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(2.24)

The solute flux across the membrane is dependent upon the concentration gradient Equation (2 .23) was integrated to yield

K

s

m .<

D

sm K

LlC .

so/utlOn (2 25)

- distribution coefficient of solute ( concentration of solute in membrane/concentration of solute in solution)

- difference in solute concentrations in solutions, kg m -' - solute

These equations for the solution-diffusion model are based on h igh membrane selectivity The solution properties and m embrane thickness are combined into the membrane constants, which must be determi ned experimentally. They are i ndependent of the solution concentrations on either side of the m embrane but are dependent on the type of solutions and the membrane (Rautenbach and Albrecht, 1 989) Lonsdale ( 1 972) reported that the effect of temperature on the permeability of cellulose acetate membranes to solutes was dependent on the anneali ng temperature used during

26 membrane manufacture. For cellulose acetate and other RO membranes solute transfer has been found to increase with increasing temperature and diffusion coefficients, and with decreasing solution viscosities ( Sourirajan, 1 970; Lonsdale, 1 972, Rautenbach and Albrecht, 1 989).

For convective flow though a porous membrane, the pore model can be used to determ ine the m embrane flux. The membrane pores are assumed to be parallel capill aries. Transport within the pore fluid is determined by both viscous flow and diffusion The total volumetric flow through a highly porous membrane can be described using Poiseuil le's law (Merton, 1 966; Cheryan and Nichols, 1 992; F ield, 1 993), where the pore velocity is equivalent to the water flux

v == e v

w w

Vw - volumetric water flux, m3 m -2 S -I

VW

- pore velocity of water, m S -1 flw - viscosity of water, kg m -I S -I r - pore radius, m e - porosity of m embrane 1: - tortuosity

A

- m embrane thickness, m (2.26)

The tortuosity factor, 't, accounts for twisting of the pores and increases in effective pore l ength. The solute flux is given by the combined viscous flow and diffusion in the pores

Ins Csm

de In _!) = e v _sm

- D �

_{sw d}

y - m ass flux rate of solute within pores, kg m -2 S I - concentration of solute in the membrane, kg m -3

- diffusion coefficient of solute in aqueous solution, m2 S -1 V - velocity of fluid in the pores, m s -\

( Merton, 1 966)

2 . 4.3. Resistance model

(2 27)

Beaudry and Lampi ( l 990(a)) found that the rate of water removal in DOC was prop0l1ionai to the difference in osmotic pressures of the OA and the dilute solution during juice concentration. The proportionality constant was approximated as the inverse of the sum of four resistances. Four resistances were encountered by the water as it

diffused from the dilute solution through a fouling layer at the membrane surface, across the membrane and into the �A.

RJ is the resistance in the juice (dilute solution) for the water to diffuse to the fouling layer on the membrane.

R( is the resistance of any foul i ng layer to the penneation of water to the membrane. Rm is the resistance of the membrane to the permeation of water.

ROA is the resistance in the OA solution for the water to diffuse from the membrane (Beaudry and Lampi, 1 990(a» .

The rate of water removal from the j ui ce was expressed as •

v _w

Vw - volumetric water flux rate, m3 m -2 S -1 11:0A - osmotic pressure of OA solution, 11:J - osmotic pressure of juice,

- 11: ) J _{(2 28)}

Beaudry and Lampi ( 1 990(a» found it simpler to approximate the flux rate from the difference in the soluble solids concentration or °Brix , where

Vw ==

k

(BOA

k

- proportionality constant, m s -1 CBrixr1 BOA - soluble solids concentration i n the OA, °Brix BJ - soluble sol ids concentration i n the juice,

(229)

Values for the proportionality constant k were determined from experimental data when the concentration difference and operating conditions were kept constant (Beaudry et aI, 1 989� Beaudry and Lampi, 1 990(a»). The main factors affecting the proportionality constant were temperature, solution concentrations, fluid velocities, membrane thickness and membrane molecular weight cut-off Beaudry Lampi ( 1 990(a» found that the first two factors influence viscosities and diffusion coefficients in the sol utions. W ith constant operating conditions and concentration difference across the membrane Beaudry and Lampi ( l 990(a» observed that the proportionality coefficient did not decline with time, concluding that fou ling was not occurring and the resistance RJ had a minimal effect They found this model to agree with what they observed during juice concentration where flux rates decreased correspondingly with the reduction in the concentration difference between the OA and the j ui ce

28

2.5. Diffusion in liq u ids

Fundamental to the operation of a membrane concentration system is the diffusion of solutes and solvents across boundary layers. The first law of diffusion is F ick's law

�1

==

-CDAB V'X�I

J1

-

molar diffusion flux o f species A, mol m -2 S-1

XA

- mole fraction of A

(2.30)

DAB

- m ass diffusivity of component A in a m ixture of A and B diffusion coefficient of component A in a mixture of A and B, m2 S-1

c - molar concentration of solution, mol m -3

v

(

_{a a a}

)

� , ay , az

vector differentiation operator (Bird et al. 1 960; Cussler, 1 984)

In a binary mixture of A and B, component A will diffuse from a h igh to a low concentration of A if a concentration gradient is present. In a binary system the diffusion coefficient

DAB

=

D8k

Diffusion coefficients in liquids are strongly concentration dependent and generally increase with temperature (Bird et aL, 1 960; Cussler, 1 984; Reid et al., 1 98 7).

The diffusion coefficient is describ ed in a variety of different ways:

DAA

-

self diffusion coefficient

D.1>

-

tracer diffusion coefficient

D IB

-

binary diffusion coefficient of A In a m ixture of A and B (inter and intra diffusion coefficients exist)

])1l1

-

b inary diffusion coefficient of B in a m ixture of A and 11

The self-diffusion coefficient,

DAA,

represents the diffusion of a molecule of component A in itself ( Reid et aL, 1 987) The

DAA

for water at 25°C is 2.299 x 10 -9 m2 s -\ (Easteal, 1 990; Menting et aI, 1 970) A special case of is the tracer diffusion coefficient. This is the diffusion coefficient of a label l ed molecule of component A in a solution of unlabelled component A (Reid et aL, 1 987). Tracer diffusion coefficients should not be compared to binary diffusion coefficients as the l abelled or tagged molecule may diffuse differently to an unlabelled or un-tagged molecule (Cuss\er, 1 984; Reid et aL, 1 987). A special case of

DAB

are the i nterdiffusion coefficients, when two solutions diffuse into each other. lntradiffusion coefficients arise when a solute is introduced to, and diffuses through a pure solution or a homogeneous mixture. The diffusion coefficient is

correlated to the or molecular diameter of the diffusing particle and to the molecul ar size ratio between the solute ilnd solvent molecules (Menting et ai, 1 970; Cussler,

1 984).

Numerous methods have been developed to experimentally measure diffusion coefficients in b inary systems (Cussler, 1 984; Tyrrell and Harris 1 984). To avoid measuring diffus ion coefficients in every diffusion situ ation studied, two theories have been proposed for diffusion in l iquids to calcul ate diffusion coefficients from solution data. One theory is based on absolute reaction rates (Glasstone et aI., 1 94 1 ) and the other b ased on hydrodynamic theory (Bird et at, 1 960).

Under the reaction rate theory, diffusion in a l iquid requires that one molecule in a l iquid layer moves from one equi librium position to another in the same l ayer. This usually i nvolves a solute molecule slipping past a solvent molecule. A molecule moves from one equi librium position to the next by overcomi ng the free energy of activation. Glasstone et al ( 1 94 1 ) proposed the "hole" theory for diffusion, during which two forms of energy are required. F i rstly energy is required to m ake one molecule jump to another position to m ake a hole. Secondly, energy i s requ ired for the diffusing molecule to jump i nto the hole formed. The total energy requi red for diffusion is dependent on the molecul ar s ize and polarity of the diffusing species. Molecular attractions (e.g. hydrogen bonding) also i nfluence the diffusion rate. Solvents attracted to a diffusing solute molecule will result in more rapid diffusion of the solute through that solvent (Menting, 1 970). W ith i ncreasing temperature the activation energy required for diffusion reduces leading to more rapid diffusion. H igh diffusion coefficients imply a small activation energy. Described qualitatively, slow d iffusing substances have to form relatively l arge holes for the molecule in the activated state, therefore, the activation energy is large (Glasstone et aL, 1 94 1 ).

When the solute molecule is l arger than the solvent, the movement of the solvent molecul e between equi librium points in the solution determines the activation energy for diffusion (Glasstone et aI ., 1 94 1 ). I n aqueous solutions the transition of water molecules between equil ibrium points is the rate determining factor in diffusion. At high water concentrations, the water molecules are bound with considerable hydration energy to the solute molecules. This hydration energy must be added to the normal activation energy for each water molecule to move from one position to the next (Gladden and Dole, 1 95 3 ) The activation energy for diffusion increases l inearly with mole fraction for sucrose and glucose solutions (Gladden and Dole, 1 95 3 ) .

3 0 Using the reaction rate theory the diffusion coefficient o f a molecule, in a n ideal solution of simi lar molecular sized species, can be determ ined using the fol lowing equation

A

k T _{(2.3 1 )}

D"III

k

- diffusion coefficient of molecule A i n a mixture of A and 13, m2

S -I

- Boltzmann constant, 1 .3 8 x 1 0 -23 J

K -I

T - tem perature,

K

11 - solution viscosity, kg m

-1 S-1

AI' �' �

- distance between two equilibrium positions, subscript 1 ,2,3 - position in x,y.Z direction away from the equilibrium position

(Glasstone et aL, 1 94 1 )

For non-ideal (concentrated) solutions, the fol lowing equation was derived for the diffusion coefficient. The diffusion coefficient in an ideal solution is represented by the self diffusion coefficient of that component

(D

AA)"

Y4 - activity coefficient of component A x,j - mole fraction of component A

+ 8 l n y A

]

8 1n xA

(2. 3 2)

!JAA - self diffusion coefficient, is estimated by the geometric average of the two self diffusion coefficients of the individual com ponents A and B, m2 s -\

(Glasstone et ai, 1 94 1 ; Cussier, 1 984; Reid et at, 1 987)"

The hydrodynamic theory was originally derived from the Nernst-Einstein equation The binary diffusion coefficient is

It l) w "' k T A k - Boltzmann constant, 1 . 3 8 x 1 0 -23 J

K -1

T - temperature,

K

(2 3 3 )

u)F -mob i lity of particle or solute A, the steady state velocity attained by the particle or solute under the action of a unit force, m s -} I N

(Bird et aI, 1 960)

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