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Chapter 4: Problem 88

Natural gas containing a mixture of methane, ethane, propane, and butane is burned in a furnace with excess air. (a) One hundred kmol/h of a gas containing 94.4 mole\% methane, 3.40\% ethane, 0.60\% propane, and \(0.50 \%\) butane is to be burned with \(17 \%\) excess air. Calculate the required molar flow rate of the air. (b) Let Derive an expression for \(\dot{n}_{\mathrm{a}}\) in terms of the other variables. Check your formula with the results of Part (a). (c) Suppose the feed rate and composition of the fuel gas are subject to periodic variations, and a process control computer is to be used to adjust the flow rate of air to maintain a constant percentage excess. A calibrated electronic flowmeter in the fuel gas line transmits a signal \(R_{\mathrm{f}}\) that is directly proportional to the flow rate \(\left(\dot{n}_{\mathrm{f}}=\alpha R_{\mathrm{f}}\right),\) with a flow rate of \(75.0 \mathrm{kmol} / \mathrm{h}\) yielding a signal \(R_{f}=60 .\) The fuel gas composition is obtained with an on-line gas chromatograph. A sample of the gas is injected into the gas chromatograph (GC), and signals \(A_{1}, A_{2}, A_{3},\) and \(A_{4},\) which are directly proportional to the moles of methane, ethane, propane, and butane, respectively, in the sample, are transmitted. (Assume the same proportionality constant for all species.) The control computer processes these data to determine the required air flow rate and then sends a signal \(R_{\mathrm{a}}\) to a control valve in the air line. The relationship between \(R_{\mathrm{a}}\) and the resulting air flow rate, \(\dot{n}_{\mathrm{a}},\) is another direct proportionality, with a signal \(R_{\mathrm{a}}=25\) leading to an air flow rate of \(550 \mathrm{kmol} / \mathrm{h}\). Write a spreadsheet or computer program to perform the following tasks: (i) Take as input the desired percentage excess and values of \(R_{\mathrm{f}}, A_{1}, A_{2}, A_{3},\) and \(A_{4}\) (ii) Calculate and print out \(\dot{n}_{\mathrm{f}}, x_{1}, x_{2}, x_{3}, x_{4}, \dot{n}_{\mathrm{a}},\) and \(R_{\mathrm{a}}\) Test your program on the data given below, assuming that \(15 \%\) excess air is required in all cases. Then explore the effects of variations in \(P_{\mathrm{xs}}\) and \(R_{\mathrm{f}}\) on \(\dot{n}_{\mathrm{a}}\) for the values of \(A_{1}-A_{4}\) given on the third line of the data table. Briefly explain your results. (d) Finally, suppose that when the system is operating as described, stack gas analysis indicates that the air feed rate is consistently too high to achieve the specified percentage excess. Give several possible explanations.

Short Answer

Expert verified
1. The molar flow rate of air is calculated using stoichiometry and factoring the excess air. 2. In an expression for this flow rate, the contribution of each hydrocarbon to the total air requirement is recognised. 3. In the case of periodic variations, a system of flow monitoring and controlling should be set up, which uses the proportionalities in the gas mix 4. If the air flow tends to be too high, it could be related to malfunctions in the control system, variations in the gas mix, or inappropriate setting of excess air.

Step by step solution

01

Calculate the required molar flow rate

Using the stoichiometry of combustion for hydrocarbons, we know that one kmol of each hydrocarbon requires a specific amount of kmol air for complete combustion. In terms of O2: Methane (CH4) requires 2, Ethane (C2H6) requires 3.5 , Propane (C3H8) requires 5 and Butane (C4H10) requires 6.5. Calculate the total kmol of air required for the 100 % combustion by multiplying the mole percent of each hydrocarbon by its individual requirement and summing up. Multiply this value by 1.17 (17 % excess air) to get the total required molar flow rate of air.
02

Derive an expression

Define \( \dot{n}_{a} \) as the molar flow rate of air. As per the procedure described in step 1, \( \dot{n}_{a} \) can be defined in terms of the mole percent of the hydrocarbons and the individual requirement of air for each. The equation could look like this: \( \dot{n}_{a} = 1.17 * (0.944*2x + 0.034*3.5x + 0.006*5x + 0.005*6.5x) \) where x is the molar flow rate of the gas mixture. Check this formula with the results of Part (a).
03

Periodic Variations

On the assumption of an onsite gas chromatograph and process control computer, the molar flow rates of the distinct species can be calculated, using the proportionality constants given in the question. First find 伪 and 尾, the proportionality constants for the fuel gas line and air line respectively, by using the given values in the question. In the control program or spreadsheet, take the desired excess and the values of \( R_f \), \( A_1 \), \( A_2 \), \( A_3 \) and \( A_4 \) as input and print out \( \dot{n}_{f} \), \( x_1 \), \( x_2 \), \( x_3 \), \( x_4 \), \( \dot{n}_{a} \) and \( R_a \) as output, calculating these parameters using the relationships given in the question and the expressions derived before.
04

Possible Explanations

If the stack gas analysis indicates that the air feed rate is consistently too high, possible reasons could be a malfunction in the control valve, inaccurate readings from the flowmeter or chromatograph, or changes in the calorific value of the gas. It should be investigated if the gas composition is not consistent or the desired percentage excess is not appropriate for the complete combustion.

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Key Concepts

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

Natural Gas Composition
In the world of combustion stoichiometry, the composition of natural gas can significantly affect the process. Natural gas is primarily composed of methane (CH鈧�), but it often includes other hydrocarbons like ethane (C鈧侶鈧�), propane (C鈧僅鈧�), and butane (C鈧凥鈧佲個). Each of these components plays a specific role in the overall combustion process, determining how much oxygen is needed to achieve complete combustion.
Understanding the natural gas composition is crucial because each component requires a different amount of oxygen. Methane requires the least, followed by ethane, propane, and then butane, which needs the most. This hierarchy affects how much air must be supplied to ensure complete combustion, which in technical terms is referred to as the stoichiometry of combustion for hydrocarbons. By knowing the mole percentage of each hydrocarbon, one can calculate the total amount of air needed efficiently.
Excess Air Calculation
Excess air calculation is a vital step in ensuring that combustion occurs efficiently and effectively. In industrial settings, it is common to supply more air than the theoretical amount required for complete combustion. This is known as excess air and is usually expressed as a percentage.

Calculating excess air involves understanding the stoichiometric amount of air required for combustion and then determining an additional amount based on a percentage excess. Typically, processes will specify a percentage such as 17% excess air, which means the total air supplied is 117% of the stoichiometric requirement.
Using this excess ensures that all fuel is combusted, minimizing unburned hydrocarbons and carbon monoxide emissions, and allowing for safer and more reliable operation. The formula for calculating required air considers the mole percentage of each hydrocarbon, their individual air requirements, and the percentage of excess air. This ensures precise control over the combustion process.
Process Control in Combustion
The control of combustion processes is vital in industrial systems to ensure efficiency, safety, and environmental compliance. Advanced process controls use sensors like gas chromatographs and flowmeters to monitor and regulate the air and fuel flow rates.

Gas chromatographs are used to analyze the composition of the fuel gas, sending signals that are proportional to the mole amounts of each component, such as methane, ethane, propane, and butane. Flowmeters measure the molar flow rate of the fuel, which is necessary to adjust the air supply accurately.
A control computer processes these signals and calculates the requisite air flow rate to maintain the correct percentage of excess air. It then sends commands to adjust valves controlling the air flow, ensuring the combustion process remains within desired parameters. Such systems are critical for adapting to variations in fuel supply, ensuring consistent and efficient combustion.
Molar Flow Rate
Understanding the molar flow rate is a cornerstone of working with chemical processes like combustion. The molar flow rate represents the amount of a substance passing through a system per unit time, expressed in kilomoles per hour (kmol/h). It is an essential factor in calculating how much air is needed for combustion.

For natural gas combustion, knowing the molar flow rate of each component helps to determine the air requirements using stoichiometry. For instance, methane, ethane, propane, and butane have specific stoichiometric ratios with oxygen, determining their air needs. When the molar flow rate is correctly managed, it helps ensure that the exact amount of air is supplied, matching the fuel gas's varying compositions.
Accurate molar flow rate calculations assist in maintaining efficiency, reducing waste, and ensuring all combusted hydrocarbons transform into carbon dioxide and water, thus achieving complete combustion.

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Most popular questions from this chapter

Seawater containing 3.50 wt\% salt passes through a series of 10 evaporators. Roughly equal quantities of water are vaporized in each of the 10 units and then condensed and combined to obtain a product stream of fresh water. The brine leaving each evaporator but the tenth is fed to the next evaporator. The brine leaving the tenth evaporator contains \(5.00 \mathrm{wt} \%\) salt. (a) Draw a flowchart of the process showing the first, fourth, and tenth evaporators. Label all the streams entering and leaving these three evaporators. (b) Write in order the set of equations you would solve to determine the fractional yield of fresh water from the process \(\left(\mathrm{kg} \mathrm{H}_{2} \mathrm{O} \text { recovered } / \mathrm{kg} \mathrm{H}_{2} \mathrm{O}\) in process feed) and the weight percent of salt in the \right. solution leaving the fourth evaporator. Each equation you write should contain no more than one previously undetermined variable. In each equation, circle the variable for which you would solve. Do not do the calculations. (c) Solve the equations derived in Part (b) for the two specified quantities. (d) The problem statement made no mention of the disposition of the 5 wt\% effluent from the tenth evaporator. Suggest two possibilities for its disposition and describe any environmental concerns that might need to be considered.

Methanol is formed from carbon monoxide and hydrogen in the gas-phase reaction The mole fractions of the reactive species at equilibrium satisfy the relation where \(P\) is the total pressure (atm), \(K_{c}\) the reaction equilibrium constant (atm \(^{-2}\) ), and \(T\) the temperature (K). The equilibrium constant \(K_{c}\) equals 10.5 at 373 K, and \(2.316 \times 10^{-4}\) at \(573 \mathrm{K}\). A semilog plot of \(K_{\mathrm{c}}\) (logarithmic scale) versus 1/ \(T\) (rectangular scale) is approximately linear between \(T=300 \mathrm{K}\) and \(T=600 \mathrm{K}\) (a) Derive a formula for \(K_{\mathrm{c}}(T),\) and use it to show that \(K_{\mathrm{e}}(450 \mathrm{K})=0.0548 \mathrm{atm}^{-2}\) (b) Write expressions for \(n_{A}, n_{B},\) and \(n_{C}\) (gram-moles of each species), and then \(y_{A}, y_{B},\) and \(y_{C},\) in terms of \(n_{\mathrm{A} 0}, n_{\mathrm{B} 0}, n_{\mathrm{C} 0},\) and \(\xi,\) the extent of reaction. Then derive an equation involving only \(n_{\mathrm{A} 0}, n_{\mathrm{B} 0}, n_{\mathrm{C} 0}, P, T,\) and \(\xi_{e},\) where \(\xi_{e}\) is the extent of reaction at equilibrium. (c) Suppose you begin with equimolar quantities of CO and \(\mathrm{H}_{2}\) and no \(\mathrm{CH}_{3} \mathrm{OH}\), and the reaction proceeds to equilibrium at 423 K and 2.00 atm. Calculate the molar composition of the product ( \(y_{\mathrm{A}}\), \(\left.y_{\mathrm{B}}, \text { and } y_{\mathrm{C}}\right)\) and the fractional conversion of \(\mathrm{CO}\) (d) The conversion of CO and \(\mathrm{H}_{2}\) can be enhanced by removing methanol from the reactor while leaving unreacted CO and \(\mathrm{H}_{2}\) in the vessel. Review the equations you derived in solving Part (c) and determine any physical constraints on \(\xi_{c}\) associated with \(n_{\mathrm{A} 0}=n_{\mathrm{B} 0}=1\) mol. Now suppose that 90\% of the methanol is removed from the reactor as it is produced; in other words, only 10\% of the methanol formed remains in the reactor. Estimate the fractional conversion of CO and the total gram moles of methanol produced in the modified operation. (e) Repeat Part (d), but now assume that \(n_{\mathrm{B} 0}=2\) mol. Explain the significant increase in fractional conversion of CO. (f) Write a set of equations for \(y_{\mathrm{A}}, y_{\mathrm{B}}, y_{\mathrm{C}},\) and \(f_{\mathrm{A}}\) (the fractional conversion of \(\mathrm{CO}\) ) in terms of \(y_{\mathrm{A} 0}, y_{\mathrm{B} 0}, T,\) and \(P(\) the reactor temperature and pressure at equilibrium). Enter the equations in an equation-solving program. Check the program by running it for the conditions of Part (c), then use it to determine the effects on \(f_{\mathrm{A}}\) (increase, decrease, or no effect) of separately increasing, (i) the fraction of \(\mathrm{CH}_{3} \mathrm{OH}\) in the feed, (ii) temperature, and (iii) pressure.

Methane reacts with chlorine to produce methyl chloride and hydrogen chloride. Once formed, the methyl chloride may undergo further chlorination to form methylene chloride ( \(\mathrm{CH}_{2} \mathrm{Cl}_{2}\) ), chloroform, and carbon tetrachloride. A methyl chloride production process consists of a reactor, a condenser, a distillation column, and an absorption column. A gas stream containing 80.0 mole \(\%\) methane and the balance chlorine is fed to the reactor. In the reactor a single-pass chlorine conversion of essentially \(100 \%\) is attained, the mole ratio of methyl chloride to methylene chloride in the product is \(5: 1,\) and negligible amounts of chloroform and carbon tetrachloride are formed. The product stream flows to the condenser. Two streams emerge from the condenser: the liquid condensate, which contains essentially all of the methyl chloride and methylene chloride in the reactor effluent, and a gas containing the methane and hydrogen chloride. The condensate goes to the distillation column in which the two component species are separated. The gas leaving the condenser flows to the absorption column where it contacts an aqueous solution. The solution absorbs essentially all of the HCl and none of the \(\mathrm{CH}_{4}\) in the feed. The liquid leaving the absorber is pumped elsewhere in the plant for further processing, and the methane is recycled to join the fresh feed to the process (a mixture of methane and chlorine). The combined stream is the feed to the reactor. (a) Choose a quantity of the reactor feed as a basis of calculation, draw and label a flowchart, and determine the degrees of freedom for the overall process and each single unit and stream mixing point. Then write in order the equations you would use to calculate the molar flow rate and molar composition of the fresh feed, the rate at which HCI must be removed in the absorber, the methyl chloride production rate, and the molar flow rate of the recycle stream. Do no calculations. (b) Calculate the quantities specified in Part (a), either manually or with an equation-solving program. (c) What molar flow rates and compositions of the fresh feed and the recycle stream are required to achieve a methyl chloride production rate of \(1000 \mathrm{kg} / \mathrm{h} ?\)

A process is carried out in which a mixture containing 25.0 wt\% methanol, \(42.5 \%\) ethanol, and the balance water is separated into two fractions. A technician draws and analyzes samples of both product streams and reports that one stream contains \(39.8 \%\) methanol and \(31.5 \%\) ethanol and the other contains 19.7\% methanol and 41.2\% ethanol. You examine the reported figures and tell the technician that they must be wrong and that stream analyses should be carried out again. (a) Prove your statement. (b) How many streams do you ask the technician to analyze? Explain.

A liquid mixture contains \(60.0 \mathrm{wt} \%\) ethanol \((\mathrm{E}), 5.0 \mathrm{wt} \%\) of a dissolved solute \((\mathrm{S}),\) and the balance water. A stream of this mixture is fed to a continuous distillation column operating at steady state. Product streams emerge at the top and bottom of the column. The column design calls for the product streams to have equal mass flow rates and for the top stream to contain 90.0 wt\% ethanol and no S. (a) Assume a basis of calculation, draw and fully label a process flowchart, do the degree-of-freedom analysis, and verify that all unknown stream flows and compositions can be calculated. (Don't do any calculations yet.) (b) Calculate (i) the mass fraction of \(S\) in the bottom stream and (ii) the fraction of the ethanol in the feed that leaves in the bottom product stream (i.e., \(\mathrm{kg} \mathrm{E}\) in bottom stream/kg \(\mathrm{E}\) in feed) if the process operates as designed. (c) An analyzer is available to determine the composition of ethanol-water mixtures. The calibration curve for the analyzer is a straight line on a plot on logarithmic axes of mass fraction of ethanol, \(x\) (kg E/kg mixture), versus analyzer reading, \(R\). The line passes through the points \((R=15, x=\) 0.100) and \((R=38, x=0.400)\). Derive an expression for \(x\) as a function of \(R(x=\cdots\) ) based on the calibration, and use it to determine the value of \(R\) that should be obtained if the top product stream from the distillation column is analyzed. (d) Suppose a sample of the top stream is taken and analyzed and the reading obtained is not the one calculated in Part (c). Assume that the calculation in Part (c) is correct and that the plant operator followed the correct procedure in doing the analysis. Give five significantly different possible causes for the deviation between \(R_{\text {measured and }} R_{\text {prediced }}\), including several assumptions made when writing the balances of Part (c). For each one, suggest something that the operator could do to check whether it is in fact the problem.
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