# Fully-developed flow and heat transfer

## Constant Wall Heat Flux

In this article, 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. (18) in Internal Forced Convection basics can be substituted into the above equation 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. (2) twice and applying the boundary conditions in eq. (3) 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. (4) 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. (4) 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. (6) and (7) and solving for the heat transfer coefficient, h, yields

$\begin{matrix}{} & h=4.364k/D \\\end{matrix}$

or in terms of the Nusselt number,

 $\begin{matrix}{}\\\end{matrix}Nu=4.364$ (8)

## Constant Surface Temperature

We begin with the energy equation (equation 29 in Internal Forced Convection Basics) 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 (equation 24 in Internal Forced Convection Basics) 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} (10)

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 [1][2][3][4].

The solution for eqs. (9) and (10) in the form of an infinite series for the temperature and Nusselt number is [2]:

 $\frac{T-{{T}_{w}}}{{{T}_{in}}-{{T}_{w}}}=\sum\limits_{m=0}^{\infty }{{{C}_{2m}}{{\left( \frac{r}{{{r}_{o}}} \right)}^{2m}}}$ (11)

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

 $\text{Nu}=\frac{1}{2}{{\lambda }^{2}}=3.657$ (12)

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:

 ${{{q}''}_{w}}=A\exp \left( \frac{1}{2}n{{x}^{+}} \right)$ (13)

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 Cite error: Closing </ref> missing for <ref> tag

 K Nuii Nuoo θi* θo* 0 ∞ 4.364 ∞ 0 0.10 11.900 4.834 1.3835 0.0562 0.25 7.735 4.904 0.7932 0.1250 0.50 6.181 5.036 0.5288 0.2160 1.00 5.384 5.384 0.3460 0.3460

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## Fully-developed flow and heat transfer in tubes of noncircular cross section

The fluid and heat transfer solution for tubes of noncircular cross section with a fully developed flow and temperature profile can be obtained by solving the energy equation for the particular geometry. The results for some of the more conventional geometries obtained by Shah and London (1974)[5] are presented in Table 2. There are three different boundary conditions in this table; “H1” which refers to the circumferentially constant wall temperature and the axial constant wall heat flux boundary condition, “H2” is for both axially and circumferentially constant heat flux at the wall, and “T” represents a constant wall temperature boundary condition. For the case of a symmetrically heated straight duct having no corners and a constant peripheral curvature, e.g. parallel planes and circular pipe, H1 and H2 boundary conditions are the same.

Table 2 Solutions for Nusselt Number and Friction Coefficient for fully developed laminar flow and temperature profile for various geometries [5]
1. Burmeister, L.C., 1993, Convective Heat Transfer, 2nd ed., John Wiley & Sons, Hoboken, NJ.
2. 2.0 2.1 Kakaç, S., Shah, R., Aung, W., 1987, Handbook of Single-Phase Convective Heat Transfer, John Wiley, New York.
3. Kays, W.M., Crawford, M.E., and Weigand, B., 2005, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NY
4. Bejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, Hoboken, NJ.
5. 5.0 5.1 Shah, R. K.; London, A. L., 1974, “Thermal Boundary Conditions and Some Solutions for Laminar Duct Flow Forced Convection,” ASME J. Heat Transfer Vol. 96, pp. 159-165.