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Chapter 4: Problem 40

A liquid mixture containing 30.0 mole \(\%\) benzene \((\mathrm{B}), 25.0 \%\) toluene \((\mathrm{T}),\) and the balance xylene \((\mathrm{X})\) is fed to a distillation column. The bottoms product contains 98.0 mole \(\% \mathrm{X}\) and no \(\mathrm{B},\) and \(96.0 \%\) of the \(\mathrm{X}\) in the feed is recovered in this stream. The overhead product is fed to a second column. The overhead product from the second column contains \(97.0 \%\) of the \(\mathrm{B}\) in the feed to this column. The composition of this stream is 94.0 mole\% B and the balance T. (a) Draw and label a flowchart of this process and do the degree-of-freedom analysis to prove that for an assumed basis of calculation, molar flow rates and compositions of all process streams can be calculated from the given information. Write in order the equations you would solve to calculate unknown process variables. In each equation (or pair of simultaneous equations), circle the variable(s) for which you would solve. Do not do the calculations. (b) Calculate (i) the percentage of the benzene in the process feed (i.e., the feed to the first column) that emerges in the overhead product from the second column and (ii) the percentage of toluene in the process feed that emerges in the bottom product from the second column.

Short Answer

Expert verified
By carefully examining the mole fractions of the given substances and applying rigorous mass balance over the whole system and individual columns, the unknown process variables can be calculated. The percentage of benzene in the process feed that emerges in the overhead product from the second column and the percentage of toluene in the process feed that emerges in the bottom product can be determined.

Step by step solution

01

Flowchart and Degrees of Freedom Analysis

Draw a flowchart with a clear indication of input and output streams of each column. The chemical species, B, T, and X would be the main component of each balance. Apply the equation for degrees of freedom, which is \(F = c - p + 2\). Here, c is the number of chemical components and p is the number of process variables. The process variables include output streams, molar flow rates, and compositions, which are required to be calculated.
02

Material Balance Equation Formulation

Apply the mass balance equation for each component over the whole system and for each column separately. The crucial assumption is that there is no loss of mass during the distillation process. Identify the unknown variable(s) in each equation or pair of simultaneous equations.
03

Calculation of Benzene in the process feed

This involves applying mass balances over the second column. Use the equation for benzene on the second column: \(F2B = F3B + B2B\), where F2B is the feed benzene to the second column (which also equals the molar flow rate of benzene in the overhead stream from the first column), F3B is the benzene in the overhead stream from the second column, and B2B is benzene in the bottom stream from the second column. Use the given percentage of benzene in the process feed to find the molar flow rate of benzene.
04

Calculation of Toluene in the process feed

The remaining benzene in the top product of the second column is mixed with toluene. By subtracting the amount of benzene from the total, the moles of toluene can be calculated. Use these values and the fact that toluene makes up the balance in this stream to calculate the percentage of toluene that emerges from the bottom product of the second column.

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Key Concepts

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

Degrees of Freedom Analysis
Understanding the degrees of freedom in a chemical process is crucial for determining the solvability of the system. For a distillation column, the degrees of freedom analysis helps to ascertain whether we have sufficient information to solve for all the unknown variables in the system, such as flow rates and compositions. This analysis involves a simple calculation: subtract the number of independent process constraints from the number of variables. In mathematical terms, it is given by the formula:
\( F = c - p + 2 \)
where \( F \) is the number of degrees of freedom, \( c \) is the number of components, and \( p \) is the number of process variables (streams, molar flow rates, compositions, etc).

In our distillation column example, we would count our components (B, T, X) and all process streams. By applying this formula, we establish the potential for a unique solution. However, if our degrees of freedom are not zero, it means that either additional information is needed or there are redundant specifications. Steps to improve a student's grasp on this concept might include practice in identifying independent equations and understanding the implication of having positive, zero, or negative degrees of freedom for a given system.
Mass Balance Equations
The mass balance equations are at the heart of material and energy balances in chemical engineering processes. These equations are based on the principle of conservation of mass, which states that mass cannot be created or destroyed. In the context of a distillation column, a mass balance would be written for each component and may look something like this:
\( \text{Input} - \text{Output} + \text{Generation} - \text{Consumption} = \text{Accumulation} \).

For a steady-state process, such as continuous distillation, the accumulation term is zero as the mass within the system is constant over time. Simplifying the equation for steady-state gives us:
\( \text{Input} = \text{Output} \).
Students might find it easier to understand these equations by breaking them down into individual component balances. This approach also simplifies complex systems into more manageable calculations. When applying this to our distillation problem, for each species, we would establish an equation that relates the mass flow of that species in various streams (inputs and outputs). The key is to clearly label all streams and apply the mass balance equation repetitively, once for each distinct component.
Chemical Engineering Processes
Chemical engineering processes like distillation are designed to separate components based on differences in volatility. Distillation columns are widely used in industry for purification tasks and consist of a series of stages where vapor-liquid equilibrium creates a separation effect. The process involves heating a mixture up to a temperature where one or more components boil and turn into a vapor. The vapor is richer in the more volatile component(s), and when it's cooled and condensed, the resultant liquid has a different composition from the feed.

To understand such complex processes, one should start by getting comfortable with the specifics of each unit operation. For students looking to improve their understanding, it's advised to study the physical and chemical principles that govern these operations, such as phase equilibria and thermodynamics. Consider the distillation columns in the problem provided: the first column is tasked with separating xylene, and the second one with separating benzene, each time refining the product streams to get closer to the desired purity. Understanding the role of each column in the context of the overall process, as well as how the columns are interconnected, is crucial for mastering chemical engineering problems.

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Most popular questions from this chapter

A stream of humid air containing 1.50 mole \(\% \mathrm{H}_{2} \mathrm{O}(\mathrm{v})\) and the balance dry air is to be humidified to a water content of 10.0 mole\% \(\mathrm{H}_{2} \mathrm{O}\). For this purpose, liquid water is fed through a flowmeter and evaporated into the air stream. The flowmeter reading, \(R\), is \(95 .\) The only available calibration data for the flowmeter are two points scribbled on a sheet of paper, indicating that readings \(R=15\) and \(R=50\) correspond to flow rates \(\dot{V}=40.0 \mathrm{ft}^{3} / \mathrm{h}\) and \(\dot{V}=96.9 \mathrm{ft}^{3} / \mathrm{h},\) respectively. (a) Assuming that the process is working as intended, draw and label the flowchart, do the degree-offreedom analysis, and estimate the molar flow rate (lb-mole/h) of the humidified (outlet) air if (i) the volumetric flow rate is a linear function of \(R\) and (ii) the reading \(R\) is a linear function of \(\dot{V}^{0.5}\) (b) Suppose the outlet air is analyzed and found to contain only \(7 \%\) water instead of the desired \(10 \%\) List as many possible reasons as you can think of for the discrepancy, concentrating on assumptions made in the calculation of Part (a) that might be violated in the real process.

Two aqueous sulfuric acid solutions containing \(20.0 \mathrm{wt} \% \mathrm{H}_{2} \mathrm{SO}_{4}(\mathrm{SG}=1.139)\) and \(60.0 \mathrm{wt} \% \mathrm{H}_{2} \mathrm{SO}_{4}\) (SG = 1.498) are mixed to form a 4.00 molar solution (SG = 1.213). (a) Calculate the mass fraction of sulfuric acid in the product solution. (b) Taking \(100 \mathrm{kg}\) of the \(20 \%\) feed solution as a basis, draw and label a flowchart of this process, labeling both masses and volumes, and do the degree-of-freedom analysis. Calculate the feed ratio (liters 20\% solution/liter 60\% solution). (c) What feed rate of the \(60 \%\) solution (L/h) would be required to produce \(1250 \mathrm{kg} / \mathrm{h}\) of the product?

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).]

The respiratory process involves hemoglobin (Hgb), an iron-containing compound found in red bloodcells. In the process, carbon dioxide diffuses from tissue cells as molecular \(\mathrm{CO}_{2}\), while \(\mathrm{O}_{2}\) simultaneously enters the tissue cells. A significant fraction of the \(\mathrm{CO}_{2}\) leaving the tissue cells enters red blood cells and reacts with hemoglobin; the \(\mathrm{CO}_{2}\) that does not enter the red blood cells ( \((\mathrm{D}\) in the figure below) remains dissolved in the blood and is transported to the lungs. Some of the \(\mathrm{CO}_{2}\) entering the red blood cells reacts with hemoglobin to form a compound (Hgb. \(\mathrm{CO}_{2} ;(\) 2) in the figure). When the red blood cells reach the lungs, the Hgb.CO_ dissociates, releasing free CO_ Meanwhile, the CO_ that enters the red blood cells but does not react with hemoglobin combines with water to form carbonic acid, \(\mathrm{H}_{2} \mathrm{CO}_{3},\) which then dissociates into hydrogen ions and bicarbonate ions ( (3) in the figure). The bicarbonate ions diffuse out of the cells ( (4) in the figure), and the ions are transported to the lungs via the bloodstream. For adult humans, every deciliter of blood transports a total of \(1.6 \times 10^{-4}\) mol of carbon dioxide in its various forms (dissolved \(\mathrm{CO}_{2}, \mathrm{Hgb} \cdot \mathrm{CO}_{2},\) and bicarbonate ions) from tissues to the lungs under normal, resting conditions. Of the total \(\mathrm{CO}_{2}, 1.1 \times 10^{-4}\) mol are transported as bicarbonate ions. In a typical resting adult human, the heart pumps approximately 5 liters of blood per minute. You have been asked to determine how many moles of \(\mathrm{CO}_{2}\) are dissolved in blood and how many moles of \(\mathrm{Hgb} \cdot \mathrm{CO}_{2}\) are transported to the lungs during an hour's worth of breathing. (a) Draw and fully label a flowchart and do a degree-of-freedom analysis. Write the chemical reactions that occur, and generate, but do not solve, a set of independent equations relating the unknown variables on the flowchart. (b) If you have enough information to obtain a unique numerical solution, do so. If you do not have enough information, identify a specific piece/pieces of information that (if known) would allow you to solve the problem, and show that you could solve the problem if that information were known. (c) When someone loses a great deal of blood due to an injury, they "go into shock": their total blood volume is low, and carbon dioxide is not efficiently transported away from tissues. The carbon dioxide reacts with water in the tissue cells to produce very high concentrations of carbonic acid, some of which can dissociate (as shown in this problem) to produce high levels of hydrogen ions. What is the likely effect of this occurrence on the blood pH near the tissue and the tissue cells? How is this likely to affect the injured person?

Methanol is formed from carbon monoxide and hydrogen in the gas-phase reaction The mole fractions of the reactive species at equilibrium satisfy the relation where \(P\) is the total pressure (atm), \(K_{c}\) the reaction equilibrium constant (atm \(^{-2}\) ), and \(T\) the temperature (K). The equilibrium constant \(K_{c}\) equals 10.5 at 373 K, and \(2.316 \times 10^{-4}\) at \(573 \mathrm{K}\). A semilog plot of \(K_{\mathrm{c}}\) (logarithmic scale) versus 1/ \(T\) (rectangular scale) is approximately linear between \(T=300 \mathrm{K}\) and \(T=600 \mathrm{K}\) (a) Derive a formula for \(K_{\mathrm{c}}(T),\) and use it to show that \(K_{\mathrm{e}}(450 \mathrm{K})=0.0548 \mathrm{atm}^{-2}\) (b) Write expressions for \(n_{A}, n_{B},\) and \(n_{C}\) (gram-moles of each species), and then \(y_{A}, y_{B},\) and \(y_{C},\) in terms of \(n_{\mathrm{A} 0}, n_{\mathrm{B} 0}, n_{\mathrm{C} 0},\) and \(\xi,\) the extent of reaction. Then derive an equation involving only \(n_{\mathrm{A} 0}, n_{\mathrm{B} 0}, n_{\mathrm{C} 0}, P, T,\) and \(\xi_{e},\) where \(\xi_{e}\) is the extent of reaction at equilibrium. (c) Suppose you begin with equimolar quantities of CO and \(\mathrm{H}_{2}\) and no \(\mathrm{CH}_{3} \mathrm{OH}\), and the reaction proceeds to equilibrium at 423 K and 2.00 atm. Calculate the molar composition of the product ( \(y_{\mathrm{A}}\), \(\left.y_{\mathrm{B}}, \text { and } y_{\mathrm{C}}\right)\) and the fractional conversion of \(\mathrm{CO}\) (d) The conversion of CO and \(\mathrm{H}_{2}\) can be enhanced by removing methanol from the reactor while leaving unreacted CO and \(\mathrm{H}_{2}\) in the vessel. Review the equations you derived in solving Part (c) and determine any physical constraints on \(\xi_{c}\) associated with \(n_{\mathrm{A} 0}=n_{\mathrm{B} 0}=1\) mol. Now suppose that 90\% of the methanol is removed from the reactor as it is produced; in other words, only 10\% of the methanol formed remains in the reactor. Estimate the fractional conversion of CO and the total gram moles of methanol produced in the modified operation. (e) Repeat Part (d), but now assume that \(n_{\mathrm{B} 0}=2\) mol. Explain the significant increase in fractional conversion of CO. (f) Write a set of equations for \(y_{\mathrm{A}}, y_{\mathrm{B}}, y_{\mathrm{C}},\) and \(f_{\mathrm{A}}\) (the fractional conversion of \(\mathrm{CO}\) ) in terms of \(y_{\mathrm{A} 0}, y_{\mathrm{B} 0}, T,\) and \(P(\) the reactor temperature and pressure at equilibrium). Enter the equations in an equation-solving program. Check the program by running it for the conditions of Part (c), then use it to determine the effects on \(f_{\mathrm{A}}\) (increase, decrease, or no effect) of separately increasing, (i) the fraction of \(\mathrm{CH}_{3} \mathrm{OH}\) in the feed, (ii) temperature, and (iii) pressure.
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