Evaporation from adiabatic walls

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Due to diffusion and convection, the temperature rises and the concentration falls from their values at the film surface to their ambient values at the edge of the boundary layer. A nonslip boundary condition exists at the film surface, and the film itself is considered stationary with respect to the gas. Assuming a steady state, constant density, and incompressible flow, the continuity equation is  
Due to diffusion and convection, the temperature rises and the concentration falls from their values at the film surface to their ambient values at the edge of the boundary layer. A nonslip boundary condition exists at the film surface, and the film itself is considered stationary with respect to the gas. Assuming a steady state, constant density, and incompressible flow, the continuity equation is  
-
<center><math>\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0</math>
+
{| class="wikitable" border="0"
-
(9.1)</center>
+
|-
 +
| width="100%" |
 +
<center><math>\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0</math></center>
 +
|{{EquationRef|(1)}}
 +
|}
 +
 
With no pressure gradients and constant viscosity assumed, the boundary layer momentum equation is written as
With no pressure gradients and constant viscosity assumed, the boundary layer momentum equation is written as
-
<center><math>u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\tilde{\nu }\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}</math>
+
{| class="wikitable" border="0"
-
    (9.2)</center>
+
|-
 +
| width="100%" |
 +
<center><math>u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\tilde{\nu }\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}</math></center>
 +
|{{EquationRef|(2)}}
 +
|}
[[Image:NewChapter9 (9).jpg|400 px|alt= Examples of film evaporators. ]]
[[Image:NewChapter9 (9).jpg|400 px|alt= Examples of film evaporators. ]]
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Neglecting viscous dissipation and assuming constant thermal diffusivity and specific heats, the boundary layer energy balance is
Neglecting viscous dissipation and assuming constant thermal diffusivity and specific heats, the boundary layer energy balance is
-
<center><math>u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\tilde{\alpha }\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}}</math>
+
{| class="wikitable" border="0"
-
    (9.3)</center>
+
|-
 +
| width="100%" |
 +
<center><math>u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\tilde{\alpha }\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}}</math></center>
 +
|{{EquationRef|(3)}}
 +
|}
where the axial conduction on the x-direction has been neglected because heat transfer occurred mainly in the y-direction.  
where the axial conduction on the x-direction has been neglected because heat transfer occurred mainly in the y-direction.  
The mass fraction of water is also accounted for by assuming a constant mass diffusivity:
The mass fraction of water is also accounted for by assuming a constant mass diffusivity:
-
<center><math>u\frac{\partial \omega }{\partial x}+v\frac{\partial \omega }{\partial y}=D\frac{{{\partial }^{2}}\omega }{\partial {{y}^{2}}}</math>
+
{| class="wikitable" border="0"
-
    (9.4)</center>
+
|-
 +
| width="100%" |
 +
<center><math>u\frac{\partial \omega }{\partial x}+v\frac{\partial \omega }{\partial y}=D\frac{{{\partial }^{2}}\omega }{\partial {{y}^{2}}}</math></center>
 +
|{{EquationRef|(4)}}
 +
|}
As <math>y\to \infty </math>the boundary conditions can be taken directly from Fig. 9.6:
As <math>y\to \infty </math>the boundary conditions can be taken directly from Fig. 9.6:
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<math>u\to {{u}_{\infty }}</math>,  
<math>u\to {{u}_{\infty }}</math>,  
<math>T\to {{T}_{\infty }}</math>,  
<math>T\to {{T}_{\infty }}</math>,  
-
<center><math>\omega \to {{\omega }_{\infty }}</math>
+
 
-
(9.5)</center>
+
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>\omega \to {{\omega }_{\infty }}</math></center>
 +
|{{EquationRef|(5)}}
 +
|}
The boundary conditions are now laid out at the film surface ''y'' = 0. There is a nonslip condition, and the mass flux <math>{\dot{m}}''</math> evaporates normal to the surface.
The boundary conditions are now laid out at the film surface ''y'' = 0. There is a nonslip condition, and the mass flux <math>{\dot{m}}''</math> evaporates normal to the surface.
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<math>\omega ={{\omega }_{\delta }}</math>,  
<math>\omega ={{\omega }_{\delta }}</math>,  
-
<center><math>T={{T}_{\delta }}={{T}_{sat}}\left( {{\omega }_{\delta }},{{p}_{\infty }} \right)</math>
+
{| class="wikitable" border="0"
-
(9.6)</center>
+
|-
 +
| width="100%" |
 +
<center><math>T={{T}_{\delta }}={{T}_{sat}}\left( {{\omega }_{\delta }},{{p}_{\infty }} \right)</math></center>
 +
|{{EquationRef|(6)}}
 +
|}
An energy balance at the interface accounts for the heat of vaporization and mass flux of the evaporating fluid. With the nonslip boundary condition, the only mode of heat transfer is conduction:
An energy balance at the interface accounts for the heat of vaporization and mass flux of the evaporating fluid. With the nonslip boundary condition, the only mode of heat transfer is conduction:
-
</center><math>{\dot{m}}''{{h}_{\ell v}}=-{q}''=\tilde{k}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}</math>
+
{| class="wikitable" border="0"
-
(9.7)<center>
+
|-
 +
| width="100%" |
 +
</center><math>{\dot{m}}''{{h}_{\ell v}}=-{q}''=\tilde{k}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}</math><center>
 +
|{{EquationRef|(7)}}
 +
|}
-
Substituting eq. (9.7) into eq. (9.6), the velocity component in the y-direction becomes  
+
Substituting eq. (7) into eq. (6), the velocity component in the y-direction becomes  
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<center><math>{{\left. v \right|}_{y=0}}=\frac{{\tilde{k}}}{\rho {{h}_{\ell v}}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}</math>
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{| class="wikitable" border="0"
-
(9.8)</center>
+
|-
 +
| width="100%" |
 +
<center><math>{{\left. v \right|}_{y=0}}=\frac{{\tilde{k}}}{\rho {{h}_{\ell v}}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}</math></center>
 +
|{{EquationRef|(8)}}
 +
|}
The mass flux at the interface is the result of both diffusion and convection. Therefore, the mass balance at the interface is written as
The mass flux at the interface is the result of both diffusion and convection. Therefore, the mass balance at the interface is written as
-
<center><math>{\dot{m}}''=-\rho D{{\left( \frac{\partial \omega }{\partial y} \right)}_{y=0}}+\rho {{\omega }_{\delta }}{{\left. v \right|}_{y=0}}</math>
+
{| class="wikitable" border="0"
-
(9.9)</center>
+
|-
 +
| width="100%" |
 +
<center><math>{\dot{m}}''=-\rho D{{\left( \frac{\partial \omega }{\partial y} \right)}_{y=0}}+\rho {{\omega }_{\delta }}{{\left. v \right|}_{y=0}}</math></center>
 +
|{{EquationRef|(9)}}
 +
|}
-
Substituting eq. (9.7) into eq. (9.9), conservation of mass at the interface becomes
+
Substituting eq. (7) into eq. (9), conservation of mass at the interface becomes
-
<center><math>-\tilde{\rho }D{{\left( \frac{\partial \omega }{\partial y} \right)}_{y=0}}+\tilde{\rho }{{\omega }_{\delta }}{{\left. v \right|}_{y=0}}=\frac{{\tilde{k}}}{{{h}_{\ell v}}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}</math>
+
{| class="wikitable" border="0"
-
(9.10)</center>
+
|-
 +
| width="100%" |
 +
<center><math>-\tilde{\rho }D{{\left( \frac{\partial \omega }{\partial y} \right)}_{y=0}}+\tilde{\rho }{{\omega }_{\delta }}{{\left. v \right|}_{y=0}}=\frac{{\tilde{k}}}{{{h}_{\ell v}}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}</math></center>
 +
|{{EquationRef|(10)}}
 +
|}
-
Equations (9.1) – (9.2) are the same as the laminar boundary layer equations for forced convection over a flat plate with blowing on the liquid surface. By defining the stream function <math>\psi </math> as follows,
+
Equations (1) – (2) are the same as the laminar boundary layer equations for forced convection over a flat plate with blowing on the liquid surface. By defining the stream function <math>\psi </math> as follows,
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
<center><math>u=\frac{\partial \psi }{\partial y}\begin{matrix}
<center><math>u=\frac{\partial \psi }{\partial y}\begin{matrix}
   {} & v=-\frac{\partial \psi }{\partial x}  \\
   {} & v=-\frac{\partial \psi }{\partial x}  \\
-
\end{matrix}</math>
+
\end{matrix}</math></center>
-
(9.11)</center>
+
|{{EquationRef|(11)}}
 +
|}
-
the continuity equation (9.1) is automatically satisfied. The momentum equation in terms of the stream function is then
+
the continuity equation (1) is automatically satisfied. The momentum equation in terms of the stream function is then
-
<center><math>\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}}}</math>
+
{| class="wikitable" border="0"
-
(9.12)</center>
+
|-
 +
| width="100%" |
 +
<center><math>\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}}}</math></center>
 +
|{{EquationRef|(12)}}
 +
|}
Introducing the following similarity variable:
Introducing the following similarity variable:
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
<center><math>\eta =y\sqrt{\frac{{{u}_{\infty }}}{\nu x}}\begin{matrix}
<center><math>\eta =y\sqrt{\frac{{{u}_{\infty }}}{\nu x}}\begin{matrix}
   , & f=\frac{\psi }{\sqrt{\nu {{u}_{\infty }}x}}  \\
   , & f=\frac{\psi }{\sqrt{\nu {{u}_{\infty }}x}}  \\
\end{matrix},\text{  }\theta =\frac{T-{{T}_{\delta }}}{{{T}_{\infty }}-{{T}_{\delta }}}\begin{matrix}
\end{matrix},\text{  }\theta =\frac{T-{{T}_{\delta }}}{{{T}_{\infty }}-{{T}_{\delta }}}\begin{matrix}
   , & \varphi =\frac{\omega -{{\omega }_{\delta }}}{{{\omega }_{\infty }}-{{\omega }_{\delta }}}  \\
   , & \varphi =\frac{\omega -{{\omega }_{\delta }}}{{{\omega }_{\infty }}-{{\omega }_{\delta }}}  \\
-
\end{matrix}</math>
+
\end{matrix}</math></center>
-
(9.13)</center>
+
|{{EquationRef|(13)}}
 +
|}
-
eq. (9.12) and eqs. (9.3) – (9.4) can be reduced to a set of ordinary differential equations:
+
eq. (12) and eqs. (3) – (4) can be reduced to a set of ordinary differential equations:
-
<center><math>{f}'''+\frac{1}{2}f{f}''=0</math>
+
{| class="wikitable" border="0"
-
(9.14)</center>
+
|-
 +
| width="100%" |
 +
<center><math>{f}'''+\frac{1}{2}f{f}''=0</math></center>
 +
|{{EquationRef|(14)}}
 +
|}
-
<center><math>{\theta }''+\frac{1}{2}\Pr f{\theta }'=0</math> (9.15)</center>
+
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>{\theta }''+\frac{1}{2}\Pr f{\theta }'=0</math></center>
 +
|{{EquationRef|(15)}}
 +
|}
-
<center><math>{\varphi }''+\frac{1}{2}\text{Sc}f{\varphi }'=0</math> (9.16)</center>
+
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>{\varphi }''+\frac{1}{2}\text{Sc}f{\varphi }'=0</math></center>
 +
|{{EquationRef|(16)}}
 +
|}
-
The velocity components in the x- and y-directions in terms of the dimensionless stream function f can be obtained by eqs. (9.11) and (9.13), i.e.,
+
The velocity components in the x- and y-directions in terms of the dimensionless stream function f can be obtained by eqs. (11) and (13), i.e.,
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
<center><math>u={{u}_{\infty }}{f}'(\eta )\begin{matrix}
<center><math>u={{u}_{\infty }}{f}'(\eta )\begin{matrix}
   , & v=\frac{1}{2}\sqrt{\frac{\nu {{u}_{\infty }}}{x}}  \\
   , & v=\frac{1}{2}\sqrt{\frac{\nu {{u}_{\infty }}}{x}}  \\
-
\end{matrix}\left( \eta {f}'-f \right)</math>
+
\end{matrix}\left( \eta {f}'-f \right)</math></center>
-
(9.17)</center>
+
|{{EquationRef|(17)}}
 +
|}
-
The boundary conditions at y→∞ represented by eq. (9.5) become
+
The boundary conditions at y→∞ represented by eq. (5) become
-
<center><math>f'(\infty )=1\begin{matrix}
+
 
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>f'(\infty )=1\begin{matrix}
   , & \theta (\infty )=1, & \varphi (\infty )=1  \\
   , & \theta (\infty )=1, & \varphi (\infty )=1  \\
-
\end{matrix}</math> (9.18)</center>
+
\end{matrix}</math></center>
 +
|{{EquationRef|(18)}}
 +
|}
-
The boundary conditions represented by eq. (9.6), except the velocity component in the y-direction, can be rewritten to  
+
The boundary conditions represented by eq. (6), except the velocity component in the y-direction, can be rewritten to  
-
+
 
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
<center><math>f'(0)=0\begin{matrix}
<center><math>f'(0)=0\begin{matrix}
   , & \theta (0)=0, & \varphi (0)=0  \\
   , & \theta (0)=0, & \varphi (0)=0  \\
-
\end{matrix}</math> (9.19)</center>
+
\end{matrix}</math></center>
 +
|{{EquationRef|(19)}}
 +
|}
-
Conservation of mass and energy at the interface, eqs. (9.8) and (9.10), can be rewritten in terms of the similarity variables, i.e.,
+
Conservation of mass and energy at the interface, eqs. (8) and (10), can be rewritten in terms of the similarity variables, i.e.,
-
+
 
-
<center><math>f(0)=-\frac{2\text{Ja}}{\Pr }{\theta }'(0)</math>
+
{| class="wikitable" border="0"
-
(9.20)</center>
+
|-
-
+
| width="100%" |
-
<center><math>\frac{{{\omega }_{\delta }}-{{\omega }_{\infty }}}{1-{{\omega }_{\delta }}}{\phi }'(0)=\frac{\text{Sc}}{\Pr }Ja{\theta }'(0)</math>
+
<center><math>f(0)=-\frac{2\text{Ja}}{\Pr }{\theta }'(0)</math></center>
-
(9.21)</center>
+
|{{EquationRef|(20)}}
 +
|}
 +
 
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>\frac{{{\omega }_{\delta }}-{{\omega }_{\infty }}}{1-{{\omega }_{\delta }}}{\phi }'(0)=\frac{\text{Sc}}{\Pr }Ja{\theta }'(0)</math></center>
 +
|{{EquationRef|(21)}}
 +
|}
where ''Ja'' is the Jakob number and is defined as
where ''Ja'' is the Jakob number and is defined as
-
+
 
-
<center><math>\text{Ja}=\frac{{{c}_{p}}\left[ {{T}_{\infty }}-{{T}_{sat}}({{\omega }_{\delta }}) \right]}{{{h}_{\ell v}}}</math>
+
{| class="wikitable" border="0"
-
(9.22)</center>
+
|-
 +
| width="100%" |
 +
<center><math>\text{Ja}=\frac{{{c}_{p}}\left[ {{T}_{\infty }}-{{T}_{sat}}({{\omega }_{\delta }}) \right]}{{{h}_{\ell v}}}</math></center>
 +
|{{EquationRef|(22)}}
 +
|}
and Sc is the Schmidt number defined as  
and Sc is the Schmidt number defined as  
-
 
-
<center><math>\text{Sc}=\frac{{\tilde{\nu }}}{D}</math>
 
-
(9.23)</center>
 
-
The evaporation problem is now described by a set of ordinary differential equations (9.14) – (9.16) subjected to boundary conditions specified by eqs. (9.18) – (9.21). Since eq. (9.14) is a third-order ordinary differential equation, it requires three boundary conditions:  in eq. (9.18), in eq. (9.19), and eq. (9.20). Equations (9.15) and (9.16) are both second-order ordinary differential equations, each of them requiring two boundary conditions, which are specified in eqs. (9.18) and (9.19). Therefore, the problem is mathematically defined with boundary conditions stated by eqs. (9.18) – (9.20), which makes the boundary condition specified in eq. (9.21) an extra boundary condition – and makes the problem overstated. This happened because the heat and mass transfer are not independent of each other and the energy balance at the interface requires that eq. (9.21) be satisfied.
+
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>\text{Sc}=\frac{{\tilde{\nu }}}{D}</math></center>
 +
|{{EquationRef|(23)}}
 +
|}
 +
 
 +
The evaporation problem is now described by a set of ordinary differential equations (14) – (16) subjected to boundary conditions specified by eqs. (18) – (21). Since eq. (14) is a third-order ordinary differential equation, it requires three boundary conditions:  in eq. (18), in eq. (19), and eq. (20). Equations (15) and (16) are both second-order ordinary differential equations, each of them requiring two boundary conditions, which are specified in eqs. (18) and (19). Therefore, the problem is mathematically defined with boundary conditions stated by eqs. (18) – (20), which makes the boundary condition specified in eq. (21) an extra boundary condition – and makes the problem overstated. This happened because the heat and mass transfer are not independent of each other and the energy balance at the interface requires that eq. (21) be satisfied.
-
The boundary value problem can be solved using the Runge-Kutta method in conjunction with a shooting method. The solution procedure begins with an assumed ω<sub>I</sub> and uses boundary conditions specified by eqs. (9.18) – (9.20) to solve eqs. (9.14) – (9.16). Once a solution is obtained, eq. (9.21) is employed to find <math>{{\omega }_{\delta }}</math>. If <math>{{\omega }_{\delta }}</math> obtained from eq. (9.21) agrees with the assumed value, the solution is complete. Otherwise, the assumed value of <math>{{\omega }_{\delta }}</math> is corrected and the solution procedure is repeated until a converged solution is obtained.
+
The boundary value problem can be solved using the Runge-Kutta method in conjunction with a shooting method. The solution procedure begins with an assumed ω<sub>I</sub> and uses boundary conditions specified by eqs. (18) – (20) to solve eqs. (14) – (16). Once a solution is obtained, eq. (21) is employed to find <math>{{\omega }_{\delta }}</math>. If <math>{{\omega }_{\delta }}</math> obtained from eq. (21) agrees with the assumed value, the solution is complete. Otherwise, the assumed value of <math>{{\omega }_{\delta }}</math> is corrected and the solution procedure is repeated until a converged solution is obtained.
Once the converged solution is obtained, the local heat transfer coefficient can be evaluated by
Once the converged solution is obtained, the local heat transfer coefficient can be evaluated by
-
+
 
-
<center><math>{{h}_{x}}=\frac{{\tilde{k}}}{{{T}_{\infty }}-{{T}_{\delta }}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}=\tilde{k}\sqrt{\frac{{{u}_{\infty }}}{\tilde{\nu }x}}{\theta }'(0)</math>
+
 
-
(9.24)</center>
+
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>{{h}_{x}}=\frac{{\tilde{k}}}{{{T}_{\infty }}-{{T}_{\delta }}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}=\tilde{k}\sqrt{\frac{{{u}_{\infty }}}{\tilde{\nu }x}}{\theta }'(0)</math></center>
 +
|{{EquationRef|(24)}}
 +
|}
The local Nusselt number is then
The local Nusselt number is then
-
+
-
<center><math>N{{u}_{x}}=\frac{{{h}_{x}}x}{k}={\theta }'(0)\operatorname{Re}_{x}^{\frac{1}{2}}</math>
+
{| class="wikitable" border="0"
-
(9.25)</center>
+
|-
 +
| width="100%" |
 +
<center><math>N{{u}_{x}}=\frac{{{h}_{x}}x}{k}={\theta }'(0)\operatorname{Re}_{x}^{\frac{1}{2}}</math></center>
 +
|{{EquationRef|(25)}}
 +
|}
The local mass transfer coefficient can be evaluated by
The local mass transfer coefficient can be evaluated by
-
+
 
-
<center><math>{{h}_{mx}}=\frac{D}{{{\omega }_{\infty }}-{{\omega }_{\delta }}}{{\left( \frac{\partial \omega }{\partial y} \right)}_{y=0}}=D\sqrt{\frac{{{u}_{\infty }}}{\nu x}}{\varphi }'(0)</math>
+
{| class="wikitable" border="0"
-
(9.26)</center>
+
|-
 +
| width="100%" |
 +
<center><math>{{h}_{mx}}=\frac{D}{{{\omega }_{\infty }}-{{\omega }_{\delta }}}{{\left( \frac{\partial \omega }{\partial y} \right)}_{y=0}}=D\sqrt{\frac{{{u}_{\infty }}}{\nu x}}{\varphi }'(0)</math></center>
 +
|{{EquationRef|(26)}}
 +
|}
and the local Sherwood number is  
and the local Sherwood number is  
 
 
-
<center><math>S{{h}_{x}}=\frac{{{h}_{m}}x}{D}={\varphi }'(0)\operatorname{Re}_{x}^{\frac{1}{2}}</math>
+
{| class="wikitable" border="0"
-
(9.27)</center>
+
|-
 +
| width="100%" |
 +
<center><math>S{{h}_{x}}=\frac{{{h}_{m}}x}{D}={\varphi }'(0)\operatorname{Re}_{x}^{\frac{1}{2}}</math></center>
 +
|{{EquationRef|(27)}}
 +
|}
where  
where  
-
<center><math>{{\operatorname{Re}}_{x}}=\frac{{{u}_{\infty }}x}{\nu }</math>
+
{| class="wikitable" border="0"
-
(9.28)</center>
+
|-
 +
| width="100%" |
 +
<center><math>{{\operatorname{Re}}_{x}}=\frac{{{u}_{\infty }}x}{\nu }</math></center>
 +
|{{EquationRef|(28)}}
 +
|}
-
Although eqs. (9.25) and (9.27) have the same form as the case of single-phase forced convective heat or mass transfer over a flat plate, the solutions of <math>{\theta }'(0)</math> and <math>{\varphi }'(0)</math> depend on ''Pr'', ''Sc'', ''Ja'', and ''ω<sub>∞</sub>''. Consequently, simple empirical correlations similar to the case of single-phase forced convective heat or mass transfer over a flat plate are very difficult to obtain.
+
Although eqs. (25) and (27) have the same form as the case of single-phase forced convective heat or mass transfer over a flat plate, the solutions of <math>{\theta }'(0)</math> and <math>{\varphi }'(0)</math> depend on ''Pr'', ''Sc'', ''Ja'', and ''ω<sub>∞</sub>''. Consequently, simple empirical correlations similar to the case of single-phase forced convective heat or mass transfer over a flat plate are very difficult to obtain.
-
The evaporation problem presented here is a coupled problem because the solutions of the momentum, energy, and species equations are coupled with eqs. (9.20) – (9.21). Since the blowing velocity at the surface of the liquid film,  determined by eq. (9.8), is usually much lower than the incoming vapor-gas mixture velocity, <math>{{u}_{\infty }}</math>, one can assume that the blowing velocity at the liquid surface is negligible. Then eq. (9.20) can be replaced with
+
The evaporation problem presented here is a coupled problem because the solutions of the momentum, energy, and species equations are coupled with eqs. (20) – (21). Since the blowing velocity at the surface of the liquid film,  determined by eq. (8), is usually much lower than the incoming vapor-gas mixture velocity, <math>{{u}_{\infty }}</math>, one can assume that the blowing velocity at the liquid surface is negligible. Then eq. (20) can be replaced with
-
+
-
<center><math>f(0)=0</math>
+
-
(9.29)</center>
+
-
The heat and mass transfer problem described by eqs. (9.14) – (9.16), (9.18) – (9.19) and (9.29) then cease to be a conjugated problem, because each equation in eq. (9.14) – (9.16) can be solved independently. Numerical solutions of these ordinary differential equations yield the following results (Kays et al., 2004):
+
{| class="wikitable" border="0"
-
+
|-
-
<center><math>{\theta }'(0)=0.332{{\Pr }^{\frac{1}{3}}}</math>
+
| width="100%" |
-
(9.30)</center>
+
<center><math>f(0)=0</math></center>
-
+
|{{EquationRef|(29)}}
-
<center><math>{\varphi }'(0)=0.332S{{c}^{\frac{1}{3}}}</math>
+
|}
-
(9.31)</center>
+
-
Substituting eqs. (9.30) and (9.31) into eqs. (9.25) and (9.27) yields
+
The heat and mass transfer problem described by eqs. (14) – (16), (18) – (19) and (29) then cease to be a conjugated problem, because each equation in eq. (14) – (16) can be solved independently. Numerical solutions of these ordinary differential equations yield the following results (Kays et al., 2004):
-
+
 
-
<center><math>N{{u}_{x}}=\frac{{{h}_{x}}x}{k}=0.332\operatorname{Re}_{x}^{\frac{1}{2}}{{\Pr }^{\frac{1}{3}}}</math>
+
{| class="wikitable" border="0"
-
(9.32) </center>
+
|-
 +
| width="100%" |
 +
<center><math>{\theta }'(0)=0.332{{\Pr }^{\frac{1}{3}}}</math></center>
 +
|{{EquationRef|(30)}}
 +
|}
-
<center><math>S{{h}_{x}}=\frac{{{h}_{m}}x}{k}=0.332\operatorname{Re}_{x}^{\frac{1}{2}}S{{c}^{\frac{1}{3}}}</math>
+
{| class="wikitable" border="0"
-
(9.33)</center>
+
|-
 +
| width="100%" |
 +
<center><math>{\varphi }'(0)=0.332S{{c}^{\frac{1}{3}}}</math></center>
 +
|{{EquationRef|(31)}}
 +
|}
 +
 
 +
Substituting eqs. (30) and (31) into eqs. (25) and (27) yields
 +
 
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>N{{u}_{x}}=\frac{{{h}_{x}}x}{k}=0.332\operatorname{Re}_{x}^{\frac{1}{2}}{{\Pr }^{\frac{1}{3}}}</math></center>
 +
|{{EquationRef|(32)}}
 +
|}
 +
 
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>S{{h}_{x}}=\frac{{{h}_{m}}x}{k}=0.332\operatorname{Re}_{x}^{\frac{1}{2}}S{{c}^{\frac{1}{3}}}</math></center>
 +
|{{EquationRef|(33)}}
 +
|}
The average heat and mass transfer characteristics are often relevant for practical applications. The average Nusselt and Sherwood numbers based on average heat and mass transfer coefficients for a liquid film with a length of ''L'' are obtained by  
The average heat and mass transfer characteristics are often relevant for practical applications. The average Nusselt and Sherwood numbers based on average heat and mass transfer coefficients for a liquid film with a length of ''L'' are obtained by  
 +
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>\overline{Nu}=\frac{\bar{h}L}{k}=0.664\operatorname{Re}_{L}^{\frac{1}{2}}{{\Pr }^{\frac{1}{3}}}</math></center>
 +
|{{EquationRef|(34)}}
 +
|}
-
<center><math>\overline{Nu}=\frac{\bar{h}L}{k}=0.664\operatorname{Re}_{L}^{\frac{1}{2}}{{\Pr }^{\frac{1}{3}}}</math>
+
{| class="wikitable" border="0"
-
(9.34)</center>
+
|-
-
+
| width="100%" |
-
<center><math>\overline{Sh}=\frac{{{{\bar{h}}}_{m}}L}{k}=0.664\operatorname{Re}_{L}^{\frac{1}{2}}S{{c}^{\frac{1}{3}}}</math>
+
<center><math>\overline{Sh}=\frac{{{{\bar{h}}}_{m}}L}{k}=0.664\operatorname{Re}_{L}^{\frac{1}{2}}S{{c}^{\frac{1}{3}}}</math></center>
-
(9.35)</center>
+
|{{EquationRef|(35)}}
 +
|}
where
where
-
<center><math>{{\operatorname{Re}}_{L}}=\frac{{{u}_{\infty }}L}{\nu }</math>
+
{| class="wikitable" border="0"
-
(9.36)</center>
+
|-
 +
| width="100%" |
 +
<center><math>{{\operatorname{Re}}_{L}}=\frac{{{u}_{\infty }}L}{\nu }</math></center>
 +
|{{EquationRef|(36)}}
 +
|}
-
The mass fraction of water at the liquid surface can be obtained by substituting eqs. (9.30) and (9.31) into eq. (9.21), i.e.,  
+
The mass fraction of water at the liquid surface can be obtained by substituting eqs. (30) and (31) into eq. (21), i.e.,  
-
<center><math>{{\omega }_{\delta }}=\frac{{{\omega }_{\infty }}+{{(\text{Sc}/\Pr )}^{2/3}}\text{Ja}}{1+{{(\text{Sc}/\Pr )}^{2/3}}\text{Ja}}</math>
+
{| class="wikitable" border="0"
-
(9.37)</center>
+
|-
 +
| width="100%" |
 +
<center><math>{{\omega }_{\delta }}=\frac{{{\omega }_{\infty }}+{{(\text{Sc}/\Pr )}^{2/3}}\text{Ja}}{1+{{(\text{Sc}/\Pr )}^{2/3}}\text{Ja}}</math></center>
 +
|{{EquationRef|(37)}}
 +
|}
The applications of the above empirical correlations can be demonstrated by the following example.
The applications of the above empirical correlations can be demonstrated by the following example.

Revision as of 18:12, 3 June 2010

Evaporation from thin liquid films occurs in many industrial and natural processes, including drying, evaporative cooling, and sweating. Figure 9.6 shows the physical model of horizontal thin film evaporation under consideration (Carey, 1992). A film wets a horizontal surface over which flows a gas of ambient temperature {{T}_{\infty }} and mass fraction of water {{\omega }_{\infty }} at a velocity of {{u}_{\infty }}. Since the solid surface underneath the liquid is adiabatic, the latent heat of vaporization is provided by the vapor-gas mixture flowing above the liquid film. The heat source (vapor-gas mixture) and the evaporating liquid are in direct contact, making this scenario an example of direct contact evaporation. The liquid is evaporated and the latent heat is absorbed in the evaporation process. The resulting vapor injects into the boundary layer and is removed by the gas flow. The boundary layer becomes thicker, and a free stream of the gas flow is displaced from the surface being cooled. While phase change is the dominant mechanism of heat transfer at lower gas temperatures, vapor injection becomes more important at higher gas temperatures that correspond to higher evaporation rates.

Due to diffusion and convection, the temperature rises and the concentration falls from their values at the film surface to their ambient values at the edge of the boundary layer. A nonslip boundary condition exists at the film surface, and the film itself is considered stationary with respect to the gas. Assuming a steady state, constant density, and incompressible flow, the continuity equation is

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

With no pressure gradients and constant viscosity assumed, the boundary layer momentum equation is written as

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

 Examples of film evaporators. Figure 9.6 Evaporation from a thin liquid film on a horizontal surface.

where ~ on top of viscosity signifies mass-averaged properties of the mixture. Neglecting viscous dissipation and assuming constant thermal diffusivity and specific heats, the boundary layer energy balance is

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

where the axial conduction on the x-direction has been neglected because heat transfer occurred mainly in the y-direction. The mass fraction of water is also accounted for by assuming a constant mass diffusivity:

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

As y\to \infty the boundary conditions can be taken directly from Fig. 9.6:

u\to {{u}_{\infty }}, T\to {{T}_{\infty }},

\omega \to {{\omega }_{\infty }}
(5)

The boundary conditions are now laid out at the film surface y = 0. There is a nonslip condition, and the mass flux {\dot{m}}'' evaporates normal to the surface.

u = 0, v=\frac{{{\dot{m}}''}}{{\tilde{\rho }}}, ω = ωδ,

T={{T}_{\delta }}={{T}_{sat}}\left( {{\omega }_{\delta }},{{p}_{\infty }} \right)
(6)

An energy balance at the interface accounts for the heat of vaporization and mass flux of the evaporating fluid. With the nonslip boundary condition, the only mode of heat transfer is conduction:

</center>{\dot{m}}''{{h}_{\ell v}}=-{q}''=\tilde{k}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}
(7)

Substituting eq. (7) into eq. (6), the velocity component in the y-direction becomes

<center>{{\left. v \right|}_{y=0}}=\frac{{\tilde{k}}}{\rho {{h}_{\ell v}}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}} (8)

The mass flux at the interface is the result of both diffusion and convection. Therefore, the mass balance at the interface is written as

{\dot{m}}''=-\rho D{{\left( \frac{\partial \omega }{\partial y} \right)}_{y=0}}+\rho {{\omega }_{\delta }}{{\left. v \right|}_{y=0}}
(9)

Substituting eq. (7) into eq. (9), conservation of mass at the interface becomes

-\tilde{\rho }D{{\left( \frac{\partial \omega }{\partial y} \right)}_{y=0}}+\tilde{\rho }{{\omega }_{\delta }}{{\left. v \right|}_{y=0}}=\frac{{\tilde{k}}}{{{h}_{\ell v}}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}
(10)

Equations (1) – (2) are the same as the laminar boundary layer equations for forced convection over a flat plate with blowing on the liquid surface. By defining the stream function ψ as follows,

u=\frac{\partial \psi }{\partial y}\begin{matrix}
   {} & v=-\frac{\partial \psi }{\partial x}  \\
\end{matrix}
(11)

the continuity equation (1) is automatically satisfied. The momentum equation in terms of the stream function is then

\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}}}
(12)

Introducing the following similarity variable:

\eta =y\sqrt{\frac{{{u}_{\infty }}}{\nu x}}\begin{matrix}
   , & f=\frac{\psi }{\sqrt{\nu {{u}_{\infty }}x}}  \\
\end{matrix},\text{  }\theta =\frac{T-{{T}_{\delta }}}{{{T}_{\infty }}-{{T}_{\delta }}}\begin{matrix}
   , & \varphi =\frac{\omega -{{\omega }_{\delta }}}{{{\omega }_{\infty }}-{{\omega }_{\delta }}}  \\
\end{matrix}
(13)

eq. (12) and eqs. (3) – (4) can be reduced to a set of ordinary differential equations:

{f}'''+\frac{1}{2}f{f}''=0
(14)
{\theta }''+\frac{1}{2}\Pr f{\theta }'=0
(15)
{\varphi }''+\frac{1}{2}\text{Sc}f{\varphi }'=0
(16)

The velocity components in the x- and y-directions in terms of the dimensionless stream function f can be obtained by eqs. (11) and (13), i.e.,

u={{u}_{\infty }}{f}'(\eta )\begin{matrix}
   , & v=\frac{1}{2}\sqrt{\frac{\nu {{u}_{\infty }}}{x}}  \\
\end{matrix}\left( \eta {f}'-f \right)
(17)

The boundary conditions at y→∞ represented by eq. (5) become

f'(\infty )=1\begin{matrix}
   , & \theta (\infty )=1, & \varphi (\infty )=1  \\
\end{matrix}
(18)

The boundary conditions represented by eq. (6), except the velocity component in the y-direction, can be rewritten to

f'(0)=0\begin{matrix}
   , & \theta (0)=0, & \varphi (0)=0  \\
\end{matrix}
(19)

Conservation of mass and energy at the interface, eqs. (8) and (10), can be rewritten in terms of the similarity variables, i.e.,

f(0)=-\frac{2\text{Ja}}{\Pr }{\theta }'(0)
(20)
\frac{{{\omega }_{\delta }}-{{\omega }_{\infty }}}{1-{{\omega }_{\delta }}}{\phi }'(0)=\frac{\text{Sc}}{\Pr }Ja{\theta }'(0)
(21)

where Ja is the Jakob number and is defined as

\text{Ja}=\frac{{{c}_{p}}\left[ {{T}_{\infty }}-{{T}_{sat}}({{\omega }_{\delta }}) \right]}{{{h}_{\ell v}}}
(22)

and Sc is the Schmidt number defined as

\text{Sc}=\frac{{\tilde{\nu }}}{D}
(23)

The evaporation problem is now described by a set of ordinary differential equations (14) – (16) subjected to boundary conditions specified by eqs. (18) – (21). Since eq. (14) is a third-order ordinary differential equation, it requires three boundary conditions: in eq. (18), in eq. (19), and eq. (20). Equations (15) and (16) are both second-order ordinary differential equations, each of them requiring two boundary conditions, which are specified in eqs. (18) and (19). Therefore, the problem is mathematically defined with boundary conditions stated by eqs. (18) – (20), which makes the boundary condition specified in eq. (21) an extra boundary condition – and makes the problem overstated. This happened because the heat and mass transfer are not independent of each other and the energy balance at the interface requires that eq. (21) be satisfied.

The boundary value problem can be solved using the Runge-Kutta method in conjunction with a shooting method. The solution procedure begins with an assumed ωI and uses boundary conditions specified by eqs. (18) – (20) to solve eqs. (14) – (16). Once a solution is obtained, eq. (21) is employed to find ωδ. If ωδ obtained from eq. (21) agrees with the assumed value, the solution is complete. Otherwise, the assumed value of ωδ is corrected and the solution procedure is repeated until a converged solution is obtained.

Once the converged solution is obtained, the local heat transfer coefficient can be evaluated by


{{h}_{x}}=\frac{{\tilde{k}}}{{{T}_{\infty }}-{{T}_{\delta }}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}=\tilde{k}\sqrt{\frac{{{u}_{\infty }}}{\tilde{\nu }x}}{\theta }'(0)
(24)

The local Nusselt number is then

N{{u}_{x}}=\frac{{{h}_{x}}x}{k}={\theta }'(0)\operatorname{Re}_{x}^{\frac{1}{2}}
(25)

The local mass transfer coefficient can be evaluated by

{{h}_{mx}}=\frac{D}{{{\omega }_{\infty }}-{{\omega }_{\delta }}}{{\left( \frac{\partial \omega }{\partial y} \right)}_{y=0}}=D\sqrt{\frac{{{u}_{\infty }}}{\nu x}}{\varphi }'(0)
(26)

and the local Sherwood number is

S{{h}_{x}}=\frac{{{h}_{m}}x}{D}={\varphi }'(0)\operatorname{Re}_{x}^{\frac{1}{2}}
(27)

where

{{\operatorname{Re}}_{x}}=\frac{{{u}_{\infty }}x}{\nu }
(28)

Although eqs. (25) and (27) have the same form as the case of single-phase forced convective heat or mass transfer over a flat plate, the solutions of θ'(0) and {\varphi }'(0) depend on Pr, Sc, Ja, and ω. Consequently, simple empirical correlations similar to the case of single-phase forced convective heat or mass transfer over a flat plate are very difficult to obtain.

The evaporation problem presented here is a coupled problem because the solutions of the momentum, energy, and species equations are coupled with eqs. (20) – (21). Since the blowing velocity at the surface of the liquid film, determined by eq. (8), is usually much lower than the incoming vapor-gas mixture velocity, {{u}_{\infty }}, one can assume that the blowing velocity at the liquid surface is negligible. Then eq. (20) can be replaced with

f(0) = 0
(29)

The heat and mass transfer problem described by eqs. (14) – (16), (18) – (19) and (29) then cease to be a conjugated problem, because each equation in eq. (14) – (16) can be solved independently. Numerical solutions of these ordinary differential equations yield the following results (Kays et al., 2004):

{\theta }'(0)=0.332{{\Pr }^{\frac{1}{3}}}
(30)
{\varphi }'(0)=0.332S{{c}^{\frac{1}{3}}}
(31)

Substituting eqs. (30) and (31) into eqs. (25) and (27) yields

N{{u}_{x}}=\frac{{{h}_{x}}x}{k}=0.332\operatorname{Re}_{x}^{\frac{1}{2}}{{\Pr }^{\frac{1}{3}}}
(32)
S{{h}_{x}}=\frac{{{h}_{m}}x}{k}=0.332\operatorname{Re}_{x}^{\frac{1}{2}}S{{c}^{\frac{1}{3}}}
(33)

The average heat and mass transfer characteristics are often relevant for practical applications. The average Nusselt and Sherwood numbers based on average heat and mass transfer coefficients for a liquid film with a length of L are obtained by

\overline{Nu}=\frac{\bar{h}L}{k}=0.664\operatorname{Re}_{L}^{\frac{1}{2}}{{\Pr }^{\frac{1}{3}}}
(34)
\overline{Sh}=\frac{{{{\bar{h}}}_{m}}L}{k}=0.664\operatorname{Re}_{L}^{\frac{1}{2}}S{{c}^{\frac{1}{3}}}
(35)

where

{{\operatorname{Re}}_{L}}=\frac{{{u}_{\infty }}L}{\nu }
(36)

The mass fraction of water at the liquid surface can be obtained by substituting eqs. (30) and (31) into eq. (21), i.e.,

{{\omega }_{\delta }}=\frac{{{\omega }_{\infty }}+{{(\text{Sc}/\Pr )}^{2/3}}\text{Ja}}{1+{{(\text{Sc}/\Pr )}^{2/3}}\text{Ja}}
(37)

The applications of the above empirical correlations can be demonstrated by the following example.

References

Carey, V.P., 1992, Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, Hemisphere Publishing Corp., Washington, D. C.