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Entropy (YAC- Ch. 6)

In this Chapter, we will: Introduce the thermodynamic property called Entropy (S) Entropy is defined using the Clausius inequality Introduce the Increase of Entropy Principle which states that

– the entropy for an isolated system (or a system plus its surroundings) is always increases or, at best, remains the same. – Second Law in terms of Entropy

Learn to use the Entropy balance equation: entropy change = entropy transfer + entropy change. Analyze entropy changes in thermodynamic process and learn how to use thermodynamic tables Examine entropy relationships (Tds relations), entropy relations for ideal gases. Property diagrams involving entropy (T-s and h-s diagrams)

Entrpy.ppt, 10/18/01

pages

Entropy – A Property

Entropy is a thermodynamic property; it can be viewed as a measure of disorder. i.e. More disorganized a system the higher its entropy. δQ Defined using Clausius inequality ≤0 ∫ T rev where δQ is the differential heat transfer & T is the absolute temperature at the boundary where the heat transfer occurs Clausius inequality is valid for all cycles, reversible and irreversible. Consider a reversible Carnot cycle: δ Q QH QL Q T Q T = , from Carnot efficiceny ηth = 1 L = 1 L , L = L ∫ T TH TL QH TH QH TH

Therefore,

δQ

∫

δQ

T

= 0 for a reversible Carnot cycle

δQ ∫ T rev = 0

Since ∫ δ = 0, i.e. it does not change if you return to the same state, T rev it must be a property, by defintion: Let's define a thermodynamic property entropy (S), such that 2 2 δQ δQ True for a Reversible

dS = T , for any reversible process ∫ dS = ∫

1 1 rev

T

= S2 S1

rev

Process only

The change of entropy can be defined based on a reversible process

Entropy (cont'd)

Since entropy is a thermodynamic property, it has fixed values at a fixed thermodynamic states. Hence, the change, S, is determined by the initial and final state. BUT.. The change is = δQ only for a Reversible Process

∫ T

∫

δQ

Consider a cycle, where Process 2-1 is reversible and 12 may or may not be reversible

δQ δQ = ∫ + ∫ ≤0 T 1 T 2 T rev δQ δQ δQ dS = 0 = ∫ =∫ + ∫ ∫ T rev 1 T rev 2 T rev

2 1 2 2

2

1

From entropy definition δQ dS= , T rev

2

T

any process

2

δQ δQ Therefore, ∫ ≤∫ = ∫ dS = S 2 S1 = S T 1 T rev 1 1 δQ S = S 2 S1 ≥ ∫ , This is valid for all processes T 1

2

1

reversible process

dS ≥

δQ

δQ δQ , since dS = , dS > T T rev T irrev

S

Increase of Entropy Principle (YAC- Ch. 6-3)

δQ S = S2 S1 ≥ ∫ , define entropy generation Sgen T 1

2

S system

δQ δQ = S2 S1 = ∫ + S gen ≥ ∫ T T 1 1

2 2

Increase of Entropy Principle

Entropy change Entropy Transfer

(due to heat transfer)

Entropy Generation

The principle states that for an isolated Or a closed adiabatic Or System + Surroundings A process can only take place such that Sgen≥ 0 where Sge = 0 for a reversible process only And Sge can never be les than zero. Implications: Entropy, unlike energy, is non-conservative since it is always increasing. The entropy of the universe is continuously increasing, in other words, it is becoming disorganized and is approaching chaotic. The entropy generation is due to the presence of irreversibilities. Therefore, the higher the entropy generation the higher the irreversibilities and, accordingly, the lower the efficiency of a device since a reversible system is the most efficient system. The above is another statement of the second law

Second Law & Entropy Balance (YAC- Ch. 6-4)

Increase of Entropy Principle is another way of stating the Second Law of Thermodynamics: Second Law : Entropy can be created but NOT destroyed (In contrast, the first law states: Energy is always conserved) Note that this does not mean that the entropy of a system cannot be reduced, it can. However, total entropy of a system + surroundings cannot be reduced Entropy Balance is used to determine the Change in entropy of a system as follows: Entropy change = Entropy Transfer + Entropy Generation where, Entropy change = S = S2 - S1 Entropy Transfer = Transfer due to Heat (Q/T) + Entropy flow due to mass flow (misi – mese) Entropy Generation = Sgen≥ 0 For a Closed System: S2 - S1 = ∑Qk /Tk + Sgen In Rate Form: dS/dt = ∑Qk /Tk + Sgen For an Open System (Control Volume):

dS cv Qk =∑ + ∑ m i s i ∑ m e s e + S gen,CV dt Tk

Similar to energy and mass conservation, the entropy balance equations can be simplified Under appropriate conditions, e.g. steady state, adiabatic….

Entropy Generation Example

Show that heat can not be transferred from the low-temperature sink to the hightemperature source based on the increase of entropy principle. Source 800 K Q=2000 kJ

S(source) = 2000/800 = 2.5 (kJ/K) S(sink) = -2000/500 = -4 (kJ/K) Sgen= S(source)+ S(sink) = -1.5(kJ/K) < 0 It is impossible based on the entropy increase principle Sgen≥0, therefore, the heat can not transfer from low-temp. to high-temp. without external work input If the process is reversed, 2000 kJ of heat is transferred from the source to the sink, Sgen=1.5 (kJ/K) > 0, and the process can occur according to the second law

Sink 500 K

If the sink temperature is increased to 700 K, how about the entropy generation? S(source) = -2000/800 = -2.5(kJ/K) S(sink) = 2000/700 = 2.86 (kJ/K) Sgen= S(source)+ S(sink) = 0.36 (kJ/K) < 1.5 (kJ/K) Entropy generation is less than when the sink temperature is 500 K, less irreversibility. Heat transfer between objects having large temperature difference generates higher degree of irreversibilities

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