# Coupled thermal and concentration entry effects

(Difference between revisions)
 Revision as of 04:48, 23 July 2010 (view source)← Older edit Current revision as of 07:34, 27 July 2010 (view source) (9 intermediate revisions not shown) Line 1: Line 1: - 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 article, coupled forced internal convection in a circular tube will be presented for both adiabatic and constant wall heat flux. + 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 phaseFaghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.. 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== Line 144: Line 144: |- |- | width="100%" |
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- $\theta =\Theta (\eta )\Gamma (\xi )$ + $\theta =\Theta (\eta )\Gamma (\xi ) |{{EquationRef|(16)}} |{{EquationRef|(16)}} Line 163: Line 163: |- |- | width="100%" | | width="100%" | - [itex]{\Gamma }'+{{\beta }^{2}}\Gamma =0$ + $\begin{matrix} {} & {} \\\end{matrix}{\Gamma }'+{{\beta }^{2}}\Gamma =0$
|{{EquationRef|(18)}} |{{EquationRef|(18)}} Line 189: Line 189: |- |- | width="100%" |
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- ${\Theta }'(0)=0$ + $\begin{matrix} {} & {} \\\end{matrix}{\Theta }'(0)=0$
|{{EquationRef|(21)}} |{{EquationRef|(21)}} Line 225: Line 225: |- |- | width="100%" |
| width="100%" |
- ${\Phi }'(0)=0$ + $\begin{matrix} {} & {} \\\end{matrix}{\Phi }'(0)=0$
|{{EquationRef|(25)}} |{{EquationRef|(25)}} Line 233: Line 233: |- |- | width="100%" |
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- $\beta =\gamma$ + $\begin{matrix} {} & {} \\\end{matrix}\beta =\gamma$
|{{EquationRef|(26)}} |{{EquationRef|(26)}} Line 334: Line 334: |} |} 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. - - [[Image:Fig5.8.png|thumb|400 px|alt=v | Nusselt and Sherwood numbers for sublimation inside an adiabatic tube.]] - The figure to right shows heat and mass transfer performance during sublimation inside an adiabatic tube. For all cases, both Nusselt and Sherwood numbers become constant when $\xi$ 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 $(a{{h}_{sv}}/{{c}_{p}})$ on the Nusselt and Sherwood numbers is relatively insignificant: both the Nusselt and Sherwood numbers increase with increasing $(a{{h}_{sv}}/{{c}_{p}})$ for Le < 1, but increasing$(a{{h}_{sv}}/{{c}_{p}})$ for Le > 1 results in decreasing Nusselt and Sherwood numbers. ==Sublimation inside a Tube Subjected to External Heating== ==Sublimation inside a Tube Subjected to External Heating== Line 417: Line 414: |} |} where ${{\varphi }_{0}}=k{{h}_{sv}}(\omega -{{\omega }_{sat,0}})/({{c}_{p}}{q}''{{r}_{o}})$ in eq. (43). where ${{\varphi }_{0}}=k{{h}_{sv}}(\omega -{{\omega }_{sat,0}})/({{c}_{p}}{q}''{{r}_{o}})$ 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 : + 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 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. 184-188.: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- | width="100%" |
| width="100%" |
- $\theta (\xi ,\eta )={{\theta }_{1}}(\xi ,\eta )+{{\theta }_{2}}(\xi ,\eta )$ + $\begin{matrix} {} & {} \\\end{matrix}\theta (\xi ,\eta )={{\theta }_{1}}(\xi ,\eta )+{{\theta }_{2}}(\xi ,\eta )$
|{{EquationRef|(47)}} |{{EquationRef|(47)}} Line 434: Line 431: |{{EquationRef|(48)}} |{{EquationRef|(48)}} |} |} - 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. (40) – (41) and (44) – (46), the corresponding homogeneous solutions of the temperature and mass fraction, ${{\theta }_{2}}(\xi ,\eta )$ and ${{\varphi }_{2}}(\xi ,\eta )$, must satisfy eqs. (40), (41), (44), and (46), as well as the following conditions: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 508: Line 505: |{{EquationRef|(57)}} |{{EquationRef|(57)}} |} |} + 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 + {|class="wikitable" border="0" {|class="wikitable" border="0" |- |- Line 517: Line 516: |} |} 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. - The Nusselt number based on the convective heat transfer coefficient is + + The Nusselt number based on the convective heat transfer coefficient is {| class="wikitable" border="0" {| class="wikitable" border="0" Line 529: Line 529: The Sherwood number is The Sherwood number is - $\text{Sh}=\frac{2{{h}_{m,x}}{{r}_{0}}}{D}=\frac{2{{r}_{0}}}{{{\omega }_{w}}-{{\omega }_{m}}}{{\left. \frac{\partial \omega }{\partial r} \right|}_{r={{r}_{o}}}}=\frac{2}{{{\varphi }_{w}}-{{\varphi }_{m}}}{{\left. \frac{\partial \varphi }{\partial \eta } \right|}_{\eta =1}}$ +
$\text{Sh}=\frac{2{{h}_{m,x}}{{r}_{0}}}{D}=\frac{2{{r}_{0}}}{{{\omega }_{w}}-{{\omega }_{m}}}{{\left. \frac{\partial \omega }{\partial r} \right|}_{r={{r}_{o}}}}=\frac{2}{{{\varphi }_{w}}-{{\varphi }_{m}}}{{\left. \frac{\partial \varphi }{\partial \eta } \right|}_{\eta =1}} Line 539: Line 539: |{{EquationRef|(60)}} |{{EquationRef|(60)}} |} |} - When the heat and mass transfer are fully developed, eqs. (5.138) – (5.140) reduce to + When the heat and mass transfer are fully developed, eqs. (58) – (60) reduce to {| class="wikitable" border="0" {| class="wikitable" border="0" Line 564: Line 564: |{{EquationRef|(63)}} |{{EquationRef|(63)}} |} |} - [[Image:Fig5.10.png|thumb|400 px|alt=Nusselt number based on total heat flux |Figure 4: Nusselt number based on total heat flux.]] + - The variations of the local Nusselt number based on total heat flux along the dimensionless location [itex]\xi$ are shown in Fig. 4. It is evident from Fig. 4(a) that Nu increases significantly with increasing $(a{{h}_{sv}}/{{c}_{p}})$.  The Lewis number has very little effect on Nux when $(a{{h}_{sv}}/{{c}_{p}})$= 0.1, but its effects become obvious in the region near the entrance when $(a{{h}_{sv}}/{{c}_{p}})$ = 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 $(a{{h}_{sv}}/{{c}_{p}})$ = 1.0, as seen in Fig. 4(b). When $(a{{h}_{sv}}/{{c}_{p}})$= 0.1, Nux increases slightly when $\xi$ is small. + ==References== - [[Image:Fig5.11.png|thumb|400 px|alt=Nusselt number based on convective heat flux and Sherwood number |Figure 5: Nusselt number based on convective heat flux and Sherwood number.]] + {{Reflist}} - 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 $(a{{h}_{sv}}/{{c}_{p}})$ and  have no effect on Shx, and Le has an insignificant effect on Shx in the entry region. +

## 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 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 (2) and (3) 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 (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:

 $\omega =aT+b\begin{matrix} , & r={{r}_{o}} \\\end{matrix}$ (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 (2) – (8) 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. (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) (16)

Substituting eq. (16) into eq. (10), 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 (17) can be rewritten as two ordinary differential equations:

 $\begin{matrix} {} & {} \\\end{matrix}{\Gamma }'+{{\beta }^{2}}\Gamma =0$ (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. (18) is

 $\Gamma ={{C}_{1}}{{e}^{-{{\beta }^{2}}\xi }}$ (20)

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

 $\begin{matrix} {} & {} \\\end{matrix}{\Theta }'(0)=0$ (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. (24) at η = 0 is

 $\begin{matrix} {} & {} \\\end{matrix}{\Phi }'(0)=0$ (25)

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

 $\begin{matrix} {} & {} \\\end{matrix}\beta =\gamma$ (26)
 $-\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{\Theta (1)}{\Phi (1)}=\text{Le}\frac{{\Theta }'(1)}{{\Phi }'(1)}$ (27)

To solve eqs. (19) and (24) 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. (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 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. (22) and (23), the solutions of eq. (10) and (11) become

 $\theta ={{C}_{1}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}$ (28)
 $\varphi ={{C}_{2}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}$ (29)

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

 $\theta =\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}}$ (30)
 $\varphi =\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}}$ (31)

where Gn and Hn can be obtained by substituting eqs. (30) and (31) into eq. (12), i.e.,

 $1=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta )}$ (32)
 $1=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta )}$ (33)

Due to the orthogonal nature of the eigenfunctions Θn and Φn, expressions of Gn and Hn can be obtained by

 ${{G}_{n}}=\frac{\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1-{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( A{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}$ (34)
 ${{H}_{n}}=\frac{A{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}$ (35)

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

 $\text{Nu}=\frac{-k{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}}{{{T}_{m}}-{{T}_{w}}}\frac{2{{r}_{o}}}{k}=-\frac{2}{{{\theta }_{m}}-{{\theta }_{w}}}\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Theta }'}}_{n}}(1)}$ (36)
 $\text{Sh}=\frac{-D{{\left. \frac{\partial \omega }{\partial r} \right|}_{r={{r}_{o}}}}}{{{\omega }_{m}}-{{\omega }_{w}}}\frac{2{{r}_{o}}}{D}=-\frac{2}{{{\varphi }_{m}}-{{\varphi }_{w}}}\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}$ (37)

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

## Sublimation inside a Tube Subjected to External Heating

Figure 3: Sublimation in a tube heated by a uniform heat flux.

When the inner wall of a tube with a sublimable-material-coated 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

 $\rho {{h}_{sv}}D\frac{\partial \omega }{\partial r}+k\frac{\partial T}{\partial r}={q}''\text{ at }r={{r}_{o}}$ (38)

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. (9), except the following:

 $\begin{matrix} \theta =\frac{k(T-{{T}_{0}})}{{q}''{{r}_{o}}}, & \varphi = \\\end{matrix}\frac{{{h}_{sv}}(\omega -{{\omega }_{sat,0}})}{{{c}_{p}}{q}''{{r}_{o}}}$ (39)

where ωsat,0 is the saturation mass fraction corresponding to the inlet temperature T0. The resulting dimensionless governing equations and boundary conditions are

 $\eta (1-{{\eta }^{2}})\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right)$ (40)
 $\eta (1-{{\eta }^{2}})\frac{\partial \varphi }{\partial \xi }=\frac{1}{\text{Le}}\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right)$ (41)
 $\theta =0\begin{matrix} , & \xi =0 \\\end{matrix}$ (42)
 $\varphi ={{\varphi }_{0}}\begin{matrix} , & \xi =0 \\\end{matrix}$ (43)
 $\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\\end{matrix}$ (44)
 $\frac{\partial \theta }{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial \varphi }{\partial \eta }=1\begin{matrix} , & \eta =1 \\\end{matrix}$ (45)
 $\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix} , & \eta =1 \\\end{matrix}$ (46)

where ${{\varphi }_{0}}=k{{h}_{sv}}(\omega -{{\omega }_{sat,0}})/({{c}_{p}}{q}''{{r}_{o}})$ 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]:

 $\begin{matrix} {} & {} \\\end{matrix}\theta (\xi ,\eta )={{\theta }_{1}}(\xi ,\eta )+{{\theta }_{2}}(\xi ,\eta )$ (47)
 $\varphi (\xi ,\eta )={{\varphi }_{1}}(\xi ,\eta )+{{\varphi }_{2}}(\xi ,\eta )$ (48)

While the fully developed solutions of temperature and mass fraction, θ1(ξ,η) and ${{\varphi }_{1}}(\xi ,\eta )$, respectively, must satisfy eqs. (40) – (41) and (44) – (46), the corresponding homogeneous solutions of the temperature and mass fraction, θ2(ξ,η) and ${{\varphi }_{2}}(\xi ,\eta )$, must satisfy eqs. (40), (41), (44), and (46), as well as the following conditions:

 ${{\theta }_{2}}=-{{\theta }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}$ (49)
 ${{\varphi }_{2}}={{\varphi }_{0}}-{{\varphi }_{1}}(\xi ,\eta )\begin{matrix} , & \xi =0 \\\end{matrix}$ (50)
 $\frac{\partial {{\theta }_{2}}}{\partial \eta }+\frac{1}{\text{Le}}\frac{\partial {{\varphi }_{2}}}{\partial \eta }=0\begin{matrix} , & \eta =1 \\\end{matrix}$ (51)

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} (52)
 \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} (53)

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

 ${{\theta }_{2}}=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}}$ (54)
 ${{\varphi }_{2}}=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}}$ (55)

where

 ${{G}_{n}}=\frac{\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\theta }_{2}}(0,\eta ){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\varphi }_{2}}(0,\eta ){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1-{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( a{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }}$ (56)
 ${{H}_{n}}=\frac{a{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}$ (57)

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} (58)

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} (59)

The Sherwood number is

$\text{Sh}=\frac{2{{h}_{m,x}}{{r}_{0}}}{D}=\frac{2{{r}_{0}}}{{{\omega }_{w}}-{{\omega }_{m}}}{{\left. \frac{\partial \omega }{\partial r} \right|}_{r={{r}_{o}}}}=\frac{2}{{{\varphi }_{w}}-{{\varphi }_{m}}}{{\left. \frac{\partial \varphi }{\partial \eta } \right|}_{\eta =1}}$

 $=\frac{2\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+2(1+\frac{a{{h}_{sv}}}{{{c}_{p}}})\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-{{\beta }_{n}}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)}}{\frac{11}{24}\text{Le}\frac{a{{h}_{sv}}}{{{c}_{p}}}+\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{-{{\beta }_{n}}^{2}\xi }}\left[ {{\Phi }_{n}}(1)+\frac{4}{\beta _{n}^{2}\text{Le}}{{{{\Phi }'}}_{n}}(1) \right]}}$ (60)

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

 $\text{Nu}=\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{48}{11}$ (61)
 $\text{N}{{\text{u}}^{*}}=\frac{48}{11}$ (62)
 $\text{Sh}=\frac{48}{11}$ (63)

## References

1. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
2. Zhang, Y., and Chen, Z.Q., 1990, “Analytical Solution of Coupled Laminar Heat-Mass 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. 341-345.
3. 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. 2512-2521 (in Japanese).
4. 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. 184-188.