# Sublimation inside an Circular Tube

In addition to the external sublimation discussed in the preceding subsection, internal sublimation is also very important. Sublimation inside an adiabatic and externally heated tube will be analyzed in this and the next 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 R 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 $u=2\bar{u}\left[ 1-{{\left( \frac{r}{R} \right)}^{2}} \right] \qquad \qquad(1)$

where $\bar{u}$ is the mean velocity of the gas flow inside the tube.

Neglecting axial conduction and diffusion, the energy and mass transfer equations are $ur\frac{\partial T}{\partial x}=\alpha \frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right) \qquad \qquad(2)$ $ur\frac{\partial \omega }{\partial x}=D\frac{\partial }{\partial r}\left( r\frac{\partial \omega }{\partial r} \right) \qquad \qquad(3)$ $T={{T}_{0}}\begin{matrix} , & x=0 \\ \end{matrix} \qquad \qquad(4)$ $\omega ={{\omega }_{0}}\begin{matrix} , & x=0 \\ \end{matrix} \qquad \qquad(5)$ $\frac{\partial T}{\partial r}=\frac{\partial \omega }{\partial r}=0\begin{matrix} , & r=0 \\ \end{matrix} \qquad \qquad(6)$ $-k\frac{\partial T}{\partial r}=\rho D{{h}_{sv}}\frac{\partial \omega }{\partial r}\begin{matrix} , & r=R \\ \end{matrix} \qquad \qquad(7)$

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

where a and b are constants.

The following nondimensional variables are then introduced: \begin{align} & \begin{matrix} \eta =\frac{r}{R} & \xi =\frac{x}{R\text{Pe}} & \text{Le}=\frac{\alpha }{D} & \operatorname{Re}=\frac{2\bar{u}R}{\nu } \\ \end{matrix} \\ & \begin{matrix} \text{Pe}=\frac{2\bar{u}R}{\alpha } & \theta =\frac{T-{{T}_{f}}}{{{T}_{0}}-{{T}_{f}}} & \varphi =\frac{\omega -{{\omega }_{f}}}{{{\omega }_{0}}-{{\omega }_{f}}} & {} \\ \end{matrix} \\ \end{align} \qquad \qquad(9)

where Tf and ωf are temperature and mass fraction of the sublimable substance, respectively, after heat and mass transfer are fully developed. Equations (2) – (8) then become $\eta (1-{{\eta }^{2}})\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right) \qquad \qquad(10)$ $\eta (1-{{\eta }^{2}})\frac{\partial \varphi }{\partial \xi }=\frac{1}{Le}\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right) \qquad \qquad(11)$ $\theta =\varphi =1\begin{matrix} , & \xi =0 \\ \end{matrix} \qquad \qquad(12)$ $\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\ \end{matrix} \qquad \qquad(13)$ $-\frac{\partial \theta }{\partial \eta }=\frac{1}{Le}\frac{\partial \varphi }{\partial \eta }\begin{matrix} , & \eta =1 \\ \end{matrix} \qquad \qquad(14)$ $\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix} , & \eta =1 \\ \end{matrix} \qquad \qquad(15)$

The heat and the mass transfer equations (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., $\theta =\Theta (\eta )\Gamma (\xi ) \qquad \qquad(16)$

Substituting eq. (16) into eq. (10), the energy equation becomes $\frac{{{\Gamma }'}}{\Gamma }=\frac{\frac{d}{d\eta }\left( \frac{d\Theta }{d\eta } \right)}{\eta (1-{{\eta }^{2}})\Theta }=-{{\beta }^{2}} \qquad \qquad(17)$

where β is the eigenvalue for the energy equation.

Equation (17) can be rewritten as the following two ordinary differential equations: ${\Gamma }'+{{\beta }^{2}}\Gamma =0 \qquad \qquad(18)$ $\frac{d}{d\eta }\left( \frac{d\Theta }{d\eta } \right)+{{\beta }^{2}}\eta (1-{{\eta }^{2}})\Theta =0 \qquad \qquad(19)$

The solution of eq. (18) is $\Gamma ={{C}_{1}}{{e}^{-{{\beta }^{2}}\xi }} \qquad \qquad(20)$

The boundary condition of eq. (19) at η = 0 is ${\Theta }'(0)=0 \qquad \qquad(21)$

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

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

where γ is the eigenvalue for the conservation of species equation, and Φ(η) satisfies $\frac{d}{d\eta }\left( \frac{d\Phi }{d\eta } \right)+Le{{\gamma }^{2}}\eta (1-{{\eta }^{2}})\Phi =0 \qquad \qquad(24)$

and the boundary condition of eq. (24) at η = 0 is ${\Phi }'(0)=0 \qquad \qquad(25)$

Substituting eqs. (22) – (23) into eqs. (14) – (15), one obtains $\beta =\gamma \qquad \qquad(26)$ $-\left( \frac{A{{h}_{sv}}}{{{c}_{p}}} \right)\frac{\Theta (1)}{\Phi (1)}=Le\frac{{\Theta }'(1)}{{\Phi }'(1)} \qquad \qquad(27)$

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 solve for eq. (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 the βn there is one set of corresponding Θn and Φn functions $(n=1,2,3,\cdots )$.

If we use any one of the eigenvalue βn and corresponding eigen functions – Θn and Φn – in eqs. (22) and (23), the solutions of eq. (10) and (11) become $\theta ={{C}_{1}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }} \qquad \qquad(28)$ $\varphi ={{C}_{2}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }} \qquad \qquad(29)$

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

where Gn and Hn can be obtained by substituting eqs. (30) and (31) into eq. (12), i.e., $1=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta )} \qquad \qquad(32)$ $1=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta )} \qquad \qquad(33)$

Due to the orthogonal nature of the eigeinfunctions Θn and Φn, expressions of Gn and Hn can be obtained by ${{G}_{n}}=\frac{\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\Theta }_{n}}(\eta )d\eta }+\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]\int_{0}^{1}{\eta (1-{{\eta }^{2}}){{\Phi }_{n}}(\eta )d\eta }}{\int_{0}^{1}{\eta (1-{{\eta }^{2}})\left\{ \Theta _{n}^{2}(\eta )+\left( A{{h}_{sv}}/{{c}_{p}} \right){{\left[ \frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)} \right]}^{2}}\Phi _{n}^{2}(\eta ) \right\}d\eta }} \qquad \qquad(34)$ ${{H}_{n}}=\frac{A{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}} \qquad \qquad(35)$

The Nusselt number due to convection and Sherwood number are $Nu=\frac{-k{{\left. \frac{\partial T}{\partial r} \right|}_{r=R}}}{\bar{T}-{{T}_{w}}}\frac{2R}{k}=-\frac{2}{\bar{\theta }-{{\theta }_{w}}}\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Theta }'}}_{n}}(1)} \qquad \qquad(36)$ $Sh=\frac{-D{{\left. \frac{\partial \omega }{\partial r} \right|}_{r=R}}}{\bar{\omega }-{{\omega }_{w}}}\frac{2R}{D}=-\frac{2}{\bar{\varphi }-{{\varphi }_{w}}}\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{e}^{-\beta _{n}^{2}\xi }}{{{{\Phi }'}}_{n}}(1)} \qquad \qquad(37)$

where $\bar{T}\text{ and }\bar{\omega }$ 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 increasing Lewis number. While the fully-developed Nusselt number increases with increasing Lewis number, the Sherwood number decreases with increasing Lewis number, because a larger Lewis number indicates larger thermal diffusivity or low mass diffusivity. The effect of (Ahsv / cp) on Nusselt and Sherwood numbers is relatively insignificant: both Nusselt and Sherwood numbers increase with increasing (Ahsv / cp) for Le < 1, but increasing (Ahsv / cp) for Le > 1 results in decreasing Nusselt and Sherwood numbers.

## Contents

### 7.2.3 Sublimation inside a Tube Subjected to External Heating

When the outer wall of a tube with a sublimable-material-coated inner 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. (1) – (8), except that the boundary condition at the inner wall of the tube is replaced by $\rho {{h}_{sv}}D\frac{\partial \omega }{\partial r}+k\frac{\partial T}{\partial r}={q}'' \qquad \qquad(38)$

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 nondimensionalized by using the dimensionless variables defined in eq. (9), except the following: $\begin{matrix} \theta =\frac{k(T-{{T}_{0}})}{{q}''R} & \varphi = \\ \end{matrix}\frac{{{h}_{sv}}(\omega -{{\omega }_{sat,0}})}{{{c}_{p}}{q}''R} \qquad \qquad(39)$

where ωsat,0 is the saturation mass fraction corresponding to the inlet temperature T0, and the resultant dimensionless governing equations and boundary conditions are $\eta (1-{{\eta }^{2}})\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right) \qquad \qquad(40)$ $\eta (1-{{\eta }^{2}})\frac{\partial \varphi }{\partial \xi }=\frac{1}{Le}\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right) \qquad \qquad(41)$ $\theta =0\begin{matrix} , & \xi =0 \\ \end{matrix} \qquad \qquad(42)$ $\varphi ={{\varphi }_{0}}\begin{matrix} , & \xi =0 \\ \end{matrix} \qquad \qquad(43)$ $\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\ \end{matrix} \qquad \qquad(44)$ $\frac{\partial \theta }{\partial \eta }+\frac{1}{Le}\frac{\partial \varphi }{\partial \eta }=1\begin{matrix} , & \eta =1 \\ \end{matrix} \qquad \qquad(45)$ $\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix} , & \eta =1 \\ \end{matrix} \qquad \qquad(46)$

where ${{\varphi }_{0}}=k{{h}_{sv}}(\omega -{{\omega }_{sat,0}})/({{c}_{p}}{q}''R)$ 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 and the solution of the corresponding homogeneous problem (Zhang and Chen, 1992), i.e., $\theta (\xi ,\eta )={{\theta }_{1}}(\xi ,\eta )+{{\theta }_{2}}(\xi ,\eta ) \qquad \qquad(47)$ $\varphi (\xi ,\eta )={{\varphi }_{1}}(\xi ,\eta )+{{\varphi }_{2}}(\xi ,\eta ) \qquad \qquad(48)$

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

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} \qquad \qquad(52) \begin{align} & {{\varphi }_{1}}=\frac{a{{h}_{sv}}/{{c}_{p}}}{1+a{{h}_{sv}}/{{c}_{p}}}\left[ 4\xi +Le{{\eta }^{2}}\left( 1-\frac{1}{4}{{\eta }^{2}} \right)+{{\varphi }_{0}} \right. \\ & \left. \text{ }-\frac{7\text{L}{{\text{e}}_{{}}}a{{h}_{sv}}/{{c}_{p}}+18Le-11}{24(1+a{{h}_{sv}}/{{c}_{p}})} \right] \\ \end{align} \qquad \qquad(53)

The solution of the corresponding homogeneous problem can be obtained by separation of variables: ${{\theta }_{2}}=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta ){{e}^{-{{\beta }_{n}}^{2}\xi }}} \qquad \qquad(54)$ ${{\varphi }_{2}}=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}} \qquad \qquad(55)$

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

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} & Nu=\frac{2{q}''R}{k({{T}_{w}}-\bar{T})}=\frac{2}{{{\theta }_{w}}-\bar{\theta }} \\ & =\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} \qquad \qquad(58)

where θw and $\bar{\theta }$ are dimensionless wall and mean temperatures, respectively.

The Nusselt number based on the convective heat transfer coefficient is \begin{align} & N{{u}^{*}}=\frac{2{{h}_{x}}R}{k}=\frac{2R}{{{T}_{w}}-\bar{T}}{{\left( \frac{\partial T}{\partial r} \right)}_{r=R}}=\frac{2}{{{\theta }_{w}}-\bar{\theta }}{{\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} \qquad \qquad(59)

The Sherwood number is \begin{align} & Sh=\frac{2{{h}_{m,x}}R}{D}=\frac{2R}{{{\omega }_{w}}-\bar{\omega }}{{\left. \frac{\partial \omega }{\partial r} \right|}_{r=R}}=\frac{2}{{{\varphi }_{w}}-\bar{\varphi }}{{\left. \frac{\partial \varphi }{\partial \eta } \right|}_{\eta =1}} \\ & =\frac{2Le\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}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}Le}{{{{\Phi }'}}_{n}}(1) \right]}} \\ \end{align} \qquad \qquad(60)

When the heat and mass transfer are fully developed, eqs. (58) – (60) reduce to $Nu=\left( 1+\frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\frac{48}{11} \qquad \qquad(61)$ $N{{u}^{*}}=\frac{48}{11} \qquad \qquad(62)$ $Sh=\frac{48}{11} \qquad \qquad(63)$

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. The effect of ${{\varphi }_{0}}$ on Nu, as is seen from Fig. 4(b), has

no apparent influence in almost the entire region when (Ahsv / cp) = 1.0. 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 ${{\varphi }_{0}}$ have no effect on Shx, but Le has an insignificant effect on Shx in the entry region.

Introducing the following nondimensional variables, \begin{align} & \begin{matrix} \eta =\frac{r}{R} & \xi =\frac{x}{RPe} & Pe=\frac{2UR}{\alpha } & Bi=\frac{{{h}_{e}}R}{k} \\ \end{matrix} \\ & \begin{matrix} \theta =\frac{{{T}_{e}}-T}{{{T}_{e}}-{{T}_{0}}} & {} & \varphi =\frac{{{\omega }_{e}}-\omega }{{{\omega }_{e}}-{{\omega }_{0}}} & {{\omega }_{e}}=a{{T}_{e}}+b \\ \end{matrix} \\ \end{align} \qquad \qquad(70)

where ωe is the saturation mass fraction corresponding to Te, the governing equations become $\frac{\eta }{2}\frac{\partial \theta }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \theta }{\partial \eta } \right) \qquad \qquad(71)$ $\frac{\eta }{2}\frac{\partial \varphi }{\partial \xi }=\frac{\partial }{\partial \eta }\left( \eta \frac{\partial \varphi }{\partial \eta } \right) \qquad \qquad(72)$ $\theta =\varphi =1\begin{matrix} , & \xi =0 \\ \end{matrix} \qquad \qquad(73)$ $\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix} , & \eta =0 \\ \end{matrix} \qquad \qquad(74)$ $\frac{A{{h}_{sv}}}{{{c}_{p}}}\frac{\partial \theta }{\partial \eta }+\frac{\partial \varphi }{\partial \eta }=-Bi{{\theta }_{w}}\begin{matrix} , & \eta =1 \\ \end{matrix} \qquad \qquad(75)$ ${{\varphi }_{w}}={{\theta }_{w}}\begin{matrix} , & \eta =1 \\ \end{matrix} \qquad \qquad(76)$

Equations (71) and (72) can be solved using separation of variables, and the resulting temperature and mass fraction distributions are (Zhang, 2002) $\theta =\varphi =\sum\limits_{n=1}^{\infty }{\frac{2{{J}_{1}}({{\beta }_{n}}){{J}_{0}}({{\beta }_{n}}\eta )}{{{\beta }_{n}}\left[ J_{0}^{2}({{\beta }_{n}})+J_{1}^{2}({{\beta }_{n}}) \right]}}{{e}^{-\beta _{n}^{2}\xi }} \qquad \qquad(77)$

where J0 and J1 are the zeroth and first order Bessel functions. The Nusselt number based on the total heat supplied by the external fluid is $Nu=\frac{2R}{k}\frac{{{h}_{e}}({{T}_{e}}-{{T}_{w}})}{({{T}_{w}}-\bar{T})}=\frac{2Bi{{\theta }_{w}}}{\bar{\theta }-{{\theta }_{w}}} \qquad \qquad(78)$

The Nusselt number based on the heat transferred to the fluid inside the tube is $N{{u}^{*}}=-\frac{2R}{({{T}_{w}}-\bar{T})}{{\left( \frac{\partial T}{\partial r} \right)}_{r=R}}=\frac{2}{\bar{\theta }-{{\theta }_{w}}}{{\left( \frac{\partial \theta }{\partial \eta } \right)}_{\eta =1}} \qquad \qquad(79)$

The Sherwood number is $Sh=-\frac{2R}{{{\omega }_{w}}-\bar{\omega }}{{\left( \frac{\partial \omega }{\partial r} \right)}_{r=R}}=\frac{2}{\bar{\varphi }-{{\varphi }_{w}}}{{\left( \frac{\partial \varphi }{\partial \eta } \right)}_{\eta =1}} \qquad \qquad(80)$

The Nusselt number based on the heat transferred to the fluid inside the tube and the Sherwood number are identical, since $\theta =\varphi$ as indicated by eq. (77). Fig. 7 shows the variation of local Nusselt number based on total heat supplied from the fluid outside the tube. The dimensionless lengths ofthe entrance slightly increase with decreasing Biot number, and the dimensionless lengths of entrance are approximately equal to 0.1. Nusselt numbers become constants after ξ is greater than 0.1. The fully-developed Nusselt number increases with decreasing Biot number. Fig. 8 shows the variation of local Nusselt number based on heat transferred to the fluid inside the tube or local Sherwood number. The variations of Nu* and Sh are similar to that of Nu in Fig. 7.

## References

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).

Zhang, Y., 2002, “Coupled Forced Convective Heat and Mass Transfer in a Circular Tube with External Convective Heating,” Progress of Computational Fluid Dynamics Journal, Vol. 2, pp. 90-96.

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.

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.