Turbulent film condensation

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Figure 1 Physical model of the condensation phenomena in contact with flowing vapor.

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

$p(\pi r^2 )+\rho_v gV_v +\rho_l gV_l = \left ( p+\frac{dp}{dx}\Delta x \right ) \pi r^2 + \tau (2\pi r\Delta x)$	(1)

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

$V v = (R - δ) 2 πΔ x$	(2)

$V l = (R - y) 2 πΔ x - V v$	(3)

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

$\tau = \frac{R-y}{2} \left ( \rho_l g -\frac{dp}{dx} \right ) - (\rho_l - \rho_v)g\frac{(R-\delta)^2}{2(R-y)}$	(4)

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

$\tau_{\delta} = \frac{R-\delta}{2} \left ( \rho_v g-\frac{dp}{dx} \right )$	(5)

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

$\tau = \frac{R-y}{r-\delta}\tau_{\delta} + (\rho_l - \rho_v )g \left [ \frac{2R(\delta - y) - \delta^2+y^2}{2(R-y)} \right ]$	(6)

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

$\tau_{\delta} = \frac{R}{2}\left ( \rho_v g - \frac{dp}{dx} \right )$	(7)

If it is further assumed that ${{\rho }_{\ell }}\gg {{\rho }_{v}}$ , eq. (6) would reduce to

$τ = τ δ + ρ l g (δ - y)$	(8)

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

$\tau = \pm \tau_{\delta} + \rho_l g(\delta -y)$	(9)

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

$\tau_w = \pm \tau_{\delta} + \rho_l g \delta$	(10)

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

$\frac{d}{dy} \left [ (v_l + \varepsilon_m )\frac{du}{dy} \right ] +g = 0$	(11)

where ε_m is the momentum eddy diffusivity and is a time-measured 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

$u = 0, y = 0$	(12)

$-\mu \frac{\partial u}{\partial y}= \pm \tau_{\delta}, y=\delta$	(13)

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:

$u=\int_{0}^{y}\frac{[g(\delta - y) \pm \tau_{\delta} / \rho_l ]}{v_l + \varepsilon_m} dy$	(14)

The liquid Reynolds number obtained by the following expression:

$Re_\delta = \frac{4\Gamma}{\mu_l} = 4\rho_l \int_{0}^{\delta} \frac{udy}{\mu_l}$	(15)

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

$\delta^+ = \frac{\delta u_f}{v_l}, \delta^* = \delta \left ( \frac{v^{2}_l}{g}\right )^{-1/3}, y^+ = \frac{yu_f}{v_l}, \tau_{\delta}^* = \frac{\tau_{\delta}(v_l g)^{-2/3}}{\rho_l}$ $u^+ = \frac{u}{u_f}, x^+ = \frac{xu_f}{v_l}, \varepsilon_{m}^+ 1+\frac{\varepsilon_m}{v_l}, D^+ = \frac{Du_f}{v_l}$	(16)

where u_f is the fractional velocity, defined as

$u_f = \left ( \frac{\tau_w}{\rho_l} \right )^{1/2}$	(17)

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

$u_{f}^3 \pm \tau_{\delta}^+ u_f (v_l g)^{2/3} - (v_l g)\delta^+ = 0$	(18)

$u^+ = \int_{0}^{y^+} \frac{(1-gv_l y^+ / u_{f}^3)}{\varepsilon_{m}^+}dy^+$	(19)

$Re_{\delta} = 4 \int_{0}^{\delta} u^+ dy^+$	(20)

It should be noted that if $\tau_{\delta}^+ =0$ and $u_{f}^3 = v_l g\delta^+$ 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 ( $q_{w}^{''}$ ). This energy balance also assumes that heat transfer across the liquid film is dominated by conduction, so the convective terms can therefore be neglected.

$\rho_l c_{pl} \left ( \frac{v_l}{Pr} + \frac{\varepsilon_m}{Pr_t} \right ) \frac {dT}{dy} = q^{''}_w = \mu_l h_{lv} \frac{d(\Gamma / \mu_l)}{dx}$	(21)

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

$\frac{d(\Gamma / \mu_l)}{dx^+} = N_T \left \{ \int^{\delta^+}_0 \left [ \frac{Pr}{Pr_t} \left ( \frac{Pr_t}{Pr} - 1+ \varepsilon_{m}^{+} \right ) \right ]^{-1} dy^+ \right \} ^{-1}$	(22)

where

$N_T = \frac{T_{sat} - T_w) c_{pl}}{h_{lv}Pr}$	(23)

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

$h_x = \frac{q^{''}_w}{T_{sat}-T_w} = \left \{ \int^{\delta}_0 \left [ \rho_l c_{pl} \left ( \frac{v_l}{Pr} + \frac{\varepsilon_m}{Pr_t} \right ) \right ]^{-1} dy \right \} ^{-1}$	(24)

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

$Nu_x = \frac{h_x \delta}{k_l} = \delta^+ \left \{ \int^{\delta^+}_0 \left [ \frac{Pr}{Pr_t} \left ( \frac{Pr_t}{Pr} - 1 + \varepsilon^{+}_m \right ) \right ]^{-1} dy^+ \right \} ^{-1}$	(25)

An average heat transfer coefficient is desirable in many practical applications. A modified Nusselt number $Nu^{+}_x$ is related to $N u x$ by

$Nu_{x}^+ = \frac{h_x}{k_l} \left ( \frac{v_l}{g} \right )^{1/3} = Nu_x \frac{u_f}{\delta^+}(v_l g)^{-1/3}$	(26)

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

$\overline{Nu^+} = \int_{0}^1 Nu^+ d\left ( \frac{x^+}{L^+} \right )$	(27)

The dimensionless shear stress at the interface can be written as

$\tau_{\delta}^* = (v_l g)^{-2/3} u_{f}^2 (u_{v}^+ + u_{l, \delta}^+ ) \left [ \frac{f_E}{2} \left ( \frac{\rho_v}{\rho_l} \right ) (u_{v}^+ + u_{l, \delta}^+ ) + \frac{d(\Gamma / \mu_l )}{dx^+} \right ]$	(28)

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 single-phase flow, f, to accommodate the two-phase nature of the flow. It is also different for ripple (Re_δ≤75) and roll wave (Re_δ>75) regimes, i.e.,

$f_E = \left\{\begin{matrix} f[1+0.045(Mg^+ - 5.9)] & Re_{\delta} \leq 75 \\ f[1+0.045Re_{v}^{-0.2}(Mg^+ -5.9)] & Re_{\delta} > 75 \end{matrix}\right \}$	(29)

where

$Mg^+ = \left \{ \begin{matrix} 0.78Re_{\delta}^{0.6} \left ( \frac{v_l}{v_v} \right ) \left ( \frac{\rho_l}{\rho_v} \frac{\tau_{\delta}}{\tau_c} \right )^{1/2} & Re_{\delta} \leq 75 \\ 0.50Re_{\delta}^{0.7} \left ( \frac{v_l}{v_v} \right ) \left ( \frac{\rho_l}{\rho_v} \frac{\tau_{\delta}}{\tau_c} \right )^{1/2} & Re_{\delta} > 75 \end{matrix} \right \}$	(30)

The characteristic stress is given by

$\frac{\tau_{\delta}}{\tau_c} = \left ( \frac{1}{3} + \frac{2}{3} \frac{\tau_w}{\tau_{\delta}} \right )^{-1}$	(31)

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 wave-like region that is directly adjacent. Faghri (1986) used a combination of the Szablewski (1968) and Van Driest models to obtain the following expression:

$\varepsilon_{m}^+ = \frac{1}{2} + \frac{1}{2} \left \{ 1+0.64y^{+^2}\frac{\tau}{\tau_w} \left [ 1 - exp\left ( -\frac{y^+}{A^+}\sqrt{\frac{\tau}{\tau_w}} \right ) \right ]^2 \left [ exp\left ( -1.66\left ( 1-\frac{\tau}{\tau_w} \right ) \right ) \right ]^2 \right \}$	(32

where A⁺ = 25.1 and $\tau/\tau_w = 1-y^+ (gv_l)/u_{f}^3$ .

This profile represents the eddy diffusivity in the inner layer closest to the wall ( $0\leq y^+ \leq 0.6\delta^+$ ), where the influence of the wall is important. In the outer layer ( $0.6\delta^+ \leq y^+ \leq \delta^+$ ) the eddy viscosity is assumed to be constant, with a continuous transition to the inner layer.

Finally, because the turbulent transport near the liquid-vapor 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:

$Pr_t = \frac{1-exp(-y^+ / A^+}{[1-exp(-y^+\sqrt{Pr} / B^+)]}$	(33)

where

$B^+ = \sum_{i=1}^5 c_i (log_10 Pr)^{i-1}$	(34)

and c₁ = 34.96; c₂ = 28.79; c₃ = 33.95; c₄ = 6.3; c₅ = -1.186.

The solution procedure begins with guessing an initial value of $\tau_{\delta}^*$ for the initial values of $\delta_{0}^+$ Re_v = 4m_v / (\pi D\mu_v )</math> and the initial vapor flow, which specifies the initial value of $R e v = 4 m v / (π D μ v)$ . Based on the values of $\tau_{\delta}^*$ and $\delta_{0}^+$ , the fractional velocity, $u f$ , is then obtained by solving eq. (18). The dimensionless eddy diffusivity, $\varepsilon_{m}^+$ , 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 two-phase 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 vapor-liquid film interface. Harley and Faghri (1994) presented a transient two-dimensional condensation in a thermosyphon that accounts for conjugate heat transfer through the wall and the falling condensate film. The complete transient two-dimensional conservation equations are solved for the vapor flow and pipe wall, and the liquid film was modeled using a quasi-steady-state Nusselt-type solution.

References

Faghri, A., 1986, “Turbulent Film Condensation in a Tube with Cocurrent and Countercurrent Vapor Flow,” AIAA Paper No. 86-1354.

Faghri, A., Chen, M. M., and Morgan, M., 1989, “Heat Transfer Characteristics in Two-Phase Closed Conventional and Concentric Annular Thermosyphons,” ASME Journal of Heat Transfer, Vol. 111, No. 3, pp. 611-618.

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 Two-Dimensional Gas-Loaded Heat Pipe Analysis,” ASME Journal of Heat Transfer, Vol. 116, pp. 716-723.

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Turbulent film condensation

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