# Heat Transfer in Fully-Developed Internal Turbulent Flow

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 Revision as of 02:58, 8 July 2010 (view source) (Created page with 'Heat transfer in fully-developed turbulent flow in a circular tube subject to constant heat flux (${{{q}''}_{w}}=$ const) will be considered in this subsection (Oosthu…')← Older edit Current revision as of 20:21, 23 July 2010 (view source) (8 intermediate revisions not shown) Line 1: Line 1: - Heat transfer in fully-developed turbulent flow in a circular tube subject to constant heat flux (${{{q}''}_{w}}=$ const) will be considered in this subsection (Oosthuizen and Naylor, 1999). When the turbulent flow in the tube is fully developed, we have $\bar{v}=0$ and the energy eq. (5.268) becomes + Heat transfer in fully-developed turbulent flow in a circular tube subject to constant heat flux (${{{q}''}_{w}}=$ const) will be considered in this article Oosthuizen, P.H., and Naylor, D., 1999, Introduction to Convective Heat Transfer Analysis, WCB/McGraw-Hill, New York.Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.. When the turbulent flow in the tube is fully developed, we have $\bar{v}=0$ and the energy equation becomes {| class="wikitable" border="0" {| class="wikitable" border="0" Line 17: Line 17: |{{EquationRef|(2)}} |{{EquationRef|(2)}} |} |} - where ${{\bar{T}}_{c}}$ is the time-averaged temperature at the centerline of the tube, and Tw is the wall temperature. Thus, + where ${{\bar{T}}_{c}}$ is the time-averaged temperature at the centerline of the tube, and ''Tw'' is the wall temperature. Thus, $({{T}_{w}}-\bar{T})/({{T}_{w}}-{{\bar{T}}_{c}})$ is a function of ''r'' only, i.e., - $({{T}_{w}}-\bar{T})/({{T}_{w}}-{{\bar{T}}_{c}})$ + - is a function of r only, i.e., + {| class="wikitable" border="0" {| class="wikitable" border="0" Line 28: Line 26: |{{EquationRef|(3)}} |{{EquationRef|(3)}} |} |} - where f is independent from x. Differentiating (5.295) yields + where ''f'' is independent from ''x''. Differentiating eq. (2) yields {| class="wikitable" border="0" {| class="wikitable" border="0" Line 46: Line 44: |{{EquationRef|(5)}} |{{EquationRef|(5)}} |} |} - Substituting eq. (5.296) into eq. (5.298), one obtains: + Substituting eq. (3) into eq. (5), one obtains: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 55: Line 53: |{{EquationRef|(6)}} |{{EquationRef|(6)}} |} |} - Since the heat flux is constant, ${{{q}''}_{w}}=$ const, it follows that + Since the heat flux is constant, ${{{q}''}_{w}}=$ const, it follows that $({{T}_{w}}-{{\bar{T}}_{c}})= Const$, i.e., - $({{T}_{w}}-{{\bar{T}}_{c}})=$ + - const., i.e., + {| class="wikitable" border="0" {| class="wikitable" border="0" Line 66: Line 62: |{{EquationRef|(7)}} |{{EquationRef|(7)}} |} |} - Therefore, eq. (5.297) becomes: + Therefore, eq. (4) becomes: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 93: Line 89: |{{EquationRef|(10)}} |{{EquationRef|(10)}} |} |} - Since ${{{q}''}_{w}}=$ const, it follows from eq. (5.302) that + Since ${{{q}''}_{w}}=$ const, it follows from eq. (9) that $({{T}_{w}}-{{\bar{T}}_{m}})= Const$, i.e., - $({{T}_{w}}-{{\bar{T}}_{m}})=$ + - const., i.e., + {| class="wikitable" border="0" {| class="wikitable" border="0" Line 104: Line 98: |{{EquationRef|(11)}} |{{EquationRef|(11)}} |} |} - Combining eqs. (5.300), (5.301) and (5.304), the following relationships are obtained: + Combining eqs. (7), (8) and (11), the following relationships are obtained: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 113: Line 107: |{{EquationRef|(12)}} |{{EquationRef|(12)}} |} |} - The time-averaged mean temperature, + The time-averaged mean temperature, ${{\bar{T}}_{m}}$, changes with ''x'' as the result of heat transfer from the tube wall. The rate of mean temperature change can be obtained as follows: - ${{\bar{T}}_{m}}$ + - , changes with x as the result of heat transfer from the tube wall. By following the same procedure as that in Example 5.2, the rate of mean temperature change can be obtained as follows: + {| class="wikitable" border="0" {| class="wikitable" border="0" Line 124: Line 116: |{{EquationRef|(13)}} |{{EquationRef|(13)}} |} |} - Substituting eq. (5.305) into eq. (5.294), the energy equation becomes: + Substituting eq. (12) into eq. (1), the energy equation becomes: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 133: Line 125: |{{EquationRef|(14)}} |{{EquationRef|(14)}} |} |} - where $y={{r}_{0}}-r$ is the distance measured from the tube wall. Equation (5.307) is subject to the following two boundary conditions: + where $y={{r}_{0}}-r$ is the distance measured from the tube wall. Equation (14) is subject to the following two boundary conditions: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 150: Line 142: |{{EquationRef|(16)}} |{{EquationRef|(16)}} |} |} - Integrating eq. (5.307) in the interval of (r0, r) and considering eq. (5.308), we have: + Integrating eq. (14) in the interval of (''r0'', ''r'') and considering eq. (15), we have: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 177: Line 169: |{{EquationRef|(19)}} |{{EquationRef|(19)}} |} |} - Integrating eq. (5.311) in the interval of (0, y) and considering eq. (5.309), one obtains: + Integrating eq. (18) in the interval of (0, ''y'') and considering eq. (16), one obtains: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 186: Line 178: |{{EquationRef|(20)}} |{{EquationRef|(20)}} |} |} - If the profiles of axial velocity and the thermal eddy diffusivity are known, eq. (5.313) can be used to obtain the correlation for internal forced convection heat transfer. With the exception of the very thin viscous sublayer, the velocity profile in the most part of the tube is fairly flat. Therefore, it is assumed that the time-averaged velocity, $\bar{u}$, in eq. (5.312) can be replaced by ${{\bar{u}}_{m}}$, and I(y) becomes: + If the profiles of axial velocity and the thermal eddy diffusivity are known, eq. (20) can be used to obtain the correlation for internal forced convection heat transfer. With the exception of the very thin viscous sublayer, the velocity profile in the most part of the tube is fairly flat. Therefore, it is assumed that the time-averaged velocity, $\bar{u}$, in eq. (19) can be replaced by ${{\bar{u}}_{m}}$, and ''I''(''y'') becomes: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 195: Line 187: |{{EquationRef|(21)}} |{{EquationRef|(21)}} |} |} - Substituting eqs. (5.314) and (5.306) into eq. (5.313) yields: + Substituting eqs. (21) and (13) into eq. (20) yields: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 213: Line 205: |{{EquationRef|(23)}} |{{EquationRef|(23)}} |} |} - where y+ is defined in eq. (5.277). + where ''y''+ is defined as ${{y}^{+}}=\frac{y{{u}_{\tau }}}{\nu }$. - To consider heat transfer in an internal turbulent flow, the entire turbulent boundary layer is divided into three regions: (1) inner region (${{y}^{+}}<5$), (2) buffer region ($5\le {{y}^{+}}\le 30$), and (3) outer region (${{y}^{+}}>30$). In the inner region + - ${{\varepsilon }_{M}}={{\varepsilon }_{H}}=0$ + To consider heat transfer in an internal turbulent flow, the entire turbulent boundary layer is divided into three regions: (1) inner region (${{y}^{+}}<5$), (2) buffer region ($5\le {{y}^{+}}\le 30$), and (3) outer region (${{y}^{+}}>30$). In the inner region ${{\varepsilon }_{M}}={{\varepsilon }_{H}}=0$ and eq. (23) becomes - and eq. (5.316) becomes + {| class="wikitable" border="0" {| class="wikitable" border="0" Line 225: Line 216: |{{EquationRef|(24)}} |{{EquationRef|(24)}} |} |} - Since the inner region is very thin, + Since the inner region is very thin, $y/{{r}_{0}}\ll 1$ and $1-y/{{r}_{0}}$ is effectively equal to 1. Therefore, the temperature profile in the inner region becomes: - $y/{{r}_{0}}\ll 1$ + - and + - $1-y/{{r}_{0}}$ + - is effectively equal to 1. Therefore, the temperature profile in the inner region becomes: + {| class="wikitable" border="0" {| class="wikitable" border="0" Line 238: Line 225: |{{EquationRef|(25)}} |{{EquationRef|(25)}} |} |} - The temperature at the boundary between the inner and buffer regions (${{y}^{+}}=5$), + The temperature at the boundary between the inner and buffer regions (${{y}^{+}}=5$), ${{\bar{T}}_{s}}$, can be obtained from eq. (25) as - ${{\bar{T}}_{s}}$ + - , can be obtained from eq. (5.318) as + {| class="wikitable" border="0" {| class="wikitable" border="0" Line 258: Line 243: |{{EquationRef|(27)}} |{{EquationRef|(27)}} |} |} - Substituting eq. (5.320) into eq. (5.316) and assuming the turbulent Prandtl number ${{\Pr }^{t}}=1$, the following expression is obtained: + Substituting eq. (27) into eq. (23) and assuming the turbulent Prandtl number ${{\Pr }^{t}}=1$, the following expression is obtained: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 267: Line 252: |{{EquationRef|(28)}} |{{EquationRef|(28)}} |} |} - Since the buffer region is also very thin, + - $1-y/{{r}_{0}}$ + Since the buffer region is also very thin, $1-y/{{r}_{0}}$ in eq. (28) is effectively equal to 1. Defining ${{T}^{+}}={(\bar{T}-{{T}_{w}})}/{\left( -\frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}}\sqrt{\frac{\rho }{{{\tau }_{w}}}} \right)}\;$ and Integrating eq. (28) yields - in eq. (5.321) is effectively equal to 1. Defining + - ${{T}^{+}}={(\bar{T}-{{T}_{w}})}/{\left( -\frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}}\sqrt{\frac{\rho }{{{\tau }_{w}}}} \right)}\;$ + - and Integrating eq. (5.321) yields + {| class="wikitable" border="0" {| class="wikitable" border="0" Line 289: Line 271: |{{EquationRef|(30)}} |{{EquationRef|(30)}} |} |} - The temperature at the top of the buffer region where ${{y}^{+}}=30$, + The temperature at the top of the buffer region where ${{y}^{+}}=30$, ${{\bar{T}}_{b}}$, becomes - ${{\bar{T}}_{b}}$ + - , becomes + {| class="wikitable" border="0" {| class="wikitable" border="0" Line 300: Line 280: |{{EquationRef|(31)}} |{{EquationRef|(31)}} |} |} - For the outer region where ${{\varepsilon }_{M}}\gg \nu \text{ and }{{\varepsilon }_{H}}\gg \alpha$,  eq. (5.316) becomes + For the outer region where ${{\varepsilon }_{M}}\gg \nu \text{ and }{{\varepsilon }_{H}}\gg \alpha$,  eq. (23) becomes {| class="wikitable" border="0" {| class="wikitable" border="0" Line 310: Line 290: |} |} where the turbulent Prandtl number is assumed to be equal to 1. where the turbulent Prandtl number is assumed to be equal to 1. - It is assumed that the Nikuradse equation (5.276) is valid in the outer region and the velocity gradient in this region becomes: + + It is assumed that the Nikuradse equation is valid in the outer region and the velocity gradient in this region becomes: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 319: Line 300: |{{EquationRef|(33)}} |{{EquationRef|(33)}} |} |} - The expression of apparent shear stress in this region, eq. (5.280) , can be non-dimensionalized using eqs. (5.277) and (5.278) as: + The expression of apparent shear stress in this region, can be non-dimensionalized as: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 328: Line 309: |{{EquationRef|(34)}} |{{EquationRef|(34)}} |} |} - Substituting eqs. (5.275) and (5.326) into eq. (5.327), the eddy diffusivity in the outer region is obtained as: + Substituting +
${{\tau }_{app}}={{\tau }_{w}}\cdot \left( \frac{r}{{{r}_{o}}} \right)={{\tau }_{w}}\cdot \left( 1-\frac{y}{{{r}_{o}}} \right)$
+ + and eq. (33) into eq. (34), the eddy diffusivity in the outer region is obtained as: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 337: Line 321: |{{EquationRef|(35)}} |{{EquationRef|(35)}} |} |} - Substituting eq. (5.328) into eq. (5.325), the temperature distribution in this region becomes: + Substituting eq. (35) into eq. (32), the temperature distribution in this region becomes: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 355: Line 339: |{{EquationRef|(37)}} |{{EquationRef|(37)}} |} |} - The temperature at the center of the tube, ${{\bar{T}}_{c}}$, can be obtained by letting + The temperature at the center of the tube, ${{\bar{T}}_{c}}$, can be obtained by letting ${{y}^{+}}=y_{c}^{+}$ in eq. (36), i.e. - ${{y}^{+}}=y_{c}^{+}$ + - in eq. (5.329), i.e. + - + {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 366: Line 348: |{{EquationRef|(38)}} |{{EquationRef|(38)}} |} |} - The overall temperature change from the wall to the center of the tube can be obtained by adding eqs. (5.319), (5.324) and (5.331): + The overall temperature change from the wall to the center of the tube can be obtained by adding eqs. (26), (31) and (38): {| class="wikitable" border="0" {| class="wikitable" border="0" Line 375: Line 357: |{{EquationRef|(39)}} |{{EquationRef|(39)}} |} |} - It follows from the definition of friction factor, eq. (5.282), that + It follows from the definition of friction factor, ${{c}_{f}}=\frac{{{\tau }_{w}}}{\rho \bar{u}_{m}^{2}/2}$, that {| class="wikitable" border="0" {| class="wikitable" border="0" Line 384: Line 366: |{{EquationRef|(40)}} |{{EquationRef|(40)}} |} |} - Substituting eq. (5.333) into eq. (5.332) and considering the definition of Reynolds number, ${{\operatorname{Re}}_{D}}={{\bar{u}}_{m}}D/\nu$, eq. (5.332) becomes: + Substituting eq. (40) into eq. (39) and considering the definition of Reynolds number, ${{\operatorname{Re}}_{D}}={{\bar{u}}_{m}}D/\nu$, eq. (39) becomes: - + + {| class="wikitable" border="0" + |- + | width="100%" |
${{T}_{w}}-{{\bar{T}}_{c}}=\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}{{{\bar{u}}}_{m}}} \right)\sqrt{\frac{2}{{{c}_{f}}}}\left[ 2.5\ln \left( \frac{{{\operatorname{Re}}_{D}}}{60}\sqrt{\frac{{{c}_{f}}}{2}} \right)+5\ln (5\Pr +1)+5\Pr \right]$ ${{T}_{w}}-{{\bar{T}}_{c}}=\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}{{{\bar{u}}}_{m}}} \right)\sqrt{\frac{2}{{{c}_{f}}}}\left[ 2.5\ln \left( \frac{{{\operatorname{Re}}_{D}}}{60}\sqrt{\frac{{{c}_{f}}}{2}} \right)+5\ln (5\Pr +1)+5\Pr \right]$ - (5.334) +
- In order to obtain the heat transfer coefficient, $h={{{q}''}_{w}}/({{T}_{w}}-{{\bar{T}}_{m}})$, the temperature difference ${{T}_{w}}-{{\bar{T}}_{m}}$ must be obtained. If the velocity profile can be approximated by eq. (5.287), and the temperature and velocity can also be approximated by the one-seventh law, i.e., + |{{EquationRef|(41)}} + |} + + In order to obtain the heat transfer coefficient, $h={{{q}''}_{w}}/({{T}_{w}}-{{\bar{T}}_{m}})$, the temperature difference ${{T}_{w}}-{{\bar{T}}_{m}}$ must be obtained. If the velocity profile can be approximated by $\frac{{\bar{u}}}{{{{\bar{u}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}$, and the temperature and velocity can also be approximated by the one-seventh law, i.e., {| class="wikitable" border="0" {| class="wikitable" border="0" Line 395: Line 383: $\frac{{{T}_{w}}-\bar{T}}{{{T}_{w}}-{{{\bar{T}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}{{,}_{_{\text{ }}}}\frac{{\bar{u}}}{{{{\bar{u}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}$ $\frac{{{T}_{w}}-\bar{T}}{{{T}_{w}}-{{{\bar{T}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}{{,}_{_{\text{ }}}}\frac{{\bar{u}}}{{{{\bar{u}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}$ - |{{EquationRef|(41)}} + |{{EquationRef|(42)}} |} |} Line 405: Line 393: ${{T}_{w}}-{{\bar{T}}_{m}}=\frac{\int_{0}^{{{r}_{o}}}{\bar{u}({{T}_{w}}-\bar{T})2\pi rdr}}{\int_{0}^{{{r}_{o}}}{\bar{u}2\pi rdr}}=\frac{5}{6}({{T}_{w}}-{{\bar{T}}_{c}})$ ${{T}_{w}}-{{\bar{T}}_{m}}=\frac{\int_{0}^{{{r}_{o}}}{\bar{u}({{T}_{w}}-\bar{T})2\pi rdr}}{\int_{0}^{{{r}_{o}}}{\bar{u}2\pi rdr}}=\frac{5}{6}({{T}_{w}}-{{\bar{T}}_{c}})$ - |{{EquationRef|(42)}} + |{{EquationRef|(43)}} |} |} - Substituting eq. (5.334) into eq. (5.336) results in: + Substituting eq. (41) into eq. (43) results in: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 414: Line 402: ${{T}_{w}}-{{\bar{T}}_{m}}=\frac{5}{6}\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}{{{\bar{u}}}_{m}}} \right)\sqrt{\frac{2}{{{c}_{f}}}}\left[ 2.5\ln \left( \frac{{{\operatorname{Re}}_{D}}}{60}\sqrt{\frac{{{c}_{f}}}{2}} \right)+5\ln (5\Pr +1)+5\Pr \right]$ ${{T}_{w}}-{{\bar{T}}_{m}}=\frac{5}{6}\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}{{{\bar{u}}}_{m}}} \right)\sqrt{\frac{2}{{{c}_{f}}}}\left[ 2.5\ln \left( \frac{{{\operatorname{Re}}_{D}}}{60}\sqrt{\frac{{{c}_{f}}}{2}} \right)+5\ln (5\Pr +1)+5\Pr \right]$ - |{{EquationRef|(43)}} + |{{EquationRef|(44)}} |} |} which can be rearranged to the following empirical correlation which can be rearranged to the following empirical correlation Line 423: Line 411: $\text{N}{{\text{u}}_{D}}=\frac{{{\operatorname{Re}}_{D}}\Pr \sqrt{\frac{{{c}_{f}}}{2}}}{\frac{5}{6}\left[ 2.5\ln \left( \frac{{{\operatorname{Re}}_{D}}}{60}\sqrt{\frac{{{c}_{f}}}{2}} \right)+5\ln (5\Pr +1)+5\Pr \right]}$ $\text{N}{{\text{u}}_{D}}=\frac{{{\operatorname{Re}}_{D}}\Pr \sqrt{\frac{{{c}_{f}}}{2}}}{\frac{5}{6}\left[ 2.5\ln \left( \frac{{{\operatorname{Re}}_{D}}}{60}\sqrt{\frac{{{c}_{f}}}{2}} \right)+5\ln (5\Pr +1)+5\Pr \right]}$ - |{{EquationRef|(44)}} + |{{EquationRef|(45)}} |} |} - which can be used together with appropriate friction coefficient discussed in the previous subsection to obtain the Nusselt number. + which can be used together with appropriate friction coefficient to obtain the Nusselt number. + + ==References== + {{Reflist}}

## Current revision as of 20:21, 23 July 2010

Heat transfer in fully-developed turbulent flow in a circular tube subject to constant heat flux (q''w = const) will be considered in this article [1][2]. When the turbulent flow in the tube is fully developed, we have $\bar{v}=0$ and the energy equation becomes

 $\bar{u}\frac{\partial \bar{T}}{\partial x}=\frac{1}{r}\frac{\partial }{\partial r}\left[ r(\alpha +{{\varepsilon }_{H}})\frac{\partial \bar{T}}{\partial r} \right]$ (1)

After the turbulent flow is hydrodynamically and thermally fully developed, the time-averaged temperature profile is no longer a function of axial distance from the inlet, i.e.,

 $\frac{\partial }{\partial x}\left( \frac{{{T}_{w}}-\bar{T}}{{{T}_{w}}-{{{\bar{T}}}_{c}}} \right)=0$ (2)

where ${{\bar{T}}_{c}}$ is the time-averaged temperature at the centerline of the tube, and Tw is the wall temperature. Thus, $({{T}_{w}}-\bar{T})/({{T}_{w}}-{{\bar{T}}_{c}})$ is a function of r only, i.e.,

 $\frac{{{T}_{w}}-\bar{T}}{{{T}_{w}}-{{{\bar{T}}}_{c}}}=f\left( r \right)$ (3)

where f is independent from x. Differentiating eq. (2) yields

 $\frac{\partial \bar{T}}{\partial x}=\frac{d{{T}_{w}}}{dx}-\left( \frac{{{T}_{w}}-\bar{T}}{{{T}_{w}}-{{{\bar{T}}}_{c}}} \right)\left( \frac{d{{T}_{w}}}{dx}-\frac{d{{{\bar{T}}}_{c}}}{dx} \right)$ (4)

At the wall, the contribution of eddy diffusivity on the heat transfer is negligible, and the heat flux at the wall becomes

 ${{{q}''}_{w}}=k{{\left. \frac{\partial \bar{T}}{\partial r} \right|}_{r={{r}_{0}}}}$ (5)

Substituting eq. (3) into eq. (5), one obtains:

 ${{{q}''}_{w}}=-k({{T}_{w}}-{{\bar{T}}_{c}}){f}'({{r}_{0}})$ (6)

Since the heat flux is constant, q''w = const, it follows that $({{T}_{w}}-{{\bar{T}}_{c}})= Const$, i.e.,

 $\frac{d{{T}_{w}}}{dx}=\frac{d{{{\bar{T}}}_{c}}}{dx}$ (7)

Therefore, eq. (4) becomes:

 $\frac{\partial \bar{T}}{\partial x}=\frac{d{{T}_{w}}}{dx}$ (8)

For fully developed flow, the local heat transfer coefficient is:

 ${{h}_{x}}=\frac{{{{{q}''}}_{w}}}{{{T}_{w}}-{{{\bar{T}}}_{m}}}=\text{const}$ (9)

where ${{\bar{T}}_{m}}$ is the time-averaged mean temperature defined as:

 ${{\bar{T}}_{m}}=\frac{2}{r_{0}^{2}}\int_{0}^{{{r}_{0}}}{\bar{u}\bar{T}rdr}$ (10)

Since q''w = const, it follows from eq. (9) that $({{T}_{w}}-{{\bar{T}}_{m}})= Const$, i.e.,

 $\frac{d{{T}_{w}}}{dx}=\frac{d{{{\bar{T}}}_{m}}}{dx}$ (11)

Combining eqs. (7), (8) and (11), the following relationships are obtained:

 $\frac{\partial \bar{T}}{\partial x}=\frac{d{{T}_{w}}}{dx}=\frac{d{{{\bar{T}}}_{c}}}{dx}=\frac{d{{{\bar{T}}}_{m}}}{dx}$ (12)

The time-averaged mean temperature, ${{\bar{T}}_{m}}$, changes with x as the result of heat transfer from the tube wall. The rate of mean temperature change can be obtained as follows:

 $\frac{d{{{\bar{T}}}_{m}}}{dx}=\frac{4\pi {{{{q}''}}_{w}}}{\rho {{c}_{p}}D{{{\bar{u}}}_{m}}}$ (13)

Substituting eq. (12) into eq. (1), the energy equation becomes:

 $\bar{u}\frac{d{{{\bar{T}}}_{m}}}{dx}=\frac{1}{{{r}_{0}}-y}\frac{\partial }{\partial y}\left[ ({{r}_{0}}-y)(\alpha +{{\varepsilon }_{H}})\frac{\partial \bar{T}}{\partial y} \right]$ (14)

where y = r0r is the distance measured from the tube wall. Equation (14) is subject to the following two boundary conditions:

 $\frac{\partial \bar{T}}{\partial y}=0,\text{ }y={{r}_{0}}$ (axisymmetric condition) (15)
 $\bar{T}={{T}_{w}}\text{ (unknown)},\text{ }y=0$ (16)

Integrating eq. (14) in the interval of (r0, r) and considering eq. (15), we have:

 $({{r}_{0}}-y)(\alpha +{{\varepsilon }_{H}})\frac{\partial \bar{T}}{\partial y}=\frac{d{{{\bar{T}}}_{m}}}{dx}\int_{{{r}_{0}}}^{y}{({{r}_{0}}-y)\bar{u}dy}$ (17)

which can be rearranged to

 $\frac{\partial \bar{T}}{\partial y}=\frac{I(y)}{({{r}_{0}}-y)(\alpha +{{\varepsilon }_{H}})}\frac{d{{{\bar{T}}}_{m}}}{dx}$ (18)

where

 $I(y)=\int_{{{r}_{o}}}^{y}{({{r}_{o}}-y)\bar{u}dy}$ (19)

Integrating eq. (18) in the interval of (0, y) and considering eq. (16), one obtains:

 $\bar{T}-{{T}_{w}}=\frac{d{{{\bar{T}}}_{m}}}{dx}\int_{0}^{y}{\frac{I(y)}{({{r}_{o}}-y)(\alpha +{{\varepsilon }_{H}})}dy}$ (20)

If the profiles of axial velocity and the thermal eddy diffusivity are known, eq. (20) can be used to obtain the correlation for internal forced convection heat transfer. With the exception of the very thin viscous sublayer, the velocity profile in the most part of the tube is fairly flat. Therefore, it is assumed that the time-averaged velocity, $\bar{u}$, in eq. (19) can be replaced by ${{\bar{u}}_{m}}$, and I(y) becomes:

 $I(y)\simeq -\frac{\bar{u}_{m}^{2}}{2}{{({{r}_{o}}-y)}^{2}}$ (21)

Substituting eqs. (21) and (13) into eq. (20) yields:

 $\bar{T}-{{T}_{w}}=-\frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}}\int_{0}^{y}{\frac{(1-y/{{r}_{o}})}{(\alpha +{{\varepsilon }_{H}})}dy}$ (22)

which can be rewritten in terms of wall coordinate

 $\bar{T}-{{T}_{w}}=-\frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}}\sqrt{\frac{\rho }{{{\tau }_{w}}}}\int_{0}^{{{y}^{+}}}{\frac{(1-y/{{r}_{o}})}{[1/\Pr +({{\varepsilon }_{M}}/\nu )/{{\Pr }^{t}}]}d{{y}^{+}}}$ (23)

where y+ is defined as ${{y}^{+}}=\frac{y{{u}_{\tau }}}{\nu }$.

To consider heat transfer in an internal turbulent flow, the entire turbulent boundary layer is divided into three regions: (1) inner region (y + < 5), (2) buffer region ($5\le {{y}^{+}}\le 30$), and (3) outer region (y + > 30). In the inner region ${{\varepsilon }_{M}}={{\varepsilon }_{H}}=0$ and eq. (23) becomes

 $\bar{T}-{{T}_{w}}=-\frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}}\sqrt{\frac{\rho }{{{\tau }_{w}}}}\Pr \int_{0}^{{{y}^{+}}}{(1-y/{{r}_{o}})d{{y}^{+}}}$ (24)

Since the inner region is very thin, $y/{{r}_{0}}\ll 1$ and 1 − y / r0 is effectively equal to 1. Therefore, the temperature profile in the inner region becomes:

 $\bar{T}-{{T}_{w}}=-\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}} \right)\sqrt{\frac{\rho }{{{\tau }_{w}}}}\Pr {{y}^{+}}$ (25)

The temperature at the boundary between the inner and buffer regions (y + = 5), ${{\bar{T}}_{s}}$, can be obtained from eq. (25) as

 ${{\bar{T}}_{s}}-{{T}_{w}}=-5\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}} \right)\sqrt{\frac{\rho }{{{\tau }_{w}}}}\Pr$ (26)

In the buffer region where $5\le {{y}^{+}}\le 30$, the eddy diffusivity in the buffer region is:

 $\frac{{{\varepsilon }_{M}}}{\nu }=\frac{{{y}^{+}}}{5}-1$ (27)

Substituting eq. (27) into eq. (23) and assuming the turbulent Prandtl number ${{\Pr }^{t}}=1$, the following expression is obtained:

 $\bar{T}-{{\bar{T}}_{s}}=-\frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}}\sqrt{\frac{\rho }{{{\tau }_{w}}}}\int_{5}^{{{y}^{+}}}{\frac{(1-y/{{r}_{o}})}{[1/\Pr +{{y}^{+}}/5-1]}d{{y}^{+}}}$ (28)

Since the buffer region is also very thin, 1 − y / r0 in eq. (28) is effectively equal to 1. Defining ${{T}^{+}}={(\bar{T}-{{T}_{w}})}/{\left( -\frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}}\sqrt{\frac{\rho }{{{\tau }_{w}}}} \right)}\;$ and Integrating eq. (28) yields

 $\int_{5\Pr }^{{{T}^{+}}}{d{{T}^{+}}}=\int_{5}^{{{y}^{+}}}{\frac{d{{y}^{+}}}{1/\Pr +({{y}^{+}}-5)/(5{{\Pr }^{t}})}}$ (29)

i.e.,

 $\bar{T}-{{\bar{T}}_{s}}=-5\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}} \right)\sqrt{\frac{\rho }{{{\tau }_{w}}}}\ln \left( \frac{{{y}^{+}}}{5}\Pr -\Pr +1 \right),\text{ }5<{{y}^{+}}<30$ (30)

The temperature at the top of the buffer region where y + = 30, ${{\bar{T}}_{b}}$, becomes

 ${{\bar{T}}_{b}}-{{\bar{T}}_{s}}=-5\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}} \right)\sqrt{\frac{\rho }{{{\tau }_{w}}}}\ln (5\Pr +1)$ (31)

For the outer region where ${{\varepsilon }_{M}}\gg \nu \text{ and }{{\varepsilon }_{H}}\gg \alpha$, eq. (23) becomes

 $\bar{T}-{{\bar{T}}_{b}}=-\frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}}\sqrt{\frac{\rho }{{{\tau }_{w}}}}\int_{30}^{{{y}^{+}}}{\frac{(1-y/{{r}_{o}})}{{{\varepsilon }_{M}}/\nu }d{{y}^{+}}}$ (32)

where the turbulent Prandtl number is assumed to be equal to 1.

It is assumed that the Nikuradse equation is valid in the outer region and the velocity gradient in this region becomes:

 $\frac{\partial {{u}^{+}}}{\partial {{y}^{+}}}=\frac{2.5}{{{y}^{+}}}$ (33)

The expression of apparent shear stress in this region, can be non-dimensionalized as:

 $\frac{{{\tau }_{app}}}{{{\tau }_{w}}}=\frac{{{\varepsilon }_{M}}}{\nu }\frac{\partial {{u}^{+}}}{\partial {{y}^{+}}}$ (34)

Substituting

${{\tau }_{app}}={{\tau }_{w}}\cdot \left( \frac{r}{{{r}_{o}}} \right)={{\tau }_{w}}\cdot \left( 1-\frac{y}{{{r}_{o}}} \right)$

and eq. (33) into eq. (34), the eddy diffusivity in the outer region is obtained as:

 $\frac{{{\varepsilon }_{M}}}{\nu }=\left( 1-\frac{y}{{{r}_{o}}} \right)\frac{{{y}^{+}}}{2.5}$ (35)

Substituting eq. (35) into eq. (32), the temperature distribution in this region becomes:

 $\bar{T}-{{\bar{T}}_{b}}=-2.5\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}} \right)\sqrt{\frac{\rho }{{{\tau }_{w}}}}\int_{30}^{{{y}^{+}}}{\frac{1}{{{y}^{+}}}d{{y}^{+}}}=-2.5\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}} \right)\sqrt{\frac{\rho }{{{\tau }_{w}}}}\ln \left( \frac{{{y}^{+}}}{30} \right)$ (36)

which is valid from y + = 30 to the center of the tube where yc = r0 or

 $y_{c}^{+}=\frac{{{r}_{0}}}{\nu }\sqrt{\frac{{{\tau }_{w}}}{\rho }}$ (37)

The temperature at the center of the tube, ${{\bar{T}}_{c}}$, can be obtained by letting ${{y}^{+}}=y_{c}^{+}$ in eq. (36), i.e.

 ${{\bar{T}}_{c}}-{{\bar{T}}_{b}}=-2.5\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}} \right)\sqrt{\frac{\rho }{{{\tau }_{w}}}}\ln \left( \frac{{{r}_{0}}}{30\nu }\sqrt{\frac{{{\tau }_{w}}}{\rho }} \right)$ (38)

The overall temperature change from the wall to the center of the tube can be obtained by adding eqs. (26), (31) and (38):

 ${{T}_{w}}-{{\bar{T}}_{c}}=\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}} \right)\sqrt{\frac{\rho }{{{\tau }_{w}}}}\left[ 2.5\ln \left( \frac{{{r}_{0}}}{30\nu }\sqrt{\frac{{{\tau }_{w}}}{\rho }} \right)+5\ln (5\Pr +1)+5\Pr \right]$ (39)

It follows from the definition of friction factor, ${{c}_{f}}=\frac{{{\tau }_{w}}}{\rho \bar{u}_{m}^{2}/2}$, that

 ${{\tau }_{w}}=\frac{1}{2}{{c}_{f}}\rho \bar{u}_{m}^{2}$ (40)

Substituting eq. (40) into eq. (39) and considering the definition of Reynolds number, ${{\operatorname{Re}}_{D}}={{\bar{u}}_{m}}D/\nu$, eq. (39) becomes:

 ${{T}_{w}}-{{\bar{T}}_{c}}=\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}{{{\bar{u}}}_{m}}} \right)\sqrt{\frac{2}{{{c}_{f}}}}\left[ 2.5\ln \left( \frac{{{\operatorname{Re}}_{D}}}{60}\sqrt{\frac{{{c}_{f}}}{2}} \right)+5\ln (5\Pr +1)+5\Pr \right]$ (41)

In order to obtain the heat transfer coefficient, $h={{{q}''}_{w}}/({{T}_{w}}-{{\bar{T}}_{m}})$, the temperature difference ${{T}_{w}}-{{\bar{T}}_{m}}$ must be obtained. If the velocity profile can be approximated by $\frac{{\bar{u}}}{{{{\bar{u}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}$, and the temperature and velocity can also be approximated by the one-seventh law, i.e.,

 $\frac{{{T}_{w}}-\bar{T}}{{{T}_{w}}-{{{\bar{T}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}{{,}_{_{\text{ }}}}\frac{{\bar{u}}}{{{{\bar{u}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}$ (42)

it follows that

 ${{T}_{w}}-{{\bar{T}}_{m}}=\frac{\int_{0}^{{{r}_{o}}}{\bar{u}({{T}_{w}}-\bar{T})2\pi rdr}}{\int_{0}^{{{r}_{o}}}{\bar{u}2\pi rdr}}=\frac{5}{6}({{T}_{w}}-{{\bar{T}}_{c}})$ (43)

Substituting eq. (41) into eq. (43) results in:

 ${{T}_{w}}-{{\bar{T}}_{m}}=\frac{5}{6}\left( \frac{{{{{q}''}}_{w}}}{\rho {{c}_{p}}{{{\bar{u}}}_{m}}} \right)\sqrt{\frac{2}{{{c}_{f}}}}\left[ 2.5\ln \left( \frac{{{\operatorname{Re}}_{D}}}{60}\sqrt{\frac{{{c}_{f}}}{2}} \right)+5\ln (5\Pr +1)+5\Pr \right]$ (44)

which can be rearranged to the following empirical correlation

 $\text{N}{{\text{u}}_{D}}=\frac{{{\operatorname{Re}}_{D}}\Pr \sqrt{\frac{{{c}_{f}}}{2}}}{\frac{5}{6}\left[ 2.5\ln \left( \frac{{{\operatorname{Re}}_{D}}}{60}\sqrt{\frac{{{c}_{f}}}{2}} \right)+5\ln (5\Pr +1)+5\Pr \right]}$ (45)

which can be used together with appropriate friction coefficient to obtain the Nusselt number.

## References

1. Oosthuizen, P.H., and Naylor, D., 1999, Introduction to Convective Heat Transfer Analysis, WCB/McGraw-Hill, New York.
2. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.