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Chapter 4: Problem 29

A sedimentation process is to be used to separate pulverized coal from slate. A suspension of finely divided particles of galena (lead sulfide, SG = 7.44) in water is prepared. The overall specific gravity of the suspension is 1.48. (a) Four hundred kilograms of galena and a quantity of water are loaded into a tank and stirred to obtain a uniform suspension with the required specific gravity. Draw and label the flowchart (label both the masses and volumes of the galena and water), do the degree-of-freedom analysis, and calculate how much water ( \(\mathrm{m}^{3}\) ) must be fed to the tank. (b) A mixture of coal and slate is placed in the suspension. The coal rises to the top and is skimmed off, and the slate sinks. What can you conclude about the specific gravities of coal and slate? (c) The separation process works well for several hours, but then a region of clear liquid begins to form at the top of the cloudy suspension and the coal sinks to the bottom of this region, making skimming more difficult. What might be happening to cause this behavior and what corrective action might be taken? Now what can you say about the specific gravity of coal?

Short Answer

Expert verified
To create a suspension with a specific gravity of 1.48, approximately 466 cubic meters of water will be necessary. Based on the behavior in the suspension, the specific gravity of coal is less than 1.48 and the specific gravity of slate is more than 1.48. If a region of clear liquid forms at the top of the suspension causing coal to sink, this could be due to a decrease in the specific gravity of the suspension, which could be addressed by stirring to resuspend solids or adding more galena.

Step by step solution

01

Understanding the Principles Involved

The principle of specific gravity (SG) will be used extensively throughout this problem. The first step involves understanding that the overall specific gravity (\(SG_o\)) of a suspension is calculated from its components by the formula \(SG_o = \frac{SG_{Galena}*m_{Galena}+SG_{water}*v_{water}}{m_{Galena}+v_{water}}\). This formula implies that the overall specific gravity is the weighted average of the specific gravities of the components.
02

Calculate volume of water needed

Rearranging the formula to solve for the volume of water (\(v_{water}\)) gives \(v_{water} = \frac{SG_o*(m_{Galena}+v_{water}) - SG_{Galena}*m_{Galena}}{SG_{water}}\). Initial substitution of known values gives \(v_{water} = \frac{1.48*(400Kg+v_{water}) - 7.44*400Kg}{1}\). Now, the expression can be solved iteratively as it involves \(v_{water}\) on both sides of the equation. Assuming a starting guess value of \(v_{water}\) as 0 and working through iterations until a sufficiently accurate value is calculated.
03

Compare SG for coal and slate

As the coal rises, its specific gravity must be less than the specific gravity of the suspension. Similarly, as the slate sinks, its specific gravity must be higher than that of the suspension. Therefore \(SG_{coal}< 1.48 < SG_{slate}\).
04

Analyze the change in process

The formation of a layer of clear liquid and the sinking of coal suggest that the specific gravity of the suspension has decreased. Possible causes may include solids settling out or evaporation of the water. Corrective action may include stirring to resuspend solids or adding more galena to increase the specific gravity. The specific gravity of the coal is still less than the suspension so \(SG_{coal}< SG_{new suspension}\).

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Key Concepts

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

Degree-Of-Freedom Analysis
When designing a separation process like the sedimentation of coal from slate, engineers use a degree-of-freedom analysis to understand whether they have enough information to solve for the unknowns. This technique is essential in determining the feasibility of reaching a solution without additional data.

Degree-of-freedom (DOF) refers to the number of independent variables that can be changed without affecting the others. In this case, the specific gravities of coal and slate, the mass of galena, and the volume of water compose the known variables. However, the exact amount of water needed to achieve a specific gravity of 1.48 in the suspension is unknown.

The formula used to calculate the volume of water needed is derived from the principle of specific gravity and mass balance. The flowchart, in essence, should reflect these balances. We calculate the DOF by taking the number of equations available (mass balances for galena and water, and the specific gravity formula) and subtracting the number of unknowns (volume of water in this case). If the DOF equals zero, the system is solvable without further input. In our exercise, knowing the mass of galena and the desired overall specific gravity allows us to solve for the volume of water, confirming that our system is well-defined with zero degrees of freedom.
Suspension Properties
Understanding the suspension properties is crucial in the sedimentation process. A suspension in this context is a heterogeneous mixture containing solid particles that are substantially larger than molecules or ions and can be affected by the force of gravity.

The overall specific gravity of the suspension is a vital property as it determines the behavior of particles when mixed. This value, 1.48 in the case of our exercise, lies between that of galena and water. By using the defined specific gravity, we can infer the relative buoyancy of particles in the mixture. As coal floats and slate sinks, their specific gravities also act as a qualitative indicator of their relative densities compared to the suspension.

It is also essential to consider the settling rate, the viscosity of the fluid, and the particle size distribution within the suspension. These properties can affect the 'clarity' of the suspension over time and the efficiency of separation. For instance, if a clear layer forms on top of the mixture, it could signal that the solid particles have settled, indicating changes in the suspension properties over time.
Sedimentation Process
The sedimentation process is a method of separating particles based on their density by allowing them to settle out of a fluid. It leverages gravity to remove suspended solids from a mixture or to separate liquids with different densities.

Sedimentation is influenced by several factors: the size and density of the particles, the viscosity of the fluid, and the depth of the liquid. These factors combine to determine the settling rate of the particles. In the sedimentation process described in the exercise, galena is used to increase the specific gravity of the suspension to a point where coal can be separated from slate due to differences in their buoyancies.

As mentioned in the exercise solution, when the coal stops floating, it implies that there has been a change in the suspension's density, or possible settling of the heavier particles has occurred. Corrective actions might include stirring to keep the particles in suspension or adjusting the specific gravity to ensure coal remains buoyant. The challenge of maintaining a consistent suspension profile highlights the need for continuous monitoring and adjustment in any sedimentation process.

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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.

The fresh feed to an ammonia production process contains nitrogen and hydrogen in stoichiometric proportion, along with an inert gas (I). The feed is combined with a recycle stream containing the same three species, and the combined stream is fed to a reactor in which a low single-pass conversion of nitrogen is achieved. The reactor effluent flows to a condenser. A liquid stream containing essentially all of the ammonia formed in the reactor and a gas stream containing all the inerts and the unreacted nitrogen and hydrogen leave the condenser. The gas stream is split into two fractions with the same composition: one is removed from the process as a purge stream, and the other is the recycle stream combined with the fresh feed. In every stream containing nitrogen and hydrogen, the two species are in stoichiometric proportion. (a) Let \(x_{10}\) be the mole fraction of inerts in the fresh feed, \(f_{\mathrm{sp}}\) the single-pass conversion of nitrogen (and of hydrogen) in the reactor, and \(y_{p}\) the fraction of the gas leaving the condenser that is purged (mol purged/mol total). Taking a basis of 1 mol fresh feed, draw and fully label a process flowchart, incorporating \(x_{10}, f_{\mathrm{sp}},\) and \(y_{\mathrm{p}}\) in the labeling to the greatest possible extent. Then, assuming that the values of these three variables are given, write a set of equations for the total moles fed to the reactor \(\left(n_{\mathrm{r}}\right),\) moles of ammonia produced \(\left(n_{\mathrm{p}}\right),\) and overall nitrogen conversion \(\left(f_{\mathrm{ov}}\right) .\) Each equation should involve only one unknown variable, which should be circled. (b) Solve the equations of Part (a) for \(x_{10}=0.01, f_{\mathrm{sp}}=0.20,\) and \(y_{\mathrm{p}}=0.10\) (c) Briefly explain in your own words the reasons for including (i) the recycle stream and (ii) the purge stream in the process design. (d) Prepare a spreadsheet to perform the calculations of Part (a) for given values of \(x_{10}, f_{\mathrm{sp}},\) and \(y_{\mathrm{p}} .\) Test it with the values in Part (b). Then in successive rows of the spreadsheet, vary each of the three input variables two or three times, holding the other two constant. The first six columns and first five rows of the spreadsheet should appear as follows:Summarize the effects on ammonia production \(\left(n_{\mathrm{P}}\right)\) and reactor throughput \(\left(n_{\mathrm{r}}\right)\) of changing each of the three input variables.

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.

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).

A variation of the indicator-dilution method (see preceding problem) is used to measure total blood volume. A known amount of a tracer is injected into the bloodstream and disperses uniformly throughout the circulatory system. A blood sample is then withdrawn, the tracer concentration in the sample is measured, and the measured concentration [which equals (tracer injected)/(total blood volume) if no tracer is lost through blood vessel walls] is used to determine the total blood volume. In one such experiment, \(0.60 \mathrm{cm}^{3}\) of a solution containing \(5.00 \mathrm{mg} / \mathrm{L}\) of a dye is injected into an artery of a grown man. About 10 minutes later, after the tracer has had time to distribute itself uniformly throughout the bloodstream, a blood sample is withdrawn and placed in the sample chamber of a spectrophotometer. A beam of light passes through the chamber, and the spectrophotometer measures the intensity of the transmitted beam and displays the value of the solution absorbance (a quantity that increases with the amount of light absorbed by the sample). The value displayed is 0.18. A calibration curve of absorbance \(A\) versus tracer concentration \(C\) (micrograms dye/liter blood) is a straight line through the origin and the point \((A=0.9, C=3 \mu \mathrm{g} / \mathrm{L}) .\) Estimate the patient's total blood volume from these data.
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