Evaporation of a Liquid Jet in a Pure Vapor

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The physical model of the problem under consideration is shown in Fig. 9.17 [[#References|(Lock, 1994)]]. A liquid jet flows out from a nozzle with a radius of r0 and is surrounded by pure vapor at saturation temperature. At the exit of the nozzle, the velocity and temperature are uniformly T0 and u0, respectively. It is further assumed that the velocity in the jet remains uniformly equal to u0 as it continues flowing; then the jet can be treated as slug flow. This is possible when the friction between the liquid jet and the surrounding vapor is negligible. The temperature of the jet will be affected by the hot gas as soon as it exits the nozzle. Because evaporation occurs on the surface of the liquid jet, its surface temperature is equal to the saturation temperature corresponding to the vapor pressure.  
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The physical model of the problem under consideration is shown in Fig. 9.17 [[#References|(Lock, 1994)]]. A liquid jet flows out from a nozzle with a radius of ''r<sub>0</sub>'' and is surrounded by pure vapor at saturation temperature. At the exit of the nozzle, the velocity and temperature are uniformly ''T<sub>0</sub>'' and ''u<sub>0</sub>'', respectively. It is further assumed that the velocity in the jet remains uniformly equal to u0 as it continues flowing; then the jet can be treated as slug flow. This is possible when the friction between the liquid jet and the surrounding vapor is negligible. The temperature of the jet will be affected by the hot gas as soon as it exits the nozzle. Because evaporation occurs on the surface of the liquid jet, its surface temperature is equal to the saturation temperature corresponding to the vapor pressure.  
The energy equation for the liquid jet is
The energy equation for the liquid jet is
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<center><math>{{u}_{0}}\frac{\partial {{T}_{\ell }}}{\partial x}=\frac{{{\alpha }_{\ell }}}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{T}_{\ell }}}{\partial r} \right)</math>
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{| class="wikitable" border="0"
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(9.228)</center>
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|-
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| width="100%" |
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<center><math>{{u}_{0}}\frac{\partial {{T}_{\ell }}}{\partial x}=\frac{{{\alpha }_{\ell }}}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{T}_{\ell }}}{\partial r} \right)</math></center>
 +
|{{EquationRef|(1)}}
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|}
where the thermophysical properties have been assumed to be constants. The initial temperature of the jet is  
where the thermophysical properties have been assumed to be constants. The initial temperature of the jet is  
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<center><math>{{T}_{\ell }}(r,x)={{T}_{0}}\begin{matrix}
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{| class="wikitable" border="0"
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  , & t=0  \\
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|-
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\end{matrix}</math>
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| width="100%" |
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(9.229)</center>
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<center><math>{{u}_{0}}\frac{\partial {{T}_{\ell }}}{\partial x}=\frac{{{\alpha }_{\ell }}}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{T}_{\ell }}}{\partial r} \right)</math></center>
 +
|{{EquationRef|(1)}}
 +
|}
[[Image:NewChapter9 (12).jpg|thumb|400 px|alt= Evaporation from a jet surrounded by a hot gas. | Figure 9.17 Evaporation from a jet surrounded by a hot gas. ]]
[[Image:NewChapter9 (12).jpg|thumb|400 px|alt= Evaporation from a jet surrounded by a hot gas. | Figure 9.17 Evaporation from a jet surrounded by a hot gas. ]]
The boundary conditions at the center and surface of the liquid jet are
The boundary conditions at the center and surface of the liquid jet are
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 +
{| class="wikitable" border="0"
 +
|-
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| width="100%" |
<center><math>\frac{\partial {{T}_{\ell }}}{\partial r}=0\begin{matrix}
<center><math>\frac{\partial {{T}_{\ell }}}{\partial r}=0\begin{matrix}
   , & r=0  \\
   , & r=0  \\
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\end{matrix}</math>
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\end{matrix}</math></center>
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(9.230)</center>
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|{{EquationRef|(2)}}
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+
|}
 +
 
 +
{| class="wikitable" border="0"
 +
|-
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| width="100%" |
<center><math>T(r,x)={{T}_{sat}}\begin{matrix}
<center><math>T(r,x)={{T}_{sat}}\begin{matrix}
   , & r={{r}_{I}}  \\
   , & r={{r}_{I}}  \\
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\end{matrix}(x)</math>
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\end{matrix}(x)</math></center>
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(9.231)</center>
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|{{EquationRef|(3)}}
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|}
where ''r<sub>I</sub>''(x) is the radius of the liquid jet, which equals ''r<sub>0</sub>'' at ''x'' = 0 but decreases with increasing ''x''.  
where ''r<sub>I</sub>''(x) is the radius of the liquid jet, which equals ''r<sub>0</sub>'' at ''x'' = 0 but decreases with increasing ''x''.  
The energy balance at the surface of the liquid jet is
The energy balance at the surface of the liquid jet is
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 +
{| class="wikitable" border="0"
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|-
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| width="100%" |
<center><math>{{\rho }_{\ell }}{{h}_{\ell v}}{{u}_{0}}\frac{d{{r}_{I}}}{dx}={{k}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial r}\begin{matrix}
<center><math>{{\rho }_{\ell }}{{h}_{\ell v}}{{u}_{0}}\frac{d{{r}_{I}}}{dx}={{k}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial r}\begin{matrix}
   , & r={{r}_{I}}  \\
   , & r={{r}_{I}}  \\
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\end{matrix}(x)</math>
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\end{matrix}(x)</math></center>
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(9.232)</center>
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|{{EquationRef|(4)}}
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|}
which demonstrates that superheated liquid supplies the latent heat of vaporization. Since the vapor temperature equals the saturation temperature, there is no convection on the surface of the liquid jet.
which demonstrates that superheated liquid supplies the latent heat of vaporization. Since the vapor temperature equals the saturation temperature, there is no convection on the surface of the liquid jet.
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During the first stage, the effect of the curvature of the cylindrical liquid jet can be neglected because the thermal boundary layer is much thinner than the radius of the tube. The solution of the temperature distribution in the thermal boundary layer can be obtained by using the solution of heat conduction in a semi-infinite solid, i.e.,   
During the first stage, the effect of the curvature of the cylindrical liquid jet can be neglected because the thermal boundary layer is much thinner than the radius of the tube. The solution of the temperature distribution in the thermal boundary layer can be obtained by using the solution of heat conduction in a semi-infinite solid, i.e.,   
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<center><math>T(y,x)={{T}_{sat}}+({{T}_{0}}-{{T}_{sat}})\text{erf}\left( \frac{y}{2\sqrt{{{\alpha }_{\ell }}x/{{u}_{0}}}} \right)</math>
 
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(9.233)</center>
 
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where the y-coordinate originates at the jet’s surface and points toward its center. It is related to ''r'' by <math>y={{r}_{I}}-r</math>. Substituting eq. (9.233) into the energy balance equation at the interface, eq. (9.232), one obtains  
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{| class="wikitable" border="0"
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|-
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<center><math>\frac{d{{r}_{I}}}{dx}=-\frac{{{k}_{\ell }}({{T}_{0}}-{{T}_{sat}})}{{{\rho }_{\ell }}{{h}_{\ell v}}\sqrt{\pi {{\alpha }_{\ell }}{{u}_{0}}x}}</math>
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| width="100%" |
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(9.234)</center>
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<center><math>T(y,x)={{T}_{sat}}+({{T}_{0}}-{{T}_{sat}})\text{erf}\left( \frac{y}{2\sqrt{{{\alpha }_{\ell }}x/{{u}_{0}}}} \right)</math></center>
 +
|{{EquationRef|(5)}}
 +
|}
 +
 
 +
where the y-coordinate originates at the jet’s surface and points toward its center. It is related to ''r'' by <math>y={{r}_{I}}-r</math>. Substituting eq. (5) into the energy balance equation at the interface, eq. (4), one obtains  
 +
 
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>\frac{d{{r}_{I}}}{dx}=-\frac{{{k}_{\ell }}({{T}_{0}}-{{T}_{sat}})}{{{\rho }_{\ell }}{{h}_{\ell v}}\sqrt{\pi {{\alpha }_{\ell }}{{u}_{0}}x}}</math></center>
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|{{EquationRef|(6)}}
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|}
The variation in the radius of the liquid jet during the first stage can be obtained by integrating eq. (9.234), i.e.,
The variation in the radius of the liquid jet during the first stage can be obtained by integrating eq. (9.234), i.e.,
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<center><math>{{r}_{I}}={{R}_{i}}-\frac{2{{k}_{\ell }}({{T}_{0}}-{{T}_{sat}})}{{{\rho }_{\ell }}{{h}_{\ell v}}}\sqrt{\frac{x}{\pi {{\alpha }_{\ell }}{{u}_{0}}}}</math>
+
{| class="wikitable" border="0"
-
(9.235)</center>
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|-
 +
| width="100%" |
 +
<center><math>{{r}_{I}}={{R}_{i}}-\frac{2{{k}_{\ell }}({{T}_{0}}-{{T}_{sat}})}{{{\rho }_{\ell }}{{h}_{\ell v}}}\sqrt{\frac{x}{\pi {{\alpha }_{\ell }}{{u}_{0}}}}</math></center>
 +
|{{EquationRef|(7)}}
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|}
Evaporation during the second stage (<math>{{\delta }_{t}}={{r}_{I}}</math>), which occurs at the lower portion of the jet, is much slower than the first stage and ceases after the superheat in the liquid jet vanishes. The final radius of the liquid jet, <math>{{r}_{I,f}}</math>, can be obtained by a simple energy balance:
Evaporation during the second stage (<math>{{\delta }_{t}}={{r}_{I}}</math>), which occurs at the lower portion of the jet, is much slower than the first stage and ceases after the superheat in the liquid jet vanishes. The final radius of the liquid jet, <math>{{r}_{I,f}}</math>, can be obtained by a simple energy balance:
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<center><math>{{\rho }_{\ell }}{{c}_{p\ell }}\pi R_{i}^{2}({{T}_{0}}-{{T}_{sat}})={{\rho }_{\ell }}{{h}_{\ell v}}\pi (R_{i}^{2}-r_{I,f}^{2})</math>
 
-
(9.236)</center>
 
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Rearranging eq. (9.236), one obtains the final radius of the liquid jet as
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{| class="wikitable" border="0"
-
+
|-
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<center><math>{{r}_{I,f}}={{R}_{i}}\sqrt{1-\text{Ja}}</math>
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| width="100%" |
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(9.237)</center>
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<center><math>{{\rho }_{\ell }}{{c}_{p\ell }}\pi R_{i}^{2}({{T}_{0}}-{{T}_{sat}})={{\rho }_{\ell }}{{h}_{\ell v}}\pi (R_{i}^{2}-r_{I,f}^{2})</math></center>
 +
|{{EquationRef|(8)}}
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|}
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 +
Rearranging eq. (8), one obtains the final radius of the liquid jet as
 +
 
 +
{| class="wikitable" border="0"
 +
|-
 +
| width="100%" |
 +
<center><math>{{r}_{I,f}}={{R}_{i}}\sqrt{1-\text{Ja}}</math></center>
 +
|{{EquationRef|(9)}}
 +
|}
where  
where  
-
+
 
-
<center><math>\text{Ja}=\frac{{{c}_{{{p}_{\ell }}}}({{T}_{0}}-{{T}_{sat}})}{{{h}_{\ell v}}}</math>
+
{| class="wikitable" border="0"
-
(9.238)</center>
+
|-
 +
| width="100%" |
 +
<center><math>\text{Ja}=\frac{{{c}_{{{p}_{\ell }}}}({{T}_{0}}-{{T}_{sat}})}{{{h}_{\ell v}}}</math></center>
 +
|{{EquationRef|(10)}}
 +
|}
is the Jakob number.
is the Jakob number.

Revision as of 19:38, 3 June 2010

The physical model of the problem under consideration is shown in Fig. 9.17 (Lock, 1994). A liquid jet flows out from a nozzle with a radius of r0 and is surrounded by pure vapor at saturation temperature. At the exit of the nozzle, the velocity and temperature are uniformly T0 and u0, respectively. It is further assumed that the velocity in the jet remains uniformly equal to u0 as it continues flowing; then the jet can be treated as slug flow. This is possible when the friction between the liquid jet and the surrounding vapor is negligible. The temperature of the jet will be affected by the hot gas as soon as it exits the nozzle. Because evaporation occurs on the surface of the liquid jet, its surface temperature is equal to the saturation temperature corresponding to the vapor pressure.

The energy equation for the liquid jet is

{{u}_{0}}\frac{\partial {{T}_{\ell }}}{\partial x}=\frac{{{\alpha }_{\ell }}}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{T}_{\ell }}}{\partial r} \right)
(1)

where the thermophysical properties have been assumed to be constants. The initial temperature of the jet is

{{u}_{0}}\frac{\partial {{T}_{\ell }}}{\partial x}=\frac{{{\alpha }_{\ell }}}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{T}_{\ell }}}{\partial r} \right)
(1)
 Evaporation from a jet surrounded by a hot gas.
Figure 9.17 Evaporation from a jet surrounded by a hot gas.

The boundary conditions at the center and surface of the liquid jet are

\frac{\partial {{T}_{\ell }}}{\partial r}=0\begin{matrix}
   , & r=0  \\
\end{matrix}
(2)
T(r,x)={{T}_{sat}}\begin{matrix}
   , & r={{r}_{I}}  \\
\end{matrix}(x)
(3)

where rI(x) is the radius of the liquid jet, which equals r0 at x = 0 but decreases with increasing x. The energy balance at the surface of the liquid jet is

{{\rho }_{\ell }}{{h}_{\ell v}}{{u}_{0}}\frac{d{{r}_{I}}}{dx}={{k}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial r}\begin{matrix}
   , & r={{r}_{I}}  \\
\end{matrix}(x)
(4)

which demonstrates that superheated liquid supplies the latent heat of vaporization. Since the vapor temperature equals the saturation temperature, there is no convection on the surface of the liquid jet.

As the liquid jet moves downstream, a thermal boundary layer, r0 − δt < r < r0, develops in the liquid jet. The evaporation of the jet can be divided into two stages: the early stage when {{\delta }_{t}}\ll {{r}_{0}} and a later stage when δt = r0. These two stages are similar to the thermal entrance problem and fully-developed flow for forced convection in a tube.

During the first stage, the effect of the curvature of the cylindrical liquid jet can be neglected because the thermal boundary layer is much thinner than the radius of the tube. The solution of the temperature distribution in the thermal boundary layer can be obtained by using the solution of heat conduction in a semi-infinite solid, i.e.,

T(y,x)={{T}_{sat}}+({{T}_{0}}-{{T}_{sat}})\text{erf}\left( \frac{y}{2\sqrt{{{\alpha }_{\ell }}x/{{u}_{0}}}} \right)
(5)

where the y-coordinate originates at the jet’s surface and points toward its center. It is related to r by y = rIr. Substituting eq. (5) into the energy balance equation at the interface, eq. (4), one obtains

\frac{d{{r}_{I}}}{dx}=-\frac{{{k}_{\ell }}({{T}_{0}}-{{T}_{sat}})}{{{\rho }_{\ell }}{{h}_{\ell v}}\sqrt{\pi {{\alpha }_{\ell }}{{u}_{0}}x}}
(6)

The variation in the radius of the liquid jet during the first stage can be obtained by integrating eq. (9.234), i.e.,

{{r}_{I}}={{R}_{i}}-\frac{2{{k}_{\ell }}({{T}_{0}}-{{T}_{sat}})}{{{\rho }_{\ell }}{{h}_{\ell v}}}\sqrt{\frac{x}{\pi {{\alpha }_{\ell }}{{u}_{0}}}}
(7)

Evaporation during the second stage (δt = rI), which occurs at the lower portion of the jet, is much slower than the first stage and ceases after the superheat in the liquid jet vanishes. The final radius of the liquid jet, rI,f, can be obtained by a simple energy balance:

{{\rho }_{\ell }}{{c}_{p\ell }}\pi R_{i}^{2}({{T}_{0}}-{{T}_{sat}})={{\rho }_{\ell }}{{h}_{\ell v}}\pi (R_{i}^{2}-r_{I,f}^{2})
(8)

Rearranging eq. (8), one obtains the final radius of the liquid jet as

{{r}_{I,f}}={{R}_{i}}\sqrt{1-\text{Ja}}
(9)

where

\text{Ja}=\frac{{{c}_{{{p}_{\ell }}}}({{T}_{0}}-{{T}_{sat}})}{{{h}_{\ell v}}}
(10)

is the Jakob number.

References

Lock, G.S.H., 1994, Latent Heat Transfer, Oxford Science Publications, Oxford University, Oxford, UK.