Evaporation of a Liquid Jet in a Pure Vapor
From ThermalFluidsPedia
(2 intermediate revisions not shown)  
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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  +  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  
  <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>  +  { class="wikitable" border="0" 
  +    
+   width="100%"   
+  <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)}}  
+  }  
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  
  +  
+  { class="wikitable" border="0"  
+    
+   width="100%"   
<center><math>{{T}_{\ell }}(r,x)={{T}_{0}}\begin{matrix}  <center><math>{{T}_{\ell }}(r,x)={{T}_{0}}\begin{matrix}  
, & t=0 \\  , & t=0 \\  
  \end{matrix}</math>  +  \end{matrix}</math></center> 
  +  {{EquationRef(2)}}  
+  }  
  [[Image:NewChapter9 (  +  [[Image:NewChapter9 (12).jpgthumb400 pxalt= 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  
  +  
+  { class="wikitable" border="0"  
+    
+   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 \\  
  \end{matrix}</math>  +  \end{matrix}</math></center> 
  +  {{EquationRef(3)}}  
  +  }  
+  
+  { class="wikitable" border="0"  
+    
+   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}} \\  
  \end{matrix}(x)</math>  +  \end{matrix}(x)</math></center> 
  +  {{EquationRef(4)}}  
+  }  
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  
  +  
+  { class="wikitable" border="0"  
+    
+   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}} \\  
  \end{matrix}(x)</math>  +  \end{matrix}(x)</math></center> 
  +  {{EquationRef(5)}}  
+  }  
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 semiinfinite 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 semiinfinite solid, i.e.,  
  
  
  
  +  { class="wikitable" border="0"  
  +    
  <center><math>  +   width="100%"  
  +  <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(6)}}  
+  }  
  The variation in the radius of the liquid jet during the first stage can be obtained by integrating eq. (  +  where the ycoordinate 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. (6) into the energy balance equation at the interface, eq. (5), one obtains 
  +  
  <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" 
  +    
+   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>  
+  {{EquationRef(7)}}  
+  }  
+  
+  The variation in the radius of the liquid jet during the first stage can be obtained by integrating eq. (7), i.e.,  
+  
+  { class="wikitable" border="0"  
+    
+   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(8)}}  
+  }  
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:  
  
  
  
  Rearranging eq. (9  +  { class="wikitable" border="0" 
  +    
  <center><math>{{r}_{I,f}}={{R}_{i}}\sqrt{1\text{Ja}}</math>  +   width="100%"  
  +  <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(9)}}  
+  }  
+  
+  Rearranging eq. (9), 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(10)}}  
+  }  
where  where  
  +  
  <center><math>\text{Ja}=\frac{{{c}_{{{p}_{\ell }}}}({{T}_{0}}{{T}_{sat}})}{{{h}_{\ell v}}}</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%"   
+  <center><math>\text{Ja}=\frac{{{c}_{{{p}_{\ell }}}}({{T}_{0}}{{T}_{sat}})}{{{h}_{\ell v}}}</math></center>  
+  {{EquationRef(11)}}  
+  }  
is the Jakob number.  is the Jakob number. 
Current revision as of 19:42, 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 r_{0} and is surrounded by pure vapor at saturation temperature. At the exit of the nozzle, the velocity and temperature are uniformly T_{0} and u_{0}, 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

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

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


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

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, r_{0} − δ_{t} < r < r_{0}, develops in the liquid jet. The evaporation of the jet can be divided into two stages: the early stage when and a later stage when δ_{t} = r_{0}. These two stages are similar to the thermal entrance problem and fullydeveloped 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 semiinfinite solid, i.e.,

where the ycoordinate originates at the jet’s surface and points toward its center. It is related to r by y = r_{I} − r. Substituting eq. (6) into the energy balance equation at the interface, eq. (5), one obtains

The variation in the radius of the liquid jet during the first stage can be obtained by integrating eq. (7), i.e.,

Evaporation during the second stage (δ_{t} = r_{I}), 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, r_{I,f}, can be obtained by a simple energy balance:

Rearranging eq. (9), one obtains the final radius of the liquid jet as

where

is the Jakob number.
References
Lock, G.S.H., 1994, Latent Heat Transfer, Oxford Science Publications, Oxford University, Oxford, UK.