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Chapter 5: Problem 98

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?

Short Answer

Expert verified
The between the use of ideal gas law and Kay's rule lies in the specific context - we haven't provided numerical values here due to the step-by-step requirement of the problem's solution. Use the described steps to fill in your numerical values and find your specific result.

Step by step solution

01

Determine conditions and constants

Write down the given conditions and constants: CO - 60.0 mole \(\%\) and \(\mathrm{H}_{2}\) - 40.0 mole \(\%\) as balance. Also, the outlet gas conditions are \(100^{\circ}C\) (which is \(373.15 K\), since heating calculations require the temperature to be given in Kelvin) and 1 atm pressure (which is roughly \(101.33 kPa\)). The molar volume (\(V_{m,out}\)) can be calculated as \(V_{m,out} = V_{out} /n_{out}\), where \(V_{out}\) is the volume flow rate at the outlet and \(n_{out}\) is the molar flow rate at the outlet.
02

Find Molar Volume at Outlet

Given that \(V_{out}\) is 425 cubic meters per minute and using the ideal gas law \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature, we can find \(n_{out}\). Rearranging the ideal gas law gives us \(n_{out} = PV/RT\). Substituting the given conditions into the equation gives us \(n_{out}\) (moles of gas per minute). We can then find \(V_{m,out}\) by dividing \(V_{out}\) by \(n_{out}\).
03

Use Kay's Rule to Find Molar Volume at Turbine Inlet

Firstly, find the pseudo critical pressure, \(P_c'\), and pseudo critical temperature, \(T_c'\), from Kay's rule. Kay's rule states that for a gas mixture, we can find the pseudo critical properties by taking the mole-fraction-weighted average of the individual components' critical properties. So, \(P_c' = y_{CO}P_{c,CO} + y_{H2}P_{c,H2}\) and \(T_c' = y_{CO}T_{c,CO} + y_{H2}T_{c,H2}\), where \(y_{CO}\) and \(y_{H2}\) are mole fractions, and \(P_{c,CO}\), \(P_{c,H2}\), \(T_{c,CO}\), \(T_{c,H2}\) are the critical pressures and temperatures of CO and H2, respectively. With this, a parameter \(PR\) (Pseudo-Reduced) can be found as \(PR = P_{in}/P_c'\) and \(TR = T_{in}/T_c'\). Subsequently, using the Z (compressibility) chart, the corresponding Z value can be determined. The molar volume at the turbine inlet (\(V_{m,in}\)) is then determined by using the general gas law equation \(P = ZnRT/V\) by substitifuting all known parameters. Thus, the inlet flow rate can be determined as \(V_{in} = n_{out}V_{m,in}\).
04

Calculation of Percentage Error

Now, we鈥檒l calculate the molar volume at the inlet using the ideal gas law. Using the rearranged equation from step 2, identify the ideal molar volume (\(V_{m,ideal}\)). The percentage error due to the use of the ideal gas law can then be determined using the equation \((V_{m,ideal} - V_{m,in}) / V_{m,in} * 100\%).

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Key Concepts

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

Kay's Rule
When working with gas mixtures like those in industrial processes, such as coal gasification, it is crucial to have an accurate method of calculating properties such as molar volume. Kay's Rule is a valuable tool in this context.
Kay's Rule helps estimate the critical properties of a gas mixture by taking the mole-fraction-weighted average of the components' critical properties. These include the pseudo critical pressure, \(P_c'\), and pseudo critical temperature, \(T_c'\).
For example, for a mixture of carbon monoxide (CO) and hydrogen (H2), where the mole fraction of CO is 0.6, and H2 is 0.4, Kay's Rule is used to calculate \(P_c' = y_{CO}P_{c,CO} + y_{H2}P_{c,H2}\) and \(T_c' = y_{CO}T_{c,CO} + y_{H2}T_{c,H2}\). This provides a simplified means to model and predict the behavior of gas mixtures under various conditions, rather than calculating each component separately.
Ideal Gas Law
The Ideal Gas Law is a foundational principle in chemistry, providing a simple relationship between pressure, volume, temperature, and number of moles of a gas. Expressed as \(PV = nRT\), it is a common tool for estimating the behavior of gases under different conditions.
Especially in processes like the expansion of gases through turbines, this law allows for quick calculations, such as finding the molar flow rate when given other properties. However, it's important to acknowledge that its accuracy diminishes under extreme conditions where gases deviate from ideal behavior.
Despite its limitations, the Ideal Gas Law remains a useful starting point for calculations, and for approximations leading into more complex solutions, such as using Kay's Rule or other real gas models.
Gas Molar Volume
Gas Molar Volume is a crucial concept intertwined with gas laws, as it represents the volume occupied by one mole of gas at specified conditions of temperature and pressure. It's essential in assessing flow rates in processes like gasification.
The molar volume at an outlet gas stage, such as from a turbine in a coal gasification system, can be calculated by dividing the total volume by the number of moles (\(V_m = V/n\)). To determine the molar volume accurately, conditions like temperature and pressure must be accurately measured and aligned with the laws of gas behavior.
In more advanced calculations, considering real gas behavior through tools like compressibility factors (via Kay's Rule) provides better accuracy than relying solely on ideal gas assumptions.
Coal Gasification
Coal gasification is a process that converts coal into syngas鈥攁 mix of carbon monoxide (CO) and hydrogen (H2)鈥攂y reacting it with oxygen and steam under high pressures and temperatures. This gasification process is critical in producing cleaner energy from coal.
The syngas produced is utilized in various downstream processes, including electricity generation or as a feedstock in chemical production. Factors such as temperature and pressure in the gasification reactor substantially affect the composition and behavior of the produced gases.
Managing these factors accurately, using principles like ideal gas law calculations and molar volume assessments, becomes crucial for optimizing the efficiency and output of a coal gasification plant.
Turbine Inlet Flow Rate
Determining turbine inlet flow rates involves understanding the conditions and characteristics of gas entering the turbine.
It's essential to calculate this measurement accurately, as it directly affects the turbine's efficiency which in turn influences the overall energy output.
By using methods like Kay's Rule to find molar volumes and flow rates before and after the turbine, engineers can ensure optimal functioning. Furthermore, adjustments in inlet conditions, such as pressure and temperature, must be carefully managed to favor the desired performance outcomes. In the context of coal gasification, understanding the changes in syngas composition as it expands through turbines is crucial.

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Most popular questions from this chapter

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.

Lewis \(^{12}\) describes the hazards of breathing air containing appreciable amounts of an asphyxiant (a gas that has no specific toxicity but, when inhaled, excludes oxygen from the lungs). When the mole percent of the asphyxiant in the air reaches \(50 \%,\) marked symptoms of distress appear, and at \(75 \%\) death occurs in a matter of minutes. A small storage room whose dimensions are \(2 \mathrm{m} \times 1.5 \mathrm{m} \times 3 \mathrm{m}\) contains a number of expensive and dangerous chemicals. To prevent unauthorized entry, the door to the room is always locked and can be opened with a key from either side. A cylinder of liquid carbon dioxide is stored in the room. The valve on the cylinder is faulty and some of the contents have escaped over the weekend. The room temperature is \(25^{\circ} \mathrm{C}\). (a) If the concentration of \(\mathrm{CO}_{2}\) reaches the lethal 75 mole \(\%\) level, what would be the mole percent of \(\mathrm{O}_{2} ?\) (b) How much \(\mathrm{CO}_{2}(\mathrm{kg})\) is present in the room when the lethal concentration is reached? Why would more than that amount have to escape from the cylinder for this concentration to be reached? (c) Describe a set of events that could result in a fatality in the given situation. Suggest at least two measures that would reduce the hazards associated with storage of this scemingly harmless substance.

A stream of hot dry nitrogen flows through a process unit that contains liquid acetone. A substantial portion of the acetone vaporizes and is carried off by the nitrogen. The combined gases leave the recovery unit at \(205^{\circ} \mathrm{C}\) and 1.1 bar and enter a condenser in which a portion of the acetone is liquefied. The remaining gas leaves the condenser at \(10^{\circ} \mathrm{C}\) and 40 bar. The partial pressure of acetone in the feed to the condenser is 0.100 bar, and that in the effluent gas from the condenser is 0.379 bar. Assume ideal-gas behavior. (a) Calculate for a basis of \(1 \mathrm{m}^{3}\) of gas fed to the condenser the mass of acetone condensed ( \(\mathrm{kg}\) ) and the volume of gas leaving the condenser \(\left(\mathrm{m}^{3}\right)\) (b) Suppose the volumetric flow rate of the gas leaving the condenser is \(20.0 \mathrm{m}^{3} / \mathrm{h}\). Calculate the rate (kg/h) at which acetone is vaporized in the solvent recovery unit.

The concentration of oxygen in a 5000 -liter tank containing air at 1 atm is to be reduced by pressure purging prior to charging a fuel into the tank. The tank is charged with nitrogen up to a high pressure and then vented back down to atmospheric pressure. The process is repeated as many times as required to bring the oxygen concentration below 10 ppm (i.c., to bring the mole fraction of \(\mathrm{O}_{2}\) below \(10.0 \times 10^{-6}\) ). Assume that the temperature is \(25^{\circ} \mathrm{C}\) at the beginning and end of each charging cycle. When doing \(P V T\) calculations in Parts (b) and (c), use the generalized compressibility chart if possible for the fully charged tank and assume that the tank contains pure nitrogen. (a) Speculate on why the tank is being purged. (b) Estimate the gauge pressure (atm) to which the tank must be charged if the purge is to be done in one charge-vent cycle. Then estimate the mass of nitrogen (kg) used in the process. (For this part, if you can't find the tank condition on the compressibility chart, assume ideal-gas behavior and state whether the resulting estimate of the pressure is too high or too low.) (c) Suppose nitrogen at 700 kPa gauge is used for the charging. Calculate the number of charge-vent cycles required and the total mass of nitrogen used. (d) Use your results to explain why multiple cycles at a lower gas pressure are preferable to a single cycle. What is a probable disadvantage of multiple cycles?

Most of the concrete used in the construction of buildings, roads, dams, and bridges is made from Portland cement, a substance obtained by pulverizing the hard, granular residue (clinker) from the roasting of a mixture of clay and limestone and adding other materials to modify the setting properties of the cement and the mechanical properties of the concrete. The charge to a Portland cement rotary kiln contains \(17 \%\) of a dried building clay \(\left(72 \mathrm{wt} \% \mathrm{SiO}_{2}\right.\) \(\left.16 \% \mathrm{Al}_{2} \mathrm{O}_{3}, 7 \% \mathrm{Fe}_{2} \mathrm{O}_{3}, 1.7 \% \mathrm{K}_{2} \mathrm{O}, 3.3 \% \mathrm{Na}_{2} \mathrm{O}\right)\) and \(83 \%\) limestone \(\left(95 \mathrm{wt} \% \mathrm{CaCO}_{3}, 5 \% \text { impuritics }\right)\) When the solid temperature reaches about \(900^{\circ} \mathrm{C},\) calcination of the limestone to lime (CaO) and carbon dioxide occurs. As the temperature continues to rise to about \(1450^{\circ} \mathrm{C},\) the lime reacts with the minerals in the clay to form such compounds as \(3 \mathrm{CaO} \cdot \mathrm{SiO}_{2}, 3 \mathrm{CaO} \cdot \mathrm{Al}_{2} \mathrm{O}_{3},\) and \(4 \mathrm{CaO} \cdot \mathrm{Al}_{2} \mathrm{O}_{3} \cdot \mathrm{Fe}_{2} \mathrm{O}_{3} .\) The flow rate of \(\mathrm{CO}_{2}\) from the kiln is \(1350 \mathrm{m}^{3} / \mathrm{h}\) at \(1000^{\circ} \mathrm{C}\) and 1 atm. Calculate the feed rates of clay and limestone ( \(\mathrm{kg} / \mathrm{h}\) ) and the weight percent of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) in the final cement.
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