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Chapter 4: Problem 42

\- An equimolar liquid mixture of benzene and toluene is separated into two product streams by distillation. A process flowchart and a somewhat oversimplified description of what happens in the process follow: Inside the column a liquid stream flows downward and a vapor stream rises. At each point in the column some of the liquid vaporizes and some of the vapor condenses. The vapor leaving the top of the column, which contains 97 mole\% benzene, is completely condensed and split into two equal fractions: one is taken off as the overhead product stream, and the other (the reflux) is recycled to the top of the column. The overhead product stream contains \(89.2 \%\) of the benzene fed to the column. The liquid leaving the bottom of the column is fed to a partial reboiler in which \(45 \%\) of it is vaporized. The vapor generated in the reboiler (the boilup) is recycled to become the rising vapor stream in the column, and the residual reboiler liquid is taken off as the bottom product stream. The compositions of the streams leaving the reboiler are governed by the relation $$\frac{y_{\mathrm{B}} /\left(1-y_{\mathrm{B}}\right)}{x_{\mathrm{B}} /\left(1-x_{\mathrm{B}}\right)}=2.25$$ where \(y_{\mathrm{B}}\) and \(x_{\mathrm{B}}\) are the mole fractions of benzene in the vapor and liquid streams, respectively. (a) Take a basis of 100 mol fed to the column. Draw and completely label a flowchart, and for each of four systems (overall process, column, condenser, and reboiler), do the degree-of-freedom analysis and identify a system with which the process analysis might appropriately begin (one with zero degrees of freedom). (b) Write in order the equations you would solve to determine all unknown variables on the flowchart, circling the variable for which you would solve in each equation. Do not do the calculations in this part. (c) Calculate the molar amounts of the overhead and bottoms products, the mole fraction of benzene in the bottoms product, and the percentage recovery of toluene in the bottoms product \((100 \times\) moles toluene in bottoms/mole toluene in feed).

Short Answer

Expert verified
The problem involves identification of a system with zero degrees of freedom, setting up mass and component balances and solving equations to determine unknown variables. Following computations, product streams are determined, leading to the determination of toluene recovery.

Step by step solution

01

Draw and Label Flowchart

Begin by drawing a process flowchart and labelling each part - feed stream, overhead product stream, bottom product stream, reflux and boilup. Assume a feed of 100 mol. The two main outputs are the overhead and bottoms product.
02

Degree of Freedom Analysis

The important systems identified are the overall process, the condenser, the column and the reboiler. Going by the rules of the degree-of-freedom analysis which involves comparing number of variables, equations, and specs, it is the condenser system that has zero degrees of freedom. It therefore provides the best starting point for process analysis. This is because as per given data, we can identify two equations and two variables (flowrates of the overhead product and reflux), giving a degree-of-freedom count of zero.
03

Write Equations for Unknown Variables

Begin by writing mass and component balance equations for each of the systems. The mass balance over the whole system yields F = D + B, where F, D and B are the molar flow rates of the feed, distillate and bottoms, respectively. Similarly, create component balances for benzene and toluene. Note down these equations and identify potential solutions. Remember not to solve anywhere at this step.
04

Solve Equations for Unknown Variables

The most important equation will be the compositional balance for benzene. The relationship given with respect to \(y_B\) and \(x_B\) can be used as the second equation. The two above-mentioned equations give us a pair of non-linear equations that may be solved simultaneously by numerical methods, providing us with specific values for \(y_B\) and \(x_B\), the mole fractions of benzene in the vapor and liquid streams respectively.
05

Calculate Toluene Recovery

This is a similar calculation where mole fraction is needed. The molar quantities of the overhead and bottom products and the mole fraction of benzene in the bottom product can be calculated from their molar flow rates and molar flow rate ratios. With these, the calculations for the percentage recovery of toluene in the bottom product can be made.

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Key Concepts

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

Degree of Freedom Analysis
In chemical processes like distillation, understanding the degree of freedom is crucial. It allows us to determine if we have enough information to solve for all unknown variables within a system. Essentially, the degree of freedom (DoF) is calculated as the difference between the number of independent equations and the number of unknown variables. For example, in this exercise focusing on a distillation column, several subsystems need to be evaluated, such as the overall process, the condenser, the column, and the reboiler. The goal is to identify a subsystem with zero degrees of freedom, which means you have a perfectly balanced number of equations and unknowns, so you can begin finding solutions efficiently. In our case, the condenser system fits this criterion, because there are two unknowns and two pieces of information or equations available (namely, the flow rates of the overhead product and the reflux). Keep this concept in mind:
  • If DoF = 0, you can immediately start solving equations.
  • If DoF > 0, more data or assumptions are needed.
  • If DoF < 0, the system is overspecified, indicating redundant data.
This methodology ensures comprehensive system analysis and helps in organizing solutions systematically, promoting a logical approach in breaking down complex problems.
Flowchart Labeling
Labeling a flowchart correctly is vital in understanding and solving process flow problems like distillation. Flowcharts visually represent process streams and equipment, making it easier to balance material and track flow throughout the operation. In this task, drawing and labeling the process flowchart accurately helps show how 100 moles of an equimolar benzene-toluene mixture enters the distillation column.
For effective flowcharting:
  • Identify and draw all significant streams: feed stream, overhead product stream, bottom product stream.
  • Indicate paths for recycling streams such as the reflux and the boilup.
  • Label each stream with its known or assumed values, including compositions, flow rates, and mole fractions.
Such a graphical representation helps visualize how the streams interact and offers a baseline from which degree of freedom analysis and balance equations can be conducted. A well-constructed flowchart is your roadmap for organizing data and efficiently developing a solution strategy.
Mass and Component Balance Equations
Mass and component balances are essential mathematical tools used to analyze chemical processes, especially in distillation systems. They involve setting up equations based on the conservation of mass principle, ensuring input equals output plus accumulation for any substance within a system.For a system like a distillation column, master the two types of balances:
  • **Overall Mass Balance**: Starting with the equation \( F = D + B \), where \( F \) is the feed, \( D \) is the distillate (overhead) product, and \( B \) is the bottoms product.
  • **Component Balance**: Each component (benzene and toluene in this case) must satisfy their own mass balance equations. For benzene, you may have something like \( Fz_F = Dy_D + Bx_B \), where \( z_F \) is the feed composition, and \( y_D \) and \( x_B \) are mole fractions in distillate and bottoms, respectively.
Component balance equations help dissect the compositions within each stream. Applying these equations to individual components aids in solving for unknowns like mole fractions and flow quantities. By using these techniques methodically, you can achieve a detailed understanding of how the system transforms its inputs into outputs, crucial for determining recovery rates and product specifications.

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Most popular questions from this chapter

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).

Mammalian cells can be cultured for a variety of purposes, including synthesis of vaccines. They must be maintained in growth media containing all of the components required for proper cellular function to ensure their survival and propagation. Traditionally, growth media were prepared by blending a powder, such as Dulbecco's Modified Eagle Medium (DMEM) with sterile deionized water. DMEM contains glucose, buffering agents, proteins, and amino acids. Using a sterile (i.e., bacterial-, fungal-,and yeast-free) growth medium ensures proper cell growth, but sometimes the water (or powder) can become contaminated, requiring the addition of antibiotics to eliminate undesired contaminants. The culture medium is supplemented with fetal bovine serum (FBS) that contains additional growth factors required by the cells. Suppose an aqueous stream (SG = 0.90) contaminated with bacteria is split, with 75\% being fed to a mixing unit to dissolve a powdered mixture of DMEM contaminated with the same bacteria found in the water. The ratio of impure feed water to powder entering the mixer is 4.4:1. The stream leaving the mixer (containing DMEM, water, and bacteria) is combined with the remaining 25\% of the aqueous stream and fed to a filtration unit to remove all of the bacteria that have contaminated the system, a total of \(20.0 \mathrm{kg}\). Once the bacteria have been removed, the sterile medium is combined with FBS and the antibiotic cocktail PSG (Penicillin-Streptomycin-L-Glutamine) in a shaking unit to generate 5000 L of growth medium (SG = 1.2). The final composition of the growth medium is 66.0 wt\% H_O, 11.0\% FBS, 8.0\% PSG, and the balance DMEM. (a) Draw and label the process flowchart. (b) Do a degree-of-freedom analysis around each piece of equipment (mixer, filter, and shaker), the splitter, the mixing point, and the overall system. Based on the analysis, identify which system or piece of equipment should be the starting point for further calculations. (c) Calculate all of the unknown process variables. (d) Determine a value for (i) the mass ratio of sterile growth medium product to feed water and (ii) the mass ratio of bacteria in the water to bacteria in the powder. (e) Suggest two reasons why the bacteria should be removed from the system.

Draw and label the given streams and derive expressions for the indicated quantities in terms of labeled variables. The solution of Part (a) is given as an illustration. (a) A continuous stream contains 40.0 mole\% benzene and the balance toluene. Write expressions for the molar and mass flow rates of benzene, \(\dot{n}_{\mathrm{B}}\left(\operatorname{mol} \mathrm{C}_{6} \mathrm{H}_{6} / \mathrm{s}\right)\) and \(\dot{m}_{\mathrm{B}}\left(\mathrm{kg} \mathrm{C}_{6} \mathrm{H}_{6} / \mathrm{s}\right),\) in terms of the total molar flow rate of the stream, \(\dot{n}(\mathrm{mol} / \mathrm{s})\) (b) The feed to a batch process contains equimolar quantities of nitrogen and methane. Write an expression for the kilograms of nitrogen in terms of the total moles \(n(\) mol) of this mixture. (c) A stream containing ethane, propane, and butane has a mass flow rate of \(100.0 \mathrm{g} / \mathrm{s}\). Write an expression for the molar flow rate of ethane, \(\dot{n}_{\mathrm{E}}\left(\text { Ib-mole } \mathrm{C}_{2} \mathrm{H}_{6} / \mathrm{h}\right)\), in terms of the mass fraction of this species, \(x_{\mathrm{E}}\). (d) A continuous stream of humid air contains water vapor and dry air, the latter containing approximately 21 mole \(\% \mathrm{O}_{2}\) and \(79 \% \mathrm{N}_{2}\). Write expressions for the molar flow rate of \(\mathrm{O}_{2}\) and for the mole fractions of \(\mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{O}_{2}\) in the gas in terms of \(\dot{n}_{1}\left(\mathrm{lb}-\mathrm{mole} \mathrm{H}_{2} \mathrm{O} / \mathrm{s}\right)\) and \(\dot{n}_{2}(\text { lb- mole dry air/s })\) (e) The product from a batch reactor contains \(\mathrm{NO}, \mathrm{NO}_{2},\) and \(\mathrm{N}_{2} \mathrm{O}_{4} .\) The mole fraction of \(\mathrm{NO}\) is 0.400. Write an expression for the gram-moles of \(\mathrm{N}_{2} \mathrm{O}_{4}\) in terms of \(n(\mathrm{mol}\) mixture) and \(y_{\mathrm{NO}_{2}}\left(\operatorname{mol} \mathrm{NO}_{2} / \mathrm{mol}\right)\)

A liquid mixture of acetone and water contains 35 mole\% acetone. The mixture is to be partially evaporated to produce a vapor that is 75 mole \(\%\) acetone and leave a residual liquid that is 18.7 mole \(\%\) (a) Suppose the process is to be carried out continuously and at steady state with a feed rate of 10.0 kmol/h. Let \(\dot{n}_{\mathrm{v}}\) and \(\dot{n}_{1}\) be the flow rates of the vapor and liquid product streams, respectively. Draw and label a process flowchart, then write and solve balances on total moles and on acetone to determine the values of \(\dot{n}_{\mathrm{v}}\) and \(\dot{n}_{\mathrm{l}}\). For each balance, state which terms in the general balance equation (accumulation \(=\)input \(+\)generation \(-\)output\(-\)consumption ) can be discarded and why. (See Example 4.2-2.) (b) Now suppose the process is to be carried out in a closed container that initially contains 10.0 kmol of the liquid mixture. Let \(n_{\mathrm{v}}\) and \(n_{1}\) be the moles of final vapor and liquid phases, respectively. Draw and label a process flowchart, then write and solve integral balances on total moles and on acetone. For each balance, state which terms of the general balance equation can be discarded and why. (c) Returning to the continuous process, suppose the vaporization unit is built and started and the product stream flow rates and compositions are measured. The measured acetone content of the vapor stream is 75 mole \(\%\) acetone, and the product stream flow rates have the values calculated in Part (a). However, the liquid product stream is found to contain 22.3 mole \(\%\) acetone. It is possible that there is an error in the measured composition of the liquid stream, but give at least five other reasons for the discrepancy. [Think about assumptions made in obtaining the solution of Part (a).]

Water enters a \(2.00-\mathrm{m}^{3}\) tank at a rate of \(6.00 \mathrm{kg} / \mathrm{s}\) and is withdrawn at a rate of \(3.00 \mathrm{kg} / \mathrm{s}\). The tank is initially half full. (a) Is this process continuous, batch, or semibatch? Is it transient or steady state? (b) Write a mass balance for the process (see Example 4.2-1). Identify the terms of the general balance equation (Equation 4.2-1) present in your equation and state the reason for omitting any terms. (c) How long will the tank take to overflow?
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