Conservation of Mass for a Control Volume
Some Useful Equations
Conservation of Energy Principle for a Closed System
(the First Law of Thermodynamics)
E = Q – W, where E = U + KE + PE + ...
Conservation of Energy Principle for a Control Volume
(z  elevation)
W > 0, work done by the system
W < 0, work done on the system
dE / dt]cv = Q – W + in m(h + V2/2 + gz) – out m(h + V2/2 + gz)
[For transient (unsteady state) operations
(i ≡ initial state, at time = 0 and f ≡ final state, at time = t)
Work done by a closed system (mass = m)
W = ∫ W = ∫ p dV = m ∫ p dv, where p = p(v)
In a polytropic process
= const.): p = (const.) /
If n ≠ 1, W = (const.) (V21 – n – V11 – n)/(1 – n)
= (p2V2 – p1V1)/(1 – n)
If n = 1, W = (const.) ln V2/V1
mass flow rate (through an area A, with an average velocity V)
m = AV = AV/v, where  = density = 1/v, v = specific volume
Q > 0, heat transfer to the system
Q < 0, heat loss from the system
(pVn
dm / dt]cv = in m – out m
mcv, f – mcv, i = in ∫ m dt – out ∫ m dt
Vn
Ucv, f – Ucv, i = ∫ Q dt – ∫ W dt +
in ∫ mh dt – out ∫ mh dt
with Ucv(t) = mcv(t) ucv(t), neglecting KE ≈ PE ≈ 0]
The thermodynamic (Kelvin) temperature scale
For a power cycle ( ʃ dU = 0)
Wcycle = Qcycle = Qin – Qout > 0; efficiency,  = Wcycle / Qin
For refrigeration or heat pump cycles ( ʃ dU = 0)
Wcycle = Qout – Qin < 0; CoP,  = |Qsought | / |Wcycle|
h = u + pv, or H = U + pV,
where H = mh, U = mu, V = mv, etc.
Two-Phase (liquid-vapor) mixtures
Quality, x = (mvap / m) = mvap / (mliq + mvap)
Any property (per unit mass), , of a liquid-vapor mixture:
 = (1 – x)f + x gf + x (g – f)
For subcooled or compressed liquids:
v(p, T) ≈ vf(T); u(p, T) ≈ uf(T); h(p, T) ≈ hf(T)
The Compressibility Factor
Z = pv / RT , where R = gas constant, = R / M,
R = 8.314 kJ/kmol·K, M = molecular weight (kg/kmol)
(in Figures A-1, A-2, & A-3: pR = p / pc, TR = T / Tc)
dS = (Q / T)int rev ( Qint rev = ∫ T ds )
The first and the second “T dS” equations
For ideal gases (per unit mass)
ds = du / T + (R / v) dv = (cv / T) dT + (R / v) dv
ds = dh / T – (R / p) dp = (cp / T) dT – (R / p) dp
[for air, Table A-22,
s(T2, p2) – s(T1, p1) = so(T2) – so(T1) – R ln (p2 / p1)]
Isentropic processes (ds = 0)
for ideal gases with a constant specific heat ratio, cp / cv = k
T2 / T1 = (p2 / p1)(k – 1)/k ; pvk = p1v1k = p2v2k = const.
p2 / p1 = pr2 / pr1 (with air, Table A-22)
Entropy balance for a closed system with its boundary at Tb
S2 – S1 = ∫ (Q / T)b +  (   0 )
Entropy rate balance for a control volume
dS / dt]cv =  (Q / T)b + in m s – out m s + cv
Specific Heats
constant volume specific heat, cv = ( u/ T)v
constant pressure specific heat, cp = ( h/ T)p
For an ideal gas
pV = mRT)
u2 – u1 = ∫ cv dT, where cv = cv(T)
h2 – h1 = ∫ cp dT, where cp = cp(T)
cp = cv + R
For liquids and solids (incompressible substances,  ≈ constant)
cp ≈ cv = c, and H ≈ U = m c T
ia:02/04
Entropy (defined in terms of a change in)
T dS = dU + p dV
T dS = dH – V dp
Enthalpy
(Z  1) pv = RT (or,
u = u(T), cv = du/dT,
h = h(T), cp = dh/dT,
h = u + pv = u + RT;
TC / TH = (QC / QH) for Carnot cycles  max = 1 – (TC / TH )
Other Constants and Definitions
Force, Work (Energy), and Power (rate of work, or heat transfer)
1 N = 1 kg·m/s2; 1 J = 1 N·m; 1 W = 1 J/s
Pressure
1 Pa = 1 N/m2; 1 bar = 100 kPa (= 105 Pa = 0.1 MPa)
1 atm = 101.325 kPa = 1.013 25 bar (standard atmosphere)
p (gage) = p (absolute) – p (atmospheric, absolute)
Temperature
T (ºC) = T (K) – 273.15
Standard Acceleration due to the Earth’s gravity
g = 9.806 65 m/s2
SI Unit prefixes for multiplication factors
c ≡ centi = 10–2; m ≡ milli = 10–3;  ≡ micro = 10–6
k ≡ kilo = 103; M ≡ mega = 106; G ≡ giga = 109