Lecture Notes on Nuclear Power Systems: II. Thermodynamics of Nuclear Power Plants - Prof., Study notes of Mechanical Engineering

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• MANE-4400MANE 4400 Nuclear Power Systems Nuclear Power SystemsEngineeringg g Lecture NotesLecture NotesProf. Michael Z. Podowski Lecture

II. Thermodynamics of

y Nuclear Power Plants

II.1. Principles of ThermodynamicsThe First Law of ThermodynamicsIn general terms, the first law of thermodynamicsg^

y states that energy can neither be created nordestroyed.The most commonly used mathematical formulationof this law is the energy conservation equation. Fort^ (i^

t^ i^ hi h^

d open systems (i.e., systems in which mass andenergy can be transformed across the systemboundaries) this equation can be written asboundaries), this equation can be written as^2 2 m^ th^ m^ th^1 12

flow

E^ E^ E^ E

Q^ W^ E    ^ ^  ^

(II.1)

2 2 1 1

Q^^12 12 flow

(II.1)

The notation used in Eq

(II 1)^ is as follows The^ notation used in Eq.

(II.1)^ is^ as follows m^ = mechanical (potential and kinetic) energy, E^ th^ th^ = internal thermal energy (also denoted by E E 

U ),

E fl^ = flow energy, flow = net heat added into the system^ QQ = net mechanical work done by the system W^

Closed systems are usually defined as systems inwhich there is no mass transfer across the systemboundaries. For such systems there is no changei^ th^ h^ i^ l

d E^ (

II 1) b in the mechanical energy, and Eq. (

II.1)^ becomes^ (II.2) (^1) U Q^ U^^2

W ^ ^  or

(II.3)

-^2

U^ U^ U

Q^ W ^ ^



The EnthalpyC^ id^ t i^

tit^ f^ b t^

i Consider a certain quantity of a substance in avolume,^ V , and under a pressure,

P. For a given thermal energy of the substance

U^ the thermal energy of the substance,

U , the substance enthalpy is defined asIf th^ f th^

b t^ i^ it^

ifi H^ U^ PV ^ ^

(II.4)

If the mass of the substance is

m , its specific enthalpy,^ h=H/m , can be expressed in terms of thespecific internal energy

u=U/m^ and the specific specific^ internal energy,

u^ U/m , and the specific volume,^  =V/m ,^ h^ P

(II 5)

h^ u^ P^ ^ ^

(II.5)

Eq.(II.5)^ can be differentiated, to obtainwhere the symbol

d^ refers to an incremental value

(II.6)

(^ )^ P dh du d P^ du^

d^ dP ^ ^  ^ ^

^ ^  where^ the symbol,

dy , refers to an incremental value change of a given parameter from the original state, y^ to a new state^

y+dy y , to a new state,^

y+dy. For a thermodynamic process transforming a fixedmass of a substance from state “1” to state “2” themass of a substance from state

1 to state^ 2 , the resultant change in the specific enthalpy of thesubstance can be obtained from^2

2 2

2 2 2 1

2 1 h^ h^ h^ du^

Pd^ dP^ u^ u

Pd^ dP  ^  ^ ^ ^ ^ ^

^ ^ ^ ^

 ^ ^ ^

^  2 1

2 1 1 1 1

1 1 h^ h^ h^ du^

d^ d^ u^ u^

d^ d ^ 

^  ^ ^ ^

^ ^ (II.7)

Two particular cases of Eq

.(II.7)^ are given below: p^

q^ (^ )^ g (a) an isobaric process(constant pressure)

(b) an isochoric process(constant volume) (constant pressure)^

(^ ) P^1

P 2 (^1) P* 2 p^1

(^1) p (^2) 2 vv^1

v^ 2

vv*

2 1 2 1

2 1

-^ -^ * (^ -^ ) h^ h^ u^ u^ P

^   2 1 h^ h^ u^ u^ * ( P - P )^ ^ ^ ^2 1 2

Reversible Thermodynamic Processesand the Concept of Entropyand the Concept of Entropy • A^ thermodynamic

process^ is^ called A^ thermodynamic

process^ is^ called reversible (ideal) if it can reverse itselfby following exactly the same path, sothat^ the^ amount

of^ heat^ and^ energy exchanged in each case is the same.• Although all actual physical processesare^ irreversible

the^ concept

of are^ irreversible,

the^ concept

of reversibility^ provides

a^ convenient vehicle^ for^ the^

analysis^ of^ variousy thermodynamic systems.

Heat exchanged during an incremental reversibleprocess can be expressed as^ dQ^ TdS ^

(II.8)

process can be expressed as dQ^ TdS ^^ rev

(II.8)

where^ S^ is the entropy of a given substanceEquation^ (II.8)^ can also be rewritten in terms ofthe specific heat^

ˆ^ and the specific/ q^ Q^ m

the^ specific heat,^

, and the specific entropy,

/ q Q m  /

s^ S^ m  ˆ dq Tds ^ rev

(II.9)

h^ i^ h^ b^

l where^ T^ is the absolute temperature

Integrating Eq.(II.9)^ between states “1” and “2,” yields^22 ˆ q^ Tds ^12 ^1 E^ l

(II.10)

Examples:(a) an isothermal process

(b) an isentropic process T^

T T^1 T*^

T 2 T^1 2 ss^1

T^2 2 s 1 s s*^ s^

ss 12 2

1 ˆ^ * (^ -

) q^ T^ s^

ˆ^ s  12 (^0) q 

For reversible processes, the following relation holdsp^

g ˆ dq^ du^ rev

Pd^  

(II.11)

qrev Combining Eq.^ (I.11)^ with Eq.

(I.6),^ we obtain ˆ rev

dh^ dq^

dP 

^ ^

(II.12)

Also^ dh^ Tds^

dP  

(II 13)

dh^ Tds^

dP  

(II.13)

d^ Td^

Pd^

(II 14)

• du Tds Pd^ 

(II.14)