The relative rate law expressions for each reaction are:
−𝑟𝐴′ −𝑟𝐵′ 𝑟𝐶′ 𝑟𝐷′
=
= =
1
3
1
1
(𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛 1)
𝑟𝐴′
𝑟𝐵′ 𝑟𝐸′ 𝑟𝐷′
=− = =
1
1
1
1
(𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛 2)
𝑟𝐶′ 𝑟𝐺′ 𝑟𝐷′
= =
2
1
1
(𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛 3)
−
−
The final concentrations can be calculated using the following expression, where j represents the
species:
𝐶𝑗 = 𝐶𝑇𝑜
𝐹𝑗 𝑃 𝑇𝑜
𝐹𝑇 𝑃𝑜 𝑇
The Ergun equation must be expressed in terms of molar flowrates since there are multiple reactions:
𝑑𝑝
𝛼 𝑇 𝐹𝑇
=−
𝑑𝑊
2𝑝 𝑇𝑜 𝐹𝑇𝑜
The general form of an energy balance is as follows:
𝑑Êsys
= Q − Ws + ∑ 𝐹𝑖𝑜 𝐻𝑖𝑜 − ∑ 𝐹𝑖 𝐻𝑖
𝑑𝑡
Where
𝑑Êsys
𝑑𝑡
is the rate of accumulation of total energy, Q is the heat rate to the system, W is the
rate of shaft work done by the system to the surroundings, and Hi is the specific molar enthalpy
of species i. This reactor is a counter-current flow non-isothermal PBR operating at steady-state.
The energy balance in this case eventually becomes:
𝑈
𝑟𝐴′ ∆𝐻𝑟𝑥 − 𝜌𝛼 (𝑇 − 𝑇𝛼 )
𝑑𝑇
𝑏
=
∑ 𝐹𝑖 𝐶𝑝𝑖
𝑑𝑊
Where Cpi represents the specific heat capacity of species i. Since this is not a membrane reactor,
transport equations are not applicable.