Annular Space between Concentric Cylinders and Spheres

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Natural convection in spaces between long horizontal concentric cylinders [see Fig. 1 from Natural Convection in Enclosures (e)] or between spheres is very complicated. The only practical approach to analyze the problem is via numerical solution. The physical model of natural convection in annular space between concentric cylinders is shown in Fig. 1. Since the problem is axisymmetric, one only needs to study the right half of the domain. In the coordinate system shown in Fig. 1, the governing equations are


\frac{1}{r}\frac{\partial u}{\partial \theta }+\frac{\partial v}{\partial r}+\frac{v}{r}=0 \qquad \qquad(1)


\frac{u}{r}\frac{\partial u}{\partial \theta }+v\frac{\partial u}{\partial r}+\frac{uv}{r}=-\frac{1}{\rho r}\frac{\partial p}{\partial \theta }+\nu \left( \frac{1}{{{r}^{2}}}\frac{{{\partial }^{2}}u}{\partial {{\theta }^{2}}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{{{\partial }^{2}}u}{\partial {{r}^{2}}}-\frac{u}{{{r}^{2}}}+\frac{2}{{{r}^{2}}}\frac{\partial v}{\partial \theta } \right)


-g\beta (T-{{T}_{ref}})\sin \theta \qquad \qquad(2)


\frac{u}{r}\frac{\partial v}{\partial \theta }+v\frac{\partial v}{\partial r}-\frac{{{u}^{2}}}{r}=-\frac{1}{\rho }\frac{\partial p}{\partial r}+\nu \left( \frac{1}{{{r}^{2}}}\frac{{{\partial }^{2}}v}{\partial {{\theta }^{2}}}+\frac{1}{r}\frac{\partial v}{\partial r}+\frac{{{\partial }^{2}}v}{\partial {{r}^{2}}}-\frac{v}{{{r}^{2}}}-\frac{2}{{{r}^{2}}}\frac{\partial u}{\partial \theta } \right)


\text{                               +}g\beta (T-{{T}_{ref}})\sin \theta  \qquad \qquad(3)



 Natural convection in horizontal annular space between concentric cylinders
Figure 1: Natural convection in horizontal annular space between concentric cylinders


\frac{u}{r}\frac{\partial T}{\partial \theta }+v\frac{\partial T}{\partial r}=\alpha \left( \frac{1}{{{r}^{2}}}\frac{{{\partial }^{2}}T}{\partial {{\theta }^{2}}}+\frac{{{\partial }^{2}}T}{\partial {{r}^{2}}}+\frac{1}{r}\frac{\partial T}{\partial r} \right) \qquad \qquad(4)


where Tref is a reference temperature and:


{{p}_{eff}}=p+\rho g\sin \theta -\rho g\cos \theta \qquad \qquad(5)


is the effective pressure. Equations (1) – (4) are subject to the following boundary conditions:


u=0,\text{ }\frac{\partial v}{\partial \theta }=\frac{\partial T}{\partial \theta }=0,\text{ at }\theta =0\text{ or }\pi \qquad \qquad(6)


u=v=0,\text{ }T={{T}_{i}},\text{ at }r={{r}_{i}} \qquad \qquad(7)


u=v=0,\text{ }T={{T}_{o}},\text{ at }r={{r}_{o}} \qquad \qquad(8)


Date (1986) solved this problem numerically using a modified SIMPLE algorithm for δ / Di = 0.8 and 0.15. Figure 2 shows the contours of the stream functions and isotherms at Ra = 4.7\times {{10}^{4}}. The dimensionless stream function, dimensionless temperature and Rayleigh number are defined as the following

\Psi =\frac{\psi }{\alpha },\text{ }\Theta =\frac{T-{{T}_{o}}}{{{T}_{i}}-{{T}_{o}}},\text{ Ra}=\frac{g\beta \Delta T{{\delta }^{3}}}{\nu \alpha } \qquad \qquad(9)


where the stream function, ψ, is defined as

u=-\frac{\partial \psi }{\partial r},\text{  }v=\frac{1}{r}\frac{\partial \psi }{\partial \theta } \qquad \qquad(10)


It can be seen that isotherms concentrate near the lower portion of the surface of the inner cylinder and the upper portion of the surface of the outer cylinder, which are indications of the development of thermal boundaries near the heated and cooled surfaces. While the contours of the streamlines have kidney-like shapes with the center of the flow rotation moves upward due to effect of natural convection. The streamlines and isotherms shown in Fig. 2 agreed very well


 Natural convection in a horizontal annulus (Pr =0.7, δ / Di=0.8, Ra =4.7×104)
Figure 2: Natural convection in a horizontal annulus (Pr =0.7, δ / Di=0.8, Ra =4.7×104)


with results obtained by Kuehn and Goldstein (1976) using vortex-stream function method. The rate of heat transfer per unit length of the annulus can be calculated by the following correlation (Raithby and Hollands, 1975; Bejan, 2004):

{q}'\cong \frac{2.425k({{T}_{i}}-{{T}_{o}})}{{{[1+{{({{D}_{i}}/{{D}_{o}})}^{3/5}}]}^{5/4}}}{{\left( \frac{\Pr \text{R}{{\text{a}}_{{{D}_{i}}}}}{0.861+\Pr } \right)}^{1/4}} \qquad \qquad(11)


where the Rayleigh number is defined as

\text{R}{{\text{a}}_{{{D}_{i}}}}=\frac{g\beta ({{T}_{i}}-{{T}_{o}})D_{i}^{3}}{\nu \alpha } \qquad \qquad(12)


Equation (11) is valid for 0.7<\Pr <6000\text{ and Ra}<{{10}^{7}}. The thermophysical properties of the fluid should be evaluated at the mean temperature (Ti + To) / 2. The scales of the thermal boundary layer on the inner surface of the outer cylinder and on the outer surface of the inner cylinder are:


{{\delta }_{o}}\tilde{\ }{{D}_{o}}Ra_{{{D}_{o}}}^{-1/4},\text{ }{{\delta }_{i}}\tilde{\ }{{D}_{i}}Ra_{{{D}_{i}}}^{-1/4}


It is obvious that δo > δi since D0 > Di. Equation (6.238) will be valid only if the boundary layer thickness is less than the gap between the two cylinders, i.e. only if δ0 < D0D1. Under lower Rayleigh numbers, on the other hand, we have


{{D}_{o}}\text{Ra}_{{{D}_{o}}}^{-1/4}>{{D}_{o}}-{{D}_{i}} \qquad \qquad(13)


and the heat transfer mechanism between two cylinders will approach pure conduction. In this case, the heat transfer rate obtained from eq. (11) may be less than that obtained from the pure conduction model, which does not make sense. Instead of using eq. (13) to check the validity of eq. (11), an alternative method is to calculate the heat transfer rate via eq. (11) and pure conduction model, and the larger of the two heat transfer rates should be used.


For natural convection in the annulus between two concentric spheres, the trends for the evolution of the flow pattern and isotherms are similar to the concentric cylinder except the circulation between concentric spheres has the shape of a doughnut. The empirical correlation for the heat transfer rate is (Raithby and Hollands, 1975; Bejan, 2004):


q\cong \frac{2.325k{{D}_{i}}({{T}_{i}}-{{T}_{o}})}{{{[1+{{({{D}_{i}}/{{D}_{o}})}^{7/5}}]}^{5/4}}}{{\left( \frac{\Pr \text{R}{{\text{a}}_{{{D}_{i}}}}}{0.861+\Pr } \right)}^{1/4}} \qquad \qquad(14)


where the definition of Rayleigh number is the same as for eq. (12). Equation (14) is valid for 0.7<\Pr <4000\text{ and Ra}<{{10}^{4}}. The thermophysical properties of the fluid should be evaluated at the mean temperature, (Ti + To) / 2. Similar to the annulus between two cylinders, one should also make use of both eq. (14) and the pure conduction model and retain the larger of the two values obtained.

References

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

Date, A.W., 1986, “Numerical Prediction of Natural Convection Heat Transfer in Horizontal Annulus,” International Journal of Heat and Mass Transfer, Vol. 29, pp. 1457-1464.

Kuehn, T.H., and Goldstein, R.J., 1976, “An Experimental and Theoretical Study of Natural Convection in the Annulus between Horizontal Concentric Cylinders,” J. Fluid Mech., Vol. 74, pp. 695-719.

Raithby, G.D., and Hollands, K.G.T., 1975, “A General Method of Obtaining Approximate Solutions to Laminar and Turbulent Free Convection Problems,” Irvine, T.F., and Hartnett, J.P., Eds., Advances in Heat Transfer, Vol. 11, pp. 265-315, Academic Press, New York, NY.

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