The demand for biopharmaceutical products in the form of complex proteins is growing. These proteins are most often produced by cells genetically engineered to produce the protein of interest, known as a recombinant protein. The cells are grown in a liquid culture, and the protein is harvested and purified to generate the final product.

Sf9 cells obtained from the fall armyworm can be used to produce protein therapeutics. Consider the growth of Sf9 cells in a bench-top bioreactor operating at 22°C, with a liquid volume of 4.0 liters that may be assumed constant.

Oxygen required for cell growth and protein production is supplied in air fed at 22°C and 1.1 atm. During the process, the gas leaving the bioreactor at 22°C and 1 atm is analyzed continuously. The data can be used to calculate the rate at which oxygen is taken up in the culture, which in turn can be used to determine the Sf9 cell growth rate (a quantity difficult to measure directly) and consistency of the operation from batch to batch.

(a) Analysis of the exhaust gas at a time 25 hours after the process is started shows a composition of 15.5mol% O₂, 78.7% N₂, and the balance CO₂ and small amounts of other gases. Determine the value of the oxygen use rate (OUR) in mmol O₂ consumed/(L ∙ h) at that point in time. Assume that nitrogen is not absorbed by the culture.

(b) OUR is related to cell concentration, X(g cells/L), by OUR = q_O2X, where q_O2 is the specific rate of oxygen consumption. Analysis of a sample of the culture taken at t = 25 h finds that the concentration of cells is 5.0 g cells/L. What is the value of q_O2? Do not forget to include its units.

(c) Six hours after this measurement, the exhaust gas contains 14.5 mol% O₂ and the percentage of N₂ is unchanged. What is the concentration of cells, X, at that point? Assume that the specific rate of oxygen consumption does not change as long as the process temperature is constant.

(d) The growth rate of cells can be expressed as:

where μ_g is the specific growth rate, with units of h^-1. Beginning with this equation and treating μ_g as a constant, derive an expression for t(X). Use the data from the previous parts of the problem to determine μ_g (include units). Then calculate the cell-doubling time (t_d), defined as the time for the cell concentration to double.

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Exhaust gas, 22°C 0.500 L air/min 22°C, 1.1 atm