Suction at the Porous Wall

In condensation, both vapor shear and suction through a porous wall directly reduce the film thickness on the wall and, therefore, significantly increase the heat transfer. Faghri and Chow (1988) incorporated both of these effects into their investigation, which is described in detail below.

Faghri and Chow (1988) investigated a system using an annular pipe with its inner wall made of a porous material (See figure on the right). In this system, steam flows through a porous pipe and cooling liquid flows through the annular region between a solid outer pipe and the porous inner pipe. The steam condenses onto the inner wall of the porous material, because the cooling liquid keeps the temperature of the porous wall below that of the steam’s saturation temperature. A small pressure drop is maintained between the steam and the cooling flow in order to drive the condensate through the porous material to the annular region, where it mixes with the cooling liquid and is swept away to a heat exchanger.

The Nusselt analysis for laminar film condensation was extended to flow inside a tube with a constant-temperature porous wall. The shear stress at the liquid-vapor interface decreases due to condensation. Assuming that the properties of the fluid remain constant and that the curvature of the condensate film can be neglected, the following nondimensional momentum and energy boundary layer equations can be written: $\frac{\partial u{}_{\ell }^{*}}{\partial {{y}^{*}}}=-{{\Pr }_{\ell }}\frac{{{\partial }^{2}}u_{\ell }^{*}}{\partial {{y}^{{{*}^{2}}}}}$ (1) $\frac{\partial {{T}^{*}}}{\partial {{y}^{*}}}=-\frac{{{\partial }^{2}}{{T}^{*}}}{\partial {{y}^{{{*}^{2}}}}}$ (2)

where the suction velocity at wall v ≅ vw= const. The nondimensional variables in eqs. (1) and (2) are defined as follows: ${{y}^{*}}=-\frac{y{{v}_{w}}}{{{\alpha }_{\ell }}}$ ${{u}^{*}}=-\frac{{{u}_{\ell }}}{{{v}_{w}}}$ ${{T}^{*}}=\frac{T-{{T}_{w}}}{{{T}_{sat}}-{{T}_{w}}}$ (3)

The inertial and convective effects are approximated in eqs. (1) and (2), respectively; therefore the results are the asymptotic behavior of the conservation of momentum and energy equations as ${{\Pr }_{\ell }}\to\infty$.

The boundary conditions for a co-current liquid-vapor flow are as follows: At y* = 0

 u * = 0, T * = 0 (4)

At y* = δ* ${{T}^{*}}=1,\text{ }{{N}_{2}}=\frac{{{\tau }_{\delta }}{{\alpha }_{\ell }}}{{{\mu }_{\ell }}v_{w}^{2}}$ (5)

where N2 in eq. (5) is the dimensionless shear stress at the liquid-vapor interface and δ* is the dimensionless film thickness given as ${{\delta }^{*}}=\frac{-{{v}_{w}}\delta }{{{\alpha }_{\ell }}}$ (6)

Integrating eq. (1) and applying boundary conditions, eqs. (4) and (5), the dimensionless velocity profile is obtained: $u_{\ell }^{*}={{N}_{2}}\exp \left( \frac{{{\delta }^{*}}}{{{\Pr }_{\ell }}} \right){{\Pr }_{\ell }}\left[ \exp \left( \frac{-{{y}^{*}}}{{{\Pr }_{\ell }}} \right)-1 \right]$ (7)

The dimensionless temperature profile across the liquid film can be obtained by integrating eq. (2) twice and applying appropriate boundary conditions, eqs. (4) and (5), i.e., ${{T}^{*}}=\frac{{{e}^{-{{y}^{*}}}}-1}{{{e}^{-{{\delta }^{*}}}}-1}$ (8)

At the liquid-vapor interface, an energy balance is needed to equate the latent heat given off by the condensation process and the heat conducted from the interface into the liquid film, i.e., $\rho {{h}_{\ell v}}\left\{ {{v}_{w}}-\frac{d}{dx}\int_{0}^{\delta }{{{u}_{\ell }}dy} \right\}=-k{{\left. \frac{\partial T}{\partial y} \right|}_{y=\delta }}$ (9)

The assumptions inherent in eq. (9) are that the vapor is not superheated and that the condensate flow is laminar. It is also assumed that the condensate film is not subcooled. Equation (9) can be nondimensionalized as shown here: $\frac{d}{d{{x}^{*}}}\int_{0}^{{{\delta }^{*}}}{u_{\ell }^{*}d{{y}^{*}}=Ja\left[ \frac{-{{e}^{-{{\delta }^{*}}}}}{{{e}^{-{{\delta }^{*}}}}-1} \right]-1}$ (10)

where $\text{Ja}=\frac{{{c}_{p\ell }}\left( {{T}_{sat}}-{{T}_{w}} \right)}{{{h}_{\ell v}}}$ (11) ${{x}^{*}}=-\frac{x{{v}_{w}}}{{{\alpha }_{\ell }}}$ (12)

Using $u_{\ell }^{*}$ from eq. (7) to evaluate eq. (10), we obtain $\frac{d{{\phi }_{1}}}{d{{x}^{*}}}={{\phi }_{2}}$ (13)

where ${{\phi }_{1}}=\frac{{{N}_{2}}\Pr _{\ell }^{2}}{\exp \left( -{{\delta }^{*}}/{{\Pr }_{\ell }} \right)}\left\{ \exp \left( -\frac{{{\delta }^{*}}}{{{\Pr }_{\ell }}} \right)+\frac{{{\delta }^{*}}}{{{\Pr }_{\ell }}}-1 \right\}$ (14)

and ${{\phi }_{2}}=\text{Ja}\left[ \frac{{{e}^{-{{\delta }^{*}}}}}{1-{{e}^{-{{\delta }^{*}}}}} \right]-1$ (15)

Equations (13) – (15) show that δ* is a function of x*, and the integration of eq. (13) would also lead to this expression.

As a primary step, a few definitions that will be incorporated into the final numerical solution procedure to generalize this problem must be shown. To start, the mass flow rate of the vapor is related to the mass flow rate of the condensate, including the suction force at the porous wall. Performing a mass balance and writing it in terms of the Reynolds numbers of the vapor and liquid, the resulting equation is as follows: ${{\operatorname{Re}}_{v,e}}-{{\operatorname{Re}}_{v}}=\frac{{{\mu }_{\ell }}}{{{\mu }_{v}}}\left( {{\operatorname{Re}}_{\ell }}-{{\operatorname{Re}}_{\ell ,e}} \right)+4\frac{{{\mu }_{\ell }}{{x}^{*}}}{{{\mu }_{v}}{{\Pr }_{\ell }}}$ (16)

where ${{\operatorname{Re}}_{v}}=\frac{{{u}_{v}}{{D}_{h}}}{{{\nu }_{v}}}$ (17) ${{\operatorname{Re}}_{\ell }}=\frac{4\int_{0}^{\delta }{{{\rho }_{\ell }}{{u}_{\ell }}dy}}{{{\mu }_{\ell }}}=\frac{4{{\phi }_{1}}}{{{\Pr }_{\ell }}}$ (18)

and e indicates the entering condition at x = 0. Also, the shear stress at the interface is due to both the friction between the liquid and the vapor and the momentum gained by condensing liquid from the faster-moving vapor. A relationship for dimensionless shear stress at the interface, N2, is given as follows: ${{N}_{2}}=\frac{{{C}_{f}}}{2}\frac{{{\rho }_{v}}}{{{\rho }_{\ell }}}\frac{1}{{{\Pr }_{\ell }}}{{\left( u_{v}^{*}-u_{\ell ,\delta }^{*} \right)}^{2}}+\frac{Ja}{{{\Pr }_{\ell }}}\frac{{{e}^{-\delta *}}}{1-{{e}^{-\delta *}}}\left( u_{v}^{*}-u_{\ell ,\delta }^{*} \right)$ (19)

where the dimensionless vapor velocity is $u_{v}^{*}=-\frac{{{\operatorname{Re}}_{v}}}{{{\operatorname{Re}}_{w}}}\frac{{{\nu }_{v}}}{{{\nu }_{\ell }}}$ (20)

and the radial Reynolds number at the wall is given as ${{\operatorname{Re}}_{w}}=\frac{{{D}_{h}}{{v}_{w}}}{{{\nu }_{\ell }}}$ (21)

The dimensionless axial velocity of the liquid at the liquid-vapor interface is given by $u_{\ell ,\delta }^{*}=-{{N}_{2}}\exp \left( \frac{{{\delta }^{*}}}{{{\Pr }_{\ell }}} \right){{\Pr }_{\ell }}\left\{ \exp \left( -\frac{{{\delta }^{*}}}{{{\Pr }_{\ell }}} \right)-1 \right\}$ (22)

The frictional coefficient in eq. (19) depends on whether the vapor flow is laminar or turbulent, i.e., ${{C}_{f}}=\left\{ \begin{matrix} 16/{{\operatorname{Re}}_{v}}\text{ } & {{\operatorname{Re}}_{v}}\le 2300 \\ 0.046\operatorname{Re}_{v}^{-0.2}\left( 1+850F \right) & {{\operatorname{Re}}_{v}}>2300 \\ \end{matrix} \right.$ (23)

where F is an empirical coefficient recommended by Henstock and Henratty (1976) that is based on the average shear stress around the circumference of the annular flow. $F=\frac{{{[{{(0.707\operatorname{Re}_{\ell }^{0.5})}^{2.5}}+{{(0.0379\operatorname{Re}_{\ell }^{0.9})}^{2.5}}]}^{0.4}}({{\mu }_{\ell }}/{{\mu }_{v}})}{{{({{\rho }_{\ell }}/{{\rho }_{v}})}^{0.5}}\operatorname{Re}_{v}^{0.9}}$ (24)

Now assuming that ${{\delta }^{*}}/{{\Pr }_{\ell }}$ << 1, which is a very good assumption for most liquids, and also assuming that the dimensionless shear stress, N2, is constant, eqs. (14) – (15) can be approximated by ${{\phi }_{1}}=\frac{{{N}_{2}}{{\delta }^{{{*}^{2}}}}}{2}$ (25) ${{\phi }_{2}}=\frac{Ja}{{{\delta }^{*}}}-1$ (26)

Substituting eqs. (25) and (26) into eq. (15) yields ${{\delta }^{{{*}^{2}}}}\frac{d{{\delta }^{*}}}{d{{x}^{*}}}+\frac{{{\delta }^{*}}}{{{N}_{2}}}=\frac{\text{Ja}}{{{N}_{2}}}$ (27)

Integrating eq. (27) with a boundary condition of δ* = 0 at x* = 0, one obtains an expression relating δ* to x* that must be solved numerically. ${{x}^{*}}=-\frac{1}{2}{{N}_{2}}{{\delta }^{{{*}^{2}}}}-{{N}_{2}}\text{Ja}{{\delta }^{*}}+{{N}_{2}}\text{J}{{\text{a}}^{2}}\ln \left| \frac{\text{Ja}}{\text{Ja}-{{\delta }^{*}}} \right|$ (29)

Faghri and Chow (1988) presented results from the iterative calculations of the above set of equations for steam condensing at one atmosphere. Porous wall temperatures of both 85 °C and 70 °C were considered. The liquid properties were assumed to be constant along the porous wall.

References

Faghri, A. and Chow, L.C., 1988, “Forced Condensation in a Tube with Suction at the Wall: Microgravitational Application,” Journal of Heat Transfer, Vol. 110, pp. 982-985.

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.

Henstock, W.H., and Hanratty, T.S., 1976, “The Interfacial Drag and the Height of the Wall Layer in Annular Flows,” AIChE Journal, Vol. 22, pp. 990-1000.