Thermodynamics Principles on Gas power cycles

2. GAS POWER CYCLES
Def: A cycle is defined as a repeated series of operations occurring in a certain order. The cycle may be
of imaginary perfect engine or actual engine. The former is called ideal cycle and the latter actual
cycle. In ideal cycle all accidental heat losses are prevented and the working substance is assumed to
behave like a perfect working substance.
Air Standard Efficiency
This is the efficiency of the engine using air as a working medium.
Relative efficiency 𝜏𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒=
𝐴𝑐𝑡𝑢𝑎𝑙 𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦
𝐴𝑖𝑟 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦
The analysis of all air standard cycles is based upon the following assumptions.
• The gas in the engine cylinder is a perfect gas i.e. it obeys the gas laws.
• The physical contact of the gas in the cylinder are the same as those off air at moderate
temperature.
• The compression and expansion processes are adiabatic and they take place without internal friction.
• No chemical reaction takes place in the cylinder.
• The cycle is considered with the same air always remaining in the cylinder to repeat the cycle.

3. THE CARNOT CYLCE
A Carnot cycle is defined as an ideal reversible closed thermodynamic cycle.
Carnot Ideal Heat Engine
A cylinder having perfectly non-conducting walls, a perfectly conducting base, and is provided with a
perfectly non-conducting piston which moves without friction in the cylinder. The cylinder contains
one mole of perfect gas as the working substance.
Source. A reservoir maintained at a constant temperature 𝑇1 from which the engine can draw heat by
perfect conduction. It has an infinite thermal capacity, and any amount of heat can be drawn from it at
constant temperature 𝑇2.
Heat insulating stand. A perfectly non conducting platform acts as a stand for adiabatic processes.
Sink. A reservoir maintained at constant lower temperature 𝑇2 (𝑇2 > 𝑇1 ) to which the heat engine can
reject any amount of heat. The thermal capacity of the sink is infinite, so its temperature remains
constant at 𝑇2, no matter how much heat is given to it.

4. Carnot Cycles and its Stages
Line AB - Isothermal Expansion: the cylinder is first placed on the
source so that the gas acquires the temperatures 𝑇1 of the source. It
is then allowed to undergo quasi-static expansion. As the gas
expands, its temperature tends to fall. Heat passes into the cylinder
through the perfectly conducting base, which is in contact with the
source. The gas, therefore, undergoes slow isothermal expansion at
the constant temperature 𝑇1 .
𝑄1 = 𝑊1 =
𝑉1
𝑉2
𝑃𝑑𝑉 = 𝑅𝑇1 ln
𝑣2
𝑣1
Line BC – Adiabatic Expansion: the cylinder is now removed from the source and placed on the insulating
stand. The gas is allowed to undergo slow adiabatic expansion, performing external work at the expense of its
internal energy until its temperature falls to 𝑇1 , the same as that of the sink. In this process, there is no transfer
of heat, the temperature falls to 𝑇2 and it does some external work given by;
𝑊2 =
𝑉2
𝑉3
𝑃𝑑𝑉 =
𝑅(𝑇1− 𝑇2)
𝛾 − 1

5. Line CD – Isothermal Compression: the cylinder is now removed from the insulating stand and is placed on
the sink, which is at a temperature of 𝑇2. The piston is slowly moved inward so that work is done on the
gas, in this process, the substance rejects heat to the sink at 𝑇2 and work is done on the substance given
by;
𝑄3 = 𝑊3 =
𝑉3
𝑉4
𝑃𝑑𝑉 = 𝑅𝑇1 ln
𝑣4
𝑣3
Line DA – Adiabatic Compression: the cylinder is now removed from the sink and again placed on the
insulating stand. The piston is slowly moved inwards so that the gas in adiabatic compression is continued
till the gas comes back to its original condition i.e state a thus completing one full cycle. In this process,
work is done on the substance and is given by;
𝑊4 =
𝑉4
𝑉1
𝑃𝑑𝑉 =
𝑅(𝑇1− 𝑇4)
𝛾 − 1
Efficiency
The efficiency of the heat engine is the rate of the quantity of heat converted into work (useful output) per
cycle to the total amount od heat absorbed per cycle.
Efficiency 𝜂 =
𝑢𝑠𝑒𝑓𝑢𝑙 𝑜𝑢𝑡𝑝𝑢𝑡
𝑖𝑛𝑝𝑢𝑡
=
𝑄1−𝑄2
𝑄1
=
𝑅(𝑇1− 𝑇2) ln
𝑣2
𝑣1
𝑅𝑇1 ln
𝑣2
𝑣1
=
𝑇1− 𝑇2
𝑇1
= 1 −
𝑇2
𝑇1

6. Example

7. CONSTANT VOLUME (OTTO CYCLE)
The Otto cycle is an idealized thermodynamic cycle that describes the functioning of typical spark
ignition piston engine. It is the description of what happens to a gas as it is subjected to changes of
pressure, temperature, volume, addition of head and removal of heat.
The point 1 represents that the cylinder is full of air with
volume V1 , pressure p1 and absolute temperature T1.
Line 1-2 represents the adiabatic compression of air due to
which p1 , V1 and T1 change to p2 , V2 and T2 , respectively.
Line 2-3 shows the supply of heat to the air at constant
volume so that p2 and T2 change to p3 and T3 (V3 being the
same as V2 ).
Line 3-4 represents the adiabatic expansion of the air. During
expansion p3 , V3 and T3 change to a final value of p4 , V4 or
V1 and T4 , respectively.
Line 4-1 shows the rejection of heat by air at constant
volume till original state (point 1) reaches.

8. Consider 1 kg of air (working substance) :
Heat supplied at constant volume = 𝐶𝑣(𝑇3 -𝑇2).
Heat rejected at constant volume = cv (𝑇4- 𝑇1).
But, work done = Heat supplied - Heat rejected = 𝐶𝑣(𝑇3 -𝑇2) - 𝐶𝑣(𝑇4 -𝑇1).
Efficiency of the Otto cycle
Efficiency =
𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒
ℎ𝑒𝑎𝑡 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑
=
𝐶𝑣(𝑇3 −𝑇2) − 𝐶𝑣(𝑇4 −𝑇1).
𝐶𝑣(𝑇3 −𝑇2)
= 1 −
𝑇4 −𝑇1
𝑇3 −𝑇2
Or Efficiency = 1 −
1
(𝑟)𝜆−1 where 𝑟 = 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 =
𝑣1
𝑣2
and expansion ratio =
𝑣4
𝑣3
This expression is known as the air efficiency of the Otto system.
Net Work done
𝑊 =
𝑃1𝑉1
𝜆−1
(𝑟𝜆−1
− 1)(𝑟𝑝 − 1) where 𝑟𝑝 = pressure ratio given by
𝑃3
𝑃2
=
𝑃4
𝑃1
= 𝑟𝑝
Mean Effective Pressure (𝑷𝒎)
𝑷𝒎 =
𝑃1𝑟 (𝑟𝜆−1−1)(𝑟𝑝−1)
(𝜆−1)(𝑟−1)
where 𝑟𝑝 = pressure ratio given by
𝑃3
𝑃2
=
𝑃4
𝑃1
= 𝑟𝑝

9. Example

10. CONSTANT PRESSURE (DIESEL CYCLE)
This cycle was introduced by Dr. R. Diesel in 1897. It differs from Otto cycle in that heat is supplied at
constant pressure instead of at constant volume. Fig. 13.15 (a and b) shows the p-v and T-s diagrams
of this cycle respectively.
This cycle comprises of the following operations :
• 1-2: Adiabatic compression.
• 2-3: Addition of heat at constant pressure.
• 3-4: Adiabatic expansion.
• 4-1: Rejection of heat at constant volume.

11. Point 1 represents that the cylinder is full of air. Let p1 , V1 and T1 be the corresponding pressure, volume and absolute
temperature. The piston then compresses the air adiabatically (i.e., PV = constant) till the values become p2 , V2 and T2
respectively (at the end of the stroke) at point 2. Heat is then added from a hot body at a constant pressure. During this
addition of heat let volume increases from V2 to V3 and temperature T2 to T3 , corresponding to point 3. This point (3)
is called the point of cut-off. The air then expands adiabatically to the conditions p4 , V4 and T4 respectively
corresponding to point 4. Finally, the air rejects the heat to the cold body at constant volume till the point 1 where it
returns to its original state.
Heat supplied at constant pressure = 𝐶𝑝(𝑇3 − 𝑇2)
Heat rejected at constant volume = 𝐶𝑉(𝑇4 − 𝑇1)
Work done = Heat supplied – Heat rejected = 𝐶𝑝(𝑇3 − 𝑇2) - 𝐶𝑉(𝑇4 − 𝑇1)
Efficiency of the Otto cycle
Efficiency =
𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒
ℎ𝑒𝑎𝑡 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑
=
𝐶𝑝(𝑇3−𝑇2) − 𝐶𝑉(𝑇4−𝑇1)
𝐶𝑝(𝑇3−𝑇2)
= 1 −
𝑇4 −𝑇1
𝜆 𝑇3 −𝑇2
where 𝜆 =
𝐶𝑝
𝐶𝑉
Or Efficiency = 1 −
1
𝜆(𝑟)𝜆−1
𝜌𝜆−1
𝜌−1
where 𝑟 = 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 =
𝑣1
𝑣2
and cut-off ratio 𝜌 =
𝑣3
𝑣2
=
𝑣𝑜𝑢𝑚𝑒 𝑎𝑡 𝑐𝑢𝑡 𝑜𝑓𝑓
𝑐𝑙𝑒𝑎𝑟𝑎𝑛𝑐𝑒 𝑣𝑜𝑙𝑢𝑚𝑒
This expression is known as the air efficiency of the Diesel system.

12. Net Work done
𝑊 =
𝑃1𝑉1𝑟𝜆−1
𝜆−1
𝜆(𝜌 − 1)𝑟1−𝜆(𝜌𝜆 − 1) where 𝜌= cut off ratio.
Mean Effective Pressure (𝑷𝒎)
𝑷𝒎 =
𝑃1𝑟𝜆 𝜆(𝜌−1)𝑟1−𝜆(𝜌𝜆−1)
(𝜆−1)(𝑟−1)
where 𝜌= cut off ratio.

13. Example