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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 molar mass (M/)
m mass (M)
M
m n number of moles ()
E Energy or general extensive property
m
E
e Specific molar energy (energy per unit mass) or general extensive property per
unit mass
eM n
E
e Specific energy (energy per unit mole) or general extensive property per unit mole
P pressure (ML-1T-2)
V volume (L^3 ); we also have the specific volume or volume per unit mass, v (L^3 M-1) and the volume per unit mole v(L^3 -1)
T temperature (Θ)
density (ML-3); = 1/v.
x quality
U thermodynamic internal energy (ML^2 T-2); we also have the internal energy per unit mass, u (L^2 T-2), and the internal energy per unit mole, u (ML^2 T-2-1)
H = U + PV thermodynamic enthalpy (ML^2 T-2); we also have the enthalpy per unit mass, h = u
- Pv (dimensions: L^2 T-2) and the internal energy per unit mole h (ML^2 T-2-1)
S entropy (ML^2 T-2Θ-1); we also have the entropy per unit mass, s(L^2 T-2Θ-1) and the internal energy per unit mole s(ML^2 T-2Θ-1-1)
W work (ML^2 T-2)
Q heat transfer (ML^2 T-2)
W u : the useful work rate or mechanical power (ML (^2) T-3)
m : the mass flow rate (MT-1)
V^ ^2
: the kinetic energy per unit mass (L^2 T-2)
gz: the potential energy per unit mass (L^2 T-2)
Etot: the total energy = m(u + 2
V^ ^2
Q : the heat transfer rate (ML^2 T-3)
dEcv dt : the rate of change of energy for the control volume. (ML
(^2) T-3)
Finding properties in tables
For a given temperature, T, and pressure, P, the substance is a liquid if T < Tsat(P); it is a gas is the opposite holds, T > Tsat(P). These conditions may also be stated as liquid if P > Psat(T) and gas if P < Psat(T).
For a given T or P and specific property, e, where e may be v, u, h, or s, determine the state as follows: (1) find the saturation properties ef and eg from the given T or P; (2) if e > eg, the state is gas, if e < ef, the state is liquid, otherwise, ef ≤ e ≤ eg, and the state is a mixture of liquid and vapor. The quality, x, defined as the mass fraction of vapor is used to specify the mixed region state: e = ef + x(eg – ef) = ef + xefg.
The general formula for interpolation to find a value of y for a given value of x between two pairs of tabular values (x 1 , y 1 ) and (x 2 , y 2 ) is y = y 1 + (y 2 – y 1 )(x – x 1 )/(x 2 – x 1 ).
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 Win, Wout, Qin, and Qout terms used in the text we can make the following statements.
Q = Qin - Qout W = Wout - Win
These equations may be substituted for any of the heat, Q, or work, W terms below.
Heat capacities: for constant pressure process, Q = CpdT; for constant volume, Q = CvdT
Work: W = pathPdV = area under path on a P-V diagram
For a linear path, P = a + bV:^12 2 1 2
V V
P P
W
For a path PVn^ = constant:
1 ln ln 1 (^1 )
2 2 2 1
2 11
forn V
V
PV
V
V
forn W PV n
PV PV
W
General first law:
inlet
i
i i i outlet
o
o u o o
cv (^) Q W m h V gz m h V gz dt
dE 2 2
Closed-system first law: Q = U + W q = u + w
Ideal gas entropy changes
a. constant heat capacity
1
2 1
2 1
2 1
2 2 1 ln^ ln ln ln v
v R T
T
c P
P
R
T
T
s s cp v
b. variable heat capacity by integration
1
2 1
2 2 1 ln ln
2
1
2
1^ v
v R T
dT c P
P
R
T
dT s s c
T
T
v
T
T
p
c. variable heat capacity by ideal gas tables
(^)
2
1 1
2 2 1 1
2 2 1 2 1 ln^ ln v
v T
T
s T s T R P
P
s s soT soT R o o
End states of an isentropic process for an ideal gas with
a. constant heat capacity (k = cp/cv) k k k k
v
v P P v
v T T P
P
T T
2
1 2 1
1
2
1 2 1
( 1 )
1
2 2 1
b. variable heat capacity using the air tables
(^1)
2 1
2 1
2 1
2 v
v v T
v T P
P
P T
P T
r
r r
r (^)
c. variable heat capacity using general ideal gas tables
(^)
2
1 1
2 2 1 1
2 2 1 ln^ ln v
v T
T
s T R P
P
s oT soT R o
d. variable heat capacity by integration of cp(T) or cv(T) equation
^ 1
2 1
ln^2 ln
2
1
2
1^ v
v R T
dT c P
P
R
T
dT c
T
T
v
T
T
p
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: |QH| = |W| + |QL|
Engine cycle efficiency: |Q |
|W|
H
Refrigeration cycle coefficient of performance: W
Q
COP L
A Carnot cycle operates between two and only two temperature reservoirs, one at a high temperagture, TH, and the other at a low temperature, TL. The efficiency of a Carnot engine cycle and the coefficienc of performance for a Carnot refrigeration cycle are given by the following equations.
Carnot = 1 – TL/TH COPCarnot = TL/(TH – TL)
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: |w |
|w |
s
a
s
For a work input device: |w |
|w |
a
s
s
Unit conversion factors
For metric units
Basic : 1 N = 1 kg·m/s^2 ; 1 J = 1 N·m; 1 W = 1 J/s; 1 Pa = 1 N/m^2.
Others : 1 kPa·m^3 = 1 kJ; T(K) = T(oC) + 273.15; 1 L (liter) = 0.001 m^3 ; 1 m^2 /s^2 = 1 J/kg.
Prefixes (and abbreviations) : nano(n) – 10 -9; micro() – 10 -6; milli(m) – 10 -3; kilo(k) – 103 ; mega(M) – 106 ; and giga(G) – 109. A metric ton (European word: tonne) is 1000 kg.
For engineering units
Energy : 1 Btu = 5.40395 psia·ft^3 = 778.169 ft·lbf = (1 kWh)/3412.14 = (1 hp·h )/2544.5 = 25,037 lbm·ft^2 /s^2.
Pressure : 1 psia = 1 lbf/in^2 = 144 psfa = 144 lbf/ft^2.
Others : T(R) = T(oF) + 459.67; 1 lbf = 32.174 lbm·ft/s^2 ; 1 ton of refrigeration = 200 Btu/min.
