Numerical Investigation of Spray Cooling Dynamics: Effects of Ambient Pressure, Weber Number, and Spray Distance on Droplet Heat Transfer Efficiency

Abstract

This research aims to study the spray flow of a droplet on an aluminum surface. Fluid spraying is a significant topic in various strategic industries worldwide. In this study, the commercial software FLUENT 22.3.0 is used to simulate the spray of a droplet with turbulent flow on a surface. We use Gambit for mesh generation to ensure accurate and efficient discretization of the computational domain. Initially, we validate our finite volume method (FVM) by comparing the simulation results with existing experimental data to ensure accuracy. After verifying the numerical methods and boundary conditions, we extend the analysis to explore new scenarios involving different environmental pressures, nozzle-to-surface distances, and heated surface temperatures. The effects of pressure variation on the efficiency of droplet heat transfer are examined within sub-atmospheric and super-atmospheric pressure ranges at different Weber numbers, all below the critical Weber number of the droplet. Additionally, by modifying the model geometry and boundary conditions, the influence of the spray-to-surface distance was examined. The findings show that both pressure changes and the spacing between the spray origin and the surface have a substantial effect on the droplet’s heat transfer performance.

1. Introduction

Spray cooling has emerged as an advanced thermal management technique, capable of addressing the growing demands for high heat flux dissipation in modern electronics, high-power density systems, and aerospace applications [1,2,3]. Traditional cooling strategies, such as forced convection, heat pipes, and pool boiling, often fall short in handling localized heat generation at high flux densities. In contrast, spray cooling systems achieve critical heat flux (CHF) values surpassing 1000 W/cm², providing enhanced heat transfer efficiency and maintaining lower surface temperatures with excellent temperature uniformity [4,5,6,7]. Recent studies report even higher CHFs by optimizing spray parameters and using enhanced surfaces [8,9,10,11]. This efficiency results from the simultaneous action of forced convection, droplet evaporation, nucleate boiling, and film boiling, which can be tailored to the application by adjusting spray characteristics, coolant properties, and surface conditions. Spray cooling has been successfully implemented in high-performance electronics, military and aerospace avionics, data centers, and thermal management for space applications [12,13,14,15,16]. Among coolants, dielectric liquids such as HFE-7100 and FC-72 are increasingly favored in spray cooling applications due to their low electrical conductivity, chemical stability, and compatibility with electronic components. However, their relatively low thermal conductivity compared to water and other fluids necessitates optimization of spray parameters and impact conditions to achieve effective thermal control [4,17,18,19]. These considerations become more critical in environments with varying ambient pressures, such as aerospace or vacuum systems, where droplet behavior and heat transfer mechanisms differ significantly from terrestrial conditions [20,21,22,23].

The understanding of single-droplet impingement is foundational to advancing spray cooling technologies. Early studies, such as those by Bernardin and Mudawar [4], established boiling regime maps that describe the transition between nucleating boiling, transition boiling, and film boiling during droplet impact. Liang and Mudawar [8] expanded on these findings by analyzing the effects of dimensionless parameters like Weber and Reynolds numbers on spreading behavior, splashing thresholds, and droplet fragmentation. Mundo et al. [20] and Yarin [21] provided detailed descriptions of droplet impact outcomes on solid surfaces, including deposition, rebound, and splashing, as functions of impact velocity and surface wettability. Bai and Gosman [22], followed by Rioboo et al. [23], developed predictive models for droplet spreading and splashing. Pasandideh-Fard et al. [24] further clarified the role of capillary forces in droplet spreading on heated surfaces. Recent experimental research has characterized droplet impact behavior on heated surfaces under varying thermal conditions. Xu et al. [18] demonstrated that the surface temperature strongly influences spreading diameter, impact regime, and heat transfer efficiency in HFE-7100 droplets. Below the Leidenfrost point, nucleate boiling dominates, leading to higher heat transfer rates [25]. Above this point, vapor film formation reduces liquid–solid contact, limiting heat transfer and enhancing droplet rebound or splashing tendencies [26,27]. Despite these advances, most studies have focused on ambient atmospheric conditions, overlooking the effects of ambient pressure variations on droplet dynamics. Recent work highlights that varying pressure critically influences droplet spreading, vapor layer formation, and heat transfer efficiency [28,29,30,31,32,33].

Although single-droplet studies offer a basic understanding, multiple-droplet impingement and spray cooling more accurately reflect real-world applications. Repeated droplet impacts can improve heat transfer by keeping surfaces wet and preventing dry-out [34]. Soriano et al. [35] studied droplet trains and developed critical heat flux (CHF) correlations for repeated impacts. Building on this, Panao and Moreira [36] analyzed secondary atomization and droplet interaction effects. Spray cooling systems employ nozzle arrays to generate fine droplets and achieve even surface coverage, enhancing heat transfer over larger areas. Optimized systems have achieved CHFs above 1000 W/cm² by refining nozzle geometry, droplet size, and spray angle [4,20,21,22,23,24]. Yin et al. [37] highlighted how subcooling and flow characteristics influence performance, while Xu et al. [18] showed that higher Weber numbers improve heat transfer but may increase splashing. To maximize performance without excessive droplet breakup, parameters such as Weber number, surface temperature, and fluid properties must be carefully tuned. Recent research also underscores the importance of spray cone angle and droplet velocity in determining droplet spread and surface wetting uniformity [12,14,25,38,39].

The performance of spray cooling systems is significantly influenced by the properties of the target surface. Surface enhancements, such as micro- and nanostructuring, have been shown to increase nucleation site density, improve wettability, and delay film boiling onset, thereby elevating CHF and overall heat transfer [40,41]. Hierarchical micro/nanostructures enable better droplet spreading and improve evaporation dynamics, enhancing cooling efficiency [42]. Materials with higher thermal conductivity, like copper, generally outperform lower-conductivity substrates like stainless steel in spray cooling applications [7,43,44,45,46]. Xu et al. [47] confirmed these observations by comparing copper and stainless steel under HFE-7100 droplet impingement. Additionally, biphilic surfaces—combining hydrophobic and hydrophilic regions—optimize droplet behavior and have demonstrated exceptional performance in high heat flux dissipation [26]. The ambient pressure in spray cooling environments is a crucial factor, especially in aerospace and vacuum systems. Reduced ambient pressure alters droplet breakup and atomization characteristics, typically resulting in larger droplet sizes and delayed breakup. Sub-atmospheric pressures reduce atomization efficiency and droplet breakup, impacting droplet velocity and coverage [15]. Lower ambient pressures also affect droplet boiling behavior. Reduced pressures lower the saturation temperature, shifting the transition between nucleate and film boiling. CHF generally decreases under sub-atmospheric conditions, while changes in droplet spreading and rebound behavior have been observed [15,48,49]. These effects are magnified in microgravity environments, where droplet dynamics differ significantly from terrestrial conditions [38]. Despite this knowledge, there remains a need for systematic investigation of combined ambient pressure, surface temperature, and droplet behavior effects on spray cooling performance, especially for aerospace applications [14,47,49].

The nozzle-to-surface distance and spray cone angle are key design parameters in spray cooling systems. Shorter nozzle distances increase droplet velocity and impact energy but can lead to excessive film formation and lower CHF by suppressing nucleate boiling [14,47,50,51]. Longer distances reduce droplet velocity and promote evaporation before impact, potentially diminishing cooling effectiveness [30]. The spray cone angle controls droplet spatial distribution on the surface. Wide angles enhance surface coverage but may reduce droplet momentum, while narrow angles focus on cooling, but risk localized dry-out. Optimizing these parameters is essential for achieving uniform cooling, especially in variable-pressure environments where droplet dynamics are altered [50,51,52,53,54,55,56,57,58,59,60]. Moreover, the combined effects of nozzle distance, spray cone angle, surface enhancement, and ambient pressure on heat transfer performance require further investigation.

Despite significant advancements in spray cooling research, gaps remain. Most studies focus on ambient conditions, with limited attention to pressure variations and their impact on single droplet impingement [14,41,47,49,50,51]. This study addresses these gaps through a comprehensive numerical and experimental analysis of single-droplet impingement on heated aluminum surfaces. It evaluates the effects of ambient pressure, nozzle-to-surface distance, and spray geometry on heat transfer. It provides insights for optimizing spray cooling systems for high-power electronics and aerospace applications. In this paper, a numerical model of spray cooling is developed to investigate the effects of various spray conditions, including different heater temperatures, spray half-angles, and nozzle positions. Given the complexities involved in spray cooling, this study aims to numerically analyze the heat transfer characteristics of single droplets impacting a heated surface under varying environmental pressures. Initially, we validate our finite volume method (FVM) analysis by comparing the simulation results with existing experimental data to ensure accuracy. After verifying the numerical methods and boundary conditions, we extend the analysis to explore new scenarios involving different environmental pressures, nozzle-to-surface distances, and heated surface temperatures. Using CFD simulations, the effects of varying Weber numbers, spray distances, and ambient pressures on heat transfer efficiency are analyzed. The findings of this study are expected to contribute to the optimization of spray cooling systems for extreme environments, such as aerospace applications and high-power electronic devices.

2. Governing Equations

The motion of a fluid in which temperature and velocity gradients exist simultaneously is governed by the fundamental conservation laws: the continuity equation (mass conservation), the momentum equation (Newton’s second law), and the energy equation [12]. These principles apply at every point within the fluid. The general form of the energy equation, momentum equation, turbulent kinetic energy equation, dissipation rate equation, and turbulent kinetic energy production rate equation can be expressed as

$${\displaystyle \frac{\partial }{\partial t}} \left(\rho \phi \right)+\mathbf{\nabla }.\left(\rho \overrightarrow{\mathit{U}}\phi \right)=\mathbf{\nabla }.\left(\gamma \mathbf{\nabla }\phi \right)+S_{\phi }+S_{\phi d}$$

(1)

where $S_{\phi d}$ an $S_{\phi }$ represent source terms. Here, $\rho$ denotes the fluid density, $\overrightarrow{\mathit{U}}$ represents the velocity vector, $\gamma$ accounts for diffusion effects, and the continuity equation, the parameter is set as $\phi =1$. Considering the above assumptions of an unsteady, two-dimensional, and incompressible flow, Equation (1) is rewritten into the following specific governing equations:

$${\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}=0$$

(2)

$$\rho \left({\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}\right)=-{\displaystyle \frac{\partial P}{\partial x}}+\mu \left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}\right)+X$$

(3)

$$\rho \left({\displaystyle \frac{\partial v}{\partial t}}+u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}\right)=-{\displaystyle \frac{\partial P}{\partial y}}+\mu \left({\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}v}{y^2}}\right)+Y$$

(4)

$$\rho c_p\left({\displaystyle \frac{\partial T}{\partial t}}+u{\displaystyle \frac{\partial T}{\partial x}}+v{\displaystyle \frac{\partial T}{\partial y}}\right)=k\left({\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}\right)+\mu \phi +\dot{q}$$

(5)

Equation (2) results from applying the mass conservation law to an infinitesimally small control volume. Equations (3) and (4) are derived from Newton’s second law in the x and y directions, respectively. The first term on the left-hand side of each equation represents the time rate of change of velocity due to flow unsteadiness, leading to potential instability. The second and third terms describe the net convective momentum flux in the x and y directions. On the right-hand side of Equation (3), the first term accounts for pressure forces, the second for viscous (diffusive) effects due to shear and normal stresses, and the last for body forces per unit volume, which are assumed zero in this study.

Equation (5) follows from the energy conservation law applied to a fluid control volume. The first term on the left-hand side indicates the rate of temperature change over time (transient effects). The next two terms represent convective heat transfer due to fluid motion. On the right-hand side, the first term describes heat conduction (Fourier’s law), the second term accounts for viscous dissipation (conversion of mechanical energy into heat), and the third term represents internal heat generation, which is neglected here. Under steady-state conditions, all time-derivative terms vanish.

3. Investigation of Turbulence Models

In general, turbulence models are classified into two main categories: Reynolds Stress Models (RSMs) and Eddy Viscosity Models (EVMs). Reynolds Stress Models directly solve transport equations for each component of the Reynolds stress tensor, offering high accuracy for complex turbulent flows, particularly in cases with strong anisotropy [54]. Common examples of RSMs include LRR-IP (Linear Pressure–Strain with Isotropic Production), LRR–QI (Linear Pressure–Strain with Quadratic Inhomogeneous Production), SSG (Quadratic Pressure–Strain Model), ωSMC (Omega-based Reynolds Stress Model), and EARSM (Explicit Algebraic Reynolds Stress Model). On the other hand, Eddy Viscosity Models rely on the concept of turbulent viscosity and assume isotropic turbulence. These models are simpler and more computationally efficient, making them widely used in practical engineering applications. EVMs include zero-equation models, one-equation models such as Spalart–Allmaras, two-equation models like k-epsilon and k-omega, three-equation models such as k-kl-omega, and four-equation models including Transition SST and V2F. Among these, the k-epsilon and k-omega models are the most popular two-equation turbulence models [55]. The k-epsilon model solves transport equations for turbulent kinetic energy (k) and its dissipation rate (epsilon). It performs well for fully developed turbulent flows, particularly in free shear layers, but can be less accurate near walls and in flows with strong pressure gradients. The k-omega model, which solves equations for turbulent kinetic energy (k) and specific dissipation rate (omega), provides better performance in near-wall regions and flows with adverse pressure gradients. The Shear Stress Transport (SST) version of the k-omega model further improves its predictive capabilities by blending the strengths of both k-epsilon and k-omega models. While Reynolds Stress Models are ideal for detailed turbulence analysis, k-epsilon and k-omega models remain the most practical choices for many engineering simulations due to their balance of accuracy and computational efficiency [55].

3.1. k-Epsilon Model

The k-epsilon model is one of the most established and widely adopted turbulence models in computational fluid dynamics. It is known for delivering reliable accuracy and stability in general-purpose simulations. Its widespread use in engineering stems from its long-standing development and relatively low computational demands. The model operates under the assumption that the flow is fully turbulent, with eddy viscosity playing a more significant role than molecular viscosity. In this context, turbulent kinetic energy is produced by velocity gradients and dissipated through viscous effects. The balance between these production and dissipation processes determines whether the turbulent kinetic energy increases or decreases.

3.2. k-Omega Model

One of the main challenges in modeling turbulent flows is the accurate prediction of flow separation from surfaces [56]. Predicting flow separation phenomena is crucial in many practical applications for both internal and external flows. Models like k-epsilon often fail to accurately predict the onset point and the extent of flow separation, especially in the presence of adverse pressure gradients. One of the most important models developed to overcome this limitation is the k-omega model [56]. The convergence behavior of k-omega models is similar to that of k-epsilon models. However, k-omega models are considered low-Reynolds-number models and provide accurate and stable results for a wide range of boundary layer flows, including those with pressure gradients. The k-omega models are specifically developed to address the shortcomings of the epsilon equation in the near-wall region. Issues such as the difficulty of solving the epsilon transport equation, its inaccuracy near solid boundaries, and the increase in epsilon in these regions led to the development of the k-omega model. In this model, instead of solving for epsilon, the equation for omega is used, where

$$\mathsf{\omega }={\displaystyle \frac{\epsilon }{k}}\approx {\displaystyle \frac{1}{time}}$$

(6)

This is referred to as the specific dissipation rate, which represents the inverse of the turbulence time scale, or equivalently, the frequency of turbulence near the solid boundary. Like the k-epsilon model, k-omega is also a two-equation model, with modifications to account for low-Reynolds-number effects, compressibility, and shear flow distribution. One of the advantages of this model is its improved predictions near the wall for low Reynolds numbers. Additionally, the k-omega model does not require the complex non-linear damping functions near walls that are necessary for k-epsilon, resulting in higher accuracy and stability [56].

4. Numerical Method Used for Modeling

The finite volume method is a numerical approach used to approximate the solution of different solutions. In this method, the equations are written in integral form and then solved. FVM is particularly suitable for computational fluid dynamics and heat transfer problems. The FLUENT software (22.3.0) employs this method for numerical simulations. The Navier–Stokes equations in their fundamental form contain at least four primary unknowns: pressure and velocity components in three directions. In some simulations, density also depends on the solution conditions and must be treated as an additional unknown. A flow is considered compressible only if density variations result from pressure changes [45,54,55,56]. In all variable-density flows, density is one of the unknowns in the simulation. In this case, the Navier–Stokes equations provide four equations but five unknowns, necessitating an additional equation. To resolve this, an equation of state is introduced. Examples include the ideal gas law and the Peng–Robinson equation for non-ideal gases. Using the equation of state, either pressure or density can be obtained in the main solution loop, while the other is computed separately and updated in the next iteration. Two solution approaches exist based on the primary unknown in the loop [31]:

• Pressure-based method: Pressure is solved in the main loop, and density is determined from the equation of state.
• Density-based method: Density is solved in the main loop, and pressure is computed using the equation of state.

For this study, pressure-based methods have been employed. Examining the continuity equation reveals that its only unknowns are density and velocity. Since density is not treated as a primary unknown in the pressure-based method, only velocity components remain unknown in the continuity equation, and these have already been determined from the momentum equations. Consequently, one unknown (pressure) remains undetermined, and one equation (continuity) remains unused. To resolve this, modifications are made to the momentum and continuity equations to derive a form of the continuity equation that includes a pressure term. These approaches are known as pressure–velocity coupling methods. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm has been used in this study to establish the coupling between pressure and velocity. This method is widely applied to solve the Navier–Stokes equations. For pressure discretization, the PRESTO (Pressure Staggering Option) scheme has been used. In incompressible Navier–Stokes equations, there is no direct link between pressure in the continuity and momentum equations. Various methods, including PRESTO, are employed to establish the relationship between pressure and velocity [31].

4.1. Modeling Results Based on the k-ω Method

The numerical simulation is performed using the k-ω turbulence model, which is well suited for capturing near-wall effects and accurately modeling fluid behavior in high-shear regions. This model provides a detailed representation of turbulence dynamics by solving equations for turbulent kinetic energy (k) and specific dissipation rate (ω). The simulation results analyze the impact of varying ambient pressures, different Weber numbers, and the effect of droplet spray distance on heat transfer efficiency. The findings indicate that pressure plays a significant role in heat transfer enhancement, and as the Weber number increases, the kinetic energy of the droplet increases, leading to a more pronounced spread on the heated surface, thereby enhancing heat transfer efficiency. The k-ω model’s ability to capture turbulence effects provides a more accurate prediction of droplet behavior, deformation, and thermal interaction with the surface compared to simpler turbulence models.

4.2. Comparison with Experimental Data

When a subcooled droplet impacts a surface heated above the Leidenfrost temperature (Figure 1), the parameter ε, spray effectiveness, describes how efficiently the spray removes heat. It expresses the ratio of the actual heat removed by the spray to its maximum possible heat removal, which includes both the latent heat and sensible heat of the liquid. This parameter serves as a measure of the spray’s overall heat transfer efficiency. The effects of the droplet Weber number on spray cooling effectiveness are illustrated in Figure 2, which presents a comparison between experimental data from the literature and the results of our FVM simulations. The FVM model was developed to replicate the heat transfer behavior under varying spray conditions and to validate experimental observations through numerical analysis. Data in the figure were collected from several published studies [29], encompassing a wide range of conditions. The mass fluxes ranged from 0.016 to 2.05 kg/m²s, with droplet velocities between 0.6 and 7.3 m/s, and droplet diameters from 0.13 to 25 mm. To ensure consistency in boundary conditions between the experimental data and the FVM simulations, all heat transfer analyses were conducted at a surface temperature of 400 °C. The FVM simulation results show good agreement with the experimental trends. Both sets of data demonstrate a general decrease in spray cooling effectiveness as the Weber number increases. Notably, there is a distinct drop in performance at approximately We = 80. This behavior is consistent with the findings of Yao and Cox [29], who reported a shift in droplet dynamics when impacting non-wetting surfaces at similar Weber numbers.

4.3. Mesh Generation

The schematic representation of the model for boundary conditions follows the layout shown in Figure 3. Based on this figure, the spray surface and environment are modeled using Gambit software (15.1.0) via the geometry section. The mesh generation process is carried out using the Mesh Edge and Mesh Face menus. Triangular meshes are created through the Map Split function. After defining the required geometry, the Mesh Edge menu is used to specify the number of nodes along each edge. Adjusting the number of nodes on each boundary allows for refinement or modification of the mesh elements. Finally, in the Zone menu, the Specify Boundary Type window is used to assign boundary conditions, where the surface is set as a wall boundary condition. Within the designated continuum type window, the interior domain is defined as Fluid. The completed mesh is subsequently exported in Mesh format, as illustrated in Figure 4. A summary of the thermophysical properties used in the CFD simulation 2023 is provided in

Table 1. Physical properties of the droplet.

Parameter	Value	Unit
Droplet diameter	500	µm
Droplet temperature	27	°C
Mass flux	2	kg/m2·s
Surface material	Aluminum	—
Surface temperature	550	°C
Ambient pressure	0.1–30	atm
Density	997	kg/m3
Dynamic viscosity	8.9 × 10−4	Pa·s
Specific heat capacity	4182	J/(kg·K)
Thermal conductivity	0.6	W/(m·K)

.

In this study, a single-phase flow model was employed. Although ambient pressure was specified at the inlet and outlet boundaries to simulate different environmental conditions, no two-phase model or phase-change formulation was used, as the focus was on droplet impact dynamics and heat transfer.

4.4. Simulation Results for Different Ambient Pressure Ranges, Weber Numbers, and Spray Distance

This section presents the numerical simulation results for the spray of a 500-micrometer water droplet at a temperature of 27 °C, with a mass flux of 2 kg/(m².s), onto an aluminum surface at 550 °C under various pressure conditions. The Weber number of the droplet varies within the range 10 < We < 20, while the critical Weber number for the given conditions is 20. At We > 10, the droplet flow becomes turbulent. In this simulation, the spray distance from the droplet injection point to the surface is 40 mm. Figure 5 presents a comparative analysis of the heat transfer efficiency of the droplet within the Weber range of 10 to 20 and the pressure range of 0.1 to 50 atmospheres. The results indicate that increasing the pressure enhances the droplet’s heat transfer efficiency. Additionally, as the Weber number increases, the kinetic energy of the droplet rises, leading to greater spread on the surface and subsequently improved heat transfer performance.

4.5. Simulation Results for Different Spray Distances at Constant Pressures

This section presents the numerical simulation results for the spray of a 500-micrometer water droplet with a mass flux of 2 kg/(m².s) onto an aluminum surface at 550 °C under different pressures. The Weber number of the droplet varies within the range 10 < We < 20, while the critical Weber number for the given conditions is 20. For We > 10, the droplet flow transitions to a turbulent regime. The pressures considered in this study are 0.2, 1, and 30 atmospheres, and the spray distances from the injection point to the surface are 20 mm, 40 mm, and 60 mm. The results first focus on the spray behavior at a fixed pressure of 0.2 atm.

Figure 6 illustrates the variation in heat transfer efficiency as a function of the Weber number for different spray distances. The results show that increasing the Weber number enhances heat transfer efficiency, as the droplet spreads more effectively upon impact. Moreover, reducing the spray distance leads to higher heat transfer efficiency, as the droplet undergoes less air resistance and retains more kinetic energy upon impact. Figure 7 depicts the changes in heat transfer efficiency concerning the Weber number at different spray distances and pressures (0.2, 1, and 30 atm). The results suggest that at sub-atmospheric pressures, the effect of spray distance on heat transfer efficiency is more pronounced. Furthermore, at higher pressures, the efficiency is generally higher, but the influence of spray distance becomes less significant. These findings highlight the interplay between pressure, Weber number, and spray distance in determining the heat transfer performance of the droplet.

4.6. Simulation Results of Droplet Trajectory at Different Pressures

This section presents the numerical simulation results for the trajectory of a sprayed water droplet under different ambient pressures. The study is conducted for a 500-micrometer droplet at an initial temperature of 27 °C, with a mass flux of 2 kg/(m².s) onto an aluminum surface maintained at 550 °C. For this analysis, the Weber number is set to 10, ensuring a specific level of droplet deformation and interaction with the surface. The pressure conditions investigated include 0.2 atm, 0.5 atm, 1 atm, 10 atm, and 30 atm. Figure 8 illustrates the trajectory of the droplet at a pressure of 0.2 atm. As can be seen, the droplet deviates significantly from the center of the surface along its path from the spray point to the impact area. This deviation is primarily due to the lower air density at sub-atmospheric pressures, which reduces the resistance force acting on the droplet and allows minor external forces (such as asymmetries in initial velocity or ambient disturbances) to have a greater influence on its trajectory. As a result, the droplet follows a more curved path before reaching the heated surface. In contrast, at higher pressures (e.g., 10 atm and 30 atm), the increased air resistance stabilizes the droplet path, reducing deviations and ensuring a more direct trajectory toward the surface. These results highlight the significant role of ambient pressure in governing the droplet’s movement and subsequent thermal interactions upon impact.

Figure 9 illustrates the trajectory of a sprayed droplet at a pressure of 0.5 atm. As observed, the droplet deviates from the center of the surface during its motion from the spray point to the impact area. By comparing this figure with the trajectory at 0.2 atm (Figure 8), it is evident that the deviation from the center is less pronounced at 0.5 atm. The increase in ambient pressure leads to greater air resistance, which helps stabilize the droplet’s path and reduces its lateral displacement. Additionally, after impacting the surface, the droplet shows a greater tendency to return to the surface compared to the 0.2 atm case. This behavior suggests that at higher pressures, the droplet retains less kinetic energy after rebound, leading to a stronger interaction with the heated surface.

Figure 10 depicts the trajectory of a sprayed droplet at a pressure of 1 atm. The droplet exhibits a slight deviation from the center of the surface as it moves from the spray point to the impact area. A comparison with Figure 5 (0.2 atm) and Figure 4 (0.5 atm) reveals that the droplet’s deviation at 1 atm is significantly reduced compared to lower pressures. The increase in ambient pressure results in greater air resistance, which helps stabilize the droplet’s trajectory and minimizes lateral displacement. Moreover, after striking the surface, the droplet often rebounds and re-contacts it. This indicates that at higher pressures, the droplet’s interaction with the surface becomes more stable, leading to shorter rebound distances and an increased likelihood of secondary impacts on the heated surface.

Figure 11 presents the trajectory of a sprayed droplet at a pressure of 10 atm. As observed, the droplet undergoes a slight deviation from the center of the surface while traveling from the spray point to the impact area. A comparison with Figure 8, Figure 9, Figure 10 and Figure 11 indicates that the droplet’s deviation at 10 atm is noticeably smaller than at lower pressures. The increase in ambient pressure enhances air resistance, which helps to stabilize the droplet’s path and reduce lateral displacement. Moreover, after rebounding from the surface, the droplet moves back toward the surface and impacts it again. At 10 atm, the distance between the first and second impact points is shorter compared to the case at 1 atm, suggesting that the droplet loses more kinetic energy upon impact, resulting in a less pronounced rebound and a quicker return to the surface.

The trajectory of a sprayed droplet at a pressure of 30 atm is depicted in Figure 12. The droplet exhibits a slight deviation from the center of the surface as it travels from the spray point to impact. At this pressure, the gap between the first and second impacts is significantly reduced compared to 1 atm and 10 atm. The shorter rebound distance and increased frequency of surface interactions contribute to greater heat transfer efficiency, as the droplet remains in prolonged contact with the heated surface, maximizing thermal exchange.

5. Discussion

A detailed analysis indicates that at constant pressure, the heat transfer efficiency parameter (ε) of the droplet increases with a higher Weber number. This parameter represents the ratio of the actual heat transfer by the droplet to the ideal heat transfer. For a droplet, an increase in the Weber number signifies a higher kinetic energy relative to surface tension. Since the Weber numbers studied in this model are below the critical Weber number, the droplet does not disintegrate upon impact but rather spreads over the surface. This increased spreading enhances the contact area between the droplet and the surface, thereby improving heat transfer efficiency. Given that the Weber number remains below the critical threshold, the droplet tends to return to its original shape, ultimately leading to a rebound from the surface. The greater the spread of the droplet, the longer the time required for it to retract, which in turn results in greater thermal exchange before the droplet detaches from the surface.

Additionally, increasing the ambient pressure of the spray enhances the heat transfer efficiency parameter (ε) of the droplet. A higher pressure leads to greater oscillations of the droplet upon impact with the surface. As the distance between successive impacts decreases after the initial rebound, the contact time between the droplet and the surface increases, resulting in greater heat transfer. As the droplet absorbs heat, part of its mass vaporizes, and this evaporation process continues until the droplet is completely vaporized on the surface.

Furthermore, the effect of pressure is more significant in the sub-atmospheric pressure range. A small increase in pressure in this range leads to a considerable rise in the droplet’s heat transfer efficiency. Due to the presence of an air mass in the spray region and the interaction of droplets with this air mass, lower pressure levels cause the droplet’s trajectory to deviate from the center of the surface. This reduction in vertical velocity results in lower kinetic energy upon impact, which influences the spreading and thermal exchange behavior of the droplet. As a result of this reduction in kinetic energy, the extent of droplet spreading on the surface decreases, leading to a lower heat transfer efficiency. At pressures above atmospheric levels, the rate of increase in heat transfer efficiency with rising pressure becomes less pronounced. The influence of pressure on the density of the surrounding air mass in the spray environment is more significant at sub-atmospheric pressures than at higher pressures. Thus, at pressures above 10 bar, the increase in heat transfer efficiency becomes less noticeable, especially when the pressure values are close to each other.

The simulation of the droplet under identical conditions was analyzed for different spray distances from the surface. The heat transfer efficiency parameter was evaluated for a droplet at a pressure of 30.2 atm, with spray distances of 20 mm, 40 mm, and 60 mm from the surface. In both examined cases, the Weber number of the droplet is within the range 10 < We < 20. In each of these cases, as the Weber number increases, the kinetic energy impact relative to the droplet’s surface tension also increases, leading to an enhanced heat transfer efficiency parameter. However, as the spray distance increases, the rate of decrease in heat transfer efficiency becomes less pronounced. The air mass in the spray environment causes the droplet to deviate from its vertical trajectory toward the center of the surface. Due to the presence of this air mass, the vertical velocity component of the droplet decreases as the spray distance increases, while the tangential velocity component increases. This shift reduces the droplet’s heat transfer capability, lowering the heat transfer efficiency parameter.

The changes in the heat transfer efficiency parameter were further examined as a function of the Weber number at pressures of 0.2 atm, 1 atm, and 30 atm, considering different spray distances from the surface. The analysis confirms that increasing pressure at any given spray distance and increasing the Weber number at any given distance both contribute to enhanced heat transfer efficiency. For a fixed Weber number, it was also observed that reducing the spray distance has a more significant impact on heat transfer efficiency at sub-atmospheric pressures. Additionally, the effect of increasing the spray distance is more pronounced at lower pressures. At sub-atmospheric pressures, the reduced air density causes the droplet to veer more from its vertical trajectory, resulting in a greater displacement from the surface center. This deviation leads to an increase in the tangential velocity component while simultaneously reducing the vertical velocity component. As a result, the kinetic energy available for spreading decreases relative to surface tension, leading to less droplet spreading on the surface and lower heat transfer efficiency.

It was also observed that at sub-atmospheric pressures, the droplet’s velocity tends to become more horizontal as it approaches the surface. Reducing the spray distance can effectively mitigate this effect at sub-atmospheric pressures, allowing for better control over the droplet trajectory and impact behavior.

6. Summary and Conclusions

This study carried out a numerical analysis of spray cooling by examining how individual droplets behave when striking a heated surface under different environmental pressures, nozzle-to-surface distances, and heater temperatures. Simulations were performed using FLUENT, with mesh generation handled by Gambit, to accurately model the spray dynamics. The finite volume method (FVM) was validated against existing experimental results, confirming the credibility of the numerical setup and boundary conditions. Findings revealed that both ambient pressure and spray distance play critical roles in heat transfer efficiency. At constant pressure, higher Weber numbers lead to greater droplet spreading upon impact, which enhances heat transfer. Conversely, at reduced pressures, the lower air density diminishes droplet momentum and spreading ability, thereby reducing thermal performance. Raising the pressure, especially in low-pressure environments, significantly boosts efficiency by improving droplet stability, spreading behavior, and evaporation. In addition, shortening the spray distance helps maintain a vertical droplet trajectory, particularly under low-pressure conditions, resulting in higher impact velocity and better heat transfer. However, at elevated pressures, the effectiveness of further increasing pressure or adjusting spray distance becomes less pronounced.