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Chapter 4: Problem 58

Air enters a compressor operating at steady state with a pressure of \(14.7 \mathrm{lbf} /\) in. \({ }^{2}\), a temperature of \(80^{\circ} \mathrm{F}\), and a volumetric flow rate of \(18 \mathrm{ft}^{3} / \mathrm{s}\). The air exits the compressor at a pressure of \(90 \mathrm{lbf} / \mathrm{in}^{2}\) Heat transfer from the compressor to its surroundings occurs at a rate of \(9.7\) Btu per lb of air flowing. The compressor power input is \(90 \mathrm{hp}\). Neglecting kinetic and potential energy effects and modeling air as an ideal gas, determine the exit temperature, in \({ }^{\circ} \mathrm{F}\).

Short Answer

Expert verified
The exit temperature of the air is in this exercise will thus be calculated using the given energy conservation and ideal gas laws.

Step by step solution

01

Determine required formulas and given data

List down the given data and identify the formula needed. Given: - Inlet Pressure, \(P_1 = 14.7 \text{ lbf/in}^2\) - Inlet Temperature, \(T_1 = 80^{\text{掳F}}\) - Volumetric flow rate, \(V_1 = 18 \text{ ft}^3/\text{s}\) - Outlet Pressure, \(P_2 = 90 \text{ lbf/in}^2\) - Heat transfer per unit mass, \(q = 9.7 \text{ Btu/lb}\) - Power input, \(W = 90 \text{ hp}.\)
02

Convert units to consistent set

Convert the given units into a consistent set. Using: - \(1 \text{ hp} = 2545 \text{ Btu/h}\)- Convert the volumetric flow rate into mass flow rate using specific volume constants.
03

Calculate mass flow rate

Using the volumetric flow rate and specific volume of the air, calculate the mass flow rate, \( \text{m} \text{虈}\). The specific volume of air at standard conditions is approximately \(0.0749 \text{ lb/ft}^3\). Therefore: \[ \text{m} = V_1 \times \text{(specific volume of air)} \]
04

Apply the energy balance (Steady Flow Energy Equation)

Apply the Steady Flow Energy Equation (SFEV) which is: \[ \text{Q} = m(\text{膜}_2 - \text{膜}_1) + \text{W} onumber \] Where 膜 represents enthalpy.
05

Extract the required enthalpy values

Using the ideal gas model for air: \[ \text{膜} = C_p \times T onumber \] Assuming specific heat at constant pressure, \(C_p\), remains constant, find \(T_2\) after rearranging and solving for it using above data.
06

Solve for exit temperature

Rearrange the given equations with the known and unknown variables and solve for the exit temperature, \(T_2\). Use initial calculations and values to determine specific enthalpy changes and thus the exit temperature.

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Key Concepts

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

steady-state energy balance
In thermodynamics, a steady-state energy balance is essential to understand how systems like compressors operate. In a steady-state, the properties of the system do not change over time. This means that all inputs, such as energy and mass, must equal the outputs to maintain equilibrium.
For our compressor problem, it operates under steady-state conditions, which simplifies our calculations. Here, we only consider the energy that comes in and goes out of the compressor, assuming that there are no energy storages or losses within the system itself.
This steady-state assumption allows us to apply the Steady Flow Energy Equation (SFEE), where we balance the heat energy, work done by the system, and changes in enthalpies.
ideal gas model
The ideal gas model is a fundamental concept used to describe gases' behavior under various conditions. In the context of our compressor problem, air is modeled as an ideal gas. This assumption is helpful because it allows us to use simpler equations to describe the relationship among pressure, volume, temperature, and mass.
The ideal gas equation is stated as: \( PV = nRT \) where:
  • \( P \) is the pressure
  • \( V \) is the volume
  • \( n \) is the number of moles
  • \( R \) is the universal gas constant
  • \( T \) is the temperature (in absolute units like Kelvin or Rankine).
Using this model, we simplify air's behavior in the compressor, allowing us to relate the initial and final states easily and solve for unknowns like the exit temperature.
enthalpy change calculation
Enthalpy, denoted as \( H\), is a measure of total energy in a thermodynamic system. The change in enthalpy is key to solving many thermodynamics problems, such as the compressor exercise. Enthalpy change can be calculated using specific heat capacities, typically under constant pressure conditions.
For an ideal gas, the change in enthalpy \( \Delta H \) is given by: \( \Delta H = C_p \Delta T \), where:
  • \( \Delta H \) is the change in enthalpy
  • \( C_p \) is the specific heat at constant pressure
  • \( \Delta T \) is the change in temperature.
In our case, the compressor's heat transfer and work inputs affect the enthalpy change. By applying the steady flow energy equation (SFEE), we resolve the final temperature using enthalpy changes, which incorporates the heat energy added to the system and the work done on the air.
heat transfer
Heat transfer is vital in understanding how energy moves in and out of the compressor. In our problem, the heat transfer rate is specified as 9.7 Btu per pound of air. This value indicates how much heat energy leaves the compressor to the surroundings.
There are three main modes of heat transfer 鈥� conduction, convection, and radiation. In compressors, due to rapid air compression, convection often dominates.
Heat transfer impacts the energy balance in the system. By using the given heat transfer rate, we can adjust our enthalpy calculations to reflect the actual thermal energy dynamics in the compressor.
In practice, accurate heat transfer values ensure that our calculations of the final temperature and overall efficiency are precise and reliable.

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Most popular questions from this chapter

A tank of volume \(1.2 \mathrm{~m}^{3}\) initially contains steam at \(8 \mathrm{MPa}\) and \(400^{\circ} \mathrm{C}\). Steam is withdrawn slowly from the tank until the pressure drops to \(p\). Heat transfer to the tank contents maintains the temperature constant at \(400^{\circ} \mathrm{C}\). Neglecting all kinetic and potential energy effects and assuming specific enthalpy of the exiting steam is linear with the mass in the tank. (a) determine the heat transfer, in \(\mathrm{kJ}\), if \(p=2 \mathrm{MPa}\). (b) plot the heat transfer, in \(\mathrm{kJ}\), versus \(p\) ranging from \(0.5\) to \(8 \mathrm{MPa}\).

An open feedwater heater operates at steady state with liquid water entering inlet 1 at 10 bar, \(50^{\circ} \mathrm{C}\), and a mass flow rate of \(60 \mathrm{~kg} / \mathrm{s}\). A separate stream of steam enters inlet 2 at 10 bar and \(200^{\circ} \mathrm{C}\). Saturated liquid at 10 bar exits the feedwater heater at exit 3 . Ignoring heat transfer with the surroundings and neglecting kinetic and potential energy effects, determine the mass flow rate, in \(\mathrm{kg} / \mathrm{s}\), of the steam at inlet 2 .

A rigid, well-insulated tank of volume \(0.9 \mathrm{~m}^{3}\) is initially evacuated. At time \(t=0\), air from the surroundings at 1 bar, \(27^{\circ} \mathrm{C}\) begins to flow into the tank. An electric resistor transfers energy to the air in the tank at a constant rate for 5 minutes, after which time the pressure in the tank is 1 bar and the temperature is \(457^{\circ} \mathrm{C}\). Modeling air as an ideal gas, determine the power input to the tank, in \(\mathrm{kW}\).

Refrigerant 134 a enters a compressor operating at steady state as saturated vapor at \(0.12 \mathrm{MPa}\) and exits at \(1.2 \mathrm{MPa}\) and \(70^{\circ} \mathrm{C}\) at a mass flow rate of \(0.108 \mathrm{~kg} / \mathrm{s}\). As the refrigerant passes through the compressor, heat transfer to the surroundings occurs at a rate of \(0.32 \mathrm{~kJ} / \mathrm{s}\). Determine at steady state the power input to the compressor, in \(\mathrm{kW}\).

Nitrogen gas is contained in a rigid \(1-\mathrm{m}\) tank, initially at 10 bar, \(300 \mathrm{~K}\). Heat transfer to the contents of the tank occurs until the temperature has increased to \(400 \mathrm{~K}\). During the process, a pressure-relief valve allows nitrogen to escape, maintaining constant pressure in the tank. Neglecting kinetic and potential energy effects, and using the ideal gas model with constant specific heats evaluated at \(350 \mathrm{~K}\), determine the mass of nitrogen that escapes, in \(\mathrm{kg}\), and the amount of energy transfer by heat, in \(\mathrm{kJ}\).
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