Classification of Thermodynamic Cycles

 The thermodynamic cycles, in general, may be classified into the following two types;
1. Reversible Cycle or ideal cycle
2. Irreversible or natural or real cycle

Assumptions in Thermodynamic Cycles

                    The analysis of all thermodynamics cycles is based on the following assumptions:

1. The gas in the engine cylinder is a perfect gas, i.e. it obeys the gas laws and constant specific heats.
2. The physical constants of the gas in the engine cylinder are same as those of air at moderate temperature.
3. All the compression and expansion processes are adiabatic and they take place without any internal friction.
4. Heat is supplied by bringing a hot body in contact with the cylinder at appropriate points during the process. Similarly heat is rejected by bringing a cold body in contact with the cylinder at these points.
5. The cycle is considered to be a closed one and the same air is used again and again to repeat the cycle.
6. No chemical reaction, whatsoever, takes place in the engine cylinder.

Law of Equipartition of Energy

                   This Law states, "The total energy of a molecule is shared equally by the various degrees of freedom possessed by it."
                     In case of mono atomic molecules like argon and helium, the energy possessed by them is only that of transnational type, the rotational energy being negligible. We have already discussed in Art.
                              Energy of translation per molecule = 3/2*kT
                             Energy per molecule per degree freedom = 1/3*3/2kT

Degrees of Freedom

                         In order to describe completely the motion of a particle in one plane, only two quantities must be known, say its two rectangular components. Similarly, for a particle moving in space, three independent quantities must be known to describe its motion. A molecule in a rigid body can have three motions of vibration along any of three co-ordinates axis in addition to its three motions of translation. It is thus obvious that in order to completely describe the state of motion of a particle, six independent quantities must be known.
                       In general, the total number of independent quantities, which must be known for describing completely the state of motion of a body, are called its degrees of freedom.

Assumptions in the Kinetic Theory of Gases

The kinetic theory of gases is based on the following assumptions:
1. The volume of a gas consists of a large number of minutes particles called molecules. it has been experimentally found that there are about 26.8*10(power 18) molecules in 1ml of a gas at N.T.P.
2. The molecules are mere mass points. In other words, the size of a molecules is assumed to be negligible, as compared to the distance between the molecules.
3. The gas molecules are perfectly elastic sphere and exert negligible force of attraction or repulsion on one another, or on the walls of the containing vessel. Hence in a direct impact, the molecules rebound with the same velocity after each collision.
4. The molecules are continuously colliding against each other, and with the walls of the containing vessel.Between two collisions, a molecule moves in a straight line. This distance is called the free path of the molecule.
5. The time during which a collision takes place is negligible as compared to the time required to transverse the free path, i.e collisions are instantaneous.

Clausius Inequality

           The Clausius inequality state that whenever a closed system undergoes a cyclic process, the cyclic integral of &Q/T is less than zero for an irreversible cyclic process and equal to zero for a reversible cyclic process. Mathematically,
                                          infinity&Q/T < 0, for an irreversible cyclic process
                                          infinity &Q/T = 0, for a reversible cyclic process
and
   combining the equations is written as
                                                            infinity -< 0
The Clausius inequality not only gives mathematically expression to the second law of thermodynamics, but it also gives the quantitative measure of irreversibly of the system. For example, the equation for an irreversible cyclic process may be written as,
                                                   infinity &Q/T + 1 = 0
Where 1 represents the amount by which the given cyclic process irreversible. When 1 is equal to zero, then the given cyclic process will be reversible. Moreover, a cyclic process in which infinity dQ/T is more then zero, is impossible because it violates the second law of thermodynamics.

Units of Entropy

The unit of entropy depends upon the unit of heat employed and the absolute temperature. We know that
                                                           Change in entropy (dS) = &Q/T
       Therefore, if the heat supplied or rejected is in kJ and the temperature is in K, then the unit of entropy is kJ/K. The entropy may be expressed in so many units entropy without assigning any dimensional units. Since the entropy is expressed per unit mass of the working substance, it would be more correct to speak specific entropy. The absolute values of entropy cannot be determined, but only the change in entropy may be obtained by using equation.
     Theoretically, the entropy of a substance is zero at absolute zero temperature. Hence, in entropy calculation, some convenient datum should be selected from which measurement may be made.
        It may be noted that water 0degree C is assumed to have zero entropy, and changes in its entropy are reckoned from this temperature.

Available and Unavailable Heat

The heat energy of a system is considered to have the following two parts:

1.Available heat energy

2.Unavailable heat energy

       

  The available heat energy is that part of the heat energy which can be converted into mechanical work by ideal process which reduce the system in a state of equilibrium.

    

  The unavailable heat energy is that part of heat energy which can not be converted into mechanical work even by ideal process which reduce the system in a state of equilibrium.The common ideas used for unavailable heat energy, according to second Law of Thermodynamics, is the heat rejected by the system to the surrounding.

     From above, we have total energy or heat energy or heat supplied to the system,

                                                   

                                                   &Q = Available heat energy + Unavailable heat energy

                                                          = A.H.E + U.H.E = Workdone + heat rejected 

Importance of Entropy

The maximum possible efficiency obtainable by any engine working on a reversible Carnot Cycle is given by

                                                                      n = T1-T2/T1

where

                                                                      T1 = Highest absolute temperature, and 

                                                                      T2 = Lowest absolute temperature.

in general, efficiency is given by

                                                     n = Maximum work obtained/Heat supplied or absorbed = &W/&Q

For one degree temperature drop, the above expression may be written as

                                                               &W = $Q/T = dS = Change in entropy 

From This expression, it can be easily understand that

1. The change in entropy represents the maximum amount of work obtainable per degree drop in temperature.

2. The change in entropy may be regarded as a measure of the rate the availability of heat for transformation into work.

3. The increase in entropy is obtained from a given quantity of heat at a low temperature.

Relation Between Heat and Entropy

Consider the heating of a working substance by a reversible process as shown whose base represents the entropy and the vertical ordinate represents the absolute temperature.

Now Consider any point A on the curve 1-2. At this point, let a small quantity of heat (&Q) be supplied to the working substance, which will increase the entropy by dS. Let the absolute temperature at this instant be T. Then according to the definition of entropy,

&Q = TdS

From Fig we see that the terms TdS represents the area under the curve during this change of entropy. 

From Equations we get

ds = &Q/T

The total change in entropy may be obtained by integration the above expression from state 1 to state 2.

then we found

TdS = dU + pdv