Rate of uptake of gas phase substrates. Experimental errors in determination of rates. Consider a well stirred laboratory reactor of volume 2 L which is used for continuous cultivation of S. cerevisiae. The reactor is sparged with vs=1.3 L air min^-1 The volume fraction of O2 in vs f is 20.96 % . T=30^∘ C, and p=1 atm. Assume, that ν5=vg t. a. Determine the rate of oxygen transfer q0 ^t when the volume fraction of oxygen in the exhaust gas is 8.15 %. b. The oxygen tension in the reactor is measured to 25 μM. s0 ^* is 1.20 mM when πa=1 atm. Determine the mass transfer coefficient k a. c. Determine the difference between qo^' and -qc when sq f is ≅0 and v=600 mL h^-1 d. The ethanol concentration in the liquid effluent from the reactor is measured to 4.2 g L^-1. The inlet concentration is zero. Determine the rate of ethanol production ( C -moles L^-1 h^-1 ). Also determine the mole fraction of ethanol in the liquid. e. From physical chemistry the following relation is obtained between the liquid mole fraction x and the gas phase mole fraction y. yi=xi (^s a t)/(p)=()/(p) is the activity coefficient, p is the total pressure, and is the partial pressure of component i in the gas phase. ^m is the vapor pressure of ethanol at the system at the system temperature = 78.4 mm Hg at 30^∘ C The activity coefficient for ethanol in water is quite high. Rarey and thermodynamic equilibrium is established between ethanol in the liquid phase and ethanol in the gas phase determine how much ethanol is being stripped to the gas phase ( C -moles L^-1 h^-1 ) when vg=vg=1.3 L min^-1. Compare with the ethanol produced by the bioreaction (question 4). Will the evaporated ethanol have a significant effect on vz-i.e. will it invalidate the assumption that va=var ? f. π480^sar is 31.7 mm Hg at 30^∘ C. Assuming that dry air, 20.96 % oxygen is used as feed to the reactor. What is the difference between vq and vz due to water evaporation? Again assume thermodynamic equilibrium between liquid and gas phase and γmoo =1. Will the difference between vg and vg influence the answer to question a? g. In question 1 the same oxygen transfer rate would be obtained if vg t was 6.5 L min ^-1 and the exit prartial pressure of O2 was 0.1840 atm . Reconsider the answer to question 6 if we also here use the assumption vs=vw. The answer to this question explains why - One should try to have a significant difference in oxygen pressure between inlet and outlet. - The off-gas should be dried before measuring πO2. (Christensen et al., 1995) The present problem is inspired by the experimental investigation of errors in measurement of oxygen uptake rates and ethanol production rates in bioreactors performed by Duboc and von Stockar (1998). This reference deserves a careful study as a preliminary to an experimental study of bioreaction rates.

Rate of uptake of gas phase substrates. Experimental errors in determination of rates. Consider a well stirred laboratory reactor of volume 2 L which is used for continuous cultivation of $S$. cerevisiae. The reactor is sparged with $v_s=1.3 \mathrm{~L}$ air $\min ^{-1}$ The volume fraction of $\mathrm{O}_2$ in $v_{s f}$ is 20.96 $\% . \mathrm{T}=30^{\circ} \mathrm{C}$, and $\mathrm{p}=1 \mathrm{~atm}$. Assume, that $\nu_5=v_{\mathrm{g} t}$. a. Determine the rate of oxygen transfer $q_0{ }^t$ when the volume fraction of oxygen in the exhaust gas is $8.15 \%$. b. The oxygen tension in the reactor is measured to $25 \mu$ M. $s_0{ }^*$ is 1.20 mM when $\pi_{\mathrm{a}}=1 \mathrm{~atm}$. Determine the mass transfer coefficient $k a$. c. Determine the difference between $q_o^{\prime}$ and $-q_c$ when $s_{\mathrm{q} f}$ is $\cong 0$ and $v=600 \mathrm{~mL} \mathrm{~h}^{-1}$ d. The ethanol concentration in the liquid effluent from the reactor is measured to $4.2 \mathrm{~g} \mathrm{~L}^{-1}$. The inlet concentration is zero. Determine the rate of ethanol production ( C -moles $\mathrm{L}^{-1} \mathrm{~h}^{-1}$ ). Also determine the mole fraction of ethanol in the liquid. e. From physical chemistry the following relation is obtained between the liquid mole fraction $x$ and the gas phase mole fraction $y$. $$ y_i=\gamma_i x_i \frac{\pi_i^{s a t}}{p}=\frac{\pi_i}{p} $$ $\gamma_i$ is the activity coefficient, $p$ is the total pressure, and $\pi_i$ is the partial pressure of component $i$ in the gas phase. $\pi_i{ }^m$ is the vapor pressure of ethanol at the system at the system temperature $=$ 78.4 mm Hg at $30^{\circ} \mathrm{C}$ The activity coefficient for ethanol in water is quite high. Rarey and thermodynamic equilibrium is established between ethanol in the liquid phase and ethanol in the gas phase determine how much ethanol is being stripped to the gas phase ( C -moles $\mathrm{L}^{-1} \mathrm{~h}^{-1}$ ) when $v_{\mathrm{g}}=v_{\mathrm{g}}=1.3 \mathrm{~L} \mathrm{~min}^{-1}$. Compare with the ethanol produced by the bioreaction (question 4). Will the evaporated ethanol have a significant effect on $v_z$-i.e. will it invalidate the assumption that $v_{\mathrm{a}}=v_{\mathrm{ar}}$ ? f. $\pi_{480}^{\text {sar }}$ is 31.7 mm Hg at $30^{\circ} \mathrm{C}$. Assuming that dry air, $20.96 \%$ oxygen is used as feed to the reactor. What is the difference between $v_q$ and $v_z$ due to water evaporation? Again assume thermodynamic equilibrium between liquid and gas phase and $\gamma_{\text {moo }}=1$. Will the difference between $v_{\mathrm{g}}$ and $v_{\mathrm{g}}$ influence the answer to question a? g. In question 1 the same oxygen transfer rate would be obtained if $v_{\mathrm{g} t}$ was $6.5 \mathrm{~L} \mathrm{~min}{ }^{-1}$ and the exit prartial pressure of $\mathrm{O}_2$ was 0.1840 atm . Reconsider the answer to question 6 if we also here use the assumption $v_s=v_w$. The answer to this question explains why - One should try to have a significant difference in oxygen pressure between inlet and outlet. - The off-gas should be dried before measuring $\pi_{\mathrm{O}_2}$. (Christensen et al., 1995) The present problem is inspired by the experimental investigation of errors in measurement of oxygen uptake rates and ethanol production rates in bioreactors performed by Duboc and von Stockar (1998). This reference deserves a careful study as a preliminary to an experimental study of bioreaction rates.

Rate of uptake of gas phase substrates. Experimental errors in determination of rates. Consider a well stirred laboratory reactor of volume 2 L which is used for continuous cultivation of $S$. cerevisiae. The reactor is sparged with $v_s=1.3 \mathrm{~L}$ air $\min ^{-1}$ The volume fraction of $\mathrm{O}_2$ in $v_{s f}$ is 20.96 $\% . \mathrm{T}=30^{\circ} \mathrm{C}$, and $\mathrm{p}=1 \mathrm{~atm}$. Assume, that $\nu_5=v_{\mathrm{g} t}$.
a. Determine the rate of oxygen transfer $q_0{ }^t$ when the volume fraction of oxygen in the exhaust gas is $8.15 \%$.
b. The oxygen tension in the reactor is measured to $25 \mu$ M. $s_0{ }^*$ is 1.20 mM when $\pi_{\mathrm{a}}=1 \mathrm{~atm}$. Determine the mass transfer coefficient $k a$.
c. Determine the difference between $q_o^{\prime}$ and $-q_c$ when $s_{\mathrm{q} f}$ is $\cong 0$ and $v=600 \mathrm{~mL} \mathrm{~h}^{-1}$
d. The ethanol concentration in the liquid effluent from the reactor is measured to $4.2 \mathrm{~g} \mathrm{~L}^{-1}$. The inlet concentration is zero. Determine the rate of ethanol production ( C -moles $\mathrm{L}^{-1} \mathrm{~h}^{-1}$ ). Also determine the mole fraction of ethanol in the liquid.
e. From physical chemistry the following relation is obtained between the liquid mole fraction $x$ and the gas phase mole fraction $y$.

$$
y_i=\gamma_i x_i \frac{\pi_i^{s a t}}{p}=\frac{\pi_i}{p}
$$

$\gamma_i$ is the activity coefficient, $p$ is the total pressure, and $\pi_i$ is the partial pressure of component $i$ in the gas phase. $\pi_i{ }^m$ is the vapor pressure of ethanol at the system at the system temperature $=$ 78.4 mm Hg at $30^{\circ} \mathrm{C}$ The activity coefficient for ethanol in water is quite high. Rarey and thermodynamic equilibrium is established between ethanol in the liquid phase and ethanol in the gas phase determine how much ethanol is being stripped to the gas phase ( C -moles $\mathrm{L}^{-1} \mathrm{~h}^{-1}$ ) when $v_{\mathrm{g}}=v_{\mathrm{g}}=1.3 \mathrm{~L} \mathrm{~min}^{-1}$. Compare with the ethanol produced by the bioreaction (question 4). Will the evaporated ethanol have a significant effect on $v_z$-i.e. will it invalidate the assumption that $v_{\mathrm{a}}=v_{\mathrm{ar}}$ ?
f. $\pi_{480}^{\text {sar }}$ is 31.7 mm Hg at $30^{\circ} \mathrm{C}$. Assuming that dry air, $20.96 \%$ oxygen is used as feed to the reactor. What is the difference between $v_q$ and $v_z$ due to water evaporation? Again assume thermodynamic equilibrium between liquid and gas phase and $\gamma_{\text {moo }}=1$. Will the difference between $v_{\mathrm{g}}$ and $v_{\mathrm{g}}$ influence the answer to question a?
g. In question 1 the same oxygen transfer rate would be obtained if $v_{\mathrm{g} t}$ was $6.5 \mathrm{~L} \mathrm{~min}{ }^{-1}$ and the exit prartial pressure of $\mathrm{O}_2$ was 0.1840 atm .
Reconsider the answer to question 6 if we also here use the assumption $v_s=v_w$. The answer to this question explains why
- One should try to have a significant difference in oxygen pressure between inlet and outlet.
- The off-gas should be dried before measuring $\pi_{\mathrm{O}_2}$. (Christensen et al., 1995)
The present problem is inspired by the experimental investigation of errors in measurement of oxygen uptake rates and ethanol production rates in bioreactors performed by Duboc and von Stockar (1998). This reference deserves a careful study as a preliminary to an experimental study of bioreaction rates.

The bacteria acetobacter aceti convert ethanol to acetic acid in the presence of oxygen according to the reaction $$\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}+\mathrm{O}_{2} \rightarrow \mathrm{CH}_{3} \mathrm{COOH}+\mathrm{H}_{2} \mathrm{O}$$ In a continuous fermentation process, ethanol enters the top of the fermenter at a rate of $145 \mathrm{kg} / \mathrm{h}$, and the air fed to the bottom of the fermenter is $25 \%$ in excess of the amount required to consume all of the ethanol. A gas stream containing nitrogen and unreacted oxygen leaves the top of the fermenter, and a liquid stream containing acetic acid, water, and $10 \%$ of the entering ethanol leaves the bottom. Assume that none of the ethanol, water, and acetic acid in the reactor is vaporized. The fermenter operates at $30^{\circ} \mathrm{C},$ maintains a liquid $(\mathrm{SG}=0.95)$ height of $4.5 \mathrm{m},$ and is open to the atmosphere (i.e., the pressure at the top of the fermenter is 1 atm). (a) What is the volumetric flow rate of air as it enters the bottom of the fermenter? What is the volumetric flow rate of gas leaving the top of the fermenter? (b) Assume a linear relationship between the fraction of oxygen reacted and the position of gas bubbles rising through the liquid in the fermenter: for example, half of the oxygen reacted is consumed in the bottom half of the fermenter. At the vertical midpoint of the fermenter, the average bubble diameter is $1.5 \mathrm{mm}$. What is the average bubble diameter at the entry point of the air and as the gas leaves the liquid at the top of the fermenter?

Elementary Principles of Chemical Processes