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Unformatted text preview: 121 CHAPTER 12 SOLUTIONS 12.1 Nomenclature F Substrate feed rate gsubst/hr KdegPenicillin degradation rate constant hr1 Kmg/L mShr1P Penicillin concentration g/L S Substrate concentration g/L SfSubstrate feed concentration g/L V VolumeLXCellmassgcell/L YP/SPenicillin yield gPen/gsubst YX/SCell mass yield gcell/gsubst Cell growth rate gcell growth/gcellhr Penicillin growth rate gPen growth/gcellhr Mass Balances Cell Mass XVSXtXV},{d)d(=XVSXtXVtVX},{dddd=+Volume Balance fSFtV=dd; that is, fSFV=Substituting XVSXtXVSFXf},{dd=+Rearranging FVSXXSXXf=},{122 Penicillin PVKXVStPVdeg}{d)d(=PVKXVStPVtVPdeg}{dddd=+Substituting and rearranging: fSFPPVKXVSPV=deg}{Rearranging: FVSPPKXSPf=deg}{Substrate FXVSKSmYXVSXVYSXtSVmSSPSX++=//}{},{d)d(FXVSKSmYXVSXVYSXtSVtVSmSSPSX++=+//}{},{ddddSubstituting and rearranging: fmSSPSXSFSFXVSKSmYXVSXVYSXSV++=//}{},{Rearranging: fmSSPSXSFSFXVSKSmYXVSXVYSXSV++=//}{},{VFSSXSKSmXYSXYSXSfmSSPSX++=1}{},{//123 12.2 The objective function in the first printing of the second edition of the textbook is corrected to: =}{00085.68.1}{}{025.dttFVPwhere is the batch time in hours. The first term accounts for the grams of penicillin produced, which is sold at $0.025/g. The second term charges for utilization of the equipment at $1.68/hr. The third term charges $0.00085/g for the total sucrose used in a batch (g). The objective function provides an approximate estimate of the net profit before taxes, as formulated by Cuthrell and Biegler (1989). Note that the utilization term, 1.68,is very high relative the gross profit: =}{00085.}{}{025.dttFVPGPIn addition, the fifth inequality constraint in Example 12.2 (page 388) is corrected to: hr20072In this range of batch times, the utilization cost greatly exceeds the gross profit. Hence, the objective is to find the smallest negative net profit, which occurs at = 72 hr. One final correction in Example 12.2 (page 388) is that max= 0.11 hr1. This exercise has been solved by Cuthrell and Biegler using optimal control techniques (orthogonal collocation on finite elements). Because optimal control techniques are beyond the scope of this textbook, a strategy using repeated simulation is used. A profile of the sucrose feed flow rate in time is entered and the mass balances are integrated over the batch time using POLYMATH. Initially, the profile in Figure 12.4 is entered for Case 1: 124 As shown in Table 1, the gross profit is $0.343 and the net profit is $120.617. The effect of a decreasing ramp input is considered in Case 2. Because the total sucrose fed during the first 30 hr is increased, the feed rate of sucrose over the next 42 hr is reduced....
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This note was uploaded on 04/10/2008 for the course CHE 233W taught by Professor Debelak during the Spring '08 term at Vanderbilt.
 Spring '08
 Debelak
 Mass Balance

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