Coupled thermal and concentration entry effects
From ThermalFluidsPedia
Yuwen Zhang (Talk  contribs) m (moved Combined hydrodynamic and thermal entrance effect to Coupled thermal and concentration entry effects) 

(12 intermediate revisions not shown)  
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  There are many transport phenomena problems in which heat and mass transfer simultaneously occur. In some cases, such as sublimation and vapor deposition, they are coupled. These problems are usually treated as a single phase. However, coupled heat and mass transfer should both be considered even though they are modeled as being single phase.  +  There are many transport phenomena problems in which heat and mass transfer simultaneously occur. In some cases, such as sublimation and vapor deposition, they are coupled. These problems are usually treated as a single phase. However, coupled heat and mass transfer should both be considered even though they are modeled as being single phase<ref>Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.</ref>. In this article, coupled forced internal convection in a circular tube will be presented for both adiabatic and constant wall heat flux. 
==Sublimation inside an Adiabatic Tube==  ==Sublimation inside an Adiabatic Tube==  
  [[Image:Fig5.7.pngthumb400 pxalt=Sublimation in an adiabatic tube   +  [[Image:Fig5.7.pngthumb400 pxalt=Sublimation in an adiabatic tube  Sublimation in an adiabatic tube.]] 
  In addition to the external sublimation  +  
  1. The entrance mass fraction,  +  In addition to the external sublimation, internal sublimation is also very important. Sublimation inside an adiabatic and externally heated tube will be analyzed. The physical model of the problem under consideration is shown in figure to the right <ref name="ZC1990">Zhang, Y., and Chen, Z.Q., 1990, “Analytical Solution of Coupled Laminar HeatMass Transfer inside a Tube with Adiabatic External Wall,” Proceedings of the 3rd National Interuniversity Conference on Engineering Thermophysics, Xi’an Jiaotong University Press, Xi’an, China, pp. 341345.</ref>. The inner surface of a circular tube with radius ''r<sub>o</sub>'' is coated with a layer of sublimable material which will sublime when gas flows through the tube. The fullydeveloped gas enters the tube with a uniform inlet mass fraction of the sublimable substance, ''ω<sub>0</sub>'', and a uniform inlet temperature, ''T<sub>0</sub>''. Since the outer wall surface is adiabatic, the latent heat of sublimation is supplied by the gas flow inside the tube; this in turn causes the change in gas temperature inside the tube. It is assumed that the flow inside the tube is incompressible laminar flow with constant properties. In order to solve the problem analytically, the following assumptions are made:<br> 
  2. The saturation mass fraction can be expressed as a linear function of the corresponding temperature.  +  1. The entrance mass fraction, ''ω<sub>0</sub>'', is assumed to be equal to the saturation mass fraction at the entry temperature, ''T<sub>0</sub>''. <br> 
  3. The mass transfer rate is small enough that the transverse velocity components can be neglected.  +  2. The saturation mass fraction can be expressed as a linear function of the corresponding temperature.<br> 
+  3. The mass transfer rate is small enough that the transverse velocity components can be neglected.<br>  
+  
The fully developed velocity profile in the tube is  The fully developed velocity profile in the tube is  
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{{EquationRef(3)}}  {{EquationRef(3)}}  
}  }  
  where D is mass diffusivity. Equations (  +  where ''D'' is mass diffusivity. Equations (2) and (3) are subjected to the following boundary conditions: 
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{{EquationRef(7)}}  {{EquationRef(7)}}  
}  }  
  Equation (  +  Equation (7) implies that the latent heat of sublimation is supplied as the gas flows inside the tube. Another boundary condition at the tube wall is obtained by setting the mass fraction at the wall as the saturation mass fraction at the wall temperature <ref name="K1973">Kurosaki, Y., 1973, “Coupled Heat and Mass Transfer in a Flow between Parallel Flat Plate (Uniform Heat Flux),” Journal of the Japan Society of Mechanical Engineers, Part B, Vol. 39, pp. 25122521 (in Japanese).</ref>. According to the second assumption, the mass fraction and temperature at the inner wall have the following relationship: 
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{{EquationRef(8)}}  {{EquationRef(8)}}  
}  }  
  where a and b are constants.  +  where ''a'' and ''b'' are constants. 
+  
The following nondimensional variables are then introduced:  The following nondimensional variables are then introduced:  
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{{EquationRef(9)}}  {{EquationRef(9)}}  
}  }  
  where  +  where ''T<sub>f</sub>'' and ''ω<sub>f</sub>'' are temperature and mass fraction of the sublimable substance, respectively, after heat and mass transfer are fully developed, and Le is Lewis number. Equations (2) – (8) then become 
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{{EquationRef(15)}}  {{EquationRef(15)}}  
}  }  
  The heat and mass transfer eqs. (  +  The heat and mass transfer eqs. (10) and (11) are independent, but their boundary conditions are coupled by eqs. (14) and (15). The solution of eqs. (10) and (11) can be obtained via separation of variables. It is assumed that the solution of ''θ'' can be expressed as a product of the function of ''η'' and a function of ''ξ'', i.e., 
{ class="wikitable" border="0"  { class="wikitable" border="0"  
    
 width="100%" <center>   width="100%" <center>  
  <math>\theta =\Theta (\eta )\Gamma (\xi )</math>  +  <big><big><math>\theta =\Theta (\eta )\Gamma (\xi )</math></big></big> 
</center>  </center>  
{{EquationRef(16)}}  {{EquationRef(16)}}  
}  }  
  Substituting eq. (  +  Substituting eq. (16) into eq. (10), the energy equation becomes 
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}  }  
where <math>\beta </math> is the eigenvalue for the energy equation.  where <math>\beta </math> is the eigenvalue for the energy equation.  
  Equation (  +  Equation (17) can be rewritten as two ordinary differential equations: 
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 width="100%" <center>   width="100%" <center>  
  <math>{\Gamma }'+{{\beta }^{2}}\Gamma =0</math>  +  <math>\begin{matrix} {} & {} \\\end{matrix}{\Gamma }'+{{\beta }^{2}}\Gamma =0</math> 
</center>  </center>  
{{EquationRef(18)}}  {{EquationRef(18)}}  
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{{EquationRef(19)}}  {{EquationRef(19)}}  
}  }  
  The solution of eq.  +  The solution of eq. (18) is 
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{{EquationRef(20)}}  {{EquationRef(20)}}  
}  }  
  The boundary condition of eq. (  +  The boundary condition of eq. (19) at <math>\eta =0</math> is 
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 width="100%" <center>   width="100%" <center>  
  <math>{\Theta }'(0)=0</math>  +  <math>\begin{matrix} {} & {} \\\end{matrix}{\Theta }'(0)=0</math> 
</center>  </center>  
{{EquationRef(21)}}  {{EquationRef(21)}}  
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{{EquationRef(23)}}  {{EquationRef(23)}}  
}  }  
  where <math>\gamma </math> is the eigenvalue for the conservation of species equation, and  +  where <math>\gamma </math> is the eigenvalue for the conservation of species equation, and <math>\Phi (\eta )</math> satisfies 
  <math>\Phi (\eta )</math>  +  
  +  
  satisfies  +  
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{{EquationRef(24)}}  {{EquationRef(24)}}  
}  }  
  and the boundary condition of eq. (  +  and the boundary condition of eq. (24) at <math>\eta =0</math> is 
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 width="100%" <center>   width="100%" <center>  
  <math>{\Phi }'(0)=0</math>  +  <math>\begin{matrix} {} & {} \\\end{matrix}{\Phi }'(0)=0</math> 
</center>  </center>  
{{EquationRef(25)}}  {{EquationRef(25)}}  
}  }  
  Substituting eqs. (  +  Substituting eqs. (22) – (23) into eqs. (14) – (15), one obtains 
  +  
  +  
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 width="100%" <center>   width="100%" <center>  
  <math>  +  <math>\begin{matrix} {} & {} \\\end{matrix}\beta =\gamma </math> 
</center>  </center>  
{{EquationRef(26)}}  {{EquationRef(26)}}  
}  }  
  +  
  +  
  +  
  +  
  +  
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 width="100%" <center>   width="100%" <center>  
  <math>\  +  <math>\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{\Theta (1)}{\Phi (1)}=\text{Le}\frac{{\Theta }'(1)}{{\Phi }'(1)}</math> 
</center>  </center>  
{{EquationRef(27)}}  {{EquationRef(27)}}  
}  }  
+  To solve eqs. (19) and (24) using the RungeKutta method it is necessary to specify two boundary conditions for each. However, there is only one boundary condition for each: eqs. (21) and (25), respectively. Since both eqs. (19) and (24) are homogeneous, one can assume that the other boundary conditions are  
+  <math>\Theta (0)=\Phi (0)=1</math> and the solve eqs. (19) and (24) numerically. It is necessary to point out that the eigenvalue, ''β'', is still unknown at this point and must be obtained by eq. (27). There will be a series of ''β'' which satisfy eq. (27), and for each value of ''β<sub>n</sub>'' there is one set of corresponding ''Θ<sub>n</sub>'' and ''Φ<sub>n</sub>'' functions <math>(n=1,2,3,\cdots )</math>.  
+  
+  If we use any one of the eigenvalues, ''β<sub>n</sub>'', and corresponding eigenfunctions, ''Θ<sub>n</sub>'' and ''Φ<sub>n</sub>'', in eqs. (22) and (23), the solutions of eq. (10) and (11) become  
{ class="wikitable" border="0"  { class="wikitable" border="0"  
    
 width="100%" <center>   width="100%" <center>  
  <math>\  +  <math>\theta ={{C}_{1}}{{\Theta }_{n}}(\eta ){{e}^{{{\beta }_{n}}^{2}\xi }}</math> 
</center>  </center>  
{{EquationRef(28)}}  {{EquationRef(28)}}  
}  }  
  
{ class="wikitable" border="0"  { class="wikitable" border="0"  
    
 width="100%" <center>   width="100%" <center>  
  <math>\  +  <math>\varphi ={{C}_{2}}{{\Phi }_{n}}(\eta ){{e}^{\beta _{n}^{2}\xi }}</math> 
</center>  </center>  
{{EquationRef(29)}}  {{EquationRef(29)}}  
}  }  
+  which satisfy all boundary conditions except those at <math>\xi =0</math>. In order to satisfy boundary conditions at <math>\xi =0</math>, one can assume that the final solutions of eqs. (10) and (11) are  
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 width="100%" <center>   width="100%" <center>  
  <math>\  +  <math>\theta =\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{{{\beta }_{n}}^{2}\xi }}}</math> 
</center>  </center>  
{{EquationRef(30)}}  {{EquationRef(30)}}  
}  }  
  +  
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 width="100%" <center>   width="100%" <center>  
  <math>\  +  <math>\varphi =\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{\beta _{n}^{2}\xi }}}</math> 
</center>  </center>  
{{EquationRef(31)}}  {{EquationRef(31)}}  
}  }  
+  where <math>{{G}_{n}}</math> and <math>{{H}_{n}}</math> can be obtained by substituting eqs. (30) and (31) into eq. (12), i.e.,  
+  { class="wikitable" border="0"  
+    
+   width="100%" <center>  
<math>1=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta )}</math>  <math>1=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta )}</math>  
  +  </center>  
+  {{EquationRef(32)}}  
+  }  
+  
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<math>1=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta )}</math>  <math>1=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta )}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(33)}} 
}  }  
Due to the orthogonal nature of the eigenfunctions <math>{{\Theta }_{n}}</math> and <math>{{\Phi }_{n}}</math>, expressions of <math>{{G}_{n}}</math> and <math>{{H}_{n}}</math> can be obtained by  Due to the orthogonal nature of the eigenfunctions <math>{{\Theta }_{n}}</math> and <math>{{\Phi }_{n}}</math>, expressions of <math>{{G}_{n}}</math> and <math>{{H}_{n}}</math> can be obtained by  
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<math>{{G}_{n}}=\frac{\int_{0}^{1}{\eta (1{{\eta }^{2}}){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1{{\eta }^{2}}){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( A{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}</math>  <math>{{G}_{n}}=\frac{\int_{0}^{1}{\eta (1{{\eta }^{2}}){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1{{\eta }^{2}}){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( A{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(34)}} 
}  }  
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<math>{{H}_{n}}=\frac{A{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}</math>  <math>{{H}_{n}}=\frac{A{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(35)}} 
}  }  
The Nusselt number due to convection and the Sherwood number due to diffusion are  The Nusselt number due to convection and the Sherwood number due to diffusion are  
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<math>\text{Nu}=\frac{k{{\left. \frac{\partial T}{\partial r} \right}_{r={{r}_{o}}}}}{{{T}_{m}}{{T}_{w}}}\frac{2{{r}_{o}}}{k}=\frac{2}{{{\theta }_{m}}{{\theta }_{w}}}\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{\beta _{n}^{2}\xi }}{{{{\Theta }'}}_{n}}(1)}</math>  <math>\text{Nu}=\frac{k{{\left. \frac{\partial T}{\partial r} \right}_{r={{r}_{o}}}}}{{{T}_{m}}{{T}_{w}}}\frac{2{{r}_{o}}}{k}=\frac{2}{{{\theta }_{m}}{{\theta }_{w}}}\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{\beta _{n}^{2}\xi }}{{{{\Theta }'}}_{n}}(1)}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(36)}} 
}  }  
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<math>\text{Sh}=\frac{D{{\left. \frac{\partial \omega }{\partial r} \right}_{r={{r}_{o}}}}}{{{\omega }_{m}}{{\omega }_{w}}}\frac{2{{r}_{o}}}{D}=\frac{2}{{{\varphi }_{m}}{{\varphi }_{w}}}\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{\beta _{n}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}</math>  <math>\text{Sh}=\frac{D{{\left. \frac{\partial \omega }{\partial r} \right}_{r={{r}_{o}}}}}{{{\omega }_{m}}{{\omega }_{w}}}\frac{2{{r}_{o}}}{D}=\frac{2}{{{\varphi }_{m}}{{\varphi }_{w}}}\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{\beta _{n}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(37)}} 
}  }  
where <math>{{T}_{m}}\text{ and }{{\omega }_{m}}</math> are mean temperature and mean mass fraction in the tube.  where <math>{{T}_{m}}\text{ and }{{\omega }_{m}}</math> are mean temperature and mean mass fraction in the tube.  
  
  
==Sublimation inside a Tube Subjected to External Heating==  ==Sublimation inside a Tube Subjected to External Heating==  
  
  
  
[[Image:Fig5.9.pngthumb400 pxalt=Sublimation in a tube heated by a uniform heat flux Figure 3: Sublimation in a tube heated by a uniform heat flux.]]  [[Image:Fig5.9.pngthumb400 pxalt=Sublimation in a tube heated by a uniform heat flux Figure 3: Sublimation in a tube heated by a uniform heat flux.]]  
  (see  +  
+  When the inner wall of a tube with a sublimablematerialcoated outer wall is heated by a uniform heat flux, <math>{q}''</math>(see figure to the right), the latent heat will be supplied by part of the heat flux at the wall. The remaining part of the heat flux will be used to heat the gas flowing through the tube. The problem can be described by eqs. (1) – (8), except that the boundary condition at the inner wall of the tube is replaced by  
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<math>\rho {{h}_{sv}}D\frac{\partial \omega }{\partial r}+k\frac{\partial T}{\partial r}={q}''\text{ at }r={{r}_{o}}</math>  <math>\rho {{h}_{sv}}D\frac{\partial \omega }{\partial r}+k\frac{\partial T}{\partial r}={q}''\text{ at }r={{r}_{o}}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(38)}} 
}  }  
where the thermal resistance of the tube wall is neglected because the tube wall and the coated layer are very thin.  where the thermal resistance of the tube wall is neglected because the tube wall and the coated layer are very thin.  
  The governing equations for sublimation inside a tube heated by a uniform heat flux can be nondimensionalized by using the dimensionless variables defined in eq. (  +  The governing equations for sublimation inside a tube heated by a uniform heat flux can be nondimensionalized by using the dimensionless variables defined in eq. (9), except the following: 
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<math>\begin{matrix} \theta =\frac{k(T{{T}_{0}})}{{q}''{{r}_{o}}}, & \varphi = \\\end{matrix}\frac{{{h}_{sv}}(\omega {{\omega }_{sat,0}})}{{{c}_{p}}{q}''{{r}_{o}}}</math>  <math>\begin{matrix} \theta =\frac{k(T{{T}_{0}})}{{q}''{{r}_{o}}}, & \varphi = \\\end{matrix}\frac{{{h}_{sv}}(\omega {{\omega }_{sat,0}})}{{{c}_{p}}{q}''{{r}_{o}}}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(39)}} 
}  }  
  where  +  where <math>{{\omega }_{sat,0}}</math> is the saturation mass fraction corresponding to the inlet temperature ''T<sub>0</sub>''. The resulting dimensionless governing equations and boundary conditions are 
  <math>{{\omega }_{sat,0}}</math>  +  
  is the saturation mass fraction corresponding to the inlet temperature  +  
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<math>\eta (1{{\eta }^{2}})\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right)</math>  <math>\eta (1{{\eta }^{2}})\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right)</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(40)}} 
}  }  
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<math>\eta (1{{\eta }^{2}})\frac{\partial \varphi }{\partial \xi }=\frac{1}{\text{Le}}\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right)</math>  <math>\eta (1{{\eta }^{2}})\frac{\partial \varphi }{\partial \xi }=\frac{1}{\text{Le}}\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right)</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(41)}} 
}  }  
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<math>\theta =0\begin{matrix} , & \xi =0 \\\end{matrix}</math>  <math>\theta =0\begin{matrix} , & \xi =0 \\\end{matrix}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(42)}} 
}  }  
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<math>\varphi ={{\varphi }_{0}}\begin{matrix} , & \xi =0 \\\end{matrix}</math>  <math>\varphi ={{\varphi }_{0}}\begin{matrix} , & \xi =0 \\\end{matrix}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(43)}} 
}  }  
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<math>\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\\end{matrix}</math>  <math>\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\\end{matrix}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(44)}} 
}  }  
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<math>\frac{\partial \theta }{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial \varphi }{\partial \eta }=1\begin{matrix} , & \eta =1 \\\end{matrix}</math>  <math>\frac{\partial \theta }{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial \varphi }{\partial \eta }=1\begin{matrix} , & \eta =1 \\\end{matrix}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(45)}} 
}  }  
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<math>\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix} , & \eta =1 \\\end{matrix}</math>  <math>\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix} , & \eta =1 \\\end{matrix}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(46)}} 
}  }  
  where  +  where <math>{{\varphi }_{0}}=k{{h}_{sv}}(\omega {{\omega }_{sat,0}})/({{c}_{p}}{q}''{{r}_{o}})</math> in eq. (43). 
  <math>{{\varphi }_{0}}=k{{h}_{sv}}(\omega {{\omega }_{sat,0}})/({{c}_{p}}{q}''{{r}_{o}})</math>  +  The sublimation problem under consideration is not homogeneous, because eq. (45) is a nonhomogeneous boundary condition. The solution of the problem is consistent with its particular (fully developed) solution as well as the solution of the corresponding homogeneous problem <ref name="ZC1992">Zhang, Y., and Chen, Z.Q., 1992, “Analytical Solution of Coupled Laminar Heat Mass Transfer in a Tube with Uniform Heat Flux,” Journal of Thermal Science, Vol. 1, No. 3, pp. 184188.</ref>: 
  in eq. (  +  
  The sublimation problem under consideration is not homogeneous, because eq. (  +  
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 width="100%" <center>   width="100%" <center>  
  <math>\theta (\xi ,\eta )={{\theta }_{1}}(\xi ,\eta )+{{\theta }_{2}}(\xi ,\eta )</math>  +  <math>\begin{matrix} {} & {} \\\end{matrix}\theta (\xi ,\eta )={{\theta }_{1}}(\xi ,\eta )+{{\theta }_{2}}(\xi ,\eta )</math> 
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(47)}} 
}  }  
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<math>\varphi (\xi ,\eta )={{\varphi }_{1}}(\xi ,\eta )+{{\varphi }_{2}}(\xi ,\eta )</math>  <math>\varphi (\xi ,\eta )={{\varphi }_{1}}(\xi ,\eta )+{{\varphi }_{2}}(\xi ,\eta )</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(48)}} 
}  }  
  While the fully developed solutions of temperature and mass fraction, <math>{{\theta }_{1}}(\xi ,\eta )</math> and <math>{{\varphi }_{1}}(\xi ,\eta )</math>, respectively, must satisfy eqs.  +  While the fully developed solutions of temperature and mass fraction, <math>{{\theta }_{1}}(\xi ,\eta )</math> and <math>{{\varphi }_{1}}(\xi ,\eta )</math>, respectively, must satisfy eqs. (40) – (41) and (44) – (46), the corresponding homogeneous solutions of the temperature and mass fraction, <math>{{\theta }_{2}}(\xi ,\eta )</math> and <math>{{\varphi }_{2}}(\xi ,\eta )</math>, must satisfy eqs. (40), (41), (44), and (46), as well as the following conditions: 
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<math>{{\theta }_{2}}={{\theta }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}</math>  <math>{{\theta }_{2}}={{\theta }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(49)}} 
}  }  
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<math>{{\varphi }_{2}}={{\varphi }_{0}}{{\varphi }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}</math>  <math>{{\varphi }_{2}}={{\varphi }_{0}}{{\varphi }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(50)}} 
}  }  
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<math>\frac{\partial {{\theta }_{2}}}{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial {{\varphi }_{2}}}{\partial \eta }=0\begin{matrix} , & \eta =1 \\\end{matrix}</math>  <math>\frac{\partial {{\theta }_{2}}}{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial {{\varphi }_{2}}}{\partial \eta }=0\begin{matrix} , & \eta =1 \\\end{matrix}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(51)}} 
}  }  
The fully developed profiles of the temperature and mass fraction are  The fully developed profiles of the temperature and mass fraction are  
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<math>\begin{align} & {{\theta }_{1}}=\frac{1}{1+a{{h}_{sv}}/{{c}_{p}}}\left[ 4\xi +{{\eta }^{2}}\left( 1\frac{1}{4}{{\eta }^{2}} \right)+{{\varphi }_{0}} \right. \\ & \text{ }+\left. \frac{11\text{L}{{\text{e}}_{{}}}a{{h}_{sv}}/{{c}_{p}}18a{{h}_{sv}}/{{c}_{p}}7}{24(1+a{{h}_{sv}}/{{c}_{p}})} \right] \\ \end{align}</math>  <math>\begin{align} & {{\theta }_{1}}=\frac{1}{1+a{{h}_{sv}}/{{c}_{p}}}\left[ 4\xi +{{\eta }^{2}}\left( 1\frac{1}{4}{{\eta }^{2}} \right)+{{\varphi }_{0}} \right. \\ & \text{ }+\left. \frac{11\text{L}{{\text{e}}_{{}}}a{{h}_{sv}}/{{c}_{p}}18a{{h}_{sv}}/{{c}_{p}}7}{24(1+a{{h}_{sv}}/{{c}_{p}})} \right] \\ \end{align}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(52)}} 
}  }  
Line 472:  Line 469:  
<math>\begin{align} & {{\varphi }_{1}}=\frac{a{{h}_{sv}}/{{c}_{p}}}{1+a{{h}_{sv}}/{{c}_{p}}}\left[ 4\xi +\text{L}{{\text{e}}_{{}}}{{\eta }^{2}}\left( 1\frac{1}{4}{{\eta }^{2}} \right)+{{\varphi }_{0}} \right. \\ & \left. \text{ }\frac{7L{{e}_{{}}}a{{h}_{sv}}/{{c}_{p}}+18Le11}{24(1+a{{h}_{sv}}/{{c}_{p}})} \right] \\ \end{align}</math>  <math>\begin{align} & {{\varphi }_{1}}=\frac{a{{h}_{sv}}/{{c}_{p}}}{1+a{{h}_{sv}}/{{c}_{p}}}\left[ 4\xi +\text{L}{{\text{e}}_{{}}}{{\eta }^{2}}\left( 1\frac{1}{4}{{\eta }^{2}} \right)+{{\varphi }_{0}} \right. \\ & \left. \text{ }\frac{7L{{e}_{{}}}a{{h}_{sv}}/{{c}_{p}}+18Le11}{24(1+a{{h}_{sv}}/{{c}_{p}})} \right] \\ \end{align}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(53)}} 
}  }  
The solution of the corresponding homogeneous problem can be obtained by separation of variables:  The solution of the corresponding homogeneous problem can be obtained by separation of variables:  
Line 481:  Line 478:  
<math>{{\theta }_{2}}=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{{{\beta }_{n}}^{2}\xi }}}</math>  <math>{{\theta }_{2}}=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{{{\beta }_{n}}^{2}\xi }}}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(54)}} 
}  }  
Line 489:  Line 486:  
<math>{{\varphi }_{2}}=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{\beta _{n}^{2}\xi }}}</math>  <math>{{\varphi }_{2}}=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{\beta _{n}^{2}\xi }}}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(55)}} 
}  }  
where  where  
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<math>{{G}_{n}}=\frac{\int_{0}^{1}{\eta (1{{\eta }^{2}}){{\theta }_{2}}(0,\eta ){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1{{\eta }^{2}}){{\varphi }_{2}}(0,\eta ){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( a{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}</math>  <math>{{G}_{n}}=\frac{\int_{0}^{1}{\eta (1{{\eta }^{2}}){{\theta }_{2}}(0,\eta ){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1{{\eta }^{2}}){{\varphi }_{2}}(0,\eta ){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( a{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(56)}} 
}  }  
Line 506:  Line 503:  
<math>{{H}_{n}}=\frac{a{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}</math>  <math>{{H}_{n}}=\frac{a{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(57)}} 
}  }  
+  
and <math>{{\beta }_{n}}</math> is the eigenvalue of the corresponding homogeneous problem. The Nusselt number based on the total heat flux at the external wall is  and <math>{{\beta }_{n}}</math> is the eigenvalue of the corresponding homogeneous problem. The Nusselt number based on the total heat flux at the external wall is  
+  
{class="wikitable" border="0"  {class="wikitable" border="0"  
    
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<math>\begin{align} & \text{Nu}=\frac{2{q}''{{r}_{0}}}{k({{T}_{w}}{{T}_{m}})}=\frac{2}{{{\theta }_{w}}{{\theta }_{m}}} \\ & =\frac{2(1+A{{h}_{sv}}/{{c}_{p}})}{\frac{11}{24}+\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{{{\beta }_{n}}^{2}\xi }}\left[ {{\Theta }_{n}}(1)+\frac{4}{\beta _{n}^{2}}{{{{\Theta }'}}_{n}}(1) \right]}} \\ \end{align}</math>  <math>\begin{align} & \text{Nu}=\frac{2{q}''{{r}_{0}}}{k({{T}_{w}}{{T}_{m}})}=\frac{2}{{{\theta }_{w}}{{\theta }_{m}}} \\ & =\frac{2(1+A{{h}_{sv}}/{{c}_{p}})}{\frac{11}{24}+\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{{{\beta }_{n}}^{2}\xi }}\left[ {{\Theta }_{n}}(1)+\frac{4}{\beta _{n}^{2}}{{{{\Theta }'}}_{n}}(1) \right]}} \\ \end{align}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(58)}} 
}  }  
where <math>{{\theta }_{w}}</math> and <math>{{\theta }_{m}}</math> are dimensionless wall and mean temperatures, respectively.  where <math>{{\theta }_{w}}</math> and <math>{{\theta }_{m}}</math> are dimensionless wall and mean temperatures, respectively.  
  +  
+  The Nusselt number based on the convective heat transfer coefficient is  
{ class="wikitable" border="0"  { class="wikitable" border="0"  
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<math>\begin{align} & \text{N}{{\text{u}}^{*}}=\frac{2{{h}_{x}}{{r}_{o}}}{k}=\frac{2{{r}_{o}}}{{{T}_{w}}{{T}_{m}}}{{\left( \frac{\partial T}{\partial r} \right)}_{r={{r}_{o}}}}=\frac{2}{{{\theta }_{w}}{{\theta }_{m}}}{{\left( \frac{\partial \theta }{\partial \eta } \right)}_{\eta =1}} \\ & =\frac{2+2(1+a{{h}_{sv}}/{{c}_{p}})\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{{{\beta }_{n}}^{2}\xi }}{{{{\Theta }'}}_{n}}(1)}}{\frac{11}{24}+\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{{{\beta }_{n}}^{2}\xi }}\left[ {{\Theta }_{n}}(1)+\frac{4}{\beta _{n}^{2}}{{{{\Theta }'}}_{n}}(1) \right]}} \\ \end{align}</math>  <math>\begin{align} & \text{N}{{\text{u}}^{*}}=\frac{2{{h}_{x}}{{r}_{o}}}{k}=\frac{2{{r}_{o}}}{{{T}_{w}}{{T}_{m}}}{{\left( \frac{\partial T}{\partial r} \right)}_{r={{r}_{o}}}}=\frac{2}{{{\theta }_{w}}{{\theta }_{m}}}{{\left( \frac{\partial \theta }{\partial \eta } \right)}_{\eta =1}} \\ & =\frac{2+2(1+a{{h}_{sv}}/{{c}_{p}})\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{{{\beta }_{n}}^{2}\xi }}{{{{\Theta }'}}_{n}}(1)}}{\frac{11}{24}+\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{{{\beta }_{n}}^{2}\xi }}\left[ {{\Theta }_{n}}(1)+\frac{4}{\beta _{n}^{2}}{{{{\Theta }'}}_{n}}(1) \right]}} \\ \end{align}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(59)}} 
}  }  
The Sherwood number is  The Sherwood number is  
  <math>\text{Sh}=\frac{2{{h}_{m,x}}{{r}_{0}}}{D}=\frac{2{{r}_{0}}}{{{\omega }_{w}}{{\omega }_{m}}}{{\left. \frac{\partial \omega }{\partial r} \right}_{r={{r}_{o}}}}=\frac{2}{{{\varphi }_{w}}{{\varphi }_{m}}}{{\left. \frac{\partial \varphi }{\partial \eta } \right}_{\eta =1}}</math>  +  <center><math>\text{Sh}=\frac{2{{h}_{m,x}}{{r}_{0}}}{D}=\frac{2{{r}_{0}}}{{{\omega }_{w}}{{\omega }_{m}}}{{\left. \frac{\partial \omega }{\partial r} \right}_{r={{r}_{o}}}}=\frac{2}{{{\varphi }_{w}}{{\varphi }_{m}}}{{\left. \frac{\partial \varphi }{\partial \eta } \right}_{\eta =1}}</math></center> 
Line 537:  Line 537:  
<math>=\frac{2\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+2(1+\frac{a{{h}_{sv}}}{{{c}_{p}}})\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{{{\beta }_{n}}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}}{\frac{11}{24}\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{{{\beta }_{n}}^{2}\xi }}\left[ {{\Phi }_{n}}(1)+\frac{4}{\beta _{n}^{2}\text{Le}}{{{{\Phi }'}}_{n}}(1) \right]}}</math>  <math>=\frac{2\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+2(1+\frac{a{{h}_{sv}}}{{{c}_{p}}})\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{{{\beta }_{n}}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}}{\frac{11}{24}\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{{{\beta }_{n}}^{2}\xi }}\left[ {{\Phi }_{n}}(1)+\frac{4}{\beta _{n}^{2}\text{Le}}{{{{\Phi }'}}_{n}}(1) \right]}}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(60)}} 
}  }  
  When the heat and mass transfer are fully developed, eqs. (  +  When the heat and mass transfer are fully developed, eqs. (58) – (60) reduce to 
{ class="wikitable" border="0"  { class="wikitable" border="0"  
Line 546:  Line 546:  
<math>\text{Nu}=\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{48}{11}</math>  <math>\text{Nu}=\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{48}{11}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(61)}} 
}  }  
Line 554:  Line 554:  
<math>\text{N}{{\text{u}}^{*}}=\frac{48}{11}</math>  <math>\text{N}{{\text{u}}^{*}}=\frac{48}{11}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(62)}} 
}  }  
Line 562:  Line 562:  
<math>\text{Sh}=\frac{48}{11}</math>  <math>\text{Sh}=\frac{48}{11}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(63)}} 
}  }  
  +  
  +  ==References==  
  +  {{Reflist}}  
  +  
  + 
Current revision as of 07:34, 27 July 2010
There are many transport phenomena problems in which heat and mass transfer simultaneously occur. In some cases, such as sublimation and vapor deposition, they are coupled. These problems are usually treated as a single phase. However, coupled heat and mass transfer should both be considered even though they are modeled as being single phase^{[1]}. In this article, coupled forced internal convection in a circular tube will be presented for both adiabatic and constant wall heat flux.
Sublimation inside an Adiabatic Tube
In addition to the external sublimation, internal sublimation is also very important. Sublimation inside an adiabatic and externally heated tube will be analyzed. The physical model of the problem under consideration is shown in figure to the right ^{[2]}. The inner surface of a circular tube with radius r_{o} is coated with a layer of sublimable material which will sublime when gas flows through the tube. The fullydeveloped gas enters the tube with a uniform inlet mass fraction of the sublimable substance, ω_{0}, and a uniform inlet temperature, T_{0}. Since the outer wall surface is adiabatic, the latent heat of sublimation is supplied by the gas flow inside the tube; this in turn causes the change in gas temperature inside the tube. It is assumed that the flow inside the tube is incompressible laminar flow with constant properties. In order to solve the problem analytically, the following assumptions are made:
1. The entrance mass fraction, ω_{0}, is assumed to be equal to the saturation mass fraction at the entry temperature, T_{0}.
2. The saturation mass fraction can be expressed as a linear function of the corresponding temperature.
3. The mass transfer rate is small enough that the transverse velocity components can be neglected.
The fully developed velocity profile in the tube is

where u_{m} is the mean velocity of the gas flow inside the tube. Neglecting axial conduction and diffusion, the energy and mass transfer equations are


where D is mass diffusivity. Equations (2) and (3) are subjected to the following boundary conditions:




Equation (7) implies that the latent heat of sublimation is supplied as the gas flows inside the tube. Another boundary condition at the tube wall is obtained by setting the mass fraction at the wall as the saturation mass fraction at the wall temperature ^{[3]}. According to the second assumption, the mass fraction and temperature at the inner wall have the following relationship:

where a and b are constants.
The following nondimensional variables are then introduced:

where T_{f} and ω_{f} are temperature and mass fraction of the sublimable substance, respectively, after heat and mass transfer are fully developed, and Le is Lewis number. Equations (2) – (8) then become






The heat and mass transfer eqs. (10) and (11) are independent, but their boundary conditions are coupled by eqs. (14) and (15). The solution of eqs. (10) and (11) can be obtained via separation of variables. It is assumed that the solution of θ can be expressed as a product of the function of η and a function of ξ, i.e.,
θ = Θ(η)Γ(ξ) 
Substituting eq. (16) into eq. (10), the energy equation becomes

where β is the eigenvalue for the energy equation. Equation (17) can be rewritten as two ordinary differential equations:


The solution of eq. (18) is

The boundary condition of eq. (19) at η = 0 is

The dimensionless temperature is then

Similarly, the dimensionless mass fraction is

where γ is the eigenvalue for the conservation of species equation, and Φ(η) satisfies

and the boundary condition of eq. (24) at η = 0 is

Substituting eqs. (22) – (23) into eqs. (14) – (15), one obtains


To solve eqs. (19) and (24) using the RungeKutta method it is necessary to specify two boundary conditions for each. However, there is only one boundary condition for each: eqs. (21) and (25), respectively. Since both eqs. (19) and (24) are homogeneous, one can assume that the other boundary conditions are Θ(0) = Φ(0) = 1 and the solve eqs. (19) and (24) numerically. It is necessary to point out that the eigenvalue, β, is still unknown at this point and must be obtained by eq. (27). There will be a series of β which satisfy eq. (27), and for each value of β_{n} there is one set of corresponding Θ_{n} and Φ_{n} functions .
If we use any one of the eigenvalues, β_{n}, and corresponding eigenfunctions, Θ_{n} and Φ_{n}, in eqs. (22) and (23), the solutions of eq. (10) and (11) become


which satisfy all boundary conditions except those at ξ = 0. In order to satisfy boundary conditions at ξ = 0, one can assume that the final solutions of eqs. (10) and (11) are


where G_{n} and H_{n} can be obtained by substituting eqs. (30) and (31) into eq. (12), i.e.,


Due to the orthogonal nature of the eigenfunctions Θ_{n} and Φ_{n}, expressions of G_{n} and H_{n} can be obtained by


The Nusselt number due to convection and the Sherwood number due to diffusion are


where T_{m} and ω_{m} are mean temperature and mean mass fraction in the tube.
Sublimation inside a Tube Subjected to External Heating
When the inner wall of a tube with a sublimablematerialcoated outer wall is heated by a uniform heat flux, q''(see figure to the right), the latent heat will be supplied by part of the heat flux at the wall. The remaining part of the heat flux will be used to heat the gas flowing through the tube. The problem can be described by eqs. (1) – (8), except that the boundary condition at the inner wall of the tube is replaced by

where the thermal resistance of the tube wall is neglected because the tube wall and the coated layer are very thin.
The governing equations for sublimation inside a tube heated by a uniform heat flux can be nondimensionalized by using the dimensionless variables defined in eq. (9), except the following:

where ω_{sat,0} is the saturation mass fraction corresponding to the inlet temperature T_{0}. The resulting dimensionless governing equations and boundary conditions are







where in eq. (43). The sublimation problem under consideration is not homogeneous, because eq. (45) is a nonhomogeneous boundary condition. The solution of the problem is consistent with its particular (fully developed) solution as well as the solution of the corresponding homogeneous problem ^{[4]}:


While the fully developed solutions of temperature and mass fraction, θ_{1}(ξ,η) and , respectively, must satisfy eqs. (40) – (41) and (44) – (46), the corresponding homogeneous solutions of the temperature and mass fraction, θ_{2}(ξ,η) and , must satisfy eqs. (40), (41), (44), and (46), as well as the following conditions:



The fully developed profiles of the temperature and mass fraction are


The solution of the corresponding homogeneous problem can be obtained by separation of variables:


where


and β_{n} is the eigenvalue of the corresponding homogeneous problem. The Nusselt number based on the total heat flux at the external wall is

where θ_{w} and θ_{m} are dimensionless wall and mean temperatures, respectively.
The Nusselt number based on the convective heat transfer coefficient is

The Sherwood number is

When the heat and mass transfer are fully developed, eqs. (58) – (60) reduce to



References
 ↑ Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
 ↑ Zhang, Y., and Chen, Z.Q., 1990, “Analytical Solution of Coupled Laminar HeatMass Transfer inside a Tube with Adiabatic External Wall,” Proceedings of the 3rd National Interuniversity Conference on Engineering Thermophysics, Xi’an Jiaotong University Press, Xi’an, China, pp. 341345.
 ↑ Kurosaki, Y., 1973, “Coupled Heat and Mass Transfer in a Flow between Parallel Flat Plate (Uniform Heat Flux),” Journal of the Japan Society of Mechanical Engineers, Part B, Vol. 39, pp. 25122521 (in Japanese).
 ↑ Zhang, Y., and Chen, Z.Q., 1992, “Analytical Solution of Coupled Laminar Heat Mass Transfer in a Tube with Uniform Heat Flux,” Journal of Thermal Science, Vol. 1, No. 3, pp. 184188.