Predicting Mass Transfer Extraction with Steam Flow, Applying Boundary-Layer Concepts

Theory and concepts of boundary layer mass transfer is applied to correlate experimental data on extraction of essential oils from vegetable leaves and stems, using steam. From these theory, concepts and experimental data with seven systems, two correlations are developed to predict the Sherwood number and mass transfer coefficient as function of Reynolds and Schmidt numbers. From these equations, the molar flux, the amount of solute extracted, and the yield of extraction is predicted. A steam of higher temperature nor-mally improves the mass transfer and the yield. A method to estimate the en-hancement for temperature increase is proposed. The correlations developed are applied to a case with industrial size that was no part of the data for correlation generation. Theory may be applied for industrial applications.


Introduction
Essential oil from plants is used in food, pharmacy, and fragrance industries due to their organoleptic and biological properties associated with their natural characteristics. Leaves and stem from plants are the raw material for the extraction of the essential oil. The total extraction of essential oil from vegetal leaves is usually small than 5% and there are several methods to perform the extraction. Hydro distillation with water in contact with the plant [1], steam distillation, with steam (but not water) contacting the plant, mechanical pressure (squeezing) [2], soxh-of a steam distillation system and a typical diagram from experimental results taken from Cerpa's Dissertation [5].
The experimental data of yield versus time of Figure 1(b) is modeled, by example with Xavier et al. [6] approach, using the mechanistic model proposed by Cerpa, Mato and Cocero [7], or some other.
In this manuscript, the final yield is modeled by using the concepts of boundary layer that were developed first for fluid flow by Ludwig Prandtl in 1904 who develop the first differential equations to model the hydraulic phenomena. Blausius helps to solve the mathematical model. Prandtl and other researchers began to apply it to heat transfer, Chilton-Coulburn and Guilliland-Sherwood applied it to mass transfer. Latter Bird Stewart and Lightfoot developed the concept of transport phenomena. This lead to apply the boundary layer theory to experimental and industrial cases, help the field of applied chemistry to be converted on chemical engineering and get maturity as science and engineering.
Chemical engineering applies the boundary layer concepts to correlate experimental data on flow of fluids, heat transfer, and mass transfer, as function of dimensionless numbers.   Table 1 shows the systems used.

Boundary Layer Concept Applied to Mass Transfer
(Taken from [8]) A concentration gradient is formed together to the hydrodynamic and thermal one. Let C AO be the concentration of the incoming flow to a plate made of a solid that is soluble in the liquid. C AO will be the concentration also at core of the flow, far from the plate. When the liquid is in contact with the plate the equilibrium is reached instantaneously at the interface liquid-solid. The concentration of A at the fluid, at the plane of contact with the solid surface will be that of saturation (C As = C Ai ). The mass molecular diffusion at y direction will set that the concentration gradient be growing when the liquid advance in x.
Authors or researchers from Table 1, report the dynamic (yield versus time). In this study only the final yield is correlated with physical properties, geometrical characteristics, and operational parameters. Defining the variable   The continuity equation for A, if density ρ, and diffusivity D AB are constant, is: It has as boundary conditions: That is similar to the one Blasius solved, but now for mass transfer. The dependent variable is now If it is desired a mathematical expression for the flux of A at the solid surface (N A in kmol/m 2 /s), it is needed to use Fick's law to obtain ( ) , 0.332 Integrating over the plate length By analogy to the thermal boundary layer, the Schmidt number relates the diffusivities of mass and momentum, giving: Equation (4) is valid only if Sc = 1. For the cases with Sc ≠ 1 it is necessary to introduce an experimental correction factor Sc 1/3 ( ) Equation (6) has been experimentally tested.

Mass Transfer Coefficient
Equation (6) allows the calculation of the rate of mass transfer for molecular diffu-try of the system is complex, as is the case in many practical cases. For this case it is necessary to use the mass transfer coefficient, defined by Equation (7): Applying Equation (6) on Equation (7) we get: This equation may be arranged and it provide the Sherwood dimensionless number: Re Sc D = = (9) Sh is the Sherwood number, counterpart of Nusselt number in heat transfer.
At turbulent flow, and for complex geometrical systems the mass transfer coefficient k C , will be empirical.

Application to Extraction with Steam Flow
From the reported data with the sources given in Table 1 Table 1. C Ai is the concentration of solute at the surface of the vegetal leave (at y = 0). This may be taken as the solubility of the solute in kmol/m 3 . Because the steam does not contain solute: C Ao = 0. Then using Equation (7):  Table 2 shows some of the physical properties used in the Excel program.  Density and viscosity were taken from [8], diffusivities were predicted withthe correlation of Fuller et al. [15]. Solubility was taken from PubChem, National

Physical Properties
Library of Medicine, Center for Biotechnology Information, that usually is expressed as mg or gr/liter. We convert it to kg and to kmol dividing between the molecular weight and converting the volume at the denominator to cubic meters.
Most of the experimental extraction with steam used atmospheric pressure and temperature of 100˚C. Only some data from Roautby et al. [12] and Rondhame and Tizaoui [13] were at temperature above 100˚C. These data were processed in a different Excel program, to get the effect of temperature over the yield of extraction.
In the general study, steam at 100˚C was used and the physical properties    Equation (12). The prediction with Equation (11) and Equation (12) is shown in Figure 4(a) and Figure 4(b). The two correlations may be considered limits for prediction.

Results for Temperature Increase
Rouatby et al. [12] studied the extraction of essential oil of thyme by superheated steam. They used steam temperatures of 100˚C, 175˚C, and 250˚C. They found that at higher temperatures the yield of extraction increases. Because the physical   properties changed, most of the parameters changed as well. Table 4 shows some of the values obtained.
The last two rows of Table 4

Case of Study, Prediction of Yield
The data on the paper of Koul et al. [16] will be taken as example to predict the These are industrial quantities. In Table 1 And Schmidt and Sherwood numbers with Equation (16) and Equation (11) or Equation (12) vap vap AB visc Sc den D =  Figure 6. Prediction on Koul [16], using the present model.  Table 5 provides the main calculated values, and Figure 6 provides the comparison between reported and predicted values.
On Figure 6 it is observed that both predictions 1 and 2 follow the order of the reported data. Prediction 1 underpredicts 0.31, and prediction 2 overpredicts 3.32 the values of reported yield of extraction from Koul et al. [16].

Conclusions
The proposed Equation (11) and Equation (12) provide correlations to predict the yield of extraction, by first estimating the dimensionless numbers Reynolds, Schmidt, and Sherwood numbers, and using equations for the mass transfer involved in the extraction of solute from vegetable leaves to steam, using boundary layer concepts and definitions like molar flux and mass transfer coefficient.
The predicting Equation (11) and Equation (12) provide limits to experimental or reported yields and predict well the effect of steam flow. Figure 5 and Equation (21) help to predict Sherwood number for superheated steam at temperatures above 100˚C. Using steam at temperatures higher than 100˚C improves the extraction yield, but at temperatures above 200˚C, the temperature degrades some components of the mixture of essential oil.