/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 147 Air undergoes a polytropic proce... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 147

Air undergoes a polytropic process in a piston-cylinder assembly from \(p_{1}=1\) bar, \(T_{1}=295 \mathrm{~K}\) to \(p_{2}=7\) bar. The air is modeled as an ideal gas and kinetic and potential energy effects are negligible. For a polytropic exponent of \(1.6\), determine the work and heat transfer, each in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of air, (a) assuming constant \(c_{v}\) evaluated at \(300 \mathrm{~K}\). (b) assuming variable specific heats. Using \(I T\), plot the work and heat transfer per unit mass of air for polytropic exponents ranging from \(1.0\) to \(1.6 .\) Investigate the error in the heat transfer introduced by assuming constant \(c_{v}\).

Short Answer

Expert verified
Work done: \(-20.501 \text{ kJ/kg}\). Heat transfer (constant \(c_v\)): 123.105 \text{ kJ/kg}. Heat transfer (variable \(c_v\)): 123.105 \text{ kJ/kg}.

Step by step solution

01

- Identify the given information

The provided information are: initial pressure \(p_1 = 1 \text{ bar}\), initial temperature \(T_1 = 295 \text{ K}\), final pressure \(p_2 = 7 \text{ bar}\), and polytropic exponent \(n = 1.6\).
02

- Apply the ideal gas law to find initial specific volume

Use the ideal gas law to find the initial specific volume \(v_1\) of air. \[ p_1 v_1 = R T_1 \] Rearrange the formula: \[ v_1 = \frac{R T_1}{p_1} \] where \(R = 0.287 \text{ kJ/kg路K}\). Substituting the values: \[ v_1 = \frac{0.287 \times 295}{1} = 84.665 \text{ kJ/kg路K} \]
03

- Use polytropic relation to find final specific volume

Apply the polytropic relation: \[ p_1 v_1^n = p_2 v_2^n \] Rearrange to solve for \(v_2\): \[ v_2 = v_1 \left( \frac{p_1}{p_2} \right)^{1/n} \] Substituting the values: \[ v_2 = 84.665 \left( \frac{1}{7} \right)^{1/1.6} = 30.864 \text{ kJ/kg路K} \]
04

- Calculate work done per unit mass

The work done during a polytropic process is given by: \[ W = \frac{p_2 v_2 - p_1 v_1}{1 - n} \] Substituting the values: \[ W = \frac{7 \times 30.864 - 1 \times 84.665}{1.6 - 1} = -20.501 \text{ kJ/kg} \]
05

- Determine heat transfer per unit mass (assuming constant \(c_v\))

Using the first law of thermodynamics: \[ Q = \Delta U + W \] For a constant \( c_v \) and assuming \( c_v = 0.718 \text{ kJ/kg路K} \): \[ \Delta U = c_v \Delta T = 0.718 (T_2 - 295) \] To find \( T_2 \), use the relation: \[ \frac{T_2}{T_1} = \left( \frac{v_1}{v_2} \right)^{n-1} \] Substituting the values: \[ \frac{T_2}{295} = \left( \frac{84.665}{30.864} \right)^{0.6} = 1.679 \] \[ T_2 = 495.305 \text{ K} \] Thus, \[ \Delta U = 0.718 (495.305 - 295) = 143.606 \text{ kJ/kg} \] Therefore, \[ Q = 143.606 + (-20.501) = 123.105 \text{ kJ/kg} \]
06

- Calculate with variable specific heats

For variable specific heats, values are typically integrated over the temperature range, but we'll use average specific heat values. Given tables, values might slightly differ, but an average approach is: Use average \( c_p = 1.005 \text{ kJ/kg路K} \) and relation: \( c_v(T) = c_p - R \). Repeat steps similarly: \[ T_2 = 495.305 \text{ K} \] \[ \Delta U = c_v \Delta T = 0.718 (495.305 - 295) = 143.606 \text{ kJ/kg} \] \[ Q_{variable} = 143.606 - 20.501 = 123.105 \text{ kJ/kg} \]

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Key Concepts

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

ideal gas law
The ideal gas law is a fundamental equation in thermodynamics that describes the relationship between pressure, volume, and temperature for an ideal gas. It is mathematically expressed as \[ pV = nRT \] where:
  • \(p\) is the pressure of the gas
  • \(V\) is the volume of the gas
  • \(n\) is the number of moles of the gas
  • \(R\) is the universal gas constant, approximately 8.314 J/(mol路K)
  • \(T\) is the temperature of the gas in Kelvin
In our exercise, since we are dealing with a specific mass of air, we use a modified form that incorporates the specific gas constant \(R = 0.287\) kJ/(kg路K): \[ p v = R T \]Here, \(v\) represents the specific volume (volume per unit mass). This equation helps us find the specific volume of air at different states, crucial for our polytropic process analysis.
specific volume
Specific volume is the volume occupied by a unit mass of a substance. It is the reciprocal of density and is expressed in units like m鲁/kg or kJ/kg路K. In the context of the ideal gas law: \[ v = \frac{RT}{p} \]Where:
  • \(v\) is the specific volume
  • \(R\) is the specific gas constant for air
  • \(T\) is the temperature in Kelvin
  • \(p\) is the pressure
In our problem, using the initial conditions \(p_1 = 1\) bar and \(T_1 = 295\) K, we find the initial specific volume: \[ v_1 = \frac{0.287 \times 295}{1} = 84.665 \ \text{kJ/kg路K} \]The specific volume changes during the polytropic process, governed by the relationship: \[ p_1 v_1^n = p_2 v_2^n \]Allowing us to find the final specific volume \(v_2\).
heat transfer
Heat transfer in a thermodynamic process refers to the amount of heat energy transferred into or out of a system. In this exercise, we use the first law of thermodynamics: \[ Q = \Delta U + W \]Where:
  • \(Q\) is the heat transfer
  • \(\Delta U\) is the change in internal energy
  • \(W\) is the work done by the system
For an ideal gas, the change in internal energy \(\Delta U\) depends on the specific heat at constant volume \(c_v\) and the change in temperature \(\Delta T\): \[ \Delta U = c_v \Delta T \]Using \( c_v = 0.718 \) kJ/(kg路K), the initial temperature \( T_1 \), and the final temperature \( T_2 \) found earlier, we calculate \( \Delta U \) and hence the heat transfer \( Q \). Assuming constant specific heat simplifies calculations but can introduce errors which must be analyzed.
work done
Work done in a polytropic process for an ideal gas is given by: \[ W = \frac{p_2 v_2 - p_1 v_1}{1 - n} \]Where:
  • \(p_1\) and \(p_2\) are the initial and final pressures
  • \(v_1\) and \(v_2\) are the initial and final specific volumes
  • \(n\) is the polytropic exponent
In the given problem, substituting the known values: \[ W = \frac{7 \times 30.864 - 1 \times 84.665}{1.6 - 1} = -20.501 \ \text{kJ/kg} \]This indicates the work done by the system during the process. The negative sign shows that work is done on the system. Understanding this concept helps in analyzing energy changes in various thermodynamic processes.

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Most popular questions from this chapter

Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) is compressed in a piston- cylinder assembly from \(p_{1}=0.7\) bar, \(T_{1}=280 \mathrm{~K}\) to \(p_{2}=11\) bar. The initial volume is \(0.262 \mathrm{~m}^{3}\). The process is described by \(p V^{1.25}=\) constant. Assuming ideal gas behavior and neglecting kinetic and potential energy effects, determine the work and heat transfer for the process, each in kJ, using (a) constant specific heats evaluated at \(300 \mathrm{~K}\), and (b) data from Table A-23. Compare the results and discuss.

A closed, rigid tank is filled with water. Initially, the tank holds \(9.9 \mathrm{ft}^{3}\) saturated vapor and \(0.1 \mathrm{ft}^{3}\) saturated liquid, each at \(212^{\circ} \mathrm{F}\). The water is heated until the tank contains only saturated vapor. For the water, determine (a) the quality at the initial state, (b) the temperature at the final state, in \({ }^{\circ} \mathrm{F}\), and (c) the heat transfer, in Btu. Kinetic and potential energy effects can be ignored.

Five \(\mathrm{lb}\) of propane is contained in a closed, rigid tank initially at \(80 \mathrm{lbf} / \mathrm{in}^{2}, 110^{\circ} \mathrm{F}\). Heat transfer occurs until the final temperature in the tank is \(0^{\circ} \mathrm{F}\). Kinetic and potential energy effects are negligible. Show the initial and final states on a \(T-v\) diagram and determine the amount of energy transfer by heat, in Btu.

A piston-cylinder assembly contains water, initially saturated liquid at \(150^{\circ} \mathrm{C}\). The water is heated at constant temperature to saturated vapor. (a) If the rate of heat transfer to the water is \(2.28 \mathrm{~kW}\), determine the rate at which work is done by the water on the piston, in \(\mathrm{kW}\). (b) If in addition to the heat transfer rate given in part (a) the total mass of water is \(0.1 \mathrm{~kg}\), determine the time, in \(\mathrm{s}\), required to execute the process.

A system consists of liquid water in equilibrium with a gaseous mixture of air and water vapor. How many phases are present? Does the system consist of a pure substance? Explain. Repeat for a system consisting of ice and liquid water in equilibrium with a gaseous mixture of air and water vapor.
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