Homework #3. From the textbook, problems 9.1, 9.2, 9.3, 9.10, In 9.2 use q P = 0.02 g P / g cell h.

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1 Homework #3 From the textbook, problems 9.1, 9.2, 9.3, 9.10, 9.15 In 9.2 use q P = 0.02 g P / g cell h. In 9.10 the factor k s is k d, the kinetic factor for the cell death. Also, use r=0 for part (b) & C as the increased cell concentration out of the cell concentrator. 1

2 MANTZARIS, N. V., P. DAOUTIDIS, F. SRIENC, AND A. G. FREDRICKSON, "Growth Processes in a Cascade ofbioreactors: Mathematical Models," AIChE J. 45(1): , MOULTON, G. G., Fed-batch Fermentation: A Practical Guide to Scalable Recombinant Protein Reduction in Escherichia coli, Elsevier, New York, MuRPHY, S. V., AND A. ATALA, "3D Bioprinting of Tissues and Organs," Nat. Biotechnol. 32(8): , PROBLEMS 9.1. Consider a CFSTR in which biomass is being produced with glucose as the substrate. The microbial system follows a Monod relationship with l!m = 0.4 h- 1, Ks = 1.5 g/1 (an unusually high value), and the yield factor Yx 1 s = 0.5 g biomasslg substrate consumed. Normal operation is with a sterile feed containing 10 g/1 glucose at a rate of 100 1/h. a. What is the specific biomass production rate (g/1-h) at steady state? b. If recycle is used with a recycle stream of 10 llh and a recycle biomass concentration five times as large as that in the reactor exit, what would be the new specific biomass. production rate?,., c: Explain ~ny difference between the values found in parts a and b In a two-stage chemostat system, the volumes of the first and second reactors are V 1 = 500 I and V 2 = 300 I, respectively. The first reactor is used for biomass production and the second is for a secondary metabolite formation. The feed flow rate to the first reactor is F= 100 llh, and the glucose concentration in the feed iss= 5.0 gil. Use the following constants for the cells: f..lm =0.3h- 1, Ks=O.Ig/1, _ 0 4 g dw cells Yxts - g glucose a. Determine cell and glucose concentrations in the effluent of the first stage. b. Assume that growth is negligible in the second stage and the specific rate of product formation is qp = 0.2 g Pig cell h, and Yp 1 s = 0.6 g Pig. Determine the product and substrate concentrations in the effluent of the second reactor Consider the following batch growth data: Timeh Xg/1 Pg/1 dx!dt g/1-h dp!dt g/1-h < < < o o Bioreactors for Suspension and Immobilized Cultures Chap. 9

3 You have available three tanks of different volumes: 900, 600, and 300 I. Given a flow rate of 100 1/h, what configuration of tanks would maximize product formation? 9.4. Two CFSTR in series are operated for fermentation of glucose to ethanol. The reactors are operated for a certain degree of conversion with a known feed substrate concentration (i.e., S 0 and S 2 are fixed). Assume a first order fermentation kinetics (Rs = k 1 S) in both reactors. a. Write down the design equations for both reactors (8H as function of S) and the total hydraulic residence time as function of substrate concentrations. b. -Prove that total 9H is minimum when S 1 = (S 0.S 2 ) 112. c. Prove that ehj = 8H2 or the volumes of the reactors would be the same for the constant feed flow rate (F) when total eh or the total volume is minimum. d. Express the hydraulic residence times for each reactor and the total hydraulic residence time in terms of kl> S"' and S Two-stage chemostat system is used for fermentation of glucose to ethanol using Saccharomyces cerevisiae. The feed flow rate and glucose concentrations are 1 1/h and S 0 = 40 g/1, respectively. The following kinetic constants are known: t d = 5 h, Ks = I g /1, Yx 1 s = 0.1 gx/gs, Yp 1 s = 0.5 g EtOH/g glucose. Assuming 80% conversion in every reactor, determine the following: a. Substrate an'd product concenthiticins in the effiuent of reactors b. Biomass concentrations in both reactors c. Volume of each reactor d. Productivity of biomass (DX) and product (DP) for each reactor e. Specific rates of product formation (qp) and substrate consumption (q 8 ) for each reactor 9.6. Penicillin is produced by Penicillium chrysogenum in a fed-batch culture with the intermittent addition of glucose solution to the culture medium. The initial culture volume at quasi-steady state is V 0 = 500 I, and glucose-containing nutrient solution is added with a flow rate off= 50 1/h. Glucose concentration in the feed solution and initial cell concentration are S 0 = 300 g/1 and X 0 = 20 g/1, respectively. The kinetic and yield coefficients of the organism are llm = 0.2 h- 1, Ks = 0.5 g/1, and Yx 1 s = 0.3 g dw/g glucose. a. Determine the culture volume at t = 10 h. b. Determine the concentration of glucose at t = 10 hat quasi-steady state. c. Determine the concentration and total amount of cells at quasi-steady state when t = 10 h. d. If qp = 0.05 g product/g cells hand P 0 = 0.1 g/1, determine the product concentration in the vessel at t = I 0 h The bioconversion of glucose to ethanol is carried out in a packed-bed, immobilized cell bioreactor containing yeast cells entrapped in Ca-alginate beads. The rate-limiting substrate is glucose, and its concentration in the feed bulk liquid phase is S 0 ; = Sg/1. The nutrient flow rate is F = 2 1/min. The particle size of Ca-alginate beads is Dp = 0.5 em. The surface area of the alginate beads per unit volume of the reactor is a= 25 cm 2 /cm 3, and the cross-sectional area of the bed is A= 100 cm 2. Following are the rate constants for this conversion: for the following rate expression: 1 ~, = 100 mg S/cm 3 h K 5 = 10 mg S/cm 3 r,ns r= - -- ' K 5 +S Problems 319

4 Assuming a first-order reaction-kinetics (e.g., relatively low substrate concentrations), determine the required bed height for 80% conversion of glucose to ethanol at the exit stream. Hint: To calculate the effectiveness factor, we can use the following equations: 1 [ 1 I ] 11 = ~ tanh 3<1> - 3<1> where and 9.8. A fluidized-bed, immobili~~d cell bioreactor is used for the conversion of glucose to ethanol by Zymomonas mobilis cells immobilized in K-carrageenan gel beads. The dimensions of the bed are 10 em (diameter) by 200 em. Since the reactor is fed from the bottom of the column and because of C0 2 gas evolution, substrate and cell concentrations decrease with the height of the column. The average cell concentration at the bottom of the column is X 0 = 45 g/1, and the average cell concentration decreases with the column height according to the following equation:. X= X 0 ( ) Z is the column height (em). The specific rate of substrate consumption is q 5 = 2 g S/g cells h. The feed flow rate and glucose concentration in the feed are 5 1/h and 160 g glucose/1, respectively. a. Determine the substrate (glucose) concentration in the eilluent. b. Determine the ethanol concentration in the eilluent and ethanol productivity (g/1-h) if YP!s = 0.48 g ethanol/g glucose Lactic acid is produced by homolactic bacteria Streptococcus lactis from glucose solution using fed-batch operation. The following data are available: V 0 = 10 I, X 0 = 5 g/1, Yxts = 0.12 g Xlg S, YPIS =0.80 g PigS, Jlm=0.2 h- 1, Ks = 0.5 g/1, SF= 30 g/1, F= 21/h. a. How long should the sytem be operated in order to obtain S = 0.5 g/1 in the fermenter? b. Determine the biomass and product concentrations at the end of operation. c. What should be the feed flow rate in order to obtain S = 0.1 g/1 after 10 h of operation? d. Determine P and X for this operation A completely mixed bioreactor is used for ethanol fermentation of hydrolyzed starch solution containing 30 g/1 total sugar. The following data are available: Jlm = 0.15 h- 1, Ks= 0.5 g/1, F= 10 1/h, Yx 1 s= 0.1 g Xlg S, YP1s =0.5 gp/gs, ks =O.l d- 1 a. For 95% conversion of substrate to product, determine the required reactor volume and ethanol productivity when the system was operated without cell recycle. b. For the same degree of conversion, determine the required reactor volume and ethanol productivity when the system was operated with cell recycle. r = 0.4, C = X,JX= 3, a=o.l. 320 Bioreactors for Suspension and Immobilized Cultures Chap. 9

5 9.11. In a fluidized-bed biofilm reactor, cells are attached on spherical plastic particles to form biofilms of average thickness L = 0.5 mm. The bed is used to remove carbon compounds from a wastewater stream. The feed flow rate and concentration of total fermentable carbon compounds in the feed are F = 2 1/h and S = 2000 mg/1. The diameter of the column is 10 em. The kinetic constants of the microbial population are rm =50 mg S/cm 3 -h and Ks = 25 mg S/cm 3. The specific surface area of the biofilm in the reactor is 2.5 cm 2 /cm 3. Assuming first-order reaction kinetics and an average effectiveness factor of TJ = 0.7 throughout the column, determine the required height of the column for effiuent total carbon concentration of S 0 ; = 100 mg/ Glucose is converted to ethanol by immobilized yeast cells entrapped in gel beads. The specific rate of ethanol production is qp = 0.2 g ethanol/g cell-h. The effectiveness factor for an average bead is 0.8. Each bead contains 50 g/1 of cells. The voids volume in the column is 40%. Assume growth is negligible (all glucose is converted into ethanol). The feed flow rate is F= 400 1/h, and glucose concentration in the feed is S 0 ; = 150 g glucose/1. The diameter of the column is 1 m, and the yield coefficient is about 0.49 g ethanollg glucose. The column height is 4 m. a. What is the glucose conversion at the exit of the column? b. What is the ethanol concentration in the exit stream? Consider the batch growth curve \ n,f:igure 9.4 and the corresponding plots of dx!dt versus X and dp/dt versus P (Figure 9.7). You are asked to design a two-stage reactor system with continuous flow that will produce product Pat a concentration of 0.55 g/1. You wish to minimize total reactor volume. For a flow rate of /h, what size reactors (and in what order) would you recommend? Consider Figure 9.9, which applies to a fed-batch system. Assume at t = 0, V = 100 I, X= 2 g/1,!l = 1/h, S 0 = 4 g/1, and S = 0.01 g/1. Vis increased at a constant rate such that dv/dt = 20 1/h = F (or flow rate) and X is constant at all times. a. Derive a formula to relate!l to V and dv!dt. b. What is!l at t = 5 h? An industrial wastewater stream is fed to a stirred-tank reactor continuously, and the cells are recycled back to the reactor from the bottom of the sedimentation tank placed after the reactor. The following are given for the system: F = 100 1/h; S 0 = 5000 mg/1;!lm = 0.25 h- 1 ; K 5 = 200 mg/1; a (recycle ratio)= 0.6; C (cell concentration factor)= 2; Y;: 5 = 0.4. The effiuent concentration is desired to be 100 mg/1. a. Determine the required reactor volume. b. Determine the cell concentration in the reactor and in the recycle stream. c. If the residence time is 2 h in the sedimentation tank, determine the volume of the sedimentation tank and cell concentration in the effiuent of the sedimentation tank A wastewater stream is treated biologically by using a reactor containing immobilized cells in porous particles. Variation of rate of substrate removal with particle size is given in the following table ' 5 (mg/1-h) Dp(mm) Problems 321

6 a. What are the effectiveness factors ford p = 4 mm and D P = 7 mm? b. The following data were obtained for Dp = 4 mm at different substrate concentrations. Assuming no liquid film resistance, determine the rm and Ks for the microbial system. S 0 (mg/1) r (mg/1-h) A wastewater stream off= 1 m 3 /h with substrate at 2000 mg/1 is treated in an upflow packed bed containing immobilized bacteria in form of biofilm on small ceramic particles. The effiuent substrate level is desired to be 30 mg/1. The rate of substrate removal is given by the following equation: kxs r=- ' K, +S By using the following information, determine the required height of the column (H): k = 0.5 h- 1, X= 10 g/1, Ks = 200 mg/1, L = 0.2 mm, a= 100 m 2 1m\ A= 4m 2, T] = A packed-column biofilm reactor (biofilter) with HID 0 = 2.5 is used for aerobic biological treatment of wastewater of flow rate Q =5m 3 /hand S 0 = 1000 mg chemical oxygen demand/1 to obtain Se = 25 mg COD/I in the effiuent. The effiuent is recycled with R = 2. The following information are available: 1-lm = 0.12 h- 1, Ks = 150 mg/1, Y = 0.35 g Xlg S, X= 10 g X/1, L = 1.5 mm, R = 2 (Effiuent recycle ratio), a 1 = 120 m 2 /m 3, De= 10-5 cm 2 /sec. a. Determine the effectiveness factor (TJ). b. Determine the volume and dimensions (D 0 and H) of the column. c. For H= 3m, determine biofilm area per unit reactor volume (a 1 ) andd 00 when H/D 0 = Bioreactors for Suspension and Immobilized Cultures Chap. 9