# Coupled thermal and concentration entry effects

(Difference between revisions)
 Revision as of 02:07, 27 June 2010 (view source)← Older edit Revision as of 06:32, 22 July 2010 (view source)Newer edit → Line 25: Line 25: - |{{EquationRef|(1)}} + |{{EquationRef|(2)}} |} |} Line 34: Line 34: - |{{EquationRef|(1)}} + |{{EquationRef|(3)}} |} |} where D is mass diffusivity. Equations (5.82) and (5.83) are subjected to the following boundary conditions: where D is mass diffusivity. Equations (5.82) and (5.83) are subjected to the following boundary conditions: Line 43: Line 43: - |{{EquationRef|(1)}} + |{{EquationRef|(4)}} |} |} {| class="wikitable" border="0" {| class="wikitable" border="0" Line 51: Line 51: - |{{EquationRef|(1)}} + |{{EquationRef|(5)}} |} |} Line 59: Line 59: $\frac{\partial T}{\partial r}=\frac{\partial \omega }{\partial r}=0\begin{matrix} , & r=0 \\\end{matrix}$ $\frac{\partial T}{\partial r}=\frac{\partial \omega }{\partial r}=0\begin{matrix} , & r=0 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(6)}} |} |} Line 67: Line 67: $-k\frac{\partial T}{\partial r}=\rho D{{h}_{sv}}\frac{\partial \omega }{\partial r}\begin{matrix} , & r={{r}_{o}} \\\end{matrix}$ $-k\frac{\partial T}{\partial r}=\rho D{{h}_{sv}}\frac{\partial \omega }{\partial r}\begin{matrix} , & r={{r}_{o}} \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(7)}} |} |} Equation (5.87) 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 (Kurosaki, 1973). According to the second assumption, the mass fraction and temperature at the inner wall have the following relationship: Equation (5.87) 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 (Kurosaki, 1973). According to the second assumption, the mass fraction and temperature at the inner wall have the following relationship: Line 76: Line 76: $\omega =aT+b\begin{matrix} , & r={{r}_{o}} \\\end{matrix}$ $\omega =aT+b\begin{matrix} , & r={{r}_{o}} \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(8)}} |} |} where a and b are constants. where a and b are constants. Line 86: Line 86: \begin{align} & \begin{matrix} \eta =\frac{r}{{{r}_{o}}}, & \xi =\frac{x}{{{r}_{0}}\text{Pe}}, & \text{Le}=\frac{\alpha }{D}, & \operatorname{Re}=\frac{2{{u}_{m}}{{r}_{o}}}{\nu } \\\end{matrix}, \\ & \begin{matrix} \text{Pe}=\frac{2{{u}_{m}}{{r}_{0}}}{\alpha }, & \theta =\frac{T-{{T}_{f}}}{{{T}_{0}}-{{T}_{f}}}, & \varphi =\frac{\omega -{{\omega }_{f}}}{{{\omega }_{0}}-{{\omega }_{f}}} & {} \\\end{matrix} \\ \end{align} \begin{align} & \begin{matrix} \eta =\frac{r}{{{r}_{o}}}, & \xi =\frac{x}{{{r}_{0}}\text{Pe}}, & \text{Le}=\frac{\alpha }{D}, & \operatorname{Re}=\frac{2{{u}_{m}}{{r}_{o}}}{\nu } \\\end{matrix}, \\ & \begin{matrix} \text{Pe}=\frac{2{{u}_{m}}{{r}_{0}}}{\alpha }, & \theta =\frac{T-{{T}_{f}}}{{{T}_{0}}-{{T}_{f}}}, & \varphi =\frac{\omega -{{\omega }_{f}}}{{{\omega }_{0}}-{{\omega }_{f}}} & {} \\\end{matrix} \\ \end{align} - |{{EquationRef|(1)}} + |{{EquationRef|(9)}} |} |} where Tf and ωf are temperature and mass fraction of the sublimable substance, respectively, after heat and mass transfer are fully developed, and Le is Lewis number. Equations  (5.82) –  (5.88) then become where Tf and ωf are temperature and mass fraction of the sublimable substance, respectively, after heat and mass transfer are fully developed, and Le is Lewis number. Equations  (5.82) –  (5.88) then become Line 95: Line 95: $\eta (1-{{\eta }^{2}})\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right)$ $\eta (1-{{\eta }^{2}})\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right)$ - |{{EquationRef|(1)}} + |{{EquationRef|(10)}} |} |} Line 103: Line 103: $\eta (1-{{\eta }^{2}})\frac{\partial \varphi }{\partial \xi }=\frac{1}{\text{Le}}\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right)$ $\eta (1-{{\eta }^{2}})\frac{\partial \varphi }{\partial \xi }=\frac{1}{\text{Le}}\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right)$ - |{{EquationRef|(1)}} + |{{EquationRef|(11)}} |} |} {| class="wikitable" border="0" {| class="wikitable" border="0" Line 110: Line 110: $\theta =\varphi =1\begin{matrix} , & \xi =0 \\\end{matrix}$ $\theta =\varphi =1\begin{matrix} , & \xi =0 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(12)}} |} |} Line 118: Line 118: $\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\\end{matrix}$ $\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(13)}} |} |} Line 126: Line 126: $-\frac{\partial \theta }{\partial \eta }=\frac{1}{\text{Le}}\frac{\partial \varphi }{\partial \eta }\begin{matrix} , & \eta =1 \\\end{matrix}$ $-\frac{\partial \theta }{\partial \eta }=\frac{1}{\text{Le}}\frac{\partial \varphi }{\partial \eta }\begin{matrix} , & \eta =1 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(14)}} |} |} Line 134: Line 134: $\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix} , & \eta =1 \\\end{matrix}$ $\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix} , & \eta =1 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(15)}} |} |} The heat and mass transfer eqs. (5.90) and (5.91) are independent, but their boundary conditions are coupled by eqs. (5.94) and (5.95). The solution of eqs. (5.90) and (5.91) 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., The heat and mass transfer eqs. (5.90) and (5.91) are independent, but their boundary conditions are coupled by eqs. (5.94) and (5.95). The solution of eqs. (5.90) and (5.91) 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., Line 143: Line 143: $\theta =\Theta (\eta )\Gamma (\xi )$ $\theta =\Theta (\eta )\Gamma (\xi )$ - |{{EquationRef|(1)}} + |{{EquationRef|(16)}} |} |} Substituting eq. (5.96) into eq. (5.90), the energy equation becomes Substituting eq. (5.96) into eq. (5.90), the energy equation becomes Line 152: Line 152: $\frac{{{\Gamma }'}}{\Gamma }=\frac{\frac{d}{d\eta }\left( \frac{d\Theta }{d\eta } \right)}{\eta (1-{{\eta }^{2}})\Theta }=-{{\beta }^{2}}$ $\frac{{{\Gamma }'}}{\Gamma }=\frac{\frac{d}{d\eta }\left( \frac{d\Theta }{d\eta } \right)}{\eta (1-{{\eta }^{2}})\Theta }=-{{\beta }^{2}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(17)}} |} |} where $\beta$ is the eigenvalue for the energy equation. where $\beta$ is the eigenvalue for the energy equation. Line 162: Line 162: ${\Gamma }'+{{\beta }^{2}}\Gamma =0$ ${\Gamma }'+{{\beta }^{2}}\Gamma =0$ - |{{EquationRef|(1)}} + |{{EquationRef|(18)}} |} |} Line 170: Line 170: $\frac{d}{d\eta }\left( \frac{d\Theta }{d\eta } \right)+{{\beta }^{2}}\eta (1-{{\eta }^{2}})\Theta =0$ $\frac{d}{d\eta }\left( \frac{d\Theta }{d\eta } \right)+{{\beta }^{2}}\eta (1-{{\eta }^{2}})\Theta =0$ - |{{EquationRef|(1)}} + |{{EquationRef|(19)}} |} |} The solution of eq.  (5.98) is The solution of eq.  (5.98) is Line 179: Line 179: $\Gamma ={{C}_{1}}{{e}^{-{{\beta }^{2}}\xi }}$ $\Gamma ={{C}_{1}}{{e}^{-{{\beta }^{2}}\xi }}$ - |{{EquationRef|(1)}} + |{{EquationRef|(20)}} |} |} The boundary condition of eq. (5.99) at $\eta =0$ is The boundary condition of eq. (5.99) at $\eta =0$ is Line 188: Line 188: ${\Theta }'(0)=0$ ${\Theta }'(0)=0$ - |{{EquationRef|(1)}} + |{{EquationRef|(21)}} |} |} The dimensionless temperature is then The dimensionless temperature is then Line 197: Line 197: $\theta ={{C}_{1}}\Theta (\eta ){{e}^{-{{\beta }^{2}}\xi }}$ $\theta ={{C}_{1}}\Theta (\eta ){{e}^{-{{\beta }^{2}}\xi }}$ - |{{EquationRef|(1)}} + |{{EquationRef|(22)}} |} |} Similarly, the dimensionless mass fraction is Similarly, the dimensionless mass fraction is Line 206: Line 206: $\varphi ={{C}_{2}}\Phi (\eta ){{e}^{-{{\gamma }^{2}}\xi }}$ $\varphi ={{C}_{2}}\Phi (\eta ){{e}^{-{{\gamma }^{2}}\xi }}$ - |{{EquationRef|(1)}} + |{{EquationRef|(23)}} |} |} where $\gamma$ is the eigenvalue for the conservation of species equation, and where $\gamma$ is the eigenvalue for the conservation of species equation, and Line 218: Line 218: $\frac{d}{d\eta }\left( \frac{d\Phi }{d\eta } \right)+\text{Le}{{\gamma }^{2}}\eta (1-{{\eta }^{2}})\Phi =0$ $\frac{d}{d\eta }\left( \frac{d\Phi }{d\eta } \right)+\text{Le}{{\gamma }^{2}}\eta (1-{{\eta }^{2}})\Phi =0$ - |{{EquationRef|(1)}} + |{{EquationRef|(24)}} |} |} and the boundary condition of eq. (5.104) at $\eta =0$ is and the boundary condition of eq. (5.104) at $\eta =0$ is Line 227: Line 227: ${\Phi }'(0)=0$ ${\Phi }'(0)=0$ - |{{EquationRef|(1)}} + |{{EquationRef|(25)}} |} |} Substituting eqs. (5.102) – (5.103) into eqs. (5.94) – (5.95), one obtains Substituting eqs. (5.102) – (5.103) into eqs. (5.94) – (5.95), one obtains Line 237: Line 237: $-\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{\Theta (1)}{\Phi (1)}=\text{Le}\frac{{\Theta }'(1)}{{\Phi }'(1)}$ $-\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{\Theta (1)}{\Phi (1)}=\text{Le}\frac{{\Theta }'(1)}{{\Phi }'(1)}$ - |{{EquationRef|(1)}} + |{{EquationRef|(26)}} |} |} To solve eqs. (5.99) and (5.104) using the Runge-Kutta method it is necessary to specify two boundary conditions for each. However, there is only one boundary condition for each: eqs. (5.101) and (5.105), respectively. Since both eqs. (5.99) and (5.104) are homogeneous, one can assume that the other boundary conditions are To solve eqs. (5.99) and (5.104) using the Runge-Kutta method it is necessary to specify two boundary conditions for each. However, there is only one boundary condition for each: eqs. (5.101) and (5.105), respectively. Since both eqs. (5.99) and (5.104) are homogeneous, one can assume that the other boundary conditions are Line 249: Line 249: $\theta ={{C}_{1}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}$ $\theta ={{C}_{1}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}$ - |{{EquationRef|(1)}} + |{{EquationRef|(27)}} |} |} Line 257: Line 257: $\varphi ={{C}_{2}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}$ $\varphi ={{C}_{2}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}$ - |{{EquationRef|(1)}} + |{{EquationRef|(28)}} |} |} which satisfy all boundary conditions except those at $\xi =0$. In order to satisfy boundary conditions at $\xi =0$, one can assume that the final solutions of eqs. (5.90) and (5.91) are which satisfy all boundary conditions except those at $\xi =0$. In order to satisfy boundary conditions at $\xi =0$, one can assume that the final solutions of eqs. (5.90) and (5.91) are Line 266: Line 266: $\theta =\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}}$ $\theta =\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(29)}} |} |} Line 274: Line 274: $\varphi =\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}}$ $\varphi =\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(30)}} |} |} where ${{G}_{n}}$ and ${{H}_{n}}$ can be obtained by substituting eqs. (5.110) and (5.111) into eq. (5.92), i.e., where ${{G}_{n}}$ and ${{H}_{n}}$ can be obtained by substituting eqs. (5.110) and (5.111) into eq. (5.92), i.e., Line 282: Line 282: $\text{Sh}=\frac{-D{{\left. \frac{\partial \omega }{\partial r} \right|}_{r={{r}_{o}}}}}{{{\omega }_{m}}-{{\omega }_{w}}}\frac{2{{r}_{o}}}{D}=-\frac{2}{{{\varphi }_{m}}-{{\varphi }_{w}}}\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}$ $\text{Sh}=\frac{-D{{\left. \frac{\partial \omega }{\partial r} \right|}_{r={{r}_{o}}}}}{{{\omega }_{m}}-{{\omega }_{w}}}\frac{2{{r}_{o}}}{D}=-\frac{2}{{{\varphi }_{m}}-{{\varphi }_{w}}}\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}$ - |{{EquationRef|(1)}} + |{{EquationRef|(31)}} |} |} Line 292: Line 292: $1=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta )}$ $1=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta )}$ - |{{EquationRef|(1)}} + |{{EquationRef|(32)}} |} |} Due to the orthogonal nature of the eigenfunctions ${{\Theta }_{n}}$ and ${{\Phi }_{n}}$, expressions of  ${{G}_{n}}$ and ${{H}_{n}}$ can be obtained by Due to the orthogonal nature of the eigenfunctions ${{\Theta }_{n}}$ and ${{\Phi }_{n}}$, expressions of  ${{G}_{n}}$ and ${{H}_{n}}$ can be obtained by Line 301: Line 301: ${{G}_{n}}=\frac{\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1-{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( A{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}$ ${{G}_{n}}=\frac{\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1-{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( A{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}$ - |{{EquationRef|(1)}} + |{{EquationRef|(33)}} |} |} Line 309: Line 309: ${{H}_{n}}=\frac{A{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}$ ${{H}_{n}}=\frac{A{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(34)}} |} |} 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 Line 318: Line 318: $\text{Nu}=\frac{-k{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}}{{{T}_{m}}-{{T}_{w}}}\frac{2{{r}_{o}}}{k}=-\frac{2}{{{\theta }_{m}}-{{\theta }_{w}}}\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Theta }'}}_{n}}(1)}$ $\text{Nu}=\frac{-k{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}}{{{T}_{m}}-{{T}_{w}}}\frac{2{{r}_{o}}}{k}=-\frac{2}{{{\theta }_{m}}-{{\theta }_{w}}}\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Theta }'}}_{n}}(1)}$ - |{{EquationRef|(1)}} + |{{EquationRef|(35)}} |} |} Line 326: Line 326: $\text{Sh}=\frac{-D{{\left. \frac{\partial \omega }{\partial r} \right|}_{r={{r}_{o}}}}}{{{\omega }_{m}}-{{\omega }_{w}}}\frac{2{{r}_{o}}}{D}=-\frac{2}{{{\varphi }_{m}}-{{\varphi }_{w}}}\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}$ $\text{Sh}=\frac{-D{{\left. \frac{\partial \omega }{\partial r} \right|}_{r={{r}_{o}}}}}{{{\omega }_{m}}-{{\omega }_{w}}}\frac{2{{r}_{o}}}{D}=-\frac{2}{{{\varphi }_{m}}-{{\varphi }_{w}}}\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}$ - |{{EquationRef|(1)}} + |{{EquationRef|(36)}} |} |} where ${{T}_{m}}\text{ and }{{\omega }_{m}}$ are mean temperature and mean mass fraction in the tube. where ${{T}_{m}}\text{ and }{{\omega }_{m}}$ are mean temperature and mean mass fraction in the tube. Line 344: Line 344: $\rho {{h}_{sv}}D\frac{\partial \omega }{\partial r}+k\frac{\partial T}{\partial r}={q}''\text{ at }r={{r}_{o}}$ $\rho {{h}_{sv}}D\frac{\partial \omega }{\partial r}+k\frac{\partial T}{\partial r}={q}''\text{ at }r={{r}_{o}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(37)}} |} |} 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. Line 355: Line 355: $\begin{matrix} \theta =\frac{k(T-{{T}_{0}})}{{q}''{{r}_{o}}}, & \varphi = \\\end{matrix}\frac{{{h}_{sv}}(\omega -{{\omega }_{sat,0}})}{{{c}_{p}}{q}''{{r}_{o}}}$ $\begin{matrix} \theta =\frac{k(T-{{T}_{0}})}{{q}''{{r}_{o}}}, & \varphi = \\\end{matrix}\frac{{{h}_{sv}}(\omega -{{\omega }_{sat,0}})}{{{c}_{p}}{q}''{{r}_{o}}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(38)}} |} |} where where Line 366: Line 366: $\eta (1-{{\eta }^{2}})\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right)$ $\eta (1-{{\eta }^{2}})\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right)$ - |{{EquationRef|(1)}} + |{{EquationRef|(39)}} |} |} Line 374: Line 374: $\eta (1-{{\eta }^{2}})\frac{\partial \varphi }{\partial \xi }=\frac{1}{\text{Le}}\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right)$ $\eta (1-{{\eta }^{2}})\frac{\partial \varphi }{\partial \xi }=\frac{1}{\text{Le}}\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right)$ - |{{EquationRef|(1)}} + |{{EquationRef|(40)}} |} |} {| class="wikitable" border="0" {| class="wikitable" border="0" Line 381: Line 381: $\theta =0\begin{matrix} , & \xi =0 \\\end{matrix}$ $\theta =0\begin{matrix} , & \xi =0 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(41)}} |} |} {| class="wikitable" border="0" {| class="wikitable" border="0" Line 388: Line 388: $\varphi ={{\varphi }_{0}}\begin{matrix} , & \xi =0 \\\end{matrix}$ $\varphi ={{\varphi }_{0}}\begin{matrix} , & \xi =0 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(42)}} |} |} Line 396: Line 396: $\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\\end{matrix}$ $\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(43)}} |} |} Line 404: Line 404: $\frac{\partial \theta }{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial \varphi }{\partial \eta }=1\begin{matrix} , & \eta =1 \\\end{matrix}$ $\frac{\partial \theta }{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial \varphi }{\partial \eta }=1\begin{matrix} , & \eta =1 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(44)}} |} |} Line 412: Line 412: $\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix} , & \eta =1 \\\end{matrix}$ $\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix} , & \eta =1 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(45)}} |} |} where where Line 424: Line 424: $\theta (\xi ,\eta )={{\theta }_{1}}(\xi ,\eta )+{{\theta }_{2}}(\xi ,\eta )$ $\theta (\xi ,\eta )={{\theta }_{1}}(\xi ,\eta )+{{\theta }_{2}}(\xi ,\eta )$ - |{{EquationRef|(1)}} + |{{EquationRef|(46)}} |} |} Line 432: Line 432: $\varphi (\xi ,\eta )={{\varphi }_{1}}(\xi ,\eta )+{{\varphi }_{2}}(\xi ,\eta )$ $\varphi (\xi ,\eta )={{\varphi }_{1}}(\xi ,\eta )+{{\varphi }_{2}}(\xi ,\eta )$ - |{{EquationRef|(1)}} + |{{EquationRef|(47)}} |} |} While the fully developed solutions of temperature and mass fraction, ${{\theta }_{1}}(\xi ,\eta )$ and ${{\varphi }_{1}}(\xi ,\eta )$, respectively, must satisfy eqs.  (5.120) – (5.121) and (5.124) – (5.126), the corresponding homogeneous solutions of the temperature and mass fraction, ${{\theta }_{2}}(\xi ,\eta )$ and ${{\varphi }_{2}}(\xi ,\eta )$, must satisfy eqs. (5.120), (5.121), (5.124), and (5.126), as well as the following conditions: While the fully developed solutions of temperature and mass fraction, ${{\theta }_{1}}(\xi ,\eta )$ and ${{\varphi }_{1}}(\xi ,\eta )$, respectively, must satisfy eqs.  (5.120) – (5.121) and (5.124) – (5.126), the corresponding homogeneous solutions of the temperature and mass fraction, ${{\theta }_{2}}(\xi ,\eta )$ and ${{\varphi }_{2}}(\xi ,\eta )$, must satisfy eqs. (5.120), (5.121), (5.124), and (5.126), as well as the following conditions: Line 440: Line 440: ${{\theta }_{2}}=-{{\theta }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}$ ${{\theta }_{2}}=-{{\theta }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(48)}} |} |} {| class="wikitable" border="0" {| class="wikitable" border="0" Line 447: Line 447: ${{\varphi }_{2}}={{\varphi }_{0}}-{{\varphi }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}$ ${{\varphi }_{2}}={{\varphi }_{0}}-{{\varphi }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(49)}} |} |} Line 455: Line 455: $\frac{\partial {{\theta }_{2}}}{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial {{\varphi }_{2}}}{\partial \eta }=0\begin{matrix} , & \eta =1 \\\end{matrix}$ $\frac{\partial {{\theta }_{2}}}{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial {{\varphi }_{2}}}{\partial \eta }=0\begin{matrix} , & \eta =1 \\\end{matrix}$ - |{{EquationRef|(1)}} + |{{EquationRef|(50)}} |} |} The fully developed profiles of the temperature and mass fraction are The fully developed profiles of the temperature and mass fraction are Line 464: Line 464: \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} \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} - |{{EquationRef|(1)}} + |{{EquationRef|(51)}} |} |} Line 472: Line 472: \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}}+18Le-11}{24(1+a{{h}_{sv}}/{{c}_{p}})} \right] \\ \end{align} \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}}+18Le-11}{24(1+a{{h}_{sv}}/{{c}_{p}})} \right] \\ \end{align} - |{{EquationRef|(1)}} + |{{EquationRef|(52)}} |} |} 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 481: ${{\theta }_{2}}=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}}$ ${{\theta }_{2}}=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(53)}} |} |} Line 489: Line 489: ${{\varphi }_{2}}=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}}$ ${{\varphi }_{2}}=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(54)}} |} |} where where Line 498: Line 498: ${{G}_{n}}=\frac{\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\theta }_{2}}(0,\eta ){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\varphi }_{2}}(0,\eta ){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1-{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( a{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}$ ${{G}_{n}}=\frac{\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\theta }_{2}}(0,\eta ){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\varphi }_{2}}(0,\eta ){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1-{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( a{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}$ - |{{EquationRef|(1)}} + |{{EquationRef|(55)}} |} |} Line 506: Line 506: ${{H}_{n}}=\frac{a{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}$ ${{H}_{n}}=\frac{a{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(56)}} |} |} and ${{\beta }_{n}}$ is the eigenvalue of the corresponding homogeneous problem. The Nusselt number based on the total heat flux at the external wall is and ${{\beta }_{n}}$ is the eigenvalue of the corresponding homogeneous problem. The Nusselt number based on the total heat flux at the external wall is Line 514: Line 514: \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} \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} - |{{EquationRef|(1)}} + |{{EquationRef|(57)}} |} |} where ${{\theta }_{w}}$ and ${{\theta }_{m}}$ are dimensionless wall and mean temperatures, respectively. where ${{\theta }_{w}}$ and ${{\theta }_{m}}$ are dimensionless wall and mean temperatures, respectively. Line 524: Line 524: \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} \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} - |{{EquationRef|(1)}} + |{{EquationRef|(58)}} |} |} Line 537: Line 537: $=\frac{2\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+2(1+\frac{a{{h}_{sv}}}{{{c}_{p}}})\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-{{\beta }_{n}}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}}{\frac{11}{24}\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{-{{\beta }_{n}}^{2}\xi }}\left[ {{\Phi }_{n}}(1)+\frac{4}{\beta _{n}^{2}\text{Le}}{{{{\Phi }'}}_{n}}(1) \right]}}$ $=\frac{2\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+2(1+\frac{a{{h}_{sv}}}{{{c}_{p}}})\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-{{\beta }_{n}}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}}{\frac{11}{24}\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{-{{\beta }_{n}}^{2}\xi }}\left[ {{\Phi }_{n}}(1)+\frac{4}{\beta _{n}^{2}\text{Le}}{{{{\Phi }'}}_{n}}(1) \right]}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(59)}} |} |} When the heat and mass transfer are fully developed, eqs. (5.138) – (5.140) reduce to When the heat and mass transfer are fully developed, eqs. (5.138) – (5.140) reduce to Line 546: Line 546: $\text{Nu}=\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{48}{11}$ $\text{Nu}=\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{48}{11}$ - |{{EquationRef|(1)}} + |{{EquationRef|(60)}} |} |} Line 554: Line 554: $\text{N}{{\text{u}}^{*}}=\frac{48}{11}$ $\text{N}{{\text{u}}^{*}}=\frac{48}{11}$ - |{{EquationRef|(1)}} + |{{EquationRef|(61)}} |} |} Line 562: Line 562: $\text{Sh}=\frac{48}{11}$ $\text{Sh}=\frac{48}{11}$ - |{{EquationRef|(1)}} + |{{EquationRef|(62)}} |} |} [[Image:Fig5.10.png|thumb|400 px|alt=Nusselt number based on total heat flux |Figure 4: Nusselt number based on total heat flux.]] [[Image:Fig5.10.png|thumb|400 px|alt=Nusselt number based on total heat flux |Figure 4: Nusselt number based on total heat flux.]]

## Revision as of 06:32, 22 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. In this section, 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 discussed in subsection 5.6.2, internal sublimation is also very important. Sublimation inside an adiabatic and externally heated tube will be analyzed in the current and the following subsections. The physical model of the problem under consideration is shown in Fig. 1 (Zhang and Chen, 1990). The inner surface of a circular tube with radius ro is coated with a layer of sublimable material which will sublime when gas flows through the tube. The fully-developed gas enters the tube with a uniform inlet mass fraction of the sublimable substance, ω0, and a uniform inlet temperature, T0. 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, ω0, is assumed to be equal to the saturation mass fraction at the entry temperature, T0. 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 $u=2{{u}_{m}}\left[ 1-{{\left( \frac{r}{{{r}_{o}}} \right)}^{2}} \right]$ (1)

where um is the mean velocity of the gas flow inside the tube. Neglecting axial conduction and diffusion, the energy and mass transfer equations are $ur\frac{\partial T}{\partial x}=\alpha \frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right)$ (2) $ur\frac{\partial \omega }{\partial x}=D\frac{\partial }{\partial r}\left( r\frac{\partial \omega }{\partial r} \right)$ (3)

where D is mass diffusivity. Equations (5.82) and (5.83) are subjected to the following boundary conditions: $T={{T}_{0}}\begin{matrix} , & x=0 \\\end{matrix}$ (4) $\omega ={{\omega }_{0}}\begin{matrix} , & x=0 \\\end{matrix}$ (5) $\frac{\partial T}{\partial r}=\frac{\partial \omega }{\partial r}=0\begin{matrix} , & r=0 \\\end{matrix}$ (6) $-k\frac{\partial T}{\partial r}=\rho D{{h}_{sv}}\frac{\partial \omega }{\partial r}\begin{matrix} , & r={{r}_{o}} \\\end{matrix}$ (7)

Equation (5.87) 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 (Kurosaki, 1973). According to the second assumption, the mass fraction and temperature at the inner wall have the following relationship: $\omega =aT+b\begin{matrix} , & r={{r}_{o}} \\\end{matrix}$ (8)

where a and b are constants. The following non-dimensional variables are then introduced: \begin{align} & \begin{matrix} \eta =\frac{r}{{{r}_{o}}}, & \xi =\frac{x}{{{r}_{0}}\text{Pe}}, & \text{Le}=\frac{\alpha }{D}, & \operatorname{Re}=\frac{2{{u}_{m}}{{r}_{o}}}{\nu } \\\end{matrix}, \\ & \begin{matrix} \text{Pe}=\frac{2{{u}_{m}}{{r}_{0}}}{\alpha }, & \theta =\frac{T-{{T}_{f}}}{{{T}_{0}}-{{T}_{f}}}, & \varphi =\frac{\omega -{{\omega }_{f}}}{{{\omega }_{0}}-{{\omega }_{f}}} & {} \\\end{matrix} \\ \end{align} (9)

where Tf and ωf are temperature and mass fraction of the sublimable substance, respectively, after heat and mass transfer are fully developed, and Le is Lewis number. Equations (5.82) – (5.88) then become $\eta (1-{{\eta }^{2}})\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right)$ (10) $\eta (1-{{\eta }^{2}})\frac{\partial \varphi }{\partial \xi }=\frac{1}{\text{Le}}\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right)$ (11) $\theta =\varphi =1\begin{matrix} , & \xi =0 \\\end{matrix}$ (12) $\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\\end{matrix}$ (13) $-\frac{\partial \theta }{\partial \eta }=\frac{1}{\text{Le}}\frac{\partial \varphi }{\partial \eta }\begin{matrix} , & \eta =1 \\\end{matrix}$ (14) $\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix} , & \eta =1 \\\end{matrix}$ (15)

The heat and mass transfer eqs. (5.90) and (5.91) are independent, but their boundary conditions are coupled by eqs. (5.94) and (5.95). The solution of eqs. (5.90) and (5.91) 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) (16)

Substituting eq. (5.96) into eq. (5.90), the energy equation becomes $\frac{{{\Gamma }'}}{\Gamma }=\frac{\frac{d}{d\eta }\left( \frac{d\Theta }{d\eta } \right)}{\eta (1-{{\eta }^{2}})\Theta }=-{{\beta }^{2}}$ (17)

where β is the eigenvalue for the energy equation. Equation (5.97) can be rewritten as two ordinary differential equations:

 Γ' + β2Γ = 0 (18) $\frac{d}{d\eta }\left( \frac{d\Theta }{d\eta } \right)+{{\beta }^{2}}\eta (1-{{\eta }^{2}})\Theta =0$ (19)

The solution of eq. (5.98) is $\Gamma ={{C}_{1}}{{e}^{-{{\beta }^{2}}\xi }}$ (20)

The boundary condition of eq. (5.99) at η = 0 is

 Θ'(0) = 0 (21)

The dimensionless temperature is then $\theta ={{C}_{1}}\Theta (\eta ){{e}^{-{{\beta }^{2}}\xi }}$ (22)

Similarly, the dimensionless mass fraction is $\varphi ={{C}_{2}}\Phi (\eta ){{e}^{-{{\gamma }^{2}}\xi }}$ (23)

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

satisfies $\frac{d}{d\eta }\left( \frac{d\Phi }{d\eta } \right)+\text{Le}{{\gamma }^{2}}\eta (1-{{\eta }^{2}})\Phi =0$ (24)

and the boundary condition of eq. (5.104) at η = 0 is

 Φ'(0) = 0 (25)

Substituting eqs. (5.102) – (5.103) into eqs. (5.94) – (5.95), one obtains β = γ (5.106) $-\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{\Theta (1)}{\Phi (1)}=\text{Le}\frac{{\Theta }'(1)}{{\Phi }'(1)}$ (26)

To solve eqs. (5.99) and (5.104) using the Runge-Kutta method it is necessary to specify two boundary conditions for each. However, there is only one boundary condition for each: eqs. (5.101) and (5.105), respectively. Since both eqs. (5.99) and (5.104) are homogeneous, one can assume that the other boundary conditions are Θ(0) = Φ(0) = 1 and the solve eqs. (5.99) and (5.104) numerically. It is necessary to point out that the eigenvalue, β, is still unknown at this point and must be obtained by eq. (5.107). There will be a series of β which satisfy eq. (5.107), and for each value of βn there is one set of corresponding Θn and Φn functions $(n=1,2,3,\cdots )$. If we use any one of the eigenvalues, βn, and corresponding eigenfunctions, Θn and Φn, in eqs. (5.102) and (5.103), the solutions of eq. (5.90) and (5.91) become $\theta ={{C}_{1}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}$ (27) $\varphi ={{C}_{2}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}$ (28)

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. (5.90) and (5.91) are $\theta =\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}}$ (29) $\varphi =\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}}$ (30)

where Gn and Hn can be obtained by substituting eqs. (5.110) and (5.111) into eq. (5.92), i.e., $\text{Sh}=\frac{-D{{\left. \frac{\partial \omega }{\partial r} \right|}_{r={{r}_{o}}}}}{{{\omega }_{m}}-{{\omega }_{w}}}\frac{2{{r}_{o}}}{D}=-\frac{2}{{{\varphi }_{m}}-{{\varphi }_{w}}}\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}$ (31) $1=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta )}$ (5.112) $1=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta )}$ (32)

Due to the orthogonal nature of the eigenfunctions Θn and Φn, expressions of Gn and Hn can be obtained by ${{G}_{n}}=\frac{\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1-{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( A{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}$ (33) ${{H}_{n}}=\frac{A{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}$ (34)

The Nusselt number due to convection and the Sherwood number due to diffusion are $\text{Nu}=\frac{-k{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}}{{{T}_{m}}-{{T}_{w}}}\frac{2{{r}_{o}}}{k}=-\frac{2}{{{\theta }_{m}}-{{\theta }_{w}}}\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Theta }'}}_{n}}(1)}$ (35) $\text{Sh}=\frac{-D{{\left. \frac{\partial \omega }{\partial r} \right|}_{r={{r}_{o}}}}}{{{\omega }_{m}}-{{\omega }_{w}}}\frac{2{{r}_{o}}}{D}=-\frac{2}{{{\varphi }_{m}}-{{\varphi }_{w}}}\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}$ (36)

where Tm and ωm are mean temperature and mean mass fraction in the tube.

Figure 2 shows heat and mass transfer performance during sublimation inside an adiabatic tube. For all cases, both Nusselt and Sherwood numbers become constant when ξ is greater than a certain number, thus indicating that heat and mass transfer in the tube have become fully developed. The length of the entrance flow increases with an increasing Lewis number. While the fully developed Nusselt number increases with an increasing Lewis number, the Sherwood number decreases with an increasing Lewis number, because a larger Lewis number indicates larger thermal diffusivity or low mass diffusivity. The effect of (ahsv / cp) on the Nusselt and Sherwood numbers is relatively insignificant: both the Nusselt and Sherwood numbers increase with increasing for Le < 1, but increasing for Le > 1 results in decreasing Nusselt and Sherwood numbers.

## Sublimation inside a Tube Subjected to External Heating

When the inner wall of a tube with a sublimable-material-coated outer wall is heated by a uniform heat flux, q''

(see Fig. 3), 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. (5.81) – (5.88), except that the boundary condition at the inner wall of the tube is replaced by $\rho {{h}_{sv}}D\frac{\partial \omega }{\partial r}+k\frac{\partial T}{\partial r}={q}''\text{ at }r={{r}_{o}}$ (37)

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 non-dimensionalized by using the dimensionless variables defined in eq. (5.89), except the following: $\begin{matrix} \theta =\frac{k(T-{{T}_{0}})}{{q}''{{r}_{o}}}, & \varphi = \\\end{matrix}\frac{{{h}_{sv}}(\omega -{{\omega }_{sat,0}})}{{{c}_{p}}{q}''{{r}_{o}}}$ (38)

where ωsat,0 is the saturation mass fraction corresponding to the inlet temperature T0. The resulting dimensionless governing equations and boundary conditions are $\eta (1-{{\eta }^{2}})\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right)$ (39) $\eta (1-{{\eta }^{2}})\frac{\partial \varphi }{\partial \xi }=\frac{1}{\text{Le}}\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right)$ (40) $\theta =0\begin{matrix} , & \xi =0 \\\end{matrix}$ (41) $\varphi ={{\varphi }_{0}}\begin{matrix} , & \xi =0 \\\end{matrix}$ (42) $\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\\end{matrix}$ (43) $\frac{\partial \theta }{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial \varphi }{\partial \eta }=1\begin{matrix} , & \eta =1 \\\end{matrix}$ (44) $\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix} , & \eta =1 \\\end{matrix}$ (45)

where ${{\varphi }_{0}}=k{{h}_{sv}}(\omega -{{\omega }_{sat,0}})/({{c}_{p}}{q}''{{r}_{o}})$ in eq. (5.123). The sublimation problem under consideration is not homogeneous, because eq. (5.125) 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 (Zhang and Chen, 1992):

 θ(ξ,η) = θ1(ξ,η) + θ2(ξ,η) (46) $\varphi (\xi ,\eta )={{\varphi }_{1}}(\xi ,\eta )+{{\varphi }_{2}}(\xi ,\eta )$ (47)

While the fully developed solutions of temperature and mass fraction, θ1(ξ,η) and ${{\varphi }_{1}}(\xi ,\eta )$, respectively, must satisfy eqs. (5.120) – (5.121) and (5.124) – (5.126), the corresponding homogeneous solutions of the temperature and mass fraction, θ2(ξ,η) and ${{\varphi }_{2}}(\xi ,\eta )$, must satisfy eqs. (5.120), (5.121), (5.124), and (5.126), as well as the following conditions: ${{\theta }_{2}}=-{{\theta }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}$ (48) ${{\varphi }_{2}}={{\varphi }_{0}}-{{\varphi }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}$ (49) $\frac{\partial {{\theta }_{2}}}{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial {{\varphi }_{2}}}{\partial \eta }=0\begin{matrix} , & \eta =1 \\\end{matrix}$ (50)

The fully developed profiles of the temperature and mass fraction are \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} (51) \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}}+18Le-11}{24(1+a{{h}_{sv}}/{{c}_{p}})} \right] \\ \end{align} (52)

The solution of the corresponding homogeneous problem can be obtained by separation of variables: ${{\theta }_{2}}=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}}$ (53) ${{\varphi }_{2}}=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}}$ (54)

where ${{G}_{n}}=\frac{\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\theta }_{2}}(0,\eta ){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\varphi }_{2}}(0,\eta ){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1-{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( a{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}$ (55) ${{H}_{n}}=\frac{a{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}$ (56)

and βn is the eigenvalue of the corresponding homogeneous problem. The Nusselt number based on the total heat flux at the external wall is \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} (57)

where θw and θm are dimensionless wall and mean temperatures, respectively.

The Nusselt  number based on the convective heat transfer coefficient is \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} (58)

The Sherwood number is $\text{Sh}=\frac{2{{h}_{m,x}}{{r}_{0}}}{D}=\frac{2{{r}_{0}}}{{{\omega }_{w}}-{{\omega }_{m}}}{{\left. \frac{\partial \omega }{\partial r} \right|}_{r={{r}_{o}}}}=\frac{2}{{{\varphi }_{w}}-{{\varphi }_{m}}}{{\left. \frac{\partial \varphi }{\partial \eta } \right|}_{\eta =1}}$ $=\frac{2\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+2(1+\frac{a{{h}_{sv}}}{{{c}_{p}}})\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-{{\beta }_{n}}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}}{\frac{11}{24}\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{-{{\beta }_{n}}^{2}\xi }}\left[ {{\Phi }_{n}}(1)+\frac{4}{\beta _{n}^{2}\text{Le}}{{{{\Phi }'}}_{n}}(1) \right]}}$ (59)

When the heat and mass transfer are fully developed, eqs. (5.138) – (5.140) reduce to $\text{Nu}=\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{48}{11}$ (60) $\text{N}{{\text{u}}^{*}}=\frac{48}{11}$ (61) $\text{Sh}=\frac{48}{11}$ (62)

The variations of the local Nusselt number based on total heat flux along the dimensionless location ξ are shown in Fig. 4. It is evident from Fig. 4(a) that Nu increases significantly with increasing (ahsv / cp). The Lewis number has very little effect on Nux when (ahsv / cp)= 0.1, but its effects become obvious in the region near the entrance when (ahsv / cp) = 1.0 and gradually diminishes in the region near the exit. ${{\varphi }_{0}}$ has almost no influence on Nu in almost the entire region when (ahsv / cp) = 1.0, as seen in Fig. 4(b). When (ahsv / cp)= 0.1, Nux increases slightly when ξ is small.

The variation of the local Nusselt number based on convective heat flux, Nu*, is shown in Fig. 5(a). Only a single curve is obtained, which implies that Nu* remains unchanged when the mass transfer parameters are varied. The value of Nu* is exactly the same as for the process without sublimation. Figure 5(b) shows the Sherwood number for various parameters. It is evident that (ahsv / cp) and have no effect on Shx, and Le has an insignificant effect on Shx in the entry region.