Governing Equations for Laminar Film Condensation of a Binary Vapor Mixture

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Knowledge of binary vapors is of the utmost practical importance, because pure vapor is rarely present in everyday life or in industrial applications. Consider the steady laminar film condensation of a binary vapor on a smooth, vertical, cooled wall. The bulk vapor mixture flows downwards and parallels the vapor boundary layer and condensate film flows, which are also flowing downwards due to the effects of gravity. For condensation of a binary vapor near a vertical plate, x is the distance measured along the flat plate starting from the leading edge and y is the normal distance from the plate. The condensate film boundary layer directly adjacent to the wall with thickness δ = δ(x). Directly adjacent to this condensate film is the binary vapor boundary layer, with thickness $δ v = δ v (x)$ which develops between the condensate film and bulk vapor flow. The velocity components of the x- and y-directions are u and v respectively. The temperature is denoted by T and the pressure of the system is a constant p. The mass fraction of a component is denoted by . The subscripts l, v, δ, w, and ∞ denote the liquid, vapor, interface, wall, and bulk conditions, respectively. Finally, the subscript 1 denotes the component of the binary vapor with the lower boiling point.

The following assumptions were made to describe the laminar film condensation (Fujii, 1991):

1.T_w, T_δ, ω_lvδ, T_v∞, ω_lv∞, u_vδ, u_lδ, and u_v∞ are independent of x.

2.The condensate film and binary vapor boundary layers both develop from the leading edge of the vertical surface, x=0.

3.Condensation takes place only at the vapor-liquid interface. In other words, no condensation takes place within the binary vapor boundary layer in the form of a mist or fog.

4.Both temperature and velocity are continuous at the vapor-liquid interface.

5.The condensate is miscible.

6.The physical properties of the system are assumed to be constant with respect to concentration and temperature except in the case of buoyancy terms.

7.The density of the condensate liquid is assumed to be much greater than that of the binary vapor ( $ρ l ?ρ v$ ).

8.The vapor mixture can be treated as an ideal gas, and the thermal diffusion is negligible.

Before the governing equations are presented, an explanation is needed for the physical conditions that occur during this process. When a binary vapor makes contact with a cooled vertical wall, the less volatile component of the mixture (the component with the higher boiling point) begins to condense first and in greater quantity. Since the system must maintain the total mass concentration to maintain equilibrium, the volatile component becomes much denser at the liquid-vapor interface. During this process, the bulk vapor with a constant mass concentration is steadily supplying the liquid-vapor interface. Therefore, the vapor boundary layer develops with a very high concentration of the volatile component at the liquid-vapor interface, which is quickly diluted to the concentration of the bulk vapor.

The governing equations for the laminar film condensation of a binary vapor mixture can be given by taking the above assumptions into account and using boundary layer analysis.

For the condensate film, the continuity, momentum and energy equations are:

$\frac{\partial u_{l }}{\partial x}+\frac{\partial v_{l} }{\partial y}=0$	(1)

$u_{l}\frac{\partial u_{l }}{\partial x}+v_{l}\frac{\partial v_{l} }{\partial y}=v_{l}\frac{\partial^{2} u_{l}}{\partial y^{2}}+g-\frac{1}{\rho_{l}}\frac{dp}{dx}$	(2)

$u_{l}\frac{\partial u_{l}}{\partial x}+v_{l}\frac{\partial u_{l}}{\partial y}=v_{l}\frac{\partial^{2}u_{l}}{\partial y^{2}}+g-\frac{1}{\rho_{l}}\frac{dp}{dx}$	(3)

For the vapor boundary layer, the continuity, momentum, energy, and species equations are:

$\frac{\partial u_{l }}{\partial x}+\frac{\partial v_{l} }{\partial y}=0$	(4)

$u_{v}\frac{\partial u_{v}}{\partial x}+v_{v}\frac{\partial u_{v}}{\partial y} = v_{v}\frac{\partial^{2}u_{v}}{\partial y^{2}} + g\left ( 1-\frac{\rho_{v\infty }}{\rho_{v}} \right )$	(5)

$u_{v}\frac{\partial T_{v}}{\partial x}+v_{v}\frac{\partial T_{v}}{\partial y} = \alpha_{v}\frac{\partial^{2}T_{v}}{\partial y^{2}} + Dc_{p12}\frac{\partial \omega_{lv}}{\partial y}\frac{\partial T_{v}}{\partial y}$	(6)

$u_{v}\frac{\partial T_{v}}{\partial x} + v_{v}\frac{\partial \omega_{lv}}{\partial y} = D\frac{\partial^{2}\omega_{lv}}{\partial y^{2}}$	(7)

where v is the kinematic viscosity, ρ is the density, k is the thermal conductivity, and D is the diffusivity between components 1 and 2. The second term on the right-hand side of eq. (6) represents the contribution of mass concentration gradient to the energy balance, which is usually negligible. The isobaric specific heat difference of the binary vapor, c_p12, is a weighted average of the isobaric specific heats of the two individual components; it is non-dimensional as follows:

$c_{p12}=\frac{c_{p1v}-c_{p2v}}{c_{p1v}\omega_{1v}+c_{p2v}\omega_{2v}}=\frac{c_{p1v}-c_{p2v}}{c_{pv}}$	(8)

This isobaric specific heat term is assumed to be constant even though it has a larger fluctuation than the other physical properties, because overall the second term on the right-hand side of eq. (6) is much smaller than the first. Other definitions of the terms in eqs. (6) – (7) are as follows, beginning with the mass fractions ω_1v and ω_2v in the vapor phase:

$\omega_{1v}=\frac{\rho_{1v}}{\rho_{v}}$	(9)

$\omega_{2v}=\frac{\rho_{2v}}{\rho_{v}}$	(10)

where

$ρ v = ρ 1 v + ρ 2 v$	(11)

Therefore, it follows from eqs. (9) – (11) that a relationship between and can be written as follows:

$ω 1 v + ω 2 v = 1$	(12)

Note that only the continuity equation for component 1 of the vapor is given. However, it should be immediately recognized from eq. (12) that if the mass fraction of one component is known at any point in space, then the mass fraction of the other components can easily be determined.

The partial pressures of the system are determined by the following simple expressions:

$\frac{p_{1}}{p}=\left ( 1+\frac{M_{1}\omega_{2v}}{M_{2}\omega_{lv}} \right )^{-1}$	(13)

$\frac{p_{2}}{p}=\left ( 1+\frac{M_{2}\omega_{1v}}{M_{1}\omega_{2v}} \right )^{-1}$	(14)

where M1 and M2 are the molecular masses of components 1 and 2, respectively. The boundary equations for the above-generalized governing equations at the surface of the cold wall are as follows:

$u l = 0, y = 0$	(15)

$v l = 0, y = 0$	(16)

$T l = T w, y = 0$	(17)

These boundary conditions represent no-slip at the wall and the continuity of temperature at the wall. The boundary conditions at locations far from the cold wall are

$u_{v}=u_{v\infty}, y \to \infty$	(18)

$T_{v}=T_{v\infty}, y \to \infty$	(19)

$\omega_{l v}=\omega_{l v\infty}, y \to \infty$	(20)

These boundary conditions represent conditions in the constant bulk binary vapor. Finally, the boundary conditions that exist at the liquid-vapor interface () are given as follows:

$u l δ = u v δ = u δ$	(21)

$\left ( \mu_{l}\frac{\partial\mu_{l}}{\partial y}\right )_{\delta} = \left ( \mu_{v}\frac{\partial u_{v}}{\partial y}\right )_{\delta}$	(22)

$\left [ \rho_{l} \left ( u_{l}\frac{d\delta}{dx}-v_{l}\right ) \right ]_{\delta} = \left [ \rho_{v} \left (u_{v}\frac{d\delta}{dx}-v_{v} \right ) \right ]_{\delta} = m^{''}=m_{1}^{''}+m_{2}^{''}$	(23)

$T l δ = T v δ = T δ$	(24)

$\left (k_{l}\frac{\partial T_{l}}{\partial y} \right )_{\delta} = h_{l v}m^{''}+\left (k_v\frac{\partial T_{v}}{\partial y} \right )_{\delta}$	(25)

$ω l v = ω l v δ$	(26)

where μ is the dynamic viscosity, k is the thermal conductivity and h_lv is the latent heat of condensation. Finally, $m^{''}=m_{1}^{''}+m_{2}^{''}$ is the condensation mass flux perpendicular to the vertical plate; it is a dependent of the location along the x-axis. Equations (21), (24), and (26) are the boundary conditions that refer to the continuity of velocity, temperature, and mass fraction at the liquid-vapor interface, respectively. Equation (22) recognizes that the shear stresses in the liquid and vapor layers are equal at the liquid-vapor interface. Equation (25) is an energy balance at the interface, which relates the heat transferred to and from the interface with that released by the latent heat. Equation (23) is the overall condensation mass continuity at the interface. The mass flux of the noncondensable component 2 at the interface in eq. (23), is not zero because the noncondensable component 2 may dissolve into the condensate.

The mass fluxes of the vapor in the binary vapor system are (see mass transfer)

$m_{1}^{''}=-\rho D_{12}\frac{\partial \omega_{1}}{\partial y} + \omega_{1} \left ( m_{1}^{''}+m_{2}^{''} \right )$	(27)

$m_{2}^{''}=-\rho D_{21}\frac{\partial \omega_{2}}{\partial y} + \omega_{2} \left ( m_{1}^{''}+m_{2}^{''} \right )$	(28)

and the molar fluxes of the vapor in the binary vapor system are (see mass transfer)

$n_{1}^{''}=-D_{12}\frac{\partial c_{1}}{\partial y} + x_{1}n_{T}^{''}$	(29)

$n_{2}^{''}=-D_{21}\frac{\partial c_{2}}{\partial y} + x_{2}n_{T}^{''}$	(30)

where c_i is molar concentration (Kmol/m³) of component i (i =1, 2). If both components can be treated as ideal gases, the molar fraction is identical to the partial pressure obtained from eqs. (13) and (14). $n_{T}^{''}=n_{1}^{''}+n_{2}^{''}$ is the total molar flux of component 1 and 2. Equations and are often expressed in terms of partial pressure, i.e.,

$n_{1}^{''}=-\frac{D}{R_{u}T}\frac{\partial p_{1}}{\partial y}+n_{T}^{''}\frac{p_{1}}{p}$	(31)

$n_{2}^{''}=-\frac{D}{R_{u}T}\frac{\partial p_{2}}{\partial y}+n_{T}^{''}\frac{p_{2}}{p}$	(32)

where R_u is the universal gas constant. Equations (29) – (32) are valid in the binary vapor boundary layer ( $\delta \leq y \leq \delta+\Delta$ ).

Also, the mass fraction of component 1 in the condensate film can be found from the following expression:

$\omega_{1l}=\frac{m_{1x}^{''}}{m_{1x}^{''}+m_{2x}^{''}}$	(33)

If the noncondensable component cannot be dissolved into the liquid film, ω_1l will be unity.

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.

Fujii, T., 1991, Theory of Laminar Film Condensation, Springer-Verlag, New York, New York.

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Governing Equations for Laminar Film Condensation of a Binary Vapor Mixture

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