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Chapter 9: Problem 60

A fuel gas containing 85.0 mole\% methane and the balance ethane is burned completely with pure oxygen at \(25^{\circ} \mathrm{C},\) and the products are cooled to \(25^{\circ} \mathrm{C}\). (a) Suppose the reactor is continuous. Take a basis of calculation of \(1 \mathrm{mol} / \mathrm{s}\) of the fuel gas, assume some value for the percent excess oxygen fed to the reactor (the value you choose will not affect the results), and calculate \(-\dot{Q}(\mathrm{k} \mathrm{W}),\) the rate at which heat must be transferred from the reactor. (b) Now suppose the combustion takes place in a constant-volume batch reactor. Take a basis of calculation of 1 mol of the fuel gas charged into the reactor, assume any percent excess oxygen, and calculate \(-Q(\mathrm{kJ}) .\) (Hint: Recall Equation 9.1-5.) (c) Briefly explain why the results in Parts (a) and (b) do not depend on the percent excess \(\mathrm{O}_{2}\) and why they would not change if air rather than pure oxygen were fed to the reactor.

Short Answer

Expert verified
The heat transfer rate \(\dot Q\) for the continuous reactor and \(Q\) for the batch reactor can be calculated using the overall combustion reaction and the standard heat of combustion. The heat transfer does not depend on the percent excess oxygen or whether air or pure oxygen is used because these do not change the heat of combustion.

Step by step solution

01

Determine the Stoichiometry of the Reaction

Firstly, let's write the balanced chemical equation for the combustion of methane and ethane: \[CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O\] for methane, and \[C_2H_6 + 3.5O_2 \rightarrow 2CO_2 + 3H_2O\] for ethane. Given that the fuel gas is 85.0 mole% methane and the balance is ethane, the overall combustion reaction is a weighted average of these two reactions.
02

Heat Transfer Calculation for Continuous Reactor

In the continuous reactor, the heat transfer rate or \(\dot Q\) is determined by the enthalpy change per mole of fuel gas feed. The standard heat of combustion can be obtained from literature values. \(\dot Q\) is commonly given in kW, so we will convert it by dividing by 1000.
03

Heat Transfer Calculation for Batch Reactor

In the batch reactor, the heat transfer or \(Q\) is based on the total enthalpy change for the entire batch of fuel gas. We multiply the standard heat of combustion by the total moles of fuel gas supplied to the reactor to calculate the heat transfer. As the question suggests, we can use Equation 9.1-5 which is consistent with this approach.
04

Explanation of Excess Oxygen Effect

Finally, we can explain why the percent excess of \(O_2\) and the use of air or pure oxygen does not affect the results. Excess oxygen does not participate in the reaction, therefore it does not change the heat of combustion. Whether air or pure oxygen is used does not make any difference to the combustion process, since nitrogen in the air does not react in the combustion process.

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Key Concepts

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

Combustion Stoichiometry
Combustion stoichiometry involves the quantitative relationship between the reactants and products in a combustion reaction. In our exercise, we are dealing with a fuel gas that is 85% methane and 15% ethane. Each component requires a specific amount of oxygen for complete combustion.
It's crucial to write balanced chemical equations to ensure all reactants and products are accounted for.
  • For methane: \[CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O\]
  • For ethane: \[C_2H_6 + 3.5O_2 \rightarrow 2CO_2 + 3H_2O\]
In combustion stoichiometry, these equations show how oxygen (O2) is consumed to produce carbon dioxide (CO2) and water (H2O). This process releases energy, as expressed in heat of combustion values. Understanding this helps in calculating how much oxygen is needed and how much energy will be released during combustion.
Heat Transfer Calculations
Heat transfer calculations in chemical reactions help determine the energy flow in a system. In continuous reactors, where the reaction is ongoing, heat transfer is commonly described as the rate of heat energy transfer, \(\dot{Q}\).
  • We begin by calculating the enthalpy change per mole of fuel gas.
  • The values for the heat of combustion are sourced from standard chemical data.
  • Finally, the heat transfer rate, \(\dot{Q}\), is converted to kW by dividing the change in enthalpy by 1000.
In a batch reactor, the total heat transfer, \(Q\), represents the entire system's enthalpy change from the start to the finish of the reaction. You calculate \(Q\) by multiplying the number of moles of fuel gas by the standard heat of combustion. Following Equation 9.1-5 in the problem provides a reliable approach to calculate this.
Excess Oxygen Impact
Excess oxygen in combustion reactions refers to oxygen that is present in greater amounts than is chemically required to completely combust the fuel. In our scenario, we find that the amount of excess oxygen chosen doesn't alter the combustion results.
  • Excess oxygen remains unreacted and thus does not contribute additional energy.
  • This concept is important when designing reactors, as you want to avoid wasteful energy input.
  • Switching from pure oxygen to air (which includes nitrogen) also doesn't impact the combustion, as nitrogen is generally inert and does not react with the fuel or affect the enthalpy change.
Therefore, while ensuring that there is enough oxygen to complete the reaction is vital, the exact percentage of excess oxygen will not alter the combustion process or the resulting energy output.
Batch and Continuous Reactors
Batch and continuous reactors are two main types used in chemical processes. Their modes of operation differ significantly, affecting how reactions proceed and are managed.
  • Batch Reactors: These operate by mixing the reactants all at once and letting the reaction occur in a closed system. The reaction proceeds until completion, at which point the products are collected for final processing. Heat transfer calculations in batch reactors target the total system enthalpy change, providing \(Q\) in kJ for the entire batch.
  • Continuous Reactors: These, on the other hand, operate by continuously feeding reactants into the system and constantly removing products. The heat transfer is described as a rate, expressed as \(\dot{Q}\) and usually given in kW.
Understanding these operational differences is key for engineers to effectively design and optimize reactors to maximize yield while minimizing energy consumption and waste.

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Most popular questions from this chapter

A dilute aqueous solution of sulfuric acid at \(25^{\circ} \mathrm{C}\) is used to absorb ammonia in a continuous reactor, thereby producing ammonium sulfate, a fertilizer: $$2 \mathrm{NH}_{3}(\mathrm{g})+\mathrm{H}_{2} \mathrm{SO}_{4}(\mathrm{aq}) \rightarrow\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}(\mathrm{aq})$$ (a) If the ammonia enters the absorber at \(75^{\circ} \mathrm{C}\), the sulfuric acid enters at \(25^{\circ} \mathrm{C}\), and the product solution emerges at \(25^{\circ} \mathrm{C}\), how much heat must be withdrawn from the unit per mol of \(\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}\) produced? (All needed physical property data may be found in Appendix B.) (b) Estimate the final temperature if the reactor of Part (a) is adiabatic and the product of the solution contains 1.00 mole \(\%\) ammonium sulfate. Take the heat capacity of the solution to be that of pure liquid water [4.184 kJ/(kg.'C)]. (c) In a real (imperfectly insulated) reactor, would the final solution temperature be less than, equal to, or greater than the value calculated in Part (b), or is there no way to tell without more information? Briefly explain your answer.

Sulfur dioxide is oxidized to sulfur trioxide in a small pilot-plant reactor. SO \(_{2}\) and \(100 \%\) excess air are fed to the reactor at \(450^{\circ} \mathrm{C}\). The reaction proceeds to a \(65 \% \mathrm{SO}_{2}\) conversion, and the products emerge from the reactor at \(550^{\circ} \mathrm{C}\). The production rate of \(\mathrm{SO}_{3}\) is \(1.00 \times 10^{2} \mathrm{kg} / \mathrm{min}\). The reactor is surrounded by a water jacket into which water at \(25^{\circ} \mathrm{C}\) is fed. (a) Calculate the feed rates (standard cubic meters per second) of the \(\mathrm{SO}_{2}\) and air feed streams and the extent of reaction, \(\xi\) (b) Calculate the standard heat of the SO_ oxidation reaction, \(\Delta H_{\mathrm{t}}^{\mathrm{r}}(\mathrm{kJ}) .\) Then, taking molecular species at \(25^{\circ} \mathrm{C}\) as references, prepare and fill in an inlet-outlet enthalpy table and write an energy balance to calculate the necessary rate of heat transfer ( \(\mathrm{kW}\) ) from the reactor to the cooling water. (c) Calculate the minimum flow rate of the cooling water if its temperature rise is to be kept below \(15^{\circ} \mathrm{C}\) (d) Briefly state what would have been different in your calculations and results if you had taken elemental species as references in Part (b).

A mixture of air and a fine spray of gasoline at ambient (outside air) temperature is fed to a set of pistonfitted cylinders in an automobile engine. Sparks ignite the combustible mixtures in one cylinder after another, and the consequent rapid increase in temperature in the cylinders causes the combustion products to expand and drive the pistons. The back-and-forth motion of the pistons is converted to rotary motion of a crank shaft, motion that in turn is transmitted through a system of shafts and gears to propel the car. Consider a car driving on a day when the ambient temperature is 298 K and suppose that the rate of heat loss from the engine to the outside air is given by the formula $$-\dot{Q}_{1}\left(\frac{\mathrm{kJ}}{\mathrm{h}}\right) \approx \frac{15 \times 10^{6}}{T_{\mathrm{a}}(\mathrm{K})}$$ where \(T_{\mathrm{a}}\) is the ambient temperature. (a) Take gasoline to be a liquid with a specific gravity of 0.70 and a higher heating value of \(49.0 \mathrm{kJ} / \mathrm{g}\), assume complete combustion and that the combustion products leaving the engine are at \(298 \mathrm{K}\), and calculate the minimum feed rate of gasoline (gal/h) required to produce 100 hp of shaft work. (b) If the exhaust gases are well above \(298 \mathrm{K}\) (which they are), is the work delivered by the pistons more or less than the value determined in Part (a)? Explain. (c) If the ambicnt temperature is much lower than \(298 \mathrm{K}\), the work delivered by the pistons would decrease. Give two reasons.

Methane and \(30 \%\) excess air are to be fed to a combustion reactor. An inexperienced technician mistakes his instructions and charges the gases together in the required proportion into an evacuated closed tank. (The gases were supposed to be fed directly into the reactor.) The contents of the charged tank are at \(25^{\circ} \mathrm{C}\) and 4.00 atm absolute. (a) Calculate the standard internal energy of combustion of the methane combustion reaction. \(\Delta \hat{U}_{c}^{\circ}(\mathrm{kJ} / \mathrm{mol}),\) taking \(\mathrm{CO}_{2}(\mathrm{g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{v})\) as the presumed products. Then prove that if the constant-pressure heat capacity of an ideal-gas species is independent of temperature, the specific internal energy of that species at temperature \(T\left(^{\circ} \mathrm{C}\right)\) relative to the same species at \(25^{\circ} \mathrm{C}\) is given by the expression $$\hat{U}=\left(C_{p}-R\right)\left(T-25^{\circ} \mathrm{C}\right)$$ where \(R\) is the gas constant. Use this formula in the next part of the problem. (b) You wish to calculate the maximum temperature, \(T_{\max }\left(^{\circ} \mathrm{C}\right),\) and corresponding pressure, \(P_{\max }(\text { atm }),\) that the tank would have to withstand if the mixture it contains were to be accidentally ignited. Taking molecular species at \(25^{\circ} \mathrm{C}\) as references and treating all species as ideal gases, prepare an inlet-outlet internal energy table for the closed system combustion process. In deriving expressions for each \(\dot{U}_{i}\) at the final reactor condition \(\left(T_{\max }, P_{\max }\right),\) use the following approximate values for \(C_{p_{i}}\left[\mathrm{k} J /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right]: 0.033 \mathrm{for} \mathrm{O}_{2}, 0.032\) for \(\mathrm{N}_{2}, 0.052 \mathrm{for} \mathrm{CO}_{2},\) and \(0.040 \mathrm{for} \mathrm{H}_{2} \mathrm{O}(\mathrm{v}) .\) Then use an energy balance and the ideal-gas equation of state to perform the required calculations. (c) Why would the actual temperature and pressure attained in a real tank be less than the values calculated in Part (a)? (State several reasons.) (d) Think of ways that the tank contents might be accidentally ignited. The list should suggest why accepted plant safety regulations prohibit the storage of combustible vapor mixtures.

A methanol-synthesis reactor is fed with a gas stream at \(220^{\circ} \mathrm{C}\) consisting of 5.0 mole\% methane, \(25.0 \%\) CO, \(5.0 \% \mathrm{CO}_{2},\) and the remainder hydrogen. The reactor and feed stream are at \(7.5 \mathrm{MPa}\). The primary reaction occurring in the reactor and its associated equilibrium constant are $$\begin{array}{l}\mathrm{CO}+2 \mathrm{H}_{2} \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH} \\\K=\frac{y_{\mathrm{CH}, \mathrm{OH}} y_{\mathrm{H}_{2}}}{y_{\mathrm{CO}} y_{H_{2}}^{2} P^{2}}=\exp \left(\begin{array}{c}21.225+\frac{9143.6}{T}-7.492 \ln T \\ +4.076 \times 10^{-3} T-7.161 \times 10^{-8} T^{2}\end{array}\right)\end{array}$$ where \(T\) is in kelvins. The product stream may be assumed to reach equilibrium at \(250^{\circ} \mathrm{C}\). (a) Determine the composition (mole fractions) of the product stream and the percentage conversions of CO and \(\mathrm{H}_{2}\). (b) Neglecting the effect of pressure on enthalpies, estimate the amount of heat (kJ/mol feed gas) that must be added to or removed from (state which) the reactor. (c) Calculate the extent of reaction and heat removal rate (kJ/mol feed) for reactor temperatures between \(200^{\circ} \mathrm{C}\) and \(400^{\circ} \mathrm{C}\) in \(50^{\circ} \mathrm{C}\) increments. Use these results to obtain an estimate of the adiabatic reaction temperature. (d) Determine the effect of pressure on the reaction by evaluating extent of conversion and rate of heat transfer at \(1 \mathrm{MPa}\) and \(15 \mathrm{MPa}\). (e) Considering the results of your calculations in Parts (c) and (d), propose an explanation for selection of the initial reaction conditions of \(250^{\circ} \mathrm{C}\) and \(7.5 \mathrm{MPa}\).
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