Natural convection on cylinders and spheres

Vertical Cylinder

Figure 1 Boundary layer for natural convection over vertical cylinder.

Natural convection on a vertical cylinder can find its applications in the cooling of nuclear reactors, large-current bus in power plants, as well as in the human body. A vertical cylinder has three surfaces: the top surface, the bottom surface and the vertical surface. The natural convection from the top and bottom surfaces can be calculated using the approaches discussed in the previous subsection. For the vertical surface of the cylinder, the development of the boundary layer is shown in Fig. 1. If the ratio of diameter to height, D/L, is large enough, the thermal boundary layer thickness will be much smaller than the radius of the cylinder and the correlations for natural convection over a vertical flat plate can be used to calculate the natural convection over the vertical cylinder. For fluid with Prandtl number greater than 1, the condition under which the correlation for flat plate is applicable is (Bejan, 2004):

 $\frac{D}{L}>\text{Ra}_{L}^{-1/4}$ (1)

This condition is referred to as the “thick cylinder limit.” If D/L is less than $\text{Ra}_{L}^{-1/4}$, the boundary layer thickness will be comparable to the radius of the cylinder and the effect of surface curvature cannot be neglected. For this case – referred to as “thin cylinder limit” – the governing equations must be given in a cylindrical coordinate system:

 $\frac{\partial u}{\partial x}+\frac{1}{r}\frac{\partial (vr)}{\partial y}=0$ (2)
 $u\frac{\partial u}{\partial x}+\frac{v}{r}\frac{\partial (ur)}{\partial r}=\nu \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial u}{\partial r} \right)+g\beta (T-{{T}_{\infty }})$ (3)
 $u\frac{\partial T}{\partial x}+\frac{v}{r}\frac{\partial (Tr)}{\partial y}=\alpha \frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial T}{\partial r} \right)$ (4)

The boundary conditions for eqs. (2) – (4) are

 u = v = 0, T = Tw, at r = R (5)
 $u=v=0,\text{ }T={{T}_{\infty }}\text{, at }r\to \infty$ (6)

The integral momentum and energy equations can be obtained by integrating eqs. (2) – (4) with respect to r in the interval of (R, R + δ):

 $\frac{d}{dx}\int_{R}^{R+\delta }{{{u}^{2}}}rdr=-\nu R{{\left. \frac{\partial u}{\partial r} \right|}_{r=R}}+g\beta \int_{R}^{R+\delta }{(T-{{T}_{\infty }})rdr}$ (7)
 $\frac{d}{dx}\int_{R}^{R+\delta }{u(T-{{T}_{\infty }})}rdr=-\alpha R{{\left. \frac{\partial T}{\partial r} \right|}_{r=R}}$ (8)

In order to simplify the solution procedure, it is assumed that the velocity and temperature profiles that were used in the integral solution of natural convection over a vertical flat plate, eqs. (23) and (24), are still valid for natural convection over a thin cylinder – except to replace y with rR. Substituting the following velocity and temperature profiles:

$\frac{u}{U}=\frac{y}{\delta }{{\left( 1-\frac{y}{\delta } \right)}^{2}}$

$\frac{T-{{T}_{\infty }}}{{{T}_{w}}-{{T}_{\infty }}}={{\left( 1-\frac{y}{{{\delta }_{t}}} \right)}^{2}}$

into eqs. (7) and (8), one obtains the following integral equations:

 $\frac{d}{dx}\left[ (8R+3\delta )\delta {{U}^{2}} \right]=-840\nu R\frac{U}{\delta }+70g\beta ({{T}_{w}}-{{T}_{\infty }})\delta (4R+\delta )$ (9)
 $\frac{d}{dx}[(7R+2\delta )U\delta ]=\frac{420\alpha R}{\delta }$ (10)

Unlike the case of natural convection over a vertical flat plate, eqs. (9) and (10) do not suggest that U and δ can be expressed as a simple function of x. Le Fevre and Ede (1956) assumed that:

 $U={{U}_{0}}+\frac{{{U}_{1}}}{R}+\frac{{{U}_{2}}}{{{R}^{2}}}+\ldots \text{ and }\delta ={{\delta }_{0}}+\frac{{{\delta }_{1}}}{R}+\frac{{{\delta }_{2}}}{{{R}^{2}}}+\ldots$ (11)

Substituting the above expression into eqs. (9) and (10) and grouping the terms for the same order of R, equations for U0, δ0, U1, δ1, and so on, can be obtained:

 $8{{\delta }_{0}}\frac{d}{dx}(U_{0}^{2}{{\delta }_{0}})=-840\nu {{U}_{0}}+280g\beta ({{T}_{w}}-{{T}_{\infty }})\delta _{0}^{2}$ (12)
 $7{{\delta }_{0}}\frac{d}{dx}({{U}_{0}}{{\delta }_{0}})=420\alpha$ (13)
 ${{\delta }_{0}}\frac{d}{dx}(8\delta _{0}^{2}U_{0}^{2}+8{{\delta }_{1}}U_{0}^{2}+16{{\delta }_{0}}{{U}_{0}}{{U}_{1}})+8{{\delta }_{1}}\frac{d}{dx}({{\delta }_{0}}U_{0}^{2})=-840\nu {{U}_{1}}+70g\beta ({{T}_{w}}-{{T}_{\infty }})(8{{\delta }_{0}}{{\delta }_{1}}+\delta _{0}^{3})$ (14)
 ${{\delta }_{0}}\frac{d}{dx}(2\delta _{0}^{2}{{U}_{0}}+7{{\delta }_{1}}{{U}_{0}}+7{{\delta }_{0}}{{U}_{1}})+7{{\delta }_{1}}\frac{d}{dx}({{\delta }_{0}}{{U}_{0}})=0$ (15)

It is assumed that:

U0 = A0xm, δ0 = B0xn

and one can use eqs. (12) and (13) to obtain m = 1 / 2 and n = 1 / 4 and:

 ${{A}_{0}}=\frac{80\alpha }{B_{0}^{2}},\text{ }B_{0}^{4}=\frac{80(20+21\Pr ){{\alpha }^{2}}}{7g\beta ({{T}_{w}}-{{T}_{\infty }})}$ (16)

By following a similar procedure, one can use eqs. (14) and (15) to obtain:

U1 = A1x3 / 4, δ1 = B1x1 / 2

where

 ${{A}_{1}}=-\frac{4(656+315\Pr )\alpha }{7(64+63\Pr ){{B}_{0}}},\text{ }{{B}_{1}}=-\frac{(272+315\Pr )B_{0}^{2}}{35(64+63\Pr )}$ (17)

If one only considers the first two terms in eq. (10), the boundary layer thickness becomes:

 $\delta \approx {{B}_{0}}{{x}^{1/4}}+\frac{{{B}_{1}}{{x}^{1/2}}}{R}$ (18)

The local heat transfer coefficient at the surface of the vertical cylinder is:

${{h}_{x}}=-\frac{k}{{{T}_{w}}-{{T}_{\infty }}}{{\left( \frac{\partial T}{\partial r} \right)}_{r=R}}=\frac{2k}{\delta }$

and the local Nusselt number is:

$\text{N}{{\text{u}}_{x}}=\frac{{{h}_{x}}x}{k}=\frac{2x}{\delta }$

Substituting eq. (18) into the above expression, one obtains

$\text{N}{{\text{u}}_{x}}=\frac{2{{\left[ \frac{7\text{G}{{\text{r}}_{x}}\Pr }{80(20+21\Pr )} \right]}^{1/4}}}{1-{{\left[ \frac{80(20+21\Pr )}{7\text{G}{{\text{r}}_{x}}{{\Pr }^{2}}} \right]}^{3/4}}\frac{272+315\Pr }{35(64+63\Pr )}\frac{x}{R}}$

Since the second term in the denominator of the above expression is much less than 1, Le Fevre and Ede (1956) suggested that the above expression for local Nusselt number can be simplified to:

 $\text{N}{{\text{u}}_{x}}={{\left[ \frac{7\text{G}{{\text{r}}_{x}}{{\Pr }^{2}}}{5(20+21\Pr )} \right]}^{1/4}}+\frac{2(272+315\Pr )}{35(64+63\Pr )}\frac{x}{R}$ (19)

The average Nusselt number can then be obtained as:

 ${{\overline{\text{Nu}}}_{L}}=\frac{4}{3}{{\left[ \frac{7\text{R}{{\text{a}}_{L}}\Pr }{\text{5(20+21Pr)}} \right]}^{1/4}}+\frac{4(272+315\Pr )L}{35(64+63\Pr )D}$ (20)

where the characteristic length in ${{\overline{\text{Nu}}}_{L}}$ and RaL is the height of the cylinder. It can be seen that the first term on the right-hand side of eq. (20) has a form similar to eq. (28), and that the second term represents the effect of aspect ratio of the vertical cylinder. The second term does not depend on the Rayleigh number.

Horizontal Cylinder and Sphere

Figure 2 Boundary layer for natural convection over horizontal cylinder or sphere.

When a horizontal cylinder or sphere with temperature Tw is immersed in a fluid with temperature T (${{T}_{w}}>{{T}_{\infty }}$), a boundary layer develops along the curved surface. The boundary layer thickness is a function of the angle φ ($\phi ={{0}^{\circ }}$ is at the bottom of the cylinder or sphere), as shown in Fig. 2. The similarity solution that worked for the case of the vertical plate does not work for natural convection over a horizontal cylinder or sphere. Merk and Prins (1953, 1954a, b) obtained an integral solution by assuming the momentum and thermal boundary layer thicknesses are identical. The results show that the local Nusselt number is highest at the bottom where the boundary layer is thinnest. As the angle φ increases, the thickness of the boundary layer increases and the local Nusselt number decreases. Although the integral solution can yield results all the way to the top where $\phi ={{180}^{\circ }}$ and Nuφ = 0, the result beyond $\phi ={{165}^{\circ }}$ is no longer applicable because boundary layer separation occurred and plume flow takes place. Based on the integral solution, Merk and Prins (1954b) recommended the following empirical correlation for natural convection over a horizontal cylinder and sphere:

 ${{\overline{\text{Nu}}}_{D}}=C\text{R}{{\text{a}}_{D}}^{1/4}$ (21)

where the characteristic length in the average Nusselt number and Rayleigh number is the diameter of the cylinder or sphere. The constant C is a function of Prandtl number and can be found from the following table.

Constant in the empirical correlation for natural convection over horizontal cylinder and sphere

 Prandtl number 0.7 1 10 100 ∞ C Cylinder 0.436 0.456 0.52 0.523 0.523 Sphere 0.474 0.497 0.576 0.592 0.595

Practically, the empirical correlations based on experimental results are more useful. Churchill and Chu (1975) recommended the following correlation for horizontal cylinders:

 ${{\overline{\text{Nu}}}_{D}}={{\left\{ 0.6+\frac{\text{0}\text{.387Ra}_{D}^{1/6}}{{{[1+{{(0.559/\Pr )}^{9/16}}]}^{8/27}}} \right\}}^{2}}$ (22)

which has the same form as the correlation for vertical plate, eq. (29), except the characteristic length has been changed from vertical plate height to the diameter of the cylinder. Equation (22) covers all Prandtl number and Rayleigh number between 0.1 and 1012.

For natural convection over a sphere, Churchill (1983) suggested that the following correlation can best fit the available experimental data:

 ${{\overline{\text{Nu}}}_{D}}=2+\frac{\text{0}\text{.589Ra}_{D}^{1/4}}{{{[1+{{(0.469/\Pr )}^{9/16}}]}^{4/9}}}$ (23)

where the average Nusselt number, ${{\overline{\text{Nu}}}_{D}}$, and the Rayleigh number, RaD, are based on the diameter of the sphere. Equation (23) is valid for the case where $\Pr \ge 0.7$and the Rayleigh number is less than 1011.

References

Bejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, New York.

Churchill, S.W., 1983, “Free Convection around Immersed Bodies,” Heat Exchanger Design Handbook, Schlunder, E.U., ed., Hemisphere Publishing, New York, NY.

Churchill, S. W., and Chu, H.H.S., 1975, “Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate,” Int. J. Heat Mass Transfer, Vol. 18, pp. 1323-1329.

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

Le Fevre, E.J., 1956, “Laminar Free Convection from a Vertical Plane Surface,” Proceedings of the 9th International Congress of Applied Mechanics, Brussels, Vol. 4, pp. 168-174.

Merk, H.J., and Prins, J.A. 1953, Thermal Convection Laminar Boundary Layer I, Applied Scientific Research, Vol. A4, pp. 11-24.

Merk, H.J., and Prins, J.A. 1954a, Thermal Convection Laminar Boundary Layer II, Applied Scientific Research, Vol. A4, pp. 195-206.

Merk, H.J., and Prins, J.A. 1954b, Thermal Convection Laminar Boundary Layer III, Applied Scientific Research, Vol. A4, pp. 207-221.
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