Developing flow

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All of the forced convective heat and mass transfer problems considered so far assumed that the flow is fully developed, which, as previously shown, occurs at x/D approximately equal to 0.05Re for a circular tube. For forced convective heat and mass transfer with constant properties, the hydrodynamic entrance length is independent of Pr or Sc. It was also shown that 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. Analysis of these criteria for a fully developed flow and temperature profile shows that when Pr \gg 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 \gg 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 \ll 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 two-dimensional 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 Fig. 5.15.


The conservation equations with the above assumptions, as well as neglecting the viscous dissipation and assuming an incompressible Newtonian fluid, are

\text{continuity}\quad \frac{\partial u}{\partial x}+\frac{1}{r}\frac{\partial }{\partial r}\left( rv \right)=0

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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]

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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]

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\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]

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\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]

(1)

Typical boundary conditions are:

Clearly there are five partial differential equations and five unknowns (u, v, P, T, ω1). All equations are of elliptic nature (Chapter 2) and one can neglect the axial diffusion terms, \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) , 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 (1978) showed that the momentum boundary layer assumption will lead to error if Re < 400 and LH / D < 0.005Re. In these circumstances, the full Navier Stokes equation should be solved. It was also shown in Section 5.2 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.