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As shown in Appendix B, a differential change in temperature, dT, produces a corresponding change in the internal energy per unit mass, dUˆ int , dUˆ int = dHˆ = CdT (2-29) where C is the constant pressure heat capacity (assumed to be constant). The total internal energy of the liquid in the tank is: U int = ρVUˆ int (2-30) 14 Chapter 2 Model Development - II An expression for the rate of internal energy accumulation can be derived from Eqs. (2-29) and (2-30): dU int dT = ρVC (2-31) dt dt Note that this term appears in the general energy balance of Eq.

Fed-batch reactor for a bioreaction. 20 Chapter 2 Modeling Assumptions 1. The exponential cell growth stage is of interest. 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).

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. 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.

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