/*! 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 27 A fuel gas containing 95 mole\% ... [FREE SOLUTION] | 91Ó°ÊÓ

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A fuel gas containing 95 mole\% methane and the balance ethane is burned completely with 25\% excess air. The stack gas leaves the furnace at \(900^{\circ} \mathrm{C}\) and is cooled to \(450^{\circ} \mathrm{C}\) in a waste- heat boiler, a heat exchanger in which heat lost by cooling gases is used to produce steam from liquid water for heating, power generation, or process applications. (a) Taking as a basis of calculation 100 mol of the fuel gas fed to the fumace, calculate the amount of heat (kJ) that must be transferred from the gas in the waste heat boiler to accomplish the indicated cooling. (b) How much saturated steam at 50 bar can be produced from boiler feedwater at \(40^{\circ} \mathrm{C}\) for the same basis of calculation? (Assume all the heat transferred from the gas goes into the steam production.) (c) At what rate ( \(k\) mol/s) must fuel gas be burned to produce 1280 kg steam per hour (an amount required elsewhere in the plant) in the waste heat boiler? What is the volumetric flow rate \(\left(\mathrm{m}^{3} / \mathrm{s}\right)\) of the gas leaving the boiler? (d) Briefly explain how the waste-heat boiler contributes to the plant profitability. (Think about what would be required in its absence.)

Short Answer

Expert verified
a) The amount of heat transferred from the gas in the waste-heat boiler can be calculated using the specific heat capacities of methane and ethane and the difference in input and output temperatures. b) The amount of saturated steam at 50 bar that can be produced using the heat transferred from the gas is determined by dividing the heat transferred by the enthalpy of vaporization of water. c) The rate at which fuel gas needs to be burned is calculated by dividing the rate of steam production by the number of mols of steam produced per mol of fuel gas. d) The waste-heat boiler increases the plant profitability by utilizing the waste heat produced in the process, saving energy costs for heat generation.

Step by step solution

01

Calculation of heat lost during cooling in the waste heat boiler

The heat lost by the fuel gas during cooling from 900°C to 450°C in the waste heat boiler can be calculated using the formula: \[Q = n\Delta{H}\] Where, ΔH can be found using the specific heat capacities of methane and ethane and n is number of moles of fuel gas, which is 100 mol.
02

Using heat from cooling to produce steam

Determine how much steam can be produced from boiler feedwater at 40°C using the heat calculated in Step 1, by using the formula: \[n_{steam} = \frac{Q}{\Delta{H_{vap}}}\] Where, ΔHvap is the heat of vaporization of water at 50 bar.
03

Calculating the required rate of fuel gas burning

Determine the rate at which the fuel gas will need to be burned to produce the given quantity of steam (1280kg/hr). Use \[k = \frac{1280}{n_{boiler} \times molar- weight}\] Then, convert mass of steam produced per hour into mols per second.
04

Calculating volumetric flow rate of gas leaving boiler

Use the ideal gas law to find the volumetric flow rate of the gas leaving the boiler. \[V = \frac{nRT}{P}\] Where, R= universal gas constant, and T and P are the temperature and pressure of the gas, respectively.
05

Explain how the waste-heat boiler contributes to the plant profitability

Discuss how the waste heat boiler leverages heat generated in the process to produce steam, thus saving the energy that would otherwise be extensively used to produce steam in its absence.

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Key Concepts

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

Fuel Gas Combustion
Understanding the combustion of fuel gas is paramount for optimizing energy use in industrial processes. In our exercise, a fuel gas mixture containing 95 mole% methane and the rest ethane is burned with an excess air of 25%. Excess air is used to assure complete combustion, which prevents the formation of dangerous carbon monoxide and unburned hydrocarbons.

Fuel gas combustion generates heat, part of which is often unutilized and lost to surroundings. A waste-heat boiler allows the capture of this thermal energy, effectively cooling the gas while generating useful steam. The stoichiometry of the burning gas facilitates the calculation of the amount of heat released during this process. Combustion also produces stack gases, the temperature and composition of which are crucial in determining the efficiency of the heat recovery process.
Heat Transfer Calculations
In industrial settings, quantifying the transfer of heat is essential for optimizing process efficiency. Heat transfer calculations involve understanding how much energy is transferred from one system to another—in this case, from hot fuel gases to water in the waste-heat boiler.

The exercise presented requires calculating the heat transfer necessary to cool the gases from \(900^\circ \mathrm{C}\) to \(450^\circ \mathrm{C}\). Using the specific heat capacities of methane and ethane, we determine the total heat capacity of the gas mixture and then calculate the total heat loss using the number of moles and the temperature change. This heat is the energy source for steam production, highlighting the interconnection between combustion temperatures, heat capacity of the gases, and steam generation rates.
Steam Production
The production of steam is a common requirement in industrial plants, and waste-heat boilers play a critical role in generating this steam efficiently. Using the heat from fuel gas combustion, which would otherwise be discharged to the environment, we can produce steam—effectively converting waste into valuable resources.

The problem directs us to estimate the quantity of saturated steam that can be produced from the recovered energy. It involves calculating the heat transfer to the water, using the heat of vaporization, to find out how many moles of steam can be generated. Steam is not just simply generated; parameters like pressure and temperature must be precisely controlled. Knowing the properties of steam at 50 bars and corresponding to the feedwater temperature of \(40^\circ \mathrm{C}\), one can use thermodynamic tables to find the heat of vaporization and solve for the steam production.
Chemical Process Profitability
Profitability in a chemical process is heavily tied to the efficiency of the process itself. Waste-heat boilers exemplify this efficiency by harnessing thermal energy that would otherwise be wasted. In the operational scenario we've discussed, the waste-heat boiler plays a vital role by cooling the stack gases and generating steam that can be used for heating, power generation, or other process applications.

In terms of cost savings, the waste-heat boiler reduces the plant's demand for additional fuel to generate steam, translating to significant financial savings. Moreover, this recuperated energy supports the sustainability of operations and potentially qualifies the plant for energy rebates and incentives. In its absence, the plant would incur higher operational costs and energy expenditure to create steam through alternative, less efficient means. Thus, waste-heat recovery is not just a technical detail but a strategic asset to a chemical process's overall profitability and environmental impact.

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Most popular questions from this chapter

Fish and wildlife managers have determined that a sudden temperature increase greater than \(5^{\circ} \mathrm{C}\) would be harmful to the marine ecosystem of a river. Warmer waters contain less dissolved oxygen and cause organisms in a river to increase their metabolism; if the temperature increase is sudden, the organisms do not have time to adapt to the new environment and likely will die. (Changes in river temperatures of five degrees and more due to seasonal temperature variations are common, but those temperature changes are gradual.) A proposed chemical plant plans to use river water for process cooling. The river flows at a rate of \(15.0 \mathrm{m}^{3} / \mathrm{s}\) at a temperature of \(15^{\circ} \mathrm{C}\), and a fraction of it will be diverted to the plant. Preliminary calculations reveal that the cooling water will remove \(5.00 \times 10^{5} \mathrm{kJ} / \mathrm{s}\) of heat from the plant. A portion of the extracted water will evaporate from the plant into the atmosphere, and the remainder will be returned to the river at a temperature of \(35^{\circ} \mathrm{C}\). (a) Draw and completely label a flowchart of the process and prove that there is enough information available to calculate all of the unknown stream flow rates on the chart. (b) Estimate the fraction of the river flow that must be diverted to the plant and the percentage of the cooling water that evaporates. Assume that water has a constant heat capacity of \(4.19 \mathrm{kJ} /\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) and a heat of vaporization roughly that of water at the normal boiling point, and also assume that the specific enthalpy of the water vapor relative to liquid water at \(15^{\circ} \mathrm{C}\) equals the heat of vaporization. (c) Write (but don't evaluate) an expression for the enthalpy change neglected by the assumption about the specific enthalpy of the steam.

A sheet of cellulose acetate film containing 5.00 wt\% liquid acetone enters an adiabatic dryer where \(90 \%\) of the acetone evaporates into a stream of dry air flowing over the film. The film enters the dryer at \(T_{\mathrm{f} 1}=35^{\circ} \mathrm{C}\) and leaves at \(T_{\mathrm{f} 2}\left(^{\circ} \mathrm{C}\right) .\) The air enters the dryer at \(T_{\mathrm{al}}\left(^{\circ} \mathrm{C}\right)\) and 1.01 atm and exits the dryer at \(T_{\mathrm{a} 2}=49^{\circ} \mathrm{C}\) and 1 atm with a relative saturation of \(40 \% . C_{p}\) may be taken to be \(1.33 \mathrm{kJ} /\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) for dry film and \(0.129 \mathrm{kJ} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\) for liquid acetone. Make a reasonable assumption regarding the heat capacity of dry air. The heat of vaporization of acetone may be considered independent of temperature. Take a basis of \(100 \mathrm{kg}\) film fed to the dryer for the requested calculations. (a) Estimate the feed ratio [liters dry air (STP)/kg dry film]. (b) Derive an expression for \(T_{\mathrm{al}}\) in terms of the film temperature change, \(\left(T_{\mathrm{f} 2}-35\right),\) and use it to answer Parts (c) and (d). (c) Calculate the film temperature change if the inlet air temperature is \(120^{\circ} \mathrm{C}\). (d) Calculate the required value of \(T_{\mathrm{al}}\) if the film temperature falls to \(34^{\circ} \mathrm{C},\) and the value if it rises to \(36^{\circ} \mathrm{C}.\) (e) If you solved Parts (c) and (d) correctly, you found that even though the air temperature is consistently higher than the film temperature in the dryer, so that heat is always transferred from the air to the film, the film temperature can drop from the inlet to the outlet. How is this possible?

A mixture of \(n\) -hexane vapor and air leaves a solvent recovery unit and flows through a \(70-\mathrm{cm}\) diameter duct at a velocity of \(3.00 \mathrm{m} / \mathrm{s}\). At a sampling point in the duct the temperature is \(40^{\circ} \mathrm{C}\), the pressure is \(850 \mathrm{mm}\) Hg, and the dew point of the sampled gas is \(25^{\circ} \mathrm{C}\). The gas is fed to a condenser in which it is cooled at constant pressure, condensing \(70 \%\) of the hexane in the feed. (a) Perform a degree-of-freedom analysis to show that enough information is available to calculate the required condenser outlet temperature \(\left(^{\circ} \mathrm{C}\right)\) and cooling rate \((\mathrm{kW})\) (b) Perform the calculations. (c) If the feed duct diameter were \(35 \mathrm{cm}\) for the same molar flow rate of the feed gas, what would be the average gas velocity (volumetric flow rate divided by cross-sectional area)? (d) Suppose you wanted to increase the percentage condensation of hexane for the same feed stream. Which three condenser operating variables might you change, and in which direction?

The heat required to raise the temperature of \(m\) (kg) of a liquid from \(T_{1}\) to \(T_{2}\) at constant pressure is $$ Q=\Delta H=m \int_{T_{1}}^{T_{2}} C_{p}(T) d T $$ In high school and in first-year college physics courses, the formula is usually given as $$ Q=m C_{p} \Delta T=m C_{p}\left(T_{2}-T_{1}\right) $$ (a) What assumption about \(C_{p}\) is required to go from Equation 1 to Equation \(2 ?\) (b) The heat capacity \(\left(C_{p}\right)\) of liquid \(n\) -hexane is measured in a bomb calorimeter. A small reaction flask (the bomb) is placed in a well- insulated vessel containing \(2.00 \mathrm{L}\) of liquid \(n-\mathrm{C}_{6} \mathrm{H}_{14}\) at \(T=300 \mathrm{K} .\) A combustion reaction known to release \(16.73 \mathrm{kJ}\) of heat takes place in the bomb, and the subsequent temperature rise of the system contents is measured and found to be \(3.10 \mathrm{K}\). In a separate experiment, it is found that \(6.14 \mathrm{kJ}\) of heat is required to raise the temperature of everything in the system except the hexane by \(3.10 \mathrm{K}\). Use these data to estimate \(C_{p}[\mathrm{kJ} /(\mathrm{mol} \cdot \mathrm{K})]\) for liquid \(n\) -hexane at \(T \approx 300 \mathrm{K},\) assuming that the condition required for the validity of Equation 2 is satisfied. Compare your result with a tabulated value.

Ever wonder why espresso costs much more per cup than regular drip coffee? Part of the reason is the expensive equipment needed to brew a proper espresso. A high-powered burr grinder first shears the coffee beans to a fine powder without producing too much heat. (Heating the coffee in the grinding stage prematurely releases the volatile oils that give espresso its rich flavor and aroma.) The ground coffee is put into a cylindrical container called a gruppa and tamped down firmly to provide an even flow of water through it. An electrically heated boiler inside the espresso machine maintains water in a reservoir at 1.4 bar and \(109^{\circ} \mathrm{C}\). An electric pump takes cold water at \(15^{\circ} \mathrm{C}\) and 1 bar, raises its pressure to slightly above 9 bar, and feeds it into a heating coil that passes through the reservoir. Heat transferred from the reservoir through the coil wall raises the water temperature to \(96^{\circ} \mathrm{C}\). The heated water flows into the top of the gruppa at \(96^{\circ} \mathrm{C}\) and 9 bar, passes slowly through the tightly packed ground beans, and dissolves the oils and some of the solids in the beans to become espresso, which decompresses to 1 atm as it exits the machine. The water temperature and uniform flow through the bed of packed coffee in the gruppa lead to the more intense flavor of espresso relative to normal drip coffee. Water drawn directly from the reservoir is expanded to atmospheric pressure where it forms steam, which is used to heat and froth milk for lattes and cappuccinos. (a) Sketch this process, using blocks to represent the pump, reservoir, and gruppa. Label all heat and work flows in the process, including electrical energy. (b) To make a 14-oz latte, you would steam 12 ounces of cold milk (3^'C) until it reaches 71 ^ C and pour it over 2 ounces of espresso. Assume that the steam cools but none of it condenses as it bubbles through the milk. For each latte made, the heating element that maintains the reservoir temperature must supply enough energy to heat the espresso water plus enough to heat the milk, plus additional energy. Assuming $$ \left(C_{p}\right)_{\operatorname{milk}}=3.93 \frac{\mathrm{J}}{\mathrm{g} \cdot^{\circ} \mathrm{C}}, \quad \mathrm{SG}_{\mathrm{milk}}=1.03 $$ calculate the quantity of electrical energy that must be provided to the heating element to accomplish those two functions. Why would more energy than what you calculate be required? (There are several reasons.) (c) Coffee beans contain a considerable amount of trapped carbon dioxide, not all of which is released when the beans are ground. When the hot pressurized water percolates through the ground beans, some of the carbon dioxide is absorbed in the liquid. When the liquid is then dispensed at atmospheric pressure, fine \(\mathrm{CO}_{2}\) bubbles come out of solution. In addition, one of the chemical compounds formed when the coffee beans are roasted and extracted into the espresso is melanoidin, a surfactant. Surfactant molecules are asymmetrical, with one end being hydrophilic (drawn to water) and the other end hydrophobic (repelled by water). When the bubbles (thin water films containing \(\mathrm{CO}_{2}\) ) pass through the espresso liquid, the hydrophilic ends of the melanoidin molecules attach to the bubbles and the dissolved bean oils in turn attach to the hydrophobic ends. The result is that the bubbles emerge coated with the oils to form the crema , the familiar reddish brown stable foam at the surface of good espresso. Speculate on why you don't see crema in normal drip coffee. (Hint: Henry's law should show up in your explanation.) Note: All soaps and shampoos contain at least one surfactant species. (A common one is sodium lauryl sulfate.) Its presence explains why if you have greasy hands, washing with plain water may leave the grease untouched but washing with soap removes the grease. (d) Explain in your own words (i) how espresso is made, (ii) why espresso has a more intense flavor than regular drip coffee, (iii) what the crema in espresso is, how it forms, and why it doesn't appear in regular drip coffee, and (iv) why washing with plain water does not remove grease but washing with soap does. (Note: Many people automatically assume that all chemical engineers are extraordinarily intelligent. If you can explain those four things, you can help perpetuate that belief.)

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