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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

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 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).

The absolute pressure within a 35.0 -liter gas cylinder should not exceed 51.0 atm. Suppose the cylinder contains \(50.0 \mathrm{mol}\) of a gas. Use the SRK equation of state to calculate the maximum permissible cylinder temperature if the gas is (a) carbon dioxide and (b) argon. Finally, calculate the values that would be predicted by the ideal-gas equation of state.

A nitrogen rotameter is calibrated by feeding \(\mathrm{N}_{2}\) from a compressor through a pressure regulator, a needle valve, the rotameter, and a dry test meter, a device that measures the total volume of gas that passes through it. A water manometer is used to measure the gas pressure at the rotameter outlet. A flow rate is set using the needle valve, the rotameter reading, \(\phi\), is noted, and the change in the dry gas meter reading \((\Delta V)\) for a measured running time \((\Delta t)\) is recorded. The following calibration data are taken on a day when the temperature is \(23^{\circ} \mathrm{C}\) and barometric pressure is \(763 \mathrm{mm} \mathrm{Hg} .\) $$\begin{array}{rrr} \hline \phi & \Delta t(\min ) & \Delta V(\mathrm{L}) \\ \hline 5.0 & 10.0 & 1.50 \\ 9.0 & 10.0 & 2.90 \\ 12.0 & 5.0 & 2.00 \\ \hline \end{array}$$ (a) Prepare a calibration chart of \(\phi\) versus \(\dot{V}_{\text {sid }}\), the flow rate in standard \(\mathrm{cm}^{3} / \mathrm{min}\) equivalent to the actual flow rate at the measurement conditions. (b) Suppose the rotameter-valve combination is to be used to set the flow rate to 0.010 mol \(\mathrm{N}_{2} / \mathrm{min}\). What rotameter reading must be maintained by adjusting the valve?

In froth flotation, air is bubbled through an aqueous solution or slurry to which a foaming agent (soap) has been added. The air-soap bubbles carry finely dispersed solids and hydrophobic materials such as grease and oil to the surface where they can be skimmed off in the foam. An ore-containing slurry is to be processed in a froth flotation tank at a rate of 300 tons/h. The slurry consists of \(20.0 \mathrm{wt} \%\) solids (the ore, \(\mathrm{SG}=1.2\) ) and the remainder an aqueous solution with a density close to that of water. Air is sparged (blown through a nozzle designed to produce small bubbles) into the slurry at a rate of \(40.0 \mathrm{ft}^{3}\) (STP)/1000 gal of slurry. The entry point of the air is 10 \(\mathrm{ft}\) below the slurry surface. The tank contents are at \(75^{\circ} \mathrm{F}\) and the barometric pressure is 28.3 inches of Hg. The sparger design is such that the average bubble diameter on entry is \(2.0 \mathrm{mm}\). (a) What is the volumetric flow rate of the air at its entering conditions? (b) By what percentage does the average bubble diameter change between the entry point and the slurry surface?
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