# Condensation Removal by Capillary Action

(Difference between revisions)
$+ - \frac{{\partial \bar \rho }}{{\partial t}} + \frac{1}{r}\frac{\partial }{{\partial r}}\left( {\bar \rho rv} \right) + \frac{\partial }{{\partial z}}\left( {\bar \rho w} \right) = 0 + {| class="wikitable" border="0" -$
+ |- - (8.330) + | width="100%" | -
$+ [itex]\frac{\partial \bar{\rho }}{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}\left( \bar{\rho }rv \right)+\frac{\partial }{\partial z}\left( \bar{\rho }w \right)=0$
- \bar \rho \frac{{DV}}{{Dt}} = - \nabla \bar p + \frac{1}{3}\bar \mu \nabla \left( {\nabla \cdot V} \right) + \bar \mu \nabla ^2 V + |{{EquationRef|(1)}} - [/itex]
+ |} - (8.331) + -
$+ {| class="wikitable" border="0" - \rho c_p \frac{{DT}}{{Dt}} - \nabla \cdot \bar k\nabla T - \nabla \cdot \left( {\sum\limits_{j = 1}^2 {D_d c_{pj} T\nabla \rho _j } } \right) - \frac{{D\bar p}}{{Dt}} - \bar \mu \Phi = 0 + |- -$
+ | width="100%" | - (8.332) +
$\bar{\rho }\frac{DV}{Dt}=-\nabla \bar{p}+\frac{1}{3}\bar{\mu }\nabla \left( \nabla \cdot V \right)+\bar{\mu }{{\nabla }^{2}}V$
+ |{{EquationRef|(2)}} + |} + + {| class="wikitable" border="0" + |- + | width="100%" | +
$\rho {{c}_{p}}\frac{DT}{Dt}-\nabla \cdot \bar{k}\nabla T-\nabla \cdot \left( \sum\limits_{j=1}^{2}{{{D}_{d}}{{c}_{pj}}T\nabla {{\rho }_{j}}} \right)-\frac{D\bar{p}}{Dt}-\bar{\mu }\Phi =0$
+ |{{EquationRef|(3)}} + |} + where the subscript j denotes either vapor (v) or gas (g). where the subscript j denotes either vapor (v) or gas (g). -
$+ - \frac{{D\rho _g }}{{Dt}} - \nabla \cdot D_{gv} \nabla \rho _g = 0 + {| class="wikitable" border="0" -$
+ |- - (8.333) + | width="100%" | - +
$\frac{D{{\rho }_{g}}}{Dt}-\nabla \cdot {{D}_{gv}}\nabla {{\rho }_{g}}=0$
+ |{{EquationRef|(4)}} + |} + and and -
$- \Phi = 2\left [itex] {\left( {\frac{{\partial v}}{{\partial r}}} \right)^2 + \left( {\frac{v}{r}} \right)^2 + \left( {\frac{{\partial w}}{{\partial z}}} \right)^2 } \right$
+ \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 - [/itex]
- (8.334) + {| class="wikitable" border="0" - Furthermore, v and w are the radial and axial vapor velocities,
$+ |- - \bar p + | width="100%" | -$
+
$\Phi =2\left[ {{\left( \frac{\partial v}{\partial r} \right)}^{2}}+{{\left( \frac{v}{r} \right)}^{2}}+{{\left( \frac{\partial w}{\partial z} \right)}^{2}} \right]+{{\left( \frac{\partial v}{\partial z}+\frac{\partial w}{\partial r} \right)}^{2}}-\frac{2}{3}{{\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( rv \right)+\frac{\partial w}{\partial z} \right]}^{2}}$
- is the total mixture pressure,
$+ |{{EquationRef|(5)}} - \bar \mu + |} -$
+ - is the mass-fraction-weighted mixture viscosity,
$+ Furthermore, ''v'' and ''w'' are the radial and axial vapor velocities, [itex]\bar{p}$ is the total mixture pressure, $\bar{\mu }$ is the mass-fraction-weighted mixture viscosity, ${{\bar{c}}_{p}}$ is the specific heat of the mixture, $\bar{k}$ is the thermal conductivity of the mixture, Dd is the self-diffusion coefficient for both vapor and gas species, Dgv is the mass diffusion coefficient of the vapor-gas pair, ρg is the density of the noncondensable gas, and the mixture density is $\bar{\rho }={{\rho }_{g}}+{{\rho }_{v}}$.  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. - \bar c_p + - [/itex]
+ 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. - is the specific heat of the mixture,
$+ - \bar k + -$
+ - is the thermal conductivity of the mixture, Dd is the self-diffusion coefficient for both vapor and gas species, Dgv is the mass diffusion coefficient of the vapor-gas pair, ρg is the density of the noncondensable gas, and the mixture density is
$+ - \bar \rho = \rho _g + \rho _v + -$
+ - .  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 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 -
$+ - \bar p = \frac{{\bar \rho {\rm{R}}_{\rm{u}} T}}{{\bar M}} + {| class="wikitable" border="0" -$
+ |- - (8.335) + | width="100%" | - where
$+ [itex]\bar{p}=\frac{\bar{\rho }{{\text{R}}_{\text{u}}}T}{{\bar{M}}}$
- \bar M + |{{EquationRef|(6)}} - [/itex]
+ |} - is the molecular weight of the vapor-gas mixture and Ru is the universal gas constant. + - In the species equation, Dgv is a function of pressure and temperature.  For a vapor-gas mixture of sodium-argon, Harley and Faghri (1994) used the following relationship for Dgv + where $\bar{M}$ is the molecular weight of the vapor-gas mixture and Ru is the universal gas constant. -
$+ - D_{gv} = 1.3265 \times 10^{ - 3} T^{3/2} \left( {\bar p} \right)^{ - 1} + In the species equation, ''Dgv''is a function of pressure and temperature. For a vapor-gas mixture of sodium-argon, [[#References|Harley and Faghri (1994)]] used the following relationship for Dgv -$
+ - (8.336) + {| class="wikitable" border="0" - where T is in degrees Kelvin,
$+ |- - \bar p + | width="100%" | -$
+
${{D}_{gv}}=1.3265\times {{10}^{-3}}{{T}^{3/2}}{{\left( {\bar{p}} \right)}^{-1}}$
- is in N/m2, and Dgv is in m2/s.  Following a similar procedure, the variable diffusion coefficient formulation for the sodium-helium pair is + |{{EquationRef|(7)}} -
$+ |} - D_{gv} = 1.2795 \times 10^{ - 3} T^{3/2} \left( {\bar p} \right)^{ - 1} + -$
+ where ''T'' is in degrees Kelvin, $\bar{p}$ is in N/m2, and Dgv is in m2/s.  Following a similar procedure, the variable diffusion coefficient formulation for the sodium-helium pair is - (8.337) + - 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, Dd, was assumed constant at the initial temperature of the heat pipe. + {| class="wikitable" border="0" - 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
$+ | width="100%" | - \left\langle {v_\ell } \right\rangle ^\ell + [itex]{{D}_{gv}}=1.2795\times {{10}^{-3}}{{T}^{3/2}}{{\left( {\bar{p}} \right)}^{-1}}$
- [/itex]
+ |{{EquationRef|(8)}} - through the porous wick, which is assumed to be isotropic and homogeneous.  For simplicity,
$+ |} - \left\langle {} \right\rangle ^\ell + -$
+ 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, Dd, was assumed constant at the initial temperature of the heat pipe. - is dropped for velocity. + + '''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 ${{\left\langle {{v}_{\ell }} \right\rangle }^{\ell }}$ through the porous wick, which is assumed to be isotropic and homogeneous.  For simplicity, ${{\left\langle {} \right\rangle }^{\ell }}$ 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 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 -
$+ - \frac{1}{r}\frac{\partial }{{\partial r}}\left( {rv_\ell } \right) + \frac{{\partial w_\ell }}{{\partial z}} = 0 + {| class="wikitable" border="0" -$
+ |- - (8.338) + | width="100%" | -
$+ [itex]\frac{1}{r}\frac{\partial }{\partial r}\left( r{{v}_{\ell }} \right)+\frac{\partial {{w}_{\ell }}}{\partial z}=0$
- \begin{array}{l} + |{{EquationRef|(9)}} - \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{array} + {| class="wikitable" border="0" - [/itex]
+ |- - (8.339) + | width="100%" | -
+ [itex]\begin{align} - \begin{array}{l} + & \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}{\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 }}{{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] \\ - = - \frac{1}{{\rho _\ell }}\frac{{\partial p_\ell }}{{\partial r}} - \frac{{v_\ell w_\ell }}{K} + \frac{{\nu _\ell }}{\varepsilon }\left [itex] {\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}[/itex]
- \end{array} + |{{EquationRef|(10)}} - [/itex]
+ |} - (8.340) + -
$+ {| class="wikitable" border="0" - \left( {\rho c_p } \right)_{{\rm{eff}}} \frac{{\partial T_\ell }}{{\partial t}} + v_\ell \frac{{\partial T_l }}{{\partial r}} + w_\ell \frac{{\partial T_\ell }}{{\partial z}} = \frac{1}{r}\frac{\partial }{{\partial r}}\left( {rk_{{\rm{eff}}} \frac{{\partial T_\ell }}{{\partial r}}} \right) + \frac{\partial }{{\partial z}}\left( {k_{{\rm{eff}}} \frac{{\partial T_\ell }}{{\partial z}}} \right) + |- -$
+ | width="100%" | - (8.341) +
\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}
+ |{{EquationRef|(11)}} + |} + + {| class="wikitable" border="0" + |- + | width="100%" | +
${{\left( \rho {{c}_{p}} \right)}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial t}+{{v}_{\ell }}\frac{\partial {{T}_{l}}}{\partial r}+{{w}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial z}=\frac{1}{r}\frac{\partial }{\partial r}\left( r{{k}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial r} \right)+\frac{\partial }{\partial z}\left( {{k}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial z} \right)$
+ |{{EquationRef|(12)}} + |} + 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 -
$+ - \rho _w c_{pw} \frac{{\partial T}}{{\partial t}} = k_w \left [itex] {\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial T}}{{\partial r}}} \right) + \frac{{\partial ^2 T}}{{\partial z^2 }}} \right$
+ - [/itex]
+ {| class="wikitable" border="0" - (8.342) + |- + | width="100%" | +
${{\rho }_{w}}{{c}_{pw}}\frac{\partial T}{\partial t}={{k}_{w}}\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right)+\frac{{{\partial }^{2}}T}{\partial {{z}^{2}}} \right]$
+ |{{EquationRef|(13)}} + |} + 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 -
$+ - v = w = \frac{{\partial T}}{{\partial z}} = \frac{{\partial \rho _g }}{{\partial z}} = 0,{\rm{ }}z = 0 + -$
+ {| class="wikitable" border="0" - (8.343) + |- -
$+ | width="100%" | - v = w = \frac{{\partial T}}{{\partial z}} = 0,\rho _g = \rho _{g,BC} ,{\rm{ }}z = L + [itex]v=w=\frac{\partial T}{\partial z}=\frac{\partial {{\rho }_{g}}}{\partial z}=0,\text{ }z=0$
- [/itex]
+ |{{EquationRef|(14)}} - (8.344) + |} - where ρg,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: + -
$+ - v = \frac{{\partial w}}{{\partial r}} = \frac{{\partial T}}{{\partial r}} = \frac{{\partial \rho _g }}{{\partial r}} = 0,{\rm{ }}r = 0 + {| class="wikitable" border="0" -$
+ |- - (8.345) + | width="100%" | - The liquid-vapor interface
$+ [itex]v=w=\frac{\partial T}{\partial z}=0,{{\rho }_{g}}={{\rho }_{g,BC}},\text{ }z=L$
- \left( {r = R_v } \right) + |{{EquationRef|(15)}} - [/itex]
+ |} - is impermeable to the noncondensable gas + -
$+ where ρg,''BC'' is iteratively adjusted to satisfy overall conservation of noncondensable gas. This boundary condition is implemented through the calculation of the total mass of noncondensable gas. This boundary condition is implemented through the calculation of the total mass of noncondensable gas in the heat pipe. If the total mass is found to be less than the mass initially present in the pipe, the boundary value is increased by 10% of the previous value. Conversely, if the calculated mass is larger than that initially present, the boundary value is decreased. This ensures the conservation of the overall mass to within a preset tolerance, which is 1% in the present formulation. The symmetry of the cylindrical heat pipe requires that the radial vapor velocity and the gradients of the axial vapor velocity, temperature, and gas density be zero at the centerline: - \dot m_g = S_\delta AD_{gv} \nabla \rho _g + \rho _g S_\delta V = 0,{\rm{ }}r = R_v + -$
+ {| class="wikitable" border="0" - (8.346) + |- - where
$+ | width="100%" | - \dot m_{ & g} + [itex]v=\frac{\partial w}{\partial r}=\frac{\partial T}{\partial r}=\frac{\partial {{\rho }_{g}}}{\partial r}=0,\text{ }r=0$
- [/itex]
+ |{{EquationRef|(16)}} - 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
$+ |} - \dot m_g + -$
+ The liquid-vapor interface $\left( r={{R}_{v}} \right)$ is impermeable to the noncondensable gas - 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
$+ {| class="wikitable" border="0" - r = R_v + |- -$
+ | width="100%" | - for
$+ [itex]{{\dot{m}}_{g}}={{S}_{\delta }}A{{D}_{gv}}\nabla {{\rho }_{g}}+{{\rho }_{g}}{{S}_{\delta }}V=0,\text{ }r={{R}_{v}}$
- z \le L_e  + L_a + |{{EquationRef|(17)}} - [/itex]
+ |} - : + -
$+ where [itex]\dot{m}_g$ 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 ${{\dot{m}}_{g}}$ 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. - T_{{\rm{sat}}} = \left( {\frac{1}{{T_0 }} - \frac{{{\rm{R}}_{\rm{u}} }}{{M_v h_{\ell v} }}\ln \frac{{p_v }}{{p_0 }}} \right)^{ - 1} + - [/itex]
+ At $r={{R}_{v}}$ for $z\le {{L}_{e}}+{{L}_{a}}$: - (8.347) + -
$+ {| class="wikitable" border="0" - v_\delta = \frac{{\left( {k_{{\rm{eff}}} \frac{{\partial T_\ell }}{{\partial r}} - \bar k_\delta \frac{{\partial T_v }}{{\partial r}}} \right)}}{{\left( {h_{\ell v} + \bar c_{p\delta } T_{{\rm{sat}}} } \right)\bar \rho _\delta }} + |- -$
+ | width="100%" | - (8.348) +
${{T}_{\text{sat}}}={{\left( \frac{1}{{{T}_{0}}}-\frac{{{\text{R}}_{\text{u}}}}{{{M}_{v}}{{h}_{\ell v}}}\ln \frac{{{p}_{v}}}{{{p}_{0}}} \right)}^{-1}}$
-
$+ |{{EquationRef|(18)}} - w = 0 + |} -$
+ - (8.349) + {| class="wikitable" border="0" - where
$+ |- - \bar k_\delta + | width="100%" | -$
+
${{v}_{\delta }}=\frac{\left( {{k}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial r}-{{{\bar{k}}}_{\delta }}\frac{\partial {{T}_{v}}}{\partial r} \right)}{\left( {{h}_{\ell v}}+{{{\bar{c}}}_{p\delta }}{{T}_{\text{sat}}} \right){{{\bar{\rho }}}_{\delta }}}$
- ,
$+ |{{EquationRef|(19)}} - \bar c_{p\delta } + |} -$
+ - and
$+ {| class="wikitable" border="0" - \bar \rho _\delta + |- -$
+ | width="100%" | - 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: +
$w=0 - [itex] + |{{EquationRef|(20)}} - p_v = \frac{{\bar M}}{{M_v }}\bar p\left( {1 - \frac{{\rho _g }}{{\bar \rho }}} \right) + |} -$
+ - (8.350) + where ${{\bar{k}}_{\delta }}$, ${{\bar{c}}_{p\delta }}$ and ${{\bar{\rho }}_{\delta }}$ are the vapor-gas mixture properties at the liquid-vapor interface.  In eq. (18), 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: + + {| class="wikitable" border="0" + |- + | width="100%" | +
${{p}_{v}}=\frac{{\bar{M}}}{{{M}_{v}}}\bar{p}\left( 1-\frac{{{\rho }_{g}}}{{\bar{\rho }}} \right)$
+ |{{EquationRef|(21)}} + |} + 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
$+ - r = R_v + At [itex]r={{R}_{v}}$ for $z>{{L}_{e}}+{{L}_{a}}$: - [/itex]
+ - for
$+ {| class="wikitable" border="0" - z > L_e + L_a + |- -$
+ | width="100%" | - : +
${{q}_{so}}=-\left( {{h}_{\ell v}}+{{{\bar{c}}}_{p\delta }}{{T}_{\delta }} \right)\left( \bar{\rho }-{{\rho }_{g}} \right){{v}_{g}}$
-
$+ |{{EquationRef|(22)}} - q_{so} = - \left( {h_{\ell v} + \bar c_{p\delta } T_\delta } \right)\left( {\bar \rho - \rho _g } \right)v_g + |} -$
+ - (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: -
$+ - k_w \frac{{\partial T}}{{\partial r}} = k_{{\rm{eff}}} \frac{{\partial T}}{{\partial r}},{\rm{ }}r = R_w + {| class="wikitable" border="0" -$
+ |- - (8.352) + | width="100%" | +
${{k}_{w}}\frac{\partial T}{\partial r}={{k}_{\text{eff}}}\frac{\partial T}{\partial r},\text{ }r={{R}_{w}}$
+ |{{EquationRef|(23)}} + |} + 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. -
$+ - k_w \left. {\frac{{\partial T}}{{\partial r}}} \right|_{r = R_O } = \left\{ {\begin{array}{*{20}c} + {| class="wikitable" border="0" - {q''_e {\rm{ evaporator}}} \\ + |- - {0{\rm{ adiabatic}}} \\ + | width="100%" | - {\sigma \varepsilon _r \left( {T_w^4 - T_\infty ^4 } \right){\rm{ condenser}}} \\ + [itex]{{k}_{w}}{{\left. \frac{\partial T}{\partial r} \right|}_{r={{R}_{O}}}}=\left\{ \begin{matrix} - \end{array}} \right. + {{{{q}''}}_{e}}\text{ evaporator} \\ -$
+ 0\text{                            adiabatic}  \\ - (8.353) + \sigma {{\varepsilon }_{r}}\left( T_{w}^{4}-T_{\infty }^{4} \right)\text{      condenser}  \\ + \end{matrix} \right.[/itex]
+ |{{EquationRef|(24)}} + |} + 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 - 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. - + '''Initial Conditions''' - Figure 8.28 Temperature profiles for the gas-loaded heat pipe with Qin = 451W: (a) transient wall temperature profile; (b) transient centerline temperature profile (Harley and Faghri, 1994). + + 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. The gas-loaded heat pipe experimentally studied by [[#References|Ponnappan (1989)]] was simulated using the above analysis.  The results show that the wall and vapor temperatures decreased significantly in the condenser section due to the presence of the noncondensable gas. + + ==References== + + Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Elsevier, Burlington, MA + + Faghri, A., Zhang, Y., and Howell, J. R., 2010, ''Advanced  Heat and Mass Transfer'', Global Digital Press, Columbia, MO. + Harley, C., and Faghri, A., 1994, “Transient Two-Dimensional Gas-Loaded Heat Pipe Analysis,” ASME Journal of Heat Transfer, Vol. 116, pp. 716-723. - 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. + Ponnappan, R., 1989, “Studies on the Startup Transients and Performance of a Gas Loaded Sodium Heat Pipe,” WRDC-TR-89-2046, Wright-Patterson AFB, OH.

## Current revision as of 07:58, 26 July 2010

Noncondensable heat pipe configuration.

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 the figure on the right. 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

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

 $\frac{\partial \bar{\rho }}{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}\left( \bar{\rho }rv \right)+\frac{\partial }{\partial z}\left( \bar{\rho }w \right)=0$ (1)
 $\bar{\rho }\frac{DV}{Dt}=-\nabla \bar{p}+\frac{1}{3}\bar{\mu }\nabla \left( \nabla \cdot V \right)+\bar{\mu }{{\nabla }^{2}}V$ (2)
 $\rho {{c}_{p}}\frac{DT}{Dt}-\nabla \cdot \bar{k}\nabla T-\nabla \cdot \left( \sum\limits_{j=1}^{2}{{{D}_{d}}{{c}_{pj}}T\nabla {{\rho }_{j}}} \right)-\frac{D\bar{p}}{Dt}-\bar{\mu }\Phi =0$ (3)

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

 $\frac{D{{\rho }_{g}}}{Dt}-\nabla \cdot {{D}_{gv}}\nabla {{\rho }_{g}}=0$ (4)

and

 $\Phi =2\left[ {{\left( \frac{\partial v}{\partial r} \right)}^{2}}+{{\left( \frac{v}{r} \right)}^{2}}+{{\left( \frac{\partial w}{\partial z} \right)}^{2}} \right]+{{\left( \frac{\partial v}{\partial z}+\frac{\partial w}{\partial r} \right)}^{2}}-\frac{2}{3}{{\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( rv \right)+\frac{\partial w}{\partial z} \right]}^{2}}$ (5)

Furthermore, v and w are the radial and axial vapor velocities, $\bar{p}$ is the total mixture pressure, $\bar{\mu }$ is the mass-fraction-weighted mixture viscosity, ${{\bar{c}}_{p}}$ is the specific heat of the mixture, $\bar{k}$ is the thermal conductivity of the mixture, Dd is the self-diffusion coefficient for both vapor and gas species, Dgv is the mass diffusion coefficient of the vapor-gas pair, ρg is the density of the noncondensable gas, and the mixture density is $\bar{\rho }={{\rho }_{g}}+{{\rho }_{v}}$. 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

 $\bar{p}=\frac{\bar{\rho }{{\text{R}}_{\text{u}}}T}{{\bar{M}}}$ (6)

where $\bar{M}$ is the molecular weight of the vapor-gas mixture and Ru is the universal gas constant.

In the species equation, Dgvis a function of pressure and temperature. For a vapor-gas mixture of sodium-argon, Harley and Faghri (1994) used the following relationship for Dgv

 ${{D}_{gv}}=1.3265\times {{10}^{-3}}{{T}^{3/2}}{{\left( {\bar{p}} \right)}^{-1}}$ (7)

where T is in degrees Kelvin, $\bar{p}$ is in N/m2, and Dgv is in m2/s. Following a similar procedure, the variable diffusion coefficient formulation for the sodium-helium pair is

 ${{D}_{gv}}=1.2795\times {{10}^{-3}}{{T}^{3/2}}{{\left( {\bar{p}} \right)}^{-1}}$ (8)

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, Dd, 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 ${{\left\langle {{v}_{\ell }} \right\rangle }^{\ell }}$ through the porous wick, which is assumed to be isotropic and homogeneous. For simplicity, ${{\left\langle {} \right\rangle }^{\ell }}$ 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

 $\frac{1}{r}\frac{\partial }{\partial r}\left( r{{v}_{\ell }} \right)+\frac{\partial {{w}_{\ell }}}{\partial z}=0$ (9)
 \begin{align} & \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} (10)
 \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} (11)
 ${{\left( \rho {{c}_{p}} \right)}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial t}+{{v}_{\ell }}\frac{\partial {{T}_{l}}}{\partial r}+{{w}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial z}=\frac{1}{r}\frac{\partial }{\partial r}\left( r{{k}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial r} \right)+\frac{\partial }{\partial z}\left( {{k}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial z} \right)$ (12)

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

 ${{\rho }_{w}}{{c}_{pw}}\frac{\partial T}{\partial t}={{k}_{w}}\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right)+\frac{{{\partial }^{2}}T}{\partial {{z}^{2}}} \right]$ (13)

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

 $v=w=\frac{\partial T}{\partial z}=\frac{\partial {{\rho }_{g}}}{\partial z}=0,\text{ }z=0$ (14)

 $v=w=\frac{\partial T}{\partial z}=0,{{\rho }_{g}}={{\rho }_{g,BC}},\text{ }z=L$ (15)

where ρg,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:

 $v=\frac{\partial w}{\partial r}=\frac{\partial T}{\partial r}=\frac{\partial {{\rho }_{g}}}{\partial r}=0,\text{ }r=0$ (16)

The liquid-vapor interface $\left( r={{R}_{v}} \right)$ is impermeable to the noncondensable gas

 ${{\dot{m}}_{g}}={{S}_{\delta }}A{{D}_{gv}}\nabla {{\rho }_{g}}+{{\rho }_{g}}{{S}_{\delta }}V=0,\text{ }r={{R}_{v}}$ (17)

where $\dot{m}_g$ is the mass flow rate of gas, A is the cross-sectional area of the heat pipe and is the surface area of the liquid-vapor interface. This formulation of ${{\dot{m}}_{g}}$ 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 = Rv for $z\le {{L}_{e}}+{{L}_{a}}$:

 ${{T}_{\text{sat}}}={{\left( \frac{1}{{{T}_{0}}}-\frac{{{\text{R}}_{\text{u}}}}{{{M}_{v}}{{h}_{\ell v}}}\ln \frac{{{p}_{v}}}{{{p}_{0}}} \right)}^{-1}}$ (18)
 ${{v}_{\delta }}=\frac{\left( {{k}_{\text{eff}}}\frac{\partial {{T}_{\ell }}}{\partial r}-{{{\bar{k}}}_{\delta }}\frac{\partial {{T}_{v}}}{\partial r} \right)}{\left( {{h}_{\ell v}}+{{{\bar{c}}}_{p\delta }}{{T}_{\text{sat}}} \right){{{\bar{\rho }}}_{\delta }}}$ (19)
 w = 0 (20)

where ${{\bar{k}}_{\delta }}$, ${{\bar{c}}_{p\delta }}$ and ${{\bar{\rho }}_{\delta }}$ are the vapor-gas mixture properties at the liquid-vapor interface. In eq. (18), 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:

 ${{p}_{v}}=\frac{{\bar{M}}}{{{M}_{v}}}\bar{p}\left( 1-\frac{{{\rho }_{g}}}{{\bar{\rho }}} \right)$ (21)

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 = Rv for z > Le + La:

 ${{q}_{so}}=-\left( {{h}_{\ell v}}+{{{\bar{c}}}_{p\delta }}{{T}_{\delta }} \right)\left( \bar{\rho }-{{\rho }_{g}} \right){{v}_{g}}$ (22)

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:

 ${{k}_{w}}\frac{\partial T}{\partial r}={{k}_{\text{eff}}}\frac{\partial T}{\partial r},\text{ }r={{R}_{w}}$ (23)

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.

 ${{k}_{w}}{{\left. \frac{\partial T}{\partial r} \right|}_{r={{R}_{O}}}}=\left\{ \begin{matrix} {{{{q}''}}_{e}}\text{ evaporator} \\ 0\text{ adiabatic} \\ \sigma {{\varepsilon }_{r}}\left( T_{w}^{4}-T_{\infty }^{4} \right)\text{ condenser} \\ \end{matrix} \right.$ (24)

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. The gas-loaded heat pipe experimentally studied by Ponnappan (1989) was simulated using the above analysis. The results show that the wall and vapor temperatures decreased significantly in the condenser section due to the presence of the noncondensable gas.

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Harley, C., and Faghri, A., 1994, “Transient Two-Dimensional Gas-Loaded Heat Pipe Analysis,” ASME Journal of Heat Transfer, Vol. 116, pp. 716-723.

Ponnappan, R., 1989, “Studies on the Startup Transients and Performance of a Gas Loaded Sodium Heat Pipe,” WRDC-TR-89-2046, Wright-Patterson AFB, OH.