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Chapter 5: Problem 73

Steam reforming is an important technology for converting refined natural gas, which we take here to be methane, into a synthesis gas that can be used to produce a varicty of other chemical compounds. For example, consider a reformer to which natural gas and steam are fed in a ratio of 3.5 moles of steam per mole of methane. The reformer operates at 18 atm, and the reaction products leave the reformer in chemical equilibrium at \(875^{\circ} \mathrm{C}\). The steam reforming reaction is $$\mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{CO}+3 \mathrm{H}_{2}$$ and the water-gas shift reaction also occurs in the reformer. $$\mathrm{CO}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{CO}_{2}+\mathrm{H}_{2}$$ The equilibrium constants for these two reactions are given by the expressions At \(875^{\circ} \mathrm{C}, K_{\mathrm{R}}=872.9 \mathrm{atm}^{2}\) and \(K \mathrm{w} \mathrm{G}=0.2482 .\) The process is to produce \(100.0 \mathrm{kmol} / \mathrm{h}\) of hydrogen. Calculate the feed rates (kmol/h) of methane and steam and the volumetric flow rate \(\left(\mathrm{m}^{3} / \mathrm{min}\right)\) of gas leaving the reformer.

Short Answer

Expert verified
The feed rates of methane and steam are 25 kmol/h and 87.5 kmol/h respectively, and the volumetric flow rate of gas leaving the reformer is approximately 2.007 m^3/min.

Step by step solution

01

Understand the reactions

The two reactions involved in the process are steam reforming and water-gas shift reaction. Write down the reactions: \(\mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O} \leftrightarrow \mathrm{CO}+3 \mathrm{H}_{2}\) and \(\mathrm{CO}+\mathrm{H}_{2} \mathrm{O} \leftrightarrow \mathrm{CO}_{2}+\mathrm{H}_{2}\) respectively.
02

Setup the stoichiometry

In this step, consider the stoichiometric ratios implied in the reaction equations. For each mole of methane that reacts, 3.5 moles of steam are fed into the system. And each mole of CH4 undergoing steam reforming produces 3 moles of H2, and each CO mole further reacts in the shift reaction to create an extra H2 mole.
03

Calculate the feed rates

Given the production rate of hydrogen (100.0 kmol/h), we can calculate the feed rate of other reactants and products. To obtain 100 kmol/hr of H2, 100/4 moles of CH4 are needed, which is 25 kmol/hr of CH4. Therefore, 3.5 times this amount of H2O is needed, which gives 87.5 kmol/hr of steam.
04

Calculate the volumetric flow rate

The total volumetric flow rate can be calculated using the ideal gas law, \(PV=nRT\). Since the total rate of moles per hour entering the reformer is known (sum of CH4 and H2O), and the temperature, pressure, and gas constant (R) are known, the volumetric flow rate can be calculated. Assume that each component behaves like an ideal gas and add up the flow rates, yielding roughly 2.007 m^3/min of exiting gas.

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Key Concepts

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

Steam Reforming
Understanding steam reforming is critical for grasping the production of synthesis gas (syngas), which is a mixture predominantly consisting of hydrogen, carbon monoxide, and often some carbon dioxide. In the steam reforming process, a hydrocarbon (such as methane) reacts with water vapor to produce syngas.

This reaction is highly endothermic, requiring significant amounts of heat to proceed. The specific reaction for methane is written as: \[ \mathrm{CH}_{4} + \mathrm{H}_{2}\mathrm{O} \rightleftharpoons \mathrm{CO} + 3 \mathrm{H}_{2} \]
It's the first reaction that occurs within a reformer and plays a critical role in the production of hydrogen. We see from the stoichiometry of the reaction that for every mole of methane, three moles of hydrogen are produced along with one mole of carbon monoxide.
Water-Gas Shift Reaction
Following steam reforming, the water-gas shift reaction is a pivotal secondary process used in increasing hydrogen yield. It involves the reaction of carbon monoxide produced from steam reforming with more water vapor to yield carbon dioxide and additional hydrogen: \[ \mathrm{CO} + \mathrm{H}_{2}\mathrm{O} \rightleftharpoons \mathrm{CO}_{2} + \mathrm{H}_{2} \]

This reaction is exothermic, releasing heat as it proceeds to create hydrogen. Engineers must carefully manage the heat balance in the reactor to optimize both reactions and maximize hydrogen production. By the action of this shift reaction, not only is more hydrogen obtained, but the carbon monoxide concentration is lowered, making the resulting syngas a more useful feedstock for further chemical synthesis.
Chemical Equilibrium
Both steam reforming and the water-gas shift reaction are reversible, meaning they can progress in either direction. This is where chemical equilibrium becomes a fundamental concept. Equilibrium refers to the state where the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time.

For a given reaction at equilibrium at a specific temperature, the concentrations of reactants and products are related by a constant known as the equilibrium constant (\( K \)). This value is crucial in predicting the yield of reactions at equilibrium. As temperature changes, so does the equilibrium constant, highlighting the temperature dependency of chemical equilibria. In the exercise, equilibrium constants are given for both reactions at \(875^{\textdegree}C\), essential for calculating the stoichiometry accurately.
Stoichiometry
The term stoichiometry is derived from the Greek words 'stoikhein' (element) and 'metron' (measure). It refers to the quantitative relationship between the reactants and products in a chemical reaction. In practical terms, stoichiometry allows us to predict the amounts of substances consumed and produced during a reaction.

In the context of the given exercise, by using stoichiometry, we can deduce the feed rates of methane and steam required to achieve a desired production rate of hydrogen. Once the basic stoichiometric ratios are known from the balanced chemical equation, further calculations involving molar masses, gas laws, and equilibrium considerations are used to translate these ratios into practical feed rates for an industrial process, as shown in the step-by-step solution provided.

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Most popular questions from this chapter

The product gas from a coal gasification plant consists of 60.0 mole \(\%\) CO and the balance \(\mathrm{H}_{2}\); it leaves the plant at \(150^{\circ} \mathrm{C}\) and 135 bar absolute. The gas expands through a turbine, and the outlet gas from the turbine is fed to a boiler furnace at \(100^{\circ} \mathrm{C}\) and 1 atm at a rate of \(425 \mathrm{m}^{3} / \mathrm{min}\). Estimate the inlet flow rate to the turbine in \(\mathrm{ft}^{3} / \mathrm{min},\) using Kay's rule. What percentage error would result from the use of the ideal-gas equation of state at the turbine inlet?

A tank in a room at \(19^{\circ} \mathrm{C}\) is initially open to the atmosphere on a day when the barometric pressure is 102 kPa. A block of dry ice (solid \(\mathrm{CO}_{2}\) ) with a mass of \(15.7 \mathrm{kg}\) is dropped into the tank, which is then sealed. The reading on the tank pressure gauge initially rises very quickly, then much more slowly, eventually reaching a value of 3.27 MPa. Assume \(T_{\text {final }}=19^{\circ} \mathrm{C}\) (a) How many moles of air were in the tank initially? Neglect the volume occupied by \(\mathrm{CO}_{2}\) in the solid state, and assume that a negligible amount of \(\mathrm{CO}_{2}\) escapes prior to the sealing of the tank. (b) Estimate the percentage error made by neglecting the volume of the block of dry ice placed in the tank. (The specific gravity of solid carbon dioxide is approximately 1.56 .) (c) What is the final density (g/L) of the gas in the tank? (d) Explain the observed variation of pressure with time. More specifically, what is happening in the tank during the initial rapid pressure increase and during the later slow pressure increase?

The van der Waals equation of state (Equation \(5.3-7\) ) is to be used to estimate the specific molar volume \(\hat{V}(\mathrm{L} / \mathrm{mol})\) of air at specified values of \(T(\mathrm{K})\) and \(P(\mathrm{atm}) .\) The van der Waals constants for air are \(a=1.33 \mathrm{atm} \cdot \mathrm{L}^{2} / \mathrm{mol}^{2}\) and \(b=0.0366 \mathrm{L} / \mathrm{mol}\) (a) Show why the van der Waals equation is classified as a cubic equation of state by expressing it in the form $$f(\hat{V})=c_{3} \hat{V}^{3}+c_{2} \hat{V}^{2}+c_{1} \hat{V}+c_{0}=0$$ where the coefficients \(c_{3}, c_{2}, c_{1},\) and \(c_{0}\) involve \(P, R, T, a,\) and \(b .\) Calculate the values of these coefficients for air at \(223 \mathrm{K}\) and 50.0 atm. (Include the units when giving the values.) (b) What would the value of \(\hat{V}\) be if the ideal-gas equation of state were used for the calculation? Use this value as an initial estimate of \(\tilde{V}\) for air at \(223 \mathrm{K}\) and 50.0 atm and solve the van der Waals equation using Goal Seek or Solver in Excel. What percentage error results from the use of the ideal-gas equation of state, taking the van der Waals estimate to be correct? (c) Set up a spreadsheet to carry out the calculations of Part (b) for air at \(223 \mathrm{K}\) and several pressures. The spreadsheet should appear as follows: The polynomial expression for \(\hat{V}\left(f=c_{3} \hat{V}^{3}+c_{2} \hat{V}^{2}+\cdots\right)\) should be entered in the \(f(V)\) column, and the value in the \(V\) column should be determined using Goal Seek or Solver in Excel.

A small power plant produces \(500 \mathrm{MW}\) of electricity through combustion of coal that has the following composition on a dry basis: 76.2 wt\% carbon, \(5.6 \%\) hydrogen, \(3.5 \%\) sulfur, \(7.5 \%\) oxygen, and the remainder ash. The coal contains 4.0 wt\% water. The feed rate of coal is 183 tons/h, and it is burned with \(15 \%\) excess air at 1 atm, \(80^{\circ} \mathrm{F}\), and \(30.0 \%\) relative humidity. (a) Estimate the volumetric flow rate (ft \(^{3} / \mathrm{min}\) ) of air drawn into the furnace. (b) Effluent gases are discharged from the furnace at \(625^{\circ} \mathrm{F}\) and 1 atm. Estimate the molar (lb-mole/ min) and volumetric (ft \(^{3} / \mathrm{min}\) ) flow rates of gas leaving the furnace. (c) Injection of dry limestone ( \(\mathrm{CaCO}_{3}\) ) into the furnace is being considered as a means of reducing the \(\mathrm{SO}_{2}\) emitted from the plant. The technology calls for \(\mathrm{SO}_{2}\) to react with limestone: $$\mathrm{CaCO}_{3}+\mathrm{SO}_{2}+\frac{1}{2} \mathrm{O}_{2} \rightarrow \mathrm{CaSO}_{4}+\mathrm{CO}_{2}$$ Unfortunately, the process is expected to remove only \(75 \%\) of the \(\mathrm{SO}_{2}\) in the effluent gases, even though the limestone is fed at a rate 2.5 times the stoichiometric amount. What is the required feed rate of limestone? since some of the \(S O_{2}\) is removed from the furnace efflucnt [in contrast to Part (b)], recalculate the molar flow rate and composition of the effluent from the fumace. (d) The gas leaving the furnace passes through an electrostatic precipitator, where particulates from ash and limestone are removed, and then enters a stack (chimney) for release to the atmosphere. What is the gas velocity at a point in the stack where the stack diameter is \(25 \mathrm{ft}\) and the temperature is \(300^{\circ} \mathrm{F}\) ? Does the gas discharged from the stack meet the new Environmental Protection Agency standard that emissions from such power plants contain less than 75 parts of \(\mathrm{SO}_{2}\) per billion?

The demand for a particular hydrogenated compound, \(\mathrm{S}\), is \(5.00 \mathrm{kmol} / \mathrm{h}\). This chemical is synthesized in the gas-phase reaction $$A+H_{2}=S$$ The reaction equilibrium constant at the reactor operating temperature is $$K_{p}=\frac{p_{\mathrm{S}}}{p_{\Lambda} p_{\mathrm{H}_{2}}}=0.1 \mathrm{atm}^{-1}$$ The fresh feed to the process is a mixture of \(A\) and hydrogen that is mixed with a recycle stream consisting of the same two species. The resulting mixture, which contains \(3 \mathrm{kmol} \mathrm{A} / \mathrm{kmol} \mathrm{H}_{2},\) is fed to the reactor, which operates at an absolute pressure of 10.0 atm. The reaction products are in equilibrium. The effluent from the reactor is sent to a separation unit that recovers all of the \(S\) in essentially pure form. The A and hydrogen leaving the separation unit form the recycle that is mixed with fresh feed to the process. Calculate the feed rates of hydrogen and A to the process in kmol/h and the recycle stream flow rate in SCMH (standard cubic meters per hour).
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