Dividing both sides of the above equation by V, we obtain  dx/dt = (F/V)xi – (F/V)x + r1 (3)  In the chemical reaction engineering, F/V is called space velocity(s−1) and V/F is called the residence time (s). But in biochemical engineering, F/V is known as Dilution rate (Dr). Accordingly, equation(3) yields:  dx/dt = Drxi – Drx + r1 (4)  dx/dt = Dr(xi – x) + r1 (5)  For substrate balance,  Flow rate of substrate into the bioreactor = FSi  Flow of the substrate out of the bioreactor = FS  Rate of generation of substrate by reaction = –Vr2  Rate of accumulation of substrate within the reactor = d(VS)/dt  Now, apply general mass balance equation i.e.,  "Rate of Mass In – Rate of Mass Out + Rate of Generation = Accumulation"  d(VS)/dt = FSi – FS – Vr2 (6)  rearranging above equation, we get  dS/dt = Dr(Si –S ) – r2 (7)  where r2 is the rate of substrate consumption. For the chemical reaction,  A ----> P  We can write  Where,  ( –rA) = rate of disappearance of A  (rA) = rate of formation of A  k = reaction rate constant  CA = Concentration of reactant A  n = order of reaction with respect to component A  For first order reaction, n = 1 and accordingly,  –rA = k CA  The reaction kinetics involved in biochemical operations is comparatively difficult to obtain than the chemical reaction kinetics. In biochemical operations, the cell kinetics is used for the unstructured models where balanced growth condition is assumed. The following equation is used to represent the net rate of cell mass growth:  r1 = "μx (10)"  "where μ" is the specific growth rate or specific growth rate coefficient(s−1). Here, "μ" is analogous to first order rate constant k but however, "μ" is not a constant. In biochemical engineering, yield is defined as the ratio of mass or moles of product formed to the mass or moles of the reactants consumed. The yield (Y) of product (P) with respect to reactant A is defined as:  Y = (mass of P formed )/(mass of A consumed) (11)  In case of bioreactor,  Y = (mass of cells formed)/(mass of substrate consumed) (12)  Thus,  Y = r1/ r2  Or,  r2 = r1/Y  Or,  r2 = "μx/"Y ( from 10) (13)  By substituting equations (10) & (13) in equations (5) & (7) respectively, we get,  dx/dt = Dr(xi – x) + "μx" (14)  dS/dt = Dr(Si – S) – ("μx/Y)" (15)  Since we have assumed that the feed stream does not contain any biomass i.e., xi = 0, therefore, the bioreactor modelling equation finally gets the following form:  dx/dt = ("μ" – Dr)x (16)  dS/dt = Dr(Si – S) – ("μx/Y)" "(from 15)"  Thus, Equations (15) and (16) are the basic equations which are used for the modelling of any bioreactor.