 W, while the same for a 30% fill ratio is evaluated till 2.0 W. It may be recalled that a stable operation of the PHP has been possible only up to 0.5 W heat input with a 20% fill ratio and up to 2.0 W with a 30% fill ratio. Therefore, effective thermal conductivity could not be evaluated for unsteady situations.

The improvement in the PHP performance (in terms of effective thermal conductivity) with a high fill ratio is evident from Fig. 11. The thermal conductivity in most of the probed situations (except in the 20% fill ratio) shows an early rise with the heat load. When the increase in the thermal conductivity is followed by a decrease (for example, in a 30% fill ratio situation), instability sets in. However, the best performance of nearly 3400 W/mK at a 3.0 W heat load has been recorded with a 70% fill ratio in the chosen PHP configuration.

In the PHP, the heat transportation from the evaporator to the condenser, occurring through a phase-changing medium, involves both latent and sensible heat. The dimensionless Jacob number gives an insight into the fraction of heat distributed in latent and sensible heat.

where $Te$ and $Tc$ represent the average temperatures of the evaporator and the condenser sections, respectively. The thermophysical properties, such as $cp,f$ and $hfg$ are determined at the arithmetic mean temperature of $Te$ and $Tc$⁠.

Figure 12 shows the variations in the Jacob number (⁠$Ja$⁠) and its modified version with heat loads for different fill ratios.

It is evident from Fig. 12 that both the Jacob numbers increase with the heat load, but they have dissimilar dependence on the fill ratios. The Jacob number diminishes with the increase in the fill ratio, while the modified Jacob numbers are higher for larger fill ratios. Per se, following these definitions of the Jacob number, if the fill ratio (y) remains invariant, the dimensionless parameter would merely be altered by a constant factor of $(y/(1−y))$⁠, and would not change for $y=50%$⁠.

The values of the Jacob number (shown in Fig. 12), irrespective of the expressions, indicate the dominance of latent heat in the overall heat transfer process. However, this conclusion contradicts the observation made in previous studies [43,44]. Shafii et al. [43] and Groll and Khandekar [44] have shown that sensible heat should be the dominant mode of heat transfer. According to them [43,44], since the bulk fluid movement inside the PHP is slug-type flow, the cooldown of the liquid slugs releases energy to the condenser as sensible heat. Concurrently, the partial condensation of vapor (content in the slug) into liquid delivers latent heat to the condenser.

The newly defined Eq. (19) for the modified Jacob number $(Ja*)$ has been plotted for various heat loads and fill ratios in Fig. 13. It is apparent from Fig. 13 that sensible heat transfer remains the dominant mode in the overall process. This observation is in agreement with the findings of Shafii et al. [43] and Groll and Khandekar [44].

The value of the modified Jacob number $Ja*$⁠, for a given heat input, progressively increases with the rise in the fill ratio. The sizable liquid mass content corresponding to a high fill ratio would naturally signify a larger sensible heat transfer through the liquid slugs.

Furthermore, according to Fig. 13, an almost linear variation of the modified Jacob number can be seen for a given fill ratio. An increase in the heat load causes a rise in the evaporator temperature (Fig. 5). There is an augmentation in the $ΔT$ with the increase in the evaporator temperature, which is also linked to the reduction in the latent heat (⁠$hfg)$ of the fluid. Thus, the combined effects of these two enhance the modified Jacob number with the heat load.

A notable additional observation (from Fig. 13) relates to a sharp rise in the Jacob number, deviating from the linearity, with the heat input for a fill ratio of 30%, leading to thermal instability beyond 2.0 W. Similar alteration has also been noted for 40% fill ratio with an increase in the heat load from 2.0 W to 3.0 W. Predictably heat load enhancement beyond 3.0 W (not reported in this study) would cause thermal destabilization in the system.

Finally, the validity of the constant fill ratio postulation has been verified in the following paragraphs. Since the liquid and vapor densities are temperature-dependent, the fractional liquid contents (both volume and mass basis) are expected to vary with the heat load linked to temperature variations (shown in Fig. 14). Both $mf/mg$ and $Vg/Vf$ decrease with the increase in the heat load.

Figure 15 shows the deviations in two fill ratios, namely, $y=70%$ and $y=30%$⁠, from its constant value with the change in input power. Although shown for only two fill ratios, similar trends can be seen for others. It is discernible from the plot (Fig. 15) that the assumption of a constant fill ratio is invalid. Though small, variations in the fill ratio occur during the PHP operation.

A two-dimensional CFD model of a single loop PHP has been performed to study the effect of fill ratio (20%–70%) and heat input (0.5 W–3.0 W) on the thermal performance and the dominant heat transfer mode.

The major conclusions of the numerical investigation are summarized below:

• For a constant heat input to the evaporator, the higher the fill ratio, the lower the maximum initial rise of evaporator temperature, and the quicker the attainment of the stable phase (viz., the quasi-steady state after the initial startup phase). That way, the PHP operation with a higher fill ratio corresponds to a smaller rise in the average temperature of the evaporator for any given heat load.

• Instability in PHP operation with smaller fill ratios is likely even if the heat input in the evaporator is not so high. For example, a stable operation in the PHP of chosen dimensions could not be sustained with thermal energy of more than 1.0 W in the evaporator for a fill ratio of 20%.

• The pressure oscillations inside the PHP are similar to the temperature oscillations—both steadily escalate with the heat input. Moreover, the differential temperature between the evaporator and the condenser also causes a differential pressure across the two ends. However, the amplitude of the pressure differential remains invariant with the heat load.

• Unlike pressure and temperature fluctuations, the amplitude of the oscillating fluid velocity inside the PHP does not keep increasing with the applied heat load. However, during thermal instability in the PHP system, the velocity oscillation transforms into a uniform (positive flow toward the condenser).

• The effective thermal conductivity depends on the heat load and the fill ratio for a given PHP configuration. The higher the fill ratio, the larger the thermal conductivity, but it attains a steady-state once a given heat load is reached.

• The modified Jacob number has been redefined, increasing with the heat load and the fill ratio. The new definition of the Jacob number is consistent with the observation predicting the dominance of sensible heat transfer over latent heat, indicated by earlier researchers.

• The assumption of a constant fill ratio is not truly valid. Change in the heat load is associated with a slight variation in the fill ratio.

We acknowledge National Supercomputing Mission (NSM) for providing computing resources of “PARAM Shakti” at IIT Kharagpur, which is implemented by C-DAC and supported by the Ministry of Electronics and Information Technology (MeitY) and Department of Science and Technology (DST), Government of India.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$Acr$ =

cross-sectional area (m²)

$cp$ =

specific heat (J/kg K)

$Dcrit$ =

critical diameter (m)

E =

energy (J)

F =

force (N)

g =

gravitational acceleration (m/s²)

$hfg$ =

latent heat (J/kg)

$Ja$ =

Jacob number

$Ja*$ =

modified Jacob number

k =

thermal conductivity (W/m K)

l =

length (m)

m =

mass (kg)

p =

pressure (Pa)

Q =

heating power (W)

$T$ =

temperature (K)

t =

time (s)

v =

velocity (m/s)

V =

volume (m³)

y =

fill ratio

Greek Symbols

$α$ =

volume fraction of a cell

$β$ =

accommodation coefficients

$ρ$ =

density (kg/m³)

$σ$ =

surface tension (N/m)

$μ$ =

dynamic viscosity (Pa-s)

Subscripts

a =

adiabatic section

c =

condenser section

e =

evaporator section

eff =

effective

f =

liquid

g =

vapor

w =

wall