# Condensate Removal by Forced Vapor Flow in Microgravity Environment

Physical model for convective condensation in an annulus.

In condensation application, the main resistance to heat transfer comes from conduction across the condensate film. Therefore, thinning the condensate film is crucial for better heat transfer in a microgravity environment. This thinning is difficult for two main reasons: (1) no gravity is present to help flush the condensate away, and (2) lightweight designs are critical, so bulky pumps and blowers are not feasible.

As mentioned above, one mechanism of condensate removal in a microgravity environment is vapor shear at the liquid-vapor interface. Faghri and Chow (1991) proposed the use of an annular pipe as a condenser, with vapor shear as the driving force for condensate removal. The physical model they investigated is shown in the figure on the right. Here, saturated vapor enters the annular region at x = 0 while the inner and outer walls are held at constant temperatures Tw,i and Tw,0 respectively. Condensate films form on both laminar and turbulent flows. A pressure drop occurs along the x-axis due to the friction at the walls. Also, the vapor temperature, density, and viscosity were all assumed to be constant.

An annulus would be more efficient than a conventional pipe due to the larger heat transfer surface area. A shorter pipe therefore would also be possible and would help with meeting the lightweight space requirement. The vapor velocity would be greater in an annular pipe than in a conventional pipe with the same outer diameter and vapor mass flow rate, thus increasing the vapor shear at the liquid-vapor interface.

Analysis of this annular system begins by assuming that the film thicknesses, δi at inner wall and δ0 at outer wall are small compared to the inner diameter of the annulus, Di. A momentum equation, neglecting inertial term, can be written for the velocity of the liquid film on either the inner or outer wall.

 ${{\mu }_{\ell }}\frac{{{d}^{2}}{{u}_{\ell }}}{d{{y}^{2}}}-\frac{dp}{dx}=0$ (1)

The boundary conditions for the liquid film on the inner wall are

 ${{u}_{\ell ,i}}=0,\begin{matrix} {} & y=0 \\ \end{matrix}$ (2)
 $\frac{\partial {{u}_{\ell ,i}}}{\partial y}=\frac{{{\tau }_{\delta ,i}}}{{{\mu }_{\ell }}},\begin{matrix} {} & y={{\delta }_{i}} \\ \end{matrix}$ (3)

After integrating eq. (1) twice and applying the boundary conditions, the liquid-vapor interface velocity for the inner liquid film is

 ${{u}_{\ell ,i,\delta }}=\frac{{{\tau }_{\delta ,i}}}{{{\mu }_{\ell }}}{{\delta }_{i}}-\frac{1}{{{\mu }_{\ell }}}\frac{dp}{dx}\frac{\delta _{i}^{2}}{2}$ (4)

This is similarly true for the outer wall with boundary conditions:

 ${{u}_{\ell ,o}}=0\begin{matrix} , & y={{D}_{o}} \\ \end{matrix}$ (5)
 $\frac{\partial {{u}_{\ell ,o}}}{\partial y}=\frac{{{\tau }_{\delta ,o}}}{{{\mu }_{\ell }}}\begin{matrix} , & y={{D}_{o}}-{{\delta }_{o}} \\ \end{matrix}$ (6)

The outer liquid film interface velocity is

 ${{u}_{\ell ,o,\delta }}=\frac{{{\tau }_{\delta ,o}}}{{{\mu }_{\ell }}}{{\delta }_{o}}-\frac{1}{{{\mu }_{\ell }}}\frac{dp}{dx}\frac{\delta _{o}^{2}}{2}$ (7)

The inner and outer walls liquid Reynolds numbers are determined as follows:

 ${{\operatorname{Re}}_{\ell ,i}}=\frac{{{\Gamma }_{i}}}{{{\mu }_{i}}}={{\rho }_{\ell }}\int_{0}^{{{\delta }_{i}}}{\frac{{{u}_{\ell ,i}}dy}{{{\mu }_{\ell }}}=\frac{{{\rho }_{\ell }}}{\mu _{\ell }^{2}}}\left[ {{\tau }_{\delta ,i}}\frac{\delta _{i}^{2}}{2}-\frac{dp}{dx}\frac{\delta _{i}^{3}}{3} \right]$ (8)
 ${{\operatorname{Re}}_{\ell ,o}}=\frac{{{\Gamma }_{o}}}{{{\mu }_{i}}}={{\rho }_{\ell }}\int_{{{D}_{o}}-{{\delta }_{o}}}^{{{D}_{o}}}{\frac{{{u}_{\ell ,o}}dy}{{{\mu }_{\ell }}}=\frac{{{\rho }_{\ell }}}{\mu _{\ell }^{2}}}\left[ {{\tau }_{\delta ,o}}\frac{\delta _{o}^{2}}{2}-\frac{dp}{dx}\frac{\delta _{o}^{3}}{3} \right]$ (9)

where Γ is the mass flow rate per unit width.

A momentum balance is needed for the vapor region, to help in determination of the overall pressure drop in the annular region.

 $\frac{dp}{dx}=\frac{4\left( {{\tau }_{\delta ,o}}{{D}_{o}}+{{\tau }_{\delta ,i}}{{D}_{i}} \right)}{D_{o}^{2}-D_{i}^{2}}$ (10)

Neglecting convection in the liquid films allows for a simple energy balance expression for the inner and outer walls, respectively:

 $\frac{d{{\operatorname{Re}}_{\ell ,i}}}{dx}=\frac{{{k}_{\ell }}\left( {{T}_{sat}}-{{T}_{w,i}} \right)}{{{h}_{\ell v}}{{\mu }_{\ell }}{{\delta }_{i}}}=\frac{J{{a}_{i}}}{{{\Pr }_{\ell }}{{\delta }_{i}}}=\frac{{{N}_{1i}}}{{{\delta }_{i}}}$ (11)
 $\frac{d{{\operatorname{Re}}_{\ell ,o}}}{dx}=\frac{{{k}_{\ell }}\left( {{T}_{s}}-{{T}_{w,o}} \right)}{{{h}_{\ell v}}{{\mu }_{\ell }}{{\delta }_{o}}}=\frac{J{{a}_{o}}}{{{\Pr }_{\ell }}{{\delta }_{o}}}=\frac{{{N}_{1o}}}{{{\delta }_{o}}}$ (12)

where Ja is the Jacob number ${{c}_{p\ell }}({{T}_{sat}}-{{T}_{w}})/{{h}_{\ell v}}$ and N1 is the ratio of the Jacob number to the liquid Prandtl number. After the momentum and energy balances, the only balance that needs to be defined is the mass balance between the liquid and vapor.

 ${{\dot{m}}_{v,e}}={{\dot{m}}_{v}}+{{\dot{m}}_{\ell ,i}}+{{\dot{m}}_{\ell ,o}}$ (13)

where e indicates the entering condition to the annular region, x = 0. Rewriting eq. (13) in terms of the Reynolds number, the following is a definition of the mass balance:

 ${{\operatorname{Re}}_{v,h}}={{\operatorname{Re}}_{v,e,h}}-4\frac{{{\mu }_{\ell }}}{{{\mu }_{v}}}\frac{{{K}^{*}}}{1+{{K}^{*}}}{{\operatorname{Re}}_{\ell ,i}}-4\frac{{{\mu }_{\ell }}}{{{\mu }_{v}}}\frac{1}{1+{{K}^{*}}}{{\operatorname{Re}}_{\ell ,o}}$ (14)

where

 ${{K}^{*}}=\frac{{{D}_{i}}}{{{D}_{o}}}$ (15)

The total shear stress is the sum of the shear stress due to friction with no mass transfer, and the shear stress due to the liquid’s momentum gain due to the condensing of the faster-moving vapor into the slower-moving liquid. This combined shear stress drives the removal of condensate from the condenser in microgravity. The expression for the inner and outer liquid-vapor shear stress is therefore as follows:

 ${{\tau }_{\delta ,i}}=\frac{{{C}_{f}}}{2}{{\rho }_{v}}{{\left( {{u}_{v}}-{{u}_{\ell ,\delta ,i}} \right)}^{2}}+{{\mu }_{\ell }}\frac{d{{\operatorname{Re}}_{\ell ,i}}}{dx}\left( {{u}_{v}}-{{u}_{\ell ,\delta ,i}} \right)$ (16)
 ${{\tau }_{\delta ,o}}=\frac{{{C}_{f}}}{2}{{\rho }_{v}}{{\left( {{u}_{v}}-{{u}_{\ell ,\delta ,o}} \right)}^{2}}+{{\mu }_{\ell }}\frac{d{{\operatorname{Re}}_{\ell ,o}}}{dx}\left( {{u}_{v}}-{{u}_{\ell ,\delta ,o}} \right)$ (17)

As in the above section on condensation with suction at the wall, Cf / 2 for both laminar and turbulent flows is given as follows:

 $\frac{{{C}_{f}}}{2}=\left\{ \begin{matrix} \frac{A}{{{\operatorname{Re}}_{v,h}}} & {{\operatorname{Re}}_{v,h}}\le 2300 \\ 0.085\operatorname{Re}_{v,h}^{-0.25}\left( 1+850F \right) & {{\operatorname{Re}}_{v,h}}>2300 \\ \end{matrix} \right.$ (18)

where A is a constant that is a function of K*, F is an empirical coefficient based on the average shear stress around the circumference of the annular flow recommended by Henstock and Hanratty (1976), and ${{\operatorname{Re}}_{v,h}}$ is the Reynolds number based on hydraulic diameter. Defining the following normalized parameters

 $u_{v}^{+}=\frac{{{u}_{v}}}{{{u}_{v,e}}}$; (19)
 $u_{\ell }^{+}=\frac{{{u}_{\ell }}}{{{u}_{v,e}}}$; (20)
 ${{x}^{+}}=\frac{x}{{{D}_{o}}}$; (21)
 $\delta _{i}^{+}=\frac{{{\delta }_{i}}}{{{D}_{o}}}$; (22)
 $\delta _{o}^{+}=\frac{{{\delta }_{o}}}{{{D}_{o}}}$; (23)
 ${{p}^{+}}=\frac{{{D}_{o}}p}{{{\mu }_{\ell }}{{u}_{v,e}}}$; (24)
 ${{N}_{2i}}=\frac{{{\tau }_{\delta ,i}}{{D}_{o}}}{{{\mu }_{\ell }}{{u}_{v,e}}}$; (25)
 ${{N}_{2o}}=\frac{{{\tau }_{\delta ,o}}{{D}_{o}}}{{{\mu }_{\ell }}{{u}_{v,e}}}$ (26)

eqs. (4), (7) – (12), (16), and (17) can be nondimensionalized as follows.

Normalized liquid velocity profiles:

 $u_{\ell ,i,\delta }^{+}={{N}_{2i}}\delta _{i}^{2}-\frac{\delta _{i}^{{{+}^{2}}}}{2}\frac{d{{p}^{+}}}{d{{x}^{+}}}$ (27)
 $u_{\ell ,o,\delta }^{+}={{N}_{2o}}\delta _{o}^{2}-\frac{\delta _{o}^{{{+}^{2}}}}{2}\frac{d{{p}^{+}}}{d{{x}^{+}}}$ (28)

Normalized wall liquid Reynolds numbers:

 ${{\operatorname{Re}}_{\ell }}_{,i}=\frac{1}{2}\frac{{{\operatorname{Re}}_{v,e,h}}}{\left( 1-{{K}^{*}} \right)}\frac{{{v}_{v}}}{{{v}_{\ell }}}{{N}_{2i}}\delta _{i}^{{{+}^{2}}}-\frac{1}{3}\frac{{{\operatorname{Re}}_{v,e,h}}}{\left( 1-{{K}^{*}} \right)}\frac{{{v}_{v}}}{{{v}_{\ell }}}\frac{d{{P}^{+}}}{d{{x}^{+}}}\delta _{i}^{{{+}^{3}}}$ (29)
 ${{\operatorname{Re}}_{\ell ,o}}=\frac{1}{2}\frac{{{\operatorname{Re}}_{v,e,h}}}{\left( 1-{{K}^{*}} \right)}\frac{{{\nu }_{v}}}{{{\nu }_{\ell }}}{{N}_{2o}}\delta _{o}^{{{+}^{2}}}-\frac{1}{3}\frac{{{\operatorname{Re}}_{v,e,h}}}{\left( 1-{{K}^{*}} \right)}\frac{{{\nu }_{v}}}{{{\nu }_{\ell }}}\frac{d{{p}^{+}}}{d{{x}^{+}}}\delta _{o}^{{{+}^{3}}}$ (30)

 $\frac{d{{p}^{+}}}{d{{x}^{+}}}=\frac{4\left( {{N}_{2o}}+{{N}_{2i}} \right)}{\left( 1-{{K}^{{{*}^{2}}}} \right)}$ (31)

Normalized energy balance:

 $\frac{d{{\operatorname{Re}}_{\ell ,i}}}{d{{x}^{+}}}=\frac{{{N}_{1i}}}{\delta _{i}^{+}}$ (32)
 $\frac{d{{\operatorname{Re}}_{\ell ,i}}}{d{{x}^{+}}}=\frac{{{N}_{1i}}}{\delta _{i}^{+}}$ (33)

Normalized liquid-vapor shear stress:

 ${{N}_{2i}}=\frac{{{C}_{f}}}{2}\left( \frac{1}{1-{{K}^{*}}} \right)\frac{{{\mu }_{v}}}{{{\mu }_{\ell }}}{{\operatorname{Re}}_{v,e,h}}\left( u_{v}^{+}-u_{\ell ,\delta ,i}^{+} \right)+\frac{d{{\operatorname{Re}}_{l,i}}}{d{{x}^{+}}}\left( u_{v}^{+}-u_{\ell ,\delta ,i}^{+} \right)$ (34)
 ${{N}_{2o}}=\frac{{{C}_{f}}}{2}\left( \frac{1}{1-{{K}^{*}}} \right)\frac{{{\mu }_{v}}}{{{\mu }_{\ell }}}{{\operatorname{Re}}_{v,e,h}}\left( u_{v}^{+}-u_{\ell ,\delta ,o}^{+} \right)+\frac{d{{\operatorname{Re}}_{\ell ,o}}}{d{{x}^{+}}}\left( u_{v}^{+}-u_{\ell ,\delta ,o}^{+} \right)$ (35)

Equations (14) and (20) – (28) are used to find the condensate film thicknesses $\delta _{o}^{+}$ and $\delta _{i}^{+}$ , N2i,N2o, ${{\operatorname{Re}}_{\ell ,i}}$, ${{\operatorname{Re}}_{\ell ,o}}$ and Rev,h as functions of the normalized axial length x+.

The local Nusselt numbers at the inner and outer walls are respectively given as

 $N{{u}_{i}}(x)=\frac{{{h}_{i}}(x){{D}_{o}}}{{{k}_{\ell }}}=\frac{1}{\delta _{i}^{+}}$ (36)
 $N{{u}_{o}}(x)=\frac{{{h}_{o}}(x){{D}_{o}}}{{{k}_{\ell }}}=\frac{1}{\delta _{o}^{+}}$ (37)

Results of this analysis were presented by Faghri and Chow (1991) for steam condensing at one atmosphere. Here, the results were reproduced for equal inner and outer wall temperatures at 70 °C. The liquid properties were approximated as the average between the vapor temperature and the inner and outer wall temperatures. The ratio of inner to outer tube diameters was set to be K* = 0.5.

## References

Faghri, A. and Chow, L.C., 1991, “Annular Condensation Heat Transfer in a Microgravity Environment,” International Communications on Heat and Mass Transfer, Vol. 18, pp. 715-729.

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.