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21 Modeling Assumptions (continued) 7. The rate of product formation per unit volume rp can be expressed as Chapter 2 rp = YP / X rg (2-95) where the product yield coefficient YP/X is defined as: YP / X = mass of product formed mass of new cells formed (2-96) 8. The feed stream is sterile and thus contains no cells. General Form of Each Balance {Rate of accumulation} = {rate in} + {rate of formation} (2-97) 22 Individual Component Balances Chapter 2 Cells: Product: d ( XV ) = V rg dt d ( PV ) dt (2-98) = Vrp d( SV ) 1 Substrate: V rg = F Sf − dt YX / S (2-99) − 1 YP / S V rP (2-100) Overall Mass Balance Mass: d (V ) = F dt (2-101) 23 Laplace Transforms Chapter 3 • Important analytical method for solving linear ordinary differential equations.

61-64) for details. 2 (continued) Chapter 3 Recall that the ODE,  y + +6  y + 11 y + 6 y = 1, with zero initial conditions resulted in the expression Y (s) = ( 1 3 2 s s + 6 s + 11s + 6 ) (3-40) The denominator can be factored as ( ) s s 3 + 6 s 2 + 11s + 6 = s ( s + 1)( s + 2 )( s + 3) (3-50) Note: Normally, numerical techniques are required in order to calculate the roots. 1, y (t ) = 1 1 −t 1 −2t 1 −3t − e + e − e 6 2 2 6 (3-52) 18 Important Properties of Laplace Transforms 1. Final Value Theorem Chapter 3 It can be used to find the steady-state value of a closed loop system (providing that a steady-state value exists.

2. The fed-batch reactor is perfectly mixed. 3. Heat effects are small so that isothermal reactor operation can be assumed. 4. The liquid density is constant. 5. The broth in the bioreactor consists of liquid plus solid material, the mass of cells. This heterogenous mixture can be approximated as a homogenous liquid. 6. The rate of cell growth rg is given by the Monod equation in (293) and (2-94). 21 Modeling Assumptions (continued) 7. The rate of product formation per unit volume rp can be expressed as Chapter 2 rp = YP / X rg (2-95) where the product yield coefficient YP/X is defined as: YP / X = mass of product formed mass of new cells formed (2-96) 8.