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Chapter 8: Problem 40

A natural gas containing 95 mole \(\%\) methane and the balance ethane is burned with \(20.0 \%\) excess air. The stack gas, which contains no unburned hydrocarbons or carbon monoxide, leaves the furnace at \(900^{\circ} \mathrm{C}\) and \(1.2 \mathrm{atm}\) and passes through a heat exchanger. The air on its way to the furnace also passes through the heat exchanger, entering it at \(20^{\circ} \mathrm{C}\) and leaving it at \(245^{\circ} \mathrm{C}\). (a) Taking as a basis \(100 \mathrm{mol} / \mathrm{s}\) of the natural gas fed to the furnace, calculate the required molar flow rate of air, the molar flow rate and composition of the stack gas, the required rate of heat transfer in the preheater, \(\dot{Q}\) (write an energy balance on the air), and the temperature at which the stack gas leaves the preheater (write an energy balance on the stack gas). Note: The problem statement does not give you the fuel feed temperature. Make a reasonable assumption, and state why your final results should be nearly independent of what you assume. (b) What would \(\dot{Q}\) be if the actual feed rate of the natural gas were 350 SCMH [standard cubic meters per hour, \(\left.\mathrm{m}^{3}(\mathrm{STP}) / \mathrm{h}\right] ?\) Scale up the flowchart of Part (a) rather than repeating the entire calculation.

Short Answer

Expert verified
The molar flow rate of required air is found to be 1165 mol/s. The molar flow rate and composition of the stack gas is determined to be 105, 205, 4382 and 466 mol/s for CO2, H2O, N2 and Excess O2 respectively. The required rate of heat transfer in the preheater is calculated as \(3.67 x 10^6\) joules/s. The temperature at which the stack gas leaves the preheater requires iterative calculations and is dependent on the specific heat capacity of the stack gas. For the actual feed rate of 350 SCMH, the heat duty is calculated to be \(3.58 x 10^6\) joules/s.

Step by step solution

01

Determine the molar flow rate of air

We know that for every one mole of methane, we need \(2(\mathrm{O}_{2}) + 2(3.76\mathrm{N}_{2})\) of air for complete combustion. With \(20.0\%\) excess air, this comes out to be \((1+0.20)(2)(4.76)=11.424\) mol air/mol CH4. For ethane, we would need \((1+0.20)(7/2)(4.76)=16.84\) mol air/mol C2H6. Using the 95 mole% as methane and the balance as ethane in the natural gas, the molar flow rate of air can be calculated as \(0.95(11.424) + 0.05(16.84) = 11.65\) mol air/mol feed. Therefore, for 100 moles of feed, we would need \(11.65(100) = 1165\) mol/s of air.
02

Determine the molar flow rate and composition of the stack gas

For each mole of methane burned, we get one mole of CO2 and two moles of water. Similarly for ethane, we get two moles of CO2 and three moles of water. We would also have nitrogen and the excess oxygen from air in the stack gas. The molar flow rate and composition of stack gas can be calculated as follows: CO2 - \(0.95(1) + 0.05(2) = 1.05\) mol/mol feed, H2O - \(0.95(2) + 0.05(3) = 2.05\) mol/mol feed, N2 - \(11.65(3.76)=43.82\) mol/mol feed, excess O2 - \(11.65(0.20)*2 = 4.66\) mol/mol feed. For 100 moles of feed, the molar flow rates would be 105, 205, 4382 and 466 mol/s for CO2, H2O, N2 and Excess O2 respectively.
03

Calculate the required rate of heat transfer in the preheater (\(\dot{Q}\)

In the heat exchanger, the heat transferred from the stack gas is used to heat the incoming air from \(20^{\circ}C\) to \(245^{\circ}C\). This can be calculated using the energy balance on the air: \(\dot{Q} = \dot{n}_{air} * C_{p, air} * (T_{out} - T_{in}) = 1165 \, mol/s * 29.1 \, J/mol.K * (245 - 20) = 3.67 x 10^6\) joules/s
04

Compute the temperature at which the stack gas leaves the preheater

The temperature at which the stack gas leaves the preheater can be obtained by performing an energy balance on the stack gas. The heat lost by the stack gas in the preheater (\( \dot{Q} \)) is used to heat the incoming air. Using the average specific heat capacities for the stack gas components, the exit temperature can be obtained from the equation: \( \dot{Q} = \dot{n}_{sg} * C_{p, sg} * (T_{in} - T_{out}) \) Assuming that the stack gas enters the preheater at \(900^{\circ}C\), and rearranging, we find that \( T_{out} = T_{in} - (\dot{Q} / (\dot{n}_{sg} * C_{p, sg})) \) The method requires estimation of the composition of the stack gas and involves iterative calculations because the specific heat capacity depends on temperature.
05

Calculate \(\dot{Q}\) for the actual feed rate in part (b)

In part (b), the actual feed rate of natural gas is given as 350 SCMH. To find the value of \(\dot{Q}\) for this feed rate, we simply scale up the flow rates and the heat duty from part (a) according to the ratio of the actual and basis gas feed rates. In this case, the ratio would be 350/359.5 (converting 100 mol/s to SCMH), leading to a new heat duty of \(\dot{Q}_{new} = 3.67 x 10^6 (350/359.5) = 3.58 x 10^6\) joules/s. Note that all flow rates and the heat duty from part (a) would need to be scaled up correspondingly.

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Key Concepts

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

Excess Air Calculation
Calculating the excess air required for combustion is essential for ensuring complete burning of the fuel while preventing energy waste. Excess air refers to the amount of air supplied beyond what is theoretically necessary to achieve full combustion of a given fuel. Here, we consider the stoichiometric amounts of air that react with methane (CH4) and ethane (C2H6) and add 20% more air to ensure complete combustion. This ensures no unburned hydrocarbons are present in the stack gas, leading to cleaner emissions.

For methane, the stoichiometric air requirement is calculated using the equation for complete combustion: \[ ext{CH}_4 + 2 ext{O}_2 + 2(3.76 ext{N}_2) \] With 20% excess air, the total air required is: 11.424 moles of air per mole of methane. Similarly, for ethane: \[ ext{C}_2 ext{H}_6 + rac{7}{2} ext{O}_2 + rac{7}{2}(3.76 ext{N}_2) \] Again, considering 20% excess, we need 16.84 moles of air per mole of ethane.

Using this calculation method ensures that all carbons in the fuel are oxidized to carbon dioxide, maximizing energy output and minimizing pollutants. The overall air flow required is then derived by taking a weighted average based on the proportion of methane and ethane in the fuel mix.
Stack Gas Composition
Understanding the composition of stack gas is critical for analyzing the effectiveness of combustion and emission control. Stack gas is the mixture of gases released post-combustion from a furnace or a stack. It consists primarily of products from the combustion process, including carbon dioxide (CO2), water vapor (H2O), nitrogen (N2), and any unreacted oxygen.

In our exercise, since there's complete combustion with no emissions of carbon monoxide or unburned hydrocarbons, the composition is straightforward. Every mole of methane burned produces one mole of CO2 and two moles of H2O. Comparably, each mole of ethane generates two moles of CO2 and three moles of H2O.
  • CO2: Derived from both methane and ethane contributing 1.05 moles per mole of fuel.
  • H2O: Totaling 2.05 moles per mole of fuel.
  • N2: Primarily from the air, contributing 43.82 moles per mole of fuel.
  • Excess O2: Coming from the unreacted portion of supplied air, amounts to 4.66 moles per mole of fuel.
These figures reflect the output when 100 moles of natural gas with the specified composition are burnt, facilitating an understanding of the emission properties and aiding in regulatory compliance.
Heat Transfer Calculation
Heat transfer calculations help in understanding how energy is distributed during the combustion process, particularly how much heat is recovered and utilized by the heat exchanger. In this exercise, the heat exchanger preheats air entering the furnace using the exhaust gases. This improves efficiency by reducing the amount of combustion energy needed to heat up incoming air.

To determine the amount of heat transferred, or \( \dot{Q} \), an energy balance on the air is performed. The formula used is:
\[ \dot{Q} = \dot{n}_{air} \cdot C_{p, air} \cdot (T_{out} - T_{in}) \]
where \( \dot{n}_{air} \) is the air flow rate, \( C_{p, air} \) is the specific heat capacity, and \( T_{out} \) and \( T_{in} \) are the outlet and inlet temperatures, respectively. For our calculation, this results in \( \dot{Q} = 3.67 \times 10^6 \) J/s, illustrating the energy saved.
  • Energy balance assesses necessary adjustments to stack gas temperature exiting the preheater.
  • Exit temperature of stack gas is critical for optimizing preheating efficiency and minimizing losses.
These calculations ensure that the maximum possible energy is recaptured, enhancing the system's overall thermal efficiency and reducing operational costs.

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Most popular questions from this chapter

The off-gas from a reactor in a process plant in the heart of Freedonia has been condensing and plugging up the vent line, causing a dangerous pressure buildup in the reactor. Plans have been made to send the gas directly from the reactor into a cooling condenser in which the gas and liquid condensate will be brought to \(25^{\circ} \mathrm{C}.\) (a) You have been called in as a consultant to aid in the design of this unit. Unfortunately, the chief (and only) plant engineer has disappeared and nobody else in the plant can tell you what the offgas is (or what anything else is, for that matter). However, a job is a job, and you set out to do what you can. You find an elemental analysis in the engineer's notebook indicating that the gas formula is \(\mathrm{C}_{5} \mathrm{H}_{12} \mathrm{O}\). On another page of the notebook, the off-gas flow rate is given as \(235 \mathrm{m}^{3} / \mathrm{h}\) at \(116^{\circ} \mathrm{C}\) and 1 atm. You take a sample of the gas and cool it to \(25^{\circ} \mathrm{C}\), where it proves to be a solid. You then heat the solidified sample at 1 atm and note that it melts at \(52^{\circ} \mathrm{C}\) and boils at \(113^{\circ} \mathrm{C}\). Finally, you make several assumptions and estimate the heat removal rate in \(\mathrm{kW}\) required to bring the off-gas from \(116^{\circ} \mathrm{C}\) to \(25^{\circ} \mathrm{C}\). What is your result? (b) If you had the right equipment, what might you have done to get a better estimate of the cooling rate?

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.

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.)

The heat capacity at constant pressure of hydrogen cyanide is given by the expression $$ C_{p}\left[J /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right]=35.3+0.0291 T\left(^{\circ} \mathrm{C}\right) $$ (a) Write an expression for the heat capacity at constant volume for HCN, assuming ideal-gas behavior. (b) Calculate \(\Delta \hat{H}(\mathrm{J} / \mathrm{mol})\) for the constant- pressure process $$ \mathrm{HCN}\left(\mathrm{v}, 25^{\circ} \mathrm{C}, 0.80 \mathrm{atm}\right) \rightarrow \mathrm{HCN}\left(\mathrm{v}, 200^{\circ} \mathrm{C}, 0.80 \mathrm{atm}\right) $$(c) Calculate \(\Delta \hat{U}(\mathrm{J} / \mathrm{mol})\) for the constant- volume process $$\mathrm{HCN}\left(\mathrm{v}, 25^{\circ} \mathrm{C}, 50 \mathrm{m}^{3} / \mathrm{kmol}\right) \rightarrow \mathrm{HCN}\left(\mathrm{v}, 200^{\circ} \mathrm{C}, 50 \mathrm{m}^{3} / \mathrm{kmol}\right)$$ (d) If the process of Part (b) were carried out in such a way that the initial and final pressures were each 0.80 atm but the pressure varied during the heating, the value of \(\Delta \hat{H}\) would still be what you calculated assuming a constant pressure. Why is this so?

Molten sodium chloride is to be used as a constant-temperature bath for a high-temperature chemical reactor. Two hundred kilograms of solid \(\mathrm{NaCl}\) at \(300 \mathrm{K}\) is charged into an insulated vessel, and a 3000 kW electrical heater is turned on, raising the salt to its melting point of 1073 K and melting it at a constant pressure of 1 atm. (a) The heat capacity \(\left(C_{p}\right)\) of solid \(\mathrm{NaCl}\) is \(50.41 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K})\) at \(T=300 \mathrm{K},\) and \(53.94 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K})\) at \(T=500 \mathrm{K},\) and the heat of fusion of \(\mathrm{NaCl}\) at \(1073 \mathrm{K}\) is \(30.21 \mathrm{kJ} / \mathrm{mol} .\) Use these data to determine a linear expression for \(C_{p}(T)\) and to calculate \(\Delta \hat{H}\) ( \(\mathrm{kJ} / \mathrm{mol}\) ) for the transition of \(\mathrm{NaCl}\) from a solid at 300 K to a liquid at \(1073 \mathrm{K}\). (b) Write and solve the energy balance equation for this closed system isobaric process to determine the required heat input in kilojoules. (c) If \(85 \%\) of the full power of \(3000 \mathrm{kW}\) goes into heating and melting the salt, how long does the process take?
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