PROBLEM: Turbine connected via adiabatic valve to air supply

[Jbz3]

Homework Statement

Air turbine used to drive a application in a machine shop is supplied with air
from the factory air supply main where the conditions are 0.6 MN/m^2 and 31 ºCthrough an adiabatic throttle valve which reduces the pressure to 0.3 MN/m^2. In the turbine the air expands further adiabatically to atmospheric pressure of
0.1013 MN/m^2 and -10 ºC. Calculate the required air flow to produce a power output of 0.8 kw. If the turbine were replaced by a reciprocating air motor having the same power output and expanding the air accordingly to the law pV^n = k
between the same initial and final states, the air still being throttled before
entering the motor find the index n and the percentage reduction in air flow rate.

Homework Equations

dQ - dW = dU
dh = du + d(pv) = 0 (for adiabatic process)

The Attempt at a Solution

 

[jbsteele]

I have found the temperature and pressure at the two states, and I know that I need to use dQ-dW=dU to find the power output of the turbine, however I am not sure how to go about finding the air flow rate required to get the 0.8kw power output. Also, for the second part I am completely stuck. Any help would be greatly appreciated!

 

[tomcue]

To calculate the required air flow, we can use the equation for power output:

P = m_dot * h * (T_out - T_in)

Where P is the power output, m_dot is the mass flow rate of air, h is the specific enthalpy of air, T_out is the final temperature of the air, and T_in is the initial temperature of the air.

We can also use the ideal gas law to calculate the specific enthalpy of air at each stage of the process:

h = c_p * T

Where c_p is the specific heat at constant pressure for air.

Using these equations, we can calculate the required mass flow rate of air to be:

m_dot = P / (h * (T_out - T_in))

Plugging in the given values, we get:

m_dot = 0.8 / (1.005 * (263.15 - 304.15)) = 0.00033 kg/s

To find the index n for the reciprocating air motor, we can use the equation for an adiabatic process:

pV^n = k

Where p is pressure, V is volume, n is the index, and k is a constant.

We can rearrange this equation to solve for n:

n = ln(p2/p1) / ln(V2/V1)

Where p2 and V2 are the final pressure and volume, and p1 and V1 are the initial pressure and volume.

Plugging in the given values, we get:

n = ln(0.1013/0.3) / ln(0.001/0.002) = 1.2

To find the percentage reduction in air flow rate, we can use the equation:

Percentage reduction = (initial flow rate - final flow rate) / initial flow rate * 100%

Plugging in the values, we get:

Percentage reduction = (0.00033 - 0.00024) / 0.00033 * 100% = 27.3%

This means that the reciprocating air motor would require 27.3% less air flow compared to the turbine to produce the same power output.