Heat Transfer Correlations

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Table 2 summarizes the existing correlations in literature for various heat transfer modes for both single-phase and two-phase systems in different geometric configurations. It can be seen that the heat transfer coefficient depends on surface geometry, the driving force of the fluid motion, thermal properties of the fluid, and flow properties.

Table 2 Correlations for convective heat transfer for various modes and geometries
Heat transfer
mode
Geometry Nusselt number Comments and
restrictions
Dimensionless
numbers
Forced convection Flow parallel to a
flat plate
Nu_x = 0.332Re_x^{1/2}Pr^{1/3}(Pr>0.6)
\overline {Nu} = 0.664Re_L^{1/2}Pr^{1/3}(Pr>0.6)
Nu_x = 0.565Re_x^{1/2}Pr^{1/2}(Pr \le 0.05)
Isothermal surface
Re_x <5 \times 10^5
(laminar)
\begin{array}{l} {\rm{N}}{{\rm{u}}_x} = \frac{{hx}}{k} \\ {\rm{R}}{{\rm{e}}_x} = \frac{{{u_\infty }x}}{\nu } \\ \end{array}

\begin{array}{l} \overline {{\rm{Nu}}}  = \frac{{\bar hL}}{k} \\ {\rm{R}}{{\rm{e}}_L} = \frac{{{u_\infty }L}}{\nu } \\ \end{array}

\overline {Nu} = 0.037({Re_L^{0.8}}-871)Pr^{0.33}
(0.6 \le Pr \le 60)
5 \times 10^5 < {Re_L} < 10^5
(Turbulent)
Flow in a pipe
(conventional size)
\begin{array}{l} \overline {{\rm{Nu}}}  = 3.66 \\ + \frac{{0.0668(D/L){\mathop{\rm Re}} \Pr }}{{1 + 0.04{{[(D/L){\mathop{\rm Re}} \Pr ]}^{2/3}}}} \\ \end{array} Isothermal surface
{\mathop{\rm Re}}  \le 2300
Thermal entry
region
\begin{array}{l} \overline {{\rm{Nu}}}  = \frac{{\bar hD}}{k} \\ {\mathop{\rm Re}}  = \frac{{\bar uD}}{\nu } \\ \end{array}
 \bar u is mean velocity
\begin{array}{l} {\rm{Nu}} = 0.027{{\mathop{\rm Re}} ^{0.8}} \\ {\rm{           }} \times {\rm{P}}{{\rm{r}}^{0.33}}{\left( {\mu /{\mu _w}} \right)^{0.14}} \\ {\rm{       (0}}{\rm{.7}} \le \Pr  \le 16700) \\ \end{array} \begin{array}{l} L/D \ge 10 \\ {\mathop{\rm Re}}  > 10,000 \\ \end{array}
(Fully developed
turbulent)
μw is viscosity
evaluated at Tw
Flow in a pipe
(miniature)
\begin{array}{l} \overline {{\rm{Nu}}}  = (1 + F) \\ \times \frac{{(f/8)({\mathop{\rm Re}}  - 1000)\Pr }}{{1 + 12.7{{(f/8)}^{0.5}}({{\Pr }^{2/3}} - 1)}} \\ f = {[1.82\log ({\mathop{\rm Re}} ) - 1.64]^{ - 2}} \\ F = 7.6 \times {10^{ - 5}}{\mathop{\rm Re}}  \\ {\rm{       }} \times [1 - {(D/{D_0})^2}] \\ \end{array} D0=1.164 mm is
reference diameter.
Correlation was obtained for
water at D=0.102, 0.76 and 1.09 mm.
\begin{array}{l} \overline {{\rm{Nu}}} = \frac{{\bar hD}}{k} \\ {\mathop{\rm Re}}  = \frac{{\bar uD}}{\nu } \\ \end{array}
Flow between
parallel plates
\begin{array}{l} \overline {{\rm{Nu}}}  = 7.54 \\ + \frac{{0.03({D_h}/L){\mathop{\rm Re}} \Pr }}{{1 + 0.016{{[({D_h}/L){\mathop{\rm Re}} \Pr ]}^{2/3}}}} \\ \end{array} Isothermal surface
{\mathop{\rm Re}}  \le 2800
(Laminar)
\begin{array}{l} \overline {{\rm{Nu}}}  = \frac{{\bar h{D_h}}}{k} \\ {\mathop{\rm Re}}  = \frac{{\bar u{D_h}}}{\nu } \\ \end{array}
\begin{array}{l} \overline {Nu}  = 0.023{{\mathop{\rm Re}} ^{0.8}}{\Pr ^{0.33}} \\  {\rm{       (}}\Pr  > 0.5) \\ 
 \end{array} {\mathop{\rm Re}}  > 10,000
(Turbulent)
Flow across a
circular cylinder
\begin{array}{l} \overline {{\rm{Nu}}}  = 0.3 \\  + \frac{{0.62{{{\mathop{\rm Re}} }^{1/2}}{{\Pr }^{1/3}}}}{{{{[1 + {{(0.4/\Pr /)}^{2/3}}]}^{1/4}}}} \\ \times {\left[ {1 + {{\left( {\frac{{{\mathop{\rm Re}} }}{{282000}}} \right)}^{5/8}}} \right]^{4/5}} \\ \end{array} {\mathop{\rm Re}} \Pr  > 0.2
(Both laminar and
turbulent)
\begin{array}{l} \overline {{\rm{Nu}}}  = \frac{{\bar hD}}{k} \\ Re = \frac{{{u_\infty }D}}{\nu } \\ \end{array}
Flow across a
sphere
\begin{array}{l} \overline {{\rm{Nu}}}  = 2 + (0.4{{\mathop{\rm Re}} ^{0.5}} \\ + 0.06{{\mathop{\rm Re}} ^{2/3}}){\Pr ^{0.4}}{(\mu /{\mu _w})^{\frac{1}{4}}} \\ \end{array}  \begin{array}{l} 3.5 < {\mathop{\rm Re}}  \\ < 76000 \\ \end{array}
 0.71 \le \Pr  \le 380
μw is viscosity
evaluated at Tw
Flow through
a packed bed of
spheres
\overline {{\rm{Nu}}}  = 1.625{{\mathop{\rm Re}} ^{1/2}}{\Pr ^{1/3}} 15 \le {\mathop{\rm Re}}  \le 120
D – diameter of
sphere
A – bed cross-
sectional area
\begin{array}{l} \overline {{\rm{Nu}}}  = \frac{{\bar hD}}{k} \\ {\mathop{\rm Re}}  = \frac{{\dot mD}}{{A\mu }} \\ \end{array}
Free convection On a vertical
surface
\begin{array}{l} {\overline {{\rm{Nu}}} ^{1/2}} = 0.825 +  \\ \frac{{0.387{\rm{R}}{{\rm{a}}^{1/6}}}}{{{{[1 + {{(0.492/\Pr )}^{9/16}}]}^{8/27}}}} \\ \end{array} \Delta T = \left| {{T_w} - {T_\infty }} \right|
Applicable to both
laminar and
turbulent
\begin{array}{l} \overline {{\rm{Nu}}}  = \frac{{\bar hL}}{k} \\ {\rm{Ra}} = \frac{{g\beta \Delta T{L^3}}}{{\nu \alpha }} \\ \end{array}
On a horizontal
heated square
facing up
\overline {{\rm{Nu}}}  = 0.54{({\rm{Gr}}\Pr )^{1/4}} Isothermal surface
{10^5} \le {\rm{Gr}}
 \le 7 \times {10^7}
For rectangle, use
shorter side of L
\begin{array}{l} \overline {{\rm{Nu}}}  = \frac{{\bar hL}}{k} \\ {\rm{Gr}} = \frac{{g\beta \Delta T{L^3}}}{{{\nu ^2}}} \\ \end{array}
On a horizontal
heated square
facing down
\overline {{\rm{Nu}}}  = 0.27{({\rm{Gr}}\Pr )^{1/4}} Isothermal surface
\begin{array}{l} 3 \times {10^5} \le {\rm{Gr}} \\ \le 3 \times {10^{10}} \\ \end{array}
For rectangle, use
shorter side of L
\begin{array}{l} \overline {{\rm{Nu}}}  = \frac{{\bar hL}}{k} \\ {\rm{Gr}} = \frac{{g\beta \Delta T{L^3}}}{{{\nu ^2}}} \\  \end{array}
On a horizontal
cylinder
\begin{array}{l} {\overline {{\rm{Nu}}} ^{1/2}} = 0.60 +  \\  + \frac{{0.387{\rm{R}}{{\rm{a}}^{1/6}}}}{{{{[1 + {{(0.559/\Pr )}^{9/16}}]}^{8/27}}}} \\  \end{array} Ra < 1012 \begin{array}{l} \overline {{\rm{Nu}}}  = \frac{{\bar hD}}{k} \\ {\rm{Ra}} = \frac{{g\beta \Delta T{D^3}}}{{\nu \alpha }} \\ \end{array}
On a sphere
\begin{array}{l} \overline {{\rm{Nu}}}  = 2 +  \\  + \frac{{0.589{\rm{R}}{{\rm{a}}^{1/4}}}}{{{{[1 + {{(0.469/\Pr )}^{9/16}}]}^{4/9}}}} \\ \end{array} \begin{array}{l}\Delta T = {T_w} - {T_\infty } \\ {\rm{Ra}} < {10^{11}} \\ \Pr  \ge 0.7 \\  \end{array} \begin{array}{l} \overline {{\rm{Nu}}}  = \frac{{\bar hD}}{k} \\ {\rm{Ra}} = \frac{{g\beta \Delta T{D^3}}}{{\nu \alpha }} \\ \end{array}
Evaporation Falling film
evaporation
Laminar
 \begin{array}{l} {\rm{Nu}} = 1.10{\mathop{\rm Re}} _\delta ^{ - 1/3} \\ ({{\mathop{\rm Re}} _\delta } \le 30) \\ \end{array}
Nu - local Nusselt
number

Γ - mass flow rate
per unit width of
the vertical surface
\begin{array}{l} {\rm{Nu}} = \frac{{h{{(\nu _\ell ^2/g)}^{\frac{1}{3}}}}}{k} \\ {{\mathop{\rm Re}} _\delta } = \frac{{4\Gamma }}{\mu } \\ \end{array}
Wavy laminar
 \begin{array}{l} {\rm{Nu}} = 0.828{\mathop{\rm Re}} _\delta ^{ - 0.22} \\ (30 \le {{\mathop{\rm Re}} _\delta } \le 1800) \\ \end{array}
Turbulent
\begin{array}{l}{\rm{Nu}} = 0.0038{\mathop{\rm Re}} _\delta ^{0.4}{\Pr ^{0.65}} \\ {\rm{    }}({{\mathop{\rm Re}} _\delta } > 1800) \\ \end{array}
Condensation On a vertical
surface
Laminar (Nusselt)
\begin{array}{l}{\rm{Nu}} = 1.10{\mathop{\rm Re}} _\delta ^{ - 1/3} \\ ({{\mathop{\rm Re}} _\delta } \le 30) \\ \end{array}
Nu - local Nusselt
number Γ - mass flow rate
per unit width of
the vertical surface
\begin{array}{l} {\rm{Nu}} = \frac{{h{{(\nu _\ell ^2/g)}^{\frac{1}{3}}}}}{k} \\  {{\mathop{\rm Re}} _\delta } = \frac{{4\Gamma }}{\mu } \\  \end{array}
Wavy laminar
 \begin{array}{l} {\rm{Nu}} = \frac{{{{{\mathop{\rm Re}} }_\delta }}}{{{\mathop{\rm Re}} _\delta ^{1.22} - 5.22}} \\ (30 \le {{\mathop{\rm Re}} _\delta } \le 1800) \\ \end{array}
Turbulent
{\rm{Nu}} = 0.023{\mathop{\rm Re}} _\delta ^{0.25}{\Pr ^{ - 0.5}}
On tubes
\begin{array}{l} \overline {Nu}  = 0.729 \\ \times {\left[ {\frac{{{D^3}{h_{\ell v}}g\left( {{\rho _\ell } - {\rho _v}} \right)}}{{n{k_\ell }{\nu _\ell }\Delta T}}} \right]^{\frac{1}{4}}} \\ \end{array} ΔT = TsatTw
n - number of tubes
\overline {Nu}  = \frac{{\bar hD}}{{{k_\ell }}}
In microscale
channel (
Dh < 1.5mm)
{\rm{Nu}} = {\rm{W}}{{\rm{e}}^{ - {\rm{Ja}}}}{\mathop{\rm Re}} {\Pr ^Y} \begin{array}{l} Y = 1.3{\rm{ }} \\ {\rm{for }}{\mathop{\rm Re}}  \le 65 \\ Y = (0.5{D_h} - 1) \\ {\rm{     }}/(2{D_h}) \\ {\rm{for }}{\mathop{\rm Re}}  > 65 \\ \end{array} {\rm{We}} = \frac{{{\rho _\ell }{V^2}L}}{\sigma }
{\rm{Ja}} = \frac{{{c_{p\ell }}({T_{sat}} - {T_w})}}{{{h_{\ell v}}}}
{\mathop{\rm Re}}  = \frac{{\dot m''{D_h}}}{{{\mu _\ell }}}
 \dot m'' –mass flux (kg/s-m2)
Boiling Nucleate,
saturated pool
boiling
\overline {{\rm{Nu}}}  = \frac{{{\rm{Ja}}_\ell ^2}}{{{C^3}\Pr _\ell ^m}} m=2 for water
m=4.1 for other
fluids
C=0.013 water-
copper or stainless
steel
C=0.006 for water-
nickel or brass
\begin{array}{l} \overline {Nu}  = \frac{{\bar h{L_c}}}{{{k_\ell }}} \\ {L_c} = \sqrt {\frac{{{\sigma _\ell }}}{{g({\rho _\ell } - {\rho _v})}}}  \\ {\rm{J}}{{\rm{a}}_\ell } = \frac{{{c_{p,\ell }}\Delta T}}{{{h_{\ell v}}}} \\ \end{array} \Delta T = {T_w} - {T_{sat}}
Film boiling on a
horizontal plate
\begin{array}{l} \overline {{\rm{Nu}}}  = 0.425 \\ \times {\left[ {Gr{{\Pr }_v}\left( {\frac{{1 + 0.4J{a_v}}}{{J{a_v}}}} \right)} \right]^{\frac{1}{4}}} \\ \end{array} Term in parentheses
accounts for
sensible heating
effect in vapor film
\begin{array}{l} \overline {Nu}  = \frac{{\bar h{L_c}}}{{{k_v}}} \\  {L_c} = \sqrt {\frac{{{\sigma _\ell }}}{{g({\rho _\ell } - {\rho _v})}}}  \\ Gr = \frac{{g[({\rho _\ell } - {\rho _v})/{\rho _v}]L_c^3}}{{\nu _v^2}} \\ {\rm{J}}{{\rm{a}}_v} = \frac{{{c_{p,v}}\Delta T}}{{{h_{\ell v}}}} \\ \end{array}
Film boiling on a
horizontal cylinder
\begin{array}{l} \overline {{\rm{Nu}}}  = 0.62 \\ \times {\left[ {{\rm{Gr}}{{\Pr }_v}\left( {\frac{{1 + 0.4J{a_v}}}{{J{a_v}}}} \right)} \right]^{\frac{1}{4}}} \\ \end{array} D \gg film
thickness
\begin{array}{l} \overline {Nu}  = \frac{{\bar hD}}{{{k_v}}} \\ {\rm{Gr}} =  \\ \frac{{g[({\rho _\ell } - {\rho _v})/{\rho _v}]{D^3}}}{{\nu_v^2}} \\ {\rm{J}}{{\rm{a}}_v} = \frac{{{c_{p,v}}\Delta T}}{{{h_{\ell v}}}} \\ \end{array}
Film boiling on a
sphere
\begin{array}{l} \overline {{\rm{Nu}}}  = 0.4 \\ \times {\left[ {{\rm{Gr}}{{\Pr }_v}\left( {\frac{{1 + 0.4{\rm{J}}{{\rm{a}}_v}}}{{{\rm{J}}{{\rm{a}}_v}}}} \right)} \right]^{\frac{1}{3}}} \\ \end{array} D\gg film
thickness
Boiling in
microchannel
(D=1.39 – 1.69 mm)
\begin{array}{l} {\rm{Nu}} = 30{{\mathop{\rm Re}} ^{0.857}} \\ \times {\rm{B}}{{\rm{o}}^{0.714}}{(1 - x)^{ - 0.143}} \\ \end{array} Correlation
obtained by using
Freon ® 141
x is quality
\overline {Nu}  = \frac{{\bar hD}}{{{k_\ell }}} {\rm{Bo}} = \frac{{q''}}{{{h_{\ell v}}\dot m''}}
 \dot m'' – mass flux (kg/s-m2)
MeltingMelting in a rectangular cavity
\begin{array}{l}
 {\rm{Nu}} = {(2\tau )^{ - 1/2}} \\ + [{c_1}{\rm{R}}{{\rm{a}}^{1/4}} - {(2\tau )^{ - 1/2}}] \\ \times {[1 + {({c_2}{\rm{R}}{{\rm{a}}^{3/4}}{\tau ^{3/2}})^n}]^{1/n}} \\ 
 \end{array}
c1 = 0.35,c2 = 0.175
n = − 2
Nusselt number is
function of time
\begin{array}{l} \overline {Nu}  = \frac{{\bar hH}}{k} \\ {\rm{Ra}} = \frac{{g\beta \Delta T{H^3}}}{{\nu \alpha }} \\ \tau  = {\rm{SteFo}} \\ \end{array} {\rm{Fo}} = \frac{{{\alpha _\ell }t}}{{{H^2}}}
SolidificationSolidification around a horizontal tube
\overline {{\rm{Nu}}}  = 0.52{\rm{R}}{{\rm{a}}^{1/4}} D is transient
equivalent outer
diameter of the
solid Ra \le {10^9}
\begin{array}{l} \overline {Nu}  = \frac{{\bar hD}}{k} \\ {\rm{Ra}} = \frac{{g\beta \Delta T{D^3}}}{{\nu \alpha }} \\ \end{array}
Sublimation \begin{array}{l} {\rm{N}}{{\rm{u}}_x} = 0.458{\mathop{\rm Re}} _x^{1/2}{\Pr ^{1/3}} \\  {\rm{S}}{{\rm{h}}_x} = 0.459{\mathop{\rm Re}} _x^{1/2}{{\mathop{\rm Sc}} ^{1/3}} \\ \end{array} Uniform heat flux surface
{{\mathop{\rm Re}} _x} < 5 \times {10^5}
\begin{array}{l} {{\mathop{\rm Nu}} _x} = \frac{{hx}}{k} \\ {{\mathop{\rm Sh}} _x} = \frac{{{h_m}x}}{D} \\ \end{array}

References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Further Reading

External Links