LESSON - 12 LIMITATION OF FIRST LAW AND ESSENCE OF SECOND LAW, HEAT ENGINES AND THE KELVIN-PLANCK STATEMENT

 12.1. LIMITATIONS OF FIRST LAW AND NEED OF SECOND LAW

In previous lessons we have studied that any cycle/process is feasible if it obeys First Law of Thermodynamics. However, the First Law places no restrictions on the direction of flow of heat and work i.e. the process proceeds in any direction and cycle can be reversed. But experimentally there is restriction on the direction of flow of heat and work i.e. the cycle/process proceeds in a certain direction in nature as explained below:

A cup of hot coffee left in a cooler room eventually cools off by transferring heat from hot coffee to cooler room. But the reverse process in which coffee getting hotter as a result of heat transfer from a cooler room does not take place.

1. Consider heating of a room by passage of electric current through an electric resistor. Transferring of heat from room will not cause electrical energy to be generated through the wire.

2. Consider a paddle-wheel mechanism operated by fall of mass. Potential energy of mass decreases and internal energy of the fluid increases. Reverse process does not happen, although this would not violate First law.

3. Water flows down hill where by potential energy is converted into K.E. Reverse of this process does not occur in nature.

4. Fuel level in fuel tank goes down since the fuel is used as a car drives up a hill, but while going down the hill, the fuel level in the fuel tank cannot be restored to its original level.

It is clear from the above examples that the processes proceed in certain direction and not in the reverse direction even they don’t violate First Law of thermodynamics. So, mere satisfaction of First law alone does not ensure that the cycle/process will actually take place. This inadequacy of the First Law to identify whether a process can take place is remedied by introducing another general principle, the Second Law of thermodynamics. The second law of thermodynamics is based on experimental feasibility of the cycle.   Thus a cycle/process will occur only if both the First and Second laws of thermodynamics are satisfied (Fig. 12.1).

Fig. 12.1 A process occurs when it satisfy 1st and 2nd Law of thermodynamics.

 The second law of thermodynamics, in addition to identifying the direction of processes, is also used in

• determining the theoretical performance of commonly used engineering devices, such as heat engines and refrigerators.

• quantifying irreversibilities in a system and processes.

• defining a temperature scale independent of the properties of any thermometric substance.

• giving the concept of entropy.

12.2. THINGS WHICH LEAD US TO THE THOUGHT OF HEAT ENGINE AND REFRIGERATOR (HEAT PUMP)?

1) Consider a system and the surroundings as shown in Fig. 12.2 (a). Let the gas constitute the system undergo a cyclic process in which during the first process the work is done on the system by the paddle wheel by lowering the weight (process ‘1-A-2’ in Fig 12.1 (b)). Then let the cycle be completed by a process of transferring heat to the surroundings, (process ‘2-B-1’ in Fig 12.2 (b)).

Fig. 12.2. A closed system that operates in a cycle in which both heat and work transfer are negative.

But from our experience we know that we cannot reverse this cycle. That is, if we transfer heat to the gas from surroundings, as shown by the dotted arrow in Fig 12.3 (a) and process 1-B-2 in Fig 12.3 (b), the temperature of the gas will increase, but the paddle wheel will not turn and results zero work done by the system (i.e. process 2-1 is not feasible).

Fig. 12.3. A closed system that cannot operate in a cycle in which both heat and work transfer are positive.

Thus we can conclude that with the given surroundings the system can operate in a cycle in which both heat and work are negative, but it cannot operate in a cycle in which both the heat transfer and work are positive, even though this would not violate the First law.

2) Consider another cycle as shown in Fig. 12.4., we know from our experience it is actually impossible to accomplish this cycle in which we have a process of heat transfer from the low- temperature system to the high-temperature system.

Fig. 12.4. A cycle having heat transfer from a low-temperature body to high-temperature body.

The above two examples of cycles lead us to the consideration of the heat engine and refrigerator(heat pump).

• With the heat engine we can have a system that operates in a cycle and has a net positive work and a net positive heat transfer, which we were not able to do with the system and surroundings in Fig. 12.3.

• With the refrigerator/heat pump we can have a system that operates in a cycle and has heat transferred from a low-temperature body to a high-temperature body, which we were not able to do in Fig. 12.4.

12.3. HEAT ENGINE AND REFRIGERATOR (HEAT PUMP)

Detail of heat engine and refrigerators/heat pumps is given as below. Before proceeding further, the concept of a thermal reservoir is to be introduced.

Thermal reservoir: A thermal reservoir is a body to which and from which heat can be transferred indefinitely without change in the temperature of the reservoir. Thus, a thermal reservoir always remains at constant temperature.

For example: Boiler furnace, combustion chamber, nuclear reactor, sun, ocean and atmosphere.

Source: A reservoir from which heat is transferred is called a source.

Sink: A reservoir to which heat is transferred is called a sink.

12.3.1. Heat engines

The two arrangements of a simple heat engine that operates in a cycle and has a net positive work and net positive heat transfer, will be considered.

1^st arrangement: First arrangement of a heat engine, consisting of a cylinder fitted with a piston and stoppers, is shown in Fig. 12.5. Let the gas in the cylinder constitute the system. Initially the piston rests on the lower stoppers with a weight on the platform.

Let the system undergo the following states and processes to complete a cycle:

• At state ‘1’: Weight is put on the platform from weight stand.

• Process ‘1-2’: By transferring heat (+Q_H) from some high-temperature body (source) to the gas. Gas expands and raises the piston to the upper stoppers resulting in work done (+₁W₂) by a system.

• At state ‘2’: Weight is removed from the platform back on weight stand.

• Process ‘2-1’: By transferring heat (-Q_L) from the gas to a low-temperature body (sink).

• At state ‘1’: System restored to its initial state ‘1’, thus completing the cycle.

Fig. 12.5: Arrangement of a simple heat engine.

From the First law we conclude that during above cycle,

(+Q_H) + (-Q_L)  = (+W_net)

 The above device is called a heat engine in which gas is a working fluid, to which and from which heat is transferred, and which does a certain amount of work as it undergoes a cycle.

Examples: Internal combustion engine and gas turbine.

2^nd arrangement: Second arrangement of a heat engine is a simple steam power plant as shown in Fig. 12.6.   Each component in this plant may be analyzed by a steady-flow process, but considered as a whole it may be considered a heat engine in which water in liquid and vapor form (steam) acts as the working fluid.

 Fig. 12.6. A heat engine involving steady state and steady flow process.

The process in each component of this steam power plant has been discussed under section ‘Application area of thermodynamics’ in ‘Lesson 1’. An amount of heat (+Q_H), is transferred from a high-temperature body (source), which may be the products of combustion in a furnace, or a reactor. The turbine which does positive mechanical work by expansion of steam also drives the pump and thus it delivers the net work output, W _net = W_turbine - W_pump during the cycle. The quantity of heat, -Q_L is rejected to a low-temperature body (sink), which is usually cooling water in a condenser or atmospheric air.

From the First law we conclude that during the cycle,

(+Q_H) + (-Q_L) = (+W _net)

The simple steam power plant is also a heat engine, for it has steam and water working fluid, to which and from which heat is transferred, and which produces a certain amount of net positive work as it undergoes a cycle.

From the above two arrangements, A heat engine may be defined as a device that operates in a thermodynamic cycle and does a certain amount of net positive work as a result of heat transfer from a high-temperature body to a low-temperature body. In broader sense, all devices which produce work either through heat transfer or combustion is called heat engine.   The principle of heat engine is shown in Fig. 12.7. Thus, by means of a heat engine we are able to have a system that operates in a cycle and produces net work output and net heat transfer both positive, which we were not able to do with the system and surroundings shown in Fig. 12.3.

Fig 12.7. Principle of Heat Engine.

At this point it is appropriate to introduce the concept of thermal efficiency of a heat engine.

 Thermal efficiency (η_thermal)   

Problem 12.1:  Heat is transferred to a heat engine from a furnace at a rate of 80 MW. If the rate of waste heat rejection to a nearby river is 50 MW, determine the net power output and the thermal efficiency for this heat engine.

Solution:  A schematic of the heat engine is given in Fig. 12.8.

Given:

The rate of heat transfer from furnace (high-temperature reservoir) to heat engine =   = 80 MW

Rate of heat transfer from heat engine to river (low-temperature reservoir) =  = 50 MW 

(a) Determine the net power output, Wnet, out  : Formula:   Answer: The net power output of this heat engine is    = 80 − 50 = 30 MW   (b) Determine the thermal efficiency, ηth: Formula:Thermal efficiency = Answer:  Thermal efficiency = or  37.5%

Fig. 12.8. Heat engine.

12.4. STATEMENT OF THE SECOND LAW OF THERMODYNAMICS

On the basis of discussion we considered for heat engine we are now ready to state the Kelvin-Planck Statement of Second Law.

12.4.1. The Kelvin-Planck Statement 

We have seen in the previous section that a heat engine must reject some heat to a low-temperature reservoir in order to complete the cycle. That is, no heat engine can convert all the heat it receives to useful work. This limitation of heat engines forms the basis for the Kelvin–Planck statement of the Second law of thermodynamics, which is expressed as:

“It is impossible to construct a device that will operate in a cycle and produce no effect other than the raising of a weight and the exchange of heat with a single reservoir”.

This statement supports our discussion of the heat engine, and, in effect, it states that it is impossible to construct a heat engine that operates in a cycle and receives a given amount of heat from a high-temperature body and does an equal amount of work as shown in Fig. 12.9(a). The only alternative is that some heat must be transferred from the working fluid at a lower temperature to a low-temperature body as shown in Fig. 12.9(b). Thus, work can be done by the transfer of heat only if there are two temperature levels involved, and heat is transferred from the high-temperature body to the heat engine and also from the heat engine to the low-temperature body. This implies that it is impossible to build a heat engine that has a thermal efficiency of 100 per cent.

Thermal efficiency =      < 100 %

Figure 12.9 (a). (PMM-II). Impossible cycle as it satisfies only First Law and violates Second Law                      (Kelvin-Planck Statement).

Figure 12.9 (b). Possible cycle as it satisfies both First and Second Laws (Kelvin-Planck Statement).