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Chapter 5: Problem 45

Ethane at \(25^{\circ} \mathrm{C}\) and 1.1 atm (abs) flowing at a rate of \(100 \mathrm{mol} / \mathrm{s}\) is burned with \(20 \%\) excess oxygen at \(175^{\circ} \mathrm{C}\) and 1.1 atm \((\text { abs }) .\) The combustion products leave the furnace at \(800^{\circ} \mathrm{C}\) and 1 atm. (a) What is the volumetric flow rate of oxygen (L/s) fed to the furnace? (b) What should the volumetric flow rate of the combustion products be? State all assumptions you make. (c) The volumetric flow rate of the combustion products is measured and found to be different from the value calculated in Part (b). Assuming that no mistakes were made in the calculation, what could be going on that could lead to the discrepancy? Consider assumptions made in the calculations and things that can go wrong in a real system.

Short Answer

Expert verified
(a) The volumetric flow rate of oxygen fed into the furnace is approximately 15771 L/s. (b) The calculated volumetric flow rate of the combustion products should be roughly 44116 L/s. (c) Discrepancies can be due to non-ideal gas behaviour, losses during combustion, measurement errors, or system leaks.

Step by step solution

01

Write the balanced combustion reaction

The combustion reaction of ethane (C2H6) with oxygen (O2) can be written as: \[ \mathrm{C2H6 + \frac{7}{2}O2 \rightarrow 2CO2 + 3H2O }\]
02

Calculate the stoichiometric requirement of \(\mathrm{O2}\)

Given that ethane is burned with 20% excess \(\mathrm{O2}\), the molar flow rate of \(\mathrm{O2}\) required for the combustion is \(1.2 \times \frac{7}{2} = 4.2\) mol \(\mathrm{O2}\) for every 1 mol of \(\mathrm{C2H6}\). Therefore, the total molar flow rate of \(\mathrm{O2}\) is \(4.2 \times 100 = 420 \) mol/s.
03

Apply the ideal gas law

The ideal gas law is given as \[ PV=nRT \]. In this context, V is the volume of the gas, n is the number of moles, R is the gas constant and T is the temperature in Kelvin. Solving for V and substituting the given values (\(n=420 \) mol/s, \(R=0.0821\) L.atm/mol.K \(T=175+273= 448K\) and \(P=1.1\) atm), we get the volumetric flow rate of \(\mathrm{O2}\) as: \[ V = \frac{nRT}{P} = \frac{420 \times 0.0821 \times 448}{1.1} = 15770.5636 \] L/s.
04

Calculate the volumetric flow rate of the combustion products

According to the balanced reaction, 1 mol of ethane produces 2 mol of \(\mathrm{CO2}\) and 3 mol of \(\mathrm{H2O}\), that is, a total of 5 moles of gaseous products. So, in this case the total molar flow rate of the combustion products is \(5 \times 100 = 500\) mol/s. Substituting \(n=500 \) mol/s, \(R=0.0821\) L.atm/mol.K, \(T=800+273= 1073K\) and \(P=1\) atm into the ideal gas law equation, we get the volumetric flow rate of the combustion products as: \[ V = \frac{nRT}{P} = \frac{500 \times 0.0821 \times 1073}{1} = 44115.515 \] L/s.
05

Identify possible reasons for any discrepancy

The calculated values could be different from the real-world measurements due to several reasons such as deviations from the assumed ideal gas behaviour, loss of combustion products during the process, inaccurate measurement of initial conditions, or leaks in the system.

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Key Concepts

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

Ideal Gas Law
The ideal gas law is a fundamental principle in chemistry and physics that relates the pressure, volume, temperature, and the amount of an ideal gas. The relationship is given by the equation \( PV = nRT \), where \( P \) represents pressure in atmospheres, \( V \) is the volume in liters, \( n \) is the number of moles, \( R \) is the ideal gas constant (0.0821 L.atm/mol.K), and \( T \) is the temperature in Kelvin.

When dealing with real gases, especially at high temperatures and low pressures, they tend to behave similarly to an ideal gas. This assumption simplifies the complex interactions between particles in a gas, making it easier to predict and calculate properties such as volumetric flow rate in chemical processes like combustion. In the provided exercise, the ideal gas law is used to calculate the volumetric flow rate of both the oxygen fed to the furnace and the combustion products exiting the furnace.
Volumetric Flow Rate Calculation
The volumetric flow rate is a measure of the volume of fluid (in this case, a gas) that passes through a given surface per unit time. It is typically expressed in liters per second (L/s). For gases, the calculation of the volumetric flow rate often incorporates the ideal gas law, as changes in pressure, temperature, and the number of moles of gas affect the volume it occupies.

The volumetric flow rate calculation provides important information in engineering and scientific applications such as identifying the capacity of a gas stream to enter a reactor, size pipelines, and ensure the right amount of reactants in industrial processes. In the context of the exercise, after determining the molar flow rate of oxygen required for ethane combustion, using the ideal gas law helps convert this into a real-world figure, the volumetric flow rate of oxygen, which is then used similarly to find the volumetric flow rate of the combustion products.
Chemical Process Assumptions
In solving chemical engineering problems, certain assumptions are made to simplify the calculations and make them more manageable. Common assumptions include treating gases as ideal, assuming constant temperature and pressure throughout the process, and neglecting losses due to incomplete combustion or physical leaks.

In real-world applications, these assumptions may not hold true. Factors such as non-ideal gas behavior, heat losses, pressure drops, and equipment inefficiencies can lead to discrepancies between calculated and actual values. In the exercise, if the measured volumetric flow rate of combustion products does not match the calculation, it could indicate such real-world effects. This highlights the importance of understanding the limitations of theoretical models and the potential need for adjusting them to better reflect the system being studied.

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Most popular questions from this chapter

A fuel cell is an electrochemical device that reacts hydrogen with oxygen from the air to produce water and DC electricity. A proposed application is replacement of the gasoline-fueled internal combustion engine in an automobile with a \(100 \mathrm{kW}\) fuel cell. You are on a summer internship with a gas supplier planning to transport hydrogen to service stations for use in cars powered by fuel cells. The hydrogen is to be transported in tube trailers, each of which has 10 tubes of length \(10.5 \mathrm{m}\) and diameter \(0.56 \mathrm{m}\). Hydrogen in the tubes at 2600 psig and an average temperature of \(298 \mathrm{K}\) is discharged at service stations to a final pressure of 55 psig. Refueling cach fuel-cell-powered automobile is estimated to require 4.0 kg of hydrogen. (a) You and your office-mate- an intern from a different university - have been asked to estimate the number of automobiles that can be refueled by one tube-trailer load of hydrogen. He does a very quick calculation and comes up with a value of 95 cars. Speculate how he did it and provide support for your speculation. What was his mistake? (b) Do the calculation using the SRK equation of state. Instead of using Eqs. \(5.3-11\) and \(5.3-13\) for the parameter \(\alpha,\) use the following correlation developed specifically for hydrogen: \(^{23}\) \(\alpha=1.202 \exp \left(-0.3228 T_{\mathrm{r}}\right)\) (c) Do the calculation using the law of corresponding states. (d) In which of the three estimates would you have the greatest confidence, and why?

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 nitrogen rotameter is calibrated by feeding \(\mathrm{N}_{2}\) from a compressor through a pressure regulator, a needle valve, the rotameter, and a dry test meter, a device that measures the total volume of gas that passes through it. A water manometer is used to measure the gas pressure at the rotameter outlet. A flow rate is set using the needle valve, the rotameter reading, \(\phi\), is noted, and the change in the dry gas meter reading \((\Delta V)\) for a measured running time \((\Delta t)\) is recorded. The following calibration data are taken on a day when the temperature is \(23^{\circ} \mathrm{C}\) and barometric pressure is \(763 \mathrm{mm} \mathrm{Hg} .\) $$\begin{array}{rrr} \hline \phi & \Delta t(\min ) & \Delta V(\mathrm{L}) \\ \hline 5.0 & 10.0 & 1.50 \\ 9.0 & 10.0 & 2.90 \\ 12.0 & 5.0 & 2.00 \\ \hline \end{array}$$ (a) Prepare a calibration chart of \(\phi\) versus \(\dot{V}_{\text {sid }}\), the flow rate in standard \(\mathrm{cm}^{3} / \mathrm{min}\) equivalent to the actual flow rate at the measurement conditions. (b) Suppose the rotameter-valve combination is to be used to set the flow rate to 0.010 mol \(\mathrm{N}_{2} / \mathrm{min}\). What rotameter reading must be maintained by adjusting the valve?

The bacteria acetobacter aceti convert ethanol to acetic acid in the presence of oxygen according to the reaction $$\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}+\mathrm{O}_{2} \rightarrow \mathrm{CH}_{3} \mathrm{COOH}+\mathrm{H}_{2} \mathrm{O}$$ In a continuous fermentation process, ethanol enters the top of the fermenter at a rate of \(145 \mathrm{kg} / \mathrm{h}\), and the air fed to the bottom of the fermenter is \(25 \%\) in excess of the amount required to consume all of the ethanol. A gas stream containing nitrogen and unreacted oxygen leaves the top of the fermenter, and a liquid stream containing acetic acid, water, and \(10 \%\) of the entering ethanol leaves the bottom. Assume that none of the ethanol, water, and acetic acid in the reactor is vaporized. The fermenter operates at \(30^{\circ} \mathrm{C},\) maintains a liquid \((\mathrm{SG}=0.95)\) height of \(4.5 \mathrm{m},\) and is open to the atmosphere (i.e., the pressure at the top of the fermenter is 1 atm). (a) What is the volumetric flow rate of air as it enters the bottom of the fermenter? What is the volumetric flow rate of gas leaving the top of the fermenter? (b) Assume a linear relationship between the fraction of oxygen reacted and the position of gas bubbles rising through the liquid in the fermenter: for example, half of the oxygen reacted is consumed in the bottom half of the fermenter. At the vertical midpoint of the fermenter, the average bubble diameter is \(1.5 \mathrm{mm}\). What is the average bubble diameter at the entry point of the air and as the gas leaves the liquid at the top of the fermenter?

The lower flammability limit (LFL) and the upper flammability limit (UFL) of propane in air at 1 atm are, respectively, 2.3 mole \(\% \mathrm{C}_{3} \mathrm{H}_{8}\) and 9.5 mole \(\% \mathrm{C}_{3} \mathrm{H}_{8} .^{17}\) If the mole percent of propane in a propane-air mixture is between \(2.3 \%\) and \(9.5 \%,\) the gas mixture will burn explosively if exposed to a flame or spark; if the percentage is outside these limits, the mixture is safe-a match may burn in it but the flame will not spread. If the percentage of propane is below the LFL, the mixture is said to be too lean to ignite; if it is above the UFL, the mixture is too rich to ignite. (a) Which would be safer to release into the atmosphere- -a fuel-air mixture that is too lean or too rich to ignite? Explain. (b) A mixture of propane in air containing 4.03 mole \(\% \mathrm{C}_{3} \mathrm{H}_{8}\) is fed to a combustion furnace. If there is a problem in the furnace, the mixture is diluted with a stream of pure air to make sure that it cannot accidentally ignite. If propane enters the furnace at a rate of \(150 \mathrm{mol} \mathrm{C}_{3} \mathrm{H}_{8} / \mathrm{s}\) in the original fuel- air mixture, what is the minimum molar flow rate of the diluting air? (c) The actual diluting air molar flow rate is specified to be \(130 \%\) of the minimum value. Assuming the fuel mixture (4.03 mole\% \(\mathrm{C}_{3} \mathrm{H}_{8}\) ) enters the furnace at the same rate as in Part (b) at \(125^{\circ} \mathrm{C}\) and 131 kPa and the diluting air enters at \(25^{\circ} \mathrm{C}\) and \(110 \mathrm{kPa}\), calculate the ratio \(\left(\mathrm{m}^{3} \text { diluting air) } /\right.\) (m \(^{3}\) fuel gas) and the mole percent of propane in the diluted mixture. (d) Give several possible reasons for feeding air at a value greater than the calculated minimum rate.
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