Such a formula has been used in the literature for modelling drying shrinkage [67]. The model showed a very good fit to the experimental swelling data with a high value of the determination

The coefficients shown in Table 1 were calculated by fitting experimental data to four chosen empirical models. As it was stated in chapter 2.2.1, the coefficients of the Peleg model, the Pilosof-Boquet-Batholomai model, and the Singh-Kulshrestha model are connected between themselves. Table 3 shows the model parameters estimated using these interdependences.

It turned out from the comparison of the results of calculations presented in Tables 1 and 3 that the values of coefficients determined using both methods are almost the same. It can be stated therefore that there is a similarity of these three considered models and their predictive abilities for rehydration of dried apple cubes are identical. The same results have been obtained by Sopade et al. in [16] for describing water absorption of wheat starch, whey protein concen‐ trate, and whey protein isolate. Shittu et al. in [68] observed, however, differences in discussed

models predictive ability for the hydration of African breadfruit seeds.

450 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

Mass Peleg [27] 7 Pilosof-Boquet-Batholomai [36] A1=0.017407518

Volume Peleg [27] 7 Pilosof-Boquet-Batholomai [36] A1=0.020565381

Peleg [27] 7 Singh-Kulshrestha [37] A1=0.017407518

Peleg [27] 7 Pilosof-Boquet-Batholomai [36] A1=-0.044699038

Peleg [27] 7 Singh-Kulshrestha [37] A1=-0.044664539

Peleg [27] 7 Singh-Kulshrestha [37] A1=0.020585496

**Table 3.** Coefficients of the Peleg model [27], the Pilosof-Boquet-Batholomai model [36], and the Singh-Kulshrestha

**no.**

**Variable Model name Equation**

Pilosof-Boquet-Batholomai [36]

Pilosof-Boquet-Batholomai [36]

Pilosof-Boquet-Batholomai [36]

model [37] estimated using the interdependences between the coefficients

Dry matter of solid

=0.00032. The estimated value of swelling coefficient *n*=0.293.

=0.98920 and low values of root mean square error RMSE=0.01780 and reduced

**Coefficients estimated using coefficients from the following model**

15 Singh-Kulshrestha [37] A3=29.10351515

15 Singh-Kulshrestha [37] A3=-6.666785913

15 Singh-Kulshrestha [37] A3=29.71819917

**Model parameters**

A2=0.034360111

A2=0.034360111

A4=0.506619958

A2=-0.149964985

A2=-0.149997317

A4=0.29776892

A2=0.033658468

A2=0.033649414

A4=0.611763873

coefficient R2

chi-square χ2

Four empirical and one theoretical models were investigated for their suitability to describe the mass transfer kinetics during rehydration of dried apple cubes. The determination coefficient, root mean square error, and reduced chi-square method were estimated for all models considered to compare their goodness of fit the experimental rehydration data. All models described the rehydration characteristics of dried apple cubes satisfactorily (R2 >0.9852). The empirical Witrowa-Rajchert model [40] and the theoretical model based on Fick's second law can be considered as the most appropriate. Theoretical models give an explanation of the phenomena occurring during rehydration but are difficult in application compared to empirical models. Therefore, if the description of rehydration curves is only needed it is better to apply empirical models. Such need occurs especially in food industry.

The determined values of mass diffusion coefficient was found to be between 2.78·10-10 m2 s-1 and 6.60·10-10 m2 s-1. These values are within the general range for food materials.

## **Acknowledgements**

The authors are grateful for the financial support from research project No. N N313 780940 from the Polish National Science Centre.

## **Nomenclature**

```
a, b, c - equation coefficients (Eq. (2))
```

## **Subscripts**

0 - initial

a, b, c - equation coefficients (Eq. (2))

*Dcalc* - calculated diffusion coefficient ( m s-1) *Deff* - effective diffusion coefficient ( m s-1)

*h* - half of cylinder heigh, high of liquid rise (m)

*Me* - equilibrium moisture content (dry basis)

s-1)

s-1)

*Rc* - half of plane or cube thickness, cylinder radius, sphere radius (m)

*D* - mass diffusion coefficient (effective diffusivity) (m2

452 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

s-1)

*Ea* - activation energy (J mol-1)

g - gravitational constant (m s-2)

*k* - rehydration rate constant (s-1)

*L* - characteristic dimension (m) *M* - moisture content (dry basis)

n - constant (Eqs. (32), (33), and (34))

R - universal gas constant (J K-1 mol-1)

n - swelling coefficient (Eq. (65))

*r* - mean pore radius (m) *r*, *x*, *y*, *z* - coordinates (m)

Rg - constant in Eq.(29)

*T* - temperature (K)

*t* - time (s)

*V* - volume (m3

R2

RMSE - root mean square error

) α - constant (Eqs. (25) and (26)) α' - constant (Eqs. (27) and (28))

 - coefficient of determination *s* - dry matter content (kg d.m. kg-1)

*x* - water absorption ratio (Eqs. (33), (34), (35))

*k1* - constant (Eq. (64)) (m2

*k2* - constant (Eq. (64)) (m2

*m* - mass kg

## **Author details**

Krzysztof Górnicki, Agnieszka Kaleta, Radosław Winiczenko, Aneta Chojnacka and Monika Janaszek

Faculty of Production Engineering, Warsaw University of Life Sciences, Poland

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## **Chapter 17**

## **Subcritical Water Extraction**

A. Haghighi Asl and M. Khajenoori

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54993

## **1. Introduction**

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[64] Planinić M, Velić D, Tomas S, Bilić M, Bucić A. Modelling of Drying and Rehydration of Carrots Using Peleg's Model. European Food Research and Technology 2005;221(3-4)

[65] Doulia D, Tzia K, Gekas V. A Knowledge Base for the Apparent Mass Diffusion Coefficient (DEFF) of Foods. International Journal of Food Properties 2000;3(1) 1-14,

[66] Maroulis ZB, Saravacos GD, Panagiotou NM, Krokida MK. Moisture Diffusivity Data Compilation for Foodstuffs: Effect of Material Moisture Content and Temperature.

[67] Górnicki K, Kaleta A. Modelling Convection Drying of Blanched Parsley Root Slices.

[68] Shittu TA, Awonorin SO, Raji AO. Evaluating Some Empirical Models for Predicting Water Absorption in African Breadfruit (Treculia Africana) Seeds. International Journal

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vulgaris L.). Journal of Food Science 1997;62(3) 520-523, ISSN 1750-3841.

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Extraction always involves a chemical mass transfer from one phase to another. The principles of extraction are used to advantage in everyday life, for example in making juices, coffee and others. To reduce the use of organic solvent and improve the extraction methods of constituents of plant materials, new methods such as microwave assisted extraction (MAE), supercritical fluid extraction (SFE), accelerated solvent extraction (ASE) or pressurized liquid extraction (PLE) and subcritical water extraction (SWE), also called superheated water extraction or pressurized hot water extraction (PHWE), have been introduced [1-3].

SWE is a new and powerful technique at temperatures between 100 and 374o C and pressure high enough to maintain the liquid state (Fig.1) [4]. Unique properties of water are namely its disproportionately high boiling point for its mass, a high dielectric constant and high polarity [4]. As the temperature rises, there is a marked and systematic decrease in permittivity, an increase in the diffusion rate and a decrease in the viscosity and surface tension. In conse‐ quence, more polar target materials with high solubility in water at ambient conditions are extracted most efficiently at lower temperatures, whereas moderately polar and non-polar targets require a less polar medium induced by elevated temperature [5].

Based on the research works published in the recent years, it has been shown that the SWE is cleaner, faster and cheaper than the conventional extraction methods. The essential oil of *Z. multiflora* was extracted by SWE and compared with two conventional methods, including hydrodistillation and Soxhlet extraction [7]. The total extraction yields found for the total essential oil of *Z. multiflora* were 2.58, 1.51 and 2.21% (w/w) based on the dry weight for SWE, hydrodistillation and Soxhlet extraction, respectively.

The comparison among the amount of thymol and carvacrol (milligram per gram dried sample) by SWE, hydrodistillation and Soxhlet extraction is shown in Table 1 [7]. The amount of valuable oxygenated components in the SWE method is significantly higher than hydro‐

**Figure 1.** Phase diagram of water as a function of temperature and pressure (Cross-hatched area indicates the prefer‐ red region (SWE)) [4].

\* Extraction time = 150 min.

i Extraction time = 210 min.

§ Retention indices (RI) on the DB-5 column.

**Table 1.** The amount of Thymol and carvacrol (mg/g dried sample) of the essential oil of *Z.multiflora*, extracted by SWE, hydrodistillation and Soxhlet extraction [7].

distillation and Soxhlet extraction. As hexane is a nonpolar solvent, non-oxygenated compo‐ nents are enhanced compared to subcritical water. On the other hand, in general, nonoxygenated components present lower vapor pressures compared to oxygenated components, and in this sense, its content in hydrodistillated extracts are increased. Because of the significant presence of the oxygenated components, the final extract using the SWE method was relatively better and more valuable.

**Figure 2.** (a) and (b) pictures of first and second generated systems respectively and (c) Schematic diagram of SWE system, B-1: Burette, C-1: Nitrogen cylinder, EC: Extraction cell, HX-1: Heat Exchanger, OV-1: Oven, P-1, 2: Pumps, V-1: Water tank, V-2: Solvent tank, V-3: Rinsing solvent tank, WI: Water inlet, WO: Water outlet.

## **2. Equipment of subcritical water extraction**

distillation and Soxhlet extraction. As hexane is a nonpolar solvent, non-oxygenated compo‐ nents are enhanced compared to subcritical water. On the other hand, in general, nonoxygenated components present lower vapor pressures compared to oxygenated components, and in this sense, its content in hydrodistillated extracts are increased. Because of the significant presence of the oxygenated components, the final extract using the SWE method was relatively

**Table 1.** The amount of Thymol and carvacrol (mg/g dried sample) of the essential oil of *Z.multiflora*, extracted by

**Figure 1.** Phase diagram of water as a function of temperature and pressure (Cross-hatched area indicates the prefer‐

460 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

better and more valuable.

\* Extraction time = 150 min. \*\* Relative SD percent. 1' Extraction time = 180 min. i Extraction time = 210 min.

§ Retention indices (RI) on the DB-5 column.

SWE, hydrodistillation and Soxhlet extraction [7].

red region (SWE)) [4].

No commercial SWE equipment is available, but the apparatus is easy to construct in the laboratory. SWE is performed in batch or continue systems but continue system is current. In this system, extraction bed is fixed and flow direction is usually up to down for easily cleaning ofanalytes.Already,weworkedSWEof*Z.multiflora*withfirstgeneratedsystem(IranianResearch OrganizationforScienceandTechnology[IROST],Tehran,Iran)[7](Fig.2(a)).Afterit,laboratorybuilt apparatus of SWE equipment was designed by studying of different systems and have advantages versus first generation (Semnan University, Semnan, Iran) (Fig. 2(b)).

The second generated system is presented in Fig. 2(c). The main parts of a dynamic SWE unit are the following: three tanks, two pumps, extraction vessel, oven for the heating of the

**Figure 3.** Effect of temperature on the main essential oil components SWE of *Z. multiflora*. Operating conditions: sam‐ ple weight=4.0 g; flow rate=2 mL/min; particle size=0.50 mm; pressure=20 bar; and extraction time=60 min.

extraction vessel, heat Exchanger for cooling of exteract, pressure restrictor and sample collection system. One of the pumps is employed for pumping the water (and extract) and another pump is employed for flushing the tubings. Also, one of tank is employed for organic solvent that is as a main solvent or co-solvent. A pressure restrictor is needed to maintain the appropriate pressure in the equipment. It was constructed of stainless steel.

## **3. Effective parameters in subcritical water extraction**

#### **3.1. Effect of temperature**

One of the most important parameters affecting SWE efficiencies is the extraction temperature. As the temperature rises, there is a marked and systematic decrease in permittivity, an increase in the diffusion rate and a decrease in the viscosity and surface tension. SWE must be carried out at the highest permitted temperature. It should be mentioned that increasing the extraction temperature above a certain value gives rise to the degradation of the essential oil components. The maximum permitted extraction temperature must be obtained experimentally for different plant materials. Regarding the extraction of essential oils, it has been shown that temperatures between 125 and 175o C will be the best condition. The extraction temperature for *Z. multi‐ flora* was optimized in order to maximize the efficiency of thymol and carvacrol (structural isomers) as key components (more than 72%) [7]. Its influence was studied between 100 and 175o C, and the mean particle size, flow rate, extraction time and pressure were selected to be 0.5 mm, 2 ml/min, 60 min and 20 bar pressure, respectively (Fig. 3).

It was seen that the efficiency of thymol and carvacrol increased generally with increase in temperature up to 150o C. At 175o C, it decreased, and an extract with a burning smell was produced. It may be the result of degradation of some of the constituents at higher tempera‐ tures. Because of the highest efficiency of thymol and carvacrol essential oil at 150o C and the

**Figure 4.** Effect of particle size on the efficiency of a) thymol and b) carvacrol SWE of *Z.multiflora.* Operating condi‐ tions: sample weight = 4.0g; flow rate = 2 ml/min; temperature = 150oC; and pressure = 20 bar [7].

disagreeable odor of the extract at higher temperatures, further experiments were carried out at this temperature.

#### **3.2. Effect of particle size**

extraction vessel, heat Exchanger for cooling of exteract, pressure restrictor and sample collection system. One of the pumps is employed for pumping the water (and extract) and another pump is employed for flushing the tubings. Also, one of tank is employed for organic solvent that is as a main solvent or co-solvent. A pressure restrictor is needed to maintain the

**Figure 3.** Effect of temperature on the main essential oil components SWE of *Z. multiflora*. Operating conditions: sam‐ ple weight=4.0 g; flow rate=2 mL/min; particle size=0.50 mm; pressure=20 bar; and extraction time=60 min.

One of the most important parameters affecting SWE efficiencies is the extraction temperature. As the temperature rises, there is a marked and systematic decrease in permittivity, an increase in the diffusion rate and a decrease in the viscosity and surface tension. SWE must be carried out at the highest permitted temperature. It should be mentioned that increasing the extraction temperature above a certain value gives rise to the degradation of the essential oil components. The maximum permitted extraction temperature must be obtained experimentally for different plant materials. Regarding the extraction of essential oils, it has been shown that temperatures

*flora* was optimized in order to maximize the efficiency of thymol and carvacrol (structural isomers) as key components (more than 72%) [7]. Its influence was studied between 100 and

It was seen that the efficiency of thymol and carvacrol increased generally with increase in

produced. It may be the result of degradation of some of the constituents at higher tempera‐

tures. Because of the highest efficiency of thymol and carvacrol essential oil at 150o

C, and the mean particle size, flow rate, extraction time and pressure were selected to be

C will be the best condition. The extraction temperature for *Z. multi‐*

C, it decreased, and an extract with a burning smell was

C and the

appropriate pressure in the equipment. It was constructed of stainless steel.

462 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

**3. Effective parameters in subcritical water extraction**

0.5 mm, 2 ml/min, 60 min and 20 bar pressure, respectively (Fig. 3).

C. At 175o

**3.1. Effect of temperature**

between 125 and 175o

temperature up to 150o

175o

The effect of the mean particle size on the efficiency of thymol and carvacrol at 150o C temper‐ ature, 2ml/min flow rate, 20 bar pressure and 150 min extraction time is shown in Fig. 4 [7]. The mean ground leaf particles were selected to be 0.25, 0.5 and 1.0 mm. The final amount of thymol and carvacrol extracted from 0.5-mm-size particles was near to 0.25-mm particles. It shows that, at least in the selected range of mean particle sizes (0.25-0.5 mm), the extraction process may not be controlled by the mass transfer of thymol and carvacrol. It was expected that the rate of the 0.25-mm-size particles was more than the 0.5-mm-size particles, but it did not happen.

A possible explanation for this observation could be that the particles were close fitting at initial times, and the extraction was done slowly. After the expired time, the close fitting particles opened from each other, and at final extraction value of the 0.25-mm-size particles was more than the 0.5-mm-size particles. Regarding the larger 1.0-mm-size particles, the efficiency is substantially lower. It shows that the process may be controlled by the mass transfer of thymol and carvacrol for larger particle sizes.

To prevent the probable vaporization of the essential oils during the grinding of the leaves and also to make the work of the filters easier, for further experiments, the best value of mean particle size was selected as 0.50 mm.

#### **3.3. Effect of flow rate**

The effect of water flow rate on the efficiency of thymol and carvacrol at 150o C temperature, 0.5-mm-particle size, 20 bar pressure and 150 min extraction time is shown in Fig. 5 [7]. The water flow rate was studied at 1, 2 and 4 ml/min. As can be seen, the rate of the essential oil extraction was faster at the higher flow rate. The rate is slower at 2 ml/min and even slower at 1 ml/min. It is in accordance with previous works [8, 9]. It means that the mass transfer of the

**Figure 5.** Effect of flow rate on the efficiency of a) thymol and b) carvacrol SWE of *Z. multiflora.* Operating conditions: sample weight = 4.0g; particle size= 0.5 mm; temperature = 150oC; and pressure = 20 bar [7].

thymol and carvacrol components from the surface of the solid phase into the water phase regulated most of the extraction process. Increase in flow rate resulted in increase in superficial velocity, and thus, quicker mass transfer [10]. The main disadvantage of applying higher water flow rates is increasing the extract volume and consequently, lower concentration of the final extracts. In practice, the best flow rate must be selected considering two important factors, including the extraction time and the final extract concentration. It is clear that a shorter extraction time and more concentrated extracts are desirable. To prevent a slower extraction rate and longer extraction times, despite the larger amount of the final extracts, a flow rate of 2 ml/min was selected as the optimum value.

While temperature, particle size and flow rate extraction are the main parameters affecting SWE, type of analyte, extraction vessel characteristics and use of modifiers and additives are also important. Although matrix and other effects play a role, many of these are less critical in SWE than in SFE because of the harsh extraction conditions (high temperature) typical in SWE, particularly for non-polar analytes.

### **4. Extraction mechanism**

The SWE process can be proposed to have six sequential steps: (1) rapid fluid entry; (2) desorption of solutes from matrix active sites; (3) diffusion of solutes through organic mate‐ rials; (4) diffusion of solutes through static fluid in porous materials; (5) diffusion of solutes through layer of stagnant fluid outside particles; and (6) elution of solutes by the flowing bulk of fluid (Fig. 6).

As we know, the extraction rate is limited by the slowest of these three steps. The effect of step (1) is typically small and often neglected. Although the diffusion of the dissolved solute within the solid is usually the rate limiting step for most botanicals, partitioning of solute between the solid matrix and solvent have been reported as the rate-limiting mechanism for SWE of essential oil from savory [10].

**Figure 6.** Proposed schematic presentation of the extraction steps in SWE.

The plots the amount of compound extracted versus solvent flow rates and versus solvent volume can determine the relative importance of these steps. For example, if the rate of extraction is controlled by intra-particle diffusion or kinetic desorption, the increase in bulk fluid flow rate would have little effect on extraction rate. On the other hand, if the extraction is controlled by external film transfer diffusion, extraction rates increase with solvent flow rate. In the case where the extraction rate is controlled by thermodynamic partitioning, doubling the bulk fluid flow rate would double the extraction rate, while the curves of extraction efficiency versus the volume of water passed for all flow rates would overlap. In one of our previous work, four proposed models have been applied to describe the extraction mecha‐ nisms obtained with SWE of *Z. multiflora* essential oil. These were included (1) partitioning coefficient model, (2) one-site (3) two-site desorption models and (4) thermodynamic partition with external mass transfer model [11]. In other studying unsteady state mass balance of the solute in solid and subcritical water phases (two-phase model) was investigated [12]. Also Computational Fluid Dynamics (CFD) modeling of extraction was considered [13].

## **5. Modeling of SWE**

thymol and carvacrol components from the surface of the solid phase into the water phase regulated most of the extraction process. Increase in flow rate resulted in increase in superficial velocity, and thus, quicker mass transfer [10]. The main disadvantage of applying higher water flow rates is increasing the extract volume and consequently, lower concentration of the final extracts. In practice, the best flow rate must be selected considering two important factors, including the extraction time and the final extract concentration. It is clear that a shorter extraction time and more concentrated extracts are desirable. To prevent a slower extraction rate and longer extraction times, despite the larger amount of the final extracts, a flow rate of

**Figure 5.** Effect of flow rate on the efficiency of a) thymol and b) carvacrol SWE of *Z. multiflora.* Operating conditions:

sample weight = 4.0g; particle size= 0.5 mm; temperature = 150oC; and pressure = 20 bar [7].

464 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

While temperature, particle size and flow rate extraction are the main parameters affecting SWE, type of analyte, extraction vessel characteristics and use of modifiers and additives are also important. Although matrix and other effects play a role, many of these are less critical in SWE than in SFE because of the harsh extraction conditions (high temperature) typical in SWE,

The SWE process can be proposed to have six sequential steps: (1) rapid fluid entry; (2) desorption of solutes from matrix active sites; (3) diffusion of solutes through organic mate‐ rials; (4) diffusion of solutes through static fluid in porous materials; (5) diffusion of solutes through layer of stagnant fluid outside particles; and (6) elution of solutes by the flowing bulk

As we know, the extraction rate is limited by the slowest of these three steps. The effect of step (1) is typically small and often neglected. Although the diffusion of the dissolved solute within the solid is usually the rate limiting step for most botanicals, partitioning of solute between the solid matrix and solvent have been reported as the rate-limiting mechanism for SWE of

2 ml/min was selected as the optimum value.

particularly for non-polar analytes.

**4. Extraction mechanism**

essential oil from savory [10].

of fluid (Fig. 6).

#### **5.1. Thermodynamic model (Partitioning coefficient (KD) model)**

Partitioning coefficient model, adopted from Kubatova et al. [10], describes the extraction process that is controlled by partitioning of solute between matrix and solvent similar to elution of solute from a partition chromatography column. For extraction, this type of behavior occurs when the initial solute concentration in the plant matrix is small. This model assumes that the initial desorption step and the subsequent fluid-matrix partitioning is rapid. Here the ther‐ modynamics partitioning coefficient, *K*D, is defined as:

$$\mathbf{K}\_D = \frac{\text{Concentration of solute in the matrix}}{\text{Concentration of solute in the extraction fluid}} \text{; at equilibrium} \tag{1}$$

Hence the Extraction with subcritical water can be fitted using this simple thermodynamic model. The mass of analyte in each unit mass of extraction fluid and the mass of analyte remaining in the matrix at that period in the entire extraction time is based on the *K*D value determined for each compound. The thermodynamic elution of analytes from matrix was the prevailing mechanism in SWE as evidenced by the fact that extraction rate increased propor‐ tionally with the subcritical water flow rate. Therefore, if the KD model applies to a certain extraction, the shape of an extraction curve would be defined by:

$$\frac{\mathcal{M}\_b}{\mathcal{M}\_i} = \frac{\begin{pmatrix} 1 & \frac{M\_a}{s} \end{pmatrix}}{\begin{pmatrix} 1 & \frac{K\_D m}{\|V\_b \cdot V\_a\|\_p} \star 1 \end{pmatrix}} + \begin{array}{c} \mathcal{M}\_x \\ \hline \mathcal{M}\_i \end{pmatrix} \tag{2}$$

Ma: cumulative mass of the analyte extracted after certain amount of volume Va (mg/g dry sample)

Mb: cumulative mass of the analyte extracted after certain amount of volume Vb (mg/g dry sample)

Mi : total initial mass of analyte in the matrix (mg/g dry sample)

Mb/Mi and Ma/Mi : cumulative fraction of the analyte extracted by the fluid of the volume Vb and Va (ml)

KD: distribution coefficient; concentration in matrix/concentration in fluid

ρ: density of extraction fluid at given condition (mg/ml)

e: exponential function

m: mass of the extracted sample (mg dry sample).

The model eq. (1) and the experimental data for *Z. multiflora* from all volumetric flow rate, were used to determine the KD value by minimizing the errors between the measured data and the KD model using Matlab curve fitting solver. The values of KD are shown in Table 2 for different flow rates [11]. It was demonstrated that individual essential oil compounds have a range of KD values from ~4 to ~250 [10].

**Table 2.** KD values of partitioning coefficient model for different volumetric flow rates [11].

#### **5.2. Mass transfer models**

#### *5.2.1. Diffusion model*

*KD* = *Concentration of solute in the matrix*

466 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

extraction, the shape of an extraction curve would be defined by:

*Mb Mi* =

: total initial mass of analyte in the matrix (mg/g dry sample)

ρ: density of extraction fluid at given condition (mg/ml)

m: mass of the extracted sample (mg dry sample).

range of KD values from ~4 to ~250 [10].

KD: distribution coefficient; concentration in matrix/concentration in fluid

**Table 2.** KD values of partitioning coefficient model for different volumetric flow rates [11].

sample)

sample)

Mi

Mb/Mi

and Va (ml)

and Ma/Mi

e: exponential function

(1- *M a M i* )

( *K Dm* (*V b* - *V a*)*ρ* + 1) +

*Concentration of solute in the extraction fluid* ; *at equilibrium* (1)

*Ma Mi*

: cumulative fraction of the analyte extracted by the fluid of the volume Vb

(2)

Hence the Extraction with subcritical water can be fitted using this simple thermodynamic model. The mass of analyte in each unit mass of extraction fluid and the mass of analyte remaining in the matrix at that period in the entire extraction time is based on the *K*D value determined for each compound. The thermodynamic elution of analytes from matrix was the prevailing mechanism in SWE as evidenced by the fact that extraction rate increased propor‐ tionally with the subcritical water flow rate. Therefore, if the KD model applies to a certain

Ma: cumulative mass of the analyte extracted after certain amount of volume Va (mg/g dry

Mb: cumulative mass of the analyte extracted after certain amount of volume Vb (mg/g dry

The model eq. (1) and the experimental data for *Z. multiflora* from all volumetric flow rate, were used to determine the KD value by minimizing the errors between the measured data and the KD model using Matlab curve fitting solver. The values of KD are shown in Table 2 for different flow rates [11]. It was demonstrated that individual essential oil compounds have a Mass transfer can be defined as the migration of a substance through a mixture under the influence of a concentration gradient in order to reach chemical equilibrium. The diffusion coefficient (De) is the main parameter in Fick's law, and application of this mathematical model to solid foods during solid-liquid extraction is a common way to calculate the effective diffusion coefficient (Crank, 1975 [14]). However, Gekas (1992) noted, values of De can vary by several orders of magnitude for the same material which may be due to structural changes in the food material during different stages of the process [15]. Therefore, it is important to keep a constant particle size as breakage of cell wall or grinding can reduce the particle size and hence decrease the distance for solute to travel from inside to surface of particle.

Fick derived a general conservation equation for one-dimensional non-steady state diffusion when the concentration within the diffusion volume changes with respect to time, known as Fick's second law (Cussler, 1984; Mantell et al., 2002) [16, 17]:

$$\begin{aligned} \frac{\partial \mathbf{C}}{\partial t} &= \mathbf{D}\_{\epsilon} \frac{\partial^2 \mathbf{C}}{\partial r^2} \\ \text{With the initial condition:} \\ \mathbf{C}\_{(t=0)} &= \mathbf{C}\_{o} \\ \text{And boundary conditions:} \\ \frac{\partial \mathbf{C}}{\partial r}\_{(r=0)} &= \mathbf{0} \\ \mathbf{C}\_{(r=R)} &= \mathbf{0} \\ \end{aligned} \tag{3}$$

Where C is the solute concentration (mg/ml) at any location in the particle at time t (s); Co is the initial solute concentration (mg/ml); De is the effective diffusion coefficient (m2 /s) assuming that De is constant with the concentration; t is extraction time (s); r is the radial distance from the centre of a spherical particle (m); R is radius of spherical particle (m).

Various solutions of Fick's second law have been presented for the diffusion of a compound during solid-liquid extraction depending on the shape of the particle. An approximate numerical solution to Fick's second law (eq. 2) for a spherical particle was given by Crank (1975) and Cussler (1984):

$$\frac{M\_t}{M\_{\text{un}}} = \mathbf{1} - \frac{6}{\pi^2} \sum\_{n=1}^{\infty} \frac{1}{n^2} \exp\left[\frac{-D\_e n^2 \pi^2 t}{R^2}\right] \tag{4}$$

Where Mt : total amount of solute (mg/g) removed from particle after time t, M∞: maximum amount (mg/g) of solute extracted after infinite time. Mt /M∞: ratio of total migration to the maximum migration concentration, R: average radius of an extractable particle. When time becomes large, the limiting form of Eq. 3 becomes:

$$1 \quad \frac{M\_t}{M\_u} = \frac{6}{\pi^2} \exp\left[\frac{-D\_r \cdot \pi^2 t}{R^2}\right] \tag{5}$$

To determine the effective diffusion coefficient values two methods were used. The first method was a linear (graphical) solution in which De was determined from the slope of the ln (1-Mt /M∞) vs. time plot (Dibert et al., 1989) [18]. Thus, eq. 4 can be solved by taking the natural logarithm of both sides. It shows that the time to reach a given solute content will be directly proportional to the square of the particle radius and inversely proportional to De:

$$\ln\left[\mathbf{1} - \frac{M\_t}{M\_{\text{un}}}\right] = \ln\frac{6}{\pi^2} - \frac{\cdot D\_e \cdot \pi^2 \, t}{R^2} \tag{6}$$

Where slope = *π* 2 *De R* 2 .

The second method of solution used involved nonlinear regression with effective diffusivity (De) as a fitting parameter. In this method, the effective diffusivity De was estimated from eq. 4 using a Microsoft Excel Solver program. The program minimizes the mean square of deviations between the experimental and predicted ln(1-Mt /M∞) values (Tutuncu and Labuza, 1996) [19]. The first 10 terms of the series solution are taken into consideration by the program as the solution to the series becomes stable after 10 terms (n=10).

Most researchers in this area have adopted diffusion models based on solutions to Fick's second law for various defined geometrical shapes of the solid given in Crank method, usually based on infinite or semi-infinite geometries of a slab (plane sheet), a cylinder or a sphere. For example, a slab can be used to describe an apple slice or a sheet of herring muscle; a sphere for the description of coffee beans or particles of cheese curd, and a cylinder for the description of cucumber pickles. There is considerable variation in the solutions adopted by various researchers. The starting point for modeling a particular diffusion process is to consider the shape of the solid and the nature of the process itself: uptake of solute into the food, leaching of solute from the food or diffusion of solute through the food, and the experimental conditions in terms of initial and equilibrium solute concentrations.

The solution models usually consider a uniform initial solute concentration throughout the food, no resistance to mass transfer in the diffusion medium and no chemical reaction; but vary for a particular geometry depending on the solute concentrations at the surface of the solid, the volume of the solution (and therefore the relative change in solute concentration in the external solvent), and the time period of the experiment. A representative selection of solutions is given in Table 3. For example, Bressan et al. investigated solute diffusional loss from coffee beans and cottage cheese curd, respectively [20]; they chose solution models to Fick's second law from Crank which assumed the geometry of the solids approximate to that of infinite spheres. However, the former researchers considered a system in which the solute concentration in the external solvent remains effectively constant and zero throughout the extraction. This can be the case for large infinite solvent volumes and small solute solid concentrations. The latter group of researchers considered a process where the concentration of the solute in the surrounding medium changes significantly. This can be the case for small solvent volumes and high solid phase solute concentrations in the case of leaching, and high external solvent concentrations in the case of solute uptake processes; and consequently adopted a different model.

#### *5.2.2. One-site kinetic desorption model*

maximum migration concentration, R: average radius of an extractable particle. When time

*π* 2 exp - *De π* 2 *t*

To determine the effective diffusion coefficient values two methods were used. The first method was a linear (graphical) solution in which De was determined from the slope of the ln

proportional to the square of the particle radius and inversely proportional to De:

*Mt M*∞ =ln 6

/M∞) vs. time plot (Dibert et al., 1989) [18]. Thus, eq. 4 can be solved by taking the natural logarithm of both sides. It shows that the time to reach a given solute content will be directly

*π* 2 - - *De π* 2 *t*

The second method of solution used involved nonlinear regression with effective diffusivity (De) as a fitting parameter. In this method, the effective diffusivity De was estimated from eq. 4 using a Microsoft Excel Solver program. The program minimizes the mean square of

1996) [19]. The first 10 terms of the series solution are taken into consideration by the program

Most researchers in this area have adopted diffusion models based on solutions to Fick's second law for various defined geometrical shapes of the solid given in Crank method, usually based on infinite or semi-infinite geometries of a slab (plane sheet), a cylinder or a sphere. For example, a slab can be used to describe an apple slice or a sheet of herring muscle; a sphere for the description of coffee beans or particles of cheese curd, and a cylinder for the description of cucumber pickles. There is considerable variation in the solutions adopted by various researchers. The starting point for modeling a particular diffusion process is to consider the shape of the solid and the nature of the process itself: uptake of solute into the food, leaching of solute from the food or diffusion of solute through the food, and the experimental conditions

The solution models usually consider a uniform initial solute concentration throughout the food, no resistance to mass transfer in the diffusion medium and no chemical reaction; but vary for a particular geometry depending on the solute concentrations at the surface of the solid, the volume of the solution (and therefore the relative change in solute concentration in the external solvent), and the time period of the experiment. A representative selection of solutions is given in Table 3. For example, Bressan et al. investigated solute diffusional loss from coffee beans and cottage cheese curd, respectively [20]; they chose solution models to Fick's second law from Crank which assumed the geometry of the solids approximate to that of infinite spheres. However, the former researchers considered a system in which the solute

*R* 2 (5)

*R* 2 (6)

/M∞) values (Tutuncu and Labuza,

becomes large, the limiting form of Eq. 3 becomes:

(1-Mt

Where slope = *π* 2 *De*

*R* 2 .

1- *Mt M*∞ =6

468 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

*ln* 1-

deviations between the experimental and predicted ln(1-Mt

in terms of initial and equilibrium solute concentrations.

as the solution to the series becomes stable after 10 terms (n=10).

One-site kinetic desorption model describes the extractions that are controlled by intraparticle diffusion. This occurs when the flow of fluid is fast enough for the concentration of a particular solute to be well below its thermodynamically controlled limit. The one-site kinetic model was derived based on the mass transfer model that is analogous to the hot ball heat transfer model [29, 30]. The assumptions are that the compound is initially uniformly distributed within the matrix and that, as soon as extraction begins, the concentration of compound at the matrix surfaces is zero (corresponding to no solubility limitation). For a spherical matrix of uniform size, the solution for the ratio of the mass, Mr, of the compound that remains in the matrix sphere after extraction time, t, to that of the initial mass of extractable compound, Mi is given as:

$$\frac{M\_r}{M\_i} = \frac{6}{\pi^2} \sum\_{n=1}^{\infty} \frac{1}{n^2} \exp(-D\_\epsilon n^2 \pi^2 t / r^2) \tag{7}$$

In which n is an integer and De is the effective diffusion coefficient of the compound in the material of the sphere (m2 /s).

**Table 3.** Diffusion phenomena.

**Diffusion process**Diffusion

Analytes diffusion though carrot cylinders [23]

Analytes diffusion though potato tissue [24]

Analytes and pectic substances diffusion though apple tissue [25]

Analytes diffusion though apple tissue [26]

Salt and acetic acid diffusion though herring

Analytes diffusion though potato tissue [28]

[27]

equation

*E* = *C* - *CL Ci* - *CL* =

1 - 4 π 1 2 ( *Det a* 2 ) 1/2 - *Det a* 2 - 1 3π 1 2 ( *Det a* 2 ) 3/2 :

*Mt Mi* = 8 π 2 ∑ *n*=0 ∞ 1

membrane

*E* = *C* - *CL Ci* - *CL* = *Mt Mi* =

8 π 2 ∑ *n*=0 ∞ 1

*E* = *Mt Mi* = 512

*Mt M*∞ =1- ∑ *n*=0 ∞ 8

*Mt*

1 - ∑ *n*=1 ∞ 2α(1 + α)

three planes.

infinite solution.

*M*∞ = *V L CLi* - *V L CL V L CLi* - *V L C*∞ =

1 + α + α 2*n* 2 *exp* -*Den* 2*t*

attainment of equilibrium

Solution of Fick's 2nd law for solute loss or uptake for sphere geometry but for finite volume of solvent with

470 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

Solution of Fick's 2nd law for solute loss or uptake for an infinite cylinder for short time periods. Authors then

> (2*n* + 1)2 *a* 2 π 2 *t* :

> > ( *Det a* 2 ) π 2 2 :

Solution of Fick's 2nd law for diffusion from an infinite

4(*a* 2 + *b* 2 + *c* 2) *t* : Equation model adopted considers diffusion in a slab in

> (2*n* + 1)2π 2*t* 4*a* 2 :

Solution of Fick's 2nd law for diffusion from/or uptake by

*a* 2 :

Solution of Fick's 2nd law similar to flow through a

modify values to apply to finite cylinder

(2*n* + 1)2 *exp* -*De*

(2*n* + 1)2 *exp* -(2*n* + 1)2

slab developed by Newman [59]

π 6 *exp* - π 2*De*

(2*n* + 1)2π 2 *exp* -*De*

Experimental measurements

withdrawn at intervals

Carrot samples withdrawn for analysis at intervals

Samples of solvent surrounding slices withdrawn and assayed at intervals, material balance used to calculate *Mi*

and *Mt*

Apple slices withdrawn for analysis at intervals

Apple slices withdrawn for analysis at intervals

Fish withdrawn for analysis at intervals

Initial solute content of potato strip and solute contents of solvent **Calculation Diffusivity**

Dimensionless time values calculated for each data point using equation, linear regression of Fourier relationship

yields *De*

*ln Mi*

A series of dimensionless time values found for data using Newman's tables. *De* calculated by linear regression of Fourier relationship

Modification of equation model by natural logarithms enables regression of *lnE* against *t* give slope with *De* term

Experimental data fitted to equation model by successive approximations

Minimizing the residuals between experimental data and theoretical

Equation model modified by omitting terms n"/>0 and natural logarithms, by nonlinear regression of

*/Mt* against *t*

The curve for the above solution tends to become linear at longer times (generally after t > 0.5 tc), and ln (M r/Mi ) is given approximately by:

$$\ln(M\_r/M\_i) = -0.4977 - t/t\_0 \tag{8}$$

Where tc (min) is a characteristic time quantity, defined as:

$$\mathbf{r}\_c = r^2 \;/\,\pi^2 \mathbf{D}\_e \tag{9}$$

An alternative form of eq. 7, or so called a one-site kinetic desorption model, can be written for the ratio of mass of analyte removed after time *t* to the initial mass, Mi , as given by:

$$\frac{M\_t}{M\_i} = 1 - e^{-kt} \tag{10}$$

In which Mt is the mass of the analyte removed by the extraction fluid after time t (mg/g dry sample), Mi is the total initial mass of analyte in the matrix (mg/g dry sample), Mt /Mi is the fraction of the solute extracted after time t, and k is a first order rate constant describing the extraction (min-1).

Matlab curve fitting solver was used to determine the desorption rate constant, k, from the data for all flow rates. The values for *Z. multiflora* SWE are show in Table 4 [11]. As mentioned, the kinetic desorption model does not include a factor describing extraction flow rate, k should be the same value for all flow rates if the model is said to fit the experimental data. However, this was not the case (Table 4, the average error 3%-17%). The kinetic desorption rate increased for the volumetric flow rate of 1 to 4 ml/min. This indicated that the kinetic desorption model may not be suitable for describing the data at different flow rates of *Z. multiflora*.

#### *5.2.3. Two-site kinetic desorption model*

Two-site kinetic model is a simple modification of the one-site kinetic desorption model that describes extraction which occurs from the "fast" and "slow" part [10]. In such case, a certain

**Table 4.** Values of k for one-site kinetic desorption model for different volumetric flow rates [7].

fraction (F) of the analyte desorbs at a fast rate defined by k1, and the remaining fraction (1-F) desorbs at a slower rate defined by k2. The model has the following form:

$$\frac{M\_t}{M\_i} = \mathbf{1} - \left[ F e^{-k\_1 t} \right] - \left[ (\mathbf{1} - F) e^{-k\_2 t} \right] \tag{11}$$

The two site kinetic model does not include solvent volume, but relies solely on extraction time. Therefore, doubling the extractant flow rate should have little effect on the extraction efficiency when plotted as a function of time. On the contrary, the thermodynamic model is only dependent on the volume of extractant used. Therefore, the extraction rate can be varied by changing the flow rate. Hence, the mechanism of thermodynamic elution and diffusion kinetics can be compared simply by changing the flow rate in SWE. If the concentration of bioactive compounds in the extract increases proportionally with an increase in flow rate at given extraction time when the solute concentration is plotted versus extraction time, the extraction mechanism can be explained by the thermodynamic model. However, if an increase in flow rate has no significant effect on the extraction of the bioactive compounds, with the other extraction parameters being kept constant, the extraction mechanism can be modeled by the two site kinetic model [10, 31]. The mechanism of control and hence the model valid for SWE may be different depending on the raw material, the target analyte and extraction conditions.

For the two-site kinetic desorption model, the values of k1 and k2 were determined by fitting the experimental data with the two-site kinetic desorption models by minimizing the errors between the data and the model results. In the two-site model, the extraction rate should not be dependent on the flow rate. The k1 and k2 values for *Z. multiflora* SWE shown in Tables 5 and 6 demonstrated that the extraction rates were not completely independent of flow rate (the average error 11%-20%).

#### *5.2.4. Thermodynamic partition with external mass transfer resistance model*

This model describes extraction which is controlled by external mass transfer whose rate is described by resistance type model of the following form:

**Table 5.** k1 and k2 values of two-site kinetic desorption model for thymol at different flow rates [11].

fraction (F) of the analyte desorbs at a fast rate defined by k1, and the remaining fraction (1-F)

1 2 1 (1 ) *t k t k t*

The two site kinetic model does not include solvent volume, but relies solely on extraction time. Therefore, doubling the extractant flow rate should have little effect on the extraction efficiency when plotted as a function of time. On the contrary, the thermodynamic model is only dependent on the volume of extractant used. Therefore, the extraction rate can be varied by changing the flow rate. Hence, the mechanism of thermodynamic elution and diffusion kinetics can be compared simply by changing the flow rate in SWE. If the concentration of bioactive compounds in the extract increases proportionally with an increase in flow rate at given extraction time when the solute concentration is plotted versus extraction time, the extraction mechanism can be explained by the thermodynamic model. However, if an increase in flow rate has no significant effect on the extraction of the bioactive compounds, with the other extraction parameters being kept constant, the extraction mechanism can be modeled by the two site kinetic model [10, 31]. The mechanism of control and hence the model valid for SWE may be different depending on the raw material, the target analyte and extraction

For the two-site kinetic desorption model, the values of k1 and k2 were determined by fitting the experimental data with the two-site kinetic desorption models by minimizing the errors between the data and the model results. In the two-site model, the extraction rate should not be dependent on the flow rate. The k1 and k2 values for *Z. multiflora* SWE shown in Tables 5 and 6 demonstrated that the extraction rates were not completely independent of flow rate

This model describes extraction which is controlled by external mass transfer whose rate is

*5.2.4. Thermodynamic partition with external mass transfer resistance model*

described by resistance type model of the following form:

ë ûë û (11)

*M Fe F e*

desorbs at a slower rate defined by k2. The model has the following form:

**Table 4.** Values of k for one-site kinetic desorption model for different volumetric flow rates [7].

472 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

*i*

*M*

conditions.

(the average error 11%-20%).

**Table 6.** k1 and k2 values of two-site kinetic desorption model for carvacrol at different flow rates [11].

$$\frac{\partial \mathbb{C}\_s}{\partial t} = -k\_e a\_p \left[ \left( \mathbb{C}\_s / K\_D \right) - \mathbb{C} \right] \tag{12}$$

in which C is the fluid phase concentration (mol/m3 ), Cs is the solid phase concentration (mol/ m3 ), ke is the external mass transfer coefficient (m/min) and ap is specific surface area of particles (m2 /m3 ) [32]. If the concentration of the solute in the bulk fluid is assumed small and the solute concentration in the liquid at the surface of solid matrix is described by partitioning equilibri‐ um, KD, the solution of eq. 11 for the solute concentration in the solid matrix, Cs, becomes:

$$\mathcal{C}\_s = \mathcal{C}\_0 - \exp(-k\_e a\_p t / K\_D) \tag{13}$$

eq. 11 can be rewritten as the ratio of the mass of diffusing solute leaving the sample to the initial mass of solute in the sample, Mt /Mi , as given by the following equation.

$$M\_t = 1 - M\_i \exp(-k\_e a\_p t / K\_D) \tag{14}$$

Because ap is difficult to be measured accurately, ap and ke are usually determined together as keap, which is called overall volumetric mass transfer coefficient. The factors that influence the value of keap include the water flow rate through the extractor and the size and shape of plant sample.

The values for the model parameters, KD and keap in eq. (9) determined by Matlab curve fitting solver from the experimental data obtained at 150°C are summarized for *Z. multiflora* SWE in Tables 7 and 8 for different mass flow rates (Q, mg min-1 ) [11]. Linear regresion of the plot between ln(keap) and lnQ gives the following correlation for keap and Q:

$$\text{for thymol:}\qquad k\_e a\_p = 6.5748 \text{ Q}^{0.2078}\_m \tag{15}$$

$$\text{for carvacrol}: \quad k\_e a\_p = 0.1605 \, Q\_m^{0.6017} \tag{16}$$

**Table 7.** Parameters KD and keap for external mass transfer model of SWE of thymol [11].

**Table 8.** Parameters KD and keap for external mass transfer model of SWE of carvacrol [11].

To quantitatively compare the extraction models, the mean percentage errors between the experimental data and the models were considered. Based on the result in fitting from experimental data, the KD model was generally suitable for the description of extraction over all the volumetric flow rates tested. On the other hand, one-site and two-site kinetic desorption models describe the extraction data reasonably at lower volumetric flow rates. Of all the models considered, however, the thermodynamic partition with external mass transfer model could best describe the experimental data.

#### *5.2.5. Two-phase model*

A mathematical model can be developed to predict optimal operating parameters for SWE in a packed-bed extractor. Three important steps consist of diffusion of solutes through particles, diffusion and convection of solutes through layer of stagnant fluid outside particles and elution of solutes by the flowing bulk of fluid are assumed. Unsteady state mass balance of the solute in solid and subcritical water phases led to two partial differential equations. The model can be solved numerically using a linear equilibrium relationship. The model parameters were mass transfer coefficient, axial dispersion coefficient, and intraparticle diffusivity. The last parameter was selected to be the model tuning parameter. The two other parameters were predicted applying existing experimental correlations.

#### *<IP_ADDRESS>. Model description*

Tables 7 and 8 for different mass flow rates (Q, mg min-1 ) [11]. Linear regresion of the plot

0.2078 (15)

0.6017 (16)

between ln(keap) and lnQ gives the following correlation for keap and Q:

474 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

**Table 7.** Parameters KD and keap for external mass transfer model of SWE of thymol [11].

**Table 8.** Parameters KD and keap for external mass transfer model of SWE of carvacrol [11].

could best describe the experimental data.

*5.2.5. Two-phase model*

To quantitatively compare the extraction models, the mean percentage errors between the experimental data and the models were considered. Based on the result in fitting from experimental data, the KD model was generally suitable for the description of extraction over all the volumetric flow rates tested. On the other hand, one-site and two-site kinetic desorption models describe the extraction data reasonably at lower volumetric flow rates. Of all the models considered, however, the thermodynamic partition with external mass transfer model

A mathematical model can be developed to predict optimal operating parameters for SWE in a packed-bed extractor. Three important steps consist of diffusion of solutes through particles, diffusion and convection of solutes through layer of stagnant fluid outside particles and elution of solutes by the flowing bulk of fluid are assumed. Unsteady state mass balance of the solute in solid and subcritical water phases led to two partial differential equations. The model can be solved numerically using a linear equilibrium relationship. The model parameters were mass transfer coefficient, axial dispersion coefficient, and intraparticle diffusivity. The last

for thymol: *keap* =6.5748 *Qm*

for carvacrol: *k eap* =0.1605 *Qm*

The more precise method is based on differential mass balances along the extraction bed. A two-phase model comprising solid and subcritical water phases can be used. Extraction vessel is considered to be a cylinder filled by mono-sized spherical solid particles. The overall scheme of system was like Fig. 7.

**Figure 7.** Characteristic dimensions and geometry of the packed bed SWE vessel.

The major assumptions used to describe the SWE process for deriving the essential oils extraction model were:

Under these assumptions, the differential mass balance equation for any component in the particle and bulk liquid phase and associated initial and boundary conditions can be written as following dimensionless forms:

**•** Solid phase:

$$\frac{\partial \mathcal{C}\_p}{\partial \tau} = \frac{2}{P e\_p} \left. \frac{L}{R} \left( \frac{\partial^2 \mathcal{C}\_p}{\partial Y^2} + \frac{2}{Y} \frac{\partial \mathcal{C}\_p}{\partial Y} \right) \right| \tag{17}$$

$$
\pi = 0 \qquad \qquad \mathsf{C}\_p = \mathsf{C}\_{p0} \tag{18}
$$

$$\text{If } Y = 0 \qquad \qquad \frac{\partial \mathcal{L}\_p}{\partial Y} = 0 \tag{19}$$

$$Y = 1 \quad \frac{\delta C\_r}{\delta Y} = Sh \text{ (\$\mathcal{C}\_f\$} - \mathcal{C}\_{fp}\$)\tag{20}$$

**•** Subcritical water phase

$$\frac{\partial \mathcal{C}\_{f}}{\partial \tau} = \frac{1}{\mathcal{P}e\_{b}} \frac{\partial^{2} \mathcal{C}\_{f}}{\partial X^{2}} - \frac{\partial \mathcal{C}\_{f}}{\partial X} - \frac{6(1 \cdot \varepsilon)L}{\varepsilon R} \frac{\partial l}{\partial e\_{p}} \left\{ \mathcal{C}\_{f} - \mathcal{C}\_{fp} \right\} \tag{21}$$

$$
\pi = 0 \qquad \mathcal{C}\_f = 0 \tag{22}
$$

$$X = 0 \qquad C\_f \quad \text{ --} \quad \frac{1}{\rho\_{Fe\_b}} \frac{\partial C\_f}{\partial X} = 0 \tag{23}$$

$$X = 1 \quad \xrightarrow[\partial \, X]{\partial \mathcal{C}\_f} = 0 \tag{24}$$

Eqs. (17) and (21) may be solved using a linear equilibrium relationship between concentra‐ tions in the solid phase and SW phase at the interface, as follows [33]:

$$\mathbf{C}\_{fp} = k\_p \mathbf{C}\_p \big|\_{Y=1} \tag{25}$$

Where Cfp is the solute concentration in the fluid phase at the particle surface, Cp|Y=1 is the solute concentration in the solid phase at equilibrium with the fluid phase and kp is the volumetric partition coefficient of the solute between the solid and the fluid phase. Therefore, there are three eqs. (17), (21) and (25) which can be solved, simultaneously, for three unknowns Cp, Cf and Cfp.

The finite difference equations are a set of simultaneous linear algebraic equations must be solved for implicit method to obtain the concentration distribution at any time. In both cases, tridiagonal systems arise which are conveniently solved at each time step by the Thomas algorithm [34]. A computer code was written using MATLAB simulation software.

#### *<IP_ADDRESS>. Parameter identification and correlations*

The possible control of mass transfer was assayed by estimating the diffusion coefficient in the liquid. We can use the correlation proposed by Wilke and Chang (1955) to estimate this coefficient [35]:

$$D\_{AB} = \frac{7.4^\* 10^{\*8}}{\mu \,\, V\_A \,\, ^{0.6}} (\text{\textperth } \text{M}\_2)^{0.5} \,\, T \tag{26}$$

Where φ is 2.26 for water and 1.5 for ethanol and VA was estimated by the Tyn Calus equation:

$$V\_A = 0.285 \text{ (}V\_c\text{)}^{1.048} \tag{27}$$

Where Vc is the estimated by the method of Joback and Reid (1987), with the aid of Molecular Modeling Plus software (Norgwyn Montgomery Software, USA) [36]. The axial dispersion coefficient in the supercritical phase was approximated as follows [37] and it may be used for the subcritical phase:

$$D\_L = \frac{\
u \, d\_p}{e \, Pe\_{pl}} \tag{28}$$

Where the average void volume fraction of the fixed bed was ε = 0.4 and:

*τ* =0 *Cp* =*Cp*0 (18)

∂*Y* =*Sh* (*C f* - *C fp*) (20)

*τ* =0 *C f* =0 (22)

∂*Y* =0(19)

*Pe p* (*C f* - *C fp*) (21)

∂ *X* =0 (23)

∂ *X* =0(24)

*C fp* =*k pCp*|*Y* =1 (25)

0.6 (*φ M*2)0.5 *T* (26)

*Y* =0∂*Cp*

∂ *X* – 6(1 - *ε*)*L εR*

*Pe b*

Eqs. (17) and (21) may be solved using a linear equilibrium relationship between concentra‐

Where Cfp is the solute concentration in the fluid phase at the particle surface, Cp|Y=1 is the solute concentration in the solid phase at equilibrium with the fluid phase and kp is the volumetric partition coefficient of the solute between the solid and the fluid phase. Therefore, there are three eqs. (17), (21) and (25) which can be solved, simultaneously, for three unknowns

The finite difference equations are a set of simultaneous linear algebraic equations must be solved for implicit method to obtain the concentration distribution at any time. In both cases, tridiagonal systems arise which are conveniently solved at each time step by the Thomas

The possible control of mass transfer was assayed by estimating the diffusion coefficient in the liquid. We can use the correlation proposed by Wilke and Chang (1955) to estimate this

algorithm [34]. A computer code was written using MATLAB simulation software.

*DAB* = 7.4\*10-8 *μ V A*

∂*C f*

*Bi*

*Y* =1∂*Cp*

476 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

*X* =0 *C f* - 1

*X* =1∂*C f*

tions in the solid phase and SW phase at the interface, as follows [33]:

∂2*C f* ∂ *X* 2 - ∂*C f*

**•** Subcritical water phase

Cp, Cf

and Cfp.

coefficient [35]:

*<IP_ADDRESS>. Parameter identification and correlations*

∂*C f* ∂ *τ* =1 *Pe b*

$$Pe\_{pd} = 1.634 \text{ } Re^{0.265} \text{ Sc}^{-0.919} \text{ } \tag{29}$$

Intraparticle diffusivity of the essential oils in the solid phase, Dm, was selected to be the tuning parameter of the model. Mass transfer between liquids and beds of spheres, kf , can be repre‐ sented by Wilson and Geankoplis in two cases [38]:

$$\{\varepsilon \: j\_D = 0.0016 \le Re \le 55, \quad 165 \le Sc \le 70600\}\tag{30}$$

$$\text{Let } j\_D = \frac{0.25}{Re^{0.35}} \text{S5} \le Re \le 1500, \quad 165 \le \text{Sc} \le 10690 \tag{31}$$

*k f* = *J D Uo Sc*2/3 (32)

Density of water at high pressures and temperatures from 273 to 473 K was assumed to be the density of saturated water (kg). It was calculated as follows [39]:

$$
\rho = 858.03 + 1.2128 \text{ T - } 0.0025 \text{ T}^2 \tag{33}
$$

Where ρ is density (kg/m3 ) and T is temperature (K). The water viscosity at temperatures from 300 to 450 K was calculated by the following equation:

$$\mu = \exp\left(-10.2 + \frac{280970}{T^{\frac{2}{2}}}\right) \tag{34}$$

Where μ is the viscosity (Pa.s) and T is temperature (K). Viscosity of water was supposed to be independent of pressure. The model was verified successfully using the SWE data for *Z.multiflora* leaves at 20 bar and 150o C. The optimum value of 2 × 10-12 m2 /s was obtained for the intraparticle diffusivity [12].

#### *5.2.6. CFD model*

There is no scientific literature about the application of CFD modeling approach in the SWE processes. In our previous work, we have tried to do CFD modeling of essential oils of *Z. multiflora* leaves [13].

In CFD modeling in packed bed reactor, there are two cases. In first one, when the reactor to particle diameter ratio is less than10, it is needed to build the exact geometry and the location of the particles and solving the governing equations in meshes and can see the field of flow between particles inside reactor. But in another one, when the reactor to particle diameter ratio is higher than10, this system is defined as porous media. For modeling of these reactors, the Navier-stocks with one additional term, which is contained viscous force and inertia loss, in porous media is solved.

In this work, because of the reactor to particle diameter ratio is higher than10; the system can define as porous media. The momentum equation is defined as:

$$\frac{\partial}{\partial t} \left( \varepsilon \rho \vec{\nu} \vec{\nu} \right) + \nabla \cdot \left( \varepsilon \rho \vec{V} \vec{V} \right) = -\varepsilon \nabla P + \nabla \cdot \left( \varepsilon \tau \right) = \left( -\frac{\mu}{\beta} \vec{V} + \frac{1}{2} \mathsf{C} \rho \left| \vec{V} \right| \vec{V} \right) \tag{35}$$

Where ∂ ∂*t* (*ερυ*→ ) + ∇.(*ερV* →*V* →)= −*ε*∇*P* + ∇.(*ετ*)=( − *μ β V* → + 1 2 *Cρ* |*V* → |*V* →) and C and β coefficient can define as:

$$\beta = \frac{\int\_{P}^{2} \mathcal{E}}{150 \left(1 - \varepsilon\right)^{2}}\tag{36}$$

$$C = \frac{3.5(1 - \varepsilon)}{d\_P \mathfrak{E}} \tag{37}$$

In this work the porosity of bed is 0.4, so we have following value for C and β coefficients.

$$\mathbf{C} \text{=} 54687.5 \quad 1/\text{m} \tag{38}$$

$$\text{gg} \ 0.23^\*10 \quad ^{10}\text{l/m} \tag{39}$$

The energy equation is defined as:

$$\frac{\partial}{\partial t}\Big(\varepsilon \,\rho\_f E\_f + \left(1 - \varepsilon\right) \rho\_s E\_s\Big) + \nabla \cdot \left(\vec{V} \left(\rho\_f E\_f + P\right)\right) = \nabla \cdot \left[\hat{K} \,\nabla T + \left(\tau, \vec{V}\right)\right] + S \tag{40}$$

Where the Effective thermal diffusion coefficient is defined as:

$$
\hat{\mathbf{K}} = \varepsilon \,\hat{\mathbf{K}}\_f + \left(1 - \varepsilon\right) \hat{\mathbf{K}}\_s \tag{41}
$$

The spices transfer equation is defined as:

$$\frac{\partial \mathbf{\hat{c}}\_{i}}{\partial t} + \frac{\partial}{\partial \mathbf{x}} \left( \mathbf{-D\_{x}} \frac{\partial \mathbf{c}\_{i}}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{x}} \left( \mathbf{-D\_{x}} \frac{\partial \mathbf{c}\_{i}}{\partial \mathbf{x}} \right) = \frac{D\_{y}}{y} \frac{\partial \mathbf{c}\_{i}}{\partial y} - \frac{u}{\varepsilon} \frac{\partial \mathbf{c}\_{i}}{\partial \mathbf{x}} - \frac{u}{\varepsilon} \frac{\partial \mathbf{c}\_{i}}{\partial y} + \mathbf{S} \tag{42}$$

S is source term. Comparing the default of spices transfer equation with spices transfer equation of this system can define the ∂*ci* ∂*t* + ∂ ∂x (-Dx ∂*ci* ∂ *x* ) + ∂ ∂x (-Dx ∂*ci* ∂ *x* ) = *Dy y* ∂*ci* ∂ *y* − *u ε* ∂*ci* ∂ *x* − *u ε* ∂*ci* ∂ *y* + *S* as the source term of this work.

Cfs was the concentration of Thymol & Carvacrol in the fluid face at the particle surface and was defined as:

$$
\mathbf{c}\_{fs} = \mathbf{k}\_p \mathbf{c}\_{ss} \tag{43}
$$

and

*5.2.6. CFD model*

*multiflora* leaves [13].

porous media is solved.

Where ∂

can define as:

*t*

∂*t* (*ερυ*→ ) + ∇.(*ερV*

The energy equation is defined as:

 e r

*t* er eru

There is no scientific literature about the application of CFD modeling approach in the SWE processes. In our previous work, we have tried to do CFD modeling of essential oils of *Z.*

In CFD modeling in packed bed reactor, there are two cases. In first one, when the reactor to particle diameter ratio is less than10, it is needed to build the exact geometry and the location of the particles and solving the governing equations in meshes and can see the field of flow between particles inside reactor. But in another one, when the reactor to particle diameter ratio is higher than10, this system is defined as porous media. For modeling of these reactors, the Navier-stocks with one additional term, which is contained viscous force and inertia loss, in

In this work, because of the reactor to particle diameter ratio is higher than10; the system can

 et

( )

( ) 3

e

3.5 1

*d*

*P*

e

In this work the porosity of bed is 0.4, so we have following value for C and β coefficients.

( ( ) )( ) ( ) ( ) ˆ 1 . .. *f f s s f f E E V E P KT V S*

r

¶ + - +Ñ + =Ñ Ñ + + é ù ë û ¶

e

2 150 1 *dP*

2 3

e=

*VV P V C VV*

*β V* → + 1 2 *Cρ* |*V*

m

r r r rr r (35)

b

r

→ |*V*

C=54687.5 1/m (38)

10 2 1/β= 0.23\*10 1/m (39)

t

r r (40)

→) and C and β coefficient

(36)

define as porous media. The momentum equation is defined as:

478 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

 er

→*V*

( ) ( ) ( ) 1 . . 2

→)= −*ε*∇*P* + ∇.(*ετ*)=( − *μ*

b

*C*

¶ æ ö +Ñ =- Ñ +Ñ = - + ç ÷ ¶ è ø

 e

$$\mathcal{L}\_{\rm ss} = 0.8 \,\mathrm{c}\_0 \exp\left\{-0.0005 \,t\right\} \tag{44}$$

In such works, the Reynolds number is defined as:

$$\text{Re}\_d = \frac{\rho \bar{\mathcal{U}}\_p d\_p}{\mu} \tag{45}$$

Re*d* = *ρU*¯ *pd* ¯ *p μ* is mean characteristic length of porous and *d* ¯ *p* is mean velocity based on po‐ rous. Based on this Reynolds number, it can be seen four regimes:

Red<1 Darcy regime or creeping flow

1<Red<150 inertia regime

150<Red<300 unsteady laminar flow regime

Red>300 unsteady Turbulent flow regime

The superficial velocity in this packed bed is defined as dividing flow rate by cross section.

Dividing superficial velocity by porosity, real velocity can calculate.

$$
\mathcal{U} = \frac{\mathcal{Q}}{A} \tag{46}
$$

$$
\overline{\mathcal{U}}\_p = \frac{\mathcal{U}}{\mathcal{E}} \tag{47}
$$

and for calculating Dp

$$
\overline{D}\_p = \frac{\mathfrak{G}}{A\_0} \tag{48}
$$

In this work the mean particle diameter was 0.6 mm.

$$A\_0 = \frac{A}{V} = \frac{4\pi r^2}{\frac{4}{3}\pi r^3} \tag{49}$$

In this work the Reynolds number is 0.2, so there is laminar flow in this system.

The material of system was mixture of Thymol & Carvacrol and water. The inlet flow was contained water with mole fraction 1 and the outer flow was consisted of water and Thymol & Carvacrol [13]. The packed bed had 103 mm height and the diameter of bed was 16 mm. The uniform mesh was used for this domain.

The governing equations are solved by a finite volume method. At main grid points placed in the center of the control volume, volume fraction, density and spices fraction are stored. The conservation equations are integrated in space and time. This integration is performed using first order upwind differencing in space and fully implicit in time. For a first-order upwind solution, the value at the center of a cell is assumed to be an average throughout the cell. The SIMPLE algorithm is used to relate the velocity and pressure equations.

The simulation geometry is shown in Fig.8. The 2D calculation domain is divided into 22\*146 grid nodes, in the radial and axial directions, respectively. The grid and mesh size are chosen to be uniform in the two directions. The inlet of system was pure water and the outlet of system was extracted Thymol & Carvacrol.

The extraction values versus time for 150o C were shown in Fig. 9. It can be seen, the extraction values were increased as exponentially. After 60 min, the change of extraction values with time was very little.

**Figure 8.** The Schematic of extractor [13].

Dividing superficial velocity by porosity, real velocity can calculate.

480 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

In this work the mean particle diameter was 0.6 mm.

uniform mesh was used for this domain.

was extracted Thymol & Carvacrol.

was very little.

The extraction values versus time for 150o

and for calculating Dp

*Q U*

*p U U* e

0

SIMPLE algorithm is used to relate the velocity and pressure equations.

In this work the Reynolds number is 0.2, so there is laminar flow in this system.

0

4 4 3

*A r A V*

2

3

*r* p

p

The material of system was mixture of Thymol & Carvacrol and water. The inlet flow was contained water with mole fraction 1 and the outer flow was consisted of water and Thymol & Carvacrol [13]. The packed bed had 103 mm height and the diameter of bed was 16 mm. The

The governing equations are solved by a finite volume method. At main grid points placed in the center of the control volume, volume fraction, density and spices fraction are stored. The conservation equations are integrated in space and time. This integration is performed using first order upwind differencing in space and fully implicit in time. For a first-order upwind solution, the value at the center of a cell is assumed to be an average throughout the cell. The

The simulation geometry is shown in Fig.8. The 2D calculation domain is divided into 22\*146 grid nodes, in the radial and axial directions, respectively. The grid and mesh size are chosen to be uniform in the two directions. The inlet of system was pure water and the outlet of system

values were increased as exponentially. After 60 min, the change of extraction values with time

*A* = (46)

= (47)

6 *Dp A* = (48)

= = (49)

C were shown in Fig. 9. It can be seen, the extraction

**Figure 9.** Extraction values versus time for 150oC [13].

In order to investigate the applicability of the CFD model, the theoretical results are compared with experimental measurements obtained at optimum conditions (20 bars, 150˚C, and 2 ml/ min). Fig. 9 shows that the modeling extraction values profile is increasing rapidly in the period of 0-60 min and thereafter in the second region (60-120 min) the slope reduces until reaches a constant trend in the third period of 120-150 min. In the first region, because of high Thymol & Carvacrol concentrations in the *Z. multiflora* leaves and therefore, high mass transfer driving force, high desorption rate of Thymol & Carvacrol from solid matrix occurs.

#### **6. Conclusion**

It was tried to give overall view about subcritical water extraction. Effective parameters, mechanism and modeling of extraction were surveyed. Overall by considering mean average errors of models, a mathematical model base on the combination of partition coefficient (*K*D) and external mass transfer gave a good description of subcritical water extraction of *Z*. *multiflora*, while the kinetic model reasonably described the extraction behavior at lower flow rates [11].

On the other side, model was developed by introducing differential mass balances using two phase model, and applying a linear equilibrium relationship. Because of considering the effect of variation of the concentration profile in the SW phase, it seems that the proposed model is more significant from the physical point of view.

The Last model was CFD modeling of extraction from *Z. multi*fl*ora* leaves using subcritical water. It was concluded that CFD is poised to remain at the forefront of cutting edge research in the sciences of fluid dynamics and mass transfer. Also, the emergence of CFD as a practical tool in modern engineering practice is steadily attracting much interest and appeal. The results of CFD model have been agreed well with experimental data. As shown, along of extractor, Thymol was extracted and was in outflow.

## **Notations**

## **Greek symbols**

errors of models, a mathematical model base on the combination of partition coefficient (*K*D) and external mass transfer gave a good description of subcritical water extraction of *Z*. *multiflora*, while the kinetic model reasonably described the extraction behavior at lower flow

On the other side, model was developed by introducing differential mass balances using two phase model, and applying a linear equilibrium relationship. Because of considering the effect of variation of the concentration profile in the SW phase, it seems that the proposed model is

water. It was concluded that CFD is poised to remain at the forefront of cutting edge research in the sciences of fluid dynamics and mass transfer. Also, the emergence of CFD as a practical tool in modern engineering practice is steadily attracting much interest and appeal. The results of CFD model have been agreed well with experimental data. As shown, along of extractor,

*Rp* ) a or ap

*Dm* ) Bi

solute concentration in the solid phase, kmol/m3 C solute concentration in the SW phase, kmol/m3 Cf solute concentration in the fluid phase at the particle surface, kmol/m3 Cfp

initial solute concentration in the solid phase, kmol/m3 Cpo diffusivity of solute (A) in liquid (B), m2/s DAB effective diffusion coefficient (m2/s) De axial dispersion coefficient, m2/s DL diffusivity in the solid, m2/s Dm particle diameter, m dp exponential function e a certain fraction of the analyte desorbs at a fast rate by k1 F remaining fraction desorbs at a slower rate by k2 (1-F1) thermodynamics partitioning coefficient *K*D external mass transfer coefficient (m/min) ke

fl

*C po* ) Cp

*ora* leaves using subcritical

rates [11].

**Notations**

specific surface, m2/m3 ( = 3

Biot number ( = *k f Rp*

more significant from the physical point of view.

Thymol was extracted and was in outflow.

solute concentration in the solid phase, kmol/m3 ( *C p*

The Last model was CFD modeling of extraction from *Z. multi*

482 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

τ dimensionless time ( = *u t L*ε )

#### **Acknowledgements**

This research was supported by Semnan University. The authors would like to thanks the Office of Brilliant Talents at the Semnan University for financial support.

#### **Author details**

A. Haghighi Asl and M. Khajenoori

School of Chemical Gas and Petroleum Engineering, Semnan University, Semnan, I.R., Iran

### **References**

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τ dimensionless time ( = *u t*

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90o

## **Mass Transfer: Impact of Intrinsic Kinetics on the Environment**

## Engr Owabor

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/55990

## **1. Introduction**

Emerging trends in environmental engineering and biosystem analysis indicate that a clean, safe and healthy ecosystem is still possible regardless of the increased quest for technolo‐ gy and industrialization by a dynamic society where the fuel requirements of the energy sector and the chemical and petrochemical needs of the chemical and allied industries are greatly dependent on the petroleum industry in which the downstream sector is a key player. These needs range from the feedstock requirements of the chemical industries, polymeric industrial sector, the solvent sector, the cosmetics industries etc. The concern over industrial and technological expansion, energy utilization, waste generation from domestic and industrial sources viz a viz their attendant negative effects on the environ‐ ment cannot be overemphasized.

This chapter is intended to expose the reader to the role of mass transfer on the topical issue of environmental remediation. This is against the backdrop of the interrelationship between mass transfer and the major processes such as biological, chemical and phyto-oxidation acting during the removal of toxic substances from the environment and the dependence of these processes on the availability of these substances which over time compromise the integrity of the environment. Research reports advanced in this chapter is aimed at highlighting the crucial need to very seriously exploit the principles of mass transfer in addressing the critical issue of environmental degradation. The underlying mechanisms involved in the removal process and the systematic protocol developed for accelerated clean up are incorporated.

Basically, mass transfer is one of the important three transport operations commonly encoun‐ tered and employed in chemical transformations. It has been described fundamentally, as the movement of the components of a mixture from one point to another as a result of observed

© 2013 Owabor; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Owabor; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

differences in concentration [1]. The material transfer is usually in the direction of decreasing concentration gradient.

The concept of mass transfer is very important in the fields of engineering (particularly Chemical Engineering) and sciences in general where it is applied for physical processes. From the standpoint of chemical engineering, mass transfer is of prime significance because most of the unit operations involving separation of mixtures into their component parts employ mass transfer operation. The relevance of mass transfer operations is continuously a subject of discourse as specifically, there is hardly any chemical process which does not require a preliminary purification of raw materials or final separation of desired products from byproducts. In essence it can be described as the process that determines the rate at which separation will occur. Issues relating the removal of certain materials from fluid streams typically by adsorption are important industrial applications of the mass transfer operation.

The challenges of maintaining a clean environment stem primarily from the increasing pollution of waters and land by heavy metals that are difficult to decompose biologically and organic compounds which are persistent, recalcitrant and ubiquitous. This is because these substances resist the self-purification capabilities of the rivers as well as decomposition in conventional wastewater treatment plant. Consequently, conventional mechanical-biological purification no longer suffices. It thus becomes imperative to supplement an additional stage of processing and this improvement has been augmented by adsorption; an innovative treatment method based on the principle of diffusion (a mechanism of mass transfer). Litera‐ ture reports have shown that the mechanism of the process of material transfer is by diffusion and this is dependent to a large extent on the physical characteristics and attributes of the mass been considered for transfer. The transfer process which is dependent on concentration gradient is basically of two types; molecular diffusion and convective transfer. While the former is known to be involved with movement of a material between two points i.e from a high to low area of concentration, the latter relates to the flow of liquids around a solid surface in forced convection motion. Interestingly, the two transfer types are governed by Fick's first law of diffusion under the condition of unchanging concentration gradient with the passage of time. However, Fick's second law becomes applicable under unsteady-state diffusion. The principles of Fick's laws which suggest that the rate of diffusion is a linear function of the solute concentration gradient has continuously been affirmed by research results involving material transport. In general, mass transfer cannot really be separated from adsorption and desorption processes as the transport of materials from one phase to another is principally by the mech‐ anism of diffusion. Sorption is the primary process in the evaluation of availability while mass transfer is the mechanism of movement from the fluid phase to the surface of the soil particle. Various isotherm correlations have been used to describe the surface adsorption of the solutes and their subsequent desorption from the soil surface [1]. The sorption of contaminant tends to separate the direct contact between microorganisms, adsorbents and contaminants, which is necessary for biodegradation detoxification to occur. The practical effect of the adsorption and desorption rate, is that it controls the overall reaction rate. The transport (mobility) of contaminant solute therefore is significantly dependent on two possible scenarios: fast sorption/desorption and slow sorption/ desorption.

Development of surface modified activated carbon from low cost and readily available natural materials and agricultural wastes have been advanced for the generation of activated carbon with far superior adsorption capacity for the treatment of contaminated water. The removal of pollutants from aqueous waste stream by adsorption in fixed beds has therefore progres‐ sively become an important treatment process and a subject of many excellent research works.

Laboratory and field studies results abound to demonstrate the applications of mass transfer in the management of the environment in terms of pollution monitoring and control. The intrinsic kinetics of the mass transfer process has through systematic and sustained researches become a relevant area of interest which provides information on availability, mobility and toxicity as a function of the measured concentration and the mechanism of sequestration of the solutes in the containing system.

## **2. Applications of mass transfer in water treatment technology**

differences in concentration [1]. The material transfer is usually in the direction of decreasing

490 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

The concept of mass transfer is very important in the fields of engineering (particularly Chemical Engineering) and sciences in general where it is applied for physical processes. From the standpoint of chemical engineering, mass transfer is of prime significance because most of the unit operations involving separation of mixtures into their component parts employ mass transfer operation. The relevance of mass transfer operations is continuously a subject of discourse as specifically, there is hardly any chemical process which does not require a preliminary purification of raw materials or final separation of desired products from byproducts. In essence it can be described as the process that determines the rate at which separation will occur. Issues relating the removal of certain materials from fluid streams typically by adsorption are important industrial applications of the mass transfer operation.

The challenges of maintaining a clean environment stem primarily from the increasing pollution of waters and land by heavy metals that are difficult to decompose biologically and organic compounds which are persistent, recalcitrant and ubiquitous. This is because these substances resist the self-purification capabilities of the rivers as well as decomposition in conventional wastewater treatment plant. Consequently, conventional mechanical-biological purification no longer suffices. It thus becomes imperative to supplement an additional stage of processing and this improvement has been augmented by adsorption; an innovative treatment method based on the principle of diffusion (a mechanism of mass transfer). Litera‐ ture reports have shown that the mechanism of the process of material transfer is by diffusion and this is dependent to a large extent on the physical characteristics and attributes of the mass been considered for transfer. The transfer process which is dependent on concentration gradient is basically of two types; molecular diffusion and convective transfer. While the former is known to be involved with movement of a material between two points i.e from a high to low area of concentration, the latter relates to the flow of liquids around a solid surface in forced convection motion. Interestingly, the two transfer types are governed by Fick's first law of diffusion under the condition of unchanging concentration gradient with the passage of time. However, Fick's second law becomes applicable under unsteady-state diffusion. The principles of Fick's laws which suggest that the rate of diffusion is a linear function of the solute concentration gradient has continuously been affirmed by research results involving material transport. In general, mass transfer cannot really be separated from adsorption and desorption processes as the transport of materials from one phase to another is principally by the mech‐ anism of diffusion. Sorption is the primary process in the evaluation of availability while mass transfer is the mechanism of movement from the fluid phase to the surface of the soil particle. Various isotherm correlations have been used to describe the surface adsorption of the solutes and their subsequent desorption from the soil surface [1]. The sorption of contaminant tends to separate the direct contact between microorganisms, adsorbents and contaminants, which is necessary for biodegradation detoxification to occur. The practical effect of the adsorption and desorption rate, is that it controls the overall reaction rate. The transport (mobility) of contaminant solute therefore is significantly dependent on two possible scenarios: fast

sorption/desorption and slow sorption/ desorption.

concentration gradient.

The demand for clean water for fishing activities, recreation etc and its non-ready availability has caused considerable attention to be focused towards recovery and re-use of waste waters. In water treatment procedures where the process of adsorption has been effectively engaged, the use of adsorbent materials has served as the solid surface onto which molecules of the adsorbates such as organic compounds, heavy metals, attach themselves. The removal of contaminant solutes from the environment has not been effectively accomplished by tradi‐ tional methods. Hence it has now become universally recognized that adsorption technology provides a feasible and effective method for the removal of pollutants from polluted water resources and waste waters. Activated carbons are the most commonly used adsorbent in the adsorption process due to their high adsorption capacity, high surface area and high degree of surface reactivity. Adsorbents have been sourced variously from refinery residue, coal but mostly from agricultural waste materials such as coconut shell, soya bean, cotton seed, walnut and rice hulls, melon husk, orange peels etc.

The effectiveness of low density adsorbents dried (ground ripe and unripe orange peels) in the removal organics typified using naphthalene and pyrene from an aqueous stream has been extensively studied [2]. The influence of the variation in process conditions such as concen‐ tration, adsorbent dosage, agitation time and particle size, pH, and initial solute concentration were analyzed to highlight the contributions of mass transfer in the treatment of contaminated water. Under the operating conditions specified above, laboratory and experimental results showed that the removal efficiency defined by adsorption capacity increased with an increase in adsorbent dosage, contact time and initial solute concentration but decreased with an increase in the particle size. This experimental result clearly demonstrates that the transfer of any organic compound under consideration from one point to another is a linear function of the operating variables which positively improve the reliability of the process.

Following the same adsorption principle for mass transfer, the removal of mono and polycyclic aromatic hydrocarbons (PAHs) from bulk fluid stream onto adsorbents from periwinkle and coconut shell (PSC) also provides an alternative technology for effective mass transfer process in thereductionofpollutantlevelofwastewater.Naphthalene,phenanthreneandanthracenewere used as the representative PAHs and the effluent stream was simulated refinery wastewater.

The result for the characterization of the refinery effluent before and after treatment with periwinkle shell activated carbon showed that the total suspended solids (TSS) reduced from an initial value of 53 mg/l to 2 mg/l after treatment.

Further insights into the role of adsorption on the contaminated water has revealed that the biochemical and chemical oxygen demand of the wastewater reduced drastically from an initial value to specifications recommended by the environmental protection agency (EPA) using activated carbon from materials such as periwinkle shell [3]. As contact time increased, percentage removal of the organic and inorganic contaminants present in the wastewater also increased. From the bench scale results [3], the total dissolved solids and total hydrocarbon content of the wastewater treated with activated periwinkle shell carbon reduced significantly. A maximum percentage removal of 78.5% was achieved. The clarity of the effluent water was observed to be greatly improved after the first hour of treatment where the turbidity dropped from 66 NTU to 6 NTU. Again the percentage turbidity increased with contact time until 100% removal was achieved following complete clarity.

The result showed comparatively, that the treated effluent water had maximum percentage reduction of approximately 83.26%, 91.19% (BOD, COD) for periwinkle shell carbon (PSC) and commercial activated carbon (CAC) respectively. The periwinkle shell carbon displayed the capacity to reduce the turbidity of the water contaminated with organic compounds via transfer of the contaminant solute from the bulk wastewater where their concentration is high to the external surface of the activated periwinkle shell carbon acting as the adsorbent.

In the development of alternative fixed filter bed for water treatment, the application of the mass transfer process by adsorption has also been shown to be feasible by examining the influence of modified clay on the rates of adsorption. The modifiers used were inorganic and organic acids, bases and salts. The results obtained showed that equilibrium adsorption of naphthalene i.e the transfer of naphthalene molecules from the bulk solution was attained at a faster rate using modifiedclaywhencomparedwiththeunmodifiedclay.Amongstthemodifiedclayusedforthe study, acids were found to be most suitable for the modification purpose [4].

This is attributable to the fact that the solutes in the acid medium had higher diffusive mobility defined by their dispersion coefficient/diffusivity as shown in Table 1 and higher influence on the physicochemical properties (porosity, surface area, bulk and particle density).

Overall, results from this study affirm that sediment modification directly affects the availa‐ bility of the contaminant chemicals.

## **3. Applications in soil remediation**

In soil remediation, the availability of the contaminant is of prime significance and it is commonly approached from the premise that chemicals are immediately accessible for

**Table 1.** Parameter estimation for naphthalene diffusivity in modified-clay sediment\*

thereductionofpollutantlevelofwastewater.Naphthalene,phenanthreneandanthracenewere used as the representative PAHs and the effluent stream was simulated refinery wastewater.

492 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

The result for the characterization of the refinery effluent before and after treatment with periwinkle shell activated carbon showed that the total suspended solids (TSS) reduced from

Further insights into the role of adsorption on the contaminated water has revealed that the biochemical and chemical oxygen demand of the wastewater reduced drastically from an initial value to specifications recommended by the environmental protection agency (EPA) using activated carbon from materials such as periwinkle shell [3]. As contact time increased, percentage removal of the organic and inorganic contaminants present in the wastewater also increased. From the bench scale results [3], the total dissolved solids and total hydrocarbon content of the wastewater treated with activated periwinkle shell carbon reduced significantly. A maximum percentage removal of 78.5% was achieved. The clarity of the effluent water was observed to be greatly improved after the first hour of treatment where the turbidity dropped from 66 NTU to 6 NTU. Again the percentage turbidity increased with contact time until 100%

The result showed comparatively, that the treated effluent water had maximum percentage reduction of approximately 83.26%, 91.19% (BOD, COD) for periwinkle shell carbon (PSC) and commercial activated carbon (CAC) respectively. The periwinkle shell carbon displayed the capacity to reduce the turbidity of the water contaminated with organic compounds via transfer of the contaminant solute from the bulk wastewater where their concentration is high to the external surface of the activated periwinkle shell carbon acting as the adsorbent.

In the development of alternative fixed filter bed for water treatment, the application of the mass transfer process by adsorption has also been shown to be feasible by examining the influence of modified clay on the rates of adsorption. The modifiers used were inorganic and organic acids, bases and salts. The results obtained showed that equilibrium adsorption of naphthalene i.e the transfer of naphthalene molecules from the bulk solution was attained at a faster rate using modifiedclaywhencomparedwiththeunmodifiedclay.Amongstthemodifiedclayusedforthe

This is attributable to the fact that the solutes in the acid medium had higher diffusive mobility defined by their dispersion coefficient/diffusivity as shown in Table 1 and higher influence on

Overall, results from this study affirm that sediment modification directly affects the availa‐

In soil remediation, the availability of the contaminant is of prime significance and it is commonly approached from the premise that chemicals are immediately accessible for

study, acids were found to be most suitable for the modification purpose [4].

the physicochemical properties (porosity, surface area, bulk and particle density).

an initial value of 53 mg/l to 2 mg/l after treatment.

removal was achieved following complete clarity.

bility of the contaminant chemicals.

**3. Applications in soil remediation**

microbial uptake only when in aqueous solution. The rationale behind this premise is that the small pore spaces internal to aggregates of soil and sediment particles exclude microbes, such that compounds that are dissolved or sorbed within these immobile-water domains must first be transported to the external aqueous phase (i.e., bulk aqueous phase) before they can be metabolized [5]. This premise is supported by several laboratory studies that have found biodegradation to occur only, or predominantly, in the bulk aqueous phase [6], [7]. A variety of factors, including physical characteristics of the sorbent (e.g., particle shapes, sizes, and internal porosities), chemical properties of the sorbates and sorbents, and biological factors (e.g., microbial abundance and affinity for the contaminant) influence availability.

In a study conducted [8], the experimental data from a soil microcosm was analyzed using the method of temporal moments (MOM) which interprets solute transport with linear equilibri‐ um sorption and first order degradation and the analytical solutions of a transport model CXTFIT version 2.0. This was with a view to estimating the transport parameters (pore-water velocity, V and dispersion coefficient, D) using non-reactive solute and the degradation parameters (retardation factor R and first order degradation rate λ) of the contaminant PAHs.

A dimensionless parameter called the retardation factor was used to represent bioavailability (i.e the accessibility of a chemical for assimilation and possible toxicity). This parameter increased with increasing solute hydrophobicity.

The result from the study is summarized in Table 2 and it showed that naphthalene had the lowest retardation factor with a corresponding higher degradation rate constant. The observed trend is similar and comparable to previous estimates from the Michaelis-Menten kinetics. The finding from this study can be attributed to the aqueous solubility, diffusivity and mobility of each contaminant solute.

**Table 2.** Comparison of the Degradation and Transport parameters for the Contaminant PAHs\*

In general, the result is significant as it affirms that the biodegradation of the contaminant organic compound or specifically PAHs is a function of their bioavailability.

Studies have led researchers to conclude that some microorganisms are capable of degrading compounds directly from the sorbed phase. Also, for many studies where solid-phase degradation was reported, biodegradation rates were nonetheless observed to decrease with contaminant soil-water contact time. Access to contaminants is increasingly inhibited as solutes migrate deeper into submicron pores of impermeable sorption domains in soil/ sediment-water environments. From the foregoing, if solutes are degraded in the mobile aqueous phase, it then becomes pertinent to state that the rates of remediation will be reduced by sorption and/or diffusion into impermeable regions, with the overall rate controlled by the slowest process of desorption or biotransformation. For example, in batch systems where the solids have a large sorption capacity, only a small fraction of the contaminant mass may be present in the bulk water. Evolving experimental reports is thus emphasizing that sorbate *diffusion* is often the limiting step, particularly in systems involving contaminants with large organic-carbon partition coefficients (*kOC*) and large fine-pored aggregates with high organic matter content.

Studies on mass transfer effects for the evaluation of bioavailability and biodegradation parameters of contaminant solutes in aqueous solid/sediment matrix applicable for biodegra‐ dation of the environmentally persistent and recalcitrant chemicals have been extensively investigated using models [7], [9] which incorporates a two-site sorption/desorption kinetics.

The applications of mass transfer by diffusion in multi-component mixtures were carried out by comparing the adsorption and desorption behavior of polycyclic aromatic hydrocarbons and benzene, toluene, ethylbenzene and xylene (BTEX) in sand and clay sediments fractions. Using equilibrium time as a basis for argument, research result showed that the contaminants in sand attained equilibrium faster than in the clay sediment for both sorption and desorption studies as a result of the higher permeability of sand sediment. The desorption equilibrium time in both sediment types was found to be slower than adsorption an indication that it may be the limiting step in the event of mineralization.

The adsorption and desorption kinetics of naphthalene using calcined and modified clay soil fractions at ambient temperature has also been investigated to demonstrate the role of mass transfer in the removal of toxic chemicals from soil. Result of the batch experiments showed that adsorption equilibrium was attained at 24, 28 and 32 hours for modified, calcined and untreated soils; while the desorption equilibrium occurred at 46 hours for modified and 52 hours for calcined and untreated soils. Following the equilibrium time, the percentage of an initial 100mg/l of naphthalene in the slurry phase system unadsorbed was 30%, 32% and 35% for the calcined, modified and untreated soils respectively; while 12.3%, 11.2% and 9.5% of the adsorbed naphthalene resisted desorption by the calcined, modified and untreated samples respectively at equilibrium. The mass transfer rate was estimated using the Lagergren equation and found to be 0.12 mg/g.hr, 0.11mg/g.hr and 0.08mg/g.hr during adsorption while the desorption rate (kdes) were 0.06mg/g.hr, 0.05mg/g.hr and 0.07mg/g.hr for calcined, modified and untreated soil respectively. The results clearly confirm that the rate of adsorption and desorption of naphthalene differ among soil types and this directly affects its effective removal from the soil. Studies on the dynamic behavior of the adsorption of naphthalene onto natural clay with various modifying agents have shown that adsorption increased to a large extent with increasing surface area and porosity of the modified clay. The results using acids, bases and salts showed that equilibrium adsorption of naphthalene from the bulk solution were attained at a faster rate using inorganic acids. Further investigations revealed that of all the acids used, the diffusivity of naphthalene was highest in the HCl-modified clay which had the largest surface area and porosity.

**PAHs R λ (per day)**

494 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

\*Adapted from Owabor, C.N. (2007) PhD Thesis, University of Lagos, Nigeria.

be the limiting step in the event of mineralization.

matter content.

**Table 2.** Comparison of the Degradation and Transport parameters for the Contaminant PAHs\*

organic compound or specifically PAHs is a function of their bioavailability.

MOM CXTFIT ε MOM CXTFIT Ε

Naphthalene 25.77 20.23 0.21 3.54 4.22 -0.19 Anthracene 41.62 28.43 0.32 1.21 2.05 -0.69 Pyrene 35.66 25.89 0.27 2.25 3.26 -0.45

In general, the result is significant as it affirms that the biodegradation of the contaminant

Studies have led researchers to conclude that some microorganisms are capable of degrading compounds directly from the sorbed phase. Also, for many studies where solid-phase degradation was reported, biodegradation rates were nonetheless observed to decrease with contaminant soil-water contact time. Access to contaminants is increasingly inhibited as solutes migrate deeper into submicron pores of impermeable sorption domains in soil/ sediment-water environments. From the foregoing, if solutes are degraded in the mobile aqueous phase, it then becomes pertinent to state that the rates of remediation will be reduced by sorption and/or diffusion into impermeable regions, with the overall rate controlled by the slowest process of desorption or biotransformation. For example, in batch systems where the solids have a large sorption capacity, only a small fraction of the contaminant mass may be present in the bulk water. Evolving experimental reports is thus emphasizing that sorbate *diffusion* is often the limiting step, particularly in systems involving contaminants with large organic-carbon partition coefficients (*kOC*) and large fine-pored aggregates with high organic

Studies on mass transfer effects for the evaluation of bioavailability and biodegradation parameters of contaminant solutes in aqueous solid/sediment matrix applicable for biodegra‐ dation of the environmentally persistent and recalcitrant chemicals have been extensively investigated using models [7], [9] which incorporates a two-site sorption/desorption kinetics.

The applications of mass transfer by diffusion in multi-component mixtures were carried out by comparing the adsorption and desorption behavior of polycyclic aromatic hydrocarbons and benzene, toluene, ethylbenzene and xylene (BTEX) in sand and clay sediments fractions. Using equilibrium time as a basis for argument, research result showed that the contaminants in sand attained equilibrium faster than in the clay sediment for both sorption and desorption studies as a result of the higher permeability of sand sediment. The desorption equilibrium time in both sediment types was found to be slower than adsorption an indication that it may

The adsorption and desorption kinetics of naphthalene using calcined and modified clay soil fractions at ambient temperature has also been investigated to demonstrate the role of mass Comparing this result with the data from the desorption study showed that while adsorption was inversely proportional to the pH of the medium, desorption was directly proportional. The implication of the retention time obtained from the equilibrium study is significant as it provides the bench mark for the interplay between sorption and degradation for transport and transformation of contaminant solutes within the soil matrix.

The adsorption and desorption kinetics of naphthalene, anthracene, and pyrene in a soil slurry reactor at ambient conditions was also investigated with a view to ascertaining the mechanisms controlling the retention and release rates of the compounds in the soil matrix. A stirred-flow method was employed to perform the experiments [10].

Analysis of the results of the batch adsorption/desorption kinetics and equilibria indicated that the desorption rate was slower than the adsorption rate. The cumulative extent of desorption for the tested chemicals (naphthalene, anthracene and pyrene) suggested that the desorption step was also the rate limiting for biodegradation. This may not be unrelated to the fact that diffusion in the pores may have been retarded by surface adsorption effects on soil organic carbon. The observed resultant effect was the lowering of the aqueous phase concentration of the contaminant PAHs which renders them not readily available to the microorganisms. The biodegradation will thus in the long run be controlled by the slow desorptive and diffusive mass transfer into biologically active areas. The extent of partitioning for the polycyclic aromatic hydrocarbons tested was found to be dependent on their solubility and diffusivity in the aqueous phase see Table 3. Experimental results from this study affirm that diffusivity/ diffusion coefficient is a property dependent on the physical properties of a system as well as the molecularity, structural configuration and angularity of the solute [8].

\*Adapted from Zander et al., (1993), Perry's Handbook of Chemical Engineers' (1998) and Oleszczuk and Baran, (2003).

**Table 3.** Some properties of investigated PAHs\*

Overall, discussion of the subject under consideration indicates that mass transfer is assessed by the rate of adsorption/desorption and it must be emphasized that this is both the measured concentration of the solute with time and the mechanism of distributing the solutes into surfaces and pores of individual adsorbent medium. The successful predictions of the fate and transport of solutes in the environment is hinged on the availability of accurate transport parameters.

Solute transport with linear equilibrium therefore must as a matter of necessity be accounted for and be accommodated as an integral component of the mineralization of toxic chemicals.

## **Author details**

#### Engr Owabor\*

Address all correspondence to<EMAIL_ADDRESS>
Department of Chemical Engineering, University of Benin, Benin City, Nigeria

### **References**

**Properties Naphthalene Anthracene Pyrene** Molecular formula C10H8 C14H10 C16H10 Molecular weight (g/mol) 128 178 202 Density (g/cm3) 1.14 1.099 1.271 Melting point (oC) 80.5 217.5 145-148 Boiling point (oC) 218 340 404 Aqueous solubility (g/m3) 0.93 0.07 0.14

496 Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling

\*Adapted from Zander et al., (1993), Perry's Handbook of Chemical Engineers' (1998) and Oleszczuk and Baran, (2003).

Overall, discussion of the subject under consideration indicates that mass transfer is assessed by the rate of adsorption/desorption and it must be emphasized that this is both the measured concentration of the solute with time and the mechanism of distributing the solutes into surfaces and pores of individual adsorbent medium. The successful predictions of the fate and transport of solutes in the environment is hinged on the availability of accurate transport

Solute transport with linear equilibrium therefore must as a matter of necessity be accounted for and be accommodated as an integral component of the mineralization of toxic chemicals.

[2] Owabor, C. N, & Audu, J. E. (2010). Studies on the Adsorption of Naphthalene and Pyrene from aqueous medium using ripe orange peels as Adsorbent" Global Journal

[3] Owabor, C. N, & Owhiri, E. (2011). Utilization of periwinkle shell carbon in the re‐ moval of polycyclic aromatic hydrocarbons from refinery wastewater." Journal of the

**Table 3.** Some properties of investigated PAHs\*

Address all correspondence to<EMAIL_ADDRESS>
[1] Treybal, R. E. (1981). Mass-Transfer Operations. , 19-23.

of Pure and Applied Sciences. , 16(1)

Nigerian Society of Chemical Engineers, , 26

Department of Chemical Engineering, University of Benin, Benin City, Nigeria

parameters.

**Author details**

Engr Owabor\*

**References**

**Coordinated and Integrated Geomorphologic Analysis of Mass Transfers in Cold Climate Environments – The SEDIBUD (Sediment Budgets in Cold Environments) Programme**

Achim A. Beylich

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54725

## **1. Introduction**

#### **General background**

Geomorphologic processes, responsible for transferring sediments and effecting landform change, are highly dependent on climate, and it is anticipated that climate change will have a major impact on the behaviour of Earth surface systems. Research on sedimentary fluxes from source to sink in a variety of different climatic environments is represented by a substantial body of literature. Studies on source-to-sink fluxes generally refer to the development of sediment budgets. A sediment budget is an accounting of the sources and disposition of sediment as it travels from its point of origin to its eventual exit from a defined landscape unit like a drainage basin, e.g. [1]. Accordingly, the development of a sediment budget necessitates the identification of processes of erosion, transport and deposition within a defined area, and their rates and controls [1, 2, 3]. The fundamental concept underpinning source-to-sink sediment flux and sediment budget studies is the basic sediment mass balance equation:

#### I = O +-ΔS

Where inputs (I) equal outputs (O) plus changes in net storage of sediment (ΔS). Source-tosink studies permit quantification of the transport and storage of sediment in a system. A thorough understanding of the current sediment production and flux regime within a system is fundamental to predict likely effects of changes to the system, whether climatic induced or human-influenced. Source-to-sink sediment flux and sediment budget research therefore enables the prediction of changes to erosion and sedimentation rates, knowledge of where