# Fully-developed flow and heat transfer

## Constant Wall Heat Flux

In this section, we consider the case of fully developed laminar flow and constant properties in a circular tube with a fully developed temperature and concentration profiles. We first consider the case of constant heat rate per unit surface area for steady, laminar, fully developed flow. The energy equation in a circular tube, by neglecting axial heat conduction and viscous dissipation terms, is:

 $u\frac{\partial T}{\partial x}=\alpha \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right) \right]$ (1)

For a fully developed flow with constant wall heat flux, eq. (5.20) can be substituted into eq. (5.36) to obtain

 $u\frac{d{{T}_{m}}}{dx}=\alpha \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right) \right]$ (2)

The boundary conditions are

 \begin{align}& -k\frac{\partial T}{\partial r}={{{{q}''}}_{w}}\begin{matrix}{} & {} \\\end{matrix}\text{at}\ r={{r}_{o}} \\ & \frac{\partial T}{\partial r}=0\begin{matrix} {} & {} & {} \\\end{matrix}\text{at}\ r=0 \\ \end{align} (3)

Integrating eq. (5.37) twice and applying the boundary conditions in eq. (5.38) to get the temperature distribution gives us

 $T={{T}_{w}}-\frac{2{{u}_{m}}}{\alpha }\frac{d{{T}_{m}}}{dx}\left( \frac{3r_{o}^{2}}{16}-\frac{{{r}^{2}}}{4}+\frac{{{r}^{4}}}{16r_{o}^{2}} \right)$ (4)

Using the definition of mean temperature presented in the last section with the above profile for temperature, and assuming constant properties:

${{T}_{m}}=\frac{\int_{A}^{{}}{uTdA}}{\int_{A}^{{}}{udA}}=\frac{2\int_{0}^{{{r}_{o}}}{\pi ruTdr}}{\pi r_{o}^{2}{{u}_{m}}}$

Substituting eq. (5.39) into the above expression yields:

 ${{T}_{m}}={{T}_{w}}-\frac{11}{96}\left( \frac{2{{u}_{m}}}{\alpha } \right)\frac{d{{T}_{m}}}{dx}r_{o}^{2}$ (5)

The heat flux at the wall can be obtained using the above relation for Tm

 ${{q}_{w}}^{\prime \prime }=h\left( {{T}_{w}}-{{T}_{m}} \right)=h\left( \frac{11}{96} \right)\left( \frac{2{{u}_{m}}}{\alpha } \right)\left( \frac{d{{T}_{m}}}{dx} \right)r_{o}^{2}$ (6)

The heat flux at the wall can also be calculated using eq. (5.39) for the temperature profile and Fourier’s law of heat conduction

 ${{{q}''}_{w}}={{\left. -k\frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}=\rho {{c}_{p}}{{r}_{o}}\left( \frac{{{u}_{m}}}{2} \right)\left( \frac{d{{T}_{m}}}{dx} \right)$ (7)

Combining eqs. (5.41) and (5.42) and solving for the heat transfer coefficient, h, yields h = 4.364 k / D or in terms of the Nusselt number,

 Nu = 4.364 (8)

## Constant Surface Temperature

We begin with the energy eq. (5.31) and neglect the effects of axial heat conduction and viscous dissipation. We already showed that for a fully developed flow and temperature profile with constant surface temperature

$\frac{\partial T}{\partial x}=\frac{{{T}_{w}}-T}{{{T}_{w}}-{{T}_{m}}}\frac{d{{T}_{m}}}{dx}$

The energy equation and boundary conditions, using the fully developed velocity profile eq. (5.26) and the above equation, are

 $2{{u}_{m}}\left( 1-\frac{{{r}^{2}}}{r_{o}^{2}} \right)\left( \frac{{{T}_{w}}-T}{{{T}_{w}}-{{T}_{m}}} \right)\frac{d{{T}_{m}}}{dx}=\alpha \left[ \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right) \right]$ (9)

\begin{align} & r={{r}_{o}},\quad \quad T={{T}_{w}} \\ & r=0,\quad \quad \frac{\partial T}{\partial r}=0\quad \text{or}\quad T=\text{finite} \\ \end{align} (5.45) The above equation and boundary conditions have been solved by various techniques in literatures, including separation of variables and infinite series. For additional detailed information the reader should refer to Burmeister (1993), Kakac, et al. (1987), Kays et al. (2005), and Bejan (2004). The solution for eqs. (5.44) and (5.45) in the form of an infinite series for the temperature and Nusselt number is (Kakac et al. 1987):

 $\frac{T-{{T}_{w}}}{{{T}_{in}}-{{T}_{w}}}=\sum\limits_{m=0}^{\infty }{{{C}_{2m}}{{\left( \frac{r}{{{r}_{o}}} \right)}^{2m}}}$ (10)

where

\begin{align} & {{c}_{o}}=1,\quad \quad {{c}_{2}}=-\frac{1}{4}\lambda _{0}^{2}=-1.828397,\quad \quad {{c}_{2m}}=\frac{\lambda _{0}^{2}}{{{\left( 2m \right)}^{2}}}\left( {{c}_{2m-4}}-{{c}_{2m-2}} \right) \\ & {{\lambda }_{0}}=2.704364 \\ \end{align}

The Nusselt number corresponding to the above temperature distribution is

 $\text{Nu}=\frac{1}{2}{{\lambda }^{2}}=3.657$ (11)

The temperature slope for constant wall heat flux at the wall is higher than the temperature slope for constant surface temperature. This effect has resulted in a 16 percent increase in Nusselt number for constant wall flux versus constant wall temperature for a fully developed flow and temperature profile. The two cases of constant wall temperature and constant wall heat flux are special cases of a more general exponential heat flux boundary condition:

 ${{{q}''}_{w}}=A\exp \left( \frac{1}{2}n{{x}^{+}} \right)$ (12)

where A and n are both constants and n can be assumed to be either a positive or negative value, and ${{x}^{+}}=\frac{x/{{r}_{o}}}{\operatorname{Re}\Pr }.$ n = 0 corresponds to constant heat flux at the wall and n = –14.63 corresponds to constant wall temperature. Shah and London (1978) developed the following correlation, which fits the exact solution of eq. (5.48) within 3 percent for –51.36 < n < 100.

 \begin{align}& Nu=4.3573+0.0424n-2.8368\times {{10}^{-4}}{{n}^{2}}+3.6250\times {{10}^{-6}}{{n}^{3}} \\ & -7.6497\times {{10}^{-8}}{{n}^{4}}+9.1222\times {{10}^{-10}}{{n}^{5}}-3.8446\times {{10}^{-12}}{{n}^{6}} \\ \end{align} (13)