Chemical Reactor Design Module-4000 Mole Balance Equations

Process Design Engineering Document Number: C&PE-CRD-MD-0001 Document Title: Chemical Reactor Design – Theoretical Aspects Author: Engr. Anees Ahmad Revision: A1 Date: September 24, 2022 Chemical Reactor Design Module-4000: Mole Balance Equations REV-D3 (June-23-2024) Engr. Anees Ahmad Process Engineer Thermo Design Engineering Ltd., Pakistan Derivation of General Component Mole Balance Equation (GMBE) 𝑑𝑛𝑖 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑒𝑖 − 𝐶𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑒𝑖 𝑑𝑡 The generation and consumption terms can be combined into a single term in the favor of net change in moles of species “i” either by consumption or by generation or by both. Hence, 𝑑𝑛𝑖 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 ± 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑒/𝐶𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑒𝑖 𝑑𝑡 The +ve for generation and -ve for consumption sign conventions can be combined into net change term which leads to: 𝑑𝑛𝑖 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑁𝑒𝑡 𝐶ℎ𝑛𝑎𝑔𝑒 𝑖𝑛 𝑀𝑜𝑙𝑒𝑠 𝑜𝑓 𝑆𝑝𝑒𝑐𝑖𝑒𝑠 "i" 𝑑𝑢𝑒 𝑡𝑜 𝐶ℎ𝑒𝑚𝑖𝑐𝑎𝑙 𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛𝑖 𝑑𝑡 𝑑𝑛𝑖 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + 𝐺𝑒𝑖 𝑑𝑡 The net change in molar rates term arise due to chemical reaction, because chemical species “i” can generate (value of Gei +ve) or consume (value of Gei -ve) by a chemical reaction/s. Hence its value is governed by the rate of reaction. From here onwards variable “V“ is introduced and will be used to dictate the volume of chemical reaction mixture in m3. Please note that it is not reactor volume it is reaction mixture volume; reactor volume will also include additional volume of internals, dead zone, mass transfer zone (if any), equilibrium zone, vapor space etc. For reactors in which there is a perfect mixing present [e.g. Continuous Stirred Tank Reactor (CSTR) and Batch Reactor (BR)] this term is: 𝐺𝑒𝑖 = 𝑟𝑖 𝑉 For reactors in which there is not perfect mixing present [e.g. Plug Flow Reactor (PFR) and Packed Bed Reactor (PBR)] this term can be calculated by dividing the system volume into “N” number of patches each of equal volume (∆𝑉): Hence, for first volume element: 𝐺𝑒𝑖1 = 𝑟𝑖1 ∆𝑉 And thus, for the entire volume of reaction mixture (V), total change in molar generation or consumption rates is the sum of individual rates of net changes in each volume element: 𝐺𝑒𝑖1 + 𝐺𝑒𝑖2 + ⋯ + 𝐺𝑒𝑖𝑁 = 𝑟𝑖1 ∆𝑉 + 𝑟𝑖2 ∆𝑉 + ⋯ + 𝑟𝑖𝑁 ∆𝑉 𝑁 𝑁 ∑ 𝐺𝑒𝑖𝑘 = ∑ 𝑟𝑖𝑘 ∆𝑉 𝑘=1 𝑘=1 𝑁 𝐺𝑒𝑖 = ∑ 𝑟𝑖𝑘 ∆𝑉 𝑘=1 Page 2 of 10 And if N approaches to infinite to make it differential changes, then this equation will become the limits of Reman’s Sum which is equivalent to integral over the system volume by mathematical definition: 𝑁 𝑉 𝐺𝑒𝑖 = lim ∑ 𝑟𝑖𝑘 ∆𝑉 = ∫ 𝑟𝑖 𝑑𝑉 𝑛→∞ 0 𝑘=1 Hence, for PFRs and PBRs: 𝑉 𝐺𝑒𝑖 = ∫ 𝑟𝑖 𝑑𝑉 0 For CSTRs and BRs the reaction rate remains constant w.r.t the system volume (not w.r.t time) due to perfect mixing hence can be taken out of the integral: 𝑉 𝐺𝑒𝑖 = 𝑟𝑖 ∫ 𝑑𝑉 = 𝑟𝑖 𝑉 0 Hence the developed relationship for PFRs and PBRs is also applicable to CSTRs and BRs because it reduces to their relationship upon considering reaction rate as constant w.r.t system volume for CSTRs and BRs, so put it in general mole balance equation: 𝑉 𝑑𝑛𝑖 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + ∫ 𝑟𝑖 𝑑𝑉 𝑑𝑡 0 Because rate of reaction for species “i” is the net reaction rate for species “i" over all the chemical reactions (j) occurring in the system hence, 𝑟𝑖 = ∑ 𝑟𝑖𝑗 = ∑ 𝑣𝑖𝑗 𝑟𝑗 𝑗 [𝐸𝑄 − 𝑋. 1 − 1] 𝑗 GENERAL MOLE BALANCE EQUATION: 𝑉 𝑑𝑛𝑖 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + ∫ (∑ 𝑣𝑖𝑗 𝑟𝑗 )𝑑𝑉 𝑑𝑡 0 [𝐸𝑄 − 𝐺𝑀𝐵𝐸] 𝑗 • It is beneficial to write GMBE in terms of rj instead of rij. Because rj is always positive. List of Assumptions: • No assumptions (As it is generalized form of mole balance equation so it is without any practical constraint) Page 3 of 10 Derivation of Component Mole Balance Equation for Batch Reactor (BR) Recall EQ-GMBE: 𝑉 𝑑𝑛𝑖 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + ∫ (∑ 𝑣𝑖𝑗 𝑟𝑗 )𝑑𝑉 𝑑𝑡 0 𝑗 For batch reactor there is no inflow and outflow during the course of reaction hence: 𝑉 𝑑𝑛𝑖 = ∫ (∑ 𝑣𝑖𝑗 𝑟𝑗 )𝑑𝑉 𝑑𝑡 0 𝑗 For BR reaction rate remains constant over system volume (not over time) hence: 𝑉 𝑑𝑛𝑖 = (∑ 𝑣𝑖𝑗 𝑟𝑗 ) ∫ 𝑑𝑉 = (∑ 𝑣𝑖𝑗 𝑟𝑗 )𝑉 = 𝑉(∑ 𝑣𝑖𝑗 𝑟𝑗 ) 𝑑𝑡 0 𝑗 𝑗 𝑑𝑛𝑖 = 𝑉 ∑ 𝑣𝑖𝑗 𝑟𝑗 𝑑𝑡 𝑗 [𝐸𝑄 − 𝑋. 1 − 2] 𝑗 Derivation of Component Mole Balance Equation for Continuous Stirred Tank Reactor (CSTR) Recall EQ-GMBE: 𝑉 𝑑𝑛𝑖 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + ∫ (∑ 𝑣𝑖𝑗 𝑟𝑗 )𝑑𝑉 𝑑𝑡 0 𝑗 For CSTR reaction rate remains constant over system volume (not over time) hence: 𝑉 𝑑𝑛𝑖 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + (∑ 𝑣𝑖𝑗 𝑟𝑗 ) ∫ 𝑑𝑉 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + (∑ 𝑣𝑖𝑗 𝑟𝑗 )𝑉 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + 𝑉(∑ 𝑣𝑖𝑗 𝑟𝑗 ) 𝑑𝑡 0 𝑗 𝑗 𝑗 For Unsteady State Operation of CSTR: 𝑑𝑛𝑖 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + 𝑉(∑ 𝑣𝑖𝑗 𝑟𝑗 ) 𝑑𝑡 [𝐸𝑄 − 𝑋. 1 − 3𝑎] 𝑗 For Steady State Operation of CSTR: 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + 𝑉(∑ 𝑣𝑖𝑗 𝑟𝑗 ) = 0 𝑗 𝑉= 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 − ∑𝑗 𝑣𝑖𝑗 𝑟𝑗 [𝐸𝑄 − 𝑋. 1 − 3𝑏] Page 4 of 10 Derivation of Component Mole Balance Equation for Plug Flow Reactor (PFR) Recall EQ-GMBE: 𝑉 𝑑𝑛𝑖 = 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + ∫ (∑ 𝑣𝑖𝑗 𝑟𝑗 )𝑑𝑉 𝑑𝑡 0 𝑗 Differentiate this equation with respect to volume. Because for PFRs reaction rate varies over the system volume hence: 𝑑 𝑑𝑛𝑖 𝑑 𝑑 𝑑 𝑉 ( )= 𝑛̇ 𝑖0 − 𝑛̇ 𝑖 + ∫ (∑ 𝑣𝑖𝑗 𝑟𝑗 )𝑑𝑉 𝑑𝑉 𝑑𝑡 𝑑𝑉 𝑑𝑉 𝑑𝑉 0 𝑗 Inlet molar flowrate do not vary with system volume hence its derivative is zero: 𝑑 𝑑𝑛𝑖 𝑑𝑛̇ 𝑖 ( )=− + ∑ 𝑣𝑖𝑗 𝑟𝑗 𝑑𝑡 𝑑𝑉 𝑑𝑉 𝑗 And for PFR molar flowrate at other locations is the function of system volume and time hence ordinary derivates will change to partial derivatives: 𝜕 𝜕𝑛𝑖 𝜕𝑛̇ 𝑖 ( )=− + ∑ 𝑣𝑖𝑗 𝑟𝑗 𝜕𝑡 𝜕𝑉 𝜕𝑉 𝑗 We know that: 𝑐𝑖 = 𝜕𝑛𝑖 𝜕𝑉 [𝐸𝑄 − 𝑋. 1 − 4] Hence, 𝜕𝑐𝑖 𝜕𝑛̇ 𝑖 =− + ∑ 𝑣𝑖𝑗 𝑟𝑗 𝜕𝑡 𝜕𝑉 [𝐸𝑄 − 𝑋. 1 − 5] 𝑗 System volume for PFR is: 𝜕𝑉 = 𝐴𝑓 𝜕𝑧 [𝐸𝑄 − 𝑋. 1 − 6] Hence, For Unsteady State Operation of PFR: 𝜕𝑐𝑖 1 𝜕𝑛̇ 𝑖 =− + ∑ 𝑣𝑖𝑗 𝑟𝑗 𝜕𝑡 𝐴𝑓 𝜕𝑧 [𝐸𝑄 − 𝑋. 1 − 7𝑎] 𝑗 For Steady State Operation of PFR: 𝑑𝑛̇ 𝑖 = 𝐴𝑓 (∑ 𝑣𝑖𝑗 𝑟𝑗 ) 𝑑𝑧 [𝐸𝑄 − 𝑋. 1 − 7𝑏] 𝑗 Page 5 of 10 Derivation of Component Mole Balance Equation for Packed Bed Reactor (PBR) Recall EQ-X.1-5 based on actual reaction rate for heterogeneous catalytic reaction: 𝜕𝑐𝑖 𝜕𝑛̇ 𝑖 =− + ∑ 𝑣𝑖𝑗 𝑟𝑗,𝑎𝑐𝑡𝑢𝑎𝑙 𝜕𝑡 𝜕𝑉 𝑗 For PBR; the PFR equation can be directly converted using the definition of bulk density of catalyst bed: 𝜌𝑏 = 𝜕𝑚𝑐 𝜕𝑉 [𝐸𝑄 − 𝑋. 1 − 8] 𝜕𝑉 = 𝜕𝑚𝑐 𝜌𝑏 Which leads to: 𝜕𝑐𝑖 𝜕𝑛̇ 𝑖 = −𝜌𝑏 + ∑ 𝑣𝑖𝑗 𝑟𝑗,𝑎𝑐𝑡𝑢𝑎𝑙 𝜕𝑡 𝜕𝑚𝑐 𝑗 Moreover, for PBR reaction rate per unit mass of catalyst is used instead of reaction rate per unit volume using: ′ 𝑟𝑗,𝑎𝑐𝑡𝑢𝑎𝑙 = 𝜌𝑏 𝑟𝑗,𝑎𝑐𝑡𝑢𝑎𝑙 [𝐸𝑄 − 𝑋. 1 − 9] 𝜕𝑐𝑖 𝜕𝑛̇ 𝑖 ′ = −𝜌𝑏 + ∑ 𝑣𝑖𝑗 𝜌𝑏 𝑟𝑗,𝑎𝑐𝑡𝑢𝑎𝑙 𝜕𝑡 𝜕𝑚𝑐 𝑗 𝜕𝑐𝑖 𝜕𝑛̇ 𝑖 ′ = −𝜌𝑏 + 𝜌𝑏 (∑ 𝑣𝑖𝑗 𝑟𝑗,𝑎𝑐𝑡𝑢𝑎𝑙 ) 𝜕𝑡 𝜕𝑚𝑐 𝑗 1 𝜕𝑐𝑖 𝜕𝑛̇ 𝑖 ′ =− + (∑ 𝑣𝑖𝑗 𝑟𝑗,𝑎𝑐𝑡𝑢𝑎𝑙 ) 𝜌𝑏 𝜕𝑡 𝜕𝑚𝑐 [𝐸𝑄 − 𝑋. 1 − 10] 𝑗 Differential mass of catalyst (mc) can be changed to differential volume using combination of EQ-X.1-8 and EQ-X.1-6: 𝜕𝑚𝑐 = 𝜌𝑏 𝜕𝑉 = 𝜌𝑏 𝐴𝑓 𝜕𝑧 𝜕𝑐𝑖 𝜕𝑛̇ 𝑖 ′ = −𝜌𝑏 + 𝜌𝑏 (∑ 𝑣𝑖𝑗 𝑟𝑗,𝑎𝑐𝑡𝑢𝑎𝑙 ) 𝜕𝑡 𝜌𝑏 𝐴𝑓 𝜕𝑧 𝑗 Finally for Heterogeneous Catalytic Packed Bed Reactors (PBRs) the actual reaction rate is always less than the theoretical reaction rate due to ineffective diffusion and deviation from plug flow behavior hence, actual reaction rate can be found by catalyst effectiveness factor as: Page 6 of 10 𝜂𝑗 = ′ 𝑟𝑗,𝑎𝑐𝑡𝑢𝑎𝑙 ′ 𝑟𝑗,𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 = ′ 𝑟𝑗,𝑎𝑐𝑡𝑢𝑎𝑙 𝑟𝑗′ Thus: ′ 𝑟𝑗,𝑎𝑐𝑡𝑢𝑎𝑙 = 𝜂𝑗 𝑟𝑗′ If we use the same catalyst particle size on which lab scale kinetics is developed for heterogeneous catalytic reaction and we employ the same laboratory verified reaction rate law then effectiveness factor can be considered as “1” because in that case by default the rate law provides us the true rate instead of intrinsic one. But if we use that same rate law but with different catalyst particle size then its kinetic parameters need to be re-evaluated with other new catalyst particle size selected to express the true rate and to make catalyst effectiveness factor as “1”. And if we want to use the same rate parameters which are developed for different catalyst particle size instead of our selected particle size then effectiveness factor cannot be considered as “1”. We can calculate catalyst effectiveness factor by lab experiments or we can obtain it from catalyst manufacturer for the reaction and catalyst under consideration or it can also be calculated theoretically if we know the catalyst specs. By lab evaluation of the catalyst effectiveness factor, we need to conduct the experiment at some arbitrary catalyst particle size which will give us actual rate of reaction and we also need to conduct the experiment at so much reduced catalyst particle size for which reaction rate becomes independent of particle size then that rate will be theoretical reaction rate. The ratio of these two rates will give us the value of catalyst effectiveness factor. Then that value of effectiveness factor can be de-rated or enhanced for the selected catalyst particle size for use in reactor design equations. There are three other factors which need to be ensured for a successful catalytic reactor design and those are following: • • • The catalytic reactor should be designed in such a way that the design pressure drop should be 2 times the calculated pressure drop by Ergun Equation; because catalyst filling, practical flow and pressure disturbances can cause to squeeze the catalyst bed up to some extent which will increase the actual pressure drop than our calculated one. And if we keep the design pressure drop 2 times the calculated one then this effect is already incorporated into our design as recommended by the JGC Reactor Design Standard Practices. The catalyst deactivation rate should also be ensured from catalyst vendor or by laboratory techniques. If catalyst deactivation rate is too much slow that reactor can be operated for 8-12 months without maintenance and refilling of catalyst, then it is ok to use a packed bed reactor (either packed shell type or multitube packed bed type depending on feasible engineering decision of heat and mass distribution/extraction/supply). And if the catalyst deactivation rate is fast up to an extent that catalyst becomes deactivated within 24 to 48 hours and if it is possible to regenerate the catalyst then multiple packed bed reactors should be used in lead-lag configuration (one is working while others are regenerating and in cooling mode). And if the catalyst deactivation rate is so fast that the catalyst becomes deactivated within few minutes (5 to 30 minutes) then it is not recommended to use a packed bed reactor at all. Then it is necessary to move to some other sort of catalytic reactor for example “Moving Bed Reactor”, “Fluidized Bed Reactor”, “Trickle Bed Reactor”, or “Slurry Reactor” whichever is found feasible based on engineering decision. Another technique to make the packed bed reactor operation feasible for moderate catalyst deactivation rates is that select the catalyst weight 2 to 3 times greater than the calculated one. In this way as time passes the active region of catalyst which is effective for desired reaction conversion will move towards downstream of reactor with the passage of time until it reaches at such a value where desired conversion ceased to achieve then catalyst regeneration will be necessary. The utilization of this technique is commercially employed for ammonia synthesizer reactors. Due to this increase in selected catalyst weight sometime multi section packed bed catalyst reactors become necessary for effective re-distribution of reaction mixture or for reaction mixture interstage heating or cooling (if absolutely necessary based on Reaction Rate vs Reaction Equilibrium Characteristic Curves). Practical applications of Multi-Section Packed Bed Reactors in standalone configuration can be found in oxidation of ammonia to form NO2 for Nitric Acid Production, and oxidation of SO2 to SO3 for Sulfuric acid production. While lead-lag configuration packed bed reactors can be used for ammonia synthesis from its fundamental elements because in this case catalyst Page 7 of 10 deactivation rate is moderate. While for catalytic cracking processes, catalyst deactivation rate is so fast that a packed bed reactor is not used at all and in that case, we move to moving bed reactors. Multitube packed bed reactors are commonly used for hydrocarbon reforming processes where to carry out endothermic reaction, the catalyst filled reactor tubes are fitted into the direct combustion chamber. Finally, it should be kept in mind that the reaction volume calculated by the developed equations in this section in actual represents the reaction mixture volume not the reactor volume. The reactor volume will be calculated by providing additional margins for vapor space and vessel internals for BR and CSTR case or with additional tube lengths for PFR case based on standard recommended practices. And for the PBR case after the application of above specified techniques of over design the final weight of catalyst is determined and reactor packed section volume is calculated based on selected catalyst weight and bed void fraction or by catalyst bulk density. Then additional space in the catalyst tube before catalyst bed starting point and after catalyst bed ending point shall be provided. And then the final volume of reactor/tube for PBRs will be evaluated. Hence For Unsteady State Operation of PBR: 𝜕𝑐𝑖 1 𝜕𝑛̇ 𝑖 =− + 𝜌𝑏 (∑ 𝜂𝑗 𝑣𝑖𝑗 𝑟𝑗′ ) 𝜕𝑡 𝐴𝑓 𝜕𝑧 [𝐸𝑄 − 𝑋. 1 − 11𝑎] 𝑗 For Steady State Operation of PBR: 𝑑𝑛̇ 𝑖 = 𝜌𝑏 𝐴𝑓 (∑ 𝜂𝑗 𝑣𝑖𝑗 𝑟𝑗′ ) 𝑑𝑧 [𝐸𝑄 − 𝑋. 1 − 11𝑏] 𝑗 For Nomenclature, refer to Module-1000. Page 8 of 10 References 1. Jmith, J.M., Introduction to Chemical Engineering Thermodynamics. 9th Ed. 2022: McGraw Hill LLC. 2. Fogler, H.S., Elements of Chemical Reaction Engineering. 4-6th. ed. 2016: Prentice Hall. 3. Bird, R.B., Transport Phenomena. 2nd Ed. 2022: John Wiley & Sons, Inc. 4. Walas, S.M., Modelling with Differential Equations in Chemical Engineering. 1st Ed. 1991: Butterworth-Heinemann 5. Hill, C.G., Introduction to Chemical Engineering Kinetics and Reactor Design. 2nd Ed. 2014: John Wiley & Sons. Page 9 of 10 Engr. Anees Ahmad Process Engineer Thermo Design Engineering Ltd., Pakistan Cell: +92-336-4082300 Email: aahmad@thermodesign.com Page 10 of 10