Evaporation from Cylindrical Pore under Low to Moderate Heat Flux
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The validity of the second assumption has been proven numerically by [[#ReferencesKhrustalev and Faghri (1995b);]] it was shown that this assumption could give an error of less than 5% when calculating the overall heat transfer coefficient during evaporation from a capillary groove. Since heat transfer during evaporation from thin films in a pore is similar to that in a capillary groove, this assumption can be justified for the present analysis.  The validity of the second assumption has been proven numerically by [[#ReferencesKhrustalev and Faghri (1995b);]] it was shown that this assumption could give an error of less than 5% when calculating the overall heat transfer coefficient during evaporation from a capillary groove. Since heat transfer during evaporation from thin films in a pore is similar to that in a capillary groove, this assumption can be justified for the present analysis.  
  +  [[Image:NewChapter9 (4).pngthumb400 pxalt= Schematic of evaporation from a cylindrical pore.  Figure 9.19 Schematic of evaporation from a cylindrical pore. ]]  
The local heat flux through the liquid film due to heat conduction is  The local heat flux through the liquid film due to heat conduction is  
  +  
  <center><math>{{{q}''}_{\ell }}={{k}_{\ell }}\frac{{{T}_{s}}{{T}_{\delta }}}{{{\delta }_{\ell }}}</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%"   
+  <center><math>{{{q}''}_{\ell }}={{k}_{\ell }}\frac{{{T}_{s}}{{T}_{\delta }}}{{{\delta }_{\ell }}}</math></center>  
+  {{EquationRef(1)}}  
+  }  
where the local thickness of the liquid layer <math>{{\delta }_{\ell }}</math> and the temperature of the free liquid film surface <math>{{T}_{\delta }}</math> are functions of the scoordinate. <math>{{T}_{\delta }}</math> is affected by the disjoining and capillary pressures. It also depends on the value of the interfacial resistance, which is defined for the case of comparatively small heat flux at the interface, <math>{{{q}''}_{\delta }}</math>, by the following relation given by the kinetic theory:  where the local thickness of the liquid layer <math>{{\delta }_{\ell }}</math> and the temperature of the free liquid film surface <math>{{T}_{\delta }}</math> are functions of the scoordinate. <math>{{T}_{\delta }}</math> is affected by the disjoining and capillary pressures. It also depends on the value of the interfacial resistance, which is defined for the case of comparatively small heat flux at the interface, <math>{{{q}''}_{\delta }}</math>, by the following relation given by the kinetic theory:  
  
  
  
  where <math>{{p}_{v\delta }}</math> and <math>{{({{p}_{sat}})}_{\delta }}</math> are the saturation pressures corresponding to  +  
+  { class="wikitable" border="0"  
+    
+   width="100%"   
+  <center><math>{{{q}''}_{\delta }}=\left( \frac{2\alpha }{2\alpha } \right)\frac{{{h}_{\ell v}}}{\sqrt{2\pi {{R}_{g}}}}\left[ \frac{{{p}_{v\delta }}}{\sqrt{{{T}_{v}}}}\frac{{{\left( {{p}_{\text{sat}}} \right)}_{\delta }}}{\sqrt{{{T}_{\delta }}}} \right]</math></center>  
+  {{EquationRef(2)}}  
+  }  
+  
+  where <math>{{p}_{v\delta }}</math> and <math>{{({{p}_{sat}})}_{\delta }}</math> are the saturation pressures corresponding to ''T<sub>v</sub>'' and at the thin liquid film interface, respectively.  
The relation between the saturation vapor pressure over the thin evaporating film, <math>{{({{p}_{sat}})}_{\delta }},</math> which is affected by the disjoining pressure, and the normal saturation pressure corresponding to <math>{{T}_{\delta }}</math>, <math>{{p}_{sat}}({{T}_{\delta }}),</math> is given by the following relation (see Chapter 5):  The relation between the saturation vapor pressure over the thin evaporating film, <math>{{({{p}_{sat}})}_{\delta }},</math> which is affected by the disjoining pressure, and the normal saturation pressure corresponding to <math>{{T}_{\delta }}</math>, <math>{{p}_{sat}}({{T}_{\delta }}),</math> is given by the following relation (see Chapter 5):  
  
  
  
  +  { class="wikitable" border="0"  
+    
+   width="100%"   
+  <center><math>{{({{p}_{sat}})}_{\delta }}={{p}_{sat}}({{T}_{\delta }})\exp \left[ \frac{{{({{p}_{sat}})}_{\delta }}{{p}_{sat}}({{T}_{\delta }})+{{p}_{d}}\sigma K}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}} \right]</math></center>  
+  {{EquationRef(3)}}  
+  }  
  +  which reflects the fact that under the influence of the disjoining and capillary pressures, the liquid free surface saturation pressure <math>{{({{p}_{sat}})}_{\delta }}</math> is different from normal saturation pressure <math>{{p}_{sat}}({{T}_{\delta }})</math>. It varies along the thin film (or scoordinate) while <math>{{p}_{v\delta }}</math> and ''T<sub>v</sub>'' are the same for any value of ''s''. This variation is also due to the fact that <math>{{T}_{\delta }}</math> changes along s. While the evaporating film thins as it approaches the point s = 0, the difference between <math>{{({{p}_{sat}})}_{\delta }},</math> given by eq. (3), and the pressure obtained for a given <math>{{T}_{\delta }}</math> using the saturation table becomes larger. This difference is the reason for the existence of the thin nonevaporating superheated film, which is in equilibrium state in spite of the fact that <math>{{T}_{\delta }}>{{T}_{sat}}</math>.  
  +  
  +  
  +  
  +  Under steadystate conditions, <math>{{{q}''}_{\ell }}={{{q}''}_{\delta }}</math>, and it follows from eqs. (1) and (2) that  
  +  { class="wikitable" border="0"  
  +    
  +   width="100%"   
  +  <center><math>{{T}_{\delta }}={{T}_{s}}+\frac{\delta }{{{k}_{\ell }}}\left( \frac{2\alpha }{2\alpha } \right)\frac{{{h}_{\ell v}}}{\sqrt{2\pi {{R}_{g}}}}\left[ \frac{{{p}_{v\delta }}}{\sqrt{{{T}_{v}}}}\frac{{{({{p}_{sat}})}_{\delta }}}{\sqrt{{{T}_{\delta }}}} \right]</math><center>  
+  {{EquationRef(4)}}  
+  }  
  where a = 1.5336 and b = 0.0243. From equations (  +  Equations (3) and (4) determine the interfacial temperature <math>{{T}_{\delta }}</math> and pressure <math>{{({{p}_{sat}})}_{\delta }}</math> for a given vapor pressure, <math>{{p}_{v\delta }}</math>, the solidliquid interface temperature <math>{{T}_{s}}</math>, and the liquid film thickness <math>{{\delta }_{\ell }}(s)</math>. 
+  
+  As the liquid film thins, the disjoining pressure, pd, and the interfacial temperature, ''T<sub>δ</sub>'', increase. Under specific conditions, a nonevaporating film thickness is present which gives the equality of the interfacial and solid surface temperatures, <math>{{T}_{\delta }}={{T}_{s}}</math>. This is the thickness of the equilibrium nonevaporating film <math>{{\delta }_{0}}</math>, For the disjoining pressure in water, the following equation for was used in the present analysis [[#References(Holm and Goplen, 1979)]]:  
+  
+  { class="wikitable" border="0"  
+    
+   width="100%"   
+  <center><math>{{p}_{d}}={{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}\ln \left[ a{{\left( \frac{\delta }{3.3} \right)}^{b}} \right]</math></center>  
+  {{EquationRef(5)}}  
+  }  
+  
+  where a = 1.5336 and b = 0.0243. From equations (3) – (5), the following expression for the thickness of the equilibrium film is given:  
  <center><math>{{\delta }_{0}}=3.3{{\left\{ \frac{1}{a}\exp \left[ \frac{{{p}_{sat}}({{T}_{w}}){{p}_{v}}\sqrt{{{T}_{w}}/{{T}_{v}}}+\sigma K}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{w}}}+\ln \left( \frac{{{p}_{v}}}{{{p}_{sat}}({{T}_{w}})}\sqrt{\frac{{{T}_{w}}}{{{T}_{v}}}} \right) \right] \right\}}^{1/b}}</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%"   
+  <center><math>{{\delta }_{0}}=3.3{{\left\{ \frac{1}{a}\exp \left[ \frac{{{p}_{sat}}({{T}_{w}}){{p}_{v}}\sqrt{{{T}_{w}}/{{T}_{v}}}+\sigma K}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{w}}}+\ln \left( \frac{{{p}_{v}}}{{{p}_{sat}}({{T}_{w}})}\sqrt{\frac{{{T}_{w}}}{{{T}_{v}}}} \right) \right] \right\}}^{1/b}}</math></center>  
+  {{EquationRef(6)}}  
+  }  
The total heat flow through a single pore is defined as  The total heat flow through a single pore is defined as  
  +  { class="wikitable" border="0"  
  +    
+   width="100%"   
+  <center><math>{{q}_{p}}=\int_{0}^{{{R}_{p}}}{\frac{{{T}_{s}}{{T}_{\delta }}}{{{\delta }_{\ell }}/{{k}_{\ell }}}}2\pi rds=\int_{0}^{{{R}_{p}}}{\frac{{{T}_{s}}{{T}_{\delta }}}{{{\delta }_{\ell }}/{{k}_{\ell }}}}2\pi {{R}_{\text{men}}}\sin \left[ \arctan \frac{{{R}_{p}}}{s+\sqrt{R_{\text{men}}^{\text{2}}R_{p}^{2}}} \right]ds</math></center>  
+  {{EquationRef(7)}}  
+  }  
The surface of the pore wall is totally covered with microroughnesses, from which the characteristic size varies, for example, Rr = 108 to 106 m. Apparently, the thin liquid film formation can be affected by some of these microroughnesses. The following approximation for the liquid film thickness was given by Khrustalev and Faghri (1995b): for <math>{{\delta }_{0}}\le {{\delta }_{\ell }}\le {{\delta }_{0}}+{{R}_{r}}</math> and <math>{{R}_{r}}\gg {{\delta }_{0}},</math>  The surface of the pore wall is totally covered with microroughnesses, from which the characteristic size varies, for example, Rr = 108 to 106 m. Apparently, the thin liquid film formation can be affected by some of these microroughnesses. The following approximation for the liquid film thickness was given by Khrustalev and Faghri (1995b): for <math>{{\delta }_{0}}\le {{\delta }_{\ell }}\le {{\delta }_{0}}+{{R}_{r}}</math> and <math>{{R}_{r}}\gg {{\delta }_{0}},</math>  
  +  
  <center><math>{{\delta }_{\ell }}={{\delta }_{0}}+{{R}_{r}}\sqrt{R_{r}^{2}{{s}^{2}}}{{R}_{\text{men}}}+{{\left( {{R}_{\text{men}}}^{2}+{{s}^{2}}+2{{R}_{\text{men}}}s\sin {{\theta }_{f}} \right)}^{1/2}}</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%"   
+  <center><math>{{\delta }_{\ell }}={{\delta }_{0}}+{{R}_{r}}\sqrt{R_{r}^{2}{{s}^{2}}}{{R}_{\text{men}}}+{{\left( {{R}_{\text{men}}}^{2}+{{s}^{2}}+2{{R}_{\text{men}}}s\sin {{\theta }_{f}} \right)}^{1/2}}</math></center>  
+  {{EquationRef(8)}}  
+  }  
where for a surface with microroughness <math>{{\theta }_{f}}=0</math>, and the liquid film thickness in the interval, <math>{{\delta }_{\ell }}\ge {{\delta }_{0}}+{{R}_{r}},</math> is  where for a surface with microroughness <math>{{\theta }_{f}}=0</math>, and the liquid film thickness in the interval, <math>{{\delta }_{\ell }}\ge {{\delta }_{0}}+{{R}_{r}},</math> is  
  
  
  
  For the smoothsurface model, <math>{{\theta }_{f}}</math> is the angle between the solidliquid and liquidvapor interfaces at the point on s where the disjoining pressure and dK/ds become zero. Note that for small values of the accommodation coefficient (for example <math>\alpha </math>= 0.05), the value of  +  { class="wikitable" border="0" 
+    
+   width="100%"   
+  <center><math>\delta ={{\delta }_{0}}+{{R}_{r}}{{R}_{\text{men}}}+{{\left( {{R}_{\text{men}}}^{2}+{{s}^{2}}+2{{R}_{\text{men}}}s\sin {{\theta }_{\text{men}}} \right)}^{1/2}}</math></center>  
+  {{EquationRef(9)}}  
+  }  
+  
+  For the smoothsurface model, <math>{{\theta }_{f}}</math> is the angle between the solidliquid and liquidvapor interfaces at the point on s where the disjoining pressure and dK/ds become zero. Note that for small values of the accommodation coefficient (for example <math>\alpha </math>= 0.05), the value of ''R<sub>r</sub>'' has not affected the total heat flow rate through the liquid film [[#References(Khrustalev and Faghri, 1995b)]].  
The interfacial radius of curvature is related to the pressure difference between the liquid and the vapor by the extended LaplaceYoung equation:  The interfacial radius of curvature is related to the pressure difference between the liquid and the vapor by the extended LaplaceYoung equation:  
  +  
  <center><math>{{p}_{v\delta }}{{p}_{\ell \delta }}=\frac{2\sigma }{{{R}_{\text{men}}}}+\frac{\rho _{v}^{2}v_{v\delta }^{2}}{{{\varepsilon }^{2}}}\left( \frac{1}{{{\rho }_{\ell }}}\frac{1}{{{\rho }_{v}}} \right)</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%"   
+  <center><math>{{p}_{v\delta }}{{p}_{\ell \delta }}=\frac{2\sigma }{{{R}_{\text{men}}}}+\frac{\rho _{v}^{2}v_{v\delta }^{2}}{{{\varepsilon }^{2}}}\left( \frac{1}{{{\rho }_{\ell }}}\frac{1}{{{\rho }_{v}}} \right)</math></center>  
+  {{EquationRef(10)}}  
+  }  
where <math>{{v}_{v\delta }}</math> is the vapor mean blowing velocity specified for a given meniscus, and <math>\varepsilon </math> is the porosity. The latter is needed in this equation because the evaporation takes place into the dry region of the porous structure. The temperature of the saturated vapor near the interface, ''T<sub>v</sub>'', is related to its pressure by the saturation conditions:  where <math>{{v}_{v\delta }}</math> is the vapor mean blowing velocity specified for a given meniscus, and <math>\varepsilon </math> is the porosity. The latter is needed in this equation because the evaporation takes place into the dry region of the porous structure. The temperature of the saturated vapor near the interface, ''T<sub>v</sub>'', is related to its pressure by the saturation conditions:  
  +  
  <center><math>{{T}_{v}}={{T}_{sat}}({{p}_{v\delta }})</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%"   
+  <center><math>{{T}_{v}}={{T}_{sat}}({{p}_{v\delta }})</math></center>  
+  {{EquationRef(11)}}  
+  }  
Then the heat transfer coefficient during evaporation from the porous surface is defined as  Then the heat transfer coefficient during evaporation from the porous surface is defined as  
  
  
  
  where <math>{{\varepsilon }_{s}}={{A}_{p}}/{{A}_{t}}</math> is the surface porosity defined as the ratio of the surface of the pores to the total surface of the porous structure for a given crosssection (it is assumed that <math>{{\varepsilon }_{s}}=\varepsilon </math>; [[#ReferencesKhrustalev and Faghri, 1995a)]]. Figure 9.20 shows that the heat transfer coefficient during evaporation from the porous surface,  +  { class="wikitable" border="0" 
+    
+   width="100%"   
+  <center><math>{{h}_{ep}}=\frac{{{\varepsilon }_{s}}{{q}_{p}}}{\pi R_{p}^{2}({{T}_{s}}{{T}_{v}})}</math></center>  
+  {{EquationRef(12)}}  
+  }  
+  
+  where <math>{{\varepsilon }_{s}}={{A}_{p}}/{{A}_{t}}</math> is the surface porosity defined as the ratio of the surface of the pores to the total surface of the porous structure for a given crosssection (it is assumed that <math>{{\varepsilon }_{s}}=\varepsilon </math>; [[#ReferencesKhrustalev and Faghri, 1995a)]]. Figure 9.20 shows that the heat transfer coefficient during evaporation from the porous surface, ''h<sub>e</sub>'', ''p'', depends significantly on the curvature of the liquid meniscus. For smaller pore sizes, the values of ''h<sub>e</sub>'', ''p'' are larger because a larger relative surface is occupied by the thin films.  
==References==  ==References==  
+  
+  Khrustalev, D. and Faghri, A., 1995a, “Heat Transfer in the Inverted Meniscus Type Evaporator at High Heat Fluxes,” International Journal of Heat and Mass Transfer, Vol. 38, pp. 30913101.  
+  
+  Khrustalev, D.K. and Faghri, A., 1995b, “Heat Transfer during Evaporation and Condensation on CapillaryGrooved Structures of Heat Pipes,” ASME Journal of Heat Transfer, Vol. 117, August, No. 3, pp. 740747. 
Current revision as of 19:48, 3 June 2010
Evaporation in a microchannel under low/moderate heat flux was studied by Khrustalev and Faghri (1995a). In the scenario under consideration, liquid evaporates from the surface of the menisci situated at the liquidvapor interface. Figure 9.19 shows the schematic of the cylindrical pore (microchannel) and liquid meniscus. A description of the heat transfer during evaporation from a pore is given here with the two following main assumptions.
1. The temperature of the solidliquid interface Ts can be considered constant along the scoordinate for small s.
2. The curvature of the axisymmetrical liquidvapor surface of the meniscus is defined by the main radius of curvature K = 2 / R_{men} and is independent of s.
The validity of the second assumption has been proven numerically by Khrustalev and Faghri (1995b); it was shown that this assumption could give an error of less than 5% when calculating the overall heat transfer coefficient during evaporation from a capillary groove. Since heat transfer during evaporation from thin films in a pore is similar to that in a capillary groove, this assumption can be justified for the present analysis.
The local heat flux through the liquid film due to heat conduction is

where the local thickness of the liquid layer and the temperature of the free liquid film surface T_{δ} are functions of the scoordinate. T_{δ} is affected by the disjoining and capillary pressures. It also depends on the value of the interfacial resistance, which is defined for the case of comparatively small heat flux at the interface, q''_{δ}, by the following relation given by the kinetic theory:

where p_{vδ} and are the saturation pressures corresponding to T_{v} and at the thin liquid film interface, respectively.
The relation between the saturation vapor pressure over the thin evaporating film, which is affected by the disjoining pressure, and the normal saturation pressure corresponding to T_{δ}, p_{sat}(T_{δ}), is given by the following relation (see Chapter 5):

which reflects the fact that under the influence of the disjoining and capillary pressures, the liquid free surface saturation pressure is different from normal saturation pressure p_{sat}(T_{δ}). It varies along the thin film (or scoordinate) while p_{vδ} and T_{v} are the same for any value of s. This variation is also due to the fact that T_{δ} changes along s. While the evaporating film thins as it approaches the point s = 0, the difference between given by eq. (3), and the pressure obtained for a given T_{δ} using the saturation table becomes larger. This difference is the reason for the existence of the thin nonevaporating superheated film, which is in equilibrium state in spite of the fact that T_{δ} > T_{sat}.
Under steadystate conditions, , and it follows from eqs. (1) and (2) that

Equations (3) and (4) determine the interfacial temperature T_{δ} and pressure for a given vapor pressure, p_{vδ}, the solidliquid interface temperature T_{s}, and the liquid film thickness .
As the liquid film thins, the disjoining pressure, pd, and the interfacial temperature, T_{δ}, increase. Under specific conditions, a nonevaporating film thickness is present which gives the equality of the interfacial and solid surface temperatures, T_{δ} = T_{s}. This is the thickness of the equilibrium nonevaporating film δ_{0}, For the disjoining pressure in water, the following equation for was used in the present analysis (Holm and Goplen, 1979):
<center> 
where a = 1.5336 and b = 0.0243. From equations (3) – (5), the following expression for the thickness of the equilibrium film is given:

The total heat flow through a single pore is defined as

The surface of the pore wall is totally covered with microroughnesses, from which the characteristic size varies, for example, Rr = 108 to 106 m. Apparently, the thin liquid film formation can be affected by some of these microroughnesses. The following approximation for the liquid film thickness was given by Khrustalev and Faghri (1995b): for and

where for a surface with microroughness θ_{f} = 0, and the liquid film thickness in the interval, is

For the smoothsurface model, θ_{f} is the angle between the solidliquid and liquidvapor interfaces at the point on s where the disjoining pressure and dK/ds become zero. Note that for small values of the accommodation coefficient (for example α= 0.05), the value of R_{r} has not affected the total heat flow rate through the liquid film (Khrustalev and Faghri, 1995b).
The interfacial radius of curvature is related to the pressure difference between the liquid and the vapor by the extended LaplaceYoung equation:

where v_{vδ} is the vapor mean blowing velocity specified for a given meniscus, and is the porosity. The latter is needed in this equation because the evaporation takes place into the dry region of the porous structure. The temperature of the saturated vapor near the interface, T_{v}, is related to its pressure by the saturation conditions:

Then the heat transfer coefficient during evaporation from the porous surface is defined as

where is the surface porosity defined as the ratio of the surface of the pores to the total surface of the porous structure for a given crosssection (it is assumed that ; Khrustalev and Faghri, 1995a). Figure 9.20 shows that the heat transfer coefficient during evaporation from the porous surface, h_{e}, p, depends significantly on the curvature of the liquid meniscus. For smaller pore sizes, the values of h_{e}, p are larger because a larger relative surface is occupied by the thin films.
References
Khrustalev, D. and Faghri, A., 1995a, “Heat Transfer in the Inverted Meniscus Type Evaporator at High Heat Fluxes,” International Journal of Heat and Mass Transfer, Vol. 38, pp. 30913101.
Khrustalev, D.K. and Faghri, A., 1995b, “Heat Transfer during Evaporation and Condensation on CapillaryGrooved Structures of Heat Pipes,” ASME Journal of Heat Transfer, Vol. 117, August, No. 3, pp. 740747.