Analytical solution of the non-uniform heat exchange in a reactor cooling coil with constant fluid flow

An analytical solution of the partial differential equation describing the enthalpy balance of the cooling fluid in a batch exothermic reactor has been developed for a constant flow. Thanks to an analytical integration on position, the simulation of the process can be performed with one independant variable (time) and without any approximation on the profile of the cooling fluid temperature. Comparisons are made with the well-known approximate method of segmentation into finite volumes at uniform temperature. The proposed method requires the integration of one ordinary differential equation (ODE) instead of the system of N + 1 ODEs used in the N finite volumes method. It is shown in simulations that a small number of finite elements (one to six) leads to a large inaccuracy in the results, however, it is commonly used. Finally the analytic method is applied as a zeroth order approximation to the case of a time varying flow. Comparisons are also made with the finite elements method and shown very good simulation results.