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Chapter 4: Problem 45

An evaporation-crystallization process of the type described in Example \(4.5-2\) is used to obtain solid potassium sulfate from an aqueous solution of this salt. The fresh feed to the process contains 19.6 wt\% \(\mathrm{K}_{2} \mathrm{SO}_{4}\). The wet filter cake consists of solid \(\mathrm{K}_{2} \mathrm{SO}_{4}\) crystals and a \(40.0 \mathrm{wt} \% \mathrm{K}_{2} \mathrm{SO}_{4}\) solution, in a ratio \(10 \mathrm{kg}\) crystals/kg solution. The filtrate, also a \(40.0 \%\) solution, is recycled to join the fresh feed. Of the water fed to the evaporator, 45.0\% is evaporated. The evaporator has a maximum capacity of 175 kg water evaporated/s. (a) Assume the process is operating at maximum capacity. Draw and label a flowchart and do the degree-of-freedom analysis for the overall system, the recycle-fresh feed mixing point, the evaporator, and the crystallizer. Then write in an efficient order (minimizing simultaneous equations) the equations you would solve to determine all unknown stream variables. In each equation, circle the variable for which you would solve, but don't do the calculations. (b) Calculate the maximum production rate of solid \(\mathrm{K}_{2} \mathrm{SO}_{4}\), the rate at which fresh feed must be supplied to achieve this production rate, and the ratio kg recycle/kg fresh feed. (c) Calculate the composition and feed rate of the stream entering the crystallizer if the process is scaled to 75\% of its maximum capacity. (d) The wet filter cake is subjected to another operation after leaving the filter. Suggest what it might be. Also, list what you think the principal operating costs for this process might be. (e) Use an equation-solving computer program to solve the equations derived in Part (a). Verify that you get the same solutions determined in Part (b).

Short Answer

Expert verified
The problem is solved by applying concepts of mass balance, degree-of-freedom analysis, and using a mathematical approach to formulate and solve equations. The production rate, fresh feed rate, and recycle ratio are calculated accordingly. The feed rate after scaling down is also determined. Drying could be a possible additional operation for the wet filter cake. A software program could be used to verify manual calculations.

Step by step solution

01

Drawing Flow Diagram and Degree of Freedom Analysis

Before starting calculations, one should carefully read the problem and draw a process flow diagram. This will visualize the process, indicating all streams, their compositions, and flow rates (if provided). Then perform the degree-of-freedom analysis for the overall system, mixing point, evaporator and crystallizer respectively, following the equation; Degree of Freedom = Unknowns - Independent Equations. After finding the equations, organize them systematically to minimize simultaneous equations.
02

Writing Balance Equations

Based on the problem and the diagram, write down all the necessary material balances. The equations will be: \n- Overall balance: Into plants = Out of plants. \n- Balance over mixing point: Input = Output. \n- Balance over evaporator: Input = Output + Evaporated. \n- Balance over crystallizer: Input = Output + Evaporated. For each equation, circle the variable to solve later. It's important not to calculate these unknowns yet as instructed by the problem.
03

Calculating Maximum Production Rate

Now calculate the maximum production rate of solid \(\mathrm{K}_{2}\mathrm{SO}_{4}\) using the given evaporator capacity and overall mass balances. Next, find the rate at which the fresh feed must be supplied to achieve this production rate, considering that the filtrate is recycled to join the fresh feed.
04

Scaling Down The Process

The exercise then asks to scale down the process to 75\% of its maximum capacity. For this part calculate the composition and feed rate of the stream entering the crystallizer. This would be the scaling fraction of the previously calculated value.
05

Post-Filter Operation and Cost Assumptions

Based on the composition of the wet filter cake (solid crystals and solution), suggest a potential additional operation after leaving the filter. Possible answer could be a drying process to further remove the solution and purify the solid \(\mathrm{K}_{2}\mathrm{SO}_{4}\). On the topic of the principal operating costs, one should consider variable costs (raw materials, energy etc.), fixed costs (labor, maintenance etc.), and possibly capital costs (equipment purchase, upgrades etc.).
06

Computer Program for Solutions Verification

Lastly, use an equation-solving computer program like MATLAB or Python to solve the equations derived in Part (a). Verify these calculations with the solutions determined in Part (b). The aim is to confirm the accuracy of the manual calculations by comparing them to the results from software.

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Key Concepts

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

Degree of Freedom Analysis

Understanding the Degree of Freedom (DoF) analysis is crucial for engineers and students working on process systems. The DoF analysis aids in identifying whether sufficient information is available to solve for all unknown variables in a system. The underlying principle is quite simple: Degree of Freedom = Unknowns - Independent Equations. This equation is applied to different parts of the process鈥攐verall system, mixing point, evaporator, and crystallizer鈥攊n the given exercise. By conducting a DoF analysis, we can organize the system equations efficiently, aiming to minimize simultaneous equations. This organization streamlines the problem-solving process as it helps to single out one variable at a time.

For ease of understanding, let's take a metaphorical detour - imagine you're trying to solve a jigsaw puzzle. The DoF analysis would be akin to sorting out the edge pieces first. This method does not complete the puzzle, but it does give you a framework to work within, simplifying the subsequent steps. In the context of a chemical process, figuring out the 'edge pieces' involves deducing which variables can be directly solved from each equation and the order in which to tackle them.

It's imperative to note, as per the exercise improvement advice, that drawing a clear and detailed flowchart is an invaluable first step. It visualizes the process and naturally leads to a better understanding of the DoF analysis. The flowchart shows where materials enter, exit, and are recycled, which are the keys to identifying the unknowns and the equations required to solve for them.

Material Balance Equations

Material balance equations are the foundation of process engineering, and when applied to an evaporation-crystallization process, they dictate the relationship between feed, output, and losses or gains in the cycle. These equations embody the principle of mass conservation, implying matter cannot be created nor destroyed within the process. In the context of this exercise, the main equations for material balance are: 1) the overall balance, 2) balance over the mixing point, 3) balance over the evaporator, and 4) balance over the crystallizer. Simplified, these equations can be expressed as:

Input = Output + Accumulation - Consumption,
where Accumulation and Consumption terms can be zero in steady state processes like this one.

When solving these equations, it's crucial to circle the variable you're solving for and systematically work through the equations in a logical sequence. Keep in mind that understanding the process flow is essential when setting up these balances, as you need to account for all possible entries and exits of material, including any recycles or by-products.

As suggested in the exercise improvement advice, it's important before solving the equations to have a full comprehension of the process, reflected in a detailed flow diagram. This allows for a visual representation of the mass flows and stream compositions which are integral when formulating and solving the material balance equations.

Maximum Production Rate Calculation

The maximum production rate calculation tells us how efficiently a system can operate under given constraints. In our exercise, the maximum capacity of the evaporator limits the system. To calculate the maximum production rate of solid potassium sulfate (K_{2}SO_{4}), we rely on the known maximum water evaporation rate of the evaporator and the intricacies of the material balances laid out earlier.

The key step is to apply these balances in the context of the system's maximum capacity, transitioning from theoretical formulas to tangible figures that describe the system's optimal throughput. By pinpointing the maximum water evaporation, we indirectly find the maximum amount of K_{2}SO_{4} that can be produced.

The calculation also helps in determining the necessary rate of fresh feed supply and the proportion of recycled filtrate in the mix. Moreover, following the advice of improving exercises, understanding the dynamics between fresh feed and recycle rates is fundamental, as the ratio affects the process economics and efficiency. When scaling down the process, as in part (c) of the problem, this relationship becomes even more apparent as we adjust to a target capacity, 75% of the maximum in this case.

It's important to note that these calculations are crucial for planning and optimizing real-world operations, providing insights into how changes in one part of the process can influence the entire system.

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Most popular questions from this chapter

Methane and oxygen react in the presence of a catalyst to form formaldehyde. In a parallel reaction, methane is oxidized to carbon dioxide and water: $$\begin{aligned} \mathrm{CH}_{4}+\mathrm{O}_{2} & \rightarrow \mathrm{HCHO}+\mathrm{H}_{2} \mathrm{O} \\ \mathrm{CH}_{4}+2 \mathrm{O}_{2} & \rightarrow \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O} \end{aligned}$$ The feed to the reactor contains equimolar amounts of methane and oxygen. Assume a basis of \(100 \mathrm{mol}\) feed/s. (a) Draw and label a flowchart. Use a degree-of-freedom analysis based on extents of reaction to determine how many process variable values must be specified for the remaining variable values to be calculated. (b) Use Equation 4.6-7 to derive expressions for the product stream component flow rates in terms of the two extents of reaction, \(\xi_{1}\) and \(\xi_{2}\) (c) The fractional conversion of methane is 0.900 and the fractional yield of formaldehyde is 0.855 . Calculate the molar composition of the reactor output stream and the selectivity of formaldehyde production relative to carbon dioxide production. (d) A classmate of yours makes the following observation: "If you add the stoichiometric equations for the two reactions, you get the balanced equation $$2 \mathrm{CH}_{4}+3 \mathrm{O}_{2} \rightarrow \mathrm{HCHO}+\mathrm{CO}_{2}+3 \mathrm{H}_{2} \mathrm{O}$$ The reactor output must therefore contain one mole of \(\mathrm{CO}_{2}\) for every mole of HCHO, so the selectivity of formaldehyde to carbon dioxide must be \(1.0 .\) Doing it the way the book said to do it, \(I\) got a different selectivity. Which way is right, and why is the other way wrong?" What is your response?

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?

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 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.

The popularity of orange juice, especially as a breakfast drink, makes this beverage an important factor in the economy of orange-growing regions. Most marketed juice is concentrated and frozen and then reconstituted before consumption, and some is "not-from-concentrate." Although concentrated juices are less popular in the United States than they were at one time, they still have a major segment of the market for orange juice. The approaches to concentrating orange juice include evaporation, freeze concentration, and reverse osmosis. Here we examine the evaporation process by focusing only on two constituents in the juice: solids and water. Fresh orange juice contains approximately 10 wt\% solids (sugar, citric acid, and other ingredients) and frozen concentrate contains approximately 42 wt\% solids. The frozen concentrate is obtained by evaporating water from the fresh juice to produce a mixture that is approximately 65 wt\% solids. However, so that the flavor of the concentrate will closely approximate that of fresh juice, the concentrate from the evaporator is blended with fresh orange juice (and other additives) to produce a final concentrate that is approximately 42 wt\% solids. (a) Draw and label a flowchart of this process, neglecting the vaporization of everything in the juice but water. First prove that the subsystem containing the point where the bypass stream splits off from the evaporator feed has one degree of freedom. (If you think it has zero degrees, try determining the unknown variables associated with this system.) Then perform the degree- offreedom analysis for the overall system, the evaporator, and the bypass- evaporator product mixing point, and write in order the equations you would solve to determine all unknown stream variables. In each equation, circle the variable for which you would solve, but don't do any calculations. (b) Calculate the amount of product (42\% concentrate) produced per 100 kg fresh juice fed to the process and the fraction of the feed that bypasses the evaporator. (c) Most of the volatile ingredients that provide the taste of the concentrate are contained in the fresh juice that bypasses the evaporator. You could get more of these ingredients in the final product by evaporating to (say) 90\% solids instead of 65\%; you could then bypass a greater fraction of the fresh juice and thereby obtain an even better tasting product. Suggest possible drawbacks to this proposal.
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