Heat Pipe Analysis and Simulation
|Historical Development of Heat Pipes
Heat Pipe Analysis and Simulation
Numerical and analytical simulation of heat pipes has progressed significantly in the last several decades. The state of the art has been advanced in steady state, continuum transient, and frozen start up simulation for high-, intermediate-, and low-temperature heat pipes of conventional and nonconventional geometries.
In determining the heat capacity transmitted through a heat pipe or heat pipe performance characteristics, it is necessary to know the liquid and vapor pressure losses in the separate segments. The thermal fluid phenomena occurring within a heat pipe can be divided into four basic categories: (1) vapor flow in the core region, (2) liquid flow in the wick, (3) interaction between the liquid and vapor flows, and (4) heat conduction in the wall. Most of the detailed analytical and numerical research on heat pipes has been done on the vapor core region and wall heat conduction, since the liquid flow is difficult to describe with an exact theoretical model. Because of the presence of the wick structure, the analysis of categories (2) and (3) requires some empirical information that must be obtained from experimentation. Furthermore, the analyses of categories (2) and (3) are basically similar for heat pipes of various shapes. In contrast, the dynamics of vapor flow are related to the geometry and boundary specifications for nonconventional heat pipes.
The interaction between different regions can be best illustrated by following the real heat pipe operating condition, as shown by the flow chart in Fig. 1. Consider a heat pipe with an initial operating condition. If the heat load from the heat source to the heat pipe evaporator surface is increased at t ≥ 0 to a higher level, the temperature of the container wall is increased accordingly, and more heat is transferred from the container to the heat pipe wick structure. In this case, the heat pipe container wall is interacting with the heat pipe wick structure. Meanwhile, the state of the liquid-vapor interface may change due to the change in liquid pressure and temperature. The interface evaporation mass flux is also increased because more heat is transferred from the wick structure to the interface. A higher mass flux at the interface will increase the vapor velocity, and result in a change in the vapor working condition. In the above process, the liquid flow in the wick structure is interacting with vapor flow via the liquid-vapor interface. At the same time, more liquid mass may condense onto the liquid-vapor interface in the condenser section due to the higher mass flow rate from the evaporator section. This will subsequently change the interface state. The interaction between the vapor flow and the liquid-vapor interface in this case can be called feedback. These feedback interactions will also occur between the liquid-vapor interface and wick structure, and between the wick structure and the container.
In Fig. 1, regions can interact only with regions directly above or below. The container region, for example, cannot interact with the vapor region directly. It can only interact with the vapor region via the wick structure region and the liquid-vapor interface. This is due to the physical locations of the different regions, which prescribe the relation between different regions. In general, all of the regions in the flow chart should be solved as a conjugate problem. However, in the real application, approximations are often introduced due to the complex nature of the problem. Some regions in the flow chart may be neglected or combined with other regions. Regions can also be decoupled from other regions. For instance, if we neglect the liquid flow in the wick structure, and solve the region as a heat conduction problem, the region associated with the wick structure may be combined with the region associated with the heat conduction in the container. If we are only interested in the vapor flow in the heat pipe, we can neglect the three preceding regions, and directly go to the fourth region associated with vapor flow. In some applications, the liquid flow in the wick structure is of interest. Therefore, the wick region can be decoupled from other regions, and solved as a one region problem with appropriate assumptions.
For the convenience of analysis, the frozen start up process of a high-temperature heat pipe or frozen low-temperate heat pipe can be divided into several periods based on experimental evidence.
1. The working fluid in the wick is frozen at the ambient temperature. For a heat pipe without any noncondensible gas present in the vapor space, the vapor core is completely evacuated before start up.
2. Heat input is started in the evaporator section. Heat is conducted through the pipe wall and wick structure, and melting of the working fluid begins at the wall-wick interface. Because the liquid-solid interface has not reached the wick-vapor boundary, no evaporation takes place and the vapor space is still evacuated.
3. The working fluid is completely melted in the evaporator section only, and liquid is evaporated at the liquid-vapor interface. The vapor flow in the rarefied or free molecular condition due to the low vapor pressure. Some of the working fluid in the wick is still frozen in the adiabatic and condenser sections.
4. Continuum vapor flow is established in the evaporator section as the vapor pressure increases due to evaporation. In the remainder of the heat pipe, however, the vapor is still in the rarefied condition.
5. The working fluid is completely melted and continuum flow is established over the entire heat pipe length. The heat pipe temperature increases steadily until the steady state is reached.
6. The heat pipe is operating under steady state operation.
The parameters limiting heat transport in conventional heat pipes due to vapor flow are the capillary, sonic, and entrainment limitations. The sonic limitation is not greatly influenced by any aspect of the wick structure. Therefore, the choking phenomena should be of primary interest in the vapor flow analysis due to the fact that the vapor velocity becomes significant compared with the sonic velocity. Furthermore, vapor flow analyses are required to predict the capillary limitation. In each concentration, there are several levels of approximation, ranging from a simple one-dimensional vapor flow analysis to a complete three-dimensional analysis, considering the conjugate nature of the wall and wick. Important modeling considerations include vapor compressibility, heat pipe geometry, and heat input distributions.
These methods cover various approximations and approaches, such as compressible vs. incompressible, analytical and closed-form solutions vs. numerical analyses, as well as one-dimensional, two-dimensional, and three-dimensional modeling considerations in each region. Faghri (2012) reviewed the advances in analysis and simulation of the different types of heat pipes under various operating conditions, also presented in this section.
The following part introduces the developments in modeling and discusses important results for various types of heat pipes under various operating conditions.
A steady state operational performance prediction is of significant value in the design of heat pipes. Faghri (1986) modeled the steady-state two-dimensional incompressible vapor flow in an annular heat pipe. In this study, the boundary-layer approximations were used to reduce the fully elliptic conservation equations to the partially parabolic Navier-Stokes equations, which were solved using a fully implicit, marching finite-difference scheme.
This work was extended by Faghri and Parvani (1988) and Faghri (1989), where vapor compressibility was considered and the complete elliptic Navier-Stokes equations were solved. One- and two-dimensional models were developed for the vapor flow in an annular heat pipe, and the assumptions used by Faghri (1986) were verified. A major contribution of this work is the derivation of analytic expressions for the axial pressure drop within the annular heat pipe. These relations can also be simplified to describe a conventional cylindrical heat pipe and be used directly in the design process of heat pipes.
Block-heated heat pipes are often used as heat transfer mechanisms for energy conservation systems and electronic cooling. The fundamental difference between these systems and conventional heat pipe systems is the nonuniform block heating at the evaporator, which influences the temperature distributions and vapor flow patterns throughout the heat pipe. Cao et al. (1989) proposed a simplified model of this process by defining two separate governing equations for the evaporator section: one for the pipe wall under the heated block and one for the remaining unheated pipe wall. In this method, a power-law relationship was used for the boiling heat flux. The block-heated simulation was performed for the heat pipe studied experimentally by Rosenfeld (1987) , which used a narrow line heated on a low-temperature heat pipe. The circumferential wall temperature profile predicted by Cao et al. (1989) is shown in Fig. 2, with good agreement with the experimental data. Cao et al. (1989) determined that the evaporative heat transfer coefficient has a strong influence on the wall temperature profile. This is because with a large evaporation coefficient, the majority of the input heat is removed through the phase change of the working fluid. With a lower evaporation coefficient, more heat is conducted circumferentially through the pipe wall, smoothing the temperature profile.
The importance of the effect of conjugate heat transfer within the heat pipe wall and wick was shown by Chen and Faghri (1990) with a model of the two-dimensional compressible vapor flow coupled to the two-dimensional conduction model in the wall and wick. This model solves the complete elliptic Navier-Stokes equations in the vapor with a finite control volume iterative technique. The simulation was performed for the high-temperature sodium/stainless-steel heat pipe studied in Ivanovskii et al. (1982) with the results shown in Fig. 3. They used three different methods: compressible elliptic, incompressible elliptic and compressible parabolic. Both the compressible elliptic and compressible parabolic solutions gave reasonable vapor temperature distributions, while the incompressible elliptic case shows the importance of including vapor compressibility.
This methodology was extended to model a low-temperature, multiple-evaporator heat pipe by Faghri and Buchko (1991). The heat pipe was simulated using a two-dimensional compressible formulation for the vapor flow, and a two-dimensional conduction model in the wall. However, Faghri and Buchko (1991) included the two-dimensional effects of liquid flow in the wick using volume-averaged velocities in the porous media. The conservation equations were solved using an elliptic finite control volume iterative scheme with the results as shown in Fig. 4. The experimental data corresponded to a water-copper multiple-evaporator heat pipe. One case was performed with only evaporator 1 active and a heat input of 97 W. Another had evaporators 1 and 2 active, with heat inputs of 99 W and 98 W, respectively.
The block heating phenomena studied by Cao et al. (1989) was further investigated by Schmalhofer and Faghri (1993a; 1993b). The complete three-dimensional compressible Navier-Stokes equations were solved in the vapor, and the wall and wick were simulated with a three-dimensional conduction model. The results of this simulation were compared with the experimental data of Schmalhofer and Faghri (1993a)  for a water-copper heat pipe with good agreement (Schmalhofer and Faghri (1993b)). An analytical and numerical study was carried out by Zhu and Vafai (1998b) for the steady incompressible vapor and liquid flow in flat plate heat pipes. The pseudo-three-dimensional analytical model employed the boundary layer approximation to describe the vapor flow conditions. The three-dimensional effects are discussed and the results show that a three-dimensional analysis is necessary if the vapor channel width-to-height ratio is less than 2.5. Lefevre and Lallemand (2006)  developed an analytical solution for a 2D hydrodynamic model for both liquid and vapor coupled to 3D heat conduction for the wall to simulate the steady state operation of a flat miniature heat pipe. The proposed model is also capable to predict the maximum heat transfer capability for a miniature heat pipe with multiple heat sources. A detailed, three-dimensional steady-state model was developed by Xiao and Faghri (2008)  to analyze the thermal hydrodynamic behaviors of flat heat pipes and vapor chambers without empirical correlations. The model accounts for the heat conduction in the wall, fluid flow in the vapor chambers and porous wicks, and the coupled heat and mass transfer at the liquid-vapor interface. The flat heat pipes with and without vertical wick columns in the vapor channel (Fig. 5) were investigated in the model by Xiao and Faghri (2008).
Parametric effects, including the evaporative heat input and size on the thermal and hydrodynamic behavior in the heat pipes, were presented. The results showed that, the vertical wick columns in the vapor core can improve the thermal and hydrodynamic performance of the heat pipes, including thermal resistance, capillary limit, wall temperature, pressure drop, and fluid velocities due to the enhancement of the fluid/heat mechanism form the bottom condenser to the top evaporator. The results also predicted that higher evaporative heat input improves the thermal and hydrodynamic performance of the heat pipe, and shortening the size of heat pipe degrades the thermal performance of the heat pipe. Figure 6 shows a comparison of the numerical results by Xiao and Faghri (2008)  with the experimental data (Wang and Vafai, 2000) for the temperature distributions at the surface of the top and bottom walls of the flat heat pipe. The deviation in the condenser section in the top wall is mainly due to the variation of the convective heat transfer coefficient dependent of different surface temperature in the experiment, which is assumed to be constant in the numerical model.
Aghvami and Faghri (2011)  presented a steady-state analytical thermal-fluid model, including the wall and both liquid and vapor flows for flat heat pipes and vapor chambers with different heating and cooling configurations. The parametric investigations showed that the assumption of uniform evaporation and condensation in the axial direction is valid only if the wall thermal conductivity is small. Shabgard and Faghri (2011) extended the above model to cylindrical heat pipes with multiple evaporator sections. The two-dimensional heat conduction in the heat pipe’s wall was coupled with the liquid flow in the wick and the vapor hydrodynamics. It was found that neglecting the axial heat conduction through the wall can result in overestimated pressure drops up to 10% depending on the heat pipe specifications.
A first-order transient model of the vapor flow in a heat pipe was proposed by Jang et al. (1991). This model simulated heat pipe operation with one-dimensional compressible vapor flow in a porous pipe, accounting for laminar and turbulent skin friction. The results of this model are compared with the experimental data of Bowman (1987) in Fig. 7, where the transient vapor pressure at three locations along the heat pipe is shown. As in a conventional heat pipe, pressure recovery occurs in the condenser section can be seen. A major advantage of this method is the ability to capture sharp pressure gradients, such as those that would occur in supersonic flow or across shock waves.
A comprehensive transient heat pipe analysis is presented by Cao and Faghri (1990). This two-dimensional model accounts for vapor compressibility and couples the vapor flow with heat conduction in the wall and wick. For high-temperature heat pipes, it was determined that the conduction model is sufficient to describe heat transfer in the wick. This methodology allows heat pipe simulation for pulsed heat inputs with either a convective or radiative boundary condition at the outer pipe wall of the condenser. The transient vapor temperature profile for a pulsed heat input of Q = 623 W to 770 W with the convective boundary condition is shown in Fig. 8(a). The operating temperature of the heat pipe increases with time after the heat pulse. This is due to the coupling of the vapor to the wall, because the outer pipe wall temperature in the condenser must increase to reject the additional heat. The vapor temperature with the radiative boundary condition is shown in Fig. 8(b). Faghri et al. (1991b)  adapted the numerical model of Cao and Faghri (1990) to a high-temperature heat pipe with multiple heat sources and sinks. Numerical results for continuum transient and steady-state operations with multiple heat sources were compared with the experimental results of Faghri et al. (1991a)  and found to be in good agreement.
A transient model capable of simulating nonconventional geometries was proposed by Cao and Faghri (1991) Cao, Y., and Faghri, A., 1991, "Transient Multidimensional Analysis of Nonconventional Heat Pipes with Uniform and Nonuniform Heat Distributions," Journal of Heat Transfer, 113(4), 995-1002. http://dx.doi.org/10.1115/1.2911233 </ref> and Cao and Faghri (1993a) . This method solves the one-dimensional compressible vapor flow coupled to the two- or three-dimensional heat conduction in the wall and wick. The one-dimensional vapor flow is easier to solve than the two-dimensional formulation, but friction coefficient information is required for both laminar and turbulent flow regimes. For complex geometries, such as a wing leading edge or spacecraft nosecap, a multidimensional formulation is used in the wall and wick. Conduction in the wall and wick is coupled to the vapor flow through a conjugate heat transfer solution technique for the elliptic conservation equations in the vapor. In this analysis, the liquid flow in the wick was assumed to be decoupled from heat pipe operation to determine the capillary limit for nonconventional geometries. This simulation was used to investigate the use of heat pipes for cooling the leading edge of reentry vehicles and hypersonic aircraft. In this case, a line heat source was applied at the leading edge stagnation point of the wing to simulate aerodynamic heating. The results of this simulation are shown in Fig. 9, where the extreme temperature gradient in the pipe wall at the leading edge can be seen. This is due to the severe heat flux (q = 75 x 106 W/m2) at this point, which represents actual hypersonic heat loads.
A transient heat pipe model was developed by Zuo and Faghri (1997) using a transient two-dimensional wall and wick heat conduction and a quasi-steady-state one-dimensional vapor flow. The assumption of quasi-steady-state vapor was based on the fact that the vapor dynamics have a much smaller timescale than the wall conduction based on dimensional analysis. The transient two-dimensional wall heat conduction was solved by a boundary element method (BEM). A control volume finite difference method (FDM) was employed to solve the vapor flow continuity and momentum equations. Two iterative “estimate-correction” processes and a new pressure-correction method were incorporated in the vapor solution procedure. Significant reduction of the computation time was obtained by using this hybrid FDM/BEM. Comparisons with previous experimental and numerical results validated the assumption of a quasi-steady-state vapor and the solution methods.
An accurate, but simple lumped model using the “network” model was developed for transient heat pipe analysis by Zuo and Faghri (1998). The heat pipe consists of a number of components with different thermal resistances and dynamic responses in the network model. Governing equations of the transient heat pipe behavior were significantly simplified to a set of first-order, linear, ordinary differential equations. Figure 10 compares the transient network model with experimental results. For this specific case, the computational time required by the network model is about 1 minute on a standard desktop PC compared to several hours of CPU needed for a full two-dimensional numerical model. Zhu and Vafai (1998a) developed an analytical solution for quasi-steady and incompressible vapor flow with transient one-dimensional heat conduction in the wall and wick for an asymmetrical flat plate heat pipe.
A more detailed two-dimensional transient numerical analysis of flat and cylindrical screen wick heat pipes was performed with no empirical correlations by Rice and Faghri (2007) , while including the flow in the wick. Single and multiple heat sources were used as well as constant, convective, and radiative heat sinks. The numerical model by Rice and Faghri (2007)  does not fix the internal pressure references by a point, but allows it to rise and fall based on the physics of the problem. Accordingly, the capillary pressure needed in the wick to drive the flow was obtained for various heating configurations and powers. These capillary pressures, in conjunction with an analysis that predicts the maximum capillary pressure for a given heating load, were used to determine the dry-out limitations of a heat pipe. Special considerations need to be accounted for when considering the interface between the wick and the vapor region because the vapor interacts with the liquid in the wick’s pores and the solid wick itself. Figure 11 is a schematic of the vapor/wick interfacial region.
Transient models with couplings between the various layers (wall, wick, and vapor) have successfully predicted the performance of heat pipes under all operating conditions for porous wick structures by neglecting the curvature effect at the liquid vapor interface. As heat pipe size or wick thickness decreases in size in the micro ranges, mainly for high thermal conductivity wicks, details of transport at the liquid vapor interface may become important in determining the capillary limit. Several efforts (Tournier and El-Genk (1994)  , Rice and Faghri (2007)  , and Ranjan et al. (2011) ) have been made to include the effect of varying the radius of the curvature of the liquid meniscus formed in the wick pores for a screen mesh wick in the full numerical simulation of heat pipes for predicting the capillary limit.
Frozen Start Up
In some applications, heat pipes are required to transfer energy with the working fluid initially in the solid phase. As heat transfer progresses, the frozen working fluid melts, and the heat pipe undergoes a transition to normal operation. To fully understand frozen start up, significant efforts have been made toward numerical simulation of this process. Early frozen start up research classified the frozen start up process into six periods, as discussed earlier (Jang et al., 1990). These stages describe the condition of the working fluid as solid, mushy, or liquid, and the condition of the vapor flow as free molecular, partially continuum, continuum transient, and continuum steady. In the mathematical formulation by Jang et al. (1990), the heat transfer through the free molecular vapor flow was neglected and the frozen start up was solved in the wick and wall for the first two stages. During the early start up period, the vapor density in the heat pipe vapor space is extremely low and partly loses its continuum characteristics. This condition is referred to as rarefied vapor flow. During the early stage of start up, when the melting interface has reached the wick-vapor interface, the liquid-vapor interface cannot be considered to be adiabatic. Otherwise, there would be no molecular vapor accumulation in the vapor space and the vapor would never reach the continuum flow state. The rarefied vapor flow was simulated by a self-diffusion model (Cao and Faghri, 1993b), where self-diffusion refers to the interdiffusion of particles of the same species due to the density gradient. This method has the advantage of modeling the heat transfer through the rarefied vapor flow as coupled to the phase change of the working fluid in the wick. However, this model is only valid for the early stages of frozen start up. A complete numerical simulation of frozen start up was completed by Cao and Faghri (1993c) by combining the rarefied self-diffusion model of Cao and Faghri (1993b) with the continuum transient model of Cao and Faghri (1990). The model completely described frozen start up, including the effects of conjugate heat transfer within the wall. Heat transfer and fluid flow in the wick coupled to the vapor flow were simulated. This methodology was applied to simulate the multiple-evaporator high-temperature sodium/stainless-steel heat pipe studied by Faghri et al. (1991a)  shown in Fig. 12(a) and the single evaporator sodium heat pipe of Ponnappan (1990)  as shown in Fig. 12(b). The comparison with the experimental data is excellent, and the location and progression of the hot front is closely simulated as a function of time.
As mentioned previously, it may not always be possible to initiate frozen start up of a heat pipe. To address this difficulty, Cao and Faghri (1992)  derived a closed-form analytical expression to describe heat pipe operation from frozen start up to continuum steady state. Also, a criterion was derived for the frozen start up limitation. The analytical model was derived under the assumption that the temperature profile at any time can be represented by three linear temperature approximations: one each in the hot and cold section of the heat pipe and one across the hot-cold front. The results of the model are shown in Fig. 13(a) when compared with the experimental frozen start up profiles of Faghri et al. (1991a) , and in Fig. 13(b) when compared with the data of Ponnappan (1990).
It was determined that the frozen start up of a high-temperature heat pipe depends on several factors, such as the difference between the melting temperature of the working fluid and the ambient temperature, the liquid density of the working fluid, the porosity of the wick, and the physical dimensions of the heat pipe. These parameters were combined into a nondimensional parameter, which is a measure of the frozen start up ability of the heat pipe. This frozen start up limit (FSL) was validated using different experimental cases. Determination of the frozen start up limitation is one of the most significant contributions to frozen start up heat pipe analysis in recent years. The limit provides a fundamental design tool when considering heat pipe operation from a frozen state (Cao and Faghri, 1992).
The aforementioned 2D models were quite successful and accurately predicted the frozen startup of low and high temperature heat pipes under various operating conditions by including both 2D free molecular and continuum vapor core regimes and freezing and melting phenomena by assuming flat liquid-vapor interface. Other frozen startup simulation efforts (Hall et al. (1994)  and Tournier and El-Genk (1996)) were made by assuming 1-D vapor flow regimes, while accounting for the local interfacial radius of the curvature of the liquid meniscus.
Axially Grooved Heat Pipes
In the modeling of axially-grooved heat pipes (AGHP), the fluid circulation should be considered along with the heat and mass transfer processes during evaporation and condensation. Khrustalev and Faghri (1995a)  have developed a mathematical model for low-temperature axially grooved heat pipes. The Khrustalev and Faghri (1995b)  model accounts for the heat transfer in the microfilm region, which is of significant importance. Khrustalev and Faghri (1995a) modeled the heat transfer through the liquid film and the fin between the grooves (Fig. 14) in both the evaporator and condenser with respect to the disjoining pressure, interfacial thermal resistance and surface roughness, which was not accounted for by previous investigators. In this model, heat transfer processes in the heat pipe container and working fluid were considered to be one-dimensional in the radial direction, thereby neglecting the axial heat conduction. It was also assumed that no puddle flow occurred in the AGHP, no part of the condenser was blocked by the liquid working fluid, the liquid was distributed uniformly between the grooves, and the fluid flow along the axis was related to the capillary potential gradient, as described by the main radius of curvature of the liquid in a groove.
The numerical model is an iterative mathematical procedure which involved the following problems: 1. Heat transfer in the evaporating film on a rough surface. 2. Heat transfer in the condensate film on the fin top surface. 3. Heat conduction in a metallic fin and liquid meniscus. 4. Fluid circulation in the AGHP. The first three problems have been presented in detail in by Khrustalev and Faghri (1995b). These three interconnected problems are included in the mathematical procedure along with the fluid circulation equations presented by Khrustalev and Faghri (1995a).
Khrustalev and Faghri (1995a)  compared their numerical results with the experimental data by Schlitt et al. (1974) , who studied an AGHP with the following geometry: Lt = 0.914 m, Lc = 0.152 m, 0.15 ≤ Le ≤ 0.343 m, W = 0.610 mm, Dg = 1.02 mm, L = 0.430 mm, Rv = 4.43 mm, Ro = 7.95 mm, γ = 0, N = 27, and tt = 0. The working fluids were ammonia and ethane, and the container material thermal conductivity was assumed to be kw = 170 W/(m-K). The longitudinal distribution of the meniscus contact angle, which is influenced by the inclination angle and the fluid pressure variation (Δp), is shown in Fig. 15(a) for ammonia (Le = 0.343 m, Tv = 250 K). For positive values of the inclination angle φ (when the condenser end is elevated) the points of minimum and maximum liquid surface curvature (the so-called “dry” and “wet” points) were shifted from the ends of the heat pipe towards the adiabatic section. As a result, the meniscus contact angle distributions in the heat loaded sections were even more uniform than those in the horizontal case. Note that the meniscus angles in the condenser differ very slightly from the maximum value, and their values in the evaporator can be almost the same as that in the condenser for positive φ. The corresponding longitudinal distributions of the local heat transfer coefficients in the evaporator and condenser and also wall and vapor temperature variations are shown in Figs. 15(b) and 15(c). The local heat transfer coefficient in the middle of the condenser was about two times larger than that at the entrance or end cap, which resulted in the external wall temperature variation shown in Fig. 15(c). In the evaporator section, the variation of the local heat transfer coefficient was weaker (for heat loads which are not close to the maximum), so the wall temperature profile was very smooth. The temperature drop along the vapor flow was less than 0.01 K.
Advances in modeling of enhanced flat miniature heat pipes with capillary grooves were reviewed by Faghri and Khrustalev (1997) . Lefèvre et al., (2010)  extended the model by approximating the temperature along the heat pipe using different thermal resistances in their nodal model. The axial heat conduction cannot be neglected when the heat pipe wall is thick and/or the operating temperature is in the intermediate or high range. Do et al. (2008)  extended the model by Khrustalev and Faghri (1995a; 1995b)  to include the axial heat conduction along the heat pipe wall. The wall temperature profile and maximum heat transport rate obtained from the model developed by Do et al. (2008)  are compared in Figure 16 with the experimental data of Hopkins et al. (1999) for a flat miniature heat pipe with a micro rectangular axial grooved wick structure.
When applying the Nusselt theory to a closed, two-phase thermosyphon, a variable vapor condensation rate must be considered. This approach was taken by Spendel (1984) , who modeled a conventional thermosyphon condenser section with a two-dimensional incompressible formulation in the vapor space and an application of the Nusselt theory for the falling liquid film. With this model, Spendel (1984)  performed a limited parametric study on the importance of including the interfacial shear stress and vapor pressure drop in determining the falling film thickness and the resulting Nusselt number for the thermosyphon. Based on these effects, the local Nusselt number in the condenser of a thermosyphon can vary as much as 60% from the approximate formulation derived by Nusselt.
An examination of conventional and concentric annular thermosyphons was performed by Faghri et al. (1989a), by which an improved flooding limit formulation was determined. Furthermore, in this study, the effects of the empirically obtained interfacial shear stress of the counterflowing vapor on the falling liquid film were considered in the condenser section alone.
Harley and Faghri (1994b) coupled a general quasi-steady Nusselt-type solution of the falling film with the complete two-dimensional vapor solution to simulate the transient two-dimensional behavior of thermosyphons with variable properties. This model is significantly different from the previous models in that it simulates the entire thermosyphon, rather than only the condenser section. A two-dimensional transient formulation for the vapor is coupled with unsteady heat conduction in the pipe wall. Furthermore, the quasi-steady falling condensate film in the condenser, adiabatic, and evaporator sections are simulated by accounting for the variable vapor condensation rate, interfacial shear stress, and vapor pressure drop. The numerical results of Harley and Faghri (1994b) were compared with the experimental data of Mingwei et al. (1991).
The transient axial variations of the outer wall and centerline vapor temperatures are shown in Fig. 17(b), where the agreement with the steady-state experimental data of Mingwei et al. (1991)  is quite good. The transient centerline axial vapor velocities for this low-temperature case are small and the axial pressure drop is negligible. The low axial vapor velocity is due in part to both the relatively high vapor density and latent heat of water and the low heat flux input to the evaporator. Obviously, compressibility is not a factor for the low-temperature thermosyphon because the maximum Mach number is much less than 0.3. The negligible axial pressure variation is typical of low-temperature two-phase heat transfer devices due to the low vapor velocities.
The condensation process is affected by the presence of a small amount of noncondensible gas in a thermosyphon. During normal operation, the noncondensible gas is swept to the condenser by the vapor, which partially blocks the condenser section. This blockage reduces the area available for the heat transfer. Gas-loaded thermosyphons were studied by Hijikata et al. (1984) , where a steady two-dimensional model that neglected heat conduction through the wall and falling liquid film was developed. However, since only the condenser section of the thermosyphon was modeled, a complex system of boundary conditions with several different empirical parameters was required. The behavior of the noncondensible gas was described by a modified two-dimensional steady-state species equation.
A more complete model of the condenser section of a gas-loaded thermosyphon was developed by Kobayashi and Matsumoto (1987) . This two-dimensional steady-state model accounts for diffusion across the vapor-gas interface but neglects the falling film and axial heat conduction through the pipe wall.
Peterson and Tien (1989)  extended the steady-state model of Hijikata et al. (1984)  by accounting for the effect of heat conduction in the pipe wall. Due to the integral simplification of the governing equations, however, this model can only be used in cases where heat conduction in the wall is either dominant or negligible. Furthermore, this formulation only models the condenser section where all properties were assumed to be constant.
Harley and Faghri (1994a)  coupled a general quasi-steady Nusselt-type solution of the falling film with the complete two-dimensional vapor-gas formulation to simulate the transient two-dimensional behavior of gas-loaded thermosyphons with variable properties. This formulation is significantly different from the previous models in that it simulates the entire thermosyphon rather than only the condenser section, using a two-dimensional transient formulation for the vapor and gas that is coupled to unsteady heat conduction in the pipe wall. Furthermore, this formulation models the quasi-steady falling condensate film in the condenser, adiabatic, and evaporator sections that accounts for the variable condensation rate, interfacial shear stress, and vapor pressure drop.
Zuo and Gunnerson (1995)  presented a numerical model of inclined thermosyphon performance. Liquid-vapor interfacial shear stress and the effects of working fluid inventory at various inclination angles were included in the model. They analyzed limiting mechanisms of the dryout and flooding and demonstrated that the model is capable of predicting the performance of an inclined thermosyphon. A one-dimensional steady-state mathematical model describing natural circulation two-phase flow in a thermosyphon with a tube separator was developed by Lin and Faghri (1997b) . Void fraction distributions along the thermosyphon were obtained under various operating conditions. Liquid fill ratios for steady flow were suggested. Lin and Faghri (1998b)  numerically studied the hydrodynamic stability of natural circulation two-phase flow in a high performance thermosyphon with tube separator. The simulation results showed that the operating temperature, the heat rate and thermosyphon inclination angle have significant influence on the flow instability.
El-Genk and Saber (1999)  developed a one-dimensional, steady-state model to determine the operation envelopes of closed, two-phase thermosyphons in terms of dimensions, type, vapor temperature of working fluid, and power throughput. The thermosyphon operation-envelope was an enclosure with three critical boundaries, related to dryout, boiling, and flooding limits. The calculations showed that an increase in the thermosyphon diameter, evaporator length, or vapor temperature expanded the operation-envelope, while an increase in the length of either the condenser or the adiabatic section only slightly changed the envelope’s upper and lower boundaries. Pan (2001)  presented a condensation model for a two-phase, closed thermosyphon by considering the interfacial shear stress due to the mass transfer and interfacial velocity. The relative velocity ratio and the momentum transfer factor greatly affect the condensation heat transfer in the thermosyphon. A sub-flooding limit was proposed to capture the interaction between the condensation and evaporation in the thermosyphon.
Jiao et al. (2008)  developed a model to investigate the effect of the filling ratio on the distribution of the liquid film and liquid pool. The total heat transfer rate of the liquid pool, including natural convection and nucleate boiling, was calculated by combining their effective areas and heat transfer coefficients. The correlation for effective area was obtained based on experimental results. The range for the filling ratio was proposed for the steady and effective operation of the thermosyphon based on analysis and comparison. The effects of heat input, operating pressure, and geometries of the thermosyphon on the range of the filling ratio were also discussed. Jiao et al. (2012)  further developed the model presented by Jiao et al. (2008)  to investigate the effect of the filling ratio on dryout, flooding and boiling limits. The experiments were conducted using nitrogen as working fluid and the experimental results were compared with the calculations. The maximum filling ratio was introduced, beyond which the heat transfer performance could be declined due to the accumulation of the liquid in condenser end. The sensitivity of the operation range to the operating pressure and geometries were also analyzed.
Rotating Heat Pipes
The rotating heat pipe concept was first proposed by Gray (1969) , who showed that such a rotating heat pipe can transfer significantly more heat than a similar stationary heat pipe. Faghri et al. (1993)  performed a rotating heat pipe vapor flow analysis by using a two-dimensional axisymmetric formulation to determine the influence of the rotation rate on the vapor pressure drop and interfacial shear stresses. They found that pressure drop and shear stress correlations for conventional heat pipes do not apply in rotating heat pipes as the vapor flow profiles are significantly different from the patterns seen in stationary heat pipes. However, that model did not consider heat transfer in the pipe wall or the counterflowing liquid film. Harley and Faghri (1995)  developed a detailed transient numerical simulation of rotating heat pipes, including the thin liquid condensable film, vapor flow, and heat conduction in the pipe wall.
The complete rotating heat pipe simulation was done by Harley and Faghri (1995)  for the heat pipe experimentally studied by Daniels and Al-Juimaily (1975) for a heat input of Q = 800 W and a rotational rate of ω = 600 rpm. The dimensions of this heat pipe were Lt = 0.325 m, Le = 0.05 m, La = 0.123 m, Lc = 0.152 m, Rv = 0.02125 m, Ro = 0.02650 m, and α = 2 deg. The heat pipe wall was copper and the working fluid was Freon-113. A true comparison with actual data was impossible due to the assumption for the outer heat transfer coefficient. An approximate heat transfer coefficient (600 W/m2-K) was determined from an examination of the wall temperature profile given by Daniels and Al-Juimaily (1975) . Because both the wall temperature and total heat input were given, the assumption of an ambient temperature is needed to obtain the external convective heat transfer coefficient.
The axial variation of the outer wall temperature is shown in Fig. 18(a). This assumed heat transfer coefficient resulted in the isothermal operation of the rotating heat pipe, which can be seen in Fig. 18(b), where the centerline vapor temperature is nearly constant along the length of the rotating heat pipe. Daniels and Al-Jumaily (1975) determined the vapor temperature with two pressure transducers located on the centerline at the two end caps, where the vapor was assumed to be saturated. As can be seen in Fig. 18(b), the assumption of a saturated vapor is valid because the vapor velocity is low enough to neglect compressibility effects.
Lin and Faghri (1997a)  investigated the flow behavior and the related heat transfer characteristics of stratified flow in axially rotating heat pipes with cylindrical and stepped wall configurations using theoretical and semi-empirical models. The predicted results and experimental data showed a good agreement. Lin and Faghri (1997c)  presented a mathematical model of the hydrodynamic performance of a rotating miniature heat pipe (RMHP) with a grooved inner wall surface. Influences of operating temperature, rotational speed, and liquid-vapor interfacial shear stress on the maximum performance and optimum liquid fill amount were discussed. Pressure drops of the axial liquid flow and vapor flow were demonstrated.
The region of hysteretic annular flow in rotating stepped wall heat pipes was experimentally determined by Lin and Faghri (1998a) . A model for predicting the condensation heat transfer coefficient was also proposed and the theoretical results were compared with experimental data. Lin and Faghri (1999)  developed a theoretical model that describes the evaporating film flow in a rotating miniature heat pipe with an axial triangular grooved internal surface. The theory of thin liquid film vaporization heat transfer was used to predict the evaporation heat transfer in the micro region. The effects of disjoining pressure, surface tension, and centrifugal force on the flow are discussed. It was concluded that the influence of rotational speed on the evaporation heat transfer in the micro region can be neglected. Therefore, the heat transfer analysis in the micro region at zero rotational speed can be applied to the case of rotational operation. Cao (2010)  developed numerical and analytical analysis for miniature high temperature rotating heat pipes for applications in gas turbine cooling.
All of the aforementioned analyses focused on a pure vapor flow with no noncondensible gases in the rotating heat pipe core. Harley and Faghri (2000)  accounted for the noncondensible gas, the vapor pressure drop, the thin liquid film, the interfacial shear stress, and the heat conduction in the pipe wall of a rotating heat pipe without the need for specifying condenser inlet conditions. The problem is solved as a conjugate coupled heat transfer problem. A parametric study was also presented to determine the influence of rotational speed, heat input and output, and the mass of noncondensible gas on a rotating heat pipe.
Loop Heat Pipes
Various numerical and analytical models have been developed in order to understand the fundamental mechanisms of thermal and hydrodynamic couplings between LHP components, as well as to quantitatively predict the LHP operational characteristics. Most of the models are based on energy and pressure balances written for each component of the LHP. Some examples of steady-state LHP modeling studies are Maydanik et al. (1994) , Kaya and Hoang (1999), Hoang and Kaya (1999) Cite error: Closing </ref> missing for <ref> tag, Muraoka et al. (2001), Kaya and Ku (2003), Hamdan (2003), Chuang (2003) , Furukawa (2006) , Kaya and Goldak (2006) , and Launay et al. (2007c; 2008). Transient LHP modeling efforts include Cullimore and Bauman (2000), Hoang and Ku (2003), Launay et al. (2007a; 2007b), Kaya et al. (2008), Chernysheva and Maydanik (2008) , and Khrustalev (2010). LHP transient modeling of two-phase systems with capillary evaporators using a Thermal Desktop was pioneered by Cullimore and Bauman (2000) and extended by Khrustalev (2010)  to upgraded loop heat pipe systems with complex radiators, multiple components, and varying environmental conditions. Launay et al. (2008) proposed separate closed-form solutions for two distinct LHP operating modes: the variable conductance mode (VCM), and the fixed conductance mode (FCM). These modes are exhibited on an experimentally observed LHP operating curve, depicting the operating temperature (usually represented by the compensation chamber temperature or evaporator temperature) versus the heat input. These models are also linked to the distribution of the condenser heat transfer area: one part is used for condensation, and the other is for liquid subcooling.
In terms of the closed-form solutions, the LHP operating temperature can be easily calculated as a function of the heat input. In Figs. 19 and 20, predictions by full numerical simulation and simplified closed-form solutions, are compared with experimental measurements for two LHP designs (Chuang, 2003; Boo and Chung, 2004)  using acetone and ammonia as the working fluids, respectively. It appears that the relative difference between the full numerical simulation and closed-form solutions peaks at the transition between the VCM and FCM; however, the maximum difference is less than 15%. The general agreement of the modeling predictions with the experimental data validates both the model and closed-form solutions as useful tools for possible LHP designs. A recent review of the fundamentals, operation, and design of loop heat pipes was given by Ambirajan et al. (2012).
Capillary Pumped Loop Heat Pipes
There are three approaches related to the analysis of CPLs. The first approach is to examine the heat transfer without including the fluid flow analysis in the wick or vapor regions of the evaporator (Kiper et al., 1990). Since an analytical approach was used for solving the energy equation instead of a numerical approach, approximations were made in the solution procedure. It should be noted that the fluid flow problem is of significant importance in the evaporator and no meaningful conclusions can be made without a combined fluid flow and heat transfer analysis. This approach also included a lumped analysis of the evaporator and subcooler, in which an exponential temperature profile was assumed in the analysis. A second approach included a detailed and accurate analysis of the evaporator component for steady and transient performance, based on solving the complete differential forms of the momentum and energy equations (Cao and Faghri, 1994b; 1994c). This methodology is the most appropriate technique for analyzing the CPL system. The third approach is related to an overall system simulation using existing semi-empirical correlations under 1 g conditions to calculate the pressure drop and heat transfer coefficients in different parts of the CPL (Kroliczek et al., 1984; Ku et al., 1986a; 1986b; 1987a; 1987b; Chalmers et al., 1988; Ku et al., 1988; Benner et al., 1989; Schweickart and Buchko, 1991; Ku, 1993). Some analytical studies concerning the complex transient phenomena of startup have been conducted (Cullimore, 1991).
Micro Heat Pipes
Khrustalev and Faghri (1994)  developed a detailed mathematical model which allows the examination of the heat and mass transfer processes in a micro heat pipe (MHP). The model gave the distribution of the liquid in a micro heat pipe and its thermal characteristics as a function of the liquid charge and the applied heat load. The liquid flow in the triangular-shaped corners of a micro heat pipe with a polygonal cross section is considered by accounting for the variation of the curvature of the free liquid surface and the interfacial shear stresses due to liquid-vapor interaction. The predicted results are compared to existing experimental data. The importance of the liquid fill, minimum wetting contact angle, and the shear stresses at the liquid-vapor interface in predicting the maximum heat transfer capacity and thermal resistance of the micro heat pipe is demonstrated.
The comparison of the numerical results (Khrustalev and Faghri, 1994) and the experimental data reported by Wu and Peterson (1991)  for the maximum heat transfer capacity are shown in Fig. 21 for micro heat pipes with copper and silver casings. While the data for the minimum contact angles are contradictory and can be influenced by many physical factors, the numerical results are presented for θmen,min= 33°, 38° and 45°. The experimental results of the onset of dryout and the present numerical prediction show a good agreement. Neglecting the shear stress at the free surface of the liquid due to liquid-vapor frictional interaction (“no shear stress” in Fig. 21) can lead to an overestimation of the maximum heat transfer capacity. The shear stress at the free liquid surface influences the liquid distribution along the heat pipe, which can result in an increase of the liquid locking of the condenser end, compared to the case of neglecting this shear stress. Sartre et al. (2000)  and Suman and Kumar (2005)  extended Khrustalev and Faghri’s model (1994)  to include heat conduction in the wall to simulate micro heat pipes of polygonal shapes. Wang and Peterson (2002)  and Launay et al. (2004b)  have improved Khrustalev and Faghri’s model (1994) to simulate arrays of micro heat pipes made of several alluminim wires bounded between two aluminum sheets.
Pulsating Heat Pipes
Since slug flow is the primary flow pattern in PHPs, most existing efforts on modeling have focused on slug flow. Miyazaki and Akachi (1996) proposed a simple analytical model of self-exciting oscillation based on an oscillating feature observed during experimentation. The reciprocal excitation of pressure oscillation due to changes in the heat transfer rate caused by the oscillation of the void fraction was investigated. Oscillation of the void fraction was out of phase with the pressure oscillation by -π/2. This model indicates that an optimal charge ratio exists for a particular PHP. If the charge ratio is too high, the PHP will experience a gradual pressure increase followed by a sudden drop. Insufficient charging will cause chaotic pressure fluctuation. However, proper charging will generate a symmetrical pressure wave. Miyazaki and Akachi (1998) derived the wave equation of pressure oscillation in the PHP based on the self-excited oscillation, in which the reciprocal excitation between the pressure oscillation and void fraction was accounted for. Miyazaki and Arikawa (1999) investigated the oscillatory flow in the PHP and then measured the wave velocity, which fairly agreed with the predictions of Miyazaki and Akachi’s model (1998). Hosoda et al. (1999)  reported a simplified numerical model of a PHP, in which temperature and pressure were calculated by solving the momentum and energy equations for two-dimensional, two-phase flow. However, the thin liquid film that surrounds a vapor plug on the tube wall and the friction between the tube and the working fluid were neglected. Experimental results were used as initial conditions for the model. The numerical results for pressure in the PHP were higher than the experimental results; however, the model showed that the propagation of vapor plugs induced fluid flow in the capillary tubes.
Zuo et al. (Zuo et al., 1999 ; 2001) attempted to model the PHP by comparing it to an equivalent single spring-mass-damper system, with the parameters of the system affected by heat transfer. Zuo et al. (2001)  showed that the spring stiffness increases with increasing time, and therefore, the amplitude of oscillation must decrease with increasing time; this is in contradiction with steady oscillations observed in PHP operation. Wong et al. (1999) modeled an open-loop PHP by considering it as a multiple spring-mass-damper system with the flow modeled under adiabatic conditions for the entire PHP. A sudden pressure pulse was applied to simulate local heat input into a vapor plug.
Shafii et al. (2001) developed a theoretical model to simulate the behavior of liquid slugs and vapor plugs in both closed- and open-loop PHPs with two turns (see Fig. 22). The model solved for the pressure, temperature, plug position and heat transfer rates. The most significant conclusion was that the majority of the heat transfer (95%) is due to sensible heat; not due to the latent heat of vaporization. Latent heat served only to drive the oscillating flow. Sakulchangsatjatai et al. (2004) applied Shafii et al.’s model (2001) to model closed-end and closed-loop PHPs as an oscillating two-phase heat and mass transfer in a straight pipe, while neglecting the thin liquid film between the vapor plug and the pipe wall. Zhang et al. (2002) analytically investigated oscillatory flow in a U-shaped miniature channel-a building block of PHPs. A significant difference between this model and other mathematical models is the nondimensionalizing of the governing equations. Flow in the tube was described by two dimensionless parameters, the non-dimensional temperature difference and the evaporation and condensation heat transfer coefficients. It was found that both the initial displacement of the liquid slug and gravity had no effect on the amplitude and angular frequency of the oscillation. In addition, the amplitude and frequency of oscillation were increased by increasing the dimensionless temperature difference. The amplitude and frequency of oscillation were correlated to the heat transfer coefficients and temperature difference.
Zhang and Faghri (2003)  investigated oscillatory flow in a closed-end pulsating heat pipe with an arbitrary number of turns (see Fig. 23). The results showed that for a PHP with few turns (less than 6), the amplitude and frequency of oscillation were independent of the number of turns. The motion of the vapor plugs was identical for odd numbered plugs once a steady state had been reached. Even numbered plugs also exhibited identical motion. Odd and even numbered plugs had the same amplitude, but were out of phase by a factor of π. As the number of turns increased above 6, the odd and even numbered plugs no longer showed identical oscillation. Each plug lagged slightly behind the next. Each plug remained out of phase by π, however. (see Fig. 24).
Dobson and Harms (1999) investigated a PHP with two open ends. The open ends were parallel and pointed in the same direction. These ends were submerged in water, while the evaporator section was coiled and attached to a float so that it remained out of the water. The evaporator was heated and the oscillatory fluid motion produced a net thrust. A numerical solution of the energy equation and the equation of motion for a vapor plug was presented to predict the plug’s temperature, position and velocity. Oscillatory motion in the PHP generated a net average thrust of 0.0027N. Heat transfer due to sensible heat was not taken into account. Dobson (2004; 2005) proposed the use of open-ended PHP in conjunction with two check valves to pump water; however, the maximum attainable mass flow rates are on the order of mg/s, which is hardly enough to irrigate fields. An improved model for liquid slug oscillation that considered pressure difference, friction, gravity, and surface tension was also presented.
Zhang and Faghri (2002) proposed models for heat transfer in the evaporator and condenser sections of a PHP with one open end by analyzing thin film evaporation and condensation. Heat transfer in the evaporator is the sum of evaporative heat transfer in the thin liquid film and at the meniscus. Heat transfer in the condenser is similarly calculated, and sensible heat transfer to the liquid slug is also considered. It is found that the overall heat transfer is dominated by the exchange of sensible heat, not by the exchange of latent heat. Shafii et al. (2002)  further developed their earlier numerical model (Shafii et al., 2001 ) by including an analysis of the evaporation and condensation heat transfer in the thin liquid film separating the liquid and vapor plugs. Both open- and closed-loop PHPs were considered, and each displayed similar results. As shown in Fig. 25, the total heat transfer is due mainly to the exchange of sensible heat (~95%). Total heat transfer slightly increases as the surface tension of the working fluid increases. The total heat transfer significantly decreases with decreased heating section wall temperature. Increasing the diameter of the tube results in a higher total heat transfer.
Liang and Ma (2004)  presented a mathematical model describing the oscillation characteristics of slug flow in a capillary tube. In addition to the modeling of oscillating motion, numerical results indicated that the isentropic bulk modulus generates stronger oscillations than the isothermal bulk modulus. While it demonstrated that the capillary pipe diameter, bubble size, length and the number of unit cells determine the oscillation, the capillary force, gravitational force, and initial pressure distribution of the working fluid significantly affects the frequency and amplitude of oscillating motion in the capillary tube. Ma et al. (2002; 2006) analytically described the oscillating motion by performing force balances of the thermally driven, capillary, frictional, and elastic restoring forces on a liquid slug. Pressure differences between the evaporator and the condenser were related to the temperature difference between the evaporator and the condenser by the Clapeyron-Clausius equation. The temperature difference between the evaporator and the condenser of the PHP was utilized as a driving force of the oscillating motion. Ma et al. (2006)  concluded that oscillating motion depends on the charge ratio, total characteristic length, diameter, temperature difference between the evaporation and condenser sections, and the working fluid and operating temperature. When compared to experimental results, the mathematical model under-predicted the temperature difference between the evaporator and condenser when compared to experimental results (Ma et al., 2002).
Holley and Faghri (2005)  presented a numerical model for a PHP with a sintered copper capillary wick and with flow channels that have different diameters. The effects of the varying channel diameters, inclination angles, and the number of parallel channels were presented. When a channel was of a smaller diameter, it induced circulation of the fluid, which in turn increased the heat load capability of the PHP. The modeled PHP performed better in bottom heat mode (smaller temperature differential) than in top heat mode. Varying the mean Nusselt number had little effect on the PHP performance. As the number of parallel channels increased, its heat load capability increased, and the PHPs sensitivity to gravity decreased. Khandekar et al. (2002)  used an Artificial Neural Network (ANN) to predict the PHP performance. The ANN was of the fully connected feed forward configuration, and was trained using 52 sets of experimental data from a closed-loop PHP. The ANN was fed the heat input and fill ratio of each data set and then calculated the effective thermal resistance of the PHP. The ANN model predicted the thermal performance for this type of PHP but neglected many parameters that affect PHP performance, including tube diameter, number of parallel channels, length of the PHP, inclination angle, and properties of the working fluid. If the ANN had more input nodes with which to consider these parameters, it would be a more effective model. Even then it would require considerably well-organized experimental data for the ANN to be effective. Khandekar and Gupta (2007)  modeled heat transfer in a radiator plate with an embedded PHP using a commercial package FLUENT. However, oscillatory flow and heat transfer of the PHP were not modeled in this investigation. The contribution of the PHP on the heat transfer in the radiator plate was considered using an effective thermal conductivity obtained from the experiment. Zhang and Faghri (2008) made a detailed review of pulsating heat pipe analysis. Khandekar et al. (2010) presented the experimental and theoretical methodologies to predict hydrodynamic characteristics in unidirectional two-phase Taylor bubble flows and recommended directions for further research on modeling of pulsating heat pipes.
Gas-Loaded Heat Pipes
Noncondensible gas-buffered heat pipes (NCHPs) offer isothermal operation for varying heat loads by changing the overall thermal resistance of the heat pipe. As the heat load increases, the vapor temperature and total pressure increase in the heat pipe. This increase in total pressure compresses the noncondensible gas in the condenser, thereby increasing the surface area available for heat transfer. This maintains a constant heat flux and temperature. NCHP operation can also be examined through the partial vapor pressure. As the heat input increases, the total pressure increases. However, because the total pressure is comprised of both the partial vapor and partial gas pressures, the increase in total pressure is accounted for by an increase in the partial gas pressure, maintaining a constant vapor pressure, and thus a constant vapor temperature. Noncondensible gas-buffered heat pipes have been modeled previously with several different levels of approximation. The classic flat-front analytical model of Marcus and Fleishman (1970)  neglected all diffusion across the vapor-gas interface. Later studies modeled one-dimensional steady diffusion (Edwards and Marcus, 1972)  and one-dimensional transient diffusion (Shukla, 1981). Rohani and Tien (1973)  studied the steady-state two-dimensional diffusion process in a noncondensible gas-buffered heat pipe. The importance of modeling radial diffusion in a NCHP was demonstrated where the noncondensible gas tends to accumulate at the liquid-vapor interface, which retards vapor condensation. However, Rohani and Tien (1973)  neglected the effect of conjugate heat transfer through the wall as well as transient effects. The importance of the transient response of a NCHP was shown by Shukla (1981), as transient thermal overshoots were discovered. This study, however, did not include the effects of radial heat conduction in the wall or wick or the radial diffusion effects of the noncondensible gas.
Harley and Faghri (1994c) studied the two-dimensional, transient NCHP operation, including the effects of conjugate heat transfer through the wall, and is modeled through a solution of the general differential conservation equations. This procedure was used to simulate the high-temperature heat pipe experimentally studied both with and without noncondensible gases. The major advantage over previous models was that this model treats the noncondensible gas as a separate entity that is described by mass transport phenomena. Additionally, the energy transport through the wall is coupled to the transient operation of the heat pipe through the use of a conjugate solution technique. The complete behavior of the heat pipe, along with the location and two-dimensional shape of the noncondensible gas front, are modeled from the initial continuum-flow, liquid-state start up to steady-state conditions. The proposed model by Harley and Faghri (1994c)  predicted the existing experimental data for the operation of high-temperature heat pipes with and without noncondensible gases. The gas-loaded heat pipe experimentally studied by Ponnappan (1989) was simulated (Case 2), with results shown in Figs. 26 and 27. This case has a higher radiative emissivity than Case 1, which results in a decrease in the thermal resistance at the outer condenser surface. In the experiment performed by Ponnappan (1989), the emissivity was increased when the noncondensible gas was added, so that a near-constant operationg temperature (compared with Case 1) could be maintained. In Fig. 26, the wall and vapor temperatures decreased significantly in the condenser section due to the presence of the noncondensible gas. The gas density increased in the condenser during transient operation, as shown in Fig. 27(a). The steady-state wall temperature is in good agreement with the data by Ponnappan (1989).
A transient lumped heat pipe formulation for conventional heat pipes is presented, and the lumped analytical solutions for different boundary conditions at the evaporator and condenser are given by Faghri and Harley (1994). For high temperature heat pipes with a radiative boundary condition at the condenser, a nonlinear ordinary differential equation is solved. In an attempt to reduce computational demands, a transient lumped conductive model was developed for noncondensible gas-loaded heat pipes. The lumped flat-front transient model was extended by accounting for axial heat conduction across the sharp vapor-gas interface. The analytic solutions for conventional and gas-loaded heat pipes were compared with the corresponding numerical results of the full two-dimensional conservation equations and experimental data, showing a good agreement.
- ↑ Faghri, A., 2012, "Review and Advances in Heat Pipe Science and Technology," Journal of Heat Transfer, 134(12), 123001. http://dx.doi.org/10.1115/1.4007407
- ↑ 2.0 2.1 Faghri, A., 1986, "Vapor Flow Analysis in a Double-Walled Concentric Heat Pipe," Numerical Heat Transfer, 10(6), 583-595.
- ↑ 3.0 3.1 3.2 3.3 3.4 Cao, Y., Faghri, A., and Mahefkey, E. T., 1989, "The Thermal Performance of Heat Pipes with Localized Heat Input," International Journal of Heat and Mass Transfer, 32(7), 1279-1287. http://dx.doi.org/10.1016/0017-9310(89)90028-8
- ↑ 4.0 4.1 Rosenfeld, J. H., 1987, "Modeling of Heat Transfer into a Heat Pipe for a Localized Heat Input Zone," AIChE Symposium Series, 83, 71-76.
- ↑ Faghri, A., and Parvani, S., 1988, "Numerical Analysis of Laminar Flow in a Double-Walled Annular Heat Pipe," Journal of Thermophysics and Heat Transfer, 2(3), 165-171. http://dx.doi.org/10.2514/3.81
- ↑ Faghri, A., 1989, "Performance Characteristics of a Concentric Annular Heat Pipe: Part II-Vapor Flow Analysis," Journal of Heat Transfer, 111(4), 851-857. http://dx.doi.org/10.1115/1.3250796
- ↑ 7.0 7.1 7.2 7.3 7.4 7.5 Chen, M. M., and Faghri, A., 1990, "An Analysis of the Vapor Flow and the Heat Conduction through the Liquid-Wick and Pipe Wall in a Heat Pipe with Single Or Multiple Heat Sources," International Journal of Heat and Mass Transfer, 33(9), 1945-1955. http://dx.doi.org/10.1016/0017-9310(90)90226-K
- ↑ Ivanovskii, M. N., Sorokin, V. P., and Yagodkin, I. V., 1982, The Physical Principles of Heat Pipes, Clarendon Press, Oxford.
- ↑ 9.0 9.1 9.2 Faghri, A., and Buchko, M., 1991, "Experimental and Numerical Analysis of Low-Temperature Heat Pipes with Multiple Heat Sources," Journal of Heat Transfer, 113(3), 728-734. http://dx.doi.org/10.1115/1.2910624
- ↑ 10.0 10.1 10.2 10.3 10.4 Xiao, B., and Faghri, A., 2008, "A Three-Dimensional Thermal-Fluid Analysis of Flat Heat Pipes," International Journal of Heat and Mass Transfer, 51(11-12), 3113-3126. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2007.08.023
- ↑ 11.0 11.1 Schmalhofer, J., and Faghri, A., 1993a, "A Study of Circumferentially-Heated and Block-Heated Heat Pipes-I. Experimental Analysis and Generalized Analytical Prediction of Capillary Limits," International Journal of Heat and Mass Transfer, 36(1), 201-212. http://dx.doi.org/10.1016/0017-9310(93)80080-E
- ↑ 12.0 12.1 Schmalhofer, J., and Faghri, A., 1993a, "A Study of Circumferentially-Heated and Block-Heated Heat Pipes-I. Experimental Analysis and Generalized Analytical Prediction of Capillary Limits," International Journal of Heat and Mass Transfer, 36(1), 201-212. http://dx.doi.org/10.1016/0017-9310(93)80080-E
- ↑ Zhu, N., and Vafai, K., 1998b, "Vapor and Liquid Flow in an Asymmetrical Flat Plate Heat Pipe: A Three-Dimensional Analytical and Numerical Investigation," International Journal of Heat and Mass Transfer, 41(1), 159-174. http://dx.doi.org/10.1016/S0017-9310(97)00075-6
- ↑ Lefèvre, F., and Lallemand, M., 2006, "Coupled Thermal and Hydrodynamic Models of Flat Micro Heat Pipes for the Cooling of Multiple Electronic Components," International Journal of Heat and Mass Transfer, 49(7-8), 1375-1383. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2005.10.001
- ↑ Wang, Y., and Vafai, K., 2000, "An Experimental Investigation of the Transient Characteristics on a Flat-Plate Heat Pipe during Startup and Shutdown Operations," Journal of Heat Transfer, 122(3), 525-535. http://dx.doi.org/10.1115/1.1287725
- ↑ Aghvami, M., and Faghri, A., 2011, "Analysis of Flat Heat Pipes with various Heating and Cooling Configurations," Applied Thermal Engineering, 31(14-15), 2645-2655. http://dx.doi.org/10.1016/j.applthermaleng.2011.04.034
- ↑ Shabgard, H., and Faghri, A., 2011, "Performance Characteristics of Cylindrical Heat Pipes with Multiple Heat Sources," Applied Thermal Engineering, 31(16), 3410-3419. http://dx.doi.org/10.1016/j.applthermaleng.2011.06.026
- ↑ 18.0 18.1 Jang, J. H., Faghri, A., and Chang, W. S., 1991, "Analysis of the One-Dimensional Transient Compressible Vapor Flow in Heat Pipes," International Journal of Heat and Mass Transfer, 34(8), 2029-2037. http://dx.doi.org/10.1016/0017-9310(91)90214-Y
- ↑ 19.0 19.1 Bowman, W. J., 1987, Simulated Heat Pipe Vapor Dynamics, Ph.D. Dissertation, Air Force Institute of Technology, Dayton, OH.
- ↑ Faghri, A., Buchko, M., and Cao, Y., 1991b, "A Study of High-Temperature Heat Pipes with Multiple Heat Sources and Sinks: Part II - Analysis of Continuum Transient and Steady- State Experimental Data with Numerical Predictions," Transactions of the ASME Journal of Heat Transfer, 113(4), 1010-1016. http://dx.doi.org/10.1115/1.2911194
- ↑ 21.0 21.1 21.2 21.3 21.4 Faghri, A., Buchko, M., and Cao, Y., 1991a, "A Study of High-Temperature Heat Pipes with Multiple Heat Sources and Sinks: Part I - Experimental Methodology and Frozen Startup Profiles," Journal of Heat Transfer, 113(4), 1003-1009. http://dx.doi.org/10.1115/1.2911193
- ↑ 22.0 22.1 Cao, Y., and Faghri, A., 1993a, "Conjugate Modeling of High-Temperature Nosecap and Wing Leading-Edge Heat Pipes," Journal of Heat Transfer, 115(3), 819-822. http://dx.doi.org/10.1115/1.2910765
- ↑ 23.0 23.1 23.2 23.3 Rice, J., and Faghri, A., 2007, "Analysis of Porous Wick Heat Pipes, Including Capillary Dry-Out Limitations," AIAA Journal of Thermophysics and Heat Transfer, 21(3), 475-486. http://dx.doi.org/10.2514/1.24809
- ↑ Tournier, J. M., and El-Genk, M. S., 1994, "A Heat Pipe Transient Analysis Model," International Journal of Heat and Mass Transfer, 37(5), 753-762. http://dx.doi.org/10.1016/0017-9310(94)90113-9
- ↑ Ranjan, R., Murthy, J. Y., Garimella, S. V., and Vadakkan, U., 2011, "A Numerical Model for Transport in Flat Heat Pipes Considering Wick Microstructure Effects," International Journal of Heat and Mass Transfer, 54(1-3), 153-168. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2010.09.057
- ↑ 26.0 26.1 Jang, J. H., Faghri, A., Chang, W. S., and Mahefkey, E. T., 1990, "Mathematical Modeling and Analysis of Heat Pipe Start-Up from the Frozen State," Journal of Heat Transfer, 112(3), 586-594. http://dx.doi.org/10.1115/1.2910427
- ↑ 27.0 27.1 Cao, Y., and Faghri, A., 1993b, "Simulation of the Early Startup Period of High-Temperature Heat Pipes from the Frozen State by a Rarefied Vapor Self-Diffusion Model," Journal of Heat Transfer, 115(1), 239-246. http://dx.doi.org/10.1115/1.2910655
- ↑ 28.0 28.1 28.2 Cao, Y., and Faghri, A., 1993c, "A Numerical Analysis of High-Temperature Heat Pipe Startup from the Frozen State," Journal of Heat Transfer, 115(1), 247-254. http://dx.doi.org/10.1115/1.2910657
- ↑ 29.0 29.1 29.2 29.3 Ponnappan, R., 1990, "Comparison of Vacuum and Gas Loaded Mode Performances of a LMHP," AIAA-90-1755, Proceedings of AIAA/ASME 5th Joint Thermophysics and Heat Transfer Conference, Seattle, WA.
- ↑ 30.0 30.1 30.2 30.3 Cao, Y., and Faghri, A., 1992, "Closed-Form Analytical Solutions of High-Temperature Heat Pipe Startup and Frozen Startup Limitation," Journal of Heat Transfer, 114(4), 1028-1035. http://dx.doi.org/10.1115/1.2911873
- ↑ Hall, M. L., Merrigan, M. A., and Reid, R. S., 1994, "Status Report on the THROHPUT Transient Heat Pipe Modeling Code," AIP-CP-301, Proceedings of the 11th Symposium on Space Nuclear Power and Propulsion, American Institute of Physics, New York, NY, 965-970.
- ↑ Tournier, J. M., and El-Genk, M. S., 1996, "A Vapor Flow Model for Analysis of Liquid-Metal Heat Pipe Startup from a Frozen State," International Journal of Heat and Mass Transfer, 39(18), 3767-3780. http://dx.doi.org/10.1016/0017-9310(96)00066-X
- ↑ 33.0 33.1 33.2 33.3 33.4 33.5 Khrustalev, D., and Faghri, A., 1995a, "Thermal Characteristics of Conventional and Flat Miniature Axially-Grooved Heat Pipes," Journal of Heat Transfer, 117(4), 1048-1054. http://dx.doi.org/10.1115/1.2836280
- ↑ 34.0 34.1 34.2 Khrustalev, D., and Faghri, A., 1995b, "Heat Transfer during Evaporation on Capillary-Grooved Structures of Heat Pipes," Journal of Heat Transfer, 117(3), 740-747. http://dx.doi.org/10.1115/1.2822638
- ↑ Schlitt, K. R., Kirkpatrick, J. P., and Brennan, P. J., 1974, "Parametric Performance of Extruded Axial Grooved Heat Pipes from 100 K to 300 K," AIAA-74-724, Proceedings of AIAA/ASME Thermophysics and Heat Transfer Conference.
- ↑ Faghri, A., and Khrustalev, D., 1997, "Advances in Modeling of Enhanced Flat Miniature Heat Pipes with Capillary Grooves," Journal of Enhanced Heat Transfer, 4(2), 99-109.
- ↑ Lef¨¨vre, F., Lips, S., and Bonjour, J., 2010, "Investigation of Evaporation and Condensation Processes Specific to Grooved Flat Heat Pipes," Frontiers in Heat Pipes (FHP), 1, 023001. http://dx.doi.org/10.5098/fhp.v1.2.3001
- ↑ 38.0 38.1 38.2 38.3 38.4 38.5 Do, K. H., Kim, S. J., and Garimella, S. V., 2008, "A Mathematical Model for Analyzing the Thermal Characteristics of a Flat Micro Heat Pipe with a Grooved Wick," International Journal of Heat and Mass Transfer, 51(19-20), 4637-4650. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2008.02.039
- ↑ 39.0 39.1 Spendel, T., 1984, "Laminar Film Condensation Heat Transfer in Closed Two-Phase Thermosyphons," Proceedings of 5th International Heat Pipe Conference, Tsukuba, Japan.
- ↑ Faghri, A., Chen, M. M., and Morgan, M., 1989a, "Heat Transfer Characteristics in Two-Phase Closed Conventional and Concentric Annular Thermosyphons," Journal of Heat Transfer, 111(3), 611-618. http://dx.doi.org/10.1115/1.3250726
- ↑ 41.0 41.1 41.2 Harley, C., and Faghri, A., 1994b, "Complete Transient Two-Dimensional Analysis of Two-Phase Closed Thermosyphons Including the Falling Condensate Film," Journal of Heat Transfer, 116(2), 418-426. http://dx.doi.org/10.1115/1.2911414
- ↑ 42.0 42.1 42.2 Mingwei, T., Li, C., Hongji, Z., and Qingming, T., 1991, "Start-Up Behavior of the Frozen Two-Phase Closed Steel-Water Thermosyphons," Proceedings of the 8th International Heat Pipe Conference.
- ↑ 43.0 43.1 Hijikata, K., Chen, S. J., and Tien, C. L., 1984, "Noncondensible Gas Effect on Condensation in a Two-Phase Closed Thermosyphon," International Journal of Heat and Mass Transfer, 27(8), 1319-1325. http://dx.doi.org/10.1016/0017-9310(84)90059-0
- ↑ Kobayashi, Y., and Matsumoto, T., 1987, "Vapor Condensation in the Presence of Non-Condensible Gas in the Gravity Assisted Thermosyphon," Proceedings of 6th International Heat Pipe Conference, Grenoble, France.
- ↑ Peterson, P. F., and Tien, C. L., 1989, "Numerical and Analytical Solutions for Two-Dimensional Gas Distribution in Gas-Loaded Heat Pipes," Journal of Heat Transfer, 111(3), 598-604. http://dx.doi.org/10.1115/1.3250724
- ↑ Harley, C., and Faghri, A., 1994a, "Transient Gas-Loaded Thermosyphon Analysis," Proceedings of the 10th International Heat Transfer Conference, Brighton, England.
- ↑ Zuo, Z. J., and Gunnerson, F. S., 1995, "Heat Transfer Analysis of an Inclined Two-Phase Closed Thermosyphon," Journal of Heat Transfer, 117(4), 1073-1075. http://dx.doi.org/10.1115/1.2836287
- ↑ Lin, L., and Faghri, A., 1997b, "Steady-State Performance in a Thermosyphon with Tube Separator," Applied Thermal Engineering, 17(7), 667-679. http://dx.doi.org/10.1016/S1359-4311(96)00084-1
- ↑ Lin, L., and Faghri, A., 1998b, "An Analysis of Two-Phase Flow Stability in a Thermosyphon with Tube Separator," Applied Thermal Engineering, 18(6), 441-455. http://dx.doi.org/10.1016/S1359-4311(97)00046-X
- ↑ El-Genk, M. S., and Saber, H. H., 1999, "Determination of Operation Envelopes for Closed, Two-Phase Thermosyphons," International Journal of Heat and Mass Transfer, 42(5), 889-903. http://dx.doi.org/10.1016/S0017-9310(98)00212-9
- ↑ Pan, Y., 2001, "Condensation Heat Transfer Characteristics and Concept of Sub-Flooding Limit in a Two-Phase Closed Thermosyphon," International Communications in Heat and Mass Transfer, 28(3), 311-322. http://dx.doi.org/10.1016/S0735-1933(01)00237-8
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- ↑ Jiao, B., Qiu, L. M., Gan, Z. H., and Zhang, X. B., 2012, "Determination of the Operation Range of a Vertical Two-Phase Closed Thermosyphon," Heat and Mass Transfer, 48(6), 1043-1055. http://dx.doi.org/10.1007/s00231-011-0954-x
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- ↑ Gray, V. H., 1969, "The Rotating Heat Pipe—A Wickless, Hollow Shaft for Transferring High Heat Fluxes," Proceedings of ASME/AIChE Heat Transfer Conference, Minneapolis, MN, 1-5.
- ↑ Faghri, A., Gogineni, S., and Thomas, S., 1993, "Vapor Flow Analysis of an Axially Rotating Heat Pipe," International Journal of Heat and Mass Transfer, 36(9), 2293-2303. http://dx.doi.org/10.1016/S0017-9310(05)80114-0
- ↑ 57.0 57.1 57.2 Daniels, T. C., and Al-Jumaily, F. K., 1975, "Investigations of the Factors Affecting the Performance of a Rotating Heat Pipe," International Journal of Heat and Mass Transfer, 18(7-8), 961-973. http://dx.doi.org/10.1016/0017-9310(75)90190-8
- ↑ Lin, L., and Faghri, A., 1997a, "Heat Transfer Analysis of Stratified Flow in Rotating Heat Pipes with Cylindrical and Stepped Walls," International Journal of Heat and Mass Transfer, 40(18), 4393-4404. http://dx.doi.org/10.1016/S0017-9310(97)00060-4
- ↑ Lin, L., and Faghri, A., 1997c, "Steady-State Performance of a Rotating Miniature Heat Pipe," Journal of Thermophysics and Heat Transfer, 11(4), 513-518. http://dx.doi.org/10.2514/2.6292
- ↑ Lin, L. C., and Faghri, A., 1998a, "Condensation in Rotating Stepped Wall Heat Pipes with Hysteretic Annular Flow," Journal of Thermophysics and Heat Transfer, 12(1), 94-99. http://dx.doi.org/10.2514/2.6307
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- ↑ Cao, Y., 2010, "Miniature High-Temperature Rotating Heat Pipes and their Applications in Gas Turbine Cooling," Frontiers in Heat Pipes (FHP), 1, 023002. http://dx.doi.org/10.5098/fhp.v1.2.3002
- ↑ Harley, C., and Faghri, A., 2000, "Transient Gas-Loaded Rotating Heat Pipes," Proceedings of the 15th National and 4th ISHMT/ASME Heat and Mass Transfer Conference, Pune, India.
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- ↑ Launay, S., Platel, V., Dutour, S., and Joly, J. L., 2007b, "Transient Modeling of Loop Heat Pipes for the Oscillating Behavior Study," Journal of Thermophysics and Heat Transfer, 21(3), 487-495. http://dx.doi.org/10.2514/1.26854
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- ↑ 130.0 130.1 130.2 Harley, C., and Faghri, A., 1994c, "Transient Two-Dimensional Gas-Loaded Heat Pipe Analysis," Journal of Heat Transfer, 116(3), 716-723. http://dx.doi.org/10.1115/1.2910927
- ↑ 131.0 131.1 Ponnappan, R., 1989, "Studies on the Startup Transients and Performance of a Gas Loaded Sodium Heat Pipe," Technical Report, WRDC-TR-89-2046.
- ↑ Faghri, A., and Harley, C., 1994, "Transient Lumped Heat Pipe Analyses," Heat Recovery Systems and CHP, 14(4), 351-363. http://dx.doi.org/10.1016/0890-4332(94)90039-6