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

(1)

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]

(1)

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]

(1)

\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]

(1)

\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: \text{Axial velocity at wall} u\left( x,{{r}_{o}} \right)=0               \text{no slip boundary condition} \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.

\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.

Inlet condition atx = 0 \left\{ \begin{align}
  & T={{T}_{in}} \\ 
 & {{\omega }_{1}}={{\omega }_{1,}}_{in} \\ 
 & u={{u}_{in}} \\ 
\end{align} \right.

\text{Outlet condition at} x=L\begin{matrix}
   {} & {} & {} & {} & {}  \\
\end{matrix}\left\{ \begin{align}
  & T=? \\ 
 & {{\omega }_{1}}=? \\ 
 & u=? \\ 
 & P=? \\ 
\end{align} \right.

Alternative text Alternative text

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. 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 (1942) and Hornbeck (1965) obtained approximate solutions for the momentum equation for circular tubes by solving the linearized momentum equation. Hornbeck (1965) 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 (Bahrami, 2009).

Local and average Nusselt numbers for the entrance region of a circular tube with constant wall temperature
Figure 1: 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 heat flux
Figure 2: 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 Figs. 1 and 2, respectively. The local and average Nusselt numbers for different Prandtl numbers and boundary conditions are also presented in Tables 1 and 2. Heaton et al. (1964) 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. Table 3 summarizes the results for parallel plates and circular annulus.

Table 1. Local and average Nusselt number for the entrance region of a circular tube with constant wall temperature Table 1

Table 2. Local and mean Nusselt number for the entrance region of a circular tube with constant wall heat flux Table 2

Table 3. Local Nusselt number for the entrance region of a group of circular Tube Annulus with Constant Wall Heat Flux Table 3