1. Unit 61: Engineering Thermodynamics
Lesson 13: General Formulation for Control Volumes

2. Objective
The purpose of this lesson is to consider compressors.

3. Cycle
Several cycles utilise a gas as the working substance, the most common being the Otto cycle and the diesel cycle used in the internal combustion engine.
The word ‘cycle’ used in reference to an internal combustion engine is technically incorrect since the working fluid does not undergo a thermodynamic cycle: air enters the engine, mixes with a fuel, undergoes combustion and exits the engine as exhaust gases.
This is often referred to as an ‘open cycle’.

4. Cycle
In a thermodynamic cycle this does not really occur; the engine itself operates in what we could call a mechanical cycle
We analyze the internal combustion engine as though the working fluid operated on a cycle – it is an approximation that allows us to predict influences of engine design on such quantities as efficiency and fuel consumption.

5. Gas Compressors . . A gas compressor can be found in refrigeration.
The control volume energy equation relates the power input to the enthalpy change…
Wcomp = m(he – hi)
Where he and hi are the exit and inlet enthalpies respectively.
In this form we model the compressor as a fixed volume into which and from which a gas flows . .

6. Gas Compressors
We assume that negligible heat transfer occurs from the compressor and ignore the kinetic and potential energy changes.
There are three general types of compressor - reciprocating, centrifugal and axial flow.
Reciprocating compressors are especially useful for producing high pressures, but are limited to relatively low flow rates: upper limits of about 200MPa with inlet flow rates of 160m3/min are achievable with a two stage unit.

7. Gas Compressors
For high flow rates with relatively low pressure rise, a centrifugal or axial-flow compressor would be selected: a pressure rise of several MPa for an inlet flow rate of over 10,000 m3/min is possible.

8. The Reciprocating Compressor
4->1
3->4
1->2
2->3 P 1 4 3 2 P 1 4 3 2
The Ideal Cycle
The Actual Cycle v v

9. The Reciprocating Compressor
The intake and exhaust valves are closed when state 1 is reached.
An isentropic compression follows as the piston travels inward until maximum pressure at state 2 is reached.
The exhaust valve then opens and the piston continues its inward motion while the air is exhausted until state 3 is reached at TDC

10. The Reciprocating Compressor
The exhaust valve then closes and the piston begins its outward motion with an isentropic expansion process until state 4 is reached.
At this point the intake valve opens and the piston moves outward during the intake process until the cycle is complete.

11. The Reciprocating Compressor
Note: the actual cycle resembles that if the continuous curve:
The intake and exhaust valves don’t open instantaneously, the air flow around the valves results in pressure gradients during intake and exhaust strokes, losses occur due to the valves and some heat transfer may take place.
The ideal cycle does however allow us to predict the influence of proposed design changes on work requirements, maximum pressure, flow rate and other quantities of interest.

12. The Reciprocating Compressor
The effectiveness of a compressor is partially measured by the volumetric efficiency which is defined as the volume of gas drawn into the cylinder divided by the displacement volume.
ηvol = (V1 – V4)/(V1 – V3)
The higher the volumetric efficiency, the greater the volume of air in as a percentage of the displacement volume.

13. The Reciprocating Compressor
To improve the performance of a reciprocating compressor we can remove heat from the compressor during the compression process 1-to-2
It thus becomes polytropic

14. The Reciprocating Compressor1 4 3 2’ 2
Isentropic: n= k ; Q = 0
Polytropic:
Q < 0 V

15. The Reciprocating Compressor
The temperature of state 2’ would be significantly lower than that of state 2 and the work requirement for the complete cycle would be less since the area under the P-v diagram would increase.
The required work is for an adiabatic compressor…
wcomp = h2 – h1 = Cp(T2 – T1)

16. The Reciprocating Compressor
This is assuming an ideal gas with constant specific heat.
For an isentropic compression between inlet and outlet, we know that…
T2 = T1(P2/P1)(k-1)/k
Thus…
wcomp = (kR/(k-1))T1[(P2/P1)(k-1)/k - 1]

17. The Reciprocating Compressor
For a polytropic process we can replace k with n… wcomp = (nR/(n-1))T1[(P2/P1)(n-1)/n - 1]

18. The Reciprocating Compressor
Consider a 2-stage compressor with a single intercooler…
1st Stage
2nd Stage
Intercooler Q 1 4 2 3
Low P
High P

19. The Reciprocating Compressor
P2 = P3 1 4 3 2 T 4
s1 = s2
s3 = s4 T1 2 3 1 v

20. The Reciprocating Compressor
Thus…
wcomp = CpT1[(P2/P1)(k-1)/k-1] + CpT3[(P4/P3)(k-1)/k-1]
= CpT1[(P2/P1)(k-1)/k + CpT3[(P4/P2)(k-1)/k - 2]
Where P2 = P3 and T1 = T3 for an ideal intercooler.

21. The Reciprocating Compressor
To determine the intercooler pressure that minimizes work, dwcomp/dp =0. Thus
P2 = (P1P4)1/2
or P2/P1 = P4/P3
That is the pressure ration is the same across each stage.

22. The Reciprocating Compressor
If three stages were used, the same analysis would lead to a low pressure of…
P2 = (P12P6)1/3
and a high pressure of
P4 = (P1P62)1/3
Where P6 is the highest pressure.

23. The Reciprocating Compressor

24. Schematic of a Power Plant
Boiler
Steam high energy)
Qin
Turbine
Wout
Steam
Low-energy
water
System Boundary
water
Condenser
Win
Pump
Qout