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8.2. Theoretical background 117(see Figure 8.2). As opposed to the bulk mass transfer model (see Chapters 3 and 4)we include a diffusion term in the mass transfer term UoT (= -Uw T ), which can be readilyincorporated in the two-phase flow model. The term UoT follows from a description ofsimple diffusion between both phases within the capillary bundle system, and is given by[140],UoT 4Dϕ= − ( ) (˜ρ o − ˜ρ eq1 + So−1/2o ) , (8.14)r 2 wwhere D is an effective diffusion coefficient, r w is the average pore radius, and ˜ρ o is thedensity of TMOS in the oleic phase, which is equal to ρ o wo T . The equilibrium coefficient˜ρ eqo follows from the partitioning function( ) ε˜ρ eqo = ˜ρ eqw exp , (8.15)R g Tand the total density of TMOS ˜ρ T , given by˜ρ T = ˜ρ o + ˜ρ w . (8.16)In Eq. 8.15 R g is the gas constant, T is the temperature, and ˜ρ w is equal to ρ w w T w. Theinteraction parameter ε is a function of the methanol concentration in water. Due to thepoor solubility of TMOS in water the interaction parameter is on the order of R g T butdecreases as the concentration of methanol is increased due to the hydrolysis of TMOS inwater. As a first order approximation, the parameter ε is chosen as (see also Chapter 3)ε = ε 0(1 − awMw), (8.17)where ε 0 is the interaction parameter in absence of methanol and a is an arbitrary scalefactor greater than zero.8.2.5 Reaction termsIn case of full hydrolysis a TMOS molecule reacts with water according toSi(OCH 3 ) 4 + 4H 2 O → Si(OH) 4+ 4CH 3 OH. (8.18)The reaction terms RαX only exist for the aqueous phase, α = w, and are related to asingle molar reaction term R byR T w = −M T R ; R W w = −4M W R ; R SAw = M T R ; R M w = 4M T R, (8.19)where M X is the molar weight of species X. The molar reaction term is given byR = k ′ h (C′T w ) β (C w ′W ) γ=k ′ h(M T ) β (M W ) γ (C T w) β (C W w ) γ= k h (C T w) β (C W w ) γ(8.20)= k h (ρ w ) β+γ (w T w) β (w W w ) γ ,