# Fully Developed Flow with Developing Temperature Profile

(Difference between revisions)
 Revision as of 07:01, 23 July 2010 (view source)← Older edit Current revision as of 09:05, 27 July 2010 (view source) (One intermediate revision not shown) Line 3: Line 3: ==Uniform Temperature== ==Uniform Temperature== - The energy equation assuming fully developed flow, including viscous dissipation and neglecting axial conduction, is + The energy equation assuming fully developed flow, including viscous dissipation and neglecting axial conductionFaghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO., is
$u\frac{\partial T}{\partial x}=\frac{\alpha }{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right)+\frac{\nu }{{{c}_{p}}}{{\left( \frac{\partial u}{\partial r} \right)}^{2}}$
$u\frac{\partial T}{\partial x}=\frac{\alpha }{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right)+\frac{\nu }{{{c}_{p}}}{{\left( \frac{\partial u}{\partial r} \right)}^{2}}$
Line 82: Line 82: |{{EquationRef|(9)}} |{{EquationRef|(9)}} |} |} - where ''ΔT'' is the difference between the temperature of the fluid at the wall, ''T''s'''', and at the tube entrance, ''T0'', i.e. $\Delta T={{T}_{0}}-{{T}_{s}}$. + where ''ΔT'' is the difference between the temperature of the fluid at the wall, ''Ts'', and at the tube entrance, ''T0'', i.e. $\Delta T={{T}_{0}}-{{T}_{s}}$. ==Uniform Heat Flux== ==Uniform Heat Flux== Line 104: Line 104: |{{EquationRef|(11)}} |{{EquationRef|(11)}} |} |} - - [[Image:Fig5.28.png|thumb|400 px|alt=The effect of viscous dissipation on Nusselt number for constant wall temperature. |The effect of viscous dissipation on Nusselt number for constant wall temperature .]] - - [[Image:Fig5.29.png|thumb|400 px|alt=The effect of viscous dissipation on Nusselt number for uniform wall heat flux |The effect of viscous dissipation on Nusselt number for uniform wall heat flux .]] where the  centerline symmetric and uniform inlet temperature conditions are the same as eqs. (7) and (8), respectively. However, the boundary condition at the wall is given by $\partial \theta /\partial {{r}^{+}}=1$. where the  centerline symmetric and uniform inlet temperature conditions are the same as eqs. (7) and (8), respectively. However, the boundary condition at the wall is given by $\partial \theta /\partial {{r}^{+}}=1$. An integral transform technique based on separation of variables was used by Tunc and Bayazitoglu to solve this problem.  An appropriate integral transform pair was developed.  Under the transformation, the variable ''x+'' was eliminated from the partial differential governing equation, which transformed the governing equation into an ordinary differential equation. An integral transform technique based on separation of variables was used by Tunc and Bayazitoglu to solve this problem.  An appropriate integral transform pair was developed.  Under the transformation, the variable ''x+'' was eliminated from the partial differential governing equation, which transformed the governing equation into an ordinary differential equation. - + [[Image:Fig5.22.png|thumb|400 px|alt=Configuration for the slip boundary condition in a microchannel. |Configuration for the slip boundary condition in a microchannel.]] - The effect of viscous heating is presented in the first figure for the constant wall temperature case, where Kn = 0.04 and Pr = 0.7. The inclusion of viscous dissipation causes an increase in Nu.  The Nusselt number first reaches the fully developed condition as if there was no viscous dissipation, and then makes a jump to its final value for a given Br. + - + - The second figure to the right shows the effect of viscous heating on the Nusselt number for a uniform wall heat flux.  Since the definition of the Brinkman number is different for the uniform wall heat flux boundary condition case, a positive Br means that the heat is being transferred to the fluid from the wall, as opposed to the uniform wall temperature case. For constant heat flux at the wall, the Nusselt number decreases as Br (> 0) increases. + Jeong and Jeong Jeong, H.E., Jeong, J.T., 2006, “Extended Graetz Problem including Streamwise Conduction and Viscous Dissipation in Microchannel,” Int. J. Heat Mass Transfer, Vol.  49, pp. 2151-2157. extended the analysis for application to microchannels of rectangular cross-section, including axial conduction and viscous dissipation. Jeong and Jeong Jeong, H.E., Jeong, J.T., 2006, “Extended Graetz Problem including Streamwise Conduction and Viscous Dissipation in Microchannel,” Int. J. Heat Mass Transfer, Vol.  49, pp. 2151-2157. extended the analysis for application to microchannels of rectangular cross-section, including axial conduction and viscous dissipation. - The configuration for developed flow with developing temperature profile for rectangular microchannels is similar to [https://www.thermalfluidscentral.org/e-encyclopedia/index.php/File:Fig5.22.png#filelinks| this picture]. The fluid temperature changes from the value ''T0'' at the entrance, to the value ''Ts'' on the walls.  The governing energy equation and boundary conditions, including axial conduction and viscous dissipation, for laminar flow are: + The configuration for developed flow with developing temperature profile for rectangular microchannels is similar to the figure on the right. The fluid temperature changes from the value ''T0'' at the entrance, to the value ''Ts'' on the walls.  The governing energy equation and boundary conditions, including axial conduction and viscous dissipation, for laminar flow are: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 132: Line 125: |- |- | width="100%" |
| width="100%" |
- $T={{T}_{0}}\text{ at }x=0$ + $T={{T}_{0}}\text{ at }x=0 |{{EquationRef|(13)}} |{{EquationRef|(13)}} Line 234: Line 227: [itex]{{C}_{2}}=\frac{2-{{F}_{t}}}{{{F}_{t}}}\frac{2\gamma }{\gamma +1}\frac{\text{Kn}}{\text{Pr}}$
${{C}_{2}}=\frac{2-{{F}_{t}}}{{{F}_{t}}}\frac{2\gamma }{\gamma +1}\frac{\text{Kn}}{\text{Pr}}$
- [[Image:Fig5.30.png|thumb|400 px|alt=Nusselt number distribution for a constant temperature boundary condition for rectangular microchannels. |Nusselt number distribution for a constant temperature boundary condition for rectangular microchannels.]] + The effects of the Knudsen number on the Nusselt number variation along a rectangular microchannel neglecting axial conduction and viscous dissipation are studied by Jeong and Jeong. For Kn = 0, the fully developed Nusselt number is approximately 7.54, which is the result for a pipe of conventional size (classic Graetz problem). The Nusselt number decreases as Kn increases due to the temperature jump at the wall. - + - [[Image:Fig5.31.png|thumb|400 px|alt=Nusselt number distribution for a constant heat flux boundary condition for rectangular microchannels. |Nusselt number distribution for a constant heat flux boundary condition for rectangular microchannels.]] + - + - The effects of the Knudsen number on the Nusselt number variation along a rectangular microchannel neglecting axial conduction and viscous dissipation are shown in figure to the right. For Kn = 0, the fully developed Nusselt number is approximately 7.54, which is the result for a pipe of conventional size (classic Graetz problem). The Nusselt number decreases as Kn increases due to the temperature jump at the wall. + - The effect of the Knudsen number on the Nusselt number distribution in a rectangular microchannel with constant wall heat flux is presented in the second figure. + When the channel is subjected to a constant wall temperature, as ${{x}^{+}}\to \infty$, the Nusselt number for $\text{Br}\ne 0$ is independent of Br and different from that for $\text{Br}=0$. The thermally fully developed Nusselt number was obtained from the fully developed temperature profile for constant wall temperature and heat flux by . When the channel is subjected to a constant wall temperature, as ${{x}^{+}}\to \infty$, the Nusselt number for $\text{Br}\ne 0$ is independent of Br and different from that for $\text{Br}=0$. The thermally fully developed Nusselt number was obtained from the fully developed temperature profile for constant wall temperature and heat flux by .

## Current revision as of 09:05, 27 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 [1][2] 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[3], is

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

The boundary and inlet conditions for constant wall temperature are:

 $\begin{matrix} {} & {} \\\end{matrix}T={{T}_{s}}\text{ at }r={{r}_{o}}$ (1)
 $\frac{\partial T}{\partial r}=0\text{ at }r=0$ (2)
 $\begin{matrix} {} & {} \\\end{matrix}T={{T}_{0}}\text{ at }x=0$ (3)

The fully developed velocity profile with slip boundary condition is used. The slip boundary condition 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+):

 $\theta =\frac{T-{{T}_{s}}}{{{T}_{0}}-{{T}_{s}}},\text{ }{{r}^{+}}=\frac{r}{{{r}_{o}}},\text{ }{{x}^{+}}=\frac{x}{L},\text{ }{{u}^{+}}=\frac{u}{{{u}_{m}}}$ (4)

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

 $\frac{\text{Gz}\left( 1-{{r}^{+}}^{2}+4\text{Kn} \right)}{2\left( 1+8\text{Kn} \right)}\frac{\partial \theta }{\partial {{x}^{+}}}=\frac{1}{{{r}^{+}}}\frac{\partial }{\partial {{r}^{+}}}\left( {{r}^{+}}\frac{\partial \theta }{\partial {{r}^{+}}} \right)+\frac{16\text{Br}}{{{\left( 1+8\text{Kn} \right)}^{2}}}{{r}^{+}}^{2}$ (5)
 $\begin{matrix} {} & {} \\\end{matrix}\theta =0\text{ at }{{r}^{+}}=1$ (6)
 $\frac{\partial \theta }{\partial {{r}^{+}}}=0\text{ at }{{r}^{+}}=0$ (7)
 $\begin{matrix} {} & {} \\\end{matrix}\theta =1\text{ at }{{x}^{+}}=0$ (8)

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

 $\text{Gz}=\frac{\text{RePr}D}{L}\text{ and Br}=\frac{\mu u_{m}^{2}}{k\Delta T}$ (9)

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.

## Uniform Heat Flux

For the case of constant heat flux at the wall, the following non-dimensional variables are used:

 $\theta =\frac{T-{{T}_{0}}}{{{{{q}''}}_{w}}{{r}_{0}}/k}\text{ and Br}=\frac{\mu u_{m}^{2}}{{{{{q}''}}_{w}}D}$ (10)

Upon use of the above non-dimensional variables, the non-dimensional energy equation for constant wall heat flux can be obtained:

 $\frac{\text{Gz}\left( 1-{{r}^{+2}}+4\text{Kn} \right)}{2\left( 1+8\text{Kn} \right)}\frac{\partial \theta }{\partial {{x}^{+}}}=\frac{1}{{{r}^{+}}}\frac{\partial }{\partial {{r}^{+}}}\left( {{r}^{+}}\frac{\partial \theta }{\partial {{r}^{+}}} \right)+\frac{16\text{Br}}{{{\left( 1+8\text{Kn} \right)}^{2}}}{{r}^{+2}}$ (11)

where the centerline symmetric and uniform inlet temperature conditions are the same as eqs. (7) and (8), respectively. However, the boundary condition at the wall is given by $\partial \theta /\partial {{r}^{+}}=1$.

An integral transform technique based on separation of variables was used by Tunc and Bayazitoglu [1] to solve this problem. An appropriate integral transform pair was developed. Under the transformation, the variable x+ was eliminated from the partial differential governing equation, which transformed the governing equation into an ordinary differential equation.

Configuration for the slip boundary condition in a microchannel.

Jeong and Jeong [4] extended the analysis for application to microchannels of rectangular cross-section, including axial conduction and viscous dissipation.

The configuration for developed flow with developing temperature profile for rectangular microchannels is similar to the figure on the right. The fluid temperature changes from the value T0 at the entrance, to the value Ts on the walls. The governing energy equation and boundary conditions, including axial conduction and viscous dissipation, for laminar flow are:

 $\rho {{c}_{p}}u\frac{\partial T}{\partial x}=k\left( \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}} \right)+\mu {{\left( \frac{\partial u}{\partial y} \right)}^{2}}$ (12)
 T = T0 at x = 0 (13)
 $T-{{T}_{w}}=\mp \frac{2-{{F}_{t}}}{{{F}_{t}}}\frac{2\gamma }{\gamma +1}\frac{\lambda }{Pr}\frac{\partial T}{\partial y}\text{ at }y=\pm H$ (14)
 $\frac{\partial T}{\partial y}=0\text{ at }y=0$ (15)

where H is half the microchannel height, and the wall length in the x-direction is L. The fully developed velocity profile in the rectangular microchannel is:

 $u\left( y \right)=-\frac{{{H}^{2}}}{2\mu }\frac{dP}{dx}\left[ 1-{{\left( \frac{y}{H} \right)}^{2}}+8\frac{2-F}{F}\text{Kn} \right]$ (16)

which satisfies the slip boundary condition:

 ${{u}_{s}}=\mp \frac{2-F}{F}\lambda \left( \frac{\partial u}{\partial y} \right)\text{ at }y=\pm H$ (17)

Defining the following dimensionless variables:

 $\theta =\frac{T-{{T}_{w}}}{{{T}_{0}}-{{T}_{w}}}\text{, }{{x}^{+}}=\frac{x}{\text{RePr}H}\text{, }{{y}^{+}}=\frac{y}{H}\text{, Br}=\frac{\mu u_{m}^{2}}{k\left( {{T}_{0}}-{{T}_{w}} \right)},\text{ Kn}=\frac{\lambda }{{{D}_{h}}}$ (18)

Equations (16) and (20) are respectively non-dimensionalized as:

 $\frac{1}{4}{{u}^{+}}\frac{\partial \theta }{\partial {{x}^{+}}}=\frac{1}{P{{e}^{2}}}\frac{{{\partial }^{2}}\theta }{\partial {{x}^{+}}^{2}}+\frac{{{\partial }^{2}}\theta }{\partial {{y}^{+}}^{2}}+\text{Br}{{\left( \frac{\partial {{u}^{+}}}{\partial {{y}^{+}}} \right)}^{2}}$ (19)

and

 ${{u}^{+}}=\frac{u}{{{u}_{m}}}=\frac{3}{2}\frac{\left( 1-{{y}^{+}}^{2}+8\frac{2-F}{F}\text{Kn} \right)}{{{C}_{1}}}$ (20)

where

 ${{C}_{1}}=1+12\frac{2-F}{F}\text{Kn}$ (21)

The non-dimensional boundary conditions are:

 $\begin{matrix} {} & {} \\\end{matrix}\theta =1\text{ at }{{x}^{+}}=0$ (22)
 $\theta =-4{{C}_{2}}\frac{\partial \theta }{\partial y}\text{ at }{{y}^{+}}=1$ (23)
 $\frac{\partial \theta }{\partial y}=0\text{ at }{{y}^{+}}=0$ (24)

where

${{C}_{2}}=\frac{2-{{F}_{t}}}{{{F}_{t}}}\frac{2\gamma }{\gamma +1}\frac{\text{Kn}}{\text{Pr}}$

The effects of the Knudsen number on the Nusselt number variation along a rectangular microchannel neglecting axial conduction and viscous dissipation are studied by Jeong and Jeong[4]. For Kn = 0, the fully developed Nusselt number is approximately 7.54, which is the result for a pipe of conventional size (classic Graetz problem). The Nusselt number decreases as Kn increases due to the temperature jump at the wall.

When the channel is subjected to a constant wall temperature, as ${{x}^{+}}\to \infty$, the Nusselt number for $\text{Br}\ne 0$ is independent of Br and different from that for Br = 0. The thermally fully developed Nusselt number was obtained from the fully developed temperature profile for constant wall temperature and heat flux by [4]. For constant wall temperature the Nusselt number as ${{x}^{+}}\to \infty$ ($\text{Br}\ne 0$) is

 $\text{N}{{\text{u}}_{\infty }}=\frac{140{{C}_{1}}}{1+7{{C}_{1}}+140{{C}_{1}}{{C}_{2}}}$ (25)

And for constant heat flux

 $\text{N}{{\text{u}}_{\infty }}=\frac{420C_{1}^{4}}{C_{1}^{2}\left( 35C_{1}^{2}+14{{C}_{1}}+2+420C_{1}^{2}{{C}_{2}} \right)+Br\left( 42C_{1}^{2}+33{{C}_{1}}+6 \right)}$ (26)

where

${{C}_{1}}=1+12\left( \frac{2-F}{F} \right)\text{Kn}$
${{C}_{2}}=\frac{2-{{F}_{t}}}{{{F}_{t}}}\frac{2\gamma }{\gamma +1}\frac{\text{Kn}}{\text{Pr}}$.

Unless Br is a large negative number, Nu, is always positive.

## References

1. 1.0 1.1 Tunc, G., and Bayazitoglu, Y., 2001, “Heat Transfer in Microtubes with Viscous Dissipation,” Int. J. Heat Mass Transfer, Vol. 44, pp. 2395-2403.
2. Tunc, G., and Bayazitoglu, Y., 2002, “Heat Transfer in Rectangular Microchannels,” Int. J. Heat Mass Transfer, Vol. 45, pp. 765-773.
3. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
4. 4.0 4.1 4.2 Jeong, H.E., Jeong, J.T., 2006, “Extended Graetz Problem including Streamwise Conduction and Viscous Dissipation in Microchannel,” Int. J. Heat Mass Transfer, Vol. 49, pp. 2151-2157.