^{1}

^{*}

^{2}

^{1}

^{3}

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 normally improves the mass transfer and the yield. A method to estimate the enhancement 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.

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 [_{2}, and microwave extraction [

The extraction of essential oil by steam distillation uses a cylindrical column filled with vegetal leaves and stems. The steam flow through the leaves and stems, first heating them and then dissolving on it the essential oil, and taking out from the column with the flowing steam. The flowing steam that leaves the column is passing to a condenser where the oil and water usually form two different liquid phases and are separated in equipment called Florentine.

The experimental data of yield versus time of

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.

For mass transfer: Reynolds R e = U e f ρ V D c μ V , Schmidt S c = μ V ρ V D A B , and Sherwood S h = k C D C D A B . Basically:

S h = α 1 R e D c α 2 S c V α 3 (1)

Several series of experimental and reported data is used to get variation on

operational, physical properties, and geometrical parameters, to generate correlations to predict Sherwood number., and to calculate mass transfer, and yield of extraction.

(Taken from [_{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

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.

Author(s) | Journal | Plant | Solute | Dc (m) | Zc (m) | Q_{V} (L/h) |
---|---|---|---|---|---|---|

Cerpa et al. 2007, 2008 [ | Ph.D. (2007) Dissertation, AIChE J. 2008 | Lavender | Linalool and linalyl acetate | 0.35 | 0.42 | 0.6 - 2.10 |

Masango, 2005 [ | J. of Cleaner Production | Artemisia | Camphor-L | 0.09 | 0.34 | 0.15 - 1.19 |

Soto-Armenta et al. 2017 [ | J. of Essential Oil Bearing Plants | Lippia graveolens (oregano) | Carvacrol and Thymol | 0.10 | 0.76 | 1.4 - 6.98 |

Malekydozzadeh, 2012 [ | Iranian J. of Chem. Eng. | Rosemary | a-pynene, 1, 8 Cineole, Camphor | 0.06 | 0.20 | 240 - 420 |

Roautby et al. 2007 [ | J. of food Engineering | Thyme | Thymol P Cymene | 0.02 | 0.10 | 2678 - 4179 |

Romdhane and Tizaoui, 2005 [ | J. of Chemical Technology and biotechnology | Aniseed | Anethol | 0.26 | 0.90 | 7318 |

Ozek, 2012 [ | Record of Natural Products | Laurel | 1.8 cineole | 0.68 | 1.36 | 44,143 - 88,830 |

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/m^{2}/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 _{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/m^{2}/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 (m^{3}) of extractor between the volumetric flows of steam (m^{3}/s).

A program was developed with Excel software and applied to each series of data in _{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):

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_{c}D/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 (α_{1}, α_{2}, and α_{3}).

Author(s) | Solute | Density (kg/m^{3}) | Viscosity (kg/m/s) | Diffusivity (m^{2}/s) | Molecular weight (kg/kmol) | Solubility (kmol/m^{3}) |
---|---|---|---|---|---|---|

Cerpa et al. 2007, 2008 [ | Linalool and linalyl acetate | 0.555 | 1.32E−5 | 1.10E−5 | 175 | 0.080000 |

Masango, 2005 [ | Camphor-L | 0.555 | 1.32E−5 | 1.17E−5 | 152 | 0.080000 |

Soto-Armenta et al. 2017 [ | Carvacrol and Thymol | 0.555 | 1.32E−5 | 1.18E−5 | 150 | 0.007160 |

Malekydozzadeh, 2012 [ | a-pynene, 1, 8 Cineole, Camphor | 0.555 | 1.32E−5 | 1.17E−5 | 152 | 0.007600 |

Roautby et al. 2007 [ | Thymol P Cymene | 0.597 (100˚C) 4.515 (175˚C) 19.984 (250˚C) | 1.30E−5 1.50E−5 1.80E−5 | 1.18E−5 1.63E−5 2.14E−5 | 148 | 0.005900 0.005900 0.005900 |

Romdhane and Tizaoui, 2005 [ | Anethol | 0.597 | 1.32E−5 | 9.06E−6 | 148 | 0.000750 |

Ozek, 2012 [ | 1.8 cineole | 0.555 | 1.32E−5 | 1.16E−5 | 154 | 0.000023 |

Density and viscosity were taken from [

Most of the experimental extraction with steam used atmospheric pressure and temperature of 100˚C. Only some data from Roautby et al. [

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 α_{3} 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: α_{3} = 0.33.

^{(1/3)}) versus Ln(Re_{Dc}) obtained when α_{3} = (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

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. [

Author(s) | Re_{Dc} | m_{A} kg | N_{A} kmol/m^{2}/s | K_{c} m/s | Sh_{α}_{1}_{α}_{2}α_{1}α_{2} | Sh_{α}_{1}_{α}_{2}α_{1}α_{2} | Sh_{α}_{1}_{α}_{2}α_{1}α_{2} |
---|---|---|---|---|---|---|---|

Cerpa et al. 2007, 2008 [ | 0.032/0.124 | 0.026/0.036 | 6.41E−9/ 3.09E−8 | 8.01E−8/ 3.86E−7 | 2.55E−3/ 1.23E−2 | 0.0801 | 1.0672 |

Masango, 2005 [ | 0.035/0.274 | 2.33E−4/ 7.13E−4 | 1.68E−8/ 4.34E−8 | 2.10E−7/ 5.43E−7 | 1.54E−3/ 3.99E−3 | 0.0057 | 0.4547 |

Soto-Armenta et al. 2017 [ | 0.288/1.440 | 3.92E−3/ 9.87E−3 | 4.69E−7/ 1.96E−6 | 6.54E−5/ 2.73E−4 | 5.55E−1/ 2.32E0 | 1.1763 | 0.6177 |

Malekydozzadeh, 2012 [ | 79.3/178.4 | 8.30E−4/ 1.55E−3 | 1.26E−4/ 5.63E−4 | 1.65E−2/ 7.41E−2 | 8.48E1/ 3.80E2 | 1.0550 | 1.0447 |

Roautby et al. [ | 2092.1/4525.7 | 0.000231/ 0.000420 | 5.54E−3/ 2.44E−1 | 1.80/ 78.6 | 1671.4/ 133,160.6 | 1.5600 | 1.2265 |

Romdhane and Tizaoui [ | 787.7/812.4 | 0.036/0.098 | 1.51E−4/ 3.51E−4 | 2.02E−1/ 4.69E−1 | 6.99E3/ 1.88E4 | 0.3470 | 1.3527 |

Ozek, 2012 [ | 1287.1/2590.1 | 0.350/1.380 | 2.73E−4/ 8.06E−4 | 12.0/35.5 | 7.04E5/ 2.08E6 | 1.201 | 1.8199 |

All prediction 1 | 0.032/4525.7 | 2.31E−4/ 1.38E0 | 6.41E−9/ 2.33E−1 | 8.01E−8/ 3.96E1 | 1.54E−3/ 2.08E6 | 0.2752 | 1.5338 |

All prediction 2 | 0.032/4525.7 | 2.31E−4/ 1.38E0 | 6.41E−9/ 2.33E−1 | 8.01E−8/ 3.96E1 | 1.54E−3/ 2.08E6 | 2.7182 | 1.2964 |

properties changed, most of the parameters changed as well.

The last two rows of

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,

Roautby et al. [ | min | sec | m | m | m^{3}/m^{3} | kg | m^{3}/s | % |
---|---|---|---|---|---|---|---|---|

t_{min} | t | D_{c} | Z | E | M_{o} | Q | Y | |

m = 1.6 k/h, Q = 0.000744, 100˚C | 40.0 | 2400.00 | 0.020000 | 0.10 | 0.750 | 0.007 | 0.000744 | 3.30 |

m = 2.5, Q = 0.001163, 100˚C | 40.0 | 2400.00 | 0.020000 | 0.10 | 0.750 | 0.007 | 0.001161 | 4.20 |

m = 1.6 k/h, Q = 0.000098, 175˚C | 40.0 | 2400.00 | 0.020000 | 0.10 | 0.750 | 0.007 | 0.000098 | 4.10 |

m = 2.5, Q = 0.000153, 175˚C | 40.0 | 2400.00 | 0.020000 | 0.10 | 0.750 | 0.007 | 0.000153 | 4.90 |

m = 1.6 k/h, Q = 0.000022, 250˚C | 40.0 | 2400.00 | 0.020000 | 0.10 | 0.750 | 0.007 | 0.000022 | 5.00 |

m = 2.5, Q = 0.000034, 250˚C | 40.0 | 2400.00 | 0.020000 | 0.10 | 0.750 | 0.007 | 0.000034 | 6.00 |

x_{5}/x_{3} | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.132 | 1.242 |

x_{7}/x_{3} | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.030 | 1.515 |

x_{6}/x_{4} | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.132 | 1.167 |

x_{8}/x_{4} | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.029 | 1.429 |

(x_{5}/x_{3} + x_{7}/x_{3})/2 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.132 | 1.205 |

(X_{6}/X_{4} + X_{8}/X_{4})/2 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.030 | 1.472 |

m^{2} | m^{3} | m/s | m/s | s | kg/m^{3} | kg/m/s | m^{2}/s | kg/kmol | kmol/m^{3} |
---|---|---|---|---|---|---|---|---|---|

A | V | Us | Uef | tR | den | vis | DAB | PMA | Cai-solub |

0.000314 | 0.000031 | 2.368220 | 3.157627 | 0.042 | 0.597 | 1.30E−05 | 1.18E−05 | 1.42E+02 | 0.003 |

0.000314 | 0.000031 | 3.695569 | 4.927426 | 0.027 | 0.597 | 1.30E−05 | 1.18E−05 | 1.42E+02 | 0.003 |

0.000314 | 0.000031 | 0.313216 | 0.417622 | 0.319 | 4.515 | 1.50E−05 | 1.63E−05 | 1.42E+02 | 0.003 |

0.000314 | 0.000031 | 0.487013 | 0.649351 | 0.205 | 4.515 | 1.50E−05 | 1.63E−05 | 1.42E+02 | 0.003 |

0.000314 | 0.000031 | 0.070665 | 0.094220 | 1.415 | 19.984 | 1.80E−05 | 2.14E−05 | 1.42E+02 | 0.003 |

0.000314 | 0.000031 | 0.108225 | 0.144300 | 0.924 | 19.984 | 1.80E−05 | 2.14E−05 | 1.42E+02 | 0.003 |

1.000 | 1.000 | 0.132 | 0.132 | 7.561 | 7.563 | 1.154 | 1.381 | 1.000 | 1.000 |

1.000 | 1.000 | 0.030 | 0.030 | 33.514 | 33.474 | 1.385 | 1.814 | 1.000 | 1.000 |

1.000 | 1.000 | 0.132 | 0.132 | 7.588 | 7.563 | 1.154 | 1.381 | 1.000 | 1.000 |

1.000 | 1.000 | 0.029 | 0.029 | 34.147 | 33.474 | 1.385 | 1.814 | 1.000 | 1.000 |

1.000 | 1.000 | 0.132 | 0.132 | 7.575 | 7.563 | 1.154 | 1.381 | 1.000 | 1.000 |

1.000 | 1.000 | 0.030 | 0.030 | 33.830 | 33.474 | 1.385 | 1.814 | 1.000 | 1.000 |

U*den*Dc/vis | den*vis/DAB | kg | kmol/m^{2}/s | m/s | Kc*Dc/DAB | Roautby et al. [ | |
---|---|---|---|---|---|---|---|

Re-Dc | Sc | mA | NA | Kc | Sh-Dc | Sh-Dc -p1 | |

2900.16 | 1.845 | 0.000231 | 0.123 | 39.56 | 67,047.3 | 21,248.9 | m = 1.6 k/h, Q = 0.000744, 100˚C |

4525.65 | 1.845 | 0.000294 | 0.244 | 78.56 | 133,160.6 | 84,452.8 | m = 2.5, Q = 0.001163, 100˚C |

2514.08 | 0.204 | 0.000287 | 0.020 | 6.50 | 7975.7 | 6546.4 | m = 1.6 k/h, Q = 0.000098, 175˚C |

3909.09 | 0.204 | 0.000343 | 0.037 | 12.08 | 14,821.0 | 25,729.6 | m = 2.5, Q = 0.000153, 175˚C |

2092.09 | 0.042 | 0.000350 | 0.006 | 1.79 | 1671.4 | 2188.8 | m = 1.6 k/h, Q = 0.000022, 250˚C |

3204.10 | 0.042 | 0.000420 | 0.010 | 3.29 | 3071.8 | 8208.3 | m = 2.5, Q = 0.000034, 250˚C |

0.867 | 0.110 | 1.242 | 0.164 | 0.164 | 0.119 | 0.308 | x_{5}/x_{3} |

0.721 | 0.023 | 1.515 | 0.045 | 0.045 | 0.025 | 0.103 | x_{7}/x_{3} |

0.864 | 0.110 | 1.167 | 0.154 | 0.154 | 0.111 | 0.305 | x_{6}/x_{4} |

0.708 | 0.023 | 1.429 | 0.042 | 0.042 | 0.023 | 0.097 | x_{8}/x_{4} |

0.865 | 0.110 | 1.205 | 0.159 | 0.159 | 0.115 | 0.306 | (x_{5}/x_{3} + x_{7}/x_{3})/2 |

0.715 | 0.023 | 1.472 | 0.044 | 0.044 | 0.024 | 0.100 | (X_{6}/X_{4} + X_{8}/X_{4})/2 |

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)

The data on the paper of Koul et al. [_{10}H_{16}O), that has the same formula than camphor then (D_{AB} = 1.17E−5 m^{2}/s) but the solubility is C_{Ai} = 0.003289 kmol/m^{3}.

The first two experiments used m_{o}= 70 kg, and the last three used 1000 kg. These are industrial quantities. In ^{3}) and the highest quantity of vegetal leaves (24 - 38 kg). We era going to use the data of Ozek to estimate the volume of cylinder needed for Koul experiments. Ozek [^{3} for 35 kg of mass. Then for Koul:

V 1 = 0.5 × 100 / 35 = 1.428 m 3 and V 2 = 0.5 × 1000 / 35 = 14.286 m 3

Choosing a diameter of Dc = 2.0 m, A= 3.1416*Dc^{2}/4 = 3.1416 m^{2}, and V = A*z, then z_{1} = V_{1}/A = 1.428/3.1416 = 0.46 m = z_{1}, and z_{2} = V_{2}/A = 14.286/3.1416 = 4.55 m = z_{2}.

The steam flow rate Q_{i} in m^{3}/s used in Koul experiments were: 15, 12, 160, 125 and 140 L/h (0.015, 0.012, 0.125, 0.140 m^{3}/h, or 4.166E−6, 3.33 E−6, 4.44E−5, 3.472E−5, 3.88E−5 m^{3}/s).

For the yield calculation, experiments 3 - 7 were: 385, 330, 5725, 5215, 5315 mL of citral oil at 5 h = 300 minutes = 18,000 seconds. With a density of citral of 0.9 gr/ml or 900 kg/m^{3} and the 70 kg of lemon grass for experiments 3 and 4 and 1000 kg of lemon grass for experiments 5 - 7, we get: y_{3} = 0.495, y_{4} = 0.424, y_{5} = 0.515, y_{6} = 0.469, y_{7} = 0.478.

With the cylinder dimensions Dc = 2 m y z_{1} = 0.46, and z_{2} = 4.55 m, we get A = pi*Dc^{2}/4 = 3.1416 m^{2}, and V_{1} = 1.44 m^{3}, V_{2} = 14.2 m^{3}. Dividing volumetric flow rate between area we get Superficial velocities, and dividing these between void fraction, we get effective velocities U_{ef}_{3} = 0.0000018, U_{ef}_{4} = 0.0000014, U_{ef}_{5} = 0.0000188, U_{ef}_{6} = 0.0000147, U_{ef}_{7} = 0.0000165 m/s.

Residence time of steam may be calculated dividing volume between steam volumetric flow rate, and we get: t_{r}_{3} = 346,888.1, t_{r}_{4} = 433,974.8, t_{r}_{5} = 321,943.2, t_{r}_{6} = 411,701.6, t_{r}_{7} = 368,409.3 seconds.

Now, we can calculate Reynolds numbers with Equation (15)

R e i = U e f i d e n v a p D c V i s c v a p (15)

And Schmidt and Sherwood numbers with Equation (16) and Equation (11) or Equation (12)

S c = v i s c v a p d e n v a p D A B (16)

Re | k_{C}_{1} | k_{C}_{2} | N_{A}_{1} | N_{A}_{2} | m_{A}_{1} | m_{A}_{2} | y_{Ap}_{-1} | y_{Ap}_{-2} | y_{A-Exp} | |
---|---|---|---|---|---|---|---|---|---|---|

E−3 | 0.15 | 1.098E−07 | 3.545E−06 | 3.610E−08 | 5.601E−09 | 0.060 | 0.929 | 0.09 | 1.33 | 0.50 |

E−4 | 0.12 | 7.785E−08 | 2.652E−06 | 2.561E−10 | 4.189E−09 | 0.053 | 0.869 | 0.08 | 1.24 | 0.42 |

E−5 | 1.58 | 4.137E−06 | 7.619E−05 | 1.361E−08 | 1.204E−07 | 2.095 | 18.533 | 0.21 | 1.85 | 0.52 |

E−6 | 1.24 | 2.837E−06 | 5.539E−05 | 9.332E−09 | 8.751E−08 | 1.837 | 17.230 | 0.18 | 1.72 | 0.47 |

E−7 | 1.38 | 3.364E−06 | 6.397E−05 | 1.107E−07 | 1.011E−07 | 1.950 | 17.807 | 0.19 | 1.78 | 0.48 |

With Sherwood number, we can get the mass transfer coefficient k_{c}, and from this, the flux N_{A}, then, m_{A}, and finally the yield y_{A} with Equations (17)-(20). Because the steam does not have solute C_{Ao} = 0

k C = S h D A B D C (17)

N A = k c ( C A i − C A o ) (18)

m A = N A × A × t R × P M A (19)

y a i = ( m A m o ) × 100 (20)

On

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.

S h ( T ˚ C ) S h ( 100 ˚ C ) = 4.591 exp − 0.015 × T ( ˚ C ) (21)

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Rocha-Uribe, J.A., Soto-Armenta, L.C., Hernandez-Ruiz, A.R. and Jimenez-Ocaña, J.C. (2021) Predicting Mass Transfer Extraction with Steam Flow, Applying Boundary-Layer Concepts. Journal of Materials Science and Chemical Engineering, 9, 46-58. https://doi.org/10.4236/msce.2021.96004