Fully Developed Laminar Flow and Temperature Profile
From ThermalFluidsPedia
(→Fully Developed Velocity Distribution) 

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<math>\frac{\partial u}{\partial x}=0</math>  <math>\frac{\partial u}{\partial x}=0</math>  
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  {{EquationRef(  +  {{EquationRef(2)}} 
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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.  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.  
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<math>\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}</math>  <math>\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}</math>  
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  {{EquationRef(  +  {{EquationRef(3)}} 
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For fully developed, steady flow of incompressible fluid, it reduces to:  For fully developed, steady flow of incompressible fluid, it reduces to:  
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<math>\frac{dp}{dx}+\frac{\mu }{r}\frac{d}{dr}\left( r\frac{du}{dr} \right)=0</math>  <math>\frac{dp}{dx}+\frac{\mu }{r}\frac{d}{dr}\left( r\frac{du}{dr} \right)=0</math>  
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As the velocity is independent of x, a constant C1 can be defined as:  As the velocity is independent of x, a constant C1 can be defined as:  
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<math>{{C}_{1}}=\frac{1}{4\mu }\frac{dp}{dx}</math>  <math>{{C}_{1}}=\frac{1}{4\mu }\frac{dp}{dx}</math>  
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Therefore, the momentum equation reduces to:  Therefore, the momentum equation reduces to:  
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<math>4{{C}_{1}}+\frac{1}{r}\frac{d}{dr}\left( r\frac{du}{dr} \right)=0</math>  <math>4{{C}_{1}}+\frac{1}{r}\frac{d}{dr}\left( r\frac{du}{dr} \right)=0</math>  
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The above equation is integrated twice to yield:  The above equation is integrated twice to yield:  
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<math>u={{C}_{2}}\ln r+{{C}_{3}}{{C}_{1}}{{r}^{2}}</math>  <math>u={{C}_{2}}\ln r+{{C}_{3}}{{C}_{1}}{{r}^{2}}</math>  
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  {{EquationRef(  +  {{EquationRef(7)}} 
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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:  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:  
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<math>u({{r}_{o}})={{u}_{s}}={{u}_{c}}{{C}_{1}}r_{o}^{2}</math>  <math>u({{r}_{o}})={{u}_{s}}={{u}_{c}}{{C}_{1}}r_{o}^{2}</math>  
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  {{EquationRef(  +  {{EquationRef(8)}} 
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Rearranging and solving for C1 we have:  Rearranging and solving for C1 we have:  
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<math>{{C}_{1}}=\frac{{{u}_{c}}{{u}_{s}}}{r_{0}^{2}}=\frac{1}{4\mu }\frac{dp}{dx}</math>  <math>{{C}_{1}}=\frac{{{u}_{c}}{{u}_{s}}}{r_{0}^{2}}=\frac{1}{4\mu }\frac{dp}{dx}</math>  
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  {{EquationRef(  +  {{EquationRef(9)}} 
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The velocity profile is obtained as follows:  The velocity profile is obtained as follows:  
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<math>u={{u}_{c}}({{u}_{c}}{{u}_{s}}){{(r/{{r}_{o}})}^{2}}={{u}_{c}}[1{{(r/{{r}_{o}})}^{2}}]+{{u}_{s}}{{(r/{{r}_{o}})}^{2}}</math>  <math>u={{u}_{c}}({{u}_{c}}{{u}_{s}}){{(r/{{r}_{o}})}^{2}}={{u}_{c}}[1{{(r/{{r}_{o}})}^{2}}]+{{u}_{s}}{{(r/{{r}_{o}})}^{2}}</math>  
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  {{EquationRef(  +  {{EquationRef(10)}} 
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The mean velocity in the tube, <math>{{u}_{m}}</math>, is now evaluated. The volumetric flow rate is:  The mean velocity in the tube, <math>{{u}_{m}}</math>, is now evaluated. The volumetric flow rate is:  
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<math>\pi r_{0}^{2}{{u}_{m}}=\int_{0}^{{{r}_{0}}}{2\pi rudr}</math>  <math>\pi r_{0}^{2}{{u}_{m}}=\int_{0}^{{{r}_{0}}}{2\pi rudr}</math>  
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  {{EquationRef(  +  {{EquationRef(11)}} 
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therefore,  therefore,  
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<math>{{u}_{m}}=2[{{u}_{c}}/2({{u}_{c}}{{u}_{s}})/4]=({{u}_{c}}+{{u}_{s}})/2</math>  <math>{{u}_{m}}=2[{{u}_{c}}/2({{u}_{c}}{{u}_{s}})/4]=({{u}_{c}}+{{u}_{s}})/2</math>  
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  {{EquationRef(  +  {{EquationRef(12)}} 
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The centerline velocity can be written as:  The centerline velocity can be written as:  
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<math>{{u}_{c}}=2{{u}_{m}}{{u}_{s}}</math>  <math>{{u}_{c}}=2{{u}_{m}}{{u}_{s}}</math>  
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Using the above relation, we get the following velocity profile in a microchannel:  Using the above relation, we get the following velocity profile in a microchannel:  
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<math>u=2({{u}_{m}}{{u}_{s}})(1{{r}^{+2}})+{{u}_{s}}</math>  <math>u=2({{u}_{m}}{{u}_{s}})(1{{r}^{+2}})+{{u}_{s}}</math>  
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where r+=r/ro. If the slip velocity is zero, the velocity distribution reduces to the Poiseuille flow distribution for conventional channels.  where r+=r/ro. If the slip velocity is zero, the velocity distribution reduces to the Poiseuille flow distribution for conventional channels.  
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<math>{{u}_{s}}=\frac{2F}{F}\lambda {{\left. \frac{du}{dr} \right}_{r={{r}_{o}}}}</math>  <math>{{u}_{s}}=\frac{2F}{F}\lambda {{\left. \frac{du}{dr} \right}_{r={{r}_{o}}}}</math>  
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For most applications, F has values near unity. Therefore,  For most applications, F has values near unity. Therefore,  
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<math>{{u}_{s}}=\lambda {{\left. \frac{du}{dr} \right}_{r={{r}_{o}}}}</math>  <math>{{u}_{s}}=\lambda {{\left. \frac{du}{dr} \right}_{r={{r}_{o}}}}</math>  
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<math>{{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}</math>  <math>{{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}</math>  
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The Knudsen number, <math>\text{Kn}=\lambda /D</math>, is introduced in the above relation to obtain the following equation for the slip velocity:  The Knudsen number, <math>\text{Kn}=\lambda /D</math>, is introduced in the above relation to obtain the following equation for the slip velocity:  
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<math>\frac{{{u}_{s}}}{{{u}_{m}}}=\frac{8\text{Kn}}{1+8\text{Kn}}</math>  <math>\frac{{{u}_{s}}}{{{u}_{m}}}=\frac{8\text{Kn}}{1+8\text{Kn}}</math>  
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Substituting the above into eq. (5.209), the nondimensional expression for the velocity profile is obtained:  Substituting the above into eq. (5.209), the nondimensional expression for the velocity profile is obtained:  
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<math>\frac{u}{{{u}_{m}}}=\frac{2(1{{r}^{+2}})+8\text{Kn}}{1+8\text{Kn}}</math>  <math>\frac{u}{{{u}_{m}}}=\frac{2(1{{r}^{+2}})+8\text{Kn}}{1+8\text{Kn}}</math>  
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[[Image:Fig5.23.pngthumb400 pxalt=Nondimensional, fully developed velocity profile in a microchannel as a function of Knudsen number Nondimensional, fully developed velocity profile in a microchannel as a function of Knudsen number.]]  [[Image:Fig5.23.pngthumb400 pxalt=Nondimensional, fully developed velocity profile in a microchannel as a function of Knudsen number Nondimensional, fully developed velocity profile in a microchannel as a function of Knudsen number.]]  
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<math>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}}</math>  <math>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}}</math>  
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The variation of temperature in the rdirection at the center is zero due to axisymmetric conditions:  The variation of temperature in the rdirection at the center is zero due to axisymmetric conditions:  
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<math>{{\left. \frac{\partial T}{\partial r} \right}_{r=0}}=0</math>  <math>{{\left. \frac{\partial T}{\partial r} \right}_{r=0}}=0</math>  
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Also, the temperature jump condition for the fluid at the wall is written as follows:  Also, the temperature jump condition for the fluid at the wall is written as follows:  
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<math>{{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}}}}</math>  <math>{{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}}}}</math>  
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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 crosssection, which are presented below.  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 crosssection, which are presented below. 
Revision as of 03:46, 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:

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

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 xdirection can be written in cylindrical coordinates as:

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

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

Therefore, the momentum equation reduces to:

The above equation is integrated twice to yield:
u = C_{2}lnr + C_{3} − C_{1}r^{2} 
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:

Rearranging and solving for C1 we have:

The velocity profile is obtained as follows:

The mean velocity in the tube, u_{m}, is now evaluated. The volumetric flow rate is:

therefore,
u_{m} = 2[u_{c} / 2 − (u_{c} − u_{s}) / 4] = (u_{c} + u_{s}) / 2 
The centerline velocity can be written as:
u_{c} = 2u_{m} − u_{s} 
Using the above relation, we get the following velocity profile in a microchannel:
u = 2(u_{m} − u_{s})(1 − r^{ + 2}) + u_{s} 
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)]:

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

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

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

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

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:

The variation of temperature in the rdirection at the center is zero due to axisymmetric conditions:

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

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 crosssection, which are presented below.
Constant Heat Flux at the Wall
The constant heat flux at the wall is described by:

where 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:

Substituting eq. (5.214) into eq. (5.215) and nondimensionalizing the resultant equation yield:

where

Since β_{1}
is unknown, three nondimensional boundary conditions are required for the second order ODE, eq. (5.220):

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

where

The local heat transfer coefficient, h, is:

The Nusselt number, based on is:

where the nondimensional mean temperature () and slip temperature at the wall () are given as follows:


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 crosssection 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 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 crosssection tube subjected to constant wall temperature. They assumed that the fluid temperature at the wall does not change along the tube length, i.e. dT_{s} / 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. ) is presented (Bahrami, 2009). The nondimensional temperature profile is defined as:

where T_{c} is the fluid temperature at the centerline. Substituting eq. (5.214) into eq. (5.215), neglecting the viscous dissipation, and nondimensionalizing the resultant equation give us:

where:


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

The Nusselt number based on θ is:

where is the nondimensional mean temperature, and is the slip temperature at the wall. The relation between β_{1}
and
β_{2}
can be obtained by taking the derivative of slip temperature boundary conditions along the xdirection, and the results are given bellow:
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 forthorder RungKutta procedure is employed to solve eq. (5.230).
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.
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.
Table Fully developed Nusselt numbers for microtubes with Br = 0, and constant temperature at the wall
Pr=0.6  Pr=0.65  Pr=0.7  Pr=0.75  Pr=0.8  Pr=0.85  Pr=0.9  Pr=0.95  Pr=1.0  
Kn=0  3.657  3.657  3.657  3.657  3.657  3.657  3.657  3.657  3.657 
Kn=0.02  3.432  3.462  3.488  3.512  3.532  3.550  3.566  3.580  3.593 
Kn=0.04  3.191  3.245  3.292  3.334  3.372  3.405  3.436  3.463  3.488 
Kn=0.06  2.952  3.024  3.088  3.145  3.196  3.243  3.285  3.323  3.359 
Kn=0.08  2.728  2.812  2.887  2.955  3.017  3.074  3.126  3.173  3.217 
Kn=0.1  2.522  2.614  2.697  2.773  2.843  2.907  2.967  3.022  3.073 
Kn=0.12  2.337  2.433  2.521  2.603  2.678  2.747  2.812  2.872  2.929 
Table Fully developed Nusselt numbers for microchannel with Brq = 0, and constant heat flux at the wall
Pr=0.6  Pr=0.65  Pr=0.7  Pr=0.75  Pr=0.8  Pr=0.85  Pr=0.9  Pr=0.95  Pr=1.0  
Kn=0.00  4.364  4.364  4.364  4.364  4.364  4.364  4.364  4.364  4.364 
Kn=0.02  3.981  4.029  4.071  4.108  4.141  4.171  4.197  4.221  4.243 
Kn=0.04  3.599  3.678  3.749  3.812  3.870  3.922  3.969  4.013  4.053 
Kn=0.06  3.252  3.350  3.439  3.519  3.593  3.661  3.723  3.781  3.834 
Kn=0.08  2.949  3.057  3.156  3.247  3.331  3.409  3.481  3.549  3.612 
Kn=0.10  2.687  2.799  2.904  3.000  3.091  3.175  3.254  3.327  3.397 
Kn=0.12  2.461  2.575  2.681  2.781  2.874  2.962  3.044  3.122  3.195 
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.