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Chapter 4: Problem 9

Inside a distillation column (see Problem 4.8), a downward-flowing liquid and an upward-flowing vapor maintain contact with each other. For reasons we will discuss in greater detail in Chapter \(6,\) the vapor stream becomes increasingly rich in the more volatile components of the mixture as it moves up the column, and the liquid stream is enriched in the less volatile components as it moves down. The vapor leaving the top of the column goes to a condenser. A portion of the condensate is taken off as a product (the overhead product), and the remainder (the reflux) is returned to the top of the column to begin its downward journey as the liquid stream. The condensation process can be represented as shown below: A distillation column is being used to separate a liquid mixture of ethanol (more volatile) and water (less volatile). A vapor mixture containing 89.0 mole \(\%\) ethanol and the balance water enters the overhead condenser at a rate of \(100 \mathrm{lb}\) -mole/h. The liquid condensate has a density of \(49.01 \mathrm{b}_{\mathrm{m}} / \mathrm{ft}^{3},\) and the reflux ratio is \(3 \mathrm{lb}_{\mathrm{m}}\) reflux/lb \(_{\mathrm{m}}\) overhead product. When the system is operating at steady state, the tank collecting the condensate is half full of liquid and the mean residence time in the tank (volume of liquid/volumetric flow rate of liquid) is 10.0 minutes. Determine the overhead product volumetric flow rate (ft \(^{3}\) /min) and the condenser tank volume (gal).

Short Answer

Expert verified
The overhead product volumetric flow rate is \(0.0085 \, ft^{3}/min\) and the condenser tank volume is \(2.54 \) gallons.

Step by step solution

01

Calculate Volumetric Flow Rate of the Condensate

First, convert the flow rate from lb-mole/h to lb_m/ft^3. utilizing the given density of the liquid condensate:\[ \frac{100 \, lb_{mole/h}}{49.01 \, lb_{m} / ft^{3}} = 2.04 \, ft^{3}/h \]Then, convert this to ft^3/min as:\[ \frac{2.04 \, ft^{3}}{60} = 0.034 \, ft^{3}/min \] This is the volumetric flow rate of the condensate.
02

Calculate Volumetric Flow Rate of Overhead Product

Use the given reflux ratio to find the overhead product flow rate. The reflux ratio is defined as lb_m reflux per lb_m overhead product, which is given as 3. Therefore, let's denote the flow rate of the overhead product as F_O, then\[ 0.034 \, ft^{3}/min = F_O + 3F_O \]Solving this we get:\[ F_O = 0.0085 \, ft^{3}/min \] This is the volumetric flow rate of the overhead product.
03

Calculate the Condenser Tank Volume

Use the given mean residence time in the tank (volume of liquid/volumetric flow rate of liquid) to find the tank volume. By rearranging the formula, we get\[ Volume = Residence \, time \times Flow \, rate\]Substitute the given values:\[ Volume = 10 \, min \times 0.034 \, ft^{3}/min = 0.34 \, ft^{3}\]In gallons (as 1ft^3 = 7.481 gal):\[ Volume = 0.34 \, ft^{3} \times 7.481 \, gal/ft^{3} = 2.54 \, gal \] This is the volume of the condenser tank.

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Key Concepts

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

Volumetric Flow Rate Calculation
Understanding the volumetric flow rate is crucial when analyzing distillation column operations. It indicates the volume of fluid passing through a given area in a specific period. The solution provided illustrates a straightforward method for calculating the flow rate: first by converting the mass flow rate using the fluid density, and then by adjusting the units of time for practical use.

When calculating flow rate from a given mass flow rate, you need to use the density of the liquid to find its volume, then scale the units as needed, in this case from hours to minutes. If the column's productivity were to change, or if you needed to design a new system, these basic calculations would lay the groundwork for scaling up or down the equipment and processes involved.
Reflux Ratio
The reflux ratio is a key parameter in the operation of a distillation column, representing the ratio of liquid sent back into the column compared to the liquid taken out as the product. A higher reflux ratio usually means better separation but also higher energy costs, as more liquid needs to be reheated.

In practice, this ratio is used to balance between operational costs and the desired purity of the end product. The exercise's step-by-step solution demonstrates how to use this ratio to determine the flow rate of the overhead product by setting up a simple linear equation. This example underscores the importance of the reflux ratio in managing the efficiency and economics of a separation process.
Condenser Design
The condenser design is a crucial aspect of a distillation column, affecting both product quality and energy efficiency. It cools and condenses the vapor from the top of the column into a liquid form that can be either collected as a product or returned to the column as reflux. The design needs to ensure sufficient surface area for heat exchange, and the capacity to handle the calculated volumetric flow rate of the vapor.

In the given problem, the condenser must accommodate the flow rate dictated by the reflux ratio. The size and effectiveness of the condenser determine the steady state operation, directly influencing the mean residence time and ultimate product consistency. Well-designed condensers are vital for both the economic and technical success of distillation processes.
Mean Residence Time
The mean residence time indicates how long, on average, a liquid stays in a system. It is a critical factor for designing and operating any kind of tank or reservoir within a chemical process, including the tank in this distillation column exercise. It influences not just the size of the tank but also product quality.

By multiplying the mean residence time by the flow rate of the liquid, operators can calculate the necessary volume of the tank, as shown in the solution steps. In industrial scenarios, an optimal residence time is essential for maintaining a balance between equipment size and the throughput of the distillation process.
Chemical Processes
Distillation is one of the most widely used chemical processes for separating mixtures based on differences in the volatilities of the mixture components. The process鈥檚 efficiency depends on a multitude of factors including but not limited to flow rates, reflux ratios, and equipment design. Each of these elements must be finely tuned to ensure the desired separation is achieved in the most cost-effective manner.

Chemical process principles such as those demonstrated in the distillation column operation also apply to other separation techniques and are fundamental knowledge for anyone involved in chemical engineering or process design. An intimate understanding of these principles allows for optimization and troubleshooting in various stages of the chemical process industry.

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Most popular questions from this chapter

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.

Ethane is chlorinated in a continuous reactor: $$\mathrm{C}_{2} \mathrm{H}_{6}+\mathrm{Cl}_{2} \rightarrow \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}+\mathrm{HCl}$$ Some of the product monochloroethane is further chlorinated in an undesired side reaction: $$\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}+\mathrm{Cl}_{2} \rightarrow \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Cl}_{2}+\mathrm{HCl}$$ (a) Suppose your principal objective is to maximize the selectivity of monochloroethane production relative to dichloroethane production. Would you design the reactor for a high or low conversion of ethane? Explain your answer. (Hint: If the reactor contents remained in the reactor long enough for most of the ethane in the feed to be consumed, what would the main product constituent probably be?) What additional processing steps would almost certainly be carried out to make the process economically sound? (b) Take a basis of \(100 \mathrm{mol} \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}\) produced. Assume that the feed contains only ethane and chlorine and that all of the chlorine is consumed, and carry out a degree-of-freedom analysis based on atomic species balances. (c) The reactor is designed to yield a \(15 \%\) conversion of ethane and a selectivity of \(14 \mathrm{mol} \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl} / \mathrm{mol}\) \(\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Cl}_{2},\) with a negligible amount of chlorine in the product gas. Calculate the feed ratio \(\left(\mathrm{mol} \mathrm{Cl}_{2} /\right.\) mol \(\mathrm{C}_{2} \mathrm{H}_{6}\) ) and the fractional yield of monochloroethane. (d) Suppose the reactor is built and started up and the conversion is only \(14 \% .\) Chromatographic analysis shows that there is no \(\mathrm{Cl}_{2}\) in the product but another species with a molecular weight higher than that of dichloroethane is present. Offer a likely explanation for these results.

A liquid-phase chemical reaction \(\mathrm{A} \rightarrow \mathrm{B}\) takes place in a well-stirred tank. The concentration of \(\mathrm{A}\) in the feed is \(C_{\mathrm{A} 0}\left(\operatorname{mol} / \mathrm{m}^{3}\right),\) and that in the tank and outlet stream is \(C_{\mathrm{A}}\left(\mathrm{mol} / \mathrm{m}^{3}\right) .\) Neither concentration varies with time. The volume of the tank contents is \(V\left(\mathrm{m}^{3}\right)\) and the volumetric flow rate of the inlet and outlet streams is \(\dot{V}\left(\mathrm{m}^{3} / \mathrm{s}\right)\). The reaction rate (the rate at which \(\mathrm{A}\) is consumed by reaction in the tank) is given by the expression $$r(\text { mol } A \text { consumed } / \mathrm{s})=k V C_{\mathrm{A}}$$ (a) Is this process continuous, batch, or semibatch? Is it transient or steady-state? (b) What would you expect the reactant concentration \(C_{\mathrm{A}}\) to equal if \(k=0\) (no reaction)? What should it approach if \(k \rightarrow \infty\) (infinitely rapid reaction)? (c) Write a differential balance on \(A,\) stating which terms in the general balance equation (accumulation = input + generation - output - consumption) you discarded and why you discarded them. Use the balance to derive the following relation between the inlet and outlet reactant concentrations: $$C_{\mathrm{A}}=\frac{C_{\mathrm{A} 0}}{1+k V / \dot{V}}$$ Verify that this relation predicts the results 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.

Carbon nanotubes (CNT) are among the most versatile building blocks in nanotechnology. These unique pure carbon materials resemble rolled-up sheets of graphite with diameters of several nanometers and lengths up to several micrometers. They are stronger than steel, have higher thermal conductivities than most known materials, and have electrical conductivities like that of copper but with higher currentcarrying capacity. Molecular transistors and biosensors are among their many applications. While most carbon nanotube research has been based on laboratory-scale synthesis, commercial applications involve large industrial-scale processes. In one such process, carbon monoxide saturated with an organo-metallic compound (iron penta-carbonyl) is decomposed at high temperature and pressure to form CNT, amorphous carbon, and CO_. Each "molecule" of CNT contains roughly 3000 carbon atoms. The reactions by which such molecules are formed are: In the process to be analyzed, a fresh feed of CO saturated with \(\mathrm{Fe}(\mathrm{CO})_{5}(\mathrm{v})\) contains \(19.2 \mathrm{wt} \%\) of the latter component. The feed is joined by a recycle stream of pure CO and fed to the reactor, where all of the iron penta-carbonyl decomposes. Based on laboratory data, \(20.0 \%\) of the CO fed to the reactor is converted, and the selectivity of CNT to amorphous carbon production is (9.00 kmol CNT/kmol C). The reactor effluent passes through a complex separation process that yields three product streams: one consists of solid \(\mathrm{CNT}, \mathrm{C},\) and \(\mathrm{Fe} ;\) a second is \(\mathrm{CO}_{2} ;\) and the third is the recycled \(\mathrm{CO}\). You wish to determine the flow rate of the fresh feed (SCM/h), the total CO_ generated in the process ( \(\mathrm{kg} / \mathrm{h}\) ), and the ratio (kmol CO recycled/kmol CO in fresh feed). (a) Take a basis of \(100 \mathrm{kmol}\) fresh feed. Draw and fully label a process flow chart and do degree-offreedom analyses for the overall process, the fresh-feed/recycle mixing point, the reactor, and the separation process. Base the analyses for reactive systems on atomic balances. (b) Write and solve overall balances, and then scale the process to calculate the flow rate (SCM/h) of fresh feed required to produce \(1000 \mathrm{kg} \mathrm{CNT} / \mathrm{h}\) and the mass flow rate of \(\mathrm{CO}_{2}\) that would be produced. (c) In your degree-of-freedom analysis of the reactor, you might have counted separate balances for C (atomic carbon) and O (atomic oxygen). In fact, those two balances are not independent, so one but not both of them should be counted. Revise your analysis if necessary, and then calculate the ratio (kmol CO recycled/kmol CO in fresh feed). (d) Prove that the atomic carbon and oxygen balances on the reactor are not independent equations.
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