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Chapter 4: Problem 15

Two streams flow into a 500 -gallon tank. The first stream is 10.0 wt\% ethanol and \(90.0 \%\) hexane (the mixture density, \(\rho_{1},\) is \(0.68 \mathrm{g} / \mathrm{cm}^{3}\) ) and the second is \(90.0 \mathrm{wt} \%\) ethanol, \(10.0 \%\) hexane \(\left(\rho_{2}=0.78 \mathrm{g} / \mathrm{cm}^{3}\right) .\) After the tank has been filled, which takes 22 \(\mathrm{min}\), an analysis of its contents determines that the mixture is 60.0 wt\% ethanol, \(40.0 \%\) hexane. You wish to estimate the density of the final mixture and the mass and volumetric flow rates of the two feed streams. (a) Draw and label a flowchart of the mixing process and do the degree-of- freedom analysis. (b) Perform the calculations and state what you assumed.

Short Answer

Expert verified
The solution involves creating a flowchart and carrying out degree-of-freedom analysis, formulating equations based on mass and volume conservation, and finally solving these equations to find the mass flow rates of the feed streams and the density of the resulting mixture.

Step by step solution

01

Draw and analyze the flowchart

First, a flowchart is to be drawn, showing all the inputs and outputs. There are two inputs, stream 1 and 2, and one output, the mixture in the tank. Each stream has its concentration of ethanol and hexane given. The mix in the tank also has its concentration specified. A degree-of-freedom analysis will show that there are 5 components (ethanol and hexane in both streams and the tank) and 3 independent equations (two for conservation of mass and one relating mass and volume for the mixture). Thus, there are \(5-3 = 2\) degrees of freedom, indicating there are enough data to solve the problem with 2 unknown variables.
02

Mathematical formulation for the mixture density

The mass (m1 + m2) of the mixture in the tank is found assuming volume is conserved. The volume is 500 gallons, which is equivalent to 1892,705924 Liters (or 1892705,924 cm鲁, since 1L = 1000 cm鲁). So total mass is the sum of ethanol and hexane and can be calculated using the equation \( m = V * \rho \), where \( m \) is mass, \( V \) is volume, and \( \rho \) is density. So, \( m = 1892705,924 * \rho \) where \( \rho \) is the final mixture density which is unknown.
03

Calculating Mass Flow Rates of the Feed Streams

Recognized that the mass of the mixture is made up of the streams flowing in, so \( m = m1_{ethanol} + m2_{ethanol} + m1_{hexane} + m2_{hexane} \). Given the mass fractions of ethanol and hexane in each flow streams, the mass flow rates (g/min) can be expressed. The volumetric flow rates (cm鲁/min) is found by dividing the mass flow rate by its respective density.
04

Solving the Equation

Solving the equations formed would give the values of the mass flow rates of the streams and the density of the mixture, which are the unknown variables.

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Key Concepts

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

Mass and Energy Balance
Understanding mass and energy balances is fundamental in analyzing any chemical process. It is based on the law of conservation of mass and energy, which states that, within a closed system, mass and energy can neither be created nor destroyed, only converted from one form to another. In our exercise, we performed a mass balance to determine the composition and flow rates of a mixture based on the amount of ethanol and hexane from two separate streams entering a tank.

Determining the mass balance requires accounting for all the mass entering and leaving the system. In this case, we know the contents and volume of the tank, the composition of the inputs, and the time it takes to fill. With this information, we can calculate the overall mass of each component within the tank by using the mass balance equation:

\( m_{total} = m_{inlet_1} + m_{inlet_2} - m_{outlet} \).

Since the tank is just being filled and has no outlet stream, the mass of ethanol and hexane is simply the sum of what has been introduced into the tank. This calculation is essential for quantifying the mass flow rates of individual streams.
Density Calculation
The concept of density plays a critical role in determining the characteristics of a mixture within a process. Density is defined as mass per unit volume (\( \rho = \frac{m}{V} \)). In our exercise, to compute the density of the final mixture, the total mass of the mixture was first determined. Since the volume of the tank is known (500 gallons), the density calculation can be inferred using the derived mass balance information.

To find the density (\

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Most popular questions from this chapter

Effluents from metal-finishing plants have the potential of discharging undesirable quantities of metals, such as cadmium, nickel, lead, manganese, and chromium, in forms that are detrimental to water and air quality. A local metal-finishing plant has identified a wastewater stream that contains 5.15 wt\% chromium (Cr) and devised the following approach to lowering risk and recovering the valuable metal. The wastewater stream is fed to a treatment unit that removes \(95 \%\) of the chromium in the feed and recycles it to the plant. The residual liquid stream leaving the treatment unit is sent to a waste lagoon. The treatment unit has a maximum capacity of 4500 kg wastewater/h. If wastewater leaves the finishing plant at a rate higher than the capacity of the treatment unit, the excess (anything above \(4500 \mathrm{kg} / \mathrm{h}\) ) bypasses the unit and combines with the residual liquid leaving the unit, and the combined stream goes to the waste lagoon. (a) Without assuming a basis of calculation, draw and label a flowchart of the process. (b) Wastewater leaves the finishing plant at a rate \(\dot{m}_{1}=6000 \mathrm{kg} / \mathrm{h}\). Calculate the flow rate of liquid to the waste lagoon, \(\dot{m}_{6}(\mathrm{kg} / \mathrm{h}),\) and the mass fraction of \(\mathrm{Cr}\) in this liquid, \(x_{6}(\mathrm{kg} \mathrm{Cr} / \mathrm{kg})\) (c) Calculate the flow rate of liquid to the waste lagoon and the mass fraction of Crin this liquid for \(\dot{m}_{1}\) varying from \(1000 \mathrm{kg} / \mathrm{h}\) to \(10,000 \mathrm{kg} / \mathrm{h}\) in \(1000 \mathrm{kg} / \mathrm{h}\) increments. Generate a plot of \(x_{6}\) versus \(\dot{m}_{1}\). (Suggestion: Use a spreadsheet for these calculations.) (d) The company has hired you as a consultant to help them determine whether or not to add capacity to the treatment unit to increase the recovery of chromium. What would you need to know to make this determination? (e) What concerns might need to be addressed regarding the waste lagoon?

Ethylene oxide is produced by the catalytic oxidation of ethylene: $$ 2 \mathrm{C}_{2} \mathrm{H}_{4}+\mathrm{O}_{2} \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O} $$ An undesired competing reaction is the combustion of ethylene: $$ \mathrm{C}_{2} \mathrm{H}_{4}+3 \mathrm{O}_{2} \longrightarrow 2 \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O} $$ The feed to the reactor (not the fresh feed to the process) contains 3 moles of ethylene per mole of oxygen. The single-pass conversion of ethylene is \(20 \%,\) and for every 100 moles of ethylene consumed in the reactor, 90 moles of ethylene oxide emerge in the reactor products. A multiple-unit process is used to separate the products: ethylene and oxygen are recycled to the reactor, ethylene oxide is sold as a product, and carbon dioxide and water are discarded. (a) Assume a quantity of the reactor feed stream as a basis of calculation, draw and label the flowchart, perform a degree-of-freedom analysis, and write the equations you would use to calculate (i) the molar flow rates of ethylene and oxygen in the fresh feed, (ii) the production rate of ethylene oxide, and (iii) the overall conversion of ethylene. Do no calculations. (b) Calculate the quantities specified in Part (a), either manually or with an equation-solving program. (c) Calculate the molar flow rates of ethylene and oxygen in the fresh feed needed to produce 1 ton per hour of ethylene oxide.

The hormone estrogen is produced in the ovaries of females and elsewhere in the body in men and postmenopausal women, and it is also administered in estrogen replacement therapy, a common treatment for women who have undergone a hysterectomy. Unfortunately, it also binds to estrogen receptors in breast tissue and can activate cells to become cancerous. Tamoxifen is a drug that also binds to estrogen receptors but does not activate cells, in effect blocking the receptors from access to estrogen and inhibiting the growth of breast-cancer cells. Tamoxifen is administered in tablet form. In the manufacturing process, a finely ground powder contains tamoxifen (tam) and two inactive fillers- -lactose monohydrate (lac) and corn starch (cs). The powder is mixed with a second stream containing water and suspended solid particles of polyvinylpymolidone (pvp) binder, which keeps the tablets from easily crumbling. The slurry leaving the mixer goes to a dryer, in which 94.2\% of the water fed to the process is vaporized. The wet powder leaving the dryer contains 8.80 wr\% tam, 66.8\% lac, 21.4\% cs, 2.00\% pvp, and 1.00\% water. After some additional processing, the powder is molded into tablets. To produce a hundred thousand tablets, 17.13 kg of wet powder is required. (a) Taking a basis of 100,000 tablets produced, draw and label a process flowchart, labeling masses of individual components rather than total masses and component mass fractions. It is unnecessary to label the stream between the mixer and the dryer. Carry out a degree-of-freedom analysis of the overall two-unit process. (b) Calculate the masses and compositions of the streams that must enter the mixer to make 100,000 tablets. (c) Why was it unnecessary to label the stream between the mixer and the dryer? Under what circumstances would it have been necessary? (d) Go back to the flowchart of Part (a). Without using the mass of the wet powder (17.13 kg) or any of the results from Part (b) in your calculations, determine the mass fractions of the stream components in the powder fed to the mixer and verify that they match your solution to Part (b). (Hint: Take a basis of \(100 \mathrm{kg}\) of wet powder.) (e) Suppose a student does Part (d) before Part (b), and re-labels the powder feed to the mixer on the flowchart of Part (a) with an unknown total mass ( \(m_{1}\) ) and the three now known mole fractions. (Sketch the resulting flowchart.) The student then does a degree-of-freedom analysis, counts four unknowns (the masses of the powder, pvp, and water fed to the mixer, and the mass of water evaporated in the dryer), and six equations (five material balances for five species and the percentage evaporation), for a net of -2 degrees of freedom. since there are more equations than unknowns, it should not be possible to get a unique solution for the four unknowns. Nevertheless, the student writes four equations, solves for the four unknowns, and verifies that all of the balance equations are satisfied. There must have been a mistake in the degree-of-freedom calculation. What was it?

The gas-phase reaction between methanol and acetic acid to form methyl acetate and water takes place in a batch reactor. When the reaction mixture comes to equilibrium, the mole fractions of the four reactive species are related by the reaction equilibrium constant $$K_{y}=\frac{y_{C} y_{D}}{y_{A} y_{B}}=4.87$$ (a) Suppose the feed to the reactor consists of \(n_{\mathrm{A} 0}, n_{\mathrm{B} 0}, n_{\mathrm{C} 0}, n_{\mathrm{D} 0},\) and \(n_{10}\) gram-moles of \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},\) and an inert gas, I, respectively. Let \(\xi\) be the extent of reaction. Write expressions for the gram-moles of each reactive species in the final product, \(n_{\mathrm{A}}(\xi), n_{\mathrm{B}}(\xi), n_{\mathrm{C}}(\xi),\) and \(n_{\mathrm{D}}(\xi) .\) Then use these expressions and the given equilibrium constant to derive an equation for \(\xi_{c}\), the equilibrium extent of reaction, in terms of \(\left.n_{\mathrm{A} 0}, \ldots, n_{10} . \text { (see Example } 4.6-2 .\right)\) (b) If the feed to the reactor contains equimolar quantities of methanol and acetic acid and no other species, calculate the equilibrium fractional conversion. (c) It is desired to produce 70 mol of methyl acetate starting with 75 mol of methanol. If the reaction proceeds to equilibrium, how much acetic acid must be fed? What is the composition of the final product? (d) Suppose it is important to reduce the concentration of methanol by making its conversion at equilibrium as high as possible, say 99\%. Again assuming the feed to the reactor contains only methanol and acetic acid and that it is desired to produce 70 mol of methyl acetate, determine the extent of reaction and quantities of methanol and acetic acid that must be fed to the reactor. (e) If you wanted to carry out the process of Part (b) or (c) commercially, what would you need to know besides the equilibrium composition to determine whether the process would be profitable? (List several things.)

One thousand kilograms per hour of a mixture containing equal parts by mass of methanol and water is distilled. Product streams leave the top and the bottom of the distillation column. The flow rate of the bottom stream is measured and found to be \(673 \mathrm{kg} / \mathrm{h}\), and the overhead stream is analyzed and found to contain 96.0 wt\% methanol. (a) Draw and label a flowchart of the process and do the degree-of-freedom analysis. (b) Calculate the mass and mole fractions of methanol and the molar flow rates of methanol and water in the bottom product stream. (c) Suppose the bottom product stream is analyzed and the mole fraction of methanol is found to be significantly higher than the value calculated in Part (b). List as many possible reasons for the discrepancy as you can think of. Include in your list possible violations of assumptions made in Part (b).
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