Kelvi Plank and Clausius Statement Equivalence
EQUIVALENCE OF THE KELVIN-PLANK AND CLAUSIUS STATEMENTS Two statements are said to be equivalent if the truth statement implies the trust of the other, and vacation of one statement implies the violation of other.
Thus, there could be two methods to show the equivalence. Method 1: Violation of Kelvin-Plank statement by violating Clausius statement: From Fig. (2. 4. 9) (a) let us assume that a heat pump receives heat Q2 from low temperature reservoir at T2 and supplies it to high temperature sink at T1 without any external work, thus violating the Clausius statement.
A larger quantity of heat (Q1 + Q2) is supplied to heat engine (by high temperature source at T1) Which produces net work output Wnet equal to Q1 and rejects an amount of heat Q2 to low temperature reservoir. The composite of two devices is shown in Fig. (b). It shows that the heat pump helps the heat Q2 to flow from low temperature reservoir to high temperature reservoir, whereas, the heat engine supplier back heat Q2 from high temperature reservoir to low temperature thermal energy reservoir.The equivalent system receives heat Q1 from high temperature reservoir and produces an equivalent amount of work, as shown is Fig.
(c). From Fig. (c) it can be observed that a heat engine receives heat from single reservoir and produces an equivalent amount of work. It is therefore a perpetual motion machine of type II (PMM-2), which violates the Kelvin-Plank’s statement for second law of thermodynamics. Thus, violation of Clausius statement leads to violation of Kelvin Planks’s statement, and we can say, the two statements are equivalent.
Method 2: Violation of Clausius statement by violating Kelvin-Plank’s statement: From fig. (a) let us assume that a heat engine receives heat Q1 from high temperature reservoir and convert it into work rejecting any heat to sink, thus violating Kelvin Plank’s statement. Refrigerator’s receives heat Q2 from low temperature reservoir and supplies an amount (Q1 + Q2) to high temperature reservoir when W = Q1 work is supplied to it. Thus, is operates to conform Clausius statement. But, from (b) it can be observed that heat Q1 follows a loop though HE and HP and perform no function.
The equivalent system thus can be shown as figure. From fig. (c) it is obvious the heat is being transferred from low temperature, thermal energy reservoir to high temperature thermal energy reservoir, without any external work. It is therefore violation of Clausius statement. Thus, violation of Kelvin Plank’s and we can say, the two statements are equivalent.
Equivalence of the Two Statements The Kelvin-Planck and the Clausius statements are equivalent in their consequences, and either statement can be used as the expression of the second law of thermodynamics.Any device that violates the Kelvin-Planck statement also violates the Clausius statement, and vice versa. This can be demonstrated as follows: • Kelvin-Planck statement: No heat engine can have a thermal efficiency of 100 percent, or for a power plant to operate, the working fluid must exchange heat with the environment as well as the furnace. • Clausius statement: It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a lower temperature body to a higher-temperature bodyConsider the heat-engine-refrigerator combination shown in figure below left, operating between the same two reservoirs. The heat engine is assumed to have, in violation of the Kelvin-Planck statement, a thermal efficiency of 100 percent, and therefore it converts all the heat QH it receives to work W. This work is now supplied to a refrigerator that removes heat in the amount of QL from the low-temperature reservoir and rejects heat in the amount of QL + QH to the hightemperature reservoir.
During this process, the high-temperature reservoir receives a net amount of heat QL (the difference between QL + QH and QH).Thus the combination of these two devices can be viewed as a refrigerator, as shown in figure at right that transfers heat in an amount of QL from a cooler body to a warmer one without requiring any input from outside. This is clearly a violation of the Clausius statement. Therefore, a violation of the Kelvin-Planck statement results in the violation of the Clausius statement. It can also be shown in a similar manner that a violation of the Clausius statement leads 7-19 to the violation of the Kelvin-Planck statement.
Therefore, the Clausius and the Kelvin- Planck statements are two equivalent expressions of the second law of thermodynamics. Isobaric Isentropic Lines of constant pressure curve from the lower left to upper right on a T-s diagram. A constant pressure process is called an isobaric process and this type of process occurs in the combustor of a gas turbine engine. During an isentropic process there is no change in the entropy of the system and the process is reversible. An isentropic process appears as a vertical line on a T-s diagram.
The area under a process curve on a T-s diagram is related to the amount of heat transferred to the gas.