4. A batch fluidized bed polymerization reactor is used to produce spherical polyethylene (PE) particles. The reactor is filled with monosized, spherical seed PE particles to begin the process. Fluidizing gas containing excess ethylene monomer is added continuously to the reactor. The ethylene monomer reacts at the surface of the PE particles and polymerizes so that the particles increase in size by a layering process. It is reasonable to assume that the PE particles are well mixed in the fluidized bed reactor. (a) Starting from the general one dimensional population balance for a well mixed system, derive the following simplified population balance equation is to describe the growth of particles in this system. State your assumptions clearly. DO NOT SOLVE THE EQUATION. tVn(x,t)=GxVn(x,t) where: V is the process control volume n(x,t) is the number distribution x is particle size t is time G is the growth rate (7 marks) (b) Starting from the general form of the zeroth moment of the population balance, show mathematically that for this system, the total number of PE particles in the reactor will not change with time. (5 marks) (c) The linear growth rate of PE particles is found to be size independent, i.e. G=dtdx=G(x). Derive an expression for the volume based growth rate Gv=dtdv. If G has a value of 0.5mmmin1 and the size of the seed particles is 2mm, what is the initial value of Gv ? If G is constant during the process, how will Gv vary with time