ScaleUp of a Batch Reactor
This Demonstration shows why a batch reactor with a cooling jacket cannot be scaled up geometrically. That is, as the diameter and height of the reactor increases, the surface area for heat transfer divided by the volume decreases, and thus the maximum temperature increases. This example is for an exothermic, liquidphase, firstorder reaction. The diameter and height of the reactor are equal. A cooling jacket around the outside of the reactor transfers heat away from the reactor. As the reactor size increases, the maximum temperature approaches the adiabatic limit. Adiabatic reactor behavior is shown by selecting the adiabatic checkbox. This is independent of the reactor size.

Details
In this Demonstration, the height of the batch reactor is equal to the inner diameter, so reactor volume and surface area can be simplified to:
where A is the reactor surface area (cm^{2}), V is reactor volume (cm^{3}), and h and d are the height and diameter of the reactor where h = d (cm).
Mass and energy balances are done to get the temperature of the reactor and moles in the reactor:
where N_{A} are moles of A (mol), C_{A} is the concentration of A (mol/cm^{3}), the subscript 0 refers to the initial condition, T and T_{a} are the reaction and cooling liquid temperatures (K), t is time (min), r_{A} is the rate of reaction (mol/[cm^{3} min]), C_{P} is heat capacity of A (J/[mol K]), ΔH is the heat of reaction (J/mol), U is the heat transfer coefficient (J/[cm^{2} K min]), k_{o} is the preexponential factor (1/min), E_{a} is activation energy (J/mol), and R is the ideal gas constant (J/[mol K]).
A video at [1] shows a reallife example of the danger associated with the scaleup of a batch reactor.