College of Engineering and Computer Science
Mechanical Engineering Department
Mechanical Engineering 370
Thermodynamics
Spring 2003 Ticket: 57010 Instructor: Larry Caretto
Summary of Important Equations in Thermodynamics
Basic notation and definition of terms
Fundamental dimensions of mass, length, time, temperature, and amount of substance (mol) are
denoted, respectively as M, L, T. Θ, and  in the abbreviations below. For example, the
dimensions of pressure are ML-1T-2. Thermodynamic properties are said to be extensive if they
depend on the size of the system. Volume and mass are examples of intensive properties;
intensive properties, such as temperature and pressure, do not depend on the size of the system.
The ratio of two extensive properties is an intensive property. The ratio of an extensive property,
such as volume, to the mass of the system, is called the specific property; e.g., the ratio of
volume to mass is called the specific volume.
M
molecular weight (M/)
m
mass (M)
n
m
M
E
e
number of moles ()
Energy or general extensive property
E
m
Specific molar energy (energy per unit mass) or general extensive property per
unit mass
e
E
 eM Specific energy (energy per unit mole) or general extensive property per unit mole
n
P
pressure (ML-1T-2)
V
volume (L3); we also have the specific volume or volume per unit mass, v (L 3M-1)
and the volume per unit mole v (L3-1)
T
temperature (Θ)

density (ML-3);  = 1/v.
x
quality
subscripts f and v denote saturated liquid and stauarated vapor, respectively
U
thermodynamic internal energy (ML2T-2); we also have the internal energy per unit
mass, u (L2T-2), and the internal energy per unit mole, u (ML2T-2-1)
H = U + PV
thermodynamic enthalpy (ML2T-2); we also have the enthalpy per unit mass, h = u
+ Pv (dimensions: L2T-2) and the internal energy per unit mole h (ML2T-2-1)
S
entropy (ML2T-2Θ-1); we also have the entropy per unit mass, s(L2T-2Θ-1) and the
internal energy per unit mole s (ML2T-2Θ-1-1)
W
work (ML2T-2)
Engineering Building Room 1333
E-mail: lcaretto@csun.edu
Mail Code
8348
Phone: 818.677.6448
Fax: 818.677.7062
Equations of thermodynamics
ME 370, L. S. Caretto, Spring 2003
Q
heat transfer (ML2T-2)
W u :
the useful work rate or mechanical power (ML2T-3)
:
m

V2
:
2
gz:
Etot:
Q :
Page 2
the mass flow rate (MT-1)
the kinetic energy per unit mass (L2T-2)
the potential energy per unit mass (L2T-2)

V2
the total energy = m(u +
+ gz) (ML2T-2)
2
the heat transfer rate (ML2T-3)
dEcv
2 -3
dt : the rate of change of energy for the control volume. (ML T )
Mass and energy balance equations
The equations below use the standard thermodynamic sign convention for heat and work. Heat
added to a system is positive and heat rejected from a system is negative. The opposite
convention holds for work. Work done by a system is positive; work done on (added to) a system
is negative. In terms of the W in, W out, Qin, and Qout terms used in the text we can make the
following statements.
Q = Qin - Qout
W = W out - W in
These equations may be substituted for any of the heat, Q, or work, W terms below.
Relation of mass flow to velocity, specific volume and area:


VA
m  VA 
v






dE cv  
Vi2
Vi2
  m

 i hi 
 i hi 
 Q  Wu   m

gz

gz
General first law:
i
i

 inlet 

dt
2
2
outlet




Mass balance equation:
dm cv
 i  m
i
 m
dt
inlet
outlet
Steady-flow assumptions:
Steady-flow first law:
Mass balance for steady flows:
dEcv
dt = 0
and
dmcv
dt = 0
2
2




V
V
i

 Q
  m
 i hi 
 i  h i  i  gz i 
W
m

gz

u
i

 inlet 

2
2
outlet




 m
inlet
i

 m
i
outlet
Steady-flow first law with negligible KE & PE:

 Q
W
u
 m h   m h
i
outlet
i
i
inlet
i
Equations of thermodynamics
ME 370, L. S. Caretto, Spring 2003
Page 3
Second law of thermodynamics and entropy
Basic definition of entropy:
dS 
dU  PdV
T
General second law:


dScv
Q
LW
cv
 isi   m
 i s i  cv 
 m
dt
T
T
outlet
inlet
Second law inequality:

dScv
Q
 isi   m
 i s i  cv
 m
dt
T
outlet
inlet
Adiabatic steady-flow, second law inequality:
 m s   m s
i i
outlet
i i
0
inlet
Ideal-gas equations
1.
The gas constant
R is the universal gas constant with dimensions of energy divided by (moles times
R
temperature), and R 
is the engineering gas constant with dimensions of energy divided
M
by (mass times temperature). The energy dimensions are sometimes expressed as energy
units (e.g., J). However, for P-v-T calculations, the energy dimensions are expressed in
pressure units times volume units (e.g., kPa•m 3. Some values of the universal gas constant are
shown below.
8.314 kJ 8.314 kPa  m 3 1545.35 ft  lb f 10.7316 psia  ft 3
R



kmol  K
kmol  K
lbmol  R
lbmol  R
2.
P-V-T calculations with mass or moles. The example in the first equation is air (R = 0.287
kJ/kg-K). Both examples are for an ideal gas that occupies a volume of 1 m 3 at a pressure of
100 kPa and a temperature of 25 C = 298.15 K.
PV
m = RT =
n=
3.
(100 kPa) (1 m3)
1 kJ
kPa-m3
kJ
0.287 kg-K 298.15 K
= 1.169 kg
PV
(100 kPa )(1 m 3 )
=
= 0.04034 kmol
RT 8.314 kPa  m 3
(298.15 K )
kmol  K
Heat capacity definitions and relations and computations of internal energy and enthalpy
changes
a.
Basic rule: for ideal gases, the internal energy, enthalpy, and heat capacities are
functions of temperature only.
b.
Heat capacity definition and relations
Equations of thermodynamics
ME 370, L. S. Caretto, Spring 2003
 H 
CP  

 T  P
 U 
Cv  

 T  v
 h 
C
c P     P  Mc P
n
 T  P
C
 u 
c v     v  Mc v
n
 T  v
C
c
 h 
cP     P  P
m M
 T  P
C
c
 u 
cv     v  v
 T  v m M
Cp – Cv = mR = n R
c.
Page 4
cp – cv = R
cP  cv  R
Relationships for internal energy and enthalpy
T2
du = cvdT
dh = cpdT
u2 – u1 =
c
T2
v
h2 – h1 =
dT
T1
For constant heat capacity:
u2 – u1 = cv(T2 – T1)
From ideal gas tables: u2 – u1 = uo(T2) – uo(T1)
4.
and
and
constant heat capacity
variable heat capacity by integration
T2
T2
P2
v2
dT
dT

s2 - s1 =  cp T - R ln P  =  cv T + R ln v 


 1
 1
T1
T1
c.
variable heat capacity by ideal gas tables
P2
s2 - s1 = so(T2) - so(T1) - R ln P 
 1
5.
End states of an isentropic process for an ideal gas with
a.
constant heat capacity (k = cp/cv)
P2 (k-1) /k
T2 = T1P 
 1
b.
v1 k
P2 = P1v 
2
variable heat capacity using the air tables
Pr(T2)
P2
Pr(T1) = P1
c.
v1 (k-1)
T2 = T1v 
 2
dT
h2 – h1 = ho(T2) – ho(T1)
T2
P2
T2
v2
s2 - s1 = cp ln T  - R ln P  = cv ln T  - R ln v 
 1
 1
 1
 1
b.
P
h2 – h1 = cP(T2 – T1)
Entropy changes
a.
c
T1
vr(T2)
v2
vr(T1) = v1
variable heat capacity using general ideal gas tables
 
Equations of thermodynamics
ME 370, L. S. Caretto, Spring 2003
Page 5
P2
so(T2) = so(T1) - R ln P 
 1
d.
variable heat capacity by integration of cp(T) or cv(T) equation
T2
P
 cp dT = R ln  2
T

P1
T1
T2
v
 cv dT = R ln  2
T

v1
or
T1
Cycles and efficiencies
In an engine cycle a certain amount of heat, |QH|, is added to the cyclic device at a high
temperature and a certain amount of work |W| is performed by the device. The difference
between |QH| and |W| is |QL|, the heat rejected from the device at a low temperature.
In a refrigeration cycle a certain amount of heat, |QL|, is added to the cyclic device at a low
temperature and a certain amount of work |W| is performed on the device. The sum of |QL| and
|W| is |QH|, the heat rejected from the device at a high temperature.
For both devices:
Engine cycle efficiency:
Refrigeration cycle coefficient of performance:
|QH| = |W| + |QL|

|W|
| QH |
COP 
QL
W
The isentropic efficiency, s, compares the actual work, |wa|, to the ideal work that would be done
in an isentropic (reversible adiabatic) process, |ws|. Both the actual and the isentropic process
have the same initial states and the same final pressure.
For a work output device:
s 
| wa |
| ws |
For a work input device:
s 
| ws |
| wa |