(15) Also, can be utilised to develop a relation between biomass and substrate concentrations, by assuming it has a constant value: (16) where is the cell biomass concentration in the stationary phase, by substituting equation (16) in equation (7), the cell balance equation becomes: (17) By re-arranging the equations, the cell balance equation becomes: (18) By integrating equation 18 at as the initial condition, the resulting solution is; X = X0 t = 0 (19) Equation (19) is an analytical relation, but non-explicit for which given all growth related X, parameters (, and , are known. and are given, and is calculated from equation (16) X0 S0 Xm with the conditions = and = Equation 19 can be used to evaluate the growth X X0 S S0. parameters by linear regression on batch culture data. Partially integrating to solve for product formation, from equation 12; P−P0= (X-X0) +β. (20) α. The difficulty in the integration of equation 20 is the last term since the solution for is not X explicit., therefore, by integrating by parts: (21) In order to obtain an analytical solution to equation 21, it was temporarily assumed that is the t dependent variable. Integrating with respect to between and the following solution was X X X0, obtained; (22) Then, by substituting equation (22) in (20), the result is; (24) (23)
(15) Also, can be utilised to develop a relation between biomass and substrate concentrations, by assuming it has a constant value: (16) where is the cell biomass concentration in the stationary phase, by substituting equation (16) in equation (7), the cell balance equation becomes: (17) By re-arranging the equations, the cell balance equation becomes: (18) By integrating equation 18 X = X0 at t = 0 as the initial condition, the resulting solution is; (19) Equation (19) is an analytical relation, but non-explicit for X, which given all growth related parameters (, and , are known. X0 and S0 are given, and Xm is calculated from equation (16) with the conditions X = X0 and S = S0. Equation 19 can be used to evaluate the growth parameters by linear regression on batch culture data. Partially integrating to solve for product formation, from equation 12; P−P0= α. (X-X0) +β. (20) The difficulty in the integration of equation 20 is the last term since the solution for X is not explicit., therefore, by integrating by parts: (21) In order to obtain an analytical solution to equation 21, it was temporarily assumed that t is the dependent variable. Integrating with respect to X between X and X0, the following solution was obtained; (22) Then, by substituting equation (22) in (20), the result is; (23) (24)
Date_Seed_Waste_PHB_Paper_JBW_27_09_16.docx