CE30240 Advanced Principles of Chemical Engineering PDF

CE30240 Advanced Principles of Chemical Engineering TOPIC 4 UNIVERSITY OF BATH CE30240 Advanced Principles of Chemical Engineering TOPIC 4...

CE30240 Advanced Principles of Chemical Engineering TOPIC 4 UNIVERSITY OF BATH CE30240 Advanced Principles of Chemical Engineering TOPIC 4 Diffusion and reaction in porous catalysts 1. Diffusion limitations In TOPIC 3, it was assumed that every point on the surface was accessible to the same reactant concentration. In other words, the concentration is constant and homogeneous in every point from the bulk to the surface of the catalyst. However, reaction rates are affected by transport effects (or by diffusion limitations). When both, reaction and diffusion limitations, happen simultaneously, there is a gradient of concentration from the bulk to the surface as in Figure 4.1: Figure 4.1. a) Different diffusion steps from the bulk to the reaction site; b) Profile of concentration of a component A from the bulk fluid to the reaction site. cAb is the concentration of A in the fluid bulk, cAs is the concentration of A on the external surface of the catalyst, cA is the concentration of A inside the pore and will depend on the radius (r) of the catalyst particle. The diffusion limitations can be internal and external. The external diffusion limitations occur from the fluid bulk to the external surface of the catalyst (TOPIC 5). The internal diffusion limitations occur during the transport of the reactants across the catalyst pores to the reaction site. When the internal diffusion limits the catalytic process, the result is a different concentration inside the pores from the one on the surface. The concentration varies with the radius. Thus the overall reaction rate will be a function of radius. 1 AE CE30240 Advanced Principles of Chemical Engineering TOPIC 4 2. Catalyst surface area The rate of reaction in a heterogeneous catalyst is proportional to the accessible catalyst surface. Porous materials, like Figure 4.2, are usually used as a catalyst as the pores increase the total surface area of the catalyst. In a porous material: 𝑉𝑐 = 𝑉𝑠 + 𝑉𝑝 Eq. 4.1 where Vc is the total volume occupied by the catalyst, Vs the volume of the solid material and Vp the volume of the pores. There are some catalyst characteristics that indicate the surface area available for the reaction to happen: The specific area Sa is the total surface per unit of mass [m2/kg]. The effective diameter dp is the mean diameter of the pores of a porous catalyst [m]. The catalyst density ρc is the mass of catalyst per unit of volume [kg/m3]. The catalyst void fraction or porosity εs is the fraction of the catalyst that is pore and can be occupied by a fluid. However, although most of the surface of a heterogeneous catalyst is because of pores, those pores are not completely Figure 4.2. Porous materials accessible due to internal diffusion limitation. 3. Diffusion inside the catalyst pores The diffusion of a component A it is defined by the Fick’s Law, following the general equation for the molar flux of component A: 𝑑𝑐𝐴 𝑊𝐴𝑧 = −𝐷𝐴 𝑑𝑧 Eq. 4.2 where WAz is the molar flux of A through the distance z [mol m-2 s-1] and DA is the diffusivity of A [m2 s-1]. There are different diffusion mechanisms: Molecular diffusion: molecules collide with each other but not with the wall. It usually happens in large pores at high pressure. For binary diffusion (Chapman-Enskog Formula): 1 1 1⁄2 𝑇 3⁄2 (𝑀 + 𝑀 ) 𝐴 𝐵 Eq. 4.3 𝐷𝐴𝐵 = 0.0018583 𝑝𝑇 𝜎𝐴𝐵 2 𝛺𝐴𝐵 where DAB is the molecular diffusivity [m2 s-1], T is the temperature [K], pT is the pressure [atm], Mi is the molecular weight of the molecules [kg mol-1], σAB is the collision diameter (calculated as 𝜎𝐴𝐵 = (𝜎𝐴 + 𝜎𝐵 )⁄2) [Å] and ΩAB is the collision integral [-]. 2 AE CE30240 Advanced Principles of Chemical Engineering TOPIC 4 Knudsen diffusion: molecules collide with the walls of the pore but not with each other. It usually happens in small pores at low P. It is defined by: o For straight cylindrical capillary: 2 8𝑅𝑇 Eq. 4.4 𝐷𝐾 = 𝑟√ 3 𝜋𝑀 o For non-intersecting cylindrical capillaries: 16 𝜀𝑠 8𝑅𝑇 Eq. 4.5 𝐷𝐾 = √ 3 𝜌𝑐 𝑆𝑎 𝜋𝑀 where DK is the Knudsen diffusivity [m2 s-1], M is the molecular mass [kg mol-1], R is the gas constant [J mol-1 K-1], r is the pore radius [m], εs is the porosity [-], Sa is the specific area [m2 kg-1] and T is the temperature [K]. [2, 3] Diffusivity is usually a mixture of different mechanisms. The overall diffusivity can be calculated using conventional expressions for resistances in series: 1 1 1 = + Eq. 4.6 𝐷𝐴 𝐷𝐴𝐵 𝐷𝐾 The pores in the catalyst are not straight and cylindrical, so the effective diffusivity is used to calculate the diffusivity in actual conditions: 𝐷𝐴 · 𝜀𝑠 · 𝜎 Eq. 4.7 𝐷𝑒 = 𝜏 where De is the effective diffusivity [m2 s-1], DA the overall diffusivity [m2 s-1], εs is the porosity [-], σ is the pore constriction [-] and τ is the pore tortuosity [-], as explained in Figure 4.3. Figure 4.3. A) Pore constriction (σ) and B) Pore tortuosity (τ) 3 AE CE30240 Advanced Principles of Chemical Engineering TOPIC 4 Surface diffusion: molecules migrate along the pore surface because of a gradient in the surface concentration. Both the flux of species and the diffusivity are defined by: 𝐷𝐴𝑆 𝑑𝑐𝐴𝑆 Eq. 4.8 𝑊𝐴𝑧 = − 𝑆 𝜏 𝑣 𝑑𝑧 0 𝐸 𝐷𝐴𝑆 = 𝐷𝐴𝑆 𝑒𝑥𝑝 (− ) Eq. 4.9 𝑅𝑇 where WAz is the flux of species A [mol m-2 s-1], DAS the surface diffusivity [m2 s-1], cAS is the surface concentration of A [mol m-2], Sv is the specific area [m2 m-3], τ is the tortuosity [-] and E is the activation energy [J mol-1]. [4-6] 4. Reaction and diffusion effects When both the diffusion and the reaction occur simultaneously in the process, if a material balance is carried out to an infinitesimal section of a spherical catalyst particle (like the one in Figure 4.4) and assuming steady-state, then: 𝑊𝐴𝑟 · 4𝜋𝑟 2 |𝑟 − 𝑊𝐴𝑟 · 4𝜋𝑟 2 |Δ𝑟+𝑟 + 𝑟𝐴 · 4𝜋𝑟 2 Δ𝑟 = 0 Eq. 4.10 Volume of infinitesimal segment Flow of A IN Flow of A OUT Generation Accumulation term (reaction) term = 0 (steady state) where WAr is the mass transfer flux [mol m-2 s-1], rA is the reaction rate [mol m-3 s-1], r is the radial position [m] and Δr is the thickness of the infinitesimal segment [m]. (space for you to derive the equations…) – Derivation 4.1 Figure 4.4. Spherical catalyst particle with radius R and infinitesimal segment (Δr) 4 AE CE30240 Advanced Principles of Chemical Engineering TOPIC 4 to obtain: 𝑑2 𝑐𝐴 2 𝑑𝑐𝐴 𝑘𝑛 2 + ( ) − 𝑐𝐴𝑛 = 0 Eq. 4.11 𝑑𝑟 𝑟 𝑑𝑟 𝐷𝑒 where cA is the concentration of A [mol m-3] and kn the reaction rate constant [(m3/mol)n-1(s-1)]. 5 AE CE30240 Advanced Principles of Chemical Engineering TOPIC 4 5. Thiele Modulus Considering the following boundary conditions (Derivation 4.2): 1. The concentration at the centre of the pellet (r=0) is cA. 2. The concentration at the external surface (R) is cAs. If we use dimensionless parameters such as: Dimensionless concentration: 𝑐𝐴 𝜓= Eq. 4.12 𝑐𝐴𝑠 Dimensionless radial coordinates: 𝑟 𝜆= Eq. 4.13 𝑅 Eq. 4.11 can be transformed into: 𝑛−1 𝑑2 𝜓 2 𝑑𝜓 𝑘𝑛 𝑅2 𝑐𝐴𝑠 + − 𝜓𝑛 = 0 Eq. 4.14 𝑑𝜆2 𝜆 𝑑𝜆 𝐷𝑒 or: 𝑑 2 𝜓 2 𝑑𝜓 + − 𝜙𝑛2 𝜓 𝑛 = 0 Eq. 4.15 𝑑𝜆2 𝜆 𝑑𝜆 where φn is the Thiele modulus: 𝑛−1 𝑘𝑛 𝑅2 𝑐𝐴𝑠 𝜙𝑛 = √ Eq. 4.16 𝐷𝑒 The Thiele modulus represents the rate of surface reaction by the rate of diffusion. Therefore, large values of Thiele modulus will indicate that internal diffusion limits the overall rate of reaction. If the value of Thiele modulus is small, the surface reaction is the limiting step. 6. Internal Effectiveness factor Suppose you want to know how much is the overall catalytic reaction being affected by the internal diffusion limitations. In that case, you can use a new parameter known as the Effectiveness factor (η): 𝐴𝑐𝑡𝑢𝑎𝑙 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 Eq. 4.17 𝜂= 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 𝑖𝑛 𝑎𝑏𝑠𝑒𝑛𝑐𝑒 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑙𝑖𝑚𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑠 The effectiveness factor is usually a value between 0 and 1. 6 AE CE30240 Advanced Principles of Chemical Engineering TOPIC 4 In the absence of internal diffusion limitations, the reaction rate would happen when the concentration of the reactant A is the same on the surface of the catalyst than inside the pore. This condition happens when cA = cAs. Therefore effectiveness factor can also be defined as: −𝑟𝐴 𝜂= Eq. 4.18 −𝑟𝐴𝑠 To derive the effectiveness factor for a first-order reaction, it is easier to operate in mol s-1 rather than in mol m-3 s-1 so: −𝑟𝐴 · 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑐𝑎𝑡𝑎𝑙𝑦𝑠𝑡 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝜂= −𝑟𝐴𝑠 · 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑐𝑎𝑡𝑎𝑙𝑦𝑠𝑡 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 Eq. 4.19 or: 𝑑𝜓 4𝜋𝑅𝐷𝑒 𝑐𝐴𝑠 𝑑𝜆 |𝜆=1 𝜂= Eq. 4.20 4 𝑘1 𝑐𝐴𝑠 ( 𝜋𝑅 3 ) 3 (space for you to derive the equations…) – Derivation 4.4 7 AE CE30240 Advanced Principles of Chemical Engineering TOPIC 4 To obtain: 3 𝜂= (𝜙 · coth(𝜙1 ) − 1) 𝜙12 1 Eq. 4.21 Graphically, the graphs in Figure 4.5 can be obtained. Figure 4.5. A) Effectiveness factor as a function of the reaction order and B) different geometries (for first-order reaction) For first-order reaction, it can be observed that for small values of Thiele modulus (φ 1), η is ≈1. For large φ1, η = 3/φ1. From Eq. 4.18, can also be obtained for a first-order reaction: 𝑟𝐴 𝜂= → 𝑟𝐴 = 𝜂(𝑘𝐶𝐴𝑠 ) 𝑟𝐴𝑠 Eq. 4.22 So the effectiveness factor can be used to correct the reaction rate, including the effect of the mass internal mass transfer limitations. 8 AE CE30240 Advanced Principles of Chemical Engineering TOPIC 4 7. Weisz-Prater criterion for internal diffusion limitations It is possible to use the criterion to determine if there are internal diffusion limitations in the reaction. According to the criterion: −𝑟𝐴′ (𝑜𝑏𝑠)𝜌𝑐 𝑅 2 𝐶𝑊𝑃 = 𝜂 · 𝜙12 = 𝐷𝑒 𝑐𝐴𝑠 Eq. 4.23 where -rA’(obs) is the reaction rate observed (measured) and the rest of the parameters are known. If CWP > 1, then there is internal diffusion limitations. 8. Thiele modulus for other catalyst shapes and reaction orders For reaction order n and different shapes, like in Figure 4.5, the Thiele modulus can be approximated as: 𝑛−1 2 𝑛 + 1 𝑘𝑐𝐴𝑠 𝐿 𝜙𝑛 = √ Eq. 4.24 2 𝐷𝑒 where L is the characteristic length of the catalyst shape. For example: L=R/3 for sphere L=R/2 for cylinder L=L for slab 9. Non-isothermal conditions If the reaction is high exothermic, the internal catalyst temperature is higher than the temperature on the surface. The rate constant inside the pellet is much larger than at the exterior, leading to η higher than 1 as in Figure 4.6: Figure 4.6. Nonisothermal effectiveness factor 9 AE CE30240 Advanced Principles of Chemical Engineering TOPIC 4 In Figure 4.6, β is the dimensionless activation energy is calculated as: ∆𝐸𝑎 𝛾= 𝑅𝑇𝑆 Eq. 4.25 and the dimensionless heat of reaction calculated as: 𝐷𝑒 (−Δ𝐻𝑅 )𝑐𝐴𝑠 𝛽= 𝑘𝑒 𝑇𝑆 Eq. 4.26 where ΔEa is the activation energy [J mol-1], ΔHR is the heat of reaction [J mol-1], ke is the effective heat conductivity [W m-1 K-1] and Ts is the temperature at the external surface of the catalyst pellet [K]. References 1. Jiang, P. and C. Ye, Recession of Environmental Barrier Coatings under High-Temperature Water Vapour Conditions: A Theoretical Model. Materials, 2020. 13(20). 2. Szmyt, W., et al., Solving the inverse Knudsen problem: Gas diffusion in random fibrous media. Journal of Membrane Science, 2021. 620: p. 118728. 3. Thomas, J.M., Principles and practice of heterogeneous catalysis, ed. W.J. Thomas. 1997, Weinheim, London : VCH. 4. Reinecke, S.A. and B.E. Sleep, Knudsen diffusion, gas permeability, and water content in an unconsolidated porous medium. Water Resources Research, 2002. 38(12): p. 16-1-16- 15. 5. Medveď, I. and R. Černý, Surface diffusion in porous media: A critical review. Microporous and Mesoporous Materials, 2011. 142(2): p. 405-422. 6. Kovalchuk, N. and C. Hadjistassou, New insights from shale gas production at the microscopic scale. The European Physical Journal E, 2018. 41(11): p. 134. 7. Fogler, H.S., Elements of chemical reaction engineering. Fifth edition. ed. 2016, Boston, London. THE END 10 AE