Notes on non isothermal reactors

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096116 Chemical Reaction Engineering Summary of governing equations for ideal reactors in non-isothermal conditions Reference reaction 𝐴𝐴+ 𝑏𝑏 𝑎𝑎𝐵𝐵↔ 𝑐𝑐 𝑎𝑎𝐶𝐶+ 𝑑𝑑 𝑎𝑎𝐷𝐷 1. The mean heat capacity difference 𝛥𝛥𝐶𝐶𝑃𝑃� is defined as: 𝛥𝛥𝐶𝐶𝑃𝑃� = � 𝜐𝜐𝑗𝑗 𝑁𝑁 𝑗𝑗=1 𝐶𝐶𝑃𝑃� 𝑗𝑗 In particular, for the reference reaction, the mean heat capacity difference 𝛥𝛥𝐶𝐶𝑃𝑃,𝐴𝐴� per mole of A is: 𝛥𝛥𝐶𝐶𝑃𝑃,𝐴𝐴� = 𝑐𝑐 𝑎𝑎𝐶𝐶𝑃𝑃� 𝐶𝐶+ 𝑑𝑑 𝑎𝑎𝐶𝐶𝑃𝑃� 𝐷𝐷− 𝑏𝑏 𝑎𝑎𝐶𝐶𝑃𝑃� 𝐵𝐵− 𝐶𝐶𝑃𝑃� 𝐴𝐴 2. The initial heat capacity 𝐶𝐶̃𝑃𝑃 𝑖𝑖𝑖𝑖 is defined as: 𝐶𝐶̃𝑃𝑃 𝑖𝑖𝑖𝑖= � 𝜗𝜗𝑗𝑗 𝑖𝑖𝑖𝑖 𝑁𝑁 𝑗𝑗=1 𝐶𝐶𝑃𝑃� 𝑗𝑗 In particular, for the reference reaction: 𝐶𝐶̃𝑃𝑃,𝐴𝐴 𝑖𝑖𝑖𝑖= 𝐶𝐶𝑃𝑃� 𝐴𝐴+ 𝜗𝜗𝐵𝐵 𝑖𝑖𝑖𝑖𝐶𝐶𝑃𝑃� 𝐵𝐵+ 𝜗𝜗𝐶𝐶 𝑖𝑖𝑖𝑖𝐶𝐶𝑃𝑃� 𝐶𝐶+ 𝜗𝜗𝐷𝐷 𝑖𝑖𝑖𝑖𝐶𝐶𝑃𝑃� 𝐷𝐷 3. The heat of reaction at temperature 𝑇𝑇 is: ∆𝐻𝐻� 𝑅𝑅(𝑇𝑇) = � 𝜐𝜐𝑗𝑗 𝑁𝑁 𝑗𝑗=1 𝐻𝐻�𝑗𝑗(𝑇𝑇) In particular, for the reference reaction, the heat of reaction at temperature 𝑇𝑇 per mole of A is: 1 ∆𝐻𝐻� 𝑅𝑅,𝐴𝐴(𝑇𝑇) = 𝑐𝑐 𝑎𝑎𝐻𝐻�𝐶𝐶(𝑇𝑇) + 𝑑𝑑 𝑎𝑎𝐻𝐻�𝐷𝐷(𝑇𝑇) − 𝑏𝑏 𝑎𝑎𝐻𝐻�𝐵𝐵(𝑇𝑇) − 𝐻𝐻�𝐴𝐴(𝑇𝑇) When there are no phase changes, the heat of reaction at temperature 𝑇𝑇 is related to the heat of reaction at standard reference temperature 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟 ∆𝐻𝐻� 𝑅𝑅(𝑇𝑇) = � 𝜐𝜐𝑗𝑗 𝑁𝑁 𝑗𝑗=1 𝐻𝐻�𝑗𝑗 𝑟𝑟𝑟𝑟𝑟𝑟+ � � 𝜐𝜐𝑗𝑗 𝑁𝑁 𝑗𝑗=1 𝐶𝐶𝑃𝑃� 𝑗𝑗𝑑𝑑𝑇𝑇 𝑇𝑇 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟 = ∆𝐻𝐻� 𝑅𝑅 𝑟𝑟𝑟𝑟𝑟𝑟+ � 𝛥𝛥𝐶𝐶𝑃𝑃� 𝑑𝑑𝑇𝑇 𝑇𝑇 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟 In particular, if we assume 𝛥𝛥𝐶𝐶𝑃𝑃� independent of temperature (i.e. 𝛥𝛥𝐶𝐶𝑃𝑃� ~𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐): ∆𝐻𝐻� 𝑅𝑅(𝑇𝑇) = ∆𝐻𝐻� 𝑅𝑅 𝑟𝑟𝑟𝑟𝑟𝑟+ 𝛥𝛥𝐶𝐶𝑃𝑃� �𝑇𝑇− 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟� 4. For adiabatic operations of a PFR, PBR, CSTR, or batch reactor, the temperature-conversion relationship is: 𝑋𝑋= − ∫ 𝐶𝐶̃𝑃𝑃 𝑖𝑖𝑖𝑖𝑑𝑑𝑇𝑇 𝑇𝑇 𝑇𝑇𝑖𝑖𝑖𝑖 ∆𝐻𝐻� 𝑅𝑅 𝑟𝑟𝑟𝑟𝑟𝑟+ ∫ 𝛥𝛥𝐶𝐶𝑃𝑃� 𝑑𝑑𝑇𝑇 𝑇𝑇 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟 In particular, if we assume both 𝐶𝐶̃𝑃𝑃 𝑖𝑖𝑖𝑖 and 𝛥𝛥𝐶𝐶𝑃𝑃� independent of temperature (i.e. both constant): 𝑋𝑋= − 𝐶𝐶̃𝑃𝑃 𝑖𝑖𝑖𝑖(𝑇𝑇− 𝑇𝑇𝑖𝑖𝑖𝑖) ∆𝐻𝐻� 𝑅𝑅 𝑟𝑟𝑟𝑟𝑟𝑟+ 𝛥𝛥𝐶𝐶𝑃𝑃� �𝑇𝑇− 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟� or, in terms of temperature, 𝑇𝑇= −𝑋𝑋∆𝐻𝐻� 𝑅𝑅 𝑟𝑟𝑟𝑟𝑟𝑟+ 𝐶𝐶̃𝑃𝑃 𝑖𝑖𝑖𝑖𝑇𝑇𝑖𝑖𝑖𝑖+ 𝑋𝑋𝛥𝛥𝐶𝐶𝑃𝑃� 𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟 𝐶𝐶̃𝑃𝑃…