Sublimation over a Flat Plate

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Sublimation over a flat plate can find its application in analogy between heat and mass transfer (Kurosaki, 1973; Zhang et al., 1996). Figure 1 shows the physical model of the sublimation problem considered by Zhang et al. (1996). A flat plate is coated with a layer of sublimable material and is subject to constant heat flux heating underneath. A gas with the ambient temperature ${{T}_{\infty }}$ and mass fraction of sublimable material ${{\omega }_{\infty }}$ flows over the flat plate at a velocity of ${{u}_{\infty }}$ . The heat flux applied from the bottom of the flat plate will be divided into two parts: one part is used to supply the latent heat of sublimation, and another part is transferred to the gas through convection. The sublimated vapor is injected into the boundary layer and is removed by the gas flow. The following assumptions are made in order to solve the problem:

1. The flat plate is very thin, and so the thermal resistance of the flat plate can be neglected.

2. The gas is incompressible, with no internal heat source in the gas.

3. The sublimation problem is two-dimensional steady state.

The governing equations for mass, momentum, energy and species of the problem are

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \qquad \qquad(1)$

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \frac{{{\partial }^{2}}u}{\partial {{y}^{2}}} \qquad \qquad(2)$

$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \frac{{{\partial }^{2}}T}{\partial {{y}^{2}}} \qquad \qquad(3)$

Figure 1: Sublimation on a flat plate with constant heat flux.

$u\frac{\partial \omega }{\partial x}+v\frac{\partial \omega }{\partial y}=D\frac{{{\partial }^{2}}\omega }{\partial {{y}^{2}}} \qquad \qquad(4)$

Nonslip condition at the surface of the flat plat require that

$u=0\begin{matrix} , & y=0 \\ \end{matrix} \qquad \qquad(5)$

For a binary mixture that contains the vapor sublimable substance and gas, the molar flux of the sublimable substance at the surface of the flat plate is [see eq. (1.102)]

${\dot{m}}''=-\frac{\rho D}{1-\omega }\frac{\partial \omega }{\partial y}\begin{matrix} , & y=0 \\ \end{matrix} \qquad \qquad(6)$

Since the mass fraction of the sublimable substance in the mixture is very low, i.e., $\omega \ll 1$ , the mass flux at the wall can be simplified to

${\dot{m}}''=-\rho D\frac{\partial \omega }{\partial y}\begin{matrix} , & y=0 \\ \end{matrix} \qquad \qquad(7)$

Sublimation at the surface causes a normal blowing velocity, ${{v}_{w}}=\dot{{m}''}/\rho$ , at the surface. The normal velocity at the surface of the flat plate is therefore

$v={{v}_{w}}=-\rho D{{\left. \frac{\partial \omega }{\partial y} \right|}_{y=0}}\begin{matrix} , & y=0 \\ \end{matrix} \qquad \qquad(8)$

The energy balance at the surface of the flat plate is

$-k\frac{\partial T}{\partial y}-\rho {{h}_{sv}}D\frac{\partial \omega }{\partial y}={{{q}''}_{w}}\begin{matrix} , & y=0 \\ \end{matrix} \qquad \qquad(9)$

Another reasonable, practical, representable boundary condition at the surface of the flat plate emerges by setting the mass fraction at the wall as the saturation mass fraction at the wall temperature. The mass fraction and the temperature at the surface of the flat plate have the following relationship (Kurosaki, 1973, 1974):

$\omega =aT+b\begin{matrix} , & y=0 \\ \end{matrix} \qquad \qquad(10)$

where $a$ and $b$ are constants that depend on the sublimable material and its temperature.

As $y\to \infty ,$ the boundary conditions are $u\to {{u}_{\infty }}$ , $T\to {{T}_{\infty }}$ ,

$\omega \to {{\omega }_{\infty }} \qquad \qquad(11)$

Introducing the stream function $ψ$ ,

$u=\frac{\partial \psi }{\partial y}\begin{matrix} {} & v=-\frac{\partial \psi }{\partial x} \\ \end{matrix} \qquad \qquad(12)$

the continuity equation (1) is automatically satisfied, and the momentum equation in terms of the stream function becomes

$\frac{\partial \psi }{\partial y}\frac{{{\partial }^{2}}\psi }{\partial x\partial y}-\frac{\partial \psi }{\partial x}\frac{{{\partial }^{2}}\psi }{\partial {{y}^{2}}}=\nu \frac{{{\partial }^{3}}\psi }{\partial {{y}^{3}}} \qquad \qquad(13)$

Similarity solutions for eq. (13) do not exist unless the injection velocity $v w$ is proportional to $x 1 / 2$ , and the incoming mass fraction of the sublimable substance, ${{\omega}_{\infty}}$ , is equal to the saturation mass fraction corresponding to the incoming temperature ${T_{\infty}}$ (Kurosaki, 1974; Zhang et al., 1996). The governing equations cannot be reduced to ordinary differential equations. The local nonsimilarity solution proposed by Zhang et al. (1996) will be presented here.

Defining the following similarity variables:

$\begin{align} & \xi =\frac{x}{L},\text{ }\eta =y\sqrt{\frac{{{u}_{\infty }}}{2\nu L\xi }}\begin{matrix} , & f=\frac{\psi }{\sqrt{2\nu {{u}_{\infty }}L\xi }} \\ \end{matrix} \\ & \theta =\frac{k(T-{{T}_{\infty }})}{{{{{q}''}}_{w}}\sqrt{2\nu L\xi /{{u}_{\infty }}}}\begin{matrix} , & \varphi =\frac{\rho {{h}_{\ell v}}D(\omega -{{\omega }_{\infty }})}{{{{{q}''}}_{w}}\sqrt{2\nu L\xi /{{u}_{\infty }}}} \\ \end{matrix} \\ \end{align} \qquad \qquad(14)$

eqs. (13) and (3) – (4) become

${f}'''+f{f}''=2\xi \left( {f}'{F}'-{f}''F \right) \qquad \qquad(15)$

${\theta }''+\Pr (f{\theta }'-{f}'\theta )=2\Pr \xi \left( {f}'\Theta -{\theta }'F \right) \qquad \qquad(16)$

${\varphi }''+\operatorname{Sc}(f{\varphi }'-{f}'\varphi )=2\operatorname{Sc}\xi \left( {f}'\Phi -{\varphi }'F \right) \qquad \qquad(17)$

where prime ' represents partial derivative with respect to $η$ , and all upper case variables represent partial derivative of primary similarity variable with respect to $ξ$ .

$F=\frac{\partial f}{\partial \xi }\begin{matrix} , & \Theta =\frac{\partial \theta }{\partial \xi }\begin{matrix} , & \Phi =\frac{\partial \varphi }{\partial \xi } \\ \end{matrix} \\ \end{matrix} \qquad \qquad(18)$

It can be seen from eqs. (15) – (17) that the similarity solution exists only if $F = Θ = Φ = 0$ . In order to use eqs. (15) – (17) to obtain a solution for the sublimation problem, the supplemental equations about $F,Θ,$ and $Φ$ must be obtained. Taking partial derivatives of eqs. (15) – (17) with respect to $ξ$ and neglecting the higher order term, one obtains

${F}'''+F{f}''+{F}''f=2\left( {f}'{F}'-{f}''F \right) \qquad \qquad(19)$

${\Theta }''+\Pr (F{\theta }'+f{\Theta }'-{F}'\theta -{f}'\Theta )=2\Pr \left( {f}'\Theta -{\theta }'F \right)\qquad\qquad (20)$

${\Phi }''+\operatorname{Sc}(F{\varphi }'+f{\Phi }'-{F}'\varphi -{f}'\Phi )=2\operatorname{Sc}\left( {f}'\Phi -{\varphi }'F \right) \qquad \qquad(21)$

The boundary conditions of eqs. (15) – (17) and eqs. (19) – (21) are

${f}'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ \end{matrix} \qquad \qquad(22)$

$f(\xi ,0)=-\frac{2}{3}B\left[ {{\xi }^{1/2}}{\varphi }'(\xi ,0)-{{\xi }^{3/2}}{\Phi }'(\xi ,0) \right]\begin{matrix} , & \eta =0 \\ \end{matrix} \qquad \qquad(23)$

${f}'(\xi ,\infty )=1\begin{matrix} , & \eta =\infty \\ \end{matrix} \qquad \qquad(24)$

${F}'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ \end{matrix} \qquad \qquad(25)$

$F(\xi ,0)=-\frac{1}{3}B\left[ \frac{1}{2}{{\xi }^{-1/2}}{\varphi }'(\xi ,0)-{{\xi }^{1/2}}{\Phi }'(\xi ,0) \right]\begin{matrix} , & \eta =0 \\ \end{matrix} \qquad \qquad(26)$

${F}'(\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix} \qquad \qquad(27)$

${\theta }'(\xi ,0)+{\varphi }'(\xi ,0)=-1\begin{matrix} , & \eta =0 \\ \end{matrix} \qquad \qquad(28)$

$\theta (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix} \qquad \qquad(29)$

${\Theta }'(\xi ,0)+{\Phi }'(\xi ,0)=0\begin{matrix} , & \eta =0 \\ \end{matrix} \qquad \qquad(30)$

$\Theta (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix} \qquad \qquad(31)$

$\varphi (\xi ,0)=\frac{a{{h}_{sv}}}{{{c}_{p}}}\frac{1}{Le}\theta (\xi ,0)+{{\varphi }_{s}}{{\xi }^{-1/2}}\begin{matrix} , & \eta =0 \\ \end{matrix} \qquad \qquad(32)$

$\varphi (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix} \qquad \qquad(33)$

$\Phi (\xi ,0)=\frac{a{{h}_{sv}}}{{{c}_{p}}}\frac{1}{Le}\Theta (\xi ,0)-\frac{{{\varphi }_{s}}}{2{{\xi }^{3/2}}}\begin{matrix} , & \eta =0 \\ \end{matrix} \qquad \qquad(34)$

$\Phi (\xi ,\infty )=0\begin{matrix} , & \eta =\infty \\ \end{matrix} \qquad \qquad(35)$

where

$B=\frac{{{{{q}''}}_{w}}}{\rho {{h}_{sv}}\nu }\sqrt{\frac{2\nu L}{{{u}_{\infty }}}} \qquad \qquad(36)$

reflects the effect of injection velocity at the surface due to sublimation, and

${{\varphi }_{s}}=\frac{\rho {{h}_{sv}}D({{\omega }_{sat,\infty }}-{{\omega }_{\infty }})}{{{{{q}''}}_{w}}\sqrt{2\nu L/{{u}_{\infty }}}} \qquad \qquad(37)$

represents the effect of the mass fraction of the sublimable substance in the incoming flow. ${{\omega }_{sat,\infty }}$ is saturation mass fraction corresponding to the incoming temperature:

${{\omega }_{sat,\infty }}=a{{T}_{\infty }}+b \qquad \qquad(38)$

The set of ordinary differential equations (15) – (17) and (19) – (21) with boundary conditions specified by eqs. (22) – (35) are boundary value

$Temperature and mass fraction distributions$

Figure 2: Temperature and mass fraction distributions

Figure 3: Nusselt number based on convection and Sherwood number

problems that can be solved using a shooting method (Zhang et al., 1996). Figure 2 shows typical dimensionless temperature and mass fraction profiles obtained by numerical solution. It can be seen that the dimensionless temperature and mass fraction at different $ξ$ are also different, which is further evidence that a similarity solution does not exist.

Once the converged solution is obtained, the local Nusselt number based on the total heat flux at the bottom of the flat plate is

$N{{u}_{x}}=\frac{{{h}_{w}}x}{k}=\frac{[{{{{q}''}}_{w}}/({{T}_{w}}-{{T}_{\infty }})]x}{k}=\frac{\operatorname{Re}_{x}^{1/2}}{\sqrt{2}\theta (\xi ,0)} \qquad \qquad(39)$

and the Nusselt number based on convective heat transfer is

$Nu_{x}^{*}=\frac{{{h}_{x}}x}{k}=\frac{x}{{{T}_{w}}-{{T}_{\infty }}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}=-\frac{{\theta }'(\xi ,0)}{\sqrt{2}\theta (\xi ,0)}\operatorname{Re}_{x}^{1/2} \qquad \qquad(40)$

The Sherwood number is

$S{{h}_{x}}=\frac{{{h}_{m}}x}{D}=\frac{x}{{{\omega }_{w}}-{{\omega }_{\infty }}}{{\left. \frac{\partial \omega }{\partial y} \right|}_{y=0}}=-\frac{{\varphi }'(\xi ,0)}{\sqrt{2}\varphi \theta (\xi ,0)}\operatorname{Re}_{x}^{1/2} \qquad \qquad(41)$

Figure 3 shows the effect of blowing velocity on the Nusselt number based on convective heat transfer and the Sherwood number for ${{\varphi }_{sat,\infty }}=0$ , i.e., the mass fraction of sublimable substance is equal to the saturation mass fraction corresponding to the incoming temperature. It can be seen that the effect of blowing velocity on mass transfer is stronger than that on heat transfer.

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).

Kurosaki, Y., 1974, “Coupled Heat-Mass Transfer of a Flat Plate with Uniform Heat Flux in a Laminar Parallel Flow,” Journal of the Japan Society of Mechanical Engineers, Part B, Vol. 40, pp. 1066-1072 (in Japanese).

Zhang, Y., Chen, Z.Q., and Chen, M., 1996, “Local Non-Similarity Solution of Coupled Heat-Mass Transfer of a Flat Plate with Uniform Heat Flux in a Laminar Parallel Flow,” Journal of Thermal Science, Vol. 5, No. 2, pp. 112-116.

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Sublimation over a Flat Plate

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