Condensation Removal by a Centrifugal Field

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Constant temperature cooled rotating disk in a large quiescent body of saturated vapor.

Another method to create artificial gravity involves use of a centrifugal field that is, a rotating disk. The problem studied here addresses a cooled rotating disk in a large quiescent body of pure saturated vapor, as shown in the figure on the right. The liquid forms a continuous film on the disk, and the fluid in this film will move radially outward due to the centrifugal force. Sparrow and Gregg (1959) investigated this problem, which is presented below.

Conservation equations for mass, momentum in the r-, θ-, and z-directions and energy for an incompressible, constant-property liquid are derived below.

$\frac{1}{r}+\frac{\partial }{\partial r}\left( r{{V}_{r}} \right)+\frac{1}{r}\frac{\partial {{V}_{\phi }}}{\partial \phi }+\frac{\partial {{V}_{z}}}{\partial z}=0$	(1)

$\rho \left( \frac{D{{V}_{r}}}{Dt}-\frac{V_{\phi }^{2}}{r} \right)=-\frac{\partial p}{\partial r}+\mu \left( {{\nabla }^{2}}{{V}_{r}}-\frac{2}{{{r}^{2}}}\frac{\partial {{V}_{\phi }}}{\partial \phi }-\frac{{{V}_{r}}}{{{r}^{2}}} \right)$	(2)

$\rho \left( \frac{D{{V}_{\phi }}}{Dt}+\frac{{{V}_{r}}{{V}_{\phi }}}{r} \right)=-\frac{1}{r}\frac{\partial p}{\partial \phi }+\mu \left( {{\nabla }^{2}}{{V}_{\phi }}+\frac{2}{{{r}^{2}}}\frac{\partial {{V}_{r}}}{\partial \phi }-\frac{{{V}_{r}}}{{{r}^{2}}} \right)$	(3)

$\rho \frac{D{{V}_{z}}}{Dt}=-\frac{\partial p}{\partial z}+\mu {{\nabla }^{2}}{{V}_{z}}$	(4)

$\rho {{c}_{p}}\frac{DT}{Dt}=k{{\nabla }^{2}}T$	(5)

where

$\frac{D}{Dt}={{V}_{r}}\frac{\partial }{\partial r}+\frac{{{V}_{\phi }}}{r}\frac{\partial }{\partial \phi }+{{V}_{z}}\frac{\partial }{\partial z}$	(6)

and

${{\nabla }^{2}}=\frac{{{\partial }^{2}}}{\partial {{r}^{2}}}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{{{r}^{2}}}\frac{{{\partial }^{2}}}{\partial {{\phi }^{2}}}+\frac{{{\partial }^{2}}}{\partial {{z}^{2}}}$	(7)

The boundary conditions at the wall, z = 0, are

$V r = 0$ , $V φ = r ω$ , $V z = 0$ , $T = T w$	(8)

where ω is the angular velocity.

The boundary conditions at the liquid-vapor interface, z = δ, are

$τ z r = 0$ , $τ z φ = 0$ , $T = T sat$	(9)

The governing equations are transformed from partial differential equations into ordinary differential equations using similarity transformation variables. The new independent variable is

$\eta ={{\left( \frac{\omega }{v} \right)}^{1/2}}z$	(10)

and the new dependent variables are

$F\left( \eta \right)=\frac{{{V}_{r}}}{r\omega }$	(11)

$G\left( \eta \right)=\frac{{{V}_{\phi }}}{r\omega }$	(12)

$H\left( \eta \right)=\frac{{{V}_{z}}}{{{\left( \omega v \right)}^{1/2}}}$	(13)

$P\left( \eta \right)=\frac{p}{\mu \omega }$	(14)

$\theta \left( \eta \right)=\frac{{{T}_{\text{sat}}}-T}{{{T}_{\text{sat}}}-{{T}_{w}}}=\frac{{{T}_{\text{sat}}}-T}{\Delta T}$	(15)

Transforming eqs. (1) – (5) using the above variables, they become

$H' = - 2 F$	(16)

$F'' = H F' + F 2 - G 2$	(17)

$G'' = H G' + 2 F G$	(18)

$P' = H'' - H H'$	(19)

${\theta }''=\left( \Pr \right)H{\theta }'$	(20)

Combining eqs. (16), (17), and (18) results in

${H}'''=H{H}''-{{\left( {{H}'} \right)}^{2}}/2+2{{G}^{2}}$	(21)

$G'' = H G' - H' G$	(22)

The boundary conditions in terms of the new variables are given as

$H = H' = G = θ = 0, η = 0$	(23)

$H'' = G' = θ = 0, η = η δ$	(24)

To relate n_δ (dimensionless condensate layer thickness) to physical quantities, an energy balance is created:

${{h}_{lv}}\int_{0}^{\delta }{\rho 2\pi r{{V}_{r}}dz+\int_{0}^{\delta }{\rho 2\pi r{{V}_{r}}{{c}_{p}}\left( {{T}_{\text{sat}}}-T \right)dz}}=k{{\left( \frac{\partial T}{\partial z} \right)}_{z=0}}\pi {{r}^{2}}$	(25)

The first term on the left-hand side represents energy released as latent heat; the second term is the energy dissipated by subcooling of the condensate. The right-hand side is the heat transferred from the condensate to the disk over a span of r = 0 to r = r. In terms of the defined variables, the energy balance becomes

< $\frac{{{c}_{p}}\Delta T}{{{h}_{lv}}}=\Pr \frac{H\left( {{\eta }_{\delta }} \right)}{{\theta }'\left( {{\eta }_{\delta }} \right)}$	(26)

Local heat flux to the disk may be computed from Fourier’s law

${q}''={{\left( k\frac{\partial T}{\partial z} \right)}_{z=0}}$	(27)

In terms of the transformed variables, the equation for $q''$ becomes

${q}''=-k\left( {{T}_{\text{sat}}}-{{T}_{w}} \right){{\left( \frac{\omega }{v} \right)}^{1/2}}{{\left( {\partial \theta }/{\partial \eta }\; \right)}_{\eta =0}}$	(28)

The definition of the local heat transfer coefficient is

$h\equiv \frac{{{q}''}}{{{T}_{\text{sat}}}-{{T}_{w}}}$	(29)

Heat-transfer results for high Prandtl number fluids.

Substituting eq. (28) into eq. (29) and rearranging gives

$\frac{h{{\left( \frac{v}{\omega } \right)}^{1/2}}}{k}=-{{\left( {\partial \theta }/{\partial \eta }\; \right)}_{\eta =0}}$	(30)

Evaluating eqs. (26) and (30) from numerical solutions, the heat-transfer results for high Prandtl numbers have been plotted in the figure on the right.

Inspection of the figure reveals that for small values of ${{c}_{p}}\Delta T/{{h}_{\ell v}}<0.1,$ the results are represented by

$\frac{h{{\left( \frac{v}{\omega } \right)}^{1/2}}}{k}=0.904{{\left( \frac{\Pr }{{{{c}_{p}}\Delta T}/{{{h}_{\ell v}}}\;} \right)}^{1/4}}$	(31)

For the limiting case of negligible inertia and heat convective effects, the following limiting relationship is derived

$\delta {{\left( \frac{\omega }{v} \right)}^{1/2}}=1.107{{\left( \frac{{{c}_{p}}\Delta T/{{h}_{\ell v}}}{\Pr } \right)}^{1/2}}$	(32)

Laminar flow of the condensate is expected when $\operatorname{Re}={{r}^{2}}\omega /v\le 3\times {{10}^{5}}$ . The heat transfer coefficient for a rotating disk, h_rot, is related to the heat transfer coefficient for a vertical plate, h_vert, only under the influence of gravity by

$\frac{{{h}_{\text{rot}}}}{{{h}_{\text{vert}}}}={{\left( \frac{8x{{\omega }^{2}}}{3g} \right)}^{{1}/{4}\;}}$	(33)

Condensation on a rotating cone was studied by Sparrow and Hartnett (1961). The uppermost apex was found to have close to the same coefficient as a rotating disk since

$\frac{{{h}_{\text{cone}}}}{{{h}_{\text{disk}}}}={{\left( \sin \phi \right)}^{{1}/{2}\;}}$	(34)

where $φ$ is the half-angle of the cone. A vertical tube spinning about its own axis was experimentally studied by Nicol and Gacesa (1970). At low angular velocity, the overall heat-transfer coefficient measurements were correlated by:

$\frac{\text{Nu}}{{{\left[ g{{L}^{3}}{{h}_{\ell v}}{{\rho }_{\ell }}/{{v}_{\ell }}{{k}_{\ell }}\left( {{T}_{\text{sat}}}-{{T}_{w}} \right) \right]}^{{1}/{4}\;}}}=\left\{ \begin{matrix} 0.0943, & \text{We}\le \text{250} \\ 0.00923\text{W}{{\text{e}}^{0.39}}, & \text{We}>250 \\ \end{matrix} \right.$	(35)

for L/D = 10, where $\text{We}={{\rho }_{\ell }}{{\omega }^{2}}{{D}^{3}}/4\sigma$ is the Weber number and $\text{Nu}=\bar{h}D/k$ . This correlation holds until the Nusselt number is tripled. At high angular velocities, the effect of gravity is not as important and the correlation becomes

$Nu = 12.26We 0.496$	(36)

References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

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

Nicol, A.A. and Gaceda, M., 1970, “Condensation of Steam on a Rotating Vertical Cylinder,” ASME Journal of Heat Transfer, Vol. 92, pp. 144-152.

Sparrow, E.M. and Gregg, J.L., 1959, “A Theory of Rotating Condensation,” Transactions of ASME, Vol. 81, 113-120.

Sparrow, E. M. and Hartnett, J. P., 1961, “Condensation on a Rotating Cone,” ASME Journal of Heat Mass Transfer, Vol. 83, pp. 101-102.

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Condensation Removal by a Centrifugal Field

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