Forced Flows Correlations

Flat Plate

Circular Channel

Laminar:

\mathrm{Re}_D\lesssim2100

Transition:

2100<\mathrm{Re}_D<10^4

Turbulent:

\mathrm{Re}_D>10^4

Entrance Lengths

Laminar Flow

Chen (1973): Smooth surface

\frac {l_{\mathrm{ent,hy}}}D=\frac {0.60}{0.035\mathrm{Re}_D+1}+0.056\mathrm{Re}_D\tag{4.2.12}

Analytical: Smooth surface

\frac {l_{\mathrm{ent,th}}}D\approx0.05\mathrm{Re}_D\mathrm{Pr}\tag{4.5.30}

Turbulent Flow

Experimental: For \mathrm{Pr}\approx1 or \mathrm{Sc}\approx1 fluids

\begin{aligned} \frac {l_{\mathrm{ent,hy}}}D & \approx10 \\ \frac {l_{\mathrm{ent,th}}}D & \approx10 \\ \frac {l_{\mathrm{ent,ma}}}D & \approx10 \end{aligned}\tag{7.1.4}

Wang (1982): Smooth surface, flat inlet velocity profile, \frac 17 power law velocity profile

\frac {l_{\mathrm{ent,hy}}}D=1.3590\mathrm{Re}_D^{1/4}\tag{7.2.3}

Fully Developed Flow

Profiles

Laminar Flow

Hagen-Poiseuille (1840): Smooth surface

\begin{aligned} u(r) & =2U_m\left[1-\left(\frac r{R_0}\right)^2\right] \\ U_m & =\frac {R_0^2}{8\mu}\left(-\frac {\mathrm dP}{\mathrm dx}\right) \end{aligned}\tag{4.3.2}

Turbulent Flow

Wang (1982): Smooth surface, flat inlet velocity profile, \frac 17 power law velocity profile

\frac {x/D}{\mathrm{Re}_D^{1/4}}=1.4039\left(\frac {\delta}{R_0}\right)^{5/4}\left[1+0.1577\left(\frac {\delta}{R_0}\right)-0.1793\left(\frac {\delta}{R_0}\right)^2-0.0168\left(\frac {\delta}{R_0}\right)^3+0.0064\left(\frac {\delta}{R_0}\right)^4\right]\tag{7.2.2}

Nikuradse (1932): Smooth surface, n power law velocity profile, valid for y^+>30

\begin{aligned} \frac {\overline u}{U_{\mathrm{max}}} & =\left(\frac y{R_0}\right)^{1/n} \\ \frac {U_m}{U_{\mathrm{max}}} & =\frac {2n^2}{(n+1)(2n+1)} \end{aligned}\tag{7.2.6}

where

\mathrm{Re}_D	4000	2.3\times10^3	1.1\times10^5	1.1\times10^6	2.0\times10^6	3.2\times10^6
n	6	6.6	7	8.8	10	10

Prandtl (1933): Smooth surface, valid for y^+>30

\frac {U_{\mathrm{max}}-\overline u}{U_{\tau}}=2.5\ln\left(\frac {R_0}y\right)\tag{7.2.8}

Wang (1946): Smooth surface

\frac {U_{\mathrm{max}}-\overline u}{U_{\tau}}=2.5\left[\ln\left(\frac {1+\sqrt{1-\frac y{R_0}}}{1-\sqrt{1-\frac y{R_0}}}\right)-2\arctan\left(\sqrt{1-\frac y{R_0}}\right)-0.572\ln\left(\frac {2.53-\frac y{R_0}+1.75\sqrt{1-\frac y{R_0}}}{2.53-\frac y{R_0}-1.75\sqrt{1-\frac y{R_0}}}\right)+1.143\arctan\left(\frac {1.75\sqrt{1-\frac y{R_0}}}{0.53+\frac y{R_0}}\right)\right]\tag{7.2.9}

Yu et al. (2001): Smooth surface, valid for 150<R_0^+<50,000 where R_0^+=\frac {U_{\tau}R_0}{\nu} and U_{\tau}=\sqrt{\frac {\tau_s}{\rho}}

U_m^+=3.2-\frac {227}{R_0^+}+\left(\frac {50}{R_0^+}\right)^2+\frac 1{0.436}\ln R_0^+\tag{7.2.34}

Yu et al. (2001): Smooth surface, valid for R_0^+>500 where R_0^+=\frac {U_{\tau}R_0}{\nu} and U_{\tau}=\sqrt{\frac {\tau_s}{\rho}}

U_{\mathrm{CL}}^+=7.52+\frac 1{0.436}\ln R_0^+

White (2006): Smooth surface, \kappa=0.4 and B=5.5, valid for R_0^+>30

U_m^+=\frac 1{\kappa}\ln R_0^++B-\frac 3{2\kappa}

Friction Factor

Laminar Flow

Hagen-Poiseuille (1840): Smooth surface

f=4C_f=\frac {64}{\mathrm{Re}_D}\tag{4.3.7}

Shah and London (1978): Smooth surface, valid for all x^*=\frac x{D\mathrm{Re}_D} range

C_{f,\mathrm{app},x}\mathrm{Re}_D=\frac {3.44}{\sqrt{x^*}}+\frac 1{1+2.1\times10^{-4}\left(x^*\right)^{-2}}\left(\frac {1.25}{4x^*}+16-\frac {3.44}{\sqrt{x^*}}\right)\tag{4.2.13}

Turbulent Flow

Wang (1982): Smooth surface, flat inlet velocity profile, \frac 17 power law velocity profile

C_{f,\mathrm{app},x}\mathrm{Re}_D^{1/4}=\frac {(U_{\mathrm{max}}/U_m)^2-1}{4x/\left(4x\mathrm{Re}_D^{1/4}\right)}\tag{7.2.4}

where

\frac {U_m}{U_{\mathrm{max}}}=1-\frac 14\left(\frac {\delta}{R_0}\right)+\frac 1{15}\left(\frac {\delta}{R_0}\right)^2\tag{7.2.5}

White (2006): Smooth surface, valid for all R_0^+=\frac {U_{\tau}R_0}{\nu} where U_{\tau}=\sqrt{\frac {\tau_s}{\rho}}

\frac 1{\sqrt{C_f}}=1.7272\ln\left(\mathrm{Re}_D\sqrt{C_f}\right)-0.395\tag{7.2.37}

Blasius (1913): Smooth surface, \frac 17 power law velocity profile, valid for \mathrm{Re}_D\lesssim10^5

C_f=0.079\mathrm{Re}_D^{-1/4}\tag{7.2.38}

White (2006): Fully rough surface

\frac 1{\sqrt f}=2.0\log_{10}\left(\frac {\mathrm{Re}_D\sqrt f}{1+0.1\mathrm{Re}_D\sqrt f\varepsilon_s/D}\right)-0.8\tag{7.2.40}

Colebrook (1939): Fully rough surface, valid for 5\leq\varepsilon_s^+\leq70

\frac 1{\sqrt f}=-2.0\log_{10}\left(\frac {\varepsilon_s/D}{3.7}+\frac {2.51}{\mathrm{Re}_D\sqrt f}\right)\tag{7.2.41}

Haaland (1983): Fully rough surface

\frac 1{\sqrt f}=-1.8\log_{10}\left[\left(\frac {\varepsilon_s/D}{3.7}\right)^{1.11}+\frac {6.9}{\mathrm{Re}_D}\right]\tag{7.2.42}

Heat Transfer Coefficient

Laminar Flow

Analytical: Smooth surface, negligible axial conduction, negligible viscous dissipation

\mathrm{Nu}_{D,\mathrm{UHF}}=\frac {48}{11}\approx4.364\tag{4.4.6}

Shah and London (1978): With viscous dissipation and volumetric energy generation, if q_v^*=\frac {\dot q_vD}{q_s''} and \mathrm{Br}'=\frac {\mu U_m^2}{q_s'' D}

\mathrm{Nu}_{D,\mathrm{UHF}}=\frac {48}{11}\frac 1{1+\frac 3{44}q_v^*+\frac {48}{11}\mathrm{Br}'}\tag{4.4.9}

Analytical: Smooth surface, negligible axial conduction, if \lambda_0=2.70436442

\mathrm{Nu}_{D,\mathrm{UWT}}=\frac {\lambda_0^2}2=3.6568\tag{4.4.22}

Analytical: Smooth surface, negligible axial conduction, negligible viscous dissipation, if x^*=\frac x{D\mathrm{Re}\mathrm{Pr}}>0.0335, temperature profile is

\frac {T_m-T_s}{T_{\mathrm{in}}-T_s}=0.81905\exp\left(-2\lambda_0^2x^*\right)\tag{4.4.23}

Michelsen and Villadsen (1974): Smooth surface, axial conduction, if \mathrm{Pe}=\mathrm{Re}_D\mathrm{Pr}

\mathrm{Nu}_{D,\mathrm{UWT}}=\begin{cases} 3.6568\left(1+\frac {1.227}{\mathrm{Pe}^2}+\cdots\right) & \qquad\qquad\mathrm{Pe}>5 \\ 4.1807(1-0.0439\mathrm{Pe}+\cdots) & \qquad\qquad\mathrm{Pe}<1.55 \end{cases}

Turbulent Flow

Norris (1970): Rough surfacess (accounted for in C_f)

\frac {\mathrm{Nu}_{D_H}}{\mathrm{Nu}_{D_H,\mathrm{smooth}}}=\min\left[\left(\frac {C_f}{C_{f,\mathrm{smooth}}}\right)^n,4^n\right]\tag{7.1.1}

where

n=\begin{cases} 0.68\mathrm{Pr}^{0.215} & \qquad\qquad\mathrm{Pr}<6 \\\\ 1 & \qquad\qquad\mathrm{Pr}>6 \end{cases}\tag{7.1.2}

Yu et al. (2001): Smooth surface, all thermal boundary conditions, valid for R_0^+>500 and \mathrm{Pr}>\mathrm{Pr}_{\mathrm{tu}}

\begin{aligned} \mathrm{Nu}_{D_H} & =\frac 1{\frac 1{\mathrm{Nu}_{D_H,1}}\frac {\mathrm{Pr}_{\mathrm{tu}}}{\mathrm{Pr}}+\frac 1{\mathrm{Nu}_{D_H,\infty}}\left[1-\left(\frac {\mathrm{Pr}_{\mathrm{tu}}}{\mathrm{Pr}}\right)^{2/3}\right]} \\ \mathrm{Nu}_{D_H,\infty} & =0.07343\mathrm{Re}_{D_H}\left(\frac {\mathrm{Pr}}{\mathrm{Pr}_{\mathrm{tu}}}\right)^{1/3}\left(\frac {C_f}2\right)^{1/2} \\ \mathrm{Pr}_{\mathrm{tu}} & =0.85+0.015\mathrm{Pr}^{-1} \end{aligned}

For uniform wall temperature or uniform heat flux boundaries, \mathrm{Nu}_{D_H,1} is

\begin{aligned} \mathrm{Nu}_{D_H,1,\mathrm{UWT}} & =\frac {\mathrm{Re}_{D_H}C_f/2}{1+145\left(U_m^+\right)^{-2/5}} \\ \mathrm{Nu}_{D_H,1,\mathrm{UHF}} & =\frac {\mathrm{Re}_{D_H}C_f/2}{1+195\left(U_m^+\right)^{-2.7}} \end{aligned}

Chilton and Colburn (1934): Smooth surface, valid for 10^4<\mathrm{Re}_D<3\times10^5 and 0.6<\mathrm{Pr}<100

j_{\mathrm{th}}=\mathrm{St}\mathrm{Pr}^{2/3}=\frac {\mathrm{Nu}_l}{\mathrm{Re}_l\mathrm{Pr}^{1/3}}=\frac {C_f}2\tag{9.7.8}

Mass Transfer

Turbulent Flow

Chilton and Colburn (1934): Smooth surface, valid for 2000<\mathrm{Re}_D<3\times10^5 and 0.6<\mathrm{Sc}<2500

j_{\mathrm{ma}}=\mathrm{St}\mathrm{Sc}^{2/3}=\frac {\mathrm{Sh}_l}{\mathrm{Re}_l\mathrm{Sc}^{1/3}}=\frac {C_f}2\tag{9.7.9}

Developed Hydrodynamic, Thermal Entrance Region

This section covers developed hydrodynamic, thermally developing flows.

Heat Transfer Coefficient

Laminar Flow

Analytical: Smooth surface, negligible axial conduction, negligible viscous dissipation

\begin{aligned} \mathrm{Nu}_{D,\mathrm{UWT}} & =1.077\left(\frac {2R_0}x\right)^{1/3}(\mathrm{Re}_D\mathrm{Pr})^{1/3} \\ \langle\mathrm{Nu}_{D,\mathrm{UWT}}\rangle & =1.615\left(\frac {2R_0}x\right)^{1/3}(\mathrm{Re}_D\mathrm{Pr})^{1/3} \end{aligned}\tag{4.5.43a}

Muzychka et al. (2011): Smooth surface, negligible axial conduction, negligible viscous dissipation

\langle\mathrm{Nu}_{D,\mathrm{UWT}}\rangle_x=\left[\left(1.615\left(\frac {2R_0}x\right)^{1/3}(\mathrm{Re}_D\mathrm{Pr})^{1/3}\right)^5+3.65^5\right]^{1/5}\tag{4.5.43b}

Muzychka et al. (2011): Smooth surface, negligible axial conduction, negligible viscous dissipation, \mathrm{Pe}\to0 (i.e., slug flow)

\langle\mathrm{Nu}_{D,\mathrm{UWT}}\rangle_x=\left[\left(1.128\left(\frac {2R_0}x\right)^{0.5}(\mathrm{Re}_D\mathrm{Pr})^{0.5}\right)^2+5.78^2\right]^{1/2}\tag{4.5.43c}

Gnielinski (2010): Smooth surface, negligible axial conduction, negligible viscous dissipation, uniform heat flux, if x^*=\frac x{D\mathrm{Re}_D\mathrm{Pr}}\ll1

\begin{aligned} \mathrm{Nu}_{D,\mathrm{UHF}}(x) & =\left[\mathrm{Nu}_{D,\mathrm{UHF},1}^3+1+(\mathrm{Nu}_{D,\mathrm{UHF},2}-1)^3\right]^{1/3} \\ \mathrm{Nu}_{D,\mathrm{UHF},1} & =4.364 \\ \mathrm{Nu}_{D,\mathrm{UHF},2} & =1.302(x^*)^{-1/3} \end{aligned}\tag{4.5.139}

Gnielinski (2010): Smooth surface, negligible axial conduction, negligible viscous dissipation, uniform heat flux, if x^*=\frac x{D\mathrm{Re}_D\mathrm{Pr}}\ll1

\begin{aligned} \langle\mathrm{Nu}_{D,\mathrm{UHF}}\rangle_x & =\left[\mathrm{Nu}_{D,\mathrm{UHF},1}^3+0.6^3+\left(\langle\mathrm{Nu}_{D,\mathrm{UHF},2}\rangle_x-0.6\right)^3\right]^{1/3} \\ \mathrm{Nu}_{D,\mathrm{UHF},1} & =4.364 \\ \langle\mathrm{Nu}_{D,\mathrm{UHF},2}\rangle_x & =1.953(x^*)^{-1/3} \end{aligned}\tag{4.5.141}

Gnielinski (2010): Smooth surface, negligible axial conduction, negligible viscous dissipation, uniform wall temperature, if x^*=\frac x{D\mathrm{Re}_D\mathrm{Pr}}\ll1

\begin{aligned} \mathrm{Nu}_{D,\mathrm{UWT}}(x) & =\left[\mathrm{Nu}_{D,\mathrm{UWT},1}^3+0.7^3+(\mathrm{Nu}_{D,\mathrm{UWT},2}-0.7)^3\right]^{1/3} \\ \mathrm{Nu}_{D,\mathrm{UWT},1} & =3.6568 \\ \mathrm{Nu}_{D,\mathrm{UWT},2} & =1.077(x^*)^{-1/3} \end{aligned}\tag{4.5.135}

Gnielinski (2010): Smooth surface, negligible axial conduction, negligible viscous dissipation, uniform wall temperature, if x^*=\frac x{D\mathrm{Re}_D\mathrm{Pr}}\ll1

\begin{aligned} \langle\mathrm{Nu}_{D,\mathrm{UWT}}\rangle_x & =\left[\mathrm{Nu}_{D,\mathrm{UWT},1}^3+0.7^3+\left(\langle\mathrm{Nu}_{D,\mathrm{UWT},2}\rangle_x-0.7\right)^3\right]^{1/3} \\ \mathrm{Nu}_{D,\mathrm{UWT},1} & =3.6568 \\ \langle\mathrm{Nu}_{D,\mathrm{UWT},2}\rangle_x & =1.615(x^*)^{-1/3} \end{aligned}\tag{4.5.137}

Combined Entrance Region

Rectangular Channel

Entrance Lengths

Fully Developed Flow

Hagen-Poiseuille Flow

For smooth surfaces, if a is longer duct side length to center plane

u(y,z)=\frac {16a^2}{\pi^3\mu}\left(-\frac {\mathrm dP}{\mathrm dx}\right)\sum\limits_{j\text{ odd}}^{\infty}(-1)^{(j-1)/2}\left[1-\frac {\cosh\left(\frac {j\pi y}{2a}\right)}{\cosh\left(\frac {j\pi b}{2a}\right)}\right]\cos\left(\frac {j\pi z}{2a}\right)\tag{4.3.14}

Mean velocity, U_m, is

U_m=\frac {a^2}{3\mu}\left(-\frac {\mathrm dP}{\mathrm dx}\right)\left[1-\frac {192a}{\pi^5b}\sum\limits_{j\text{ odd}}^{\infty}\frac 1{j^5}\tanh\left(\frac {j\pi b}{2a}\right)\right]\tag{4.3.15}

If \mathrm{Re}_{D_H}=\frac {U_mD_H}{\nu} and \alpha^*=\frac ba\leq1

C_f\mathrm{Re}_{D_H}=\frac {24}{\left(1+\frac 1{\alpha^*}\right)^2\left[1-\frac {192}{\pi^5\alpha^*}\sum\limits_{j\text{ odd}}\frac 1{j^5}\tanh\left(\frac {j\pi\alpha^*}2\right)\right]}\tag{4.3.16}

Shah and Bhatti (1987) give curve fit as

C_f\mathrm{Re}_{D_H}\approx24\left[1-1.3553\alpha^*+1.9467(\alpha^*)^2-1.7012(\alpha^*)^3+0.9564(\alpha^*)^4-0.2537(\alpha^*)^5\right]\tag{4.3.17}

Heat Transfer Coefficient

Turbulent Flow

Yu et al. (2001): Smooth surface, all thermal boundary conditions, valid for R_0^+>500 and \mathrm{Pr}>\mathrm{Pr}_{\mathrm{tu}}

The quantity \mathrm{Nu}_{D_H,1} should be calculated based off whether the boundary is a uniform wall temperature or uniform heat flux boundary.

Flat Channel

Entrance Lengths

Laminar Flow

Chen (1973): Smooth surface, Reynolds number defined as \mathrm{Re}_{D_H}=\frac {U_mD_H}{\nu}

\frac {l_{\mathrm{ent,hy}}}{D_H}=0.011\mathrm{Re}_{D_H}+\frac {0.315}{1+0.0175\mathrm{Re}_{D_H}}\tag{4.2.15}

Analytical: Smooth surface

\frac {l_{\mathrm{ent,th,UHF}}}{D_H}=0.0115439\mathrm{Re}_{D_H}\mathrm{Pr}\tag{4.5.100}

Analytical: Smooth surface

\frac {l_{\mathrm{ent,th,UWT}}}{D_H}=0.00797\mathrm{Re}_{D_H}\mathrm{Pr}

Analytical: Smooth surface

\frac {l_{\mathrm{ent,ma,UMF}}}{D_H}=0.0115\mathrm{Re}_{D_H}\mathrm{Sc}\tag{4.5.107}

Turbulent Flow

Fully Developed Flow

Hagen-Poiseuille Flow

Center defined as midplane, both surfaces are a distance b from the center line. If a smooth surface and U_m=\frac 1{3\mu}\left(-\frac {\mathrm dP}{\mathrm dx}\right)

u(y)=\frac {b^2}{2\mu}\left(-\frac {\mathrm dP}{\mathrm dx}\right)\left[1-\left(\frac yb\right)\right]\tag{4.3.11}

If \mathrm{Re}_{D_H}=\frac {U_mD_H}{\nu}

C_f\mathrm{Re}_{D_H}=24\tag{4.3.13}

Friction Factor

Laminar Flow

Shah and London (1978): Smooth surface, Reynolds number defined as \mathrm{Re}_{D_H}=\frac {U_mD_H}{\nu}

C_{f,\mathrm{app},x}\mathrm{Re}_{D_H}=\frac {3.44}{(x^*)^{1/2}}+\frac {24+\frac {0.674}{4x^*}-\frac {3.44}{(x^*)^{1/2}}}{1+0.000029(x^*)^{-2}}\tag{4.2.16}

Hagen-Poiseuille (1840): Smooth surface, Reynolds number defined as \mathrm{Re}_{D_H}=\frac {U_mD_H}{\nu}

C_f\mathrm{Re}_{D_H}=24\tag{4.3.13}

Turbulent Flow

Heat Transfer Coefficient

Refer to the five wall boundary conditions for flat channel: (a) equal and constant surface temperature, (b) unequal and constant surface temperature, (c) equal and constant heat flux, (d) unequal and constant heat flux, (e) one surface constant temperature and the other constant heat flux

Laminar Flow

Analytical: Smooth surface, negligible axial conduction, uniform wall heat flux (Case C in Fig. 1)

\mathrm{Nu}_{D_H,\mathrm{UHF}}=\frac {140}{17}\approx8.235\tag{4.4.42}

Tyagi, (1966), Shah and London, (1978): With viscous dissipation and volumetric energy generation, if q_v^*=\frac {\dot q_vD_H}{q_s''} and \mathrm{Br}'=\frac {\mu U_m^2}{q_s''D_H}

\mathrm{Nu}_{D_H,\mathrm{UHF}}=\frac {140}{17}\frac 1{1+\frac 3{68}q_v^*+\frac {108}{17}\mathrm{Br}'}\tag{4.4.43}

Analytical: Smooth surface, negligible axial conduction, unequal and constant heat flux (Case C in Fig. 1)

\begin{aligned} \mathrm{Nu}_{D_H,1} & =\frac {140}{26-9\frac {q_{s2}''}{q_{s1}''}} \\ \mathrm{Nu}_{D_H,2} & =\frac {140}{26-9\frac {q_{s1}''}{q_{s2}''}} \end{aligned}\tag{4.4.47}

Analytical: Smooth surface, negligible axial conduction, negligible viscous dissipation, one surface constant temperature and the other constant heat flux (Case E in Fig. 1)

\begin{aligned} \mathrm{Nu}_{D_H,T} & =4 \\ \mathrm{Nu}_{D_H,q''} & =4 \end{aligned}\tag{4.4.49}

Cheng and Wu (1976): Smooth surface, axial conduction, viscous dissipation, one surface constant temperature and the other constant heat flux (Case E in Fig. 1), assume T_{s1}>T_{s2}, if \mathrm{Br}=\frac {2\mu U_m^2}{k\left[\frac 12(T_{s1}+T_{s2})-T_m\right]}

\begin{aligned} \mathrm{Nu}_{D_H,1} & =\frac {4(1-6\mathrm{Br})}{1-\frac {48}{35}\mathrm{Br}} \\ \mathrm{Nu}_{D_H,2} & =\frac {4(1+6\mathrm{Br})}{1-\frac {48}{35}\mathrm{Br}} \end{aligned}\tag{4.4.56}

Analytical: Smooth surface, negligible axial condution, uniform wall temperature (Case A in Fig. 1)

\mathrm{Nu}_{D_H,\mathrm{UWT}}=7.5407\tag{4.4.53}

Pahor and Strand (1961): Smooth surface, axial conduction, if \mathrm{Pe}=\mathrm{Re}_D\mathrm{Pr}

\mathrm{Nu}_{D_H,\mathrm{UWT}}=\begin{cases} 7.540\left(1+\frac {3.79}{\mathrm{Pe}^2}+\cdots\right) & \qquad\qquad\mathrm{Pe}\gg1 \\ 8.118(1-0.031\mathrm{Pe}+\cdots) & \qquad\qquad\mathrm{Pe}\ll1 \end{cases}

Turbulent Flow

Developed Hydrodynamic, Thermal Entrance Region

Heat Transfer Coefficient

Laminar Flow

Shah and London (1978): Smooth surface, negligible axial conduction, negligible viscous dissipation, if x^*=\frac x{D_H\mathrm{Re}_{D_H}\mathrm{Pr}}

\begin{aligned} \mathrm{Nu}_{D_H,\mathrm{UHF}}(x^*) & =\begin{cases} 1.490\left(x^*\right)^{-1/3} & \qquad x^*\leq2\times10^{-4} \\ 1.490\left(x^*\right)^{-1/3}-0.4 & \qquad2\times10^{-4}<x^*\leq10^{-3} \\ 8.235+8.68\left(x^*\times10^3\right)^{-0.506}e^{-164x^*} & \qquad x^*>10^{-3} \end{cases} \\ \langle\mathrm{Nu}_{D_H,\mathrm{UHF}}\rangle_x & =\begin{cases} 2.236\left(x^*\right)^{-1/3} & \qquad\qquad\qquad\qquad x^*\leq10^{-3} \\ 2.236\left(x^*\right)^{-1/3}+0.9 & \qquad\qquad\qquad\qquad10^{-3}<x^*\leq0.01 \\ 8.235+0.0364\left(x^*\right)^{-1} & \qquad\qquad\qquad\qquad x^*>0.01 \end{cases} \end{aligned}\tag{4.5.101}

Analytical: Smooth surface, if x^*=\frac x{D_H\mathrm{Re}_{D_H}\mathrm{Pr}}

\mathrm{Nu}_{D_H,\mathrm{UWT}}(x^*)=\frac 2{\Gamma\left(\frac 43\right)\left(6x^*\right)^{1/3}}\tag{4.5.127}

Shah and London (1978): Smooth surface, if x^*=\frac x{D_H\mathrm{Re}_{D_H}\mathrm{Pr}}

\begin{aligned} \mathrm{Nu}_{D_H,\mathrm{UWT}}(x^*) & =\begin{cases} 1.233\left(x^*\right)^{-1/3}+0.4 & \qquad x^*\leq10^{-3} \\ 7.541+6.874\left(x^*\times10^{-3}\right)^{-0.488}e^{-245x^*} & \qquad x^*>10^{-3} \end{cases} \\ \langle\mathrm{Nu}_{D_H,\mathrm{UWT}}\rangle_x & =\begin{cases} 1.849\left(x^*\right)^{-1/3} & \qquad\qquad\qquad\qquad\quad x^*\leq5\times10^{-4} \\ 1.849\left(x^*\right)^{-1/3}+0.6 & \qquad\qquad\qquad\quad\qquad5\times10^{-4}<x^*\leq0.006 \\ 7.541+0.0235\left(x^*\right)^{-1} & \qquad\qquad\qquad\qquad\quad x^*>0.006 \end{cases} \end{aligned}\tag{4.5.128}

Gnielinski (2010): Smooth surface, negligible axial conduction, negligible viscous dissipation, \frac {D_H\mathrm{Re}_{D_H}\mathrm{Pr}}x\gg1

\begin{aligned} \langle\mathrm{Nu}_{D_H,\mathrm{UWT}}\rangle_x & =\left[\mathrm{Nu}_{D_H,\mathrm{UWT},1}^3+\mathrm{Nu}_{D_H,\mathrm{UWT},2}^3(x)\right]^{1/3} \\ \mathrm{Nu}_{D_H,\mathrm{UWT},1} & =7.5407 \\ \mathrm{Nu}_{D_H,\mathrm{UWT},2}(x) & =1.841\left(\frac {D_H\mathrm{Re}_{D_H}\mathrm{Pr}}x\right)^{1/3} \end{aligned}\tag{4.5.143}

Turbulent Flow

Concentric Annular Duct

Fully Developed Flow

Hagen-Poiseuille Flow

Smooth surface, if R_i and R_0 are the inner and outer radii respectively

u(r)=\left[R_0^2-r^2+(R_0^2-R_i^2)\frac {\ln\left(r/R_0\right)}{\ln\left(R_0/R_i\right)}\right]\left(-\frac 1{4\mu}\frac {\mathrm dP}{\mathrm dx}\right)\tag{4.3.31}

Mean velocity

General Duct Geometries

Friction Factor

Laminar Flow

Muzychka and Yovanovich (2004): Smooth surface, Reynolds number defined as \mathrm{Re}_{\sqrt A}=\frac {U_m\sqrt A}{\nu}, dimensionless distance defined as x^*=\frac x{\sqrt A\mathrm{Re}_{\sqrt A}}, accurate to within \pm10\%

C_{f,\mathrm{app},x}\mathrm{Re}_{\sqrt A}=\sqrt{\left(\frac {12}{\sqrt{\alpha^*}(1+\alpha^*)\left[1-\frac {192\alpha^*}{\pi^5}\tanh\left(\frac {\pi}{2\alpha^*}\right)\right]}\right)^2+\left(\frac {3.44}{\sqrt{x^*}}\right)^2}\tag{4.2.17}