Table 1 shows the systems used.

(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 γ = C A i − C A (analogous to temperature in excess), the concentration gradient will be that of Figure 2. The dotted line is laminar underlayer.

The edge of the concentration boundary layer is then the geometric place where γ A = 0.99 γ A o , being γ A o = C A i − C A o the maximum value de γ.

The continuity equation for A, if density ρ, and diffusivity D_AB are constant, is:

υ x ∂ γ A ∂ x + υ y ∂ γ A ∂ y = D A B ( ∂ 2 γ A ∂ y 2 ) (2)

It has as boundary conditions: γ A ( x = 0 , y ) = γ A o ; γ A ( x , y = 0 ) = 0 ; γ A ( x , y = ∞ ) = γ A o

That is similar to the one Blasius solved, but now for mass transfer. The dependent variable is now ∅ = γ A γ A o = C A i − C A C A i − C A o that is function of the dimensionless variable η = y x R e x .

If it is desired a mathematical expression for the flux of A at the solid surface (N_A in kmol/m²/s), it is needed to use Fick’s law to obtain

N A , x = 0.332 D A B ( C A i − C A o ) R e x x (3)

Integrating over the plate length

N A = 0.664 D A B ( C A i − C A o ) L R e L (4)

By analogy to the thermal boundary layer, the Schmidt number relates the diffusivities of mass and momentum, giving:

S c ≡ ν D A B = μ ρ D A B (5)

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

N A = 0.664 D A B ( C A i − C A o ) L R e L S c 1 / 3 (6)

Equation (6) has been experimentally tested.

Equation (6) allows the calculation of the rate of mass transfer for molecular diffusion at forced convection for laminar flow. If the flow is turbulent or the geometry 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):

k c ≡ N A Δ C A = N A C A i − C A o (7)

Applying Equation (6) on Equation (7) we get:

k c = 0.664 D A B R e L L S c 1 / 3 (8)

This equation may be arranged and it provide the Sherwood dimensionless number:

S h = k c L D A B = 0.664 R e L S c 1 / 3 (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.

From the reported data with the sources given in Table 1, we may calculate the amount of mass extracted with m_extracted = m_o yieldx 100 (kg) and convert it to kmol dividing it between the molecular weight of the solute extracted. Then the flux of A (N_A, kmol/m²/s) will be this kmol divided between the transversal area to flow and also divided by the residence time of the flow. This last parameter may be calculated dividing the volume (m³) of extractor between the volumetric flows of steam (m³/s).

A program was developed with Excel software and applied to each series of data in 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/m³. Because the steam does not contain solute: C_Ao = 0. Then using Equation (7):

k c = N A C A i − C A o = kmol m 2 ⋅ s kmol m 3 = m s (10)

Then with physical properties, we calculate Sherwood number (k_cD/D_AB), and calculating Reynolds number R e = U e f d e n V D C v i s c V = m s ⋅ kg m 3 ⋅ m kg m ⋅ s and Schmidt number S c = v i s c d e n D A B = kg m ⋅ s kg m 3 ⋅ m 2 s and using a reasonable number of data points we can correlate coefficient and exponents (α₁, α₂, and α₃).

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 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 density and viscosity keep constant values. Diffusivity varies a little depending of the solute.

The exponent α₃ in Equation (1) keep a constant value of 1/3 or 0.33 for the Schmidt number in the hydrodynamic bounder layer as well as for the Prandtl thermal boundary. Then, here for mass transfer, we are going to take this exponent constant: α₃ = 0.33.

Table 3 shows some of the results of the Excel program applied to the data at 100˚C. It is noted that the Reynolds number is less than unity for the systems of Cerpa et al. and Masango et al., and is bigger for all other systems, reaching values above 2000 for the systems of Roautby et al. and Ozek.

Figure 3 shows the plot of Ln(Sh/Sc^(1/3)) versus Ln(Re_Dc) obtained when α₃ = (1/3) for all data for the several systems used. From this data, Equation (11) provides the first correlation equation.

S h = α 1 R e D c α 2 S c α 3 = 0.2754 R e D c 1.5338 S c 0.333 (11)

If we pass the line at the intersection Sh/Sc^0.33 = 1.0 to get Re_Dc = 0, we get the equation y = 1.2964 x + 1 and from this equation we get the second correlation 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.

S h = α 1 R e D c α 2 S c α 3 = 2.7182 R e D c 1.2964 S c 0.333 (12)

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 provide the ratio of the property or parameter at 175˚C/100˚C, and 250˚C/100˚C. When that ratio is unity, the values are similar for the three temperatures.

When the ratio is fractional, by example the volumetric flow of steam: Q 175 Q 100 = 0.132 means that the volumetric flow rate of steam at 175˚C is 0.132 times the volumetric flow rate at 100˚C. This happen because the density of steam at 175˚C is 7.56 times higher than at 100˚C.

For the flow at 250˚C the ratio is Q 250 Q 100 = 0.030 this means that the flow rate of steam at 250˚C is about 3% of the volumetric flow rate of steam at 100˚C.

When the ratio is higher than unity, by example Y 175 Y 100 = 1.205 this means that the yield of extraction is 1.205 higher at 175˚C than at 100˚C.

Note that the superficial and effective velocities are higher at low temperatures (100˚C) than at 175˚C or 250˚C, residence time, density, viscosity, diffusivity, and the mass extracted are higher for bigger temperatures. Reynolds, Schmidt,

and Sherwood numbers, as well as the molar flux and the mass transfer coefficient are lower for the higher temperatures. For predicted Sherwood number:

S h 175 S h 100 = 0.306 (13)

S h 250 S h 100 = 0.100 (14)

Figure 5 provides a method to estimate the Sherwood number at different temperatures of steam.