Chapter 1 We now define an effectiveness factor to compare the effect of diffusion with that of reaction within the catalyst particle actual converstion rate WA actual ideal conversion rate WA ideal 1 4 3 For steady state the actual conversion rate WA actual within the particle can be determined from either of the following expressions WA actual 4 R2 DA dC A dr R 4 r 2 kCA dr r R 1 4 4 0 The actual conversion rate is determined using the first expression of Eq 1 4 4 CA CR R sinh r r sinh R dCA CR R dr sinh R dC A dr dC A dr 1 R cosh R R 2 sinh R CR R sinh R CR R coth R 1 R r R r R 1 r 2 sinh r r cosh r 1 2 kR 2 Let R Thiele modulus for a first order reaction Ignoring the minus sign DA the actual conversion of species A is then WA actual 4 R2 DA CR coth 1 4 R DACR coth 1 R 1 4 5 The ideal conversion rate WA actual is the maximum conversion rate that can be achieved when the mass transfer rate is much larger than the reaction rate In this condition the concentration of species A everywhere within the bead is the same as the concentration at the 4 surface CR Therefore the total conversion rate is simply the total volume R 3 time the 3 conversion rate per unit volume kCR The effectiveness factor is introduced since it is 1 35 more convenient to cast mathematical models in dimensionless format to indicate the minimum number of dimensionless groups that affect the physics This knowledge is useful in guiding the design of experiments and in the correlation of data The Thiele modulus is essentially the ratio of reaction to diffusion rates The effectiveness factor is then WA actual WA ideal 4 RDACR coth 1 4 R 3 kCR 3 3 coth 1 3 2 coth 1 2 kR DA 1 4 6 We should choose a value of R so that the effectiveness is close to one This means that the reaction rate is not limited by the diffusion of species A into the catalyst particle The following Matlab statements are used to plot equation 1 4 6 3 Matlab program to plot 2 coth 1 Effectiveness as a function of Thiele modulus phi1 0 1 1 1 phi2 2 10 phi3 20 10 100 phi phi1 phi2 phi3 ena 3 phi coth phi 1 phi 2 loglog phi ena grid on xlabel Thiele Modulus ylabel Effectiveness Factor Figure 1 4 1 plots the effectiveness factor versus the Thiele modulus We see that has almost reaches its maximum value as soon as is less than unity There is no need to reduce R any further once 1 The radius of the spherical catalyst particle can then be determined from the reaction rate constant and the diffusivity of A 1 2 kR 2 DA R 1 Rdesign k DA 1 2 1 36 Effectiveness Factor 10 10 10 0 1 2 1 10 10 0 1 10 2 10 Thiele Modulus Figure 1 4 1 The effectiveness factor for first order reaction in spherical catalyst particle This is the design Rdesign we should use to minimize mass transfer effect For any R Rdesign the conversion of urea will be limited by diffusion and for R Rdesign we will find a conversion rate that is independent of R and independent of DA For R Rdesign 1 4 Conversion rate R 3 kCR 3 4 3 R kCR 3 Example 1 4 5 A liquid is in contact with a well mixed gas containing substance A to be absorbed Near the surface of the liquid there is a film of thickness across which A diffuses steadily while being consumed by a first order homogeneous chemical reaction with a rate constant k1 At the gas liquid interface the liquid solution is in equilibrium with the gas and its concentration is cAi at the other side of the film its concentration is virtually zero Assuming dilute solutions derive an expression for the ratio of the absorption flux with chemical reaction to the corresponding flux without a chemical reaction 1 Solution The molar flux of A is given by 1 Benitez J Principle and Modern Applications of Mass Transfer Operations Wiley 2009 p 66 1 37 NA z cDAB dx A xA NA z NB z dz For dilute solution xA is much less than one so that c can be considered to be constant and the bulk motion contribution to the flux is negligible Therefore NA z cDAB dx A dc DAB A dz dz Well mixed gas NA z NA z z z Dilute liquid solution Making a mole balance around the control volume S z gives S NA z z S NA z z z 0 Dividing the equation by S z and letting z 0 yields dN A z 0 dz Substituting NA z DAB DAB E 1 dc A into equation E 1 we obtain dz d 2cA d 2cA 0 0 dz 2 dz 2 E 2 Integrating Eq E 2 we obtain c A K1z K2 E 3 Applying the two boundary conditions cA z 0 cAi and cA z 0 we have z cA cAi 1 E 4 The absorption flux without chemical reaction is then 1 38 NA z 0 NA z DAB dc A c D Ai AB dz E 5 Making a mole balance around the control volume S z with chemical reaction gives S NA z z S NA z z z S z rA 0 Dividing the equation by S z and letting z 0 yields dN A z rA k1cA dz Substituting NA z DAB DAB E 6 dc A into equation E 6 we obtain dz d 2cA d 2cA k k c 1 cA 1 A 2 2 dz dz DAB E 7 The solution to the homogeneous equation E 7 in terms of the hyperbolic functions is k1 k1 z B2cosh z cA B1sinh DAB DAB At z 0 cA cAi B2 k1 k1 cAi cosh At z cA 0 B1sinh D D AB AB Therefore B1 cAi k1 cosh DAB k1 sinh D AB Eq E 8 becomes cA cAi k1 cosh D AB sinh k1 z c cosh k1 z Ai DAB DAB k1 sinh DAB 1 39 E 8 cA cAi k1 k1 k1 k1 sinh cosh z sinh z cosh DAB DAB DAB DAB k1 sinh DAB Using the identity sinh A B sinh A cosh B sinh B cosh B we have cA cAi k1 sinh z DAB k1 sinh DAB E 9 The absorption flux with chemical reaction is then NA z 0 rxn DAB NA z 0 rxn cAi DAB dc A z 0 dz k1 k1 cosh z DAB DAB z 0 k1 sinh DAB k 2 NA z 0 rxn cAi k1DAB 1 2coth 1 DAB E 10 Therefore the ratio of absorption flux with chemical reaction to absorption flux without reaction is given by N A …