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Chapter 5: Problem 28

In froth flotation, air is bubbled through an aqueous solution or slurry to which a foaming agent (soap) has been added. The air-soap bubbles carry finely dispersed solids and hydrophobic materials such as grease and oil to the surface where they can be skimmed off in the foam. An ore-containing slurry is to be processed in a froth flotation tank at a rate of 300 tons/h. The slurry consists of \(20.0 \mathrm{wt} \%\) solids (the ore, \(\mathrm{SG}=1.2\) ) and the remainder an aqueous solution with a density close to that of water. Air is sparged (blown through a nozzle designed to produce small bubbles) into the slurry at a rate of \(40.0 \mathrm{ft}^{3}\) (STP)/1000 gal of slurry. The entry point of the air is 10 \(\mathrm{ft}\) below the slurry surface. The tank contents are at \(75^{\circ} \mathrm{F}\) and the barometric pressure is 28.3 inches of Hg. The sparger design is such that the average bubble diameter on entry is \(2.0 \mathrm{mm}\). (a) What is the volumetric flow rate of the air at its entering conditions? (b) By what percentage does the average bubble diameter change between the entry point and the slurry surface?

Short Answer

Expert verified
a) Volumetric flow rate of air at its entering conditions is calculated. b) Percentage change in average bubble diameter from the entry point to the slurry surface is calculated.

Step by step solution

01

Calculate the Volumetric Flow Rate of the Air at Its Entering Conditions

The volumetric flow rate at STP (Standard Temperature and Pressure) is given as \(40.0 \mathrm{ft}^{3}\) per 1000 gallons of slurry. It's important to note that standard conditions refer to a temperature of \(0^{\circ}C\) or \(273.15K\), and a pressure of \(1 atm\). To determine the volumetric flow at the entering conditions (i.e., \(75^{\circ} F\) and 28.3 inches of Hg), we need to use the ideal gas law that states \(PV=nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant and T is the temperature. As the number of moles remains unchanged, the law can be rewritten as \(P_{1}V_{1}/T_{1}=P_{2}V_{2}/T_{2}\) where subscripts 1 and 2 denote different sets of P, V and T. By substituting these inputs and performing the calculation, we get the volumetric flow rate at the entering conditions.
02

Calculate the Change in Bubble Diameter from Entry Point to Surface

The average bubble diameter changes from the entry point to the slurry surface due to the decrease in pressure with increasing height. We can start this step using the ideal gas law, which states that a gas expands as it rises. Assuming that the bubble keeps a spherical shape during its rise, the diameter can be equated with \(d=\sqrt[3]{6V/\pi}\) where V is the volume. Since both the initial and final volumes will change due to pressure changes, this formula can be applied twice to find the initial and final diameter. Having the initial (entry point) and final (surface point) diameters lets us calculate the percentage change.

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Key Concepts

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

Volumetric Flow Rate
Understanding the volumetric flow rate is crucial in processes like froth flotation, where the amount of air introduced impacts the efficiency of the separation. The volumetric flow rate is the volume of fluid (in this case, air) that passes through a given area per unit time. It is usually measured in units such as cubic feet per minute (cfm) or liters per second (L/s).

For the given exercise, the flow rate of air at standard conditions is provided, but we need the rate at the specific conditions in the flotation tank. This is where the ideal gas law comes into play. By using the relationship between temperature, pressure, and volume, we can adjust the STP flow rate to the actual conditions in the tank, providing a more accurate measurement of the air flow rate for the froth flotation process. This calculation is imperative to maintain the proper air to slurry ratio and thereby, optimizing the recovery rates of valuable minerals.
Ideal Gas Law
The ideal gas law, represented by the equation \(PV=nRT\), explains the relationship among pressure \(P\), volume \(V\), amount of gas in moles \(n\), the ideal gas constant \(R\), and temperature \(T\). This principle is vital in understanding how gasses behave under varying conditions of temperature and pressure, which directly applies to our froth flotation scenario.

In the froth flotation process, air bubbles are introduced at a specific pressure and temperature. The condition in the flotation tank is different from the standard conditions. To find the correct volumetric flow rate at tank conditions, we employ the ideal gas law to relate the conditions at which the volumetric flow rate is given (STP) to the conditions in the flotation tank. Since the quantity of air (in moles) remains the same, the ideal gas law can be manipulated into a formula that equates the states before and after the change in environmental conditions. This results in the corrected flow rate which is critical for our calculation of air introduction into the slurry.
Bubble Size Dynamics
Bubble size dynamics play a key role in froth flotation, as the size of the air bubbles affects the recovery and grade of minerals. When air is introduced into the slurry, it forms bubbles that attach to the hydrophobic particles and carry them to the surface. The size of these bubbles is a result of the sparger (the device used to generate them) design and the operating conditions, such as pressure and temperature.

As the bubbles rise to the surface, they experience a decrease in hydrostatic pressure, which causes them to expand as predicted by the ideal gas law. This expansion of gas volume implies an increase in the diameter of the bubbles, assuming uniform bubble shape and no coalescence. This change in size affects how well the bubbles can carry the hydrophobic materials to the surface. Calculating the percentage change in bubble diameter from the bottom to the top of the tank requires understanding that the bubble volume is proportional to the cube of its diameter, as well as being able to calculate the volume at both the entry and surface levels considering the ideal gas law. This aspect of the flotation process is pivotal to optimizing the entire operation.

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Most popular questions from this chapter

The ultimate analysis of a No. 4 fuel oil is 86.47 wt\% carbon, \(11.65 \%\) hydrogen, \(1.35 \%\) sulfur, and the balance noncombustible inerts. This oil is burned in a steam-generating furnace with \(15 \%\) excess air. The air is preheated to \(175^{\circ} \mathrm{C}\) and enters the furnace at a gauge pressure of \(180 \mathrm{mm}\) Hg. The sulfur and hydrogen in the fuel are completely oxidized to \(\mathrm{SO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O} ; 5 \%\) of the carbon is oxidized to \(\mathrm{CO}\), and the balance forms \(\mathrm{CO}_{2}\) (a) Calculate the feed ratio ( \(\mathrm{m}^{3}\) air) \(/(\mathrm{kg} \text { oil })\) (b) Calculate the mole fractions (dry basis) and ppm (parts per million on a wet basis, or moles contained in \(10^{6}\) moles of the wet stack gas) of the stack-gas species that might be considered environmental hazards.

A fuel gas containing \(86 \%\) methane, \(8 \%\) ethane, and \(6 \%\) propane by volume flows to a furnace at a rate of \(1450 \mathrm{m}^{3} / \mathrm{h}\) at \(15^{\circ} \mathrm{C}\) and \(150 \mathrm{kPa}\) (gauge), where it is burned with \(8 \%\) excess air. Calculate the required flow rate of air in SCMH (standard cubic meters per hour).

Many references give the specific gravity of gases with reference to air. For example, the specific gravity of carbon dioxide is 1.53 relative to air at the same temperature and pressure. Show that this value is correct as long as the ideal-gas equation of state applies.

A tank in a room at \(19^{\circ} \mathrm{C}\) is initially open to the atmosphere on a day when the barometric pressure is 102 kPa. A block of dry ice (solid \(\mathrm{CO}_{2}\) ) with a mass of \(15.7 \mathrm{kg}\) is dropped into the tank, which is then sealed. The reading on the tank pressure gauge initially rises very quickly, then much more slowly, eventually reaching a value of 3.27 MPa. Assume \(T_{\text {final }}=19^{\circ} \mathrm{C}\) (a) How many moles of air were in the tank initially? Neglect the volume occupied by \(\mathrm{CO}_{2}\) in the solid state, and assume that a negligible amount of \(\mathrm{CO}_{2}\) escapes prior to the sealing of the tank. (b) Estimate the percentage error made by neglecting the volume of the block of dry ice placed in the tank. (The specific gravity of solid carbon dioxide is approximately 1.56 .) (c) What is the final density (g/L) of the gas in the tank? (d) Explain the observed variation of pressure with time. More specifically, what is happening in the tank during the initial rapid pressure increase and during the later slow pressure increase?

Steam reforming is an important technology for converting refined natural gas, which we take here to be methane, into a synthesis gas that can be used to produce a varicty of other chemical compounds. For example, consider a reformer to which natural gas and steam are fed in a ratio of 3.5 moles of steam per mole of methane. The reformer operates at 18 atm, and the reaction products leave the reformer in chemical equilibrium at \(875^{\circ} \mathrm{C}\). The steam reforming reaction is $$\mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{CO}+3 \mathrm{H}_{2}$$ and the water-gas shift reaction also occurs in the reformer. $$\mathrm{CO}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{CO}_{2}+\mathrm{H}_{2}$$ The equilibrium constants for these two reactions are given by the expressions At \(875^{\circ} \mathrm{C}, K_{\mathrm{R}}=872.9 \mathrm{atm}^{2}\) and \(K \mathrm{w} \mathrm{G}=0.2482 .\) The process is to produce \(100.0 \mathrm{kmol} / \mathrm{h}\) of hydrogen. Calculate the feed rates (kmol/h) of methane and steam and the volumetric flow rate \(\left(\mathrm{m}^{3} / \mathrm{min}\right)\) of gas leaving the reformer.
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