course no: me 209 thermodynamics lecture 21: property ...me 209 thermodynamics lecture 21 u. n....
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THERMODYNAMICS
Course No: ME 209
Department: Mechanical Engineering
Instructor: U. N. Gaitonde
Lecture 21: Property Relations
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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Lecture 21: Property Relations
• Relations between properties of a
simple compressible (fluid) system
• Often, we assume a unit mass of the system,
so we work with specific properties
• Tools:
– Calculus of exact differentials
– Calculus of partial derivatives
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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Energy Functions
• First law defines U ,
the thermal internal energy
• Second law defines S, entropy
• The basic property relation relates the two:
TdS = dU + pdV
• We will also use the first law:
dQ = dU + dW
• and the second law:
TdS ≥ dQ or TdS ≥ dU + dW
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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Internal Energy, U
dQ = dU + dW = dU + pdV + dWother
∴ dQv = dU + dWother
Enthalpy, H
H ≡ U + pV
∴ dH = dU + pdV + V dp
∴ dQ = dU + pdV + dWother = dH − V dp + dWother
∴ dQp = dH + dWother
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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The Helmholtz Function, A
A ≡ U − TS
∴ dA = dU − TdS − SdT
∴ dA = dQ− dW − TdS − SdT
∴ dA + dW + SdT = dQ− TdS ≤ 0∴ dW ≤ −dA− SdT
∴ dWT ≤ −dAT
Thus, A : a potential, the decrease in which represents the
maximum work that can be obtained in an isothermal process.
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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The Gibbs Function, G
G ≡ U + pV − TS = H − TS
∴ dG = dU + pdV + V dp− TdS − SdT
∴ dG = dQ− dWother + V dp− TdS − SdT
∴ dG + dWother − V dp + SdT = dQ− TdS ≤ 0∴ dWother ≤ −dG + V dp− SdT
∴ dWother,p,T ≤ −dGp,T
Thus, G : a potential, the decrease in which represents the
maximum work (other than expansion) that can be obtained in an
isobaric-cum-isothermal process.
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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Energy FunctionsWe now look at the properties of the four
‘energy functions’, U , H , A, and G.
We use the specific-property version,
so u, h, a, and g.
We use the property relation, and the following from calculus:
If z(x, y) and dz = Mdx + Ndy is an exact differential, then
M =(
∂z
∂x
)y
and N =(
∂z
∂y
)x
also:
(∂M
∂y
)x
=(
∂N
∂x
)y
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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Internal Energy, u∵ du = Tds− pdv,
we consider u(s, v)
T =(
∂u
∂s
)v
and p = −(
∂u
∂v
)s
and (∂T
∂v
)s
= −(
∂p
∂s
)v
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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Enthalpy, h = u + pv
dh = Tds− pdv + pdv + vdp
= Tds + vdp
We consider h(s, p)
T =(
∂h
∂s
)p
and v =(
∂h
∂p
)s
and (∂T
∂p
)s
=(
∂v
∂s
)p
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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Helmholtz Functiona = u− Tsda = Tds− pdv − Tds− sdT
= −sdT − pdv
We consider a(T, v)
s = −(
∂a
∂T
)v
and p = −(
∂a
∂v
)T
and (∂s
∂v
)T
=(
∂p
∂T
)v
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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Gibbs Functiong = h− Ts
dg = Tds + vdp− Tds− sdT
= −sdT + vdp
We consider g(T, p)
s = −(
∂g
∂T
)p
and v =(
∂g
∂p
)T
and (∂s
∂p
)T
= −(
∂v
∂T
)p
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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Maxwell’s Relations(∂T
∂v
)s
= −(
∂p
∂s
)v(
∂T
∂p
)s
=(
∂v
∂s
)p(
∂s
∂v
)T
=(
∂p
∂T
)v(
∂s
∂p
)T
= −(
∂v
∂T
)p
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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Maxwell’s Relations (contd)
• These relations (and their ‘reciprocals’),
known as Maxwell’s Relations,
are very useful property relations.
• The third and fourth ones relate entropy variation to purely
p-v-T (Equation-of-state or EoS) data.
• They help reduce the requirement of cp (or cv) data for
mapping the state space.
• How do you remember them? (No cogsheets, please!)
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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The Area RelationConsider a reversible cycle.
Represent it on a p-v diagram.
Also on a T -s diagram.p
v
T
s
Apv ATs
Are the two areas equal?
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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The Area Relation (contd)Yes! They are. Thus,‹
dp dv =‹
dT ds
For any reversible cycle (of a simple compressible system).
This means that the Jacobian of the transformation is 1.
∂(T, s)∂(p, v)
=∂(p, v)∂(T, s)
= 1
Now, we can use the properties of the Jacobian for the calculus
of properties.
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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The JacobianSome properties:
∂(u, v)∂(x, y)
∂(x, y)∂(u, v)
= 1
∂(p, q)∂(r, s)
∂(r, s)∂(t, u)
=∂(p, q)∂(t, u)
∂(u, y)∂(x, y)
=∂(y, u)∂(y, x)
= −∂(y, u)∂(x, y)
= −∂(u, y)∂(y, x)
=(
∂u
∂x
)y
Using these (and other) properties of the Jacobian we can
manage the Maxwell’s relations
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
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An Example(∂T
∂v
)s
=∂(T, s)∂(v, s)
=∂(T, s)∂(v, s)
∂(p, v)∂(T, s)
=∂(p, v)∂(v, s)
= −(
∂p
∂s
)v
In a similar manner all the other relations can be derived.
ME 209 THERMODYNAMICS Lecture 21 U. N. Gaitonde
