Thermodynamics. Turbine, hwo to use enthelpy to find mass flow?

[eventob]

Homework Statement

The problem (tried my best to translate it):
A small high speed turbine is operating on compressed air. It deliveres dW/dT=100 W. At the inlet, the pressure is 400 kPa and the temperature 50*C.

At the exit, the pressure is 150 kPa and the temperature -30*C.

Neglect the velocity and assume an adiabatic process. Find the necessary mass flow of air through the turbine.

Homework Equations

First law for control volume
Definition ofg enthalpy

The Attempt at a Solution

I have derived the first law for a control volume:
dE/dt=(dQ/dt)-(dW/dt)+∑m_i (h_i+0.5v_i^2+gz_i)-∑m_e (h_e+0.5v_e^2+gz_e)

Where t is time, and m_i og m_e is rate of change of mass flow at the inlet and exit, respecitively.
Assumed steady state: dE/dt=0.
Adibatic dQ/dt=0.
Also m_e´=m_i´=m´

By neglecting kinetic and potential energy associated with gravity, i end up with:

dW/dt=m(h_i-h_e) <=> m=(dW/dt) / (h_i-h_e)

So far so good, but now I need to find the change of enthalphy. We were supposed to solve this task without the use of steam tables. I have tried to use the definition of constant volume heat capacity, but no luck so far. Any input?

Thanks in advance. :)

 

[Andrew Mason]

I think all you have to do is apply the first law to an adiabatic process (ΔQ=0) so:

ΔU = -W => dU/dt = -dW/dt

So start by determining the internal energy change of 1 mole of air going from 323K to 243K. [This is not a reversible process so the change in pressure does not help you determine the work done]. Assume air is an ideal diatomic gas with a Cv of 5R/2.

AM

 

[LabGuy330]

Let me ask, are you using temperature/pressure tables for air?

If so then with the known temp/pressure at the inlet (T1/P1) and the known temp/pressure at the outlet (T2/P2) then you can refer to the tables to find the inlet and outlet enthalpies, h1 and h2 respectively.

Thus Q = m(dot)*(h1 - h2)

Solve for m(dot)

Not sure if this is what your looking or but that is my two cents

 

[Andrew Mason]

Let me ask, are you using temperature/pressure tables for air?

If so then with the known temp/pressure at the inlet (T1/P1) and the known temp/pressure at the outlet (T2/P2) then you can refer to the tables to find the inlet and outlet enthalpies, h1 and h2 respectively.

Thus Q = m(dot)*(h1 - h2)

Solve for m(dot)

Not sure if this is what your looking or but that is my two cents

My suggestion to use change in internal energy is not correct because the work done on the turbine by the gas is not the total work that is done. Since the volume of the air doubles there is also the expansion work done against the atmosphere. This is taken into account in the change in enthalpy but not with the change in internal energy.

Since atmospheric pressure is constant you can determine the work done on the atmosphere: W_atm = P_atmΔV

By the first law,

ΔQ = 0 = ΔU + W where W is the total work done by the air = W_turbine + W_atm

ΔU = -W_turbine + -W_atm → W_turbine = - (ΔU + W_atm) = -(ΔU + P_atmΔV)

AM

 

[asteriatic]

Thermodynamics is a branch of science that deals with the study of energy and its transformations. It is a fundamental concept in understanding the behavior of various systems, including the operation of a turbine.

In this problem, we are given the necessary information to determine the mass flow of air through the turbine. To find this, we can use the first law of thermodynamics for a control volume, which takes into account the energy balance of the system.

We can also use the concept of enthalpy, which is defined as the sum of the internal energy and the product of pressure and volume. In an adiabatic process, where there is no heat transfer, the change in enthalpy is equal to the work done by the system.

To find the necessary mass flow, we can use the equation m = (dW/dt) / (h_i - h_e), where m is the mass flow, dW/dt is the power delivered by the turbine, and h_i and h_e are the enthalpies at the inlet and exit, respectively.

To find the enthalpies, we can use the definition of enthalpy and the ideal gas law. Since we are given the pressure and temperature at both the inlet and exit, we can use the ideal gas law to calculate the specific volume of air at those conditions. Then, by using the definition of enthalpy, we can calculate the enthalpies at the inlet and exit.

By substituting the values into the equation for mass flow, we can determine the necessary mass flow of air through the turbine. It is important to note that this solution assumes an adiabatic process and neglects the effects of kinetic and potential energy. However, in a real system, these factors may need to be taken into account for a more accurate calculation.

I hope this explanation helps in understanding how to use enthalpy to find mass flow in a turbine. Remember to always consider the assumptions and limitations of your calculations to ensure the accuracy of your results.