Developing flow
From ThermalFluidsPedia
Yuwen Zhang (Talk  contribs) 

(23 intermediate revisions not shown)  
Line 1:  Line 1:  
  +  For forced convective heat and mass transfer with constant properties, the hydrodynamic entrance length is independent of Pr or Sc. When assuming fully developed flow, the point at which the temperature profile becomes fully developed for forced convection in tubes is linearly proportional to RePr.<ref>Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.</ref>  
+  
Analysis of these criteria for a fully developed flow and temperature profile shows that when Pr <math>\gg </math> 1, as is the case with fluids with high viscosities such as oils, the temperature profile takes a longer distance to completely develop. In these circumstances (Pr <math>\gg </math> 1), it makes sense to assume fully developed velocity since the thermal entrance is much longer than the hydrodynamic entrance.  Analysis of these criteria for a fully developed flow and temperature profile shows that when Pr <math>\gg </math> 1, as is the case with fluids with high viscosities such as oils, the temperature profile takes a longer distance to completely develop. In these circumstances (Pr <math>\gg </math> 1), it makes sense to assume fully developed velocity since the thermal entrance is much longer than the hydrodynamic entrance.  
+  
Obviously, from the definition of Prandtl number and the above criteria, one expects that when Pr ≈ 1 for fluids such as gases, the temperature and velocity develop at the same rate. When Pr <math>\ll </math> 1, as in the case of liquid metals, the temperature profile will develop much faster than the velocity profile, and therefore a uniform velocity assumption (slug flow) is appropriate.  Obviously, from the definition of Prandtl number and the above criteria, one expects that when Pr ≈ 1 for fluids such as gases, the temperature and velocity develop at the same rate. When Pr <math>\ll </math> 1, as in the case of liquid metals, the temperature profile will develop much faster than the velocity profile, and therefore a uniform velocity assumption (slug flow) is appropriate.  
+  
Similar analysis and conclusions can be made with the Schmidt number, Sc, relative to mass transfer problems concerning the entrance effects due to mass diffusion. If one needs to get detailed information concerning the hydrodynamic, thermal or concentration entrance effects, the conservation equations should be solved without a fully developed velocity, concentration, or temperature profile.  Similar analysis and conclusions can be made with the Schmidt number, Sc, relative to mass transfer problems concerning the entrance effects due to mass diffusion. If one needs to get detailed information concerning the hydrodynamic, thermal or concentration entrance effects, the conservation equations should be solved without a fully developed velocity, concentration, or temperature profile.  
  
+  [[Image:Fig5.15.pngthumb400 pxalt=Geometry and coordinate system flow for forced convective heat and mass transfer in a circular tube. Geometry and coordinate system flow for forced convective heat and mass transfer in a circular tube.]]  
+  
+  Consider laminar forced convective heat and mass transfer in a circular tube for the case of steady twodimensional constant properties. The inlet velocity, temperature and concentration are uniform at the entrance with the possibility of mass transfer between the wall and fluid, as shown in figure to the right.  
The conservation equations with the above assumptions, as well as neglecting the viscous dissipation and assuming an incompressible Newtonian fluid, are  The conservation equations with the above assumptions, as well as neglecting the viscous dissipation and assuming an incompressible Newtonian fluid, are  
Line 21:  Line 26:  
<math>x\text{momentum}\quad u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}=\frac{1}{\rho }\frac{\partial p}{\partial x}+\nu \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial u}{\partial r} \right)+\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}} \right]</math>  <math>x\text{momentum}\quad u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}=\frac{1}{\rho }\frac{\partial p}{\partial x}+\nu \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial u}{\partial r} \right)+\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}} \right]</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(2)}} 
}  }  
Line 29:  Line 34:  
<math>r\text{momentum}\quad u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial r}=\frac{1}{\rho }\frac{\partial p}{\partial r}+\nu \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial v}{\partial r} \right)+\frac{{{\partial }^{2}}v}{\partial {{x}^{2}}} \right]</math>  <math>r\text{momentum}\quad u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial r}=\frac{1}{\rho }\frac{\partial p}{\partial r}+\nu \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial v}{\partial r} \right)+\frac{{{\partial }^{2}}v}{\partial {{x}^{2}}} \right]</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(3)}} 
}  }  
Line 37:  Line 42:  
<math>\text{energy}\quad u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\alpha \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right)+\frac{{{\partial }^{2}}T}{\partial {{x}^{2}}} \right]</math>  <math>\text{energy}\quad u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\alpha \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right)+\frac{{{\partial }^{2}}T}{\partial {{x}^{2}}} \right]</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(4)}} 
}  }  
Line 45:  Line 50:  
<math>\text{species}\quad u\frac{\partial {{\omega }_{1}}}{\partial x}+v\frac{\partial {{\omega }_{1}}}{\partial r}=D\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{\omega }_{1}}}{\partial r} \right)+\frac{{{\partial }^{2}}{{\omega }_{1}}}{\partial {{x}^{2}}} \right]</math>  <math>\text{species}\quad u\frac{\partial {{\omega }_{1}}}{\partial x}+v\frac{\partial {{\omega }_{1}}}{\partial r}=D\left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial {{\omega }_{1}}}{\partial r} \right)+\frac{{{\partial }^{2}}{{\omega }_{1}}}{\partial {{x}^{2}}} \right]</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(5)}} 
}  }  
  Typical boundary conditions are:  +  Typical boundary conditions are: 
  [  +  
  +  <center><math>\text{Axial velocity at wall} u\left( x,{{r}_{o}} \right)=0 \text{no slip boundary condition}</math></center>  
+  <center><math>\text{Radial velocity at wall} v\left( x,{{r}_{o}} \right)= \left\{ \begin{align}  
+  & {{v}_{w}}=0 \text{impermeable wall} \\  
+  & {{v}_{w}}>0 \text{injection and} {{v}_{w}}<0 \text{suction} \\  
+  & {{{\dot{m}}}_{w}}^{\prime \prime }= \text{mass flux due to diffusion} \\  
+  & \quad \ \ \ =\rho \left[ {{\omega }_{1,w}}{{v}_{w}}{{D}_{12}}{{\left. \frac{\partial {{\omega }_{1}}}{\partial r} \right}_{r={{r}_{o}}}} \right] \\  
+  \end{align} \right.</math></center>  
+  <center><math>\text{Thermal condition on wall at} \left( r={{r}_{o}} \right)\quad \quad \left\{ \begin{align}  
+  & {{T}_{w}}= \text{const}\text{.} \\  
+  & {{q}_{w}}^{\prime \prime }=k{{\left. \frac{\partial T}{\partial r} \right}_{r={{r}_{o}}}}= \text{const}\text{. or} \\  
+  & {{T}_{w}}=f\left( x \right) \text{or} \\  
+  & {{q}_{w}}^{\prime \prime }=g\left( x \right) \\  
+  \end{align} \right.</math></center>  
+  <center><math>\begin{matrix}  
+  {} & {} \\  
+  \end{matrix}\text{Inlet condition at} x=0</math></center>  
+  <center><math>\left\{ \begin{align}  
+  & T={{T}_{in}} \\  
+  & {{\omega }_{1}}={{\omega }_{1,}}_{in} \\  
+  & u={{u}_{in}} \\  
+  \end{align} \right.</math></center>  
+  <center><math>\text{Outlet condition at} x=L\begin{matrix}  
+  {} & {} & {} & {} & {} \\  
+  \end{matrix}\left\{ \begin{align}  
+  & T=? \\  
+  & {{\omega }_{1}}=? \\  
+  & u=? \\  
+  & P=? \\  
+  \end{align} \right.</math></center>  
  Clearly there are five partial differential equations and five unknowns (u, v, P, T,  +  Clearly there are five partial differential equations and five unknowns (''u'', ''v'', ''P'', ''T'', ''ω<sub>1</sub>''). All equations are of elliptic nature and one can neglect the axial diffusion terms, <math>\left( \frac{{{\partial }^{2}}u}{\partial {{x}^{2}}},\ \frac{{{\partial }^{2}}v}{\partial {{x}^{2}}},\ \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}},\ \frac{{{\partial }^{2}}{{\omega }_{1}}}{\partial {{x}^{2}}} \right)</math>, under some circumstances in order to make the conservation equations of parabolic nature. These axial diffusion terms can also be neglected under boundary layer assumptions. 
  <math>\left( \frac{{{\partial }^{2}}u}{\partial {{x}^{2}}},\ \frac{{{\partial }^{2}}v}{\partial {{x}^{2}}},\ \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}},\ \frac{{{\partial }^{2}}{{\omega }_{1}}}{\partial {{x}^{2}}} \right)</math>  +  
  , under some circumstances in order to make the conservation equations of parabolic nature. These axial diffusion terms can also be neglected under boundary layer assumptions.  +  Making boundary layer assumptions makes the result invalid very close to the tube entrance where the Reynolds number is very small. Shah and London <ref name="SL1978">Shah, R. K.; London, A. L., 1978, “Laminar Flow Convection in Ducts,” Advances in Heat Transfer, Supplement 1, Irvine, T. F. and Harnett, J.P., Eds., Academic Press, San Diego, CA.</ref> showed that the momentum boundary layer assumption will lead to error if Re < 400 and ''L<sub>H</sub>'' / ''D'' < 0.005Re. In these circumstances, the full Navier Stokes equation should be solved. 
  Making boundary layer assumptions makes the result invalid very close to the tube entrance where the Reynolds number is very small. Shah and London  +  
  It was also shown in  +  It was also shown in [[Basics of Internal Forced Convection]] that there are circumstances other than boundary layer assumptions where axial diffusion terms, such as the axial conduction term, can be neglected. However, as we showed in the case of the energy equation, one cannot neglect axial conduction for a very low Prandtl number despite the thermal boundary layer assumption. 
+  
In general, elliptic equations are more complex to solve analytically or numerically than parabolic equations. Furthermore, to solve the equations as elliptic you need pertinent information at the outlet as well, which in some cases is unknown. The momentum equation is nonlinear while the energy equation is linear under the constant property assumption.  In general, elliptic equations are more complex to solve analytically or numerically than parabolic equations. Furthermore, to solve the equations as elliptic you need pertinent information at the outlet as well, which in some cases is unknown. The momentum equation is nonlinear while the energy equation is linear under the constant property assumption.  
  In most cases, the momentum, energy, and species equations are uncoupled, except under the following circumstances which make the equations coupled.  +  
  1. Variable properties, such as density variation as a function of temperature in natural convection problems.  +  In most cases, the momentum, energy, and species equations are uncoupled, except under the following circumstances which make the equations coupled.<br> 
  2. Coupled governing equations and/or boundary conditions in phase change problems, such as absorption or dissolution problems.  +  1. Variable properties, such as density variation as a function of temperature in natural convection problems.<br> 
  3. Existence of a source term in one conservation equation that is a function of the dependent variable in another conservation equation.  +  2. Coupled governing equations and/or boundary conditions in phase change problems, such as absorption or dissolution problems.<br> 
  Langhaar  +  3. Existence of a source term in one conservation equation that is a function of the dependent variable in another conservation equation.<br> 
  [[Image:Fig5.16.jpgthumb400 pxalt=Local and average Nusselt numbers for the entrance region of a circular tube with constant wall temperature   +  
  [[Image:Fig5.17.jpgthumb400 pxalt=Local and average Nusselt numbers for the entrance region of a circular tube with constant heat flux   +  Langhaar <ref name="L1942">Langhaar, H.L., 1942, Steady Flow in the Transition Length of a Straight Tube,” J. Appl. Mech., Vol. 9, pp. A55A58.</ref> and Hornbeck <ref name="H1965">Hornbeck, R.W., 1965, “An Allnumerical Method for Heat Transfer in the Inlet of a Tube,” ASME Paper No. 65WA HT36.</ref> obtained approximate solutions for the momentum equation for circular tubes by solving the linearized momentum equation. Hornbeck <ref name="H1965"/> solved the momentum equation numerically by making boundary layer assumptions (parabolic form). Several investigators solved the energy equation either using Langhaar’s approximate velocity profile, or solving the momentum and energy equations numerically for both constant wall temperature and constant wall heat flux in circular tubes. Heat transfer in hydrodynamic and thermal entrance region has been solved numerically based on full elliptic governing equations <ref name="B2009">Bahrami, H., 2009, Personal Communication, Storrs, CT.</ref>. 
  Variations of local and average Nusselt numbers for different Prandtl numbers under constant wall temperature and constant heat flux using full elliptic governing equations are shown in  +  
  Heaton et al.  +  [[Image:Fig5.16.jpgthumb400 pxalt=Local and average Nusselt numbers for the entrance region of a circular tube with constant wall temperature  Local and average Nusselt numbers for the entrance region of a circular tube with constant wall temperature.]] 
+  [[Image:Fig5.17.jpgthumb400 pxalt=Local and average Nusselt numbers for the entrance region of a circular tube with constant heat flux  Local and average Nusselt numbers for the entrance region of a circular tube with constant heat flux.]]  
+  
+  Variations of local and average Nusselt numbers for different Prandtl numbers under constant wall temperature and constant heat flux using full elliptic governing equations are shown in the two figures to the right, respectively. The local and average Nusselt numbers for different Prandtl numbers and boundary conditions are also presented in the following two tables.  
+  
+  <center>  
+  <div style="display:inline;">  
+  '''Table 1''' Local and average Nusselt number for the entrance region of a circular tube with constant wall temperature  
+  { class="wikitable" border="1"  
+   align="center" style="background:#f0f0f0;" width="5%"  n  
+   colspan="3" align="center" style="background:#f0f0f0;" width="10%"  ''Nu<sub>x</sub>''  
+   colspan="3" align="center" style="background:#f0f0f0;" width="10%"  ''Nu<sub>m</sub>''''  
+    
+  ''x<sup>+</sup>''  
+  Pr = 0.7  
+  Pr = 2  
+  Pr = 5  
+  Pr = 0.7  
+  Pr = 2  
+  Pr = 5  
+    
+  0.001  
+  17.0  
+  12.5  
+  10.6  
+  60.7  
+  34.9  
+  24.9  
+    
+  0.002  
+  11.6  
+  8.86  
+  8.04  
+  37.3  
+  22.6  
+  17.0  
+    
+  0.004  
+  8.29  
+  6.64  
+  6.30  
+  23.3  
+  15.1  
+  12.0  
+    
+  0.008  
+  6.29  
+  5.23  
+  5.09  
+  14.9  
+  10.4  
+  8.80  
+    
+  0.01  
+  5.81  
+  4.89  
+  4.80  
+  13.1  
+  9.36  
+  8.03  
+    
+  0.02  
+  4.69  
+  4.12  
+  4.12  
+  8.9  
+  6.89  
+  6.21  
+    
+  0.04  
+  4.01  
+  3.75  
+  3.76  
+  6.46  
+  5.39  
+  5.06  
+    
+  0.06  
+  3.79  
+  3.67  
+  3.68  
+  5.56  
+  4.83  
+  4.61  
+    
+  0.08  
+  3.71  
+  3.66  
+  3.66  
+  5.09  
+  4.54  
+  4.37  
+    
+  0.1  
+  3.68  
+  3.66  
+  3.66  
+  4.81  
+  4.36  
+  4.23  
+    
+  0.12  
+  3.66  
+  3.66  
+  3.66  
+  4.62  
+  4.24  
+  4.14  
+    
+  <math>\infty </math>  
+  3.66  
+  3.66  
+  3.66  
+  3.66  
+  3.66  
+  3.66  
+  }  
+  </div></center>  
+  
+  <center><div style="display:inline;">  
+  '''Table 2''' Local and mean Nusselt number for the entrance region of a circular tube with constant wall heat flux  
+  { class="wikitable" border="1"  
+   align="center" style="background:#f0f0f0;" width="5%"  n  
+   colspan="3" align="center" style="background:#f0f0f0;" width="10%"  ''Nu<sub>x</sub>''  
+   colspan="3" align="center" style="background:#f0f0f0;" width="10%"  ''Nu<sub>m</sub>''''  
+    
+  ''x<sup>+</sup>''  
+  Pr = 0.7  
+  Pr = 2  
+  Pr = 5  
+  Pr = 0.7  
+  Pr = 2  
+  Pr = 5  
+    
+  0.001  
+  23.0  
+  17.8  
+  15.0  
+  61.0  
+  43.4  
+  34.2  
+    
+  0.002  
+  15.6  
+  12.5  
+  11.1  
+  39.8  
+  29.0  
+  23.5  
+    
+  0.004  
+  10.9  
+  9.16  
+  8.44  
+  26.3  
+  19.8  
+  16.5  
+    
+  0.008  
+  7.92  
+  7.01  
+  6.65  
+  17.7  
+  13.8  
+  11.9  
+    
+  0.01  
+  7.24  
+  6.49  
+  6.22  
+  15.7  
+  12.4  
+  10.8  
+    
+  0.02  
+  5.69  
+  5.30  
+  5.21  
+  11.0  
+  9.11  
+  8.23  
+    
+  0.04  
+  4.81  
+  4.64  
+  4.62  
+  8.09  
+  7.00  
+  6.54  
+    
+  0.06  
+  4.53  
+  4.46  
+  4.45  
+  6.94  
+  6.18  
+  5.87  
+    
+  0.08  
+  4.43  
+  4.39  
+  4.39  
+  6.33  
+  5.74  
+  5.51  
+    
+  0.1  
+  4.39  
+  4.37  
+  4.37  
+  5.95  
+  5.47  
+  5.29  
+    
+  0.12  
+  4.37  
+  4.36  
+  4.36  
+  5.69  
+  5.29  
+  5.13  
+    
+  0.16  
+  4.36  
+  4.36  
+  4.36  
+  5.36  
+  5.23  
+  4.94  
+    
+  <math>\infty </math>  
+  4.36  
+  4.36  
+  4.36  
+  4.36  
+  4.36  
+  4.36  
+  }  
+  </div></center>  
+  
+  Heaton et al. <ref name="H1964">Heaton, H. S., Reynolds, W. C. and Kays, W. M., 1964, “Heat Transfer in Annular Passages: Simultaneous Development of Velocity and Temperature Fields in Laminar Flow”, Int. J. Heat Mass Transfer, Vol. 7, pp. 763781.</ref> approximated the result for linearized momentum and energy equations using the energy equation for constant wall heat flux for a group of circular tube annulus for several Prandtl numbers. The following table summarizes the results for parallel plates and circular annulus.  
+  
+  <center>  
+  <div style="display:inline;">  
+  '''Table 3''' Local Nusselt number for the entrance region of a group of circular Tube Annulus with Constant Wall Heat Flux  
+  { class="wikitable" border="1"  
+   align="center" style="background:#f0f0f0;" width="5%"   
+   colspan="3" align="center" style="background:#f0f0f0;" width="10%"  Parallel planes  
+   colspan="3" align="center" style="background:#f0f0f0;" width="10%"  Circulartube annulus K= 0.50  
+    
+  Pr  
+  ''Nu<sub>11</sub>''  
+  ''θ<sub>1</sub>''<small>*</small>  
+  ''Nu<sub>ii</sub>''  
+  ''Nu<sub>oo</sub>''  
+  ''θ<sub>i</sub>''<small>*</small>  
+  ''θ<sub>i</sub>''<small>*</small>  
+    
+  rowspan="6"0.01  
+  24.2  
+  0.048  
+     
+  24.2  
+     
+  0.0322  
+    
+  11.7  
+  0.117  
+     
+  11.8  
+     
+  0.0786  
+    
+  8.8  
+  0.176  
+  9.43  
+  8.9  
+  0.252  
+  0.118  
+    
+  5.77  
+  0.378  
+  6.4  
+  5.88  
+  0.525  
+  0.231  
+    
+  5.53  
+  0.376  
+  6.22  
+  5.6  
+  0.532  
+  0.238  
+    
+  5.39  
+  0.346  
+  6.18  
+  5.04  
+  0.528  
+  0.216  
+    
+  rowspan="6"0.7  
+  18.5  
+  0.037  
+  19.22  
+  18.3  
+  0.0513  
+  0.0243  
+    
+  9.62  
+  0.096  
+  10.47  
+  9.45  
+  0.139  
+  0.063  
+    
+  7.68  
+  0.154  
+  8.52  
+  7.5  
+  0.228  
+  0.0998  
+    
+  5.55  
+  0.327  
+  6.35  
+  5.27  
+  0.498  
+  0.207  
+    
+  5.4  
+  0.345  
+  6.19  
+  5.06  
+  0.527  
+  0.215  
+    
+  5.39  
+  0.346  
+  6.18  
+  5.04  
+  0.528  
+  0.216  
+    
+  rowspan="6"10  
+  15.6  
+  0.0311  
+  16.86  
+  15.14  
+  0.045  
+  0.0201  
+    
+  9.2  
+  0.092  
+  10.2  
+  8.75  
+  0.136  
+  0.0583  
+    
+  7.49  
+  0.149  
+  8.43  
+  7.09  
+  0.224  
+  0.0943  
+    
+  5.55  
+  0.327  
+  6.35  
+  5.2  
+  0.498  
+  0.204  
+    
+  5.4  
+  0.345  
+  6.19  
+  5.05  
+  0.527  
+  0.215  
+    
+  5.39  
+  0.346  
+  6.18  
+  5.04  
+  0.528  
+  0.216  
+  }  
+  </div></center>  
+  
+  ==References==  
+  {{Reflist}} 
Current revision as of 08:07, 27 July 2010
For forced convective heat and mass transfer with constant properties, the hydrodynamic entrance length is independent of Pr or Sc. When assuming fully developed flow, the point at which the temperature profile becomes fully developed for forced convection in tubes is linearly proportional to RePr.^{[1]}
Analysis of these criteria for a fully developed flow and temperature profile shows that when Pr 1, as is the case with fluids with high viscosities such as oils, the temperature profile takes a longer distance to completely develop. In these circumstances (Pr 1), it makes sense to assume fully developed velocity since the thermal entrance is much longer than the hydrodynamic entrance.
Obviously, from the definition of Prandtl number and the above criteria, one expects that when Pr ≈ 1 for fluids such as gases, the temperature and velocity develop at the same rate. When Pr 1, as in the case of liquid metals, the temperature profile will develop much faster than the velocity profile, and therefore a uniform velocity assumption (slug flow) is appropriate.
Similar analysis and conclusions can be made with the Schmidt number, Sc, relative to mass transfer problems concerning the entrance effects due to mass diffusion. If one needs to get detailed information concerning the hydrodynamic, thermal or concentration entrance effects, the conservation equations should be solved without a fully developed velocity, concentration, or temperature profile.
Consider laminar forced convective heat and mass transfer in a circular tube for the case of steady twodimensional constant properties. The inlet velocity, temperature and concentration are uniform at the entrance with the possibility of mass transfer between the wall and fluid, as shown in figure to the right.
The conservation equations with the above assumptions, as well as neglecting the viscous dissipation and assuming an incompressible Newtonian fluid, are





Typical boundary conditions are:
Clearly there are five partial differential equations and five unknowns (u, v, P, T, ω_{1}). All equations are of elliptic nature and one can neglect the axial diffusion terms, , under some circumstances in order to make the conservation equations of parabolic nature. These axial diffusion terms can also be neglected under boundary layer assumptions.
Making boundary layer assumptions makes the result invalid very close to the tube entrance where the Reynolds number is very small. Shah and London ^{[2]} showed that the momentum boundary layer assumption will lead to error if Re < 400 and L_{H} / D < 0.005Re. In these circumstances, the full Navier Stokes equation should be solved.
It was also shown in Basics of Internal Forced Convection that there are circumstances other than boundary layer assumptions where axial diffusion terms, such as the axial conduction term, can be neglected. However, as we showed in the case of the energy equation, one cannot neglect axial conduction for a very low Prandtl number despite the thermal boundary layer assumption.
In general, elliptic equations are more complex to solve analytically or numerically than parabolic equations. Furthermore, to solve the equations as elliptic you need pertinent information at the outlet as well, which in some cases is unknown. The momentum equation is nonlinear while the energy equation is linear under the constant property assumption.
In most cases, the momentum, energy, and species equations are uncoupled, except under the following circumstances which make the equations coupled.
1. Variable properties, such as density variation as a function of temperature in natural convection problems.
2. Coupled governing equations and/or boundary conditions in phase change problems, such as absorption or dissolution problems.
3. Existence of a source term in one conservation equation that is a function of the dependent variable in another conservation equation.
Langhaar ^{[3]} and Hornbeck ^{[4]} obtained approximate solutions for the momentum equation for circular tubes by solving the linearized momentum equation. Hornbeck ^{[4]} solved the momentum equation numerically by making boundary layer assumptions (parabolic form). Several investigators solved the energy equation either using Langhaar’s approximate velocity profile, or solving the momentum and energy equations numerically for both constant wall temperature and constant wall heat flux in circular tubes. Heat transfer in hydrodynamic and thermal entrance region has been solved numerically based on full elliptic governing equations ^{[5]}.
Variations of local and average Nusselt numbers for different Prandtl numbers under constant wall temperature and constant heat flux using full elliptic governing equations are shown in the two figures to the right, respectively. The local and average Nusselt numbers for different Prandtl numbers and boundary conditions are also presented in the following two tables.
Table 1 Local and average Nusselt number for the entrance region of a circular tube with constant wall temperature
n  Nu_{x}  Nu_{m}''  
x^{+}  Pr = 0.7  Pr = 2  Pr = 5  Pr = 0.7  Pr = 2  Pr = 5 
0.001  17.0  12.5  10.6  60.7  34.9  24.9 
0.002  11.6  8.86  8.04  37.3  22.6  17.0 
0.004  8.29  6.64  6.30  23.3  15.1  12.0 
0.008  6.29  5.23  5.09  14.9  10.4  8.80 
0.01  5.81  4.89  4.80  13.1  9.36  8.03 
0.02  4.69  4.12  4.12  8.9  6.89  6.21 
0.04  4.01  3.75  3.76  6.46  5.39  5.06 
0.06  3.79  3.67  3.68  5.56  4.83  4.61 
0.08  3.71  3.66  3.66  5.09  4.54  4.37 
0.1  3.68  3.66  3.66  4.81  4.36  4.23 
0.12  3.66  3.66  3.66  4.62  4.24  4.14 
3.66  3.66  3.66  3.66  3.66  3.66 
Table 2 Local and mean Nusselt number for the entrance region of a circular tube with constant wall heat flux
n  Nu_{x}  Nu_{m}''  
x^{+}  Pr = 0.7  Pr = 2  Pr = 5  Pr = 0.7  Pr = 2  Pr = 5 
0.001  23.0  17.8  15.0  61.0  43.4  34.2 
0.002  15.6  12.5  11.1  39.8  29.0  23.5 
0.004  10.9  9.16  8.44  26.3  19.8  16.5 
0.008  7.92  7.01  6.65  17.7  13.8  11.9 
0.01  7.24  6.49  6.22  15.7  12.4  10.8 
0.02  5.69  5.30  5.21  11.0  9.11  8.23 
0.04  4.81  4.64  4.62  8.09  7.00  6.54 
0.06  4.53  4.46  4.45  6.94  6.18  5.87 
0.08  4.43  4.39  4.39  6.33  5.74  5.51 
0.1  4.39  4.37  4.37  5.95  5.47  5.29 
0.12  4.37  4.36  4.36  5.69  5.29  5.13 
0.16  4.36  4.36  4.36  5.36  5.23  4.94 
4.36  4.36  4.36  4.36  4.36  4.36 
Heaton et al. ^{[6]} approximated the result for linearized momentum and energy equations using the energy equation for constant wall heat flux for a group of circular tube annulus for several Prandtl numbers. The following table summarizes the results for parallel plates and circular annulus.
Table 3 Local Nusselt number for the entrance region of a group of circular Tube Annulus with Constant Wall Heat Flux
Parallel planes  Circulartube annulus K= 0.50  
Pr  Nu_{11}  θ_{1}*  Nu_{ii}  Nu_{oo}  θ_{i}*  θ_{i}* 
0.01  24.2  0.048    24.2    0.0322 
11.7  0.117    11.8    0.0786  
8.8  0.176  9.43  8.9  0.252  0.118  
5.77  0.378  6.4  5.88  0.525  0.231  
5.53  0.376  6.22  5.6  0.532  0.238  
5.39  0.346  6.18  5.04  0.528  0.216  
0.7  18.5  0.037  19.22  18.3  0.0513  0.0243 
9.62  0.096  10.47  9.45  0.139  0.063  
7.68  0.154  8.52  7.5  0.228  0.0998  
5.55  0.327  6.35  5.27  0.498  0.207  
5.4  0.345  6.19  5.06  0.527  0.215  
5.39  0.346  6.18  5.04  0.528  0.216  
10  15.6  0.0311  16.86  15.14  0.045  0.0201 
9.2  0.092  10.2  8.75  0.136  0.0583  
7.49  0.149  8.43  7.09  0.224  0.0943  
5.55  0.327  6.35  5.2  0.498  0.204  
5.4  0.345  6.19  5.05  0.527  0.215  
5.39  0.346  6.18  5.04  0.528  0.216 
References
 ↑ Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
 ↑ Shah, R. K.; London, A. L., 1978, “Laminar Flow Convection in Ducts,” Advances in Heat Transfer, Supplement 1, Irvine, T. F. and Harnett, J.P., Eds., Academic Press, San Diego, CA.
 ↑ Langhaar, H.L., 1942, Steady Flow in the Transition Length of a Straight Tube,” J. Appl. Mech., Vol. 9, pp. A55A58.
 ↑ ^{4.0} ^{4.1} Hornbeck, R.W., 1965, “An Allnumerical Method for Heat Transfer in the Inlet of a Tube,” ASME Paper No. 65WA HT36.
 ↑ Bahrami, H., 2009, Personal Communication, Storrs, CT.
 ↑ Heaton, H. S., Reynolds, W. C. and Kays, W. M., 1964, “Heat Transfer in Annular Passages: Simultaneous Development of Velocity and Temperature Fields in Laminar Flow”, Int. J. Heat Mass Transfer, Vol. 7, pp. 763781.