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Received for review June 16, 1987 Reuised manuscript received December 14, 1987 Accepted January 5, 1988
Transport Processes in Thin Liquid Films during High-Vacuum Distillation Maruti Bhandarkar and John R. Ferron* Department of Chemical Engineering, University of Rochester, Rochester, New York 14627
A description of the liquid-phase transfer processes in high-vacuum distillation is presented. T h e thin liquid film on a conical, rotating evaporator is modeled by the mass, momentum, and energy balance equations. T h e equations are solved by a finite-difference method. Effects of various parameters upon the distillation rates and the separation efficiencies are studied for binary systems. A simple procedure to scale-up the process is suggested. Experimental data for evaporation of pure compounds and distillation of a binary mixture are compared with the present theoretical analysis assuming t h e Langmuir (i.e., maximum) rate of evaporation. On the basis of experimental data, we may conclude that the present analysis provides the first step toward the complete characterization of distillation under vacuum and t h a t the analyses of vapor-phase and condenser processes need also t o be included for a fuller treatment. Commercial as well as laboratory distillation under high vacuum (pressures ranging from 0.1 to 15 dyn/cm2) is typically carried out in the so-called centrifugal molecular still, in which the distilland flows over a rotating, truncated conical surface. We have studied the transport processes occurring in thin liquid films formed in such situations as part of a wider study of the vacuum distillation process. The flow of liquid as a thin film over a rotating disk may be considered as a special case where the cone half-angle is 90°. Greenberg (1972) considered an approximate model of the flow of a pure liquid over a rotating conical surface (see Figure 1). The heat-transfer analysis was limited to the cases in which the feed and mean distilland residue temperatures were roughly equal. This is valid when the feed is preheated and the evaporative heat losses are small. Greenberg’s theoretical analysis agreed with his experimental data for the evaporation of dibutyl phthalate, dibutyl sebacate, isooctyl phthalate, and isooctyl sebacate. However, for glycerol the calculated rates were higher. In all cases the maximum evaporation rate (Langmuir, 1913) calculated for the liquid surface temperature was used to determine the rate. Analysis of the centrifugal still during distillation of a binary liquid mixture was carried out by Ruckenstein et al. (1983) with a view to studying the effect of diffusional resistance among other things. The heat transport analysis was again simplified by assuming that the feed is a t the distillation temperature and that convective heat transfer is negligible compared to conductive heat transfer. For the relatively large diffusion coefficient used (4 x cm2/s at 500 K), diffusional effects were negligible. Most liquid solutions subjected to high-vacuum distillation have diffwivities in the range of 104-106 cm2/s, for which cases
mass-transfer resistances would be expected to be quite large. The most recent analysis of the evaporation of a pure liquid in high vacuum was by Kaplon et al. (1986). The analysis was for situations in which the liquid is preheated and flows over an insulated rotating disc. All the heat necessary for evaporation came from feed itself. The model describing heat transfer was considerably simplified and can lead to an overestimation of evaporation rates as we show later. In commercial as well as laboratory high-vacuum distillation using the centrifugal molecular still, we usually encounter feeds relatively colder than the distillation temperature. This means that the distilland heats up as it travels across the evaporator, often with wide variations in the material properties, and previous analyses assuming that the feed is at distillation temperature are inadequate. It would be useful from a design point of view to simulate situations involving a variety of operating parameters. In this work, a general treatment of the heat and mass transport in the liquid flowing over rotating surfaces and distilling under vacuum is presented. The analysis includes evaporators shaped as rotary cones as well as disks. Situations in which the feed is preheated, in which the rotary surface is heated, or combinations of these are considered. Theoretical results are compared with literature data for pure materials and with binary data obtained in our laboratory. Finally, a simple procedure has been described for scaling up the results from a laboratory still to a larger, commercial still. Throughout the treatment we assume that the evaporative processes are independent of the vapor-phase properties and the condenser behavior. These assumptions are valid in the limit of very low pressures; hence, of small
0888-5885/88/2627- 1016$01.50/0 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1017 The equation describing mass transfer in a binary mixture is written as
Evaporator surface
a2c, 1 aC cot 4 ac, ac, + u pac, = D a b -+-a_-
U -
ax
xO 2
aY
x ax
ay2
ay
x
where C, is the mass fraction of component a in the mixture of a and b. The appropriate boundary conditions are C, = CaO
at x = xo (feed location) at y = 0
aC,/ay = 0
(9) (10)
and X
Figure 1. Sketch of the evaporator and its coordinate systems.
evaporation rate, and for a cold, highly efficient condenser. An objective of the paper is to scout the limits of these assumptions.
Mathematical Model The equation of continuity for the liquid flowing as a thin film in the ( x , y ) system in Figure 1 is
(11)
Here, E , and Eb are the evaporative fluxes (g/cm2/s) of a and b, respectively. We neglect free convection in both mass and heat transfer, again assuming small temperature and concentration changes through the film. The energy balance equation is u-+u-=cy ax ay aT aT
where u and u are velocities in x and y directions, respectively. Axial symmetry has been assumed here. (Equation 1may be obtained from a cylindrical form based on the (z,r) system by rewriting the differential operators in the (x,y) system. Here x = z cos 4 + r sin 4 and y = z sin 4 - r cos 4.) The boundary conditions are
u = uo
a t x = xo (feed location)
(2)
at y = O
(3)
and u=O
The equation of motion is simplified by assuming that (1) the liquid film has a fully developed, steady-state velocity profile for x > xo; (2) liquid pressure gradients are negligible; (3) gravitational effects are negligible; (4)variations of temperature and composition, hence of mixture viscosity, through the film (in the y direction) are small; we employ the average kinematic viscosity,
(Even though there are small variations of concentration and temperature over the film, gradients are substantial because the film is thin. Densities and transport properties other than viscosity are assumed to be relatively insensitive to changes in composition and temperature. The validity of this assumption was verified for special cases.) Certain simplifications (Bhandarkar, 1985) also result from considering that the liquid film is very thin compared to the size of the evaporator (6(x)
