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Chapter 5: Problem 90

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?

Short Answer

Expert verified
The answer depends on the correctness and appropriateness of calculations. If done correctly, the number of cars can vary when applying different concepts: the ideal gas law, SRK equation and the law of corresponding states. The most reliable approach depends on both theoretical appropriateness and practical feasibility.

Step by step solution

01

Identify the Volume of the Tube-Trailer

To calculate the volume of the tube-trailer carrying the hydrogen, utilize the formula for the volume of a cylinder, which is \( V= \pi r^2 h\). The radius (r) is half of the diameter, and the height (h) is the length of the tube. By plugging the numbers, the volume of a tube is obtained. Since the trailer contains 10 tubes, multiply the volume of a single tube by 10 to find the total volume of the tube-trailer.
02

Convert Pressure Terms

The given pressure units are in psig, which stands for pounds per square inch gauge. However, the ideal gas law requires pressure to be stated in absolute terms. To get this, one needs to convert psig to psi absolute (psia) by adding atmospheric pressure (14.7 psi), to both the beginning and final pressures.
03

Determine the Amount of Hydrogen in the Tube-Trailer

Now, use the ideal gas law, \( PV=nRT\), to calculate the number of moles (n) of hydrogen in the tube-trailer upon start and at the end. Here, P is the pressure, V is the volume, R is the ideal gas constant, and T is the temperature. After getting n in start and at the end, subtract these to find the number of moles discharged.
04

Calculate the Number of Cars that can be Refueled

It's given that refueling each fuel cell powered car requires 4.0 kg of hydrogen, this is equivalent to \( \frac{4.0 \mathrm{kg}}{2.02 \mathrm{kg/ mol}} \) moles of hydrogen. Subsequently, calculate the number of cars that can be refueled by dividing the total number of moles of hydrogen discharged by the moles of hydrogen needed per car.
05

Apply SRK Equation of State

Calculate the number of moles of hydrogen according to the SRK equation of state, and then determine the number of cars that could be refueled. Use the original volume of the tube-trailer and the given equation for the parameter \(\alpha\) in the calculation. Be cautious that the temperature is in reduced temperature \(\left(T_{\mathrm{r}}\right)\) in the equation for \(\alpha\).
06

Apply the Law of Corresponding States

Perform the similar calculation using the law of corresponding states. The critical properties of hydrogen, the critical pressure, temperature and volume, are needed for this step. After obtaining the number of moles of hydrogen, calculate the number of cars that can be refueled.
07

Evaluate the Estimations

After implementing the ideal gas law, SRK equation of state, and the law of corresponding states, compare all three estimations. Decide which estimation instills the most confidence based on the underlying assumptions of each method.

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Key Concepts

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

Fuel Cell Technology
Fuel cells are fascinating devices that convert chemical energy directly into electrical energy through a chemical reaction. This process involves hydrogen and oxygen, producing water and electricity as by-products.
Unlike traditional combustion engines that burn fuel to produce energy, fuel cells rely on electrochemical reactions. This makes them more efficient and environmentally friendly.
Benefits of fuel cells include:
  • Higher energy efficiency compared to internal combustion engines.
  • Reduction in greenhouse gas emissions.
  • Lower noise pollution.
In the context of automobiles, fuel cells present a promising alternative to gasoline engines, providing a cleaner, sustainable energy source.
SRK Equation of State
The SRK Equation of State is an essential tool in chemical engineering for predicting the behavior of gases and liquids. This model is particularly useful when dealing with real gases.
The SRK equation is formulated as:
\[ P = \frac{{RT}}{{V_m - b}} - \frac{{a \cdot \alpha}}{{V_m(V_m + b)}} \]
In this context, P is the pressure, R is the ideal gas constant, T is the temperature, V_m is the molar volume, and a, b, and \alpha are specific constants for the substance under consideration.
These expressions help predict how substances react under various pressures and temperatures. By using the adapted formula for hydrogen, the SRK model can refine hydrogen storage calculations, leading to more accurate outcomes for applications like fuel cells.
Law of Corresponding States
The Law of Corresponding States is a profound concept, stating that all gases behave similarly if they share the same reduced properties. These properties include reduced pressure, volume, and temperature.
The principle provides a universal framework, offering insights into the behaviors of different substances without detailed experimental data.
To apply this law, physicists rely on:
  • Critical pressure
  • Critical temperature
  • Critical volume
By knowing these critical values, hydrogen's behavior can be evaluated and compared with other gases, aiding in better predictions for chemical processes like those in hydrogen transport and fuel cells.
Ideal Gas Law
The Ideal Gas Law, represented by the formula \( PV = nRT \), is a fundamental equation in chemistry to describe the behavior of an ideal gas.
Variables in the equation include pressure \( P \), volume \( V \), number of moles \( n \), ideal gas constant \( R \), and temperature \( T \).
This simple yet versatile law is often the starting point for calculations in gas-related scenarios. It assumes gases are composed of point particles that interact minimally.
While it is not perfect for real gases under high pressures or low temperatures, it provides a starting point for further, more complex calculations in chemical engineering processes, such as those involving hydrogen in fuel cells.

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Most popular questions from this chapter

The volume of a dry box (a closed chamber with dry nitrogen flowing through it) is \(2.0 \mathrm{m}^{3}\). The dry box is maintained at a slight positive gauge pressure of \(10 \mathrm{cm} \mathrm{H}_{2} \mathrm{O}\) and room temperature \(\left(25^{\circ} \mathrm{C}\right) .\) If the contents of the box are to be replaced every five minutes, calculate the required mass flow rate of nitrogen in \(g / \min\) by (a) direct solution of the ideal-gas equation of state and (b) conversion from standard conditions. You may assume the gas in the dry box is well mixed.

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.

Magnesium sulfate has a number of uses, some of which are related to the ability of the anhydrate form to remove water from air and others based on the high solubility of the heptahydrate \(\left(\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}\right)\) form, also known as Epsom salt. The densities of the anhydrate and heptahydrate crystalline forms are 2.66 and \(1.68 \mathrm{g} / \mathrm{mL},\) respectively. Suppose you wish to form a 20.0 wt\% \(\mathrm{MgSO}_{4}\) aqueous solution by simply pouring crystals of one of the forms into a tank of water while the temperature is held constant at \(30^{\circ} \mathrm{C}\). The specific gravity of the 20.0 wt\% solution at \(30^{\circ} \mathrm{C}\) is \(1.22 .\) Answer the following questions for both forms of the \(\mathrm{MgSO}_{4}\) crystals: (a) What volume of water should be in the tank before crystals are added if the final product is to be 1000 kg of the 20 wt\% solution? (b) Suppose the tank diameter is \(0.30 \mathrm{m}\). What is the height of liquid in the tank before the crystals are added? (c) What is the height of the water in the tank after addition of the crystals but before they begin to dissolve? (d) What is the height of liquid in the tank after all the MgSO \(_{4}\) has dissolved?

Spray drying is a process in which a liquid containing dissolved or suspended solids is injected into a chamber through a spray nozzle or centrifugal disk atomizer. The resulting mist is contacted with hot air, which evaporates most or all of the liquid, leaving the dried solids to fall to a conveyor belt at the bottom of the chamber. Powdered milk is produced in a spray dryer \(6 \mathrm{m}\) in diameter by \(6 \mathrm{m}\) high. Air enters at \(167^{\circ} \mathrm{C}\) and \(-40 \mathrm{cm} \mathrm{H}_{2} \mathrm{O}\). The milk fed to the atomizer contains \(70 \%\) water by mass, all of which evaporates. The outlet gas contains 12 mole \(\%\) water and leaves the chamber at \(83^{\circ} \mathrm{C}\) and \(1 \mathrm{atm}\) (absolute) at a rate of \(311 \mathrm{m}^{3} / \mathrm{min}\). (a) Calculate the production rate of dried milk and the volumetric flow rate of the inlet air. Estimate the upward velocity of air (m/s) at the bottom of the dryer. (b) Engineers often face the challenge of what to do to a process when demand for a product increases (or decreases). Suppose in the present case production must be doubled. (i) Why is it unlikely that the flow rates of feed and air can simply be increased to achieve the new production rate? (ii) An obvious option is to buy another dryer like the existing one and operate the two in parallel. Give two advantages and two disadvantages of this option. (iii) Still another possibility is to buy a larger dryer to replace the original unit. Give two advantages and two disadvantages of doing so. Estimate the approximate dimensions of the larger unit.

The current global reliance on fossil fuels for heating, transportation, and electric power generation raises concems regarding the release of \(\mathrm{CO}_{2}\) and \(\mathrm{CH}_{4},\) which are greenhouse gases thought to lead to climate change, and NO, which contributes to smog. One potential solution to these problems is to produce transportation fuels from renewable biomass. You have been asked to evaluate a proposed process for converting forest residues to alcohols that may be used as transportation fuels. In the first stage of the process, steam and dry wood from hybrid poplar trees (which grow between five and eight feet a year and can be harvested roughly every five years) are fed to a gasifier in which the biomass is converted to light gases in the following reactions: $$\begin{aligned} \mathrm{C}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}+\mathrm{H}_{2} \\\ \mathrm{CO}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}_{2}+\mathrm{H}_{2} \\ \mathrm{C}+\mathrm{CO}_{2} & \rightarrow 2 \mathrm{CO} \\ \mathrm{C}+2 \mathrm{H}_{2} & \rightarrow \mathrm{CH}_{4} \\ \mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}+3 \mathrm{H}_{2} \end{aligned}$$ The effluents from the reactor are a gas stream containing \(\mathrm{H}_{2}, \mathrm{CO}, \mathrm{CO}_{2}, \mathrm{CH}_{4},\) and \(\mathrm{H}_{2} \mathrm{O},\) and a solid char stream that contains only carbon and hydrogen. The char is discarded and the gases go through additional steps in which the hydrogen and carbon monoxide are converted to mixed alcohols. This problem only concerns the gasifier. \(\cdot\) Elemental composition of biomass: 51.9 mass \(\%\) C \(, 6.3 \%\) H, and \(41.8 \%\) O \(\cdot\) Pressure and temperature of entering steam: \(155^{\circ} \mathrm{C}, 4.4 \mathrm{atm}\) \(\cdot\) Feed ratio of steam to biomass: 1.1 kg steam/kg biomass \(\cdot\) Yield and dry-basis composition of product gas: 1.35 kg dry gas/kg biomass at \(700^{\circ} \mathrm{C}, 1.2\) atm; 50.7 mol\% \(\mathrm{H}_{2}, 23.8 \%\) CO, \(18.0 \% \mathrm{CO}_{2}, 7.5 \% \mathrm{CH}_{4}\) (a) Taking a basis of \(100 \mathrm{kg}\) of biomass fed, draw and completely label a flowchart for the gasifier incorporating the given data, labeling the volumes of the steam fed and the gases produced. Perform a degree-of-freedom analysis. (b) Calculate the mass and mass composition of the char and the volumes of the steam feed and product gas streams. (c) List advantages and possible drawbacks of using biomass rather than petroleum as a fuel source.
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