Fully Developed Laminar Flow and Temperature Profile

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[[Image:Fig5.23.png|thumb|400 px|alt=Non-dimensional, fully developed velocity profile in a microchannel as a function of Knudsen number |Non-dimensional, fully developed velocity profile in a microchannel as a function of Knudsen number.]]
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.
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.
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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.
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.
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[[Image:Fig5.24.png|thumb|400 px|alt=The variation of the Nusselt number for microchannel, with the Knudsen number, for different values of the Brinkman number for constant heat flux at the wall |The variation of the Nusselt number for microchannel, with the Knudsen number, for different values of the Brinkman number for constant heat flux at the wall.]]
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 (
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 (
<math>{{{q}''}_{w}}>0</math>
<math>{{{q}''}_{w}}>0</math>
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A closed form solution for Nu cannot be obtained for this case.  However, the solution of θ can be obtained by using an iterative procedure.  The temperature profile for the constant heat flux at the wall can be used as the first approximation, and eq. (5.230) is then integrated to obtain θ.  This iterative procedure is repeated until an acceptable convergence is obtained. However, in order to get a very good accuracy, a forth-order Rung-Kutta procedure is employed to solve eq. (5.230).
A closed form solution for Nu cannot be obtained for this case.  However, the solution of θ can be obtained by using an iterative procedure.  The temperature profile for the constant heat flux at the wall can be used as the first approximation, and eq. (5.230) is then integrated to obtain θ.  This iterative procedure is repeated until an acceptable convergence is obtained. However, in order to get a very good accuracy, a forth-order Rung-Kutta procedure is employed to solve eq. (5.230).
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[[Image:Fig5.25.png|thumb|400 px|alt=Variation of the fully developed Nusselt number for microchannel with the Knudsen number for constant wall temperature and neglecting viscous dissipation effects |Variation of the fully developed Nusselt number for microchannel with the Knudsen number for constant wall temperature and neglecting viscous dissipation effects.]]
Figure 5.25 presents Nu versus Kn for the case of constant wall temperature (Bahrami, 2009). Similar trends to those obtained for the case of constant heat flux at the wall are observed.  Nu values for the case of constant wall temperature are, for the same Kn, lower than those Nu values for the case of constant heat flux at the wall.
Figure 5.25 presents Nu versus Kn for the case of constant wall temperature (Bahrami, 2009). Similar trends to those obtained for the case of constant heat flux at the wall are observed.  Nu values for the case of constant wall temperature are, for the same Kn, lower than those Nu values for the case of constant heat flux at the wall.
 +
[[Image:Fig5.26.png|thumb|400 px|alt=Effect of temperature jump on fully developed Nu in microchannel for different Brq when the wall is subjected to constant heat flux |Effect of temperature jump on fully developed Nu in microchannel for different Brq when the wall is subjected to constant heat flux.]]
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[[Image:Fig5.27.png|thumb|400 px|alt=The effect of temperature jump on fully developed Nu for microchannel when the wall is subjected to constant temperature |The effect of temperature jump on fully developed Nu for microchannel when the wall is subjected to constant temperature.]]
The fully developed Nusselt numbers for <math>0<\text{Kn}<0.12</math> and no viscous dissipation are shown in Tables 5.10 and 5.11 for constant wall temperature and heat flux, respectively (Bahrami, 2009). The fully developed Nusselt number decreases as Kn increases.
The fully developed Nusselt numbers for <math>0<\text{Kn}<0.12</math> and no viscous dissipation are shown in Tables 5.10 and 5.11 for constant wall temperature and heat flux, respectively (Bahrami, 2009). The fully developed Nusselt number decreases as Kn increases.
The effect of temperature jump on the Nusselt number is shown in Figs. 5.26 and 5.27. The solid and dashed lines represent the results obtained for considering the temperature jump and neglecting the temperature jump conditions, respectively.  When the temperature jump condition is not accounted for, i.e. only the velocity slip condition is taken into consideration, the Nusselt number increases with increasing Kn, which indicates that the velocity slip and temperature jump have opposite effects on the Nusselt number.
The effect of temperature jump on the Nusselt number is shown in Figs. 5.26 and 5.27. The solid and dashed lines represent the results obtained for considering the temperature jump and neglecting the temperature jump conditions, respectively.  When the temperature jump condition is not accounted for, i.e. only the velocity slip condition is taken into consideration, the Nusselt number increases with increasing Kn, which indicates that the velocity slip and temperature jump have opposite effects on the Nusselt number.

Revision as of 03:18, 7 July 2010

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 + C3C1r2

(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 − (ucus) / 4] = (uc + us) / 2

(1)

The centerline velocity can be written as:

uc = 2umus

(1)

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

u = 2(umus)(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)
Non-dimensional, fully developed velocity profile in a microchannel as a function of Knudsen number
Non-dimensional, fully developed velocity profile in a microchannel as a function of Knudsen number.

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.

The variation of the Nusselt number for microchannel, with the Knudsen number, for different values of the Brinkman number for constant heat flux at the wall
The variation of the Nusselt number for microchannel, with the Knudsen number, for different values of the Brinkman number for constant heat flux at the wall.

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.

Constant Wall Temperature

Aydin and Avci (2006) investigated the effects of the Brinkman and Knudsen numbers for both fully developed flow and temperature profile in a microchannel with circular cross-section tube subjected to constant wall temperature. They assumed that the fluid temperature at the wall does not change along the tube length, i.e. dTs / dx = 0. However, their analysis for the viscous dissipation effect on Nusselt number was found to be inconsistent with other researchers’ results (Hooman, 2008). In the following the effect of the Knudsen number for both fully developed flow and temperature profile in a microchannel subjected to constant wall temperature considering variation of fluid temperature at the wall (i.e. d{{T}_{s}}/dx\ne 0) is presented (Bahrami, 2009). The non-dimensional temperature profile is defined as:

\theta =\frac{{{T}_{s}}-T}{{{T}_{s}}-{{T}_{c}}}\text{   }

(1)

where Tc is the fluid temperature at the centerline. Substituting eq. (5.214) into eq. (5.215), neglecting the viscous dissipation, and non-dimensionalizing the resultant equation give us:

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

(1)

where:

{{\beta }_{1}}=\frac{2{{u}_{m}}r_{o}^{2}}{\left( 1+8\text{Kn} \right)\alpha \left( {{T}_{c}}-{{T}_{s}} \right)}\frac{d{{T}_{c}}}{dx}

(1)

{{\beta }_{2}}=\frac{2{{u}_{m}}r_{o}^{2}}{\left( 1+8\text{Kn} \right)\alpha \left( {{T}_{c}}-{{T}_{s}} \right)}\frac{d{{T}_{s}}}{dx}

(1)

Equation (5.230) is subject to the following boundary conditions:

\begin{align}  & \frac{\partial \theta }{\partial {{r}^{+}}}=0\text{  and  }\theta =1\text{   at  }{{r}^{+}}=0 \\  & \theta =0\text{   at  }{{r}^{+}}=1 \\ \end{align}

(1)

The Nusselt number based on θ is:

\text{Nu}=\frac{hD}{k}=\frac{-2{{\left( \frac{\partial \theta }{\partial {{r}^{+}}} \right)}_{{{r}^{+}}=1}}}{\theta _{m}^{{}}-\theta _{w}^{{}}}

(1)

where \theta _{m}^{{}} is the non-dimensional mean temperature, and \theta _{w}^{{}} is the slip temperature at the wall. The relation between β1

and 

β2

can be obtained by taking the derivative of slip temperature boundary conditions along the x-direction, and the results are given bellow:

\frac{d{{T}_{c}}}{dx}=\frac{\xi }{\xi +1}\frac{d{{T}_{s}}}{dx}\text{,   }\xi =-\frac{2-{{F}_{t}}}{{{F}_{t}}}\frac{4\gamma }{1+\gamma }\frac{\text{Kn}}{\Pr }Nu\left( \theta _{m}^{{}}-\theta _{w}^{{}} \right)

A closed form solution for Nu cannot be obtained for this case. However, the solution of θ can be obtained by using an iterative procedure. The temperature profile for the constant heat flux at the wall can be used as the first approximation, and eq. (5.230) is then integrated to obtain θ. This iterative procedure is repeated until an acceptable convergence is obtained. However, in order to get a very good accuracy, a forth-order Rung-Kutta procedure is employed to solve eq. (5.230).

Variation of the fully developed Nusselt number for microchannel with the Knudsen number for constant wall temperature and neglecting viscous dissipation effects
Variation of the fully developed Nusselt number for microchannel with the Knudsen number for constant wall temperature and neglecting viscous dissipation effects.

Figure 5.25 presents Nu versus Kn for the case of constant wall temperature (Bahrami, 2009). Similar trends to those obtained for the case of constant heat flux at the wall are observed. Nu values for the case of constant wall temperature are, for the same Kn, lower than those Nu values for the case of constant heat flux at the wall.

Effect of temperature jump on fully developed Nu in microchannel for different Brq when the wall is subjected to constant heat flux
Effect of temperature jump on fully developed Nu in microchannel for different Brq when the wall is subjected to constant heat flux.
The effect of temperature jump on fully developed Nu for microchannel when the wall is subjected to constant temperature
The effect of temperature jump on fully developed Nu for microchannel when the wall is subjected to constant temperature.

The fully developed Nusselt numbers for 0 < Kn < 0.12 and no viscous dissipation are shown in Tables 5.10 and 5.11 for constant wall temperature and heat flux, respectively (Bahrami, 2009). The fully developed Nusselt number decreases as Kn increases. The effect of temperature jump on the Nusselt number is shown in Figs. 5.26 and 5.27. The solid and dashed lines represent the results obtained for considering the temperature jump and neglecting the temperature jump conditions, respectively. When the temperature jump condition is not accounted for, i.e. only the velocity slip condition is taken into consideration, the Nusselt number increases with increasing Kn, which indicates that the velocity slip and temperature jump have opposite effects on the Nusselt number.