# Fully Developed Flow with Developing Temperature Profile

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## Revision as of 02:31, 8 July 2010

Convective heat transfer for steady state, laminar, hydrodynamically developed flow and developing temperature profile in microchannels with both uniform temperature and uniform heat flux boundary conditions were solved by Tunc and Bayazitoglu (2001, 2002) using the integral transform technique that is presented below.

## Uniform Temperature

The energy equation assuming fully developed flow, including viscous dissipation and neglecting axial conduction, is the same as eq. (5.215). The boundary and inlet conditions for constant wall temperature are:

 T = Ts at r = ro (1)
 $\frac{\partial T}{\partial r}=0\text{ at }r=0$ (1)
 T = T0 at x = 0 (1)

The fully developed velocity profile with slip boundary condition given by eq. (5.214) is used. The slip boundary condition given by eq. (5.195) is also used to express the wall temperature jump. The following non-dimensional variables are introduced: for temperature (θ), radial coordinate (r+), axial coordinate (x+), and velocity (u+):

 $\theta =\frac{T-{{T}_{s}}}{{{T}_{0}}-{{T}_{s}}},\text{ }{{r}^{+}}=\frac{r}{{{r}_{o}}},\text{ }{{x}^{+}}=\frac{x}{L},\text{ }{{u}^{+}}=\frac{u}{{{u}_{m}}}$ (1)

The non-dimensional energy equation and boundary conditions are obtained through use of the above variables:

 $\frac{\text{Gz}\left( 1-{{r}^{+}}^{2}+4\text{Kn} \right)}{2\left( 1+8\text{Kn} \right)}\frac{\partial \theta }{\partial {{x}^{+}}}=\frac{1}{{{r}^{+}}}\frac{\partial }{\partial {{r}^{+}}}\left( {{r}^{+}}\frac{\partial \theta }{\partial {{r}^{+}}} \right)+\frac{16\text{Br}}{{{\left( 1+8\text{Kn} \right)}^{2}}}{{r}^{+}}^{2}$ (1)
 θ = 0 at r + = 1 (1)
 $\frac{\partial \theta }{\partial {{r}^{+}}}=0\text{ at }{{r}^{+}}=0$ (1)
 θ = 1 at x + = 0 (1)

where the Graetz number (Gz) and the Brinkman number (Br) are defined as:

 $\text{Gz}=\frac{\text{RePr}D}{L}\text{ and Br}=\frac{\mu u_{m}^{2}}{k\Delta T}$ (1)

where ΔT is the difference between the temperature of the fluid at the wall, Ts, and at the tube entrance, T0, i.e. ΔT = T0Ts.