Week 6. Energy Analysis of Closed Systems II,
Specific Heat
Objectives
1. Define the specific heat at constant volume and the specific heat at constant pressure
2. Relate the specific heats to the calculation of the changes in internal energy and enthalpy of ideal gases
3. Describe incompressible substances and determine the changes in their internal energy and enthalpy
4. Solve energy balance problems for closed (fixed mass) systems that
involve heat and work interactions for general pure substances, ideal
gases, and incompressible substances
Specific Heats
• A property that enables to explain the energy storage capabilities of various substances
• The energy required to raise the temperature of a unit mass of a substance by one degree - Specific heat at constant volume (Cv) as the volume is maintained constant
-Specific heat at constant pressure (Cp) as the pressure is maintained constant
Specific Heats (Continue)
in o u t
in o u t
V
P
e e d u
d u e
C
e d u P d V d
d
C h
T
d T
δ δ
δ δ
− =
=
− = +
=
Different Specific Heat
Temperature 20˚C 21˚C
Oh, Come on I am burnt
Piece of cake
What the heck!!
Internal Energy, Enthalpy, And Specific Heats of Ideal Gases
• For ideal gas, u is a function of temperature only according to Joule
• Since R is constant and
where
h u Pv
Pv RT h u RT
= +
=
= +
( ) u = u T
( ) ( )
V P
d u C T d T d h C T d T
=
=
( )
h = h T
Specific Heats (Continue)
• Three ways to determine the ∆u and ∆h of ideal gas -Using the tabulated u and h data
-Using the Cv or Cp relations as a function of temperature and performing the integrations.
-Using average specific heats if ∆T is small
v
( )
v, av(
2 1)
u C T dT C T T
∆ = ∫ ≅ −
P
( )
P, av(
2 1)
h C T dT C T T
∆ = ∫ ≅ −
Specific Heats (Continue)
•Specific heat relations of ideal gases
Cp = Cv + R
• Specific heat ratio k = Cp/Cv
- 1.667 for monatomic gases - 1.4 for diatomic gases
where h u Pv
Pv RT h u RT
dh du RdT dh du
dT dT R
= +
=
= +
= +
= +
Ex1) Evaluation of the ∆u of an Ideal Gas
Ex2) Heating of a Gas in a Tank by Stirring
Ex3) Heating of a Gas by a Resistance Heater
Ex4) Heating of a Gas at Constant Pressure
u, h, and c of incompressible substance
• A substance whose specific volume is constant during a process (e.g. solid, liquid)
• The constant-volume and constant-pressure specific heats are identical for incompressible substances
• The specific heat of incompressible substances depend on temperature only
• Internal Energy Changes
c c
c
p=
v=
) (
) ( ) (
2 1 1 2
T T c
dT T c u
u u
dT T c dT c du v
−
≅
=
−
=
∆
=
=
∫
u, h, and c of incompressible substance II
• Enthalpy Changes
• For a process between states 1 and 2, the last relation can be expressed as
• By taking state 2 to be the compressed liquid state at a given T and P and state 1 to be the saturated liquid state at the same temperature, the enthalpy of the compressed liquid can be expressed as
P v T c
P v u h
vdP du
Pdv vdP
du dh
avg
∆ + ∆
≅
∆ +
∆
=
∆
+
= +
+
=
Solid
∆ h = ∆ u ≅ c
avg∆ T
Liquid (if 0) : (if 0) :
P h u cavg T
T h v P
∆ = ∆ = ∆ ≅ ∆
∆ = ∆ = ∆
@ , @ @ @
@
( )
Since ( ) 0
P T f T f T sat T
sat T
h h v P P
P P
≅ + −
− <<
2 1 ( 2 1) h ≅h +v P −P
