# Condensation Removal by Capillary Action

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- | In a zero gravity environment, capillary action is one mechanism of condensate removal. Heat pipes fall under the category of capillary driven devices. Gas-loaded heat pipes have been applied in many diverse fields, and are useful when the temperature of a device must be held constant while a variable heat load is dissipated. In this section, a noncondensable gas-loaded heat pipe modeled by Harley and Faghri (1994) is presented below where the effect of capillary and noncondensable action is applied simultaneously. | + | In a zero gravity environment, capillary action is one mechanism of condensate removal. Heat pipes fall under the category of capillary driven devices. Gas-loaded heat pipes have been applied in many diverse fields, and are useful when the temperature of a device must be held constant while a variable heat load is dissipated. In this section, a noncondensable gas-loaded heat pipe modeled by [[#References|Harley and Faghri (1994)]] is presented below where the effect of capillary and noncondensable action is applied simultaneously. |

+ | |||

The physical configuration and coordinate system of the gas-loaded heat pipe studied is shown in Fig. 8.27. Gas-loaded heat pipes 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 noncondensable gas in the condenser, increasing the surface area available for heat transfer, which maintains a nearly constant heat flux and temperature. | The physical configuration and coordinate system of the gas-loaded heat pipe studied is shown in Fig. 8.27. Gas-loaded heat pipes 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 noncondensable gas in the condenser, increasing the surface area available for heat transfer, which maintains a nearly constant heat flux and temperature. | ||

- | Vapor Space | + | |

+ | ==Vapor Space== | ||

+ | |||

The conservation equations for transient, compressible, two-species flow for mass, momentum, energy, and species in vapor space are as follows: | The conservation equations for transient, compressible, two-species flow for mass, momentum, energy, and species in vapor space are as follows: | ||

- | <center><math> | + | |

- | \frac | + | <center><math>\frac{\partial \bar{\rho }}{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}\left( \bar{\rho }rv \right)+\frac{\partial }{\partial z}\left( \bar{\rho }w \right)=0</math> |

- | </math | + | (8.330)</center> |

- | (8.330) | + | |

- | <center><math> | + | <center><math>\bar{\rho }\frac{DV}{Dt}=-\nabla \bar{p}+\frac{1}{3}\bar{\mu }\nabla \left( \nabla \cdot V \right)+\bar{\mu }{{\nabla }^{2}}V</math> |

- | \bar \rho \frac | + | (8.331)</center> |

- | </math | + | |

- | (8.331) | + | <center><math>\rho {{c}_{p}}\frac{DT}{Dt}-\nabla \cdot \bar{k}\nabla T-\nabla \cdot \left( \sum\limits_{j=1}^{2}{{{D}_{d}}{{c}_{pj}}T\nabla {{\rho }_{j}}} \right)-\frac{D\bar{p}}{Dt}-\bar{\mu }\Phi =0</math> |

- | <center><math> | + | (8.332)</center> |

- | \rho | + | |

- | </math | + | |

- | (8.332) | + | |

where the subscript j denotes either vapor (v) or gas (g). | where the subscript j denotes either vapor (v) or gas (g). | ||

- | <center><math> | + | |

- | \frac | + | <center><math>\frac{D{{\rho }_{g}}}{Dt}-\nabla \cdot {{D}_{gv}}\nabla {{\rho }_{g}}=0</math> |

- | </math | + | (8.333)</center> |

- | (8.333) | + | |

and | and | ||

- | <center><math> | + | |

- | \Phi | + | <center><math>\Phi =2\left[ {{\left( \frac{\partial v}{\partial r} \right)}^{2}}+{{\left( \frac{v}{r} \right)}^{2}}+{{\left( \frac{\partial w}{\partial z} \right)}^{2}} \right]+{{\left( \frac{\partial v}{\partial z}+\frac{\partial w}{\partial r} \right)}^{2}}-\frac{2}{3}{{\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( rv \right)+\frac{\partial w}{\partial z} \right]}^{2}}</math> |

- | </math | + | |

- | (8.334) | + | (8.334)</center> |

- | Furthermore, v and w are the radial and axial vapor velocities, | + | |

- | \bar p | + | Furthermore, ''v'' and ''w'' are the radial and axial vapor velocities, <math>\bar{p}</math> is the total mixture pressure, <math>\bar{\mu }</math> is the mass-fraction-weighted mixture viscosity, <math>{{\bar{c}}_{p}}</math> is the specific heat of the mixture, <math>\bar{k}</math> is the thermal conductivity of the mixture, Dd is the self-diffusion coefficient for both vapor and gas species, D<sub>gv</sub> is the mass diffusion coefficient of the vapor-gas pair, ρg is the density of the noncondensable gas, and the mixture density is <math>\bar{\rho }={{\rho }_{g}}+{{\rho }_{v}}</math>. The partial gas density is determined from the species equation, and the vapor density is found from the ideal gas relation using the partial vapor pressure. |

- | </math> | + | |

- | + | The two choices in species conservation formulation are mass and molar fraction. Molar fraction offers the possibility of a direct simplification in the formulation by the assumption of constant molar density. The assumption is valid over a wider range of temperature and pressure than the corresponding assumption of constant mass density. However, when molar fractions are used, the momentum equation must be written in terms of molar-weighted velocities. The resulting equation cannot be written in terms of the total material derivative, and is significantly more difficult to solve. | |

- | \bar \mu | + | |

- | </math> | + | |

- | + | ||

- | \bar | + | |

- | </math> | + | |

- | + | ||

- | \bar k | + | |

- | </math> | + | |

- | + | ||

- | \bar \rho | + | |

- | </math> | + | |

- | . The partial gas density is determined from the species equation, and the vapor density is found from the ideal gas relation using the partial vapor pressure. | + | |

- | + | ||

A benefit of the general equation formulation is its allowance for variable properties. Typical of compressible gas applications, the density is related to the temperature and pressure through the equation of state | A benefit of the general equation formulation is its allowance for variable properties. Typical of compressible gas applications, the density is related to the temperature and pressure through the equation of state | ||

- | <center><math> | + | |

- | \bar p = \frac | + | <center><math>\bar{p}=\frac{\bar{\rho }{{\text{R}}_{\text{u}}}T}{{\bar{M}}}</math> |

- | </math | + | (8.335)</center> |

- | (8.335) | + | |

- | + | where <math>\bar{M}</math> is the molecular weight of the vapor-gas mixture and R<sub>u</sub> is the universal gas constant. | |

- | \bar M | + | |

- | </math> | + | In the species equation, ''D<sub>gv</sub>''is a function of pressure and temperature. For a vapor-gas mixture of sodium-argon, [[#References|Harley and Faghri (1994)]] used the following relationship for D<sub>gv</sub> |

- | + | ||

- | In the species equation, | + | <center><math>{{D}_{gv}}=1.3265\times {{10}^{-3}}{{T}^{3/2}}{{\left( {\bar{p}} \right)}^{-1}}</math> |

- | <center><math> | + | (8.336)</center> |

- | + | ||

- | </math | + | where ''T'' is in degrees Kelvin, <math>\bar{p}</math> is in N/m2, and D<sub>gv</sub> is in m2/s. Following a similar procedure, the variable diffusion coefficient formulation for the sodium-helium pair is |

- | (8.336) | + | |

- | where T is in degrees Kelvin, | + | <center><math>{{D}_{gv}}=1.2795\times {{10}^{-3}}{{T}^{3/2}}{{\left( {\bar{p}} \right)}^{-1}}</math> |

- | \bar p | + | (8.337)</center> |

- | </math> | + | |

- | + | The interspecies heat transfer that occurs through the vapor-gas mass diffusion was modeled with a self-diffusion model. In the present model, however, the self-diffusion coefficient, D<sub>d</sub>, was assumed constant at the initial temperature of the heat pipe. | |

- | <center><math> | + | |

- | + | ==Wick== | |

- | </math | + | |

- | (8.337) | + | The solid structure in the wick is saturated with the working fluid. The condensate is pumped to the evaporator through capillary action. The liquid velocity is taken to be the intrinsic phase-averaged velocity <math>{{\left\langle {{v}_{\ell }} \right\rangle }^{\ell }}</math> through the porous wick, which is assumed to be isotropic and homogeneous. For simplicity, <math>{{\left\langle {} \right\rangle }^{\ell }}</math> is dropped for velocity. |

- | The interspecies heat transfer that occurs through the vapor-gas mass diffusion was modeled with a self-diffusion model. In the present model, however, the self-diffusion coefficient, | + | |

- | Wick | + | |

- | The solid structure in the wick is saturated with the working fluid. The condensate is pumped to the evaporator through capillary action. The liquid velocity is taken to be the intrinsic phase-averaged velocity | + | |

- | \left\langle { | + | |

- | </math> | + | |

- | + | ||

- | \left\langle {} \right\rangle ^\ell | + | |

- | </math> | + | |

- | + | ||

Furthermore, the working fluid and wick structure are assumed to be in local thermal equilibrium. The continuity, momentum, and energy equations for the liquid saturated wick are | Furthermore, the working fluid and wick structure are assumed to be in local thermal equilibrium. The continuity, momentum, and energy equations for the liquid saturated wick are | ||

- | <center><math> | + | |

- | \frac{1}{r}\frac{\partial } | + | <center><math>\frac{1}{r}\frac{\partial }{\partial r}\left( r{{v}_{\ell }} \right)+\frac{\partial {{w}_{\ell }}}{\partial z}=0</math> |

- | </math | + | (8.338)</center> |

- | (8.338) | + | |

- | <center><math> | + | <center><math>\begin{align} |

- | \begin{ | + | & \frac{1}{\varepsilon }\frac{\partial {{v}_{\ell }}}{\partial t}+\frac{1}{{{\varepsilon }^{2}}}\left( {{v}_{\ell }}\frac{\partial {{v}_{\ell }}}{\partial r}+{{w}_{\ell }}\frac{\partial {{v}_{\ell }}}{\partial z} \right) \\ |

- | + | & =-\frac{1}{{{\rho }_{\ell }}}\frac{\partial {{p}_{\ell }}}{\partial r}-\frac{{{v}_{\ell }}{{v}_{\ell }}}{K}+\frac{{{\nu }_{\ell }}}{\varepsilon }\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{v}_{\ell }}}{\partial r} \right)-\frac{{{v}_{\ell }}}{{{r}^{2}}}+\frac{{{\partial }^{2}}{{v}_{\ell }}}{\partial {{z}^{2}}} \right] \\ | |

- | + | \end{align}</math> | |

- | + | ||

- | </math | + | |

(8.339) | (8.339) | ||

- | + | ||

- | \begin{ | + | <math>\begin{align} |

- | + | & \frac{1}{\varepsilon }\frac{\partial {{w}_{\ell }}}{\partial t}+\frac{1}{{{\varepsilon }^{2}}}\left( {{v}_{\ell }}\frac{\partial {{w}_{\ell }}}{\partial r}+{{w}_{\ell }}\frac{\partial {{w}_{\ell }}}{\partial z} \right) \\ | |

- | + | & =-\frac{1}{{{\rho }_{\ell }}}\frac{\partial {{p}_{\ell }}}{\partial r}-\frac{{{v}_{\ell }}{{w}_{\ell }}}{K}+\frac{{{\nu }_{\ell }}}{\varepsilon }\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{w}_{\ell }}}{\partial r} \right)-\frac{{{w}_{\ell }}}{{{r}^{2}}}+\frac{{{\partial }^{2}}{{v}_{\ell }}}{\partial {{z}^{2}}} \right] \\ | |

- | + | \end{align}</math> | |

- | </math | + | (8.340)</center> |

- | (8.340) | + | |

- | <center><math> | + | <center><math>{{\left( \rho {{c}_{p}} \right)}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial t}+{{v}_{\ell }}\frac{\partial {{T}_{l}}}{\partial r}+{{w}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial z}=\frac{1}{r}\frac{\partial }{\partial r}\left( r{{k}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial r} \right)+\frac{\partial }{\partial z}\left( {{k}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial z} \right)</math> |

- | \left( | + | (8.341)</center> |

- | </math | + | |

- | (8.341) | + | |

where ε is the porosity of the wick and K is the permeability. | where ε is the porosity of the wick and K is the permeability. | ||

- | Wall | + | |

+ | ==Wall== | ||

+ | |||

In the heat pipe wall, heat transfer is described by the transient two-dimensional conduction equation | In the heat pipe wall, heat transfer is described by the transient two-dimensional conduction equation | ||

- | <center><math> | + | |

- | \rho | + | <center><math>{{\rho }_{w}}{{c}_{pw}}\frac{\partial T}{\partial t}={{k}_{w}}\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right)+\frac{{{\partial }^{2}}T}{\partial {{z}^{2}}} \right]</math> |

- | + | (8.342)</center> | |

- | (8.342) | + | |

where the subscript w denotes the heat pipe wall material. | where the subscript w denotes the heat pipe wall material. | ||

- | Boundary Conditions | + | |

+ | ==Boundary Conditions== | ||

+ | |||

At the end caps of the heat pipe, the no-slip condition for velocity, the adiabatic conduction for temperature, and the overall gas conservation conditions are imposed | At the end caps of the heat pipe, the no-slip condition for velocity, the adiabatic conduction for temperature, and the overall gas conservation conditions are imposed | ||

- | <center><math> | + | |

- | v = w = \frac | + | <center><math>v=w=\frac{\partial T}{\partial z}=\frac{\partial {{\rho }_{g}}}{\partial z}=0,\text{ }z=0</math> |

- | </math | + | (8.343)</center> |

- | (8.343) | + | |

- | <center><math> | + | <center><math>v=w=\frac{\partial T}{\partial z}=0,{{\rho }_{g}}={{\rho }_{g,BC}},\text{ }z=L</math> |

- | v = w = \frac | + | (8.344)</center> |

- | </math | + | |

- | (8.344) | + | where ρ<sub>g</sub>,''BC'' is iteratively adjusted to satisfy overall conservation of noncondensable gas. This boundary condition is implemented through the calculation of the total mass of noncondensable gas. This boundary condition is implemented through the calculation of the total mass of noncondensable gas in the heat pipe. If the total mass is found to be less than the mass initially present in the pipe, the boundary value is increased by 10% of the previous value. Conversely, if the calculated mass is larger than that initially present, the boundary value is decreased. This ensures the conservation of the overall mass to within a preset tolerance, which is 1% in the present formulation. The symmetry of the cylindrical heat pipe requires that the radial vapor velocity and the gradients of the axial vapor velocity, temperature, and gas density be zero at the centerline: |

- | where | + | |

- | <center><math> | + | <center><math>v=\frac{\partial w}{\partial r}=\frac{\partial T}{\partial r}=\frac{\partial {{\rho }_{g}}}{\partial r}=0,\text{ }r=0</math> |

- | v = \frac | + | (8.345)</center> |

- | </math | + | |

- | (8.345) | + | The liquid-vapor interface <math>\left( r={{R}_{v}} \right)</math> is impermeable to the noncondensable gas |

- | The liquid-vapor interface | + | |

- | \left( | + | <center><math>{{\dot{m}}_{g}}={{S}_{\delta }}A{{D}_{gv}}\nabla {{\rho }_{g}}+{{\rho }_{g}}{{S}_{\delta }}V=0,\text{ }r={{R}_{v}}</math> |

- | </math> | + | (8.346)</center> |

- | + | ||

- | <center><math> | + | where <math>{{\dot{m}}_{ & g}}</math> is the mass flow rate of gas, A is the cross-sectional area of the heat pipe and Sδ is the surface area of the liquid-vapor interface. This formulation of <math>{{\dot{m}}_{g}}</math> accounts for both the convective and diffusive noncondensable gas mass fluxes at the liquid-vapor interface. To ensure saturation conditions in the evaporator section (and part of the adiabatic section since the exact transition point is determined iteratively), the Clausius-Clapeyron equation is used to determine the interface temperature as a function of pressure. The interface radial velocity is then found through the evaporation rate required to satisfy heat transfer requirements. The no-slip condition is still in effect for the axial velocity component. |

- | \dot | + | |

- | </math | + | At <math>r={{R}_{v}}</math> for <math>z\le {{L}_{e}}+{{L}_{a}}</math>: |

- | (8.346) | + | |

- | + | <center><math>{{T}_{\text{sat}}}={{\left( \frac{1}{{{T}_{0}}}-\frac{{{\text{R}}_{\text{u}}}}{{{M}_{v}}{{h}_{\ell v}}}\ln \frac{{{p}_{v}}}{{{p}_{0}}} \right)}^{-1}}</math> | |

- | \dot | + | (8.347)</center> |

- | </math> | + | |

- | + | <center><math>{{v}_{\delta }}=\frac{\left( {{k}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial r}-{{{\bar{k}}}_{\delta }}\frac{\partial {{T}_{v}}}{\partial r} \right)}{\left( {{h}_{\ell v}}+{{{\bar{c}}}_{p\delta }}{{T}_{\text{sat}}} \right){{{\bar{\rho }}}_{\delta }}}</math> | |

- | \dot | + | (8.348)</center> |

- | </math> | + | |

- | + | <center><math>w=0</math> | |

- | At | + | (8.349)</center> |

- | r = | + | |

- | </math> | + | where <math>{{\bar{k}}_{\delta }}</math>, <math>{{\bar{c}}_{p\delta }}</math> and |

- | + | <math>{{\bar{\rho }}_{\delta }}</math> are the vapor-gas mixture properties at the liquid-vapor interface. In eq. (8.347), the saturation temperature of the vapor is found from the partial vapor pressure. A solution of the momentum equation gives the total mixture pressure, but the partial vapor pressure can be found using the local gas density: | |

- | z \le | + | |

- | </math> | + | <center><math>{{p}_{v}}=\frac{{\bar{M}}}{{{M}_{v}}}\bar{p}\left( 1-\frac{{{\rho }_{g}}}{{\bar{\rho }}} \right)</math> |

- | : | + | (8.350)</center> |

- | <center><math> | + | |

- | + | ||

- | </math | + | |

- | (8.347) | + | |

- | <center><math> | + | |

- | + | ||

- | </math | + | |

- | (8.348) | + | |

- | <center><math> | + | |

- | w = 0 | + | |

- | </math | + | |

- | (8.349) | + | |

- | + | ||

- | \bar | + | |

- | </math> | + | |

- | , | + | |

- | \bar | + | |

- | </math> | + | |

- | + | ||

- | \bar \rho _\delta | + | |

- | </math> | + | |

- | + | ||

- | <center><math> | + | |

- | + | ||

- | </math | + | |

- | (8.350) | + | |

which was derived assuming a mixture of ideal gases following Dalton’s model for mixtures. | which was derived assuming a mixture of ideal gases following Dalton’s model for mixtures. | ||

+ | |||

At the liquid-vapor interface in the active portions of the condenser section, vapor condenses and releases its latent heat energy. This process is simulated by applying a heat source at the interface grids in the condenser section. The interface velocity can be obtained through a mass balance between the evaporator and condenser section, allowing for inactive sections of the condenser. | At the liquid-vapor interface in the active portions of the condenser section, vapor condenses and releases its latent heat energy. This process is simulated by applying a heat source at the interface grids in the condenser section. The interface velocity can be obtained through a mass balance between the evaporator and condenser section, allowing for inactive sections of the condenser. | ||

- | At | + | |

- | r = | + | At <math>r={{R}_{v}}</math> for <math>z>{{L}_{e}}+{{L}_{a}}</math>: |

- | </math> | + | |

- | + | <center><math>{{q}_{so}}=-\left( {{h}_{\ell v}}+{{{\bar{c}}}_{p\delta }}{{T}_{\delta }} \right)\left( \bar{\rho }-{{\rho }_{g}} \right){{v}_{g}}</math> | |

- | z > | + | (8.351)</center> |

- | </math> | + | |

- | : | + | |

- | <center><math> | + | |

- | + | ||

- | </math | + | |

- | (8.351) | + | |

Due to the conjugate nature of the solution procedure, the boundary condition between the wick and the wall is automatically satisfied. In addition to the equality of temperature, this condition requires the equality of the heat fluxes into and out of the wick-wall interface: | Due to the conjugate nature of the solution procedure, the boundary condition between the wick and the wall is automatically satisfied. In addition to the equality of temperature, this condition requires the equality of the heat fluxes into and out of the wick-wall interface: | ||

- | <center><math> | + | |

- | + | <center><math>{{k}_{w}}\frac{\partial T}{\partial r}={{k}_{\text{eff}}}\frac{\partial T}{\partial r},\text{ }r={{R}_{w}}</math> | |

- | </math | + | (8.352)</center> |

- | (8.352) | + | |

At the outer pipe wall surface, the boundary conditions depend on both the axial position and the mechanism of heat transfer being studied. In the evaporator, a constant heat flux is specified. In the condenser, a radiative boundary condition is imposed. | At the outer pipe wall surface, the boundary conditions depend on both the axial position and the mechanism of heat transfer being studied. In the evaporator, a constant heat flux is specified. In the condenser, a radiative boundary condition is imposed. | ||

- | <center><math> | + | |

- | + | <center><math>{{k}_{w}}{{\left. \frac{\partial T}{\partial r} \right|}_{r={{R}_{O}}}}=\left\{ \begin{matrix} | |

- | {q'' | + | {{{{q}''}}_{e}}\text{ evaporator} \\ |

- | + | 0\text{ adiabatic} \\ | |

- | + | \sigma {{\varepsilon }_{r}}\left( T_{w}^{4}-T_{\infty }^{4} \right)\text{ condenser} \\ | |

- | \end{ | + | \end{matrix} \right.</math> |

- | </math | + | (8.353)</center> |

- | (8.353) | + | |

where σ is the Stefan-Boltzman constant and εr is the emissivity. | where σ is the Stefan-Boltzman constant and εr is the emissivity. | ||

- | Initial Conditions | + | |

+ | ==Initial Conditions== | ||

+ | |||

There is no motion of either the gas or vapor, and the noncondensable gas is evenly distributed throughout the vapor space by diffusion. The initial temperature of the heat pipe is above the free-molecular/continuum-flow transition temperature for the specific heat pipe vapor diameter. | There is no motion of either the gas or vapor, and the noncondensable gas is evenly distributed throughout the vapor space by diffusion. The initial temperature of the heat pipe is above the free-molecular/continuum-flow transition temperature for the specific heat pipe vapor diameter. | ||

- | Figure 8.28 Temperature profiles for the gas-loaded heat pipe with | + | Figure 8.28 Temperature profiles for the gas-loaded heat pipe with ''Q<sub>in</sub>'' = 451W: (a) transient wall temperature profile; (b) transient centerline temperature profile ([[#References|Harley and Faghri, 1994]]). |

+ | |||

+ | The gas-loaded heat pipe experimentally studied by [[#References|Ponnappan (1989)]] was simulated using the above analysis, with results shown in Fig. 8.28. It can be seen that the wall and vapor temperatures decreased significantly in the condenser section due to the presence of the noncondensable gas. | ||

- | + | ==References== |

## Revision as of 17:08, 31 May 2010

In a zero gravity environment, capillary action is one mechanism of condensate removal. Heat pipes fall under the category of capillary driven devices. Gas-loaded heat pipes have been applied in many diverse fields, and are useful when the temperature of a device must be held constant while a variable heat load is dissipated. In this section, a noncondensable gas-loaded heat pipe modeled by Harley and Faghri (1994) is presented below where the effect of capillary and noncondensable action is applied simultaneously.

The physical configuration and coordinate system of the gas-loaded heat pipe studied is shown in Fig. 8.27. Gas-loaded heat pipes 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 noncondensable gas in the condenser, increasing the surface area available for heat transfer, which maintains a nearly constant heat flux and temperature.

## Contents |

## Vapor Space

The conservation equations for transient, compressible, two-species flow for mass, momentum, energy, and species in vapor space are as follows:

where the subscript j denotes either vapor (v) or gas (g).

and

Furthermore, *v* and *w* are the radial and axial vapor velocities, is the total mixture pressure, is the mass-fraction-weighted mixture viscosity, is the specific heat of the mixture, is the thermal conductivity of the mixture, Dd is the self-diffusion coefficient for both vapor and gas species, D_{gv} is the mass diffusion coefficient of the vapor-gas pair, ρg is the density of the noncondensable gas, and the mixture density is . The partial gas density is determined from the species equation, and the vapor density is found from the ideal gas relation using the partial vapor pressure.

The two choices in species conservation formulation are mass and molar fraction. Molar fraction offers the possibility of a direct simplification in the formulation by the assumption of constant molar density. The assumption is valid over a wider range of temperature and pressure than the corresponding assumption of constant mass density. However, when molar fractions are used, the momentum equation must be written in terms of molar-weighted velocities. The resulting equation cannot be written in terms of the total material derivative, and is significantly more difficult to solve. A benefit of the general equation formulation is its allowance for variable properties. Typical of compressible gas applications, the density is related to the temperature and pressure through the equation of state

where is the molecular weight of the vapor-gas mixture and R_{u} is the universal gas constant.

In the species equation, *D _{gv}*is a function of pressure and temperature. For a vapor-gas mixture of sodium-argon, Harley and Faghri (1994) used the following relationship for D

_{gv}

where *T* is in degrees Kelvin, is in N/m2, and D_{gv} is in m2/s. Following a similar procedure, the variable diffusion coefficient formulation for the sodium-helium pair is

The interspecies heat transfer that occurs through the vapor-gas mass diffusion was modeled with a self-diffusion model. In the present model, however, the self-diffusion coefficient, D_{d}, was assumed constant at the initial temperature of the heat pipe.

## Wick

The solid structure in the wick is saturated with the working fluid. The condensate is pumped to the evaporator through capillary action. The liquid velocity is taken to be the intrinsic phase-averaged velocity through the porous wick, which is assumed to be isotropic and homogeneous. For simplicity, is dropped for velocity. Furthermore, the working fluid and wick structure are assumed to be in local thermal equilibrium. The continuity, momentum, and energy equations for the liquid saturated wick are

(8.339)

(8.340)

where ε is the porosity of the wick and K is the permeability.

## Wall

In the heat pipe wall, heat transfer is described by the transient two-dimensional conduction equation

where the subscript w denotes the heat pipe wall material.

## Boundary Conditions

At the end caps of the heat pipe, the no-slip condition for velocity, the adiabatic conduction for temperature, and the overall gas conservation conditions are imposed

where ρ_{g},*BC* is iteratively adjusted to satisfy overall conservation of noncondensable gas. This boundary condition is implemented through the calculation of the total mass of noncondensable gas. This boundary condition is implemented through the calculation of the total mass of noncondensable gas in the heat pipe. If the total mass is found to be less than the mass initially present in the pipe, the boundary value is increased by 10% of the previous value. Conversely, if the calculated mass is larger than that initially present, the boundary value is decreased. This ensures the conservation of the overall mass to within a preset tolerance, which is 1% in the present formulation. The symmetry of the cylindrical heat pipe requires that the radial vapor velocity and the gradients of the axial vapor velocity, temperature, and gas density be zero at the centerline:

The liquid-vapor interface is impermeable to the noncondensable gas

where **Failed to parse (syntax error): {{\dot{m}}_{ & g}}**

is the mass flow rate of gas, A is the cross-sectional area of the heat pipe and Sδ is the surface area of the liquid-vapor interface. This formulation of accounts for both the convective and diffusive noncondensable gas mass fluxes at the liquid-vapor interface. To ensure saturation conditions in the evaporator section (and part of the adiabatic section since the exact transition point is determined iteratively), the Clausius-Clapeyron equation is used to determine the interface temperature as a function of pressure. The interface radial velocity is then found through the evaporation rate required to satisfy heat transfer requirements. The no-slip condition is still in effect for the axial velocity component.

At *r* = *R*_{v} for :

*w*= 0 (8.349)

where , and are the vapor-gas mixture properties at the liquid-vapor interface. In eq. (8.347), the saturation temperature of the vapor is found from the partial vapor pressure. A solution of the momentum equation gives the total mixture pressure, but the partial vapor pressure can be found using the local gas density:

which was derived assuming a mixture of ideal gases following Dalton’s model for mixtures.

At the liquid-vapor interface in the active portions of the condenser section, vapor condenses and releases its latent heat energy. This process is simulated by applying a heat source at the interface grids in the condenser section. The interface velocity can be obtained through a mass balance between the evaporator and condenser section, allowing for inactive sections of the condenser.

At *r* = *R*_{v} for *z* > *L*_{e} + *L*_{a}:

Due to the conjugate nature of the solution procedure, the boundary condition between the wick and the wall is automatically satisfied. In addition to the equality of temperature, this condition requires the equality of the heat fluxes into and out of the wick-wall interface:

At the outer pipe wall surface, the boundary conditions depend on both the axial position and the mechanism of heat transfer being studied. In the evaporator, a constant heat flux is specified. In the condenser, a radiative boundary condition is imposed.

where σ is the Stefan-Boltzman constant and εr is the emissivity.

## Initial Conditions

There is no motion of either the gas or vapor, and the noncondensable gas is evenly distributed throughout the vapor space by diffusion. The initial temperature of the heat pipe is above the free-molecular/continuum-flow transition temperature for the specific heat pipe vapor diameter.

Figure 8.28 Temperature profiles for the gas-loaded heat pipe with *Q _{in}* = 451W: (a) transient wall temperature profile; (b) transient centerline temperature profile (Harley and Faghri, 1994).

The gas-loaded heat pipe experimentally studied by Ponnappan (1989) was simulated using the above analysis, with results shown in Fig. 8.28. It can be seen that the wall and vapor temperatures decreased significantly in the condenser section due to the presence of the noncondensable gas.