Dimensionless Heat Transfer Coefficient Research Articles

A numerical investigation on heat transfer characteristics of a particle cluster in fluid with variable properties

This work investigates the heat transfer characteristics of particle clusters under the effects of the complex properties of supercritical water (SCW). It analyzes the heat transfer characteristics of sub-particles and the average heat transfer characteristics of particle clusters. The results reveal a phenomenon of shifting positions of high specific heat regions. It led to variations in the dimensionless heat transfer coefficient distribution. Furthermore, the results indicate that as the heat transfer process strengthens, the effects of variations in property distribution on heat transfer tends to stabilize. Based on this conclusion, the effects of variations in property distribution on heat transfer are categorized into Stable Effects Region and Non-Stable Effects Region. By utilizing the principles of fluid flow-heat transfer coupling and similarity, a heat transfer prediction model for particle clusters in SCW is established.

A dimensionless heat transfer coefficient for free convection that is more appropriate than Nusselt number

Heat transfer in flows created by buoyancy, or natural convection, is a widely studied topic across various disciplines spanning natural flows as well as those with engineering applications. The convective heat transfer rate on a surface is commonly represented by the Nusselt number (Nu), a ratio of convective to diffusive transport, expressed often as RanPrm, where Ra is the Rayleigh number, the buoyancy forcing parameter, and Pr the Prandtl number. Motivated by the observation that n∼1/3 for turbulent convection, which implies the heat flux is independent of the length scale (L, characteristic length related to the geometry), we propose an alternate and physically more meaningful non-dimensional heat transfer parameter, denoted by Cq. Cq is derived using only the near wall variables and does not contain L. For n=1/3, Cq is constant. Even for laminar convection, where n∼1/4, Cq∼Ra−1/12, a weak function of Ra. We show that for natural convection over several geometries and a wide range of Ra, the Cq values within a narrow range while the corresponding Nu values span several orders of magnitude. We also show that Cq is akin to the non-dimensional representation of wall shear stress, skin friction coefficient Cf. We believe that just like Cf, Cq will be an equally useful non-dimensional measure of heat transfer in natural convection flows.

Heat transfer measurement near injection hole of supersonic nozzle with a sonic jet injection

Heat transfer measurements were conducted to investigate surface heat transfer characteristics resulting from the interaction between a sonic jet and supersonic nozzle crossflow. A sonic jet was injected into the nozzle where the average Mach number is 2.88. Experiments were carried out for three different momentum flux ratios (J=0.5, 1.0, 1.5). A half-nozzle was introduced to measure heat transfer on the inner walls of the nozzle using Infrared thermography. Pressure measurements using PSP and oil flow visualization were also conducted to understand the heat transfer. From the oil flow visualization and heat transfer results, it was found that the area around the hole influenced by recirculation vortexes exhibited high heat transfer coefficients, showing up to 5 times higher values compared to the freestream. Furthermore, as the momentum flux of the jet increased, both the heat transfer augmentation area and the heat transfer coefficient also increased. To analyze the heat transfer characteristics between the nozzle and flat surface, comparisons were made using dimensionless heat transfer coefficients. The nozzle exhibited higher heat transfer in a smaller interaction area compared to that of a flat surface, but the average heat transfer coefficient near the hole was lower. Accordingly, distinct thermal treatment strategies should be employed for nozzles and flat surfaces, especially near injection holes. These insights could be valuable for the thermal design of rockets and other supersonic/hypersonic mobility systems.

Numerical simulation and thermodynamic test analysis of plate heat exchanger based on topology optimization

Efficient heat transfer in PHEs relies on optimizing flow distribution to minimize thermal dead zones. Existing studies have predominantly focused on single-phase fluids as working media and employed density-based methods for structural optimization of microchannel heat exchangers at small scales. In this study, the Brinkman term from the porous media flow model is introduced in an innovative manner, along with the utilization of a material density interpolation model, which builds upon the density-based approach. A novel two-phase fluid topology optimization method is proposed, specifically tailored for large-scale PHE operating conditions, resulting in the successful design of a new PHE topology. To compare the performance of the two heat exchangers, finite element simulation, visualization experiments, and thermodynamic performance testing are conducted. The simulation results reveal that the heat transfer capacity of the topological PHE is 810 W with pressure drops of 18.8 kPa, while the dimple PHE exhibits heat transfer capacity of 652 W with pressure drops of 25.9 kPa. Compared to the dimple PHE, the topological PHE demonstrates 24.2 % improvement in heat transfer performance and 27.8 % reduction in pressure drop. The dimensionless comprehensive heat transfer coefficient (j/f) indicates that the overall performance of the topological PHE significantly surpasses that of the traditional PHE. Furthermore, the visualization experiment results demonstrate that the complex texture structure of the topological PHE effectively induces fluid motion, compelling the fluid to flow along the texture direction. This significantly mitigates the issue of uneven flow within the PHE, thus enhancing heat transfer efficiency. Specifically, the topological PHE exhibits inlet–outlet temperature difference of 10.3 °C and pressure drops of 11.2 kPa, while the dimple PHE shows inlet–outlet temperature difference of 6 °C and pressure drops of 12.9 kPa. The findings of this study provide guidance for future research endeavors aimed at evaluating potential solutions to advance heat transfer in PHEs and facilitate the implementation of thermal management strategies.

A semi-analytical temperature solution for multi-segment deep coaxial borehole heat exchangers

A semi-analytical and a finite-difference scheme are presented for the simulation of temperature and the heat transfer in a multi-segment coaxial borehole heat exchanger. The single-segment solution on closed-form is extended to a semi-analytical multi-segment solution, where each segment may have unique properties. These properties are such as different casings, widths of the annulus, radius of the inner tubing, material properties, rock properties and geothermal gradients. The multi-segment model is a simple and powerful alternative to numerical methods for simulating a complex coaxial borehole heat exchanger with a constant flow rate. It is demonstrated with a deep coaxial borehole heat exchanger made of three different segments. The analytical and semi-analytical models are validated by comparison with numerical solutions obtained with an upstream finite difference scheme. The match between the solutions is excellent. The solution on a closed-form is used to study the temperature difference between the outlet and the inlet regarding two dimensionless numbers. It is found that the maximum temperature difference occurs when the dimensionless heat transfer coefficient for the casing-rock is much larger than one. A second necessary condition is that the dimensionless heat transfer coefficient for the insulator between the inner tube and the annulus must be much less than one. The power leakage from the inner tubing to the annulus is also at a maximum under these conditions.

Redefined interface error, 2D verification and validation for pure solid-gallium phase change modeling by enthalpy-porosity methodology

For simulating phase change process of pure solid gallium, previously published literature contains some key problems, including the inappropriate definition & computation for interface errors, and the unsatisfactory numerical verifications & validations. Herein, the 2D verifications and 2D validations for the enthalpy-porosity modeling of pure gallium melting starting from a vertical end wall have been reported comprehensively. The well-known and classic experimental data presented in literature are used to verify and validate the model. The groundbreaking contributions in this work are that: i) The definition and computational method of the interface-position errors between the numerical and experimental results are revisited and proven carefully; ii) The numerical schemes as well as major parameters of 2D modeling gallium melting are selected with overall interface error <9%. However, these schemes and parameters are not appropriate for the solidification modeling of gallium due to the irregular and non-reproducible interface shapes caused by highly anisotropic properties of solid crystal gallium as experimentally reported in literature; iii) The 2D validations of enthalpy-porosity modeling of gallium melting are investigated in detail, and the maximum interface error is within the limits of ±12%. Moreover, the simulated liquid volume fractions and dimensionless heat transfer coefficients are well correlated in terms of related criteria parameters, and they are compared with correlations available in literature. It is concluded that to solidly validate the modeling of pure metal melting, both the global parameters (e.g., volume-averaged liquid fraction) and the local parameters (e.g., interface position) need to be considered together.

Application of 4D two-colour LIF to explore the temperature field of laterally confined turbulent Rayleigh–Bénard convection

Recent studies show the significant effect of the third dimension and flow unsteadiness of laterally confined Rayleigh–Bénard convection (RBC). However, there are limited studies investigating 4D flow properties. The application of 4D two-colour laser-induced fluorescence to explore a laterally confined RBC at high Rayleigh number of $$Ra=9.9\times {10}^{7}$$ and Prandtl number of $$Pr=6.1$$ is investigated using a scanning laser system. A two-colour, two dyes approach was employed to resolve the laser sheet intensity variations due to refractive index variations caused mainly by the generation of thermal plumes. Two temperature-sensitive fluorescent dyes with opposite sensitivities were used to enhance the overall temperature sensitivity to 7.3%/°C. Details of the experimental procedure and optical system employed to reach such a high sensitivity are demonstrated. From the whole field temperature distribution, the dimensionless heat transfer coefficient, the Nusselt number, and its evolution were calculated for both hot and cold boundaries. Temperature field and Nusselt number obtained from 3D and 2D fields are reported to compare the results for these two scenarios. Thermal plumes were found to have a conical shape in the laterally confined RBC compared to the conventional mushroom shape. From visualization of the time-resolved 3D temperature field and 2D distribution of the Nusselt number, it was also found that only by volumetric measurement, temporal and spatial variations of the temperature and heat transfer can be evaluated. This shows that to evaluate the classic and ultimate theories, volumetric measurement is required for a coherent understanding of the physics.

Effect of non-condensable gas on submerged steam jet condensation in a narrow pipe

A visualization study was performed to investigate the effect of non-condensable gas on stable submerged steam jet condensation in water flow in a narrow pipe. The direct contact condensation (DCC) characteristics including jet plume shapes, dimensionless penetration length and average heat transfer coefficient were analyzed in a wide range of water flow subcooling, Reynolds number of water flow and steam–air mixture flow rates. As four different steam plume shapes for pure steam jet in subcooled water flow in a pipe were reported in the previous research, the effect of non-condensable gas on the four steam plume shapes was investigated by introducing a certain amount of air into steam. In the current study, the submerged steam–air mixture jet behavior was investigated through direct visualization techniques. The effect of non-condensable gas on jet plume shapes was recorded by using a high speed camera. The gas–liquid interface and the dimensionless penetration length was obtained by digital image processing method implemented by a MATLAB subroutine. Then the average heat transfer coefficient was calculated based on thermal equilibrium of gas–liquid interface. In the experiment, the gas mass flux at nozzle exit was within 320–780 kg/m2/s, with no more than 1 % non-condensable gas. The results indicated that the non-condensable gas plays an important role on the gas–liquid interface, jet penetration length and heat transfer coefficient. It was found that when 1 % non-condensable gas was mixed with steam jet, the dimensionless penetration length was extended by 16–70 %, and the average heat transfer coefficient decreased by 12–50 %. Based on the force-balance of jet plume, a new quantitative correlation for the steam–air mixture jet penetration length was proposed. The predicted jet penetration length agrees well with the experimental measured values.

An engineering model of heat transfer through a turbulent falling liquid film for a condensing vapor flow

An engineering model of heat transfer through a turbulent falling liquid film for a condensing (or evaporating) vapor flow is suggested. A simple correlation for the eddy diffusivity, based on the Barenblatt’s hypothesis of incomplete similarity, is employed for description of momentum and heat transfer across the liquid film. An equation for the dimensionless heat transfer coefficient at a substantial interfacial shear stress is obtained and successfully validated by experimental data available in open literature.

Calculation Of The Nusselt Number From Temperature And Velocity Data Obtained From Enhanced 3D Two-Colour LIF And 3D PTV In A Rayleigh-Benard Convection Cell

Experimental study of Rayleigh-Benard convection (RBC) requires 3D measurement of the velocity and temperature due to the complexity of the physics of this system. Using a scanning system, the application of the 3D particle image velocimetry (PTV) and 3D time-resolved two-color laser-induced fluorescence (LIF) is investigated on a slender RBC cell. Calculation of the out-of-plane velocity component by using the planar velocity components and applying the continuity law is discussed. From these observations that calculation of the dimensionless heat transfer coefficient, the Nusselt number from the 3D reconstructed temperature field close to the boundary of the fluid domain is investigated.

Morphological comparative assessment of a rooftop solar chimney through numerical modeling

An assessment on the performance of three different configurations at the outlet sections of a rooftop solar chimney is conducted in this work. The starting configuration corresponds to a representative dwelling-solar chimney system. On the basis of the knowledge reached in the technical bibliography, a numerical modeling is developed, considering both the wind-driving and the buoyancy-driving forces. The turbulent nature of the airflow is simulated through an enough validated two-transport equations model, including a low-Reynolds treatment at walls. Special care is taken in the numerical simulation of the wind, implementing a logarithmic speed profile to properly introduce the atmospheric boundary layer. At walls, a heat-flux heating condition (q uniform) is imposed in order to realistically simulate the heating from irradiation. Focus is posed on a comparative and systematic study of the behavior of airflow through the three considered morphologies, free outlet, covered outlet and side outlet. The more relevant differences found in the performance of each constructive shape are highlighted, both for near-calm as well as for wind-dominant conditions. A wide range of heating conditions are considered, 0.5≤q≤1000 W/m2 (corresponding to Rayleigh numbers from 1.085×1010 to 2.170×1013), as well as reference wind velocities from windless condition up to 10 m/s. Several correlations for the air ventilation rate and the dimensionless heat transfer coefficient are proposed, oriented to engineering practical applications. Wind reveals as dominant from reference velocities in the range 2-3 m/s. The conditions under which a certain better performance wall-to-wall spacing between walls forming the solar tower can be encountered, are analyzed and discussed. Illustratively, in some cases, the maximum value of the average Nusselt number reaches twice the value corresponding to aspect ratio equal to unity.

Numerical simulation of entropy transport in the oscillating fluid flow with transpiration and internal fluid heating by GGDQM

The numerical investigation of entropy generation and its depreciation in oscillating fluid flow with internal fluid heating and transpiration is presented. The current examination includes a flow and heat transfer investigation. The flow is driven by an infinite porous plate that is stretched and oscillated periodically in its plane. Boundary layer approximations (BLAs) are used to simplify the momentum and energy equations. A dimensionless nonlinear system of partial differential equations is solved for the velocity field, temperature profile, entropy generation distribution, and Bejan number using the Gear generalized differential quadrature method (GGDQM). The numerical values of skin friction coefficient and dimensionless heat transfer coefficient are tabulated against various flow governing parameters. The applied numerical scheme reliability is established by comparing numerical estimation of the skin friction coefficient and Nusselt number obtained by the GQGM and Gear Gauss-Lobato spectral method (GGLSM). A comparison of numerical data shows an excellent agreement and validates the reliability of the numerical simulation. Tables and graphs demonstrate the effects of various control parameters on the physical quantities of interest. With the rising Eckert number, the thermal and entropy profiles amplify, whereas a decrease in the Bejan number is observed. Furthermore, ininjection and suction instances, entropy and Bejan number profile arguments at and near the oscillating surface.

Comments on ‘Nonlinear thermal analysis of a convective-radiative longitudinal porous fin of functionally graded material for efficient cooling of consumer electronics'

The purpose of these comments is to alert the International Journal of Ambient Energy. Discussion is presented on the investigation by Oguntala et al. [2019. “Nonlinear Thermal Analysis of a Convective-Radiative Longitudinal Porous fin of Functionally Graded Material for Efficient Cooling of Consumer Electronics.” International Journal of Ambient Energy], where the authors presented nonlinear thermal analysis and the radiative-convective longitudinal porous fin optimisation utilising functionally graded material. First of all, Oguntala et al. (2019. “Nonlinear Thermal Analysis of a Convective-Radiative Longitudinal Porous fin of Functionally Graded Material for Efficient Cooling of Consumer Electronics.” International Journal of Ambient Energy] defined in Nomenclature H as dimensionless heat transfer coefficient at the fin base with the units of W m−2 K−1. It is known that the dimensionless parameter must have no units. Also, the units of the permeability (K) are corrected. Using the correct units of the permeability (K) leads to the Darcy number (Da) is dimensionless. In addition, the radiation heat transfer number (Nr) and the convective heat transfer number (Nc) are dimensionless in the above paper.

A Similarity Solutions Approach to Forced Convection via a Porous Medium Attached to Flat Plate

In the present study, a porous medium adjoining a heated flat plate was modelled by a similarity approach to determine the effect of porosity on the heat transfer phenomena. The momentum and energy equations for porous media transport were transformed using an appropriately determined similarity parameter. The solutions to the momentum equations proved to depend only on the porosity and not the material used. The energy equation however was additionally dependent on the combinations of fluid type and metal matrix used and was solved for four combinations; water/aluminum, air/aluminum, air/copper and SAE 20W-50/stainless. The results show that replacing the clear fluid control volume with a porous matrix altered both the velocity profiles and temperature distribution profiles at all Reynolds numbers. As the porosity of the medium decreased, it resulted in an increase in interfacial area as well as thermal diffusion in the direction normal to the plate, both of which was seen to enhance the heat transfer coefficients. Decreasing porosity also resulted in an increase in thermal storage as well a reduction in volume flow rate going through the medium, which on the other hand tend to inhibit convection. Thus, changing the porosity triggered effects on the heat transfer coefficient. This opposing trend favored convection at porosity greater than 0.5 for low Prandtl number fluids. The clear fluid condition has the lowest heat transfer coefficient and the values increased steeply as porosity changed from 0.99 through 0.7. The heat convection curve reached its maximum turning point at a porosity of 0.5 and then reversed in trend. The dimensionless heat transfer coefficient was found to fit the equation hLL/ReL0.5kfPrf=aebPrmc ( a= 0.51966;b=0.54683;c=-0.665349) . Using Stanton number representation, the relation is StReL&gt;1/2=1/3 aebPrmc , which portrays in relative terms how the convection enhancement and the opposing thermal storage effects vary with porosity. This study concluded that implementing a porous structure in a medium is feasible for enhancing heat transfer performance.

A Comparison of Circular Duct and Real Hexagonal Duct Results Using Hydraulic Diameter

To see whether the turbulent flow correlations derived for circular ducts can be used for hexagonal cross-sectional ducts using hydraulic diameter, turbulent flow in hexagonal ducts is numerically investigated under constant wall temperature boundary condition using ANSYS Fluent 17.0 software. Investigated parameters are the Reynolds number between 10×103Re50×103 and side angle of the duct varying between 30o and 90o. Standard k-ε model is used as turbulence model. General expressions are proposed for fully developed dimensionless heat transfer coefficient Nusselt number and fully developed Darcy friction factor in terms of Reynolds number and side angle for hexagonal-shaped cross-sectional duct. Results show that side angle of hexagonal duct affects the pressure drop along duct and heat transfer coefficient in duct. Results point out that regular hexagonal duct, =60o, gives minimum pressure drop and maximum Nusselt number. It is concluded that correlations given in the literature for circular ducts in turbulent flow can give 14% higher dimensionless heat transfer coefficient, Nusselt number, than that of actual hexagonal duct flow.

Determining correlations for solar chimneys in buildings with wind interference: A numerical approach

The impact of wind on the performance of a solar thermal chimney is analyzed by numerical investigation. The chosen geometry for the dwelling-solar chimney system can be considered as sufficiently generic and representative, and similar to the buildings assisted with passive systems that are currently being designed. Both the wind driving and the buoyancy forces are included in the study. The developed numerical approach takes into account the atmospheric boundary layer for specifying the acting wind, and includes a low-Reynolds treatment of turbulence at walls for obtaining a detailed description of the airflow. The procedure is validated enough, and permit us to address a systematic analysis of the influence of temperature difference (10–100 K, corresponding to Rayleigh numbers from 3.616×109 to 3.616×1010 and wind speed ranging 0–10 m/s) on the behavior of chimney in ventilation mode. Two opposite directions of the wind are studied, finding clear different impacts on the solar chimney performance. For the more favorable wind direction, a complete set of correlations for the induced mass-flow rate and the dimensionless heat transfer coefficient is proposed, aimed at practical engineering applications.

Experimental Investigation on Cooling Performance of Impingement–Effusion Full Coverage Film on Suction Surface of a Vane

Abstract This study experimentally investigated the net benefit of film cooling with six rows of impingement–effusion structures on the suction surface of a vane. The experiment obtained the film cooling effectiveness of a double-walled system on the suction surface via the pressure-sensitive paint (PSP) technique. The film cooling effectiveness obtained by the PSP technique is coupled with the transient liquid crystal (TLC) technique to determine the heat transfer coefficient. This combination of techniques reduces the time required for the experiment and improves efficiency of the experiment. Through the experimentally measured film cooling effectiveness and dimensionless heat transfer coefficient, the net heat flux reduction (NHFR) is calculated to comprehensively measure the net benefit of film cooling. At the same time, in view of the lower net benefit of film cooling of the film holes in the front of the suction surface under a higher mass flux ratio (MFR), the study improved the cylindrical holes into fan-shaped holes and proposed two improvement schemes: Vane A and Vane B. The experiment was carried out under three MFRs (0.4%, 0.8%, 1.6%), and compared the film cooling effectiveness, heat transfer coefficient ratio, and net heat flux reduction of Vane A and Vane B with the baseline vane under each mass flux ratio. The findings show that using the coupling of PSP and TLC to determine the heat transfer coefficient can yield credible results. The improvement of the fan-shaped holes makes the film cooling effectiveness and heat transfer coefficient ratio improved compared with the baseline vane. Changing cylindrical holes to fan-shaped holes does not necessarily lead to a better net benefit of film cooling. The fan-shaped holes should be arranged reasonably to obtain the better net benefit of film cooling.

Non-similar forced convection analysis of Oldroyd-B fluid flow over an exponentially stretching surface

In the study of boundary layer regions, it is in practice to dimensionalize the governing system and grouping variables together into dimensionless quantities in order to curtail the total number of variables. In similar flow phenomenon the physical parameters do not vary along the streamwise direction. However in non-similar flows the physical quantities change in the streamwise direction. In non-similar flows we are forced to non-dimensionalize the governing equations through non-similarity transformations. The forced flow of Oldroyd-B fluid is initiated as a result of stretching of a surface at an exponential rate. Flows over stretching surfaces are important because of their applications in extrusion processes. The forthright purpose of this study is to consider the non-similar aspects of forced convection from flat heated surface subjected to external viscoelastic fluid flow, described by the freely growing boundary layers enclosed by a region that involves without velocity and temperature gradients. The governing system of nonlinear partial differential equations (PDE’s) is transformed into dimensionless form by proposing new non-similar transformations. The dimensionless partial differential system is solved by using local non-similarity via bvp4c. Thermal transport analysis is conducted for distinct values of dimensionless numbers. It is revealed that heat shifting process expanded by the increase in the numerical values of Prandtl number and relaxation time. The dimensionless convective heat transfer coefficient results revealed that it is declining by expanding relaxation time constant [Formula: see text] and a boost is observed by enlarging the Pr and retardation time constant [Formula: see text]. A comparison of Nusselt number is presented.

Sensitivity Analysis on Thermal Performance of Gas Heater with Finned and Finless Tubes using Characteristics-based Method

Natural gas must be preheated to prevent phase change and gas hydrate in pressure reduction stations. This paper aims to investigate the effect of the fins of gas tubes and their configuration, arrangement, and shape on the heat transfer and thermal efficiency of gas. To conduct a parametric study, two tube cases with fins and without fins, and in the finned case for the fin’s configuration, two longitudinal and circular arrangements, and the formation of the fins, two solid and interrupted forms were analyzed. Also, three types of cross-sections, including rectangular, convergent parabolic, and divergent parabolic, for the shape of the fins have been studied. For this simulation, the three-dimensional, incompressible, and steady flow was considered, and for analysis and discretization of convective heat equations, the characteristic-based method was applied. FORTRAN software was also used to implement and solve the equations. The results show that in solid and interrupted fins and increasing the number of fins in parallel, the dimensionless heat transfer coefficient increases. Also, the dimensional heat transfer coefficient decreases with increasing the ratio of fin height to the tube’s diameter. Also, the most significant heat transfer improvement was related to the divergent parabolic cross-section.

Investigation of Ferro-nanofluid flow within a porous ribbed microchannel heat sink using single-phase and two-phase approaches in the presence of constant magnetic field

In this investigation, the entropy generation, heat transfer, and Fe3O4-water ferro-nanofluid flow within a porous ribbed microchannel heat sink in the presence of a constant magnetic field are investigated, and their impacts are analyzed for single-phase and two-phase approaches. The values of dimensionless numbers, namely Hartmann and the Reynolds numbers are in the range of 0 ≤ Ha ≤ 10 and 10 ≤ Re ≤ 80. Additionally, the porosity percentage and volume fraction of nanoparticles are in the range of 0 ≤ ε ≤ 75 % and 0 ≤ ϕ ≤ 2 %, respectively. The influences of dimensionless numbers and porosity percentage in three cases of the microchannel (Cases A.1, A.2, and A.3) are investigated. This study provides a comparison between assumed parameters. It is demonstrated that in all cases, variables such as porosity percentage, Reynolds and Hartmann numbers are directly related to the enhancement of the heat transfer coefficient. Also, in the single-phase model, the dimensionless heat transfer coefficient has a lower amount compared to the two-phase model, and the only exception is for ϕ = 0%.