Numerical solution of internal convection
From ThermalFluidsPedia
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Since the heat pipe in Fig. 5.20 is closed at both ends, it is required that the vapor which flows out of the evaporator segment enters into the condenser section. The conservation of mass, momentum, and energy equations for the compressible vapor flow region including viscous dissipation are  Since the heat pipe in Fig. 5.20 is closed at both ends, it is required that the vapor which flows out of the evaporator segment enters into the condenser section. The conservation of mass, momentum, and energy equations for the compressible vapor flow region including viscous dissipation are  
  <math>\frac{1}{r}\frac{\partial }{\partial r}({{\rho }_{v}}r{{v}_{v}})+\frac{\partial }{\partial z}({{\rho }_{v}}{{w}_{v}})=0</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%" <center>  
+  <math>\frac{1}{r}\frac{\partial }{\partial r}({{\rho }_{v}}r{{v}_{v}})+\frac{\partial }{\partial z}({{\rho }_{v}}{{w}_{v}})=0</math  
+  </center>  
+  {{EquationRef(1)}}  
+  }  
<math>\begin{align}  <math>\begin{align}  
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where the viscous dissipation is  where the viscous dissipation is  
  <math>\Phi =2\left[ {{\left( \frac{\partial {{v}_{v}}}{\partial r} \right)}^{2}}+{{\left( \frac{{{v}_{v}}}{r} \right)}^{2}}+{{\left( \frac{\partial {{w}_{v}}}{\partial z} \right)}^{2}}+\frac{1}{2}{{\left( \frac{\partial {{v}_{v}}}{\partial z}+\frac{\partial {{w}_{v}}}{\partial r} \right)}^{2}}\frac{1}{3}{{(\nabla \cdot {{\mathbf{V}}_{v}})}^{2}} \right]</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%" <center>  
+  <math>\Phi =2\left[ {{\left( \frac{\partial {{v}_{v}}}{\partial r} \right)}^{2}}+{{\left( \frac{{{v}_{v}}}{r} \right)}^{2}}+{{\left( \frac{\partial {{w}_{v}}}{\partial z} \right)}^{2}}+\frac{1}{2}{{\left( \frac{\partial {{v}_{v}}}{\partial z}+\frac{\partial {{w}_{v}}}{\partial r} \right)}^{2}}\frac{1}{3}{{(\nabla \cdot {{\mathbf{V}}_{v}})}^{2}} \right]</math  
+  </center>  
+  {{EquationRef(1)}}  
+  }  
and <math>\nabla \cdot {{\text{V}}_{v}}</math> is given by  and <math>\nabla \cdot {{\text{V}}_{v}}</math> is given by  
  <math>\nabla \cdot {{\mathbf{V}}_{v}}=\frac{1}{r}\frac{\partial }{\partial r}(r{{v}_{v}})+\frac{\partial {{w}_{v}}}{\partial z}</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%" <center>  
+  <math>\nabla \cdot {{\mathbf{V}}_{v}}=\frac{1}{r}\frac{\partial }{\partial r}(r{{v}_{v}})+\frac{\partial {{w}_{v}}}{\partial z}</math  
+  </center>  
+  {{EquationRef(1)}}  
+  }  
The ideal gas law (<math>p={{\rho }_{v}}{{R}_{g}}{{T}_{v}}</math>) is employed to account for the compressibility of the vapor.  The ideal gas law (<math>p={{\rho }_{v}}{{R}_{g}}{{T}_{v}}</math>) is employed to account for the compressibility of the vapor.  
The use of liquid capillary action is a unique feature of the heat pipe. From a fundamental point of view, the liquid capillary flow in heat pipes with screen wicks should be modeled as a flow through a porous media. It is assumed that the wicking material is isotropic and of constant thickness. In addition, the wick is saturated with liquid, and the vapor condenses and the liquid evaporates at the liquidvapor interface.  The use of liquid capillary action is a unique feature of the heat pipe. From a fundamental point of view, the liquid capillary flow in heat pipes with screen wicks should be modeled as a flow through a porous media. It is assumed that the wicking material is isotropic and of constant thickness. In addition, the wick is saturated with liquid, and the vapor condenses and the liquid evaporates at the liquidvapor interface.  
The averaging technique has been applied by many investigators to obtain the general equation which describes the conservation of momentum in a porous structure. Since the development of Darcy’s semiempirical relation, which characterizes the fluid motion under certain conditions, many researchers have tried to develop and extend Darcy’s law in order to see the effect of the inertia terms. In this respect, those who have tried to model the flow with the NavierStokes equations were the most successful (Chapter 2). The general equations of continuity, momentum and energy for steady state laminar incompressible liquid flow in porous media in terms of the volumeaveraged velocities as presented in Chapter 2 are:  The averaging technique has been applied by many investigators to obtain the general equation which describes the conservation of momentum in a porous structure. Since the development of Darcy’s semiempirical relation, which characterizes the fluid motion under certain conditions, many researchers have tried to develop and extend Darcy’s law in order to see the effect of the inertia terms. In this respect, those who have tried to model the flow with the NavierStokes equations were the most successful (Chapter 2). The general equations of continuity, momentum and energy for steady state laminar incompressible liquid flow in porous media in terms of the volumeaveraged velocities as presented in Chapter 2 are:  
  <math>\frac{\partial }{\partial z}({{\rho }_{\ell }}{{w}_{\ell }})+\frac{1}{r}\frac{\partial }{\partial r}({{\rho }_{\ell }}r{{v}_{\ell }})=0</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%" <center>  
+  <math>\frac{\partial }{\partial z}({{\rho }_{\ell }}{{w}_{\ell }})+\frac{1}{r}\frac{\partial }{\partial r}({{\rho }_{\ell }}r{{v}_{\ell }})=0</math  
+  </center>  
+  {{EquationRef(1)}}  
+  }  
  <math>\frac{1}{{{\varepsilon }^{2}}}\left( {{w}_{\ell }}\frac{\partial {{w}_{\ell }}}{\partial z}+{{v}_{\ell }}\frac{\partial {{w}_{\ell }}}{\partial r} \right)=\frac{1}{{{\rho }_{\ell }}}\frac{\partial {{p}_{\ell }}}{\partial z}\frac{{{\nu }_{\ell }}{{w}_{\ell }}}{K}+\frac{{{\nu }_{\ell }}}{\varepsilon }\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{w}_{\ell }}}{\partial z} \right)+\frac{{{\partial }^{2}}{{w}_{v}}}{\partial {{z}^{2}}} \right]</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%" <center>  
+  <math>\frac{1}{{{\varepsilon }^{2}}}\left( {{w}_{\ell }}\frac{\partial {{w}_{\ell }}}{\partial z}+{{v}_{\ell }}\frac{\partial {{w}_{\ell }}}{\partial r} \right)=\frac{1}{{{\rho }_{\ell }}}\frac{\partial {{p}_{\ell }}}{\partial z}\frac{{{\nu }_{\ell }}{{w}_{\ell }}}{K}+\frac{{{\nu }_{\ell }}}{\varepsilon }\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{w}_{\ell }}}{\partial z} \right)+\frac{{{\partial }^{2}}{{w}_{v}}}{\partial {{z}^{2}}} \right]</math  
+  </center>  
+  {{EquationRef(1)}}  
+  }  
  <math>\frac{1}{{{\varepsilon }^{2}}}\left( {{w}_{\ell }}\frac{\partial {{v}_{\ell }}}{\partial z}+{{v}_{\ell }}\frac{\partial {{v}_{\ell }}}{\partial r} \right)=\frac{1}{{{\rho }_{\ell }}}\frac{\partial {{p}_{\ell }}}{\partial r}\frac{{{\nu }_{\ell }}{{v}_{\ell }}}{K}+\frac{{{\nu }_{\ell }}}{\varepsilon }\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{v}_{\ell }}}{\partial z} \right)\frac{{{v}_{\ell }}}{{{r}^{2}}}+\frac{{{\partial }^{2}}{{v}_{v}}}{\partial {{z}^{2}}} \right]</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%" <center>  
+  <math>\frac{1}{{{\varepsilon }^{2}}}\left( {{w}_{\ell }}\frac{\partial {{v}_{\ell }}}{\partial z}+{{v}_{\ell }}\frac{\partial {{v}_{\ell }}}{\partial r} \right)=\frac{1}{{{\rho }_{\ell }}}\frac{\partial {{p}_{\ell }}}{\partial r}\frac{{{\nu }_{\ell }}{{v}_{\ell }}}{K}+\frac{{{\nu }_{\ell }}}{\varepsilon }\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{v}_{\ell }}}{\partial z} \right)\frac{{{v}_{\ell }}}{{{r}^{2}}}+\frac{{{\partial }^{2}}{{v}_{v}}}{\partial {{z}^{2}}} \right]</math  
+  </center>  
+  {{EquationRef(1)}}  
+  }  
  <math>{{\rho }_{\ell }}{{c}_{p\ell }}\left( {{w}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial z}+{{v}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial r} \right)=\frac{1}{r}\frac{\partial }{\partial r}\left( r{{k}_{eff}}\frac{\partial {{T}_{\ell }}}{\partial r} \right)+\frac{\partial }{\partial z}\left( {{k}_{eff}}\frac{\partial {{T}_{\ell }}}{\partial z} \right)</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%" <center>  
+  <math>{{\rho }_{\ell }}{{c}_{p\ell }}\left( {{w}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial z}+{{v}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial r} \right)=\frac{1}{r}\frac{\partial }{\partial r}\left( r{{k}_{eff}}\frac{\partial {{T}_{\ell }}}{\partial r} \right)+\frac{\partial }{\partial z}\left( {{k}_{eff}}\frac{\partial {{T}_{\ell }}}{\partial z} \right)</math  
+  </center>  
+  {{EquationRef(1)}}  
+  }  
where ε is the volume fraction or porosity of the wick, and K is the permeability of the wick structure.  where ε is the volume fraction or porosity of the wick, and K is the permeability of the wick structure.  
The effective thermal conductivity of the wick, keff, is related to the thermal conductivity of the solid and liquid phases  The effective thermal conductivity of the wick, keff, is related to the thermal conductivity of the solid and liquid phases  
  <math>{{k}_{eff}}=\frac{{{k}_{\ell }}[({{k}_{\ell }}+{{k}_{s}})(1\varepsilon )({{k}_{\ell }}{{k}_{s}})]}{[({{k}_{\ell }}+{{k}_{s}})+(1\varepsilon )({{k}_{\ell }}{{k}_{s}})]}</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%" <center>  
+  <math>{{k}_{eff}}=\frac{{{k}_{\ell }}[({{k}_{\ell }}+{{k}_{s}})(1\varepsilon )({{k}_{\ell }}{{k}_{s}})]}{[({{k}_{\ell }}+{{k}_{s}})+(1\varepsilon )({{k}_{\ell }}{{k}_{s}})]}</math  
+  </center>  
+  {{EquationRef(1)}}  
+  }  
The steady state energy equation that describes the temperature in the heat pipe wall is  The steady state energy equation that describes the temperature in the heat pipe wall is  
  <math>\frac{\partial }{\partial z}\left( {{k}_{w}}\frac{\partial {{T}_{w}}}{\partial z} \right)+\frac{1}{r}\frac{\partial }{\partial r}\left( {{k}_{w}}r\frac{\partial {{T}_{w}}}{\partial r} \right)=0</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%" <center>  
+  <math>\frac{\partial }{\partial z}\left( {{k}_{w}}\frac{\partial {{T}_{w}}}{\partial z} \right)+\frac{1}{r}\frac{\partial }{\partial r}\left( {{k}_{w}}r\frac{\partial {{T}_{w}}}{\partial r} \right)=0</math  
+  </center>  
+  {{EquationRef(1)}}  
+  }  
where kw is the local thermal conductivity of the heat pipe wall. The boundary conditions are  where kw is the local thermal conductivity of the heat pipe wall. The boundary conditions are  
  <math>w(0,r)=v(0,r)=0</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%" <center>  
+  <math>w(0,r)=v(0,r)=0</math  
+  </center>  
+  {{EquationRef(1)}}  
+  }  
  <math>w({{L}_{t}},r)=v({{L}_{t}},r)=0</math>  +  { class="wikitable" border="0" 
  +    
+   width="100%" <center>  
+  <math>w({{L}_{t}},r)=v({{L}_{t}},r)=0</math  
+  </center>  
+  {{EquationRef(1)}}  
+  }  
<math>w(z,{{R}_{v}})=0</math>  <math>w(z,{{R}_{v}})=0</math> 
Revision as of 03:32, 27 June 2010
The algorithms to solve the convectiondiffusion equation and flow field that were addressed in Section 4.8 are still applicable to the internal convection problems. However, special attention must be paid to proper treatment of the boundary conditions. For an internal flow and heat transfer problem, there are several different types of boundary conditions (see Fig. 5.18): (1) inflow condition, (2) axisymmetric condition, (3) impermeable solid surface, and (4) outflow condition. The inflow conditions are normally specified by a given distribution of the inlet velocity and the general variable, φ, at the inlet (x = 0). The axisymmetric condition can be implemented by setting the gradient of the velocity component in the ydirection, , and the gradient of the general variable, , equal to zero along the axisymmetric line (y = 0). In addition, the velocity component in the ydirection, v, at the axisymmetric boundary should also be zero. For the impermeable solid surface, all three kinds of boundary conditions for the general variables are possible: specified φ (the first kind), specified gradient of φ in the direction perpendicular to the impermeable surface, , (the second kind), or specified relation between φ and (the third kind). The velocity components in both the tangential and normal direction of the impermeable surface are zero, i.e., u = v = 0, due to noslip and impermeable conditions. No special treatment for the boundary conditions at inflow, axisymmetric, and impermeable surfaces is required. On the contrary, the outflow boundary condition requires special treatment as outlined below. Since the momentum equation in the internal flow direction (the xdirection in Fig. 5.18) and the conservation equation for the general variable (see eq. (4.200)) are elliptic, the second order derivatives with respect to x appeared in the partial difference equation. Mathematically, boundary conditions at both the inflow and outflow boundary need to be specified. While the inflow boundary conditions are always known, the outflow boundary conditions (at y = L in Fig. 5.16) are usually unknown. Unless the experimentally measured distribution at the outflow boundary is available, we cannot directly use the algorithms introduced in the previous chapter to solve the internal convection problem. A coordinate (such as x in Fig. 5.18, or time for a transient problem) can be either twoway or oneway depending on the nature of the problem (Patankar, 1980, 1991). If the condition at a given location, x, is influenced by changes of conditions at either side of the given location, the coordinate is said to be twoway. On the other hand, if the conditions at the given location, x, are influenced by changes of conditions from only one side, the coordinate is said to be oneway. Mathematically, if the second order derivation with respect to x appeared in the partial differential equation, the coordinate in the xdirection is twoway. For example, the spatial coordinate in the onedimensional steadystate heat conduction in a fin (see Section 3.2.1) is twoway because the temperature at any point is influenced by the temperatures at either side, and the second order derivative appears in the energy equation that describes heat conduction in the fin. On the other hand, time is a one way coordinate because the conditions at a given time are only influenced by what happened before that time, and what happen afterward will not affect the conditions at the current time. While the space coordinate in heat and mass transfer is normally always twoway, it can become oneway for some special cases. For internal flow, the change of condition at a given point will have a more profound effect on the conditions of the points downstream, while its influence on the conditions of the points upstream will be relatively weak. For the case that convection overpowers diffusion, one can assume that the condition changes at one point can only propagate downstream and the coordinate becomes oneway. The most commonly used approach to handle the outflow boundary condition is to assume the coordinate at the outflow boundary is locally oneway. Thus, the value of the general variable, φ, at the inner point, P, shown in Fig. 5.19 is not affected by the value of φ at the outflow boundary grid point, E, which is located downstream. Computationally, one can set a_{E} = 0 in the discretized equation for the point P. This simple treatment effectively avoids the problem associated with an unknown value ofφat the outflow boundary. It follows from Section 4.8 that this treatment will be more accurate for cases with a high Peclet number. Another situation that can be handled by a similar approach is the case that the unknown variable is fully developed at the outflow boundary, in which case the outflow boundary condition becomes . The implication of the fully developed condition at the outflow is exactly the same as the local oneway behavior discussed above, and can be treated using the same approach. However, it should be pointed out that for the case of fully developed heat transfer, one must not use , and the correct condition for fully developed heat transfer should be
(see Section 5.3).
Many practical problems are not similar to the cases corresponding to simple internal channel flow presented in Sections 5.15.6. In many practical applications for internal flow, one needs to deal with conjugate effects, compressibility, multidomain, and porous media, as well as nonconventional boundary conditions. To show the power of numerical simulation, the modeling of a conventional wicked heat pipe with variable heat flux (see Fig. 5.20) will be presented here, compared to the nonconventional internal flow case presented in previous sections. The problem is modeled as a twodimensional conjugate, with compressible flow including the effect of porous media and coupling between various regions.
Since the heat pipe in Fig. 5.20 is closed at both ends, it is required that the vapor which flows out of the evaporator segment enters into the condenser section. The conservation of mass, momentum, and energy equations for the compressible vapor flow region including viscous dissipation are
Failed to parse (syntax error): \frac{1}{r}\frac{\partial }{\partial r}({{\rho }_{v}}r{{v}_{v}})+\frac{\partial }{\partial z}({{\rho }_{v}}{{w}_{v}})=0</math </center> {{EquationRef(1)}} } <math>\begin{align} & {{\rho }_{v}}\left( {{w}_{v}}\frac{\partial {{w}_{v}}}{\partial z}+{{v}_{v}}\frac{\partial {{w}_{v}}}{\partial r} \right)=\frac{\partial {{p}_{v}}}{\partial z} \\ & +{{\mu }_{v}}\left[ \frac{4}{3}\frac{{{\partial }^{2}}{{w}_{v}}}{\partial {{z}^{2}}}+\frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{w}_{v}}}{\partial r} \right)+\frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{v}_{v}}}{\partial z} \right)\frac{2}{3}\frac{\partial }{\partial z}\left( \frac{1}{r}\frac{\partial }{\partial r}(r{{v}_{v}}) \right) \right] \\ \end{align} (5.167) (5.168) (5.169) where the viscous dissipation is
