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Chapter 5: Problem 47

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?

Short Answer

Expert verified
The volumetric flow rates of air at the bottom and of gas leaving at the top of the fermenter as well as the average bubble diameter at the entry point of the air and as the gas leaves the liquid at the top can be determined using stoichiometric relationships, mass balance principles and gas-liquid interactions in the fermenter. The specific values depend on the physical properties and operational conditions of the system.

Step by step solution

01

Determine stoichiometry

Given the reaction \(C_{2} H_{5} OH + O_{2} \rightarrow CH_{3} COOH + H_{2} O\), we can see that 1 mole of \(C_{2} H_{5} OH\) (ethanol) reacts with 1 mole of \(O_{2}\) (oxygen) to produce 1 mole of \(CH_{3} COOH\) (acetic acid) and 1 mole of \(H_{2} O\). So, mole to mole ratio is 1:1:1:1.
02

Find the air inflow rate

First find the molar flow rate of ethanol entering the system, then calculate the molar flow rate of oxygen needed for the reaction. Given that the air is 25% in excess of the required quantity, we can obtain the total molar flow rate of air entering the system. Convert this to volumetric flow rate using ideal gas law.
03

Find the gas outflow rate

Use the stoichiometry, air inlet rate and given information that all the ethanol is consumed to calculate the molar flow rate of oxygen leaving the fermenter. Convert this into volumetric flow rate using the ideal gas law.
04

Calculate bubble diameter at entry point, midpoint and top of the fermenter

Assuming a linear relationship between the fraction of oxygen reacted and the position of gas bubbles rising through the liquid, we determine the relationship between bubble diameter and its location. It's given that the average bubble diameter at the midpoint is 1.5 mm, therefore, the average bubble diameter at the entry point and at the top of the fermenter can be deduced.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stoichiometry
Understanding stoichiometry is crucial for professionals in chemical engineering. It's the fundamental concept that deals with the quantitative relationships between the reactants and products in a chemical reaction. In the context of the bacterial conversion of ethanol to acetic acid, stoichiometry helps us determine how much oxygen is required to react with a given amount of ethanol, and subsequently, how much acetic acid and water will be produced.

The reaction \( C_{2}H_{5}OH + O_{2} \rightarrow CH_{3}COOH + H_{2}O \) is balanced, meaning one mole of ethanol reacts with one mole of oxygen to produce one mole of acetic acid and one mole of water. To further clarify for students, a 'mole' is a standard scientific unit for measuring large quantities of very small entities such as atoms, molecules, or other specified particles. The equation also tells us that for every kilogram of ethanol consumed, there will be a proportional kilogram of acetic acid produced, since their molar ratios are equal. This is a key concept in designing and operating a fermentation process effectively and efficiently.
Volumetric Flow Rate
The volumetric flow rate is a measure of the volume of fluid that passes through a given surface per unit time. It's usually expressed in units like liters per second or cubic meters per hour. In chemical engineering education, grasping the concept of the volumetric flow rate is necessary because it impacts the design and scaling of reactors and other process equipment.

In our exercise, we first calculate the volumetric flow rate of air needed for the fermentation process. Air is being fed in excess (25%), so we must account for this surplus when calculating the volumetric flow rate. Using the stoichiometric relationship established earlier, we convert the molar flow rate of oxygen to the molar flow rate of air by accounting for the excess and then use the ideal gas law (PV = nRT) to find the volumetric flow rate, where P is pressure, V is volume, n is number of moles, R is the gas constant, and T is temperature. Similarly, the volumetric flow rate of the gas leaving the fermenter is calculated by considering the molar flow rate of the remaining oxygen, as all ethanol should be consumed. This concept ensures that there is sufficient reaction with ethanol to produce acetic acid while also being mindful of economic and environmental concerns.
Fermentation Process
The fermentation process is a metabolic pathway that converts sugar to acids, gases, or alcohol. It occurs in yeast and bacteria, and also in oxygen-starved muscle cells. In chemical engineering, fermentation is a key operation, particularly in the production of alcohols, solvents, and organic acids. Our focus lies on the production of acetic acid from ethanol by the bacteria acetobacter aceti.

Different types of fermentations, such as continuous fermentation shown in this exercise, are critical for the efficient production of chemicals. Continuous fermentation involves the constant addition of nutrients and removal of products, which is beneficial for the production of acetic acid since the process does not have to be stopped to harvest the product. This style of fermentation can be more efficient than batch processes, especially for large-scale production. The maintenance of a stable environment, such as temperature, pH, and oxygen availability, which are implied in the exercise, is also an integral part of the fermentation process. This ensures that bacteria will continuously convert ethanol into acetic acid, maintaining the reaction's efficiency.

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Most popular questions from this chapter

A fuel cell is an electrochemical device that reacts hydrogen with oxygen from the air to produce water and DC electricity. A proposed application is replacement of the gasoline-fueled internal combustion engine in an automobile with a \(100 \mathrm{kW}\) fuel cell. You are on a summer internship with a gas supplier planning to transport hydrogen to service stations for use in cars powered by fuel cells. The hydrogen is to be transported in tube trailers, each of which has 10 tubes of length \(10.5 \mathrm{m}\) and diameter \(0.56 \mathrm{m}\). Hydrogen in the tubes at 2600 psig and an average temperature of \(298 \mathrm{K}\) is discharged at service stations to a final pressure of 55 psig. Refueling cach fuel-cell-powered automobile is estimated to require 4.0 kg of hydrogen. (a) You and your office-mate- an intern from a different university - have been asked to estimate the number of automobiles that can be refueled by one tube-trailer load of hydrogen. He does a very quick calculation and comes up with a value of 95 cars. Speculate how he did it and provide support for your speculation. What was his mistake? (b) Do the calculation using the SRK equation of state. Instead of using Eqs. \(5.3-11\) and \(5.3-13\) for the parameter \(\alpha,\) use the following correlation developed specifically for hydrogen: \(^{23}\) \(\alpha=1.202 \exp \left(-0.3228 T_{\mathrm{r}}\right)\) (c) Do the calculation using the law of corresponding states. (d) In which of the three estimates would you have the greatest confidence, and why?

The van der Waals equation of state (Equation \(5.3-7\) ) is to be used to estimate the specific molar volume \(\hat{V}(\mathrm{L} / \mathrm{mol})\) of air at specified values of \(T(\mathrm{K})\) and \(P(\mathrm{atm}) .\) The van der Waals constants for air are \(a=1.33 \mathrm{atm} \cdot \mathrm{L}^{2} / \mathrm{mol}^{2}\) and \(b=0.0366 \mathrm{L} / \mathrm{mol}\) (a) Show why the van der Waals equation is classified as a cubic equation of state by expressing it in the form $$f(\hat{V})=c_{3} \hat{V}^{3}+c_{2} \hat{V}^{2}+c_{1} \hat{V}+c_{0}=0$$ where the coefficients \(c_{3}, c_{2}, c_{1},\) and \(c_{0}\) involve \(P, R, T, a,\) and \(b .\) Calculate the values of these coefficients for air at \(223 \mathrm{K}\) and 50.0 atm. (Include the units when giving the values.) (b) What would the value of \(\hat{V}\) be if the ideal-gas equation of state were used for the calculation? Use this value as an initial estimate of \(\tilde{V}\) for air at \(223 \mathrm{K}\) and 50.0 atm and solve the van der Waals equation using Goal Seek or Solver in Excel. What percentage error results from the use of the ideal-gas equation of state, taking the van der Waals estimate to be correct? (c) Set up a spreadsheet to carry out the calculations of Part (b) for air at \(223 \mathrm{K}\) and several pressures. The spreadsheet should appear as follows: The polynomial expression for \(\hat{V}\left(f=c_{3} \hat{V}^{3}+c_{2} \hat{V}^{2}+\cdots\right)\) should be entered in the \(f(V)\) column, and the value in the \(V\) column should be determined using Goal Seek or Solver in Excel.

A distillation column is being used to separate methanol and water at atmospheric pressure. The column temperature varies from approximately \(65^{\circ} \mathrm{C}\) at the top to \(100^{\circ} \mathrm{C}\) at the bottom. Liquid enters the top of the column and flows down to the bottom; vapor is generated in a reboiler at the bottom of the column, flows upward, and leaves at the top. The molar flow rate of vapor up the column may be assumed to be constant from top to bottom. The vapor velocity is kept below \(5.0 \mathrm{ft} / \mathrm{s}\) to keep the vapor from entraining liquid (suspending and carrying away liquid droplets). (a) Where in the column is the greatest risk of liquid entrainment? Explain your answer. (b) Assuming that the liquid flowing down the column and the column internals (equipment inside the column) occupy a negligible fraction of the column cross-sectional area, estimate the minimum column diameter if the vapor flow rate is 25.0 lb-mole/min. (c) Suppose the column is constructed with a diameter \(10 \%\) greater than that determined in Part (b). What are the vapor velocities at the top and bottom of the column if the vapor molar flow rate in both locations is 25.0 ib-mole/min? How much can the vapor molar flow rate be increased without causing liquid entrainment? (d) There is a need to increase process throughput, which would require the vapor molar flow rate to be doubled. It has been suggested that increasing the pressure in the column would allow that to be done without risking excessive liquid entrainment. Again applying a vapor velocity limit of \(5 \mathrm{ft} / \mathrm{s}\) what would the new pressure be?

The current global reliance on fossil fuels for heating, transportation, and electric power generation raises concems regarding the release of \(\mathrm{CO}_{2}\) and \(\mathrm{CH}_{4},\) which are greenhouse gases thought to lead to climate change, and NO, which contributes to smog. One potential solution to these problems is to produce transportation fuels from renewable biomass. You have been asked to evaluate a proposed process for converting forest residues to alcohols that may be used as transportation fuels. In the first stage of the process, steam and dry wood from hybrid poplar trees (which grow between five and eight feet a year and can be harvested roughly every five years) are fed to a gasifier in which the biomass is converted to light gases in the following reactions: $$\begin{aligned} \mathrm{C}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}+\mathrm{H}_{2} \\\ \mathrm{CO}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}_{2}+\mathrm{H}_{2} \\ \mathrm{C}+\mathrm{CO}_{2} & \rightarrow 2 \mathrm{CO} \\ \mathrm{C}+2 \mathrm{H}_{2} & \rightarrow \mathrm{CH}_{4} \\ \mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}+3 \mathrm{H}_{2} \end{aligned}$$ The effluents from the reactor are a gas stream containing \(\mathrm{H}_{2}, \mathrm{CO}, \mathrm{CO}_{2}, \mathrm{CH}_{4},\) and \(\mathrm{H}_{2} \mathrm{O},\) and a solid char stream that contains only carbon and hydrogen. The char is discarded and the gases go through additional steps in which the hydrogen and carbon monoxide are converted to mixed alcohols. This problem only concerns the gasifier. \(\cdot\) Elemental composition of biomass: 51.9 mass \(\%\) C \(, 6.3 \%\) H, and \(41.8 \%\) O \(\cdot\) Pressure and temperature of entering steam: \(155^{\circ} \mathrm{C}, 4.4 \mathrm{atm}\) \(\cdot\) Feed ratio of steam to biomass: 1.1 kg steam/kg biomass \(\cdot\) Yield and dry-basis composition of product gas: 1.35 kg dry gas/kg biomass at \(700^{\circ} \mathrm{C}, 1.2\) atm; 50.7 mol\% \(\mathrm{H}_{2}, 23.8 \%\) CO, \(18.0 \% \mathrm{CO}_{2}, 7.5 \% \mathrm{CH}_{4}\) (a) Taking a basis of \(100 \mathrm{kg}\) of biomass fed, draw and completely label a flowchart for the gasifier incorporating the given data, labeling the volumes of the steam fed and the gases produced. Perform a degree-of-freedom analysis. (b) Calculate the mass and mass composition of the char and the volumes of the steam feed and product gas streams. (c) List advantages and possible drawbacks of using biomass rather than petroleum as a fuel source.

Phosgene (CCl, O) is a colorless gas that was used as an agent of chemical warfare in World War I. It has the odor of new-mown hay (which is a good warning if you know the smell of new-mown hay). Pete Brouillette, an innovative chemical engincering student, came up with what he believed was an effective new process that utilized phosgene as a starting material. He immediately set up a reactor and a system for analyzing the reaction mixture with a gas chromatograph. To calibrate the chromatograph (i.e., to determine its response to a known quantity of phosgene), he evacuated a 15.0 cm length of tubing with an outside diameter of \(0.635 \mathrm{cm}\) and a wall thickness of \(0.559 \mathrm{mm}\), and then connected the tube to the outlet valve of a cylinder containing pure phosgene. The idea was to crack the valve, fill the tube with phosgene, close the valve, feed the tube contents into the chromatograph, and observe the instrument response. What Pete hadn't thought about (among other things) was that the phosgene was stored in the cylinder at a pressure high enough for it to be a liquid. When he opened the cylinder valve, the liquid rapidly flowed into the tube and filled it. Now he was stuck with a tube full of liquid phosgene at a pressure the tube was not designed to support. Within a minute he was reminded of a tractor ride his father had once given him through a hayfield, and he knew that the phosgene was leaking. He quickly ran out of the lab, called campus security, and told them that a toxic leak had occurred, that the building had to be evacuated, and the tube removed and disposed of properly. Personnel in air masks shortly appeared, took care of the problem, and then began an investigation that is still continuing. (a) Show why one of the reasons phosgene was an effective weapon is that it would collect in low spots soldiers often mistakenly entered for protection. (b) Pete's intention was to let the tube equilibrate at room temperature ( \(23^{\circ} \mathrm{C}\) ) and atmospheric pressure. How many gram-moles of phosgene would have been contained in the sample fed to the chromatograph if his plan had worked? (c) The laboratory in which Pete was working had a volume of \(2200 \mathrm{ft}^{3}\), the specific gravity of liquid phosgene is \(1.37,\) and Pete had read somewhere that the maximum "safe" concentration of phosgene in air is \(0.1 \mathrm{ppm}\) \(\left(0.1 \times 10^{-6} \mathrm{mol} \mathrm{CCl}_{2} \mathrm{O} / \mathrm{mol}\) air) \right. Would the "safe" concentration have been exceeded if all the liquid phosgene in the tube had evaporated into the room? Even if the limit would not have been exceeded, give several reasons why the lab would still have been unsafe. (d) List several things Pete did (or failed to do) that made his experiment unnecessarily hazardous.
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