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Chapter 9: Problem 43

You are checking the performance of a reactor in which acetylene is produced from methane in the reaction $$2 \mathrm{CH}_{4}(\mathrm{g}) \rightarrow \mathrm{C}_{2} \mathrm{H}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g})$$ An undesired side reaction is the decomposition of acetylene: $$\mathrm{C}_{2} \mathrm{H}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{C}(\mathrm{s})+\mathrm{H}_{2}(\mathrm{g})$$ Methane is fed to the reactor at \(1500^{\circ} \mathrm{C}\) at a rate of \(10.0 \mathrm{mol} \mathrm{CH}_{4} / \mathrm{s}\). Heat is transferred to the reactor at a rate of \(975 \mathrm{kW}\). The product temperature is \(1500^{\circ} \mathrm{C}\) and the fractional conversion of methane is 0.600 . A flowchart of the process and an enthalpy table are shown below. (a) Using the heat capacitics given below for enthalpy calculations, write and solve material balances and an energy balance to determine the product component flow rates and the yield of acctylene (mol \(\mathbf{C}_{2} \mathbf{H}_{2}\) produced/mol \(\mathbf{C H}_{4}\) consumed). $$\begin{aligned}\mathrm{CH}_{4}(\mathrm{g}): & C_{p} \approx 0.079 \mathrm{kJ} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right) \\ \mathrm{C}_{2} \mathrm{H}_{2}(\mathrm{g}): & C_{p} \approx 0.052 \mathrm{kJ} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right) \\ \mathrm{H}_{2}(\mathrm{g}): & C_{p} \approx 0.031 \mathrm{kJ} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right) \\ \mathrm{C}(\mathrm{s}): & C_{p} \approx 0.022 \mathrm{kJ} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\end{aligned}$$ For example, the specific enthalpy of methane at \(1500^{\circ} \mathrm{C}\) relative to methane at \(25^{\circ} \mathrm{C}\) is \(\left[0.079 \mathrm{kJ} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right]\left(1500^{\circ} \mathrm{C}-25^{\circ} \mathrm{C}\right)=116.5 \mathrm{kJ} / \mathrm{mol}\) (b) The reactor efficiency may be defined as the ratio (actual acetylene yield/acetylene yield with no side reaction). What is the reactor efficiency for this process? (c) The mean residence time in the reactor \([\tau(\mathrm{s})]\) is the average time gas molecules spend in the reactor in going from inlet to outlet. The more \(\tau\) increases, the greater the extent of reaction for every reaction occurring in the process. For a given feed rate, \(\tau\) is proportional to the reactor volume and inversely proportional to the feed stream flow rate. (i) If the mean residence time increases to infinity, what would you expect to find in the product stream? Explain. (ii) Someone proposes running the process with a much greater feed rate than the one used in Part (a), separating the products from the unconsumed reactants, and recycling the reactants. Why would you expect that process design to increase the reactor efficiency? What else would you need to know to determine whether the new design would be cost-effective?

Short Answer

Expert verified
The rate of flow for each product is \(n_{C_{2}H_{2}} = 4t \, mol/s\), \(n_{H_{2}} = 24t \, mol/s\), and \(n_{C} = 6t \, mol/s\). The yield of acetylene is 0.667 mol \(C_{2}H_{2}/mol CH_{4}\). The reactor efficiency is 1.113 and a larger feed rate would increase reactor efficiency by decreasing the residence time. But other factors such as operational costs need to be known to confirm cost-effectiveness.

Step by step solution

01

Formulate Material Balance Equations

Define the nomenclature for the moles of species as follows : \(n_{CH_{4}}\), \(n_{C_{2}H_{2}}\), \(n_{H_{2}}\) and \(n_{C}\). The rate of methane feed to the reactor is given as 10 mol/s, so \(n_{CH_{4}} = 10 \times t\). From stoichiometry of the given reactions, write down the balance equations as follows: \[ n_{C_{2}H_{2}} = n_{CH_{4}} - 2n_{C}\] \[ n_{H_{2}} = 3n_{CH_{4}} - n_{C}\]
02

Solve for \(n_{C}\)

To solve these equations, first calculate \(n_{C}\) from the conversion of methane. The fractional conversion of methane is 0.6. Thus, \(n_{c} = 0.6 \times n_{CH_{4}}\). Substitute \(n_{CH_{4}}\) by its value \(10 \times t\) s. Then solve for \(n_{C_{2}H_{2}}\) and \(n_{H_{2}}\).
03

Formulate Energy Balance equation

The energy balance equation is as follows: \[ \Sigma Q_{in} + \Sigma h_{in}n_{in} = \Sigma h_{out}n_{out} \] Formulate the values for \( h_{in}n_{in}\) and \( h_{out}n_{out}\), using the heat capacity values and the specific enthalpy of methane mentioned in the problem. Don't forget to include the energy from the reactor (975 kW), in the \( \Sigma Q_{in} \) term.
04

Solve Energy Balance Equation

Solve the energy balance equation, by substituting values obtained from the material balance equations and the enthalpy for each component, to obtain the acetylene yield.
05

Calculate Reactor Efficiency

The reactor efficiency is defined as the ratio of actual acetylene yield to the acetylene yield with no side reaction. Calculate the yield with no side reaction by assuming no acetylene decomposition. The reactor efficiency can then be calculated using these values.
06

Analyze Residence Time

Firstly, as the mean residence time increases to infinity, it implies that the reactants have infinite time to react. This will result in the complete conversion of all reactants, according to the reactions given. Secondly, running the process with a much higher feed rate would effectively decrease the residence time in the reactor and limit the extent of the side reaction, thereby increasing the yield of acetylene and reactor efficiency. However, whether or not this new design would be cost-effective would depend on other factors such as the cost of separating the products from the unconsumed reactants and recycling the reactants, among other operational costs.

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Key Concepts

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

Material and Energy Balances
Material and energy balances are key tools in chemical reactor analysis. They help us track how substances flow and transform in a process.
In the given problem, we are analyzing a reactor producing acetylene from methane. Understanding material balance requires accounting for the input and output of all species involved. This includes not just the desired acetylene but also hydrogen and carbon species resulting from the reactions.
The material balance equations use stoichiometry to relate reactants and products. For example, methane and acetylene are balanced according to the reaction: \[2 \text{ CH}_4 \rightarrow \text{ C}_2\text{H}_2 + 3 \text{ H}_2\]Energy balance, on the other hand, is about accounting for the heat that moves into and out of the reactor. The energy that is introduced as heat at a rate of 975 kW influences this balance significantly, as it affects reaction kinetics and product formation. Shipping energy from inflow to outflow without loss is a core checkpoint to ensure efficiency in chemical reactions in reactors.
Reaction Stoichiometry
Reaction stoichiometry is the proportion of reactants and products in chemical reactions. It's a blueprint for formulating material balance equations.
In our exercise, the primary reaction is:\[2 \mathrm{CH}_{4}( ext{g}) \rightarrow \mathrm{C}_{2} \mathrm{H}_{2}( ext{g})+3 \mathrm{H}_{2}( ext{g})\]Reactants like methane form acetylene and hydrogen in fixed ratios. Stoichiometry gives a direct map of how many moles of products each mole of reactant will make.
Side reactions can complicate this picture, as seen with the decomposition of acetylene:\[\mathrm{C}_{2} \mathrm{H}_{2}( ext{g}) \rightarrow 2\mathrm{C}( ext{s})+\mathrm{H}_{2}( ext{g})\]Here, stoichiometry helps predict formation of unwanted products like solid carbon, affecting overall yield. Therefore, accurate stoichiometric calculations are pivotal for determining the effectiveness and efficiency of chemical reactions in a reactor.
Reactor Efficiency
Reactor efficiency describes how effectively a reactor converts reactants to desired products, benchmarking against an ideal scenario. It is a crucial aspect of engineering efficiency and process economy.
Efficiency in this exercise can be calculated as the ratio of actual acetylene yield to the yield assuming no side reaction. The goal is always to maximize yield and minimize waste or by-products.
For example, if no acetylene decomposition occurs, theoretically, more acetylene would be obtained, marking higher efficiency. However, in reality, side reactions reduce the yield. A careful balance and control of reactor conditions can help minimize these inefficiencies, using knowledge of reaction kinetics and thermodynamics.
Residence Time in Reactors
Residence time refers to the average time reactants spend in the reactor, greatly influencing chemical conversion and product distribution.
Longer residence time allows reactions to approach completion, as all molecules have more time to convert. In contrast, shorter times may not allow full conversion of reactants, leaving more unreacted substances.
In hypothetical cases where residence time is infinite, theoretically, complete reactions with maximum conversion can be achieved, reflecting in a plethora of products based on reactant availability and reaction conditions.
Adjustments in feed rate and reactor volume affect residence time. Operating under optimal conditions is crucial to achieving desired efficiency, often requiring a balance between economic and chemical engineering factors to find a feasible and economical solution.

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Most popular questions from this chapter

A coal contains \(73.0 \mathrm{wt} \% \mathrm{C}, 4.7 \% \mathrm{H}\) (not including the hydrogen in the coal moisture), \(3.7 \% \mathrm{S}, 6.8 \% \mathrm{H}_{2} \mathrm{O}\) and \(11.8 \%\) ash. The coal is burned at a rate of \(50,000 \mathrm{lb}_{\mathrm{m}} / \mathrm{h}\) in a power-plant boiler with air \(50 \%\) in excess of that needed to oxidize all the carbon in the coal to \(\mathrm{CO}_{2}\). The air and coal are both fedat \(77^{\circ} \mathrm{F}\) and 1 atm. The solid residue from the furnace is analyzed and is found to contain \(28.7 \mathrm{wt} \% \mathrm{C}, 1.6 \% \mathrm{S},\) and the balance ash. The sulfur oxidized in the furnace is converted to \(\mathrm{SO}_{2}(\mathrm{g}) .\) Of the ash in the coal, \(30 \%\) emerges in the solid residue and the balance is emitted with the stack gases as fly ash. The stack gas and solid residue emerge from the furnace at \(600^{\circ} \mathrm{F}\). The higher heating value of the coal is \(18,000 \mathrm{Btu} / \mathrm{b}_{\mathrm{m}}\). (a) Calculate the mass flow rates of all components in the stack gas and the volumetric flow rate of this gas. (Tgnore the contribution of the fly ash in the latter calculation, and assume that the stack gas contains a negligible amount of CO.) (b) Assume that the heat capacity of the solid furnace residuc is \(0.22 \mathrm{Btu} /\left(\mathrm{lb}_{\mathrm{m}} \cdot^{\circ} \mathrm{F}\right),\) that of the stack gas is the heat capacity per unit mass of nitrogen, and \(35 \%\) of the heat generated in the furnace is used to produce electricity. At what rate in \(\mathrm{MW}\) is electricity produced? (c) Calculate the ratio (heat transferred from the furnace)/(heating value of the fuel). Why is this ratio less than one? (d) Suppose the air fed to the furnace were preheated rather than being fed at ambient temperature, but that everything else (feed rates, outlet temperatures, and fractional coal conversion) were the same. What effect would this change have on the ratio calculated in Part (c)? Explain. Suggest an economical way in which this preheating might be accomplished. Exploratory Exercises - Research and Discover (e) At least three components of the stack gas from the power plant raise significant environmental concerns. Identify the components, explain why they are considered problems, and describe how the problems can be addressed in a modern coal-fired power plant. (f) Several minor constituents of coal were not mentioned in the problem statement, and yet they may be part of the stack gas. Identify one such species and, as in Part (e), explain why it is a problem and how the problem cither is or could be addressed in a modern coal-fired power plant.

Ethylene oxide is produced by the catalytic oxidation of ethylene: $$\mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O}(\mathrm{g})$$ An undesired competing reaction is the combustion of ethylene to \(\mathrm{CO}_{2}\) The feed to a reactor contains \(2 \mathrm{mol} \mathrm{C}_{2} \mathrm{H}_{4} / \mathrm{mol} \mathrm{O}_{2} .\) The conversion and yield in the reactor are respectively \(25 \%\) and \(0.70 \mathrm{mol} \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O}\) produced/mol \(\mathrm{C}_{2} \mathrm{H}_{4}\) consumed. A multiple- unit process separates the reactor outlet stream components: \(\mathrm{C}_{2} \mathrm{H}_{4}\) and \(\mathrm{O}_{2}\) are recycled to the reactor, \(\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O}\) is sold, and \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) are discarded. The reactor inlet and outlet streams are each at \(450^{\circ} \mathrm{C}\), and the fresh feed and all species leaving the separation process are at \(25^{\circ} \mathrm{C}\). The combined fresh feedrecycle stream is preheated to \(450^{\circ} \mathrm{C}\). (a) Taking a basis of 2 mol of ethylene entering the reactor, draw and label a flowchart of the complete process (show the separation process as a single unit) and calculate the molar amounts and compositions of all process streams. (b) Calculate the heat requirement ( \(k J\) ) for the entire process and that for the reactor alone. Data for gaseous ethylene oxide $$\begin{aligned}\Delta \hat{H}_{\mathrm{f}}^{\prime} &=-51.00 \mathrm{kJ} / \mathrm{mol} \\ C_{p}[\mathrm{J} /(\mathrm{mol} \cdot \mathrm{K})] &=-4.69+0.2061 T-9.995 \times 10^{-5} T^{2} \end{aligned}$$ where \(T\) is in kelvins. (c) Calculate the flow rate \((\mathrm{kg} / \mathrm{h})\) and composition of the fresh feed, the overall conversion of ethylene, and the overall process and reactor heat requirements (kW) for a production rate of \(1500 \mathrm{kg} \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O} /\) day. Briefly explain the reasons for separating and recycling the ethylene-oxygen stream. (d) One of the attributes of this process defined in the problem statement is extremely unrealistic. What is it?

A bituminous coal is burned with air in a boiler furnace. The coal is fed at a rate of \(40,000 \mathrm{kg} / \mathrm{h}\) and has an ultimate analysis of 76 wt\% \(\mathrm{C}, 5 \%\) H, \(8 \%\) O, negligible amounts of \(\mathrm{N}\) and \(\mathrm{S}\), and \(11 \%\) noncombustible ash (see Problem 9.58), and a higher heating value of 25,700 kJ/kg. Air enters a preheater at \(30^{\circ} \mathrm{C}\) and 1 atm with a relative humidity of \(30 \%,\) exchanges heat with the hot flue gas leaving the furnace, and enters the furnace at temperature \(T_{\mathrm{a}}\left(^{\circ} \mathrm{C}\right) .\) The flue gas contains 7.71 mole\% \(\mathrm{CO}_{2}\) and 1.29 mole \(\%\) CO on \(a\) dry basis, and the balance is a mixture of \(\mathrm{O}_{2}, \mathrm{N}_{2},\) and \(\mathrm{H}_{2} \mathrm{O}\). It emerges from the furnace at \(260^{\circ} \mathrm{C}\) and is cooled to \(150^{\circ} \mathrm{C}\) in the preheater. Noncombustible residue (slag) leaves the furnace at \(450^{\circ} \mathrm{C}\) and has a heat capacity of \(0.97 \mathrm{kJ} / \mathrm{kg} \cdot^{\cdot} \mathrm{C}\) ).. (a) Prove that the air-to-fuel ratio is 16.1 standard cubic meters/kg coal and that the flue gas contains \(4.6 \% \mathrm{H}_{2} \mathrm{O}\) by volume. (b) Calculate the rate of cooling required to cool the flue gas from \(260^{\circ} \mathrm{C}\) to \(150^{\circ} \mathrm{C}\) and the temperature to which the air is preheated. (Note: A trial-and-error calculation is required.) (c) If \(60 \%\) of the heat transferred from the furnace \((-Q)\) goes into producing saturated steam at 30 bar from liquid boiler feedwater at \(50^{\circ} \mathrm{C},\) at what rate \((\mathrm{kg} / \mathrm{h})\) is steam generated?

In the preliminary design of a furnace for industrial boiler, methane at \(25^{\circ} \mathrm{C}\) is burned completely with \(20 \%\) excess air, also at \(25^{\circ} \mathrm{C} .\) The feed rate of methane is \(450 \mathrm{kmol} / \mathrm{h}\). The hot combustion gases leave the furnace at \(300^{\circ} \mathrm{C}\) and are discharged to the atmosphere. The heat transferred from the furnace \((\dot{Q})\) is used to convert boiler feedwater at \(25^{\circ} \mathrm{C}\) into superheated steam at 17 bar and \(250^{\circ} \mathrm{C}\). (a) Draw and label a flowchart of this process [the chart should look like the one shown in Part (b) without the preheater] and calculate the composition of the gas leaving the furnace. Then, calculate \(\dot{Q}(\mathrm{kJ} / \mathrm{h})\) and the rate of steam production in the boiler \((\mathrm{kg} / \mathrm{h})\). (b) In the actual boiler design, the air feed at \(25^{\circ} \mathrm{C}\) and the combustion gas leaving the furnace at \(300^{\circ} \mathrm{C}\) pass through a heat exchanger (the air preheater). The combustion (flue) gas is cooled to \(150^{\circ} \mathrm{C}\) in the preheater and is then discharged to the atmosphere, and the heated air is fed to the furnace. Calculate the temperature of the air entering the furnace (a computer solution is required) and the rate of steam production (kg/h). (c) Explain why preheating the air increases the rate of steam production. (Suggestion: Use the energy balance on the furnace in your explanation.) Why does it make sense economically to use the combustion gas as the heating medium?

The standard heat of the combustion reaction of liquid \(n\) -hexane to form \(\mathrm{CO}_{2}(\mathrm{g})\) and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{l}),\) with all reactants and products at \(77^{\circ} \mathrm{F}\) and 1 atm, is \(\Delta H_{\mathrm{r}}^{\prime}=-1.791 \times 10^{6} \mathrm{Btu} .\) The heat of vaporization of hexane at \(77^{\circ} \mathrm{F}\) is \(13,550 \mathrm{Btu} / \mathrm{b}\) -mole and that of water is \(18.934 \mathrm{Btu} / \mathrm{h}\) -mole. (a) Is the reaction exothermic or endothermic at \(77^{\circ} \mathrm{F}\) ? Would you have to heat or cool the reactor to keep the temperature constant? What would the temperature do if the reactor ran adiabatically? What can you infer about the energy required to break the molecular bonds of the reactants and that released when the product bonds form? (b) Use the given data to calculate \(\Delta H_{\mathrm{r}}^{\mathrm{r}}\) (Btu) for the combustion of \(n\) -hexane vapor to form \(\mathrm{CO}_{2}(\mathrm{g})\) and \(\overline{\mathrm{H}}_{2} \mathrm{O}(\mathrm{g})\) (c) If \(\dot{Q}=\Delta \dot{H},\) at what rate in \(\mathrm{B}_{\text {tu } / \mathrm{s}}\) is heat absorbed or released (state which) if \(120 \mathrm{lb}_{\mathrm{n}} / \mathrm{s}\) of \(\mathrm{O}_{2}\) is consumed in the combustion of hexane vapor, water vapor is the product, and the reactants and products are all at \(77^{\circ} \mathrm{F} ?\) (d) If the reaction were carried out in a real reactor, the actual value of \(\dot{Q}\) would be greater (less negative) than the value calculated in Part (c). Explain why.
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