Coupled thermal and concentration entry effects

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There are many transport phenomena problems in which heat and mass transfer simultaneously occur. In some cases, such as sublimation and vapor deposition, they are coupled. These problems are usually treated as a single phase. However, coupled heat and mass transfer should both be considered even though they are modeled as being single phase. 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. 5.7 (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)

(1)

ur\frac{\partial \omega }{\partial x}=D\frac{\partial }{\partial r}\left( r\frac{\partial \omega }{\partial r} \right)

(1)

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}

(1)

\omega ={{\omega }_{0}}\begin{matrix}   , & x=0  \\\end{matrix}

(1)

\frac{\partial T}{\partial r}=\frac{\partial \omega }{\partial r}=0\begin{matrix}   , & r=0  \\\end{matrix}

(1)

-k\frac{\partial T}{\partial r}=\rho D{{h}_{sv}}\frac{\partial \omega }{\partial r}\begin{matrix}   , & r={{r}_{o}}  \\\end{matrix}

(1)

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}

(1)

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}

(1)

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)

(1)

\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)

(1)

\theta =\varphi =1\begin{matrix}   , & \xi =0  \\\end{matrix}

(1)

\frac{\partial \theta }{\partial \eta }=\frac{\partial \varphi }{\partial \eta }=0\begin{matrix}   , & \eta =0  \\\end{matrix}

(1)

-\frac{\partial \theta }{\partial \eta }=\frac{1}{\text{Le}}\frac{\partial \varphi }{\partial \eta }\begin{matrix}   , & \eta =1  \\\end{matrix}

(1)

\varphi =\left( \frac{a{{h}_{sv}}}{{{c}_{p}}} \right)\theta \begin{matrix}   , & \eta =1  \\\end{matrix}

(1)

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)

(1)

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}}

(1)

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

Γ' + β2Γ = 0

(1)

\frac{d}{d\eta }\left( \frac{d\Theta }{d\eta } \right)+{{\beta }^{2}}\eta (1-{{\eta }^{2}})\Theta =0

(1)

The solution of eq. (5.98) is

\Gamma ={{C}_{1}}{{e}^{-{{\beta }^{2}}\xi }}

(1)

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

Θ'(0) = 0

(1)

The dimensionless temperature is then

\theta ={{C}_{1}}\Theta (\eta ){{e}^{-{{\beta }^{2}}\xi }}

(1)

Similarly, the dimensionless mass fraction is

\varphi ={{C}_{2}}\Phi (\eta ){{e}^{-{{\gamma }^{2}}\xi }}

(1)

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

(1)

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

Φ'(0) = 0

(1)

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)}

(1)

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 }}

(1)

\varphi ={{C}_{2}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}

(1)

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 }}}

(1)

\varphi =\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta ){{e}^{-\beta _{n}^{2}\xi }}}

(1)

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)}

(1)

1=\sum\limits_{n=1}^{\infty }{{{G}_{n}}{{\Theta }_{n}}(\eta )} (5.112)

1=\sum\limits_{n=1}^{\infty }{{{H}_{n}}{{\Phi }_{n}}(\eta )}

(1)

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 }}

(1)

{{H}_{n}}=\frac{A{{h}_{sv}}}{{{c}_{p}}}\frac{{{\Theta }_{n}}(1)}{{{\Phi }_{n}}(1)}{{G}_{n}}

(1)

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)}

(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)}

(1)

where Tm and ωm are mean temperature and mean mass fraction in the tube. Figure 5.8 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.