Turbulent film condensation
From ThermalFluidsPedia
We will now consider turbulent flow for a specific application – turbulent condensate flow in a circular tube, as shown in Fig. 1. Turbulent film condensation occurs at the inner surface of the circular tube. While the liquid condensate flows downward due to gravity, the vapor flows either downward (cocurrent vapor flow) or upward (countercurrent vapor flow). Faghri (1986) proposed a method of predicting the average film thickness, the local heat transfer coefficient, and the overall heat transfer coefficient for turbulent film condensation in a tube with interfacial shear stress caused by cocurrent and countercurrent vapor flow. In a fashion similar to that of Nusselt condensation, the inertia terms are neglected and the only forces included are body, pressure, and viscous forces. This particular model takes into account the decrease in the stream flow rate due to condensation.
To obtain an expression of the shear stress in the liquid film, a control volume with radius r and height Δx as shown in Fig. 1 is considered. For the case of countercurrent flow, a force balance results in

where the volume of the vapor and the liquid portion of the control volume are as follows:


Substituting eqs. (2) and (3) into eq. (1) and dividing through by Δx to solve for the shear stress, the following is obtained:

The shear stress at the liquidvapor interface can be found by letting y = δ in eq. (4), i.e.,

Substituting eq. (5) into eq. (4), the shear stress at any radius can be related to the shear stress at the liquid film surface, τ_{δ}, by

Assuming the tube radius is much greater than the condensate film thickness forming on the inner surface, the curvature of the liquid film can be neglected and the resulting analysis would be applicable to condensation between two flat plates. Taking this into account, eq. (5) reduces to

If it is further assumed that , eq. (6) would reduce to

which can be written into a generalized form that includes the case of cocurrent flow, i.e.,

where the + sign denotes downward vapor flow (cocurrent flow) and the  sign denotes upward vapor flow (countercurrent flow). The shear stress at the wall, y = 0, is

The velocity profile in the liquid film can be found from the following differential equation when all axial terms and the curvature are neglected:

where ε_{m} is the momentum eddy diffusivity and is a timemeasured flow property that adjusts the viscosity term for turbulent flow.
Assuming that the wall is impermeable and that the interfacial shear stress is known, the boundary conditions for this problem are given as


Integrating eq. (11) twice with respect to y and applying boundary conditions specified by eqs. (12) and (13), the velocity profile in the liquid film is obtained:

The liquid Reynolds number obtained by the following expression:

To generalize the problem statement, the following nondimensional variables are defined:

where u_{f} is the fractional velocity, defined as

Applying these nondimensional variables to eqs. (11), (14), and (15), their nondimensional forms are obtained as follows:



It should be noted that if and then eq. (18) would reduce to the nonsheared film case (classical Nusselt analysis).
We will now consider the thermal side of the problem. An energy balance can be written for the case of constant heat flux at the wall (). This energy balance also assumes that heat transfer across the liquid film is dominated by conduction, so the convective terms can therefore be neglected.

where Pr_{t} is the turbulent Prandtl number, which will be discussed thoroughly towards the end of this subsection. Equation (21) can be nondimensionalized to obtain

where

Finally, the local heat transfer coefficient can be found directly from eq. (21):

Nondimensionalizing eq. (24) as a Nusselt number, the following is obtained:

An average heat transfer coefficient is desirable in many practical applications. A modified Nusselt number is related to Nu_{x} by

The average modified Nusselt number is then found from the following relation:

The dimensionless shear stress at the interface can be written as

where f_{E} is the friction factor for vapor flow, which is different for upflows and downflows. The friction factor for vapor flow can be obtained by modifying the friction factor for singlephase flow, f, to accommodate the twophase nature of the flow. It is also different for ripple (Re_{δ}≤75) and roll wave (Re_{δ}>75) regimes, i.e.,

where

The characteristic stress is given by

To calculate the velocity distribution and the heat transfer coefficient, a definition is required for ε_{m} and Pr_{t} from an appropriate turbulent model. In modeling of ε_{m}, it is customary to divide the flow into two regions – the inner region, where the turbulent transport is dominated by the wall, and another wavelike region that is directly adjacent. Faghri (1986) used a combination of the Szablewski (1968) and Van Driest models to obtain the following expression:

where A^{+} = 25.1 and .
This profile represents the eddy diffusivity in the inner layer closest to the wall (), where the influence of the wall is important. In the outer layer () the eddy viscosity is assumed to be constant, with a continuous transition to the inner layer.
Finally, because the turbulent transport near the liquidvapor interface is quite different from that near the wall, the turbulent Prandtl number, Prt, cannot be assumed to be constant. Faghri (1986) used the following expression (Habib and Na, 1974) for the analysis of turbulent transfer in pipes:

where

and c_{1} = 34.96; c_{2} = 28.79; c_{3} = 33.95; c_{4} = 6.3; c_{5} = 1.186.
The solution procedure begins with guessing an initial value of for the initial values of Re_v = 4m_v / (\pi D\mu_v )</math> and the initial vapor flow, which specifies the initial value of Re_{v} = 4m_{v} / (πDμ_{v}). Based on the values of and , the fractional velocity, u_{f}, is then obtained by solving eq. (18). The dimensionless eddy diffusivity, , is obtained from eq. (17). Equations (18) and (19) are integrated numerically to obtain the velocity profile and the liquid Reynolds number. An updated dimensionless shear stress at the interface can be obtained from eq. (28). The process is repeated until the Re_{l} values between two consecutive iterations differ by less than 0.5%. The local convective heat transfer coefficient can be obtained from eqs. (26) and (27). The above procedure can be repeated for different x until heat transfer coefficients are obtained at all locations.
Heat transfer in the condenser sections of conventional and annular twophase closed thermosyphon tubes has been studied analytically by Faghri et al. (1989). The method involved extending Nusselt theory to include the variation of the shear at the vaporliquid film interface. Harley and Faghri (1994) presented a transient twodimensional condensation in a thermosyphon that accounts for conjugate heat transfer through the wall and the falling condensate film. The complete transient twodimensional conservation equations are solved for the vapor flow and pipe wall, and the liquid film was modeled using a quasisteadystate Nusselttype solution.
References
Faghri, A., 1986, “Turbulent Film Condensation in a Tube with Cocurrent and Countercurrent Vapor Flow,” AIAA Paper No. 861354.
Faghri, A., Chen, M. M., and Morgan, M., 1989, “Heat Transfer Characteristics in TwoPhase Closed Conventional and Concentric Annular Thermosyphons,” ASME Journal of Heat Transfer, Vol. 111, No. 3, pp. 611618.
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.
Harley, C., and Faghri, A., 1994, “Transient TwoDimensional GasLoaded Heat Pipe Analysis,” ASME Journal of Heat Transfer, Vol. 116, pp. 716723.