Entropy

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Entropy. Cengel & Boles, Chapter 6. Entropy. From the 1st Carnot principle: this is valid for two thermal reservoirs. The Clausius Inequality. For a system undergoing a cycle and communicating with N thermal reservoirs, it can be shown that

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### Entropy

Cengel & Boles, Chapter 6

ME 152

Entropy
• From the 1st Carnot principle:
• this is valid for two thermal reservoirs

ME 152

The Clausius Inequality
• For a system undergoing a cycle and communicating with N thermal reservoirs, it can be shown that
• This can be further generalized into the Clausius Inequality:
• where Q is the heat transfer at a particular location along the system boundary during a portion of the cycle and T is the absolute temperature at that location. This is called a cyclic integral.

ME 152

The Clausius Inequality, cont.
• If the cycle is internally reversible, then
• What does this mean? Consider:

ME 152

New Property: Entropy
• Therefore, the quantity Q/T is a property of the system in differential form when the integration is performed along an internally reversible path. This property is known as entropy (S):

ME 152

New Property: Entropy
• This integral defines a new property of the system called entropy (S):
• entropy is an extensive property with units of kJ/K; specific entropy is defined by s = S/m, with units of kJ/kg-K
• the integration will only yield entropy change when carried out along an int. rev. path
• like enthalpy, entropy is a convenient and useful property that has been introduced without physical motivation; its utility will be discovered as we learn more about its characteristics

ME 152

Entropy Change and Heat Transfer
• Suppose we have a closed system undergoing an internally reversible process with heat transfer
• if heat is added (Q>0), then S2>S1 or entropy increases
• if heat is removed (Q<0), then S2<S1 or entropy decreases
• if system is adiabatic (Q=0), then S2=S1 or entropy is constant (isentropic)

ME 152

Entropy Change and Heat Transfer, cont.
• Entropy equation can be rearranged:
• when temperature is plotted against entropy, the area under a process path is equal to the heat transfer when the process is internally reversible.

ME 152

Increase in Entropy Principle
• Consider a cycle consisting of an irreversible process followed by a reversible one:

ME 152

Increase in Entropy Principle, cont.
• The inequality can be turned into an equality by considering the “extra” contribution to the entropy change as entropy generated by the irreversibilities of the process:

ME 152

Increase in Entropy Principle, cont.
• The increase in entropy principle states that an isolated system (or an adiabatic closed system) will always experience an increase in entropy since there can be no heat transfer, i.e.,
• However, this principle does not preclude an entropy decrease, which may occur for a system that loses heat (Q < 0)

ME 152

Isentropic Processes
• A process is isentropic if S (or s) is a constant, i.e.,
• this corresponds to best performance (reversible) in the absence of heat transfer since irreversibilities are zero.
• many engineering devices such as pumps, turbines, nozzles, and diffusers are essentially adiabatic, so they perform best when isentropic.
• note that a reversible adiabatic process must be isentropic, but an isentropic process is not necessarily a reversible adiabatic process.

ME 152

The T-s Diagram
• A T-s diagram displays the phases of a substance in much the same way as a P-v or T-v diagram:
• Zero entropy is defined at sf(T = 0.01C) for H2O, sf(T = -40C) for R-134a, and s(T = 0K) for ideal gases

ME 152

The Tds Relations
• Consider a closed, stationary system containing a simple compressible substance undergoing an int. rev. process:
• substituting, we obtain:

ME 152

The Tds Equations, cont.
• Recalling H = U + PV, then
• substituting,
• these are known as the Tds relations, which allow one to evaluate entropy in terms of more familiar quantities; the equations are also valid for irreversible processes because they involve only properties and so are path-independent.

ME 152

Calculating Entropy Change
• From Tds relations,
• these equations are used to evaluate entropy for H2O and R-134a in the text tables A-4 through A-13

ME 152

Entropy Change for Ideal Gases
• For ideal gases,
• similarly,

ME 152

Variable Specific Heats (Exact Analysis)
• Define absolute entropy:
• this is tabulated as a function of T in the ideal gas tables, A-17 through A-25
• thus,

ME 152

Constant Specific Heats (Approximate Analysis)
• If Cv, Cp constant, then

ME 152

Entropy Change for Incompressible Substances
• For incompressible substances (i.e., liquids and solids), v = constant and Cv= Cp= C

ME 152

Isentropic Processes for Ideal Gases
• Recall for an ideal gas:
• If process is isentropic,
• Since adiabatic, isentropic processes are reversible and yield the best performance, solving this equation is important (e.g., finding T2 and h2)

ME 152

Variable Specific Heats (Exact Isentropic Analysis)

i) if T1, P1, and P2 are known, then T2 can be found from the ideal gas tables and

ii) if T1, P1, and T2 are known, then P2 can be found from

ME 152

Relative Pressure, Pr
• The quantity exp(so/R) is tabulated in the ideal gas tables as a function of temperature - it is called the relative pressure, Pr
• If the pressure ratio P2/P1 is known, then Pr is useful in finding the isentropic process by setting
• The relative pressure values have no physical significance - they are only used as a “shortcut” in determining an isentropic process from the ideal gas tables

ME 152

Relative Volume, vr
• From the ideal gas law,
• The quantity T/Pr is also tabulated in the ideal gas tables and is known as the relative volume, vr ; if the specific volume ratio is known, then vr is useful in finding the isentropic process by setting

ME 152

Constant Specific Heats (Approx. Isentropic Analysis)
• Recall:
• with k = Cp/Cv and Cp = Cv + R , the following relations result:
• note that Pvk = constant for isentropic processes; thus, these processes are polytropic where n = k

ME 152

• Recall relation between heat transfer and entropy for an int. rev. process:

ME 152

• From CV energy balance,
• for a reversible process,

ME 152

• Note that if wrev = 0, we have the simple form of the Bernoulli equation
• For turbines, compressors, and pumps with negligible KE, PE effects:

ME 152

Special Cases of Reversible Work

1) Pumps - fluid is incompressible (i.e., liquid) so v = constant:

2) Ideal gas (Pv = RT) compressor:

ME 152

Special Cases of Reversible Work, cont.

3) Polytropic compressor

• substituting into reversible work equation and integrating yields:

ME 152

Special Cases of Reversible Work, cont.

4) Ideal gas and polytropic compressor:

4) Ideal gas and isothermal compressor:

ME 152

• Isentropic Efficiency is a perfor-mance measure of an adiabatic device that compares an actual process to an isentropic process, both having the same exit pressure

1) Turbines:

ME 152

Isentropic Efficiencies, cont.

2) Compressors:

3) Pumps:

ME 152

Entropy Balance for Closed Systems
• From the increase in entropy principle,
• If temperature is constant where heat transfer takes place, then

ME 152

Entropy Balance for Control Volumes
• It can also be shown that the following entropy balance applies to steady-flow control volumes with single-inlet, single-exit:

ME 152