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5.2. Theory 59where α T is a coefficient indicating the rate of transfer. Since the mass transfer rate iscontrolled by the solubility of TMOS in water, and therefore by the formation of methanolin water, the coefficient α T should increase with increasing k and vice versa. From Eqs.5.1, 5.2 and 5.3 it follows thatα Tn w (t) = n t (0) [exp(−α T t) − exp(−kt)] ,k − α T(5.4){n m (t) = 4n t (0) 1 − α [(T1 + k − α )]}Texp(−α T t) − exp(−kt) .k − α T α T(5.5)It is noted that Eqs. 5.4 and 5.5 do not diverge when α T = k. For any nonzero α Tand k the profile of n w (t) resembles that of a distorted bell-shaped curve, and n m is amonotonically increasing function of t. In the following the formed amount of silicic aciddue to hydrolysis is represented by n s . The consumption of silicic acid due to condensationis not reflected in n s , so thatn s (t) = 4n m (t). (5.6)5.2.2 Population balance modelGeneral equationsLet P (x) denote a continuous distribution of molecules with weight x at a certain instant.In case of irreversible aggregation the reaction between two aggregates or larger clusters(see Figure 5.1) is represented byP (x) + P (x ′ ) kp−→ P (x + x ′ ), (5.7)where the reaction rate k p should be an increasing function of the molecular sizes orweights. A general rate kernel is defined as [111]k p (x, x ′ ) = κ(xx ′ ) ν , (5.8)where κ is a constant and ν is a value between 0 and 1. Next, let p(x, t)dx be theconcentration of molecules at time t with molecular weight in the range (x, x + dx). Thepopulation balance equation can be written as (based on Ref. [111])∂p(x, t)∂t= −2κx ν p(x)p [ν] + κ∫ x0(x ′ ) ν (x − x ′ ) ν p(x − x ′ )p(x ′ )dx ′ + S, (5.9)where the bulk moment p [n] is defined asp [n] =∫ ∞0x n p(x)dx, (5.10)and the source term S is due to the hydrolysis of TMOS into silicic acid monomers, givenbyS = kn w (t)δ(x − x m ), (5.11)