/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 The indicator-dilution method is... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 34

The indicator-dilution method is a technique used to determine flow rates of fluids in channels for which devices like rotameters and orifice meters cannot be used (e.g., rivers, blood vessels, and largediameter pipelines). A stream of an easily measured substance (the tracer) is injected into the channel at a known rate, and the tracer concentration is measured at a point far enough downstream of the injection point for the tracer to be completely mixed with the flowing fluid. The larger the flow rate of the fluid, the lower the tracer concentration at the measurement point. A gas stream that contains 1.50 mole \(\% \mathrm{CO}_{2}\) flows through a pipeline. Twenty (20.0) kilograms of \(\mathrm{CO}_{2}\) per minute is injected into the line. A sample of the gas is drawn from a point in the line 150 meters (a) Estimate the gas flow rate (kmol/min) upstream of the injection point. (b) Eighteen seconds elapse from the instant the additional \(\mathrm{CO}_{2}\) is first injected to the time the \(\mathrm{CO}_{2}\) concentration at the measurement point begins to rise. Assuming that the tracer travels at the average velocity of the gas in the pipeline (i.e., neglecting diffusion of \(\mathrm{CO}_{2}\) ), estimate the average velocity (m/s). If the molar gas density is \(0.123 \mathrm{kmol} / \mathrm{m}^{3}\), what is the pipe diameter?

Short Answer

Expert verified
The estimated gas flow rate upstream of the injection point is around 30.3 kmol/min, the estimated average velocity is roughly 8.33 m/s, and the pipe diameter comes to about 1.18 meters.

Step by step solution

01

- Gas Flow Rate Calculation

The tracer is the CO2 gas. It amounts to 1.50 molar % of the stream before injection and 20 kilograms are added each minute. By extension, the gas stream flow at that point should be \( \frac{20 \mathrm{kg CO2/min}}{0.015} \approx 1333.3 \mathrm{kg CO2/min} \). Given that the molar mass of CO2 is approximately 44.01 g/mol (or 44.01 kg/kmol), one can calculate the molar flow rate to be \( \frac{1333.3 \mathrm{kg/min}}{44.01 \mathrm{kg/kmol}} \approx 30.3 \mathrm{kmol/min} \).
02

- Pipeline Velocity Calculation

To calculate the pipeline velocity, use the formula for distance, \( \mathrm{velocity} = \frac{\mathrm{distance}}{\mathrm{time}} \). Distance is established as 150 meters and time taken is 18 seconds. Thus, the pipeline velocity is \( \frac{150 \mathrm{m}}{18 \mathrm{s}} \approx 8.33 \mathrm{m/s} \).
03

- Pipe Diameter Calculation

Pipe diameter (d) will require manipulating the pipe flow rate equation \( \mathrm{Flowrate} = \mathrm{density} \times \mathrm{area} \times \mathrm{velocity} \) to \( d = \sqrt{\frac{4 \times \mathrm{Flowrate}}{\pi \times \mathrm{density} \times \mathrm{velocity}}} \). Substituting the calculated flow rate (\(30.3 \mathrm{kmol/min} \times \frac{1 \mathrm{min}}{60 \mathrm{s}} = 0.505 \mathrm{kmol/s}\)) and velocity (8.33 m/s) and the given gas density (0.123 kmol/m3), the diameter can be computed as \( \sqrt{\frac{4 \times 0.505 \mathrm{kmol/s}}{\pi \times 0.123 \mathrm{kmol/m}^3 \times 8.33 \mathrm{m/s}}} \approx 1.18 \mathrm{m} \).

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Key Concepts

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

Flow Rate Calculation
In many technical and natural systems, the Indicator-Dilution Method is a valuable tool for determining flow rates, especially when typical measurement devices like rotameters and orifice meters are not applicable. This method involves injecting a known quantity of a tracer, such as \( ext{CO}_2\), into the pipeline.
The concentration of this injected tracer is measured further downstream, providing insights into the original flow rate.
To calculate the flow rate of the gas upstream, it's crucial to understand the correlation between the original stream concentration and the additional tracer quantity.
  • The initial mole percentage of \(\text{CO}_2\) is 1.50\%.
  • An additional 20 kg/min of \(\text{CO}_2\) is introduced.
Thus, by using the calculated molar mass of \(\text{CO}_2\) (44.01 kg/kmol), the gas flow rate upstream can be derived using the formula:\[\text{Flow Rate} = \frac{\text{Mass Rate of CO}_2}{\text{Mole Fraction}}\] This ultimately results in a molar flow rate of approximately 30.3 kmol/min.
This method allows for accurate flow rate predictions, reflecting the system's dynamics even when devices cannot be directly applied.
Pipeline Velocity
When using the Indicator-Dilution Method, determining the velocity of the tracer also helps in understanding the gas flow characteristics. By examining the time it takes for the tracer to travel a measured distance, we can calculate the pipeline velocity. The formula for velocity is:\[\text{Velocity} = \frac{\text{Distance}}{\text{Time}}\]
In the provided exercise, a tracer starts at an injection point and is measured 150 meters downstream after 18 seconds. Applying these values gives:
  • Distance = 150 meters
  • Time = 18 seconds
The resulting pipeline velocity comes out to approximately 8.33 m/s.
This velocity indicates the average speed at which the gas flows through the pipeline, impacting both the system's efficiency and the transport of the tracer.
Pipe Diameter Calculation
A fundamental principle in fluid mechanics is the relationship between flow rate, velocity, and pipe diameter. To find the diameter of a pipe, we use the continuity equation, expressed as:\[\text{Flowrate} = \text{Density} \times \text{Area} \times \text{Velocity}\]By re-arranging the formula to solve for the pipe diameter, \(d\), we get:\[d = \sqrt{\frac{4 \times \text{Flowrate}}{\pi \times \text{Density} \times \text{Velocity}}}\]
Given:
  • Flow rate = 0.505 kmol/s (converting from 30.3 kmol/min)
  • Velocity = 8.33 m/s
  • Gas density = 0.123 kmol/m鲁
Applying these values, we find that the pipe diameter is approximately 1.18 meters.
The accurate calculation of pipe diameter is essential for ensuring proper flow conditions and optimizing system design, highlighting the interplay between physical dimensions and flow characteristics.

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Most popular questions from this chapter

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.

A fuel oil is fed to a furnace and burned with \(25 \%\) excess air. The oil contains \(87.0 \mathrm{wt} \% \mathrm{C}, 10.0 \% \mathrm{H},\) and 3.0\% S. Analysis of the furnace exhaust gas shows only \(\mathrm{N}_{2}, \mathrm{O}_{2}, \mathrm{CO}_{2}, \mathrm{SO}_{2},\) and \(\mathrm{H}_{2} \mathrm{O}\). The sulfur dioxide emission rate is to be controlled by passing the exhaust gas through a scrubber, in which most of the \(\mathrm{SO}_{2}\) is absorbed in an alkaline solution. The gases leaving the scrubber (all of the \(\mathrm{N}_{2}, \mathrm{O}_{2},\) and \(\mathrm{CO}_{2}\), and some of the \(\mathrm{H}_{2} \mathrm{O}\) and \(\mathrm{SO}_{2}\) entering the unit) pass out to a stack. The scrubber has a limited capacity, however, so that a fraction of the furnace exhaust gas must be bypassed directly to the stack. At one point during the operation of the process, the scrubber removes \(90 \%\) of the \(\mathrm{SO}_{2}\) in the gas fed to it, and the combined stack gas contains 612.5 ppm (parts per million) \(\mathrm{SO}_{2}\) on a dry basis; that is, every million moles of dry stack gas contains 612.5 moles of \(\mathrm{SO}_{2}\). Calculate the fraction of the exhaust bypassing the scrubber at this moment.

Effluents from metal-finishing plants have the potential of discharging undesirable quantities of metals, such as cadmium, nickel, lead, manganese, and chromium, in forms that are detrimental to water and air quality. A local metal-finishing plant has identified a wastewater stream that contains 5.15 wt\% chromium (Cr) and devised the following approach to lowering risk and recovering the valuable metal. The wastewater stream is fed to a treatment unit that removes \(95 \%\) of the chromium in the feed and recycles it to the plant. The residual liquid stream leaving the treatment unit is sent to a waste lagoon. The treatment unit has a maximum capacity of 4500 kg wastewater/h. If wastewater leaves the finishing plant at a rate higher than the capacity of the treatment unit, the excess (anything above \(4500 \mathrm{kg} / \mathrm{h}\) ) bypasses the unit and combines with the residual liquid leaving the unit, and the combined stream goes to the waste lagoon. (a) Without assuming a basis of calculation, draw and label a flowchart of the process. (b) Wastewater leaves the finishing plant at a rate \(\dot{m}_{1}=6000 \mathrm{kg} / \mathrm{h}\). Calculate the flow rate of liquid to the waste lagoon, \(\dot{m}_{6}(\mathrm{kg} / \mathrm{h}),\) and the mass fraction of \(\mathrm{Cr}\) in this liquid, \(x_{6}(\mathrm{kg} \mathrm{Cr} / \mathrm{kg})\) (c) Calculate the flow rate of liquid to the waste lagoon and the mass fraction of Crin this liquid for \(\dot{m}_{1}\) varying from \(1000 \mathrm{kg} / \mathrm{h}\) to \(10,000 \mathrm{kg} / \mathrm{h}\) in \(1000 \mathrm{kg} / \mathrm{h}\) increments. Generate a plot of \(x_{6}\) versus \(\dot{m}_{1}\). (Suggestion: Use a spreadsheet for these calculations.) (d) The company has hired you as a consultant to help them determine whether or not to add capacity to the treatment unit to increase the recovery of chromium. What would you need to know to make this determination? (e) What concerns might need to be addressed regarding the waste lagoon?

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.

n-Pentane is burned with excess air in a continuous combustion chamber. (a) A technician runs an analysis and reports that the product gas contains 0.270 mole\% pentane, \(5.3 \%\) oxygen, \(9.1 \%\) carbon dioxide, and the balance nitrogen on \(a\) dry basis. Assume 100 mol of dry product gas as a basis of calculation, draw and label a flowchart, perform a degree-offreedom analysis based on atomic species balances, and show that the system has -1 degree of freedom. Interpret this result. (b) Use balances to prove that the reported percentages could not possibly be correct. (c) The technician reruns the analysis and reports new values of 0.304 mole\% pentane, \(5.9 \%\) oxygen, \(10.2 \%\) carbon dioxide, and the balance nitrogen. Verify that this result could be correct and, assuming that it is, calculate the percent excess air fed to the reactor and the fractional conversion of pentane. (d) It was emphasized in Part (c) that the new composition could be correct. Explain why it isn't possible to say for sure; illustrate your response by considering a set of equations with -1 degree of freedom.
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