Fully Developed Laminar Flow and Temperature Profile

Fully Developed Velocity Distribution

The velocity distribution is derived from the continuity and momentum equations along with the slip condition. The velocity profile is expressed in terms of Knudsen number. The model for the analysis of velocity distribution can be considered as the flow of a fluid in a circular tube of radius ro, as shown in Fig. 5.22. The continuity equation in cylindrical coordinates is given as:

 $\frac{\partial \rho }{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}\left( \rho rv \right)+\frac{\partial }{\partial x}\left( \rho u \right)=0$ (1)

For steady and fully developed flow of an incompressible fluid, it becomes:

 $\frac{\partial u}{\partial x}=0$ (1)

Therefore, rv = constant. Since the radial velocity at the wall is zero (impermeability condition), it can be concluded that v = 0 everywhere in the flow field. The steady parabolic momentum equation in the x-direction can be written in cylindrical coordinates as:

 $\rho u\frac{\partial u}{\partial x}+\rho v\frac{\partial u}{\partial r}=\frac{\mu }{r}\frac{\partial }{\partial r}\left( r\frac{\partial u}{\partial r} \right)-\frac{\partial p}{\partial x}$ (1)

For fully developed, steady flow of incompressible fluid, it reduces to:

 $-\frac{dp}{dx}+\frac{\mu }{r}\frac{d}{dr}\left( r\frac{du}{dr} \right)=0$ (1)

As the velocity is independent of x, a constant C1 can be defined as:

 ${{C}_{1}}=-\frac{1}{4\mu }\frac{dp}{dx}$ (1)

Therefore, the momentum equation reduces to:

 $4{{C}_{1}}+\frac{1}{r}\frac{d}{dr}\left( r\frac{du}{dr} \right)=0$ (1)

The above equation is integrated twice to yield:

 u = C2lnr + C3 − C1r2 (1)

Since the velocity, u, is finite at the center of the tube (r = 0), we conclude C2=0. At the centerline of the tube (r = 0), the velocity is equal to uc, therefore, u(0) = uc = C3. At the inner surface of the tube wall (r = ro), the velocity is not zero due to the slip flow condition, but is equal to a finite velocity us:

 $u({{r}_{o}})={{u}_{s}}={{u}_{c}}-{{C}_{1}}r_{o}^{2}$ (1)

Rearranging and solving for C1 we have:

 ${{C}_{1}}=\frac{{{u}_{c}}-{{u}_{s}}}{r_{0}^{2}}=-\frac{1}{4\mu }\frac{dp}{dx}$ (1)

The velocity profile is obtained as follows:

 $u={{u}_{c}}-({{u}_{c}}-{{u}_{s}}){{(r/{{r}_{o}})}^{2}}={{u}_{c}}[1-{{(r/{{r}_{o}})}^{2}}]+{{u}_{s}}{{(r/{{r}_{o}})}^{2}}$ (1)

The mean velocity in the tube, um, is now evaluated. The volumetric flow rate is:

 $\pi r_{0}^{2}{{u}_{m}}=\int_{0}^{{{r}_{0}}}{2\pi rudr}$ (1)

therefore,

 um = 2[uc / 2 − (uc − us) / 4] = (uc + us) / 2 (1)

The centerline velocity can be written as:

 uc = 2um − us (1)

Using the above relation, we get the following velocity profile in a microchannel:

 u = 2(um − us)(1 − r + 2) + us (1)

where r+=r/ro. If the slip velocity is zero, the velocity distribution reduces to the Poiseuille flow distribution for conventional channels. The velocity slip is given by the following condition [see eq. (5.194)]:

 ${{u}_{s}}=-\frac{2-F}{F}\lambda {{\left. \frac{du}{dr} \right|}_{r={{r}_{o}}}}$ (1)

For most applications, F has values near unity. Therefore,

 ${{u}_{s}}=-\lambda {{\left. \frac{du}{dr} \right|}_{r={{r}_{o}}}}$ (1)

The slip velocity is derived using eq. (5.209) for velocity profile:

 ${{u}_{s}}=-\frac{\lambda }{{{r}_{o}}}{{\left. \frac{du}{d{{r}^{+}}} \right|}_{{{r}^{+}}=1}}=\frac{4\lambda ({{u}_{m}}-{{u}_{s}})}{{{r}_{o}}}=\frac{8\lambda ({{u}_{m}}-{{u}_{s}})}{D}$ (1)

The Knudsen number, Kn = λ / D, is introduced in the above relation to obtain the following equation for the slip velocity:

 $\frac{{{u}_{s}}}{{{u}_{m}}}=\frac{8\text{Kn}}{1+8\text{Kn}}$ (1)

Substituting the above into eq. (5.209), the non-dimensional expression for the velocity profile is obtained:

 $\frac{u}{{{u}_{m}}}=\frac{2(1-{{r}^{+2}})+8\text{Kn}}{1+8\text{Kn}}$ (1)

The velocity distribution is shown in Fig. 5.23 for different Kn numbers. The slip velocity at the wall increases with an increasing Kn number.

Fully Developed Heat Transfer Coefficient in Microchannel The flow is considered to be steady, laminar, and fully developed both hydrodynamically and thermally. The conventional continuum approach is coupled with the two main characteristics of the microscale phenomena, the velocity slip and the temperature jump, as noted above. The energy equation including the viscous dissipation term for steady, fully developed flow in a pipe and neglecting axial conduction is:

 $u\frac{\partial T}{\partial x}=\frac{\alpha }{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right)+\frac{\nu }{{{c}_{p}}}{{\left( \frac{\partial u}{\partial r} \right)}^{2}}$ (1)

The variation of temperature in the r-direction at the center is zero due to axisymmetric conditions:

 ${{\left. \frac{\partial T}{\partial r} \right|}_{r=0}}=0$ (1)

Also, the temperature jump condition for the fluid at the wall is written as follows:

 ${{T}_{s}}-{{T}_{w}}=-\frac{2-{{F}_{t}}}{{{F}_{t}}}\frac{4\gamma }{1+\gamma }\frac{\text{Kn}}{\Pr }{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}$ (1)

Both conventional boundary conditions of constant heat flux and constant temperature at the wall are solved analytically by Aydin and Avci (2006) for microchannel with circular cross-section, which are presented below.

Constant Heat Flux at the Wall

The constant heat flux at the wall is described by:

 $k{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{0}}}}={{{q}''}_{w}}$ (1)

where $q_{w}^{''}$ is positive when its direction is to the fluid (the hot wall) and negative when its direction is from the fluid (the cold wall). For the constant wall heat flux, the following equation, similar to the analysis presented for conventional heat pipes, is applicable:

 $\frac{\partial T}{\partial x}=\frac{d{{T}_{w}}}{dx}=\frac{d{{T}_{s}}}{dx}$ (1)

Substituting eq. (5.214) into eq. (5.215) and non-dimensionalizing the resultant equation yield:

 $\frac{1}{{{r}^{+}}}\frac{d}{d{{r}^{+}}}\left( {{r}^{+}}\frac{d{{\theta }_{q}}}{d{{r}^{+}}} \right)={{\beta }_{1}}\left( 1-{{r}^{{{+}^{2}}}}+4\text{Kn} \right)+32B{{r}_{q}}\frac{{{r}^{{{+}^{2}}}}}{{{\left( 1+8\text{Kn} \right)}^{2}}}$ (1)

where

 $B{{r}_{q}}=\frac{\mu u_{m}^{2}}{Dq_{w}^{''}}\text{, }\theta _{q}^{{}}=\frac{T-{{T}_{s}}}{{{{{q}''}}_{w}}{{r}_{0}}/k},\text{ }{{\beta }_{1}}=-\frac{2\rho {{c}_{p}}{{u}_{m}}{{r}_{o}}}{\left( 1+8\text{Kn} \right){{{{q}''}}_{w}}}\frac{d{{T}_{s}}}{dx}$ (1)

Since β1

is unknown, three non-dimensional boundary conditions are required for the second order ODE, eq. (5.220):

 \begin{align} & \frac{\partial {{\theta }_{q}}}{\partial {{r}^{+}}}=0\text{ at }{{r}^{+}}=0 \\ & {{\theta }_{q}}=0\text{ and }\frac{\partial {{\theta }_{q}}}{\partial {{r}^{+}}}=-1\text{ at }{{r}^{+}}=1 \\ \end{align} (1)

The solution of eq. (5.220) using the above boundary conditions is:

 ${{\theta }_{q}}\left( {{r}^{+}} \right)={{\beta }_{1}}\left( \frac{{{r}^{{{+}^{2}}}}}{4}-\frac{{{r}^{+4}}}{16}+\text{Kn}{{r}^{{{+}^{2}}}} \right)+{{\beta }_{2}}\left( \frac{{{r}^{+4}}}{16} \right)-{{\beta }_{3}}$ (1)

where

 ${{\beta }_{2}}=\frac{32B{{r}_{q}}}{{{\left( 1+8\text{Kn} \right)}^{2}}}\text{, }{{\beta }_{1}}=-\frac{{{\beta }_{2}}+4}{1+8\text{Kn}}\text{, }{{\beta }_{3}}=\frac{{{\beta }_{2}}+{{\beta }_{1}}\left( 3+16\text{Kn} \right)}{16}$ (1)

The local heat transfer coefficient, h, is:

 $h=\frac{k}{{{T}_{w}}-{{T}_{m}}}{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}$ (1)

The Nusselt number, based on $\theta _{q}^{{}}$ is:

 $\text{Nu}=\frac{hD}{k}=\frac{2}{\theta _{q,m}^{{}}-\theta _{q,w}^{{}}}$ (1)

where the non-dimensional mean temperature ($\theta _{q,m}^{{}}$) and slip temperature at the wall ($\theta _{q,w}^{{}}$) are given as follows:

 $\theta _{q,m}^{{}}\text{=}\frac{\text{1}}{\text{4}}\text{+}\frac{5+{{\beta }_{\text{2}}}+\text{Kn}\left( 32+\text{6}{{\beta }_{\text{2}}} \right)}{24{{\left( 1+8\text{Kn} \right)}^{2}}}$ (1)
 $\theta _{q,w}^{{}}\text{=-}\frac{\text{2-}{{F}_{t}}}{{{F}_{t}}}\frac{4\gamma }{1+\gamma }\frac{\text{Kn}}{\Pr }$ (1)

Aydin and Avci (2006) investigated the effects of the Brinkman number and Knudsen number for both fully developed flow and temperature profile in a microchannel with circular cross-section using the above analytical technique. Kn = 0 represents the macroscale case, while Kn > 0 holds for the microscale case, and Brq = 0 represents the case without the effect of the viscous dissipation.

Figure 5.24 shows the variation of Nusselt number with the Knudsen number for different Brinkman numbers for the case of constant wall heat flux (Bahrami, 2009). For Brq = 0, an increase at Kn decreases Nu due to the temperature slip at the wall. Viscous dissipation significantly affects Nu. Positive values of Brq correspond to wall heating ( q''w > 0 ), while the opposite is true for negative values of Brq. With no viscous dissipation, the solution is independent of whether there is wall heating or cooling. Nu decreases with increasing Brq for the wall heating case. Increasing Brq in the negative direction increases Nu. The trend followed by Nu versus Kn for lower values of the Brinkman number, either in the case of wall heating (Brq = 0.05) or in the case of wall cooling (Brq = –0.05) is very similar to that of Brq = 0. For the wall cooling case, at Brq = –0.1, the decreasing effect of Kn on Nu is more significant. At Brq = 0.1, increasing Kn increases Nu up to $\text{Kn}\approx 0.01$ where a maximum occurs, after which Nu decreases with increasing Kn.