For ideal gases (internal energy depends only on temperature): Z 2 δu = u2 − u1 = cv (T )dT ≈ cv,avg (T2 − T1 )
(33)
1 2
Z
cp (T )dT ≈ cp,avg (T2 − T1 )
δh = h2 − h1 =
(34)
1
For ideal gases: Cp − Cv = nR
and γ =
Cp Cv
(35)
For solids or liquids =⇒ incompressible: cp ≈ cv ≈ c
(36)
Chapter 5 - MASS AND ENERGY ANALYSIS OF CONTROL VOLUMES Conservation of mass: X dm X = m ˙ in − m ˙ out dt
and δm =
X
min −
X
mout
(37)
Flow work (or flow energy): wf low = pν
(38)
h = u + wf low = u + pν
(39)
such that: First law (open systems / control volumes): X 2 2 X dEsystem vin vout ˙ ˙ =Q−W + m ˙ in hin + + gzin − + gzout m ˙ out hout + dt 2 2 out in In steady-state (that is, no change in total energy with time): X v2 v2 ˙ m ˙ steady hout − hin + out − in + gzout − gzin = Q˙ − W 2 2
(40)
(41)
Total energy of flowing fluid: θ = h + ke + pe = h +
v2 + gz 2
(42)
such that: X
˙ m ˙ steady (θout − θin ) = Q˙ − W
(43)
˙ are the NET heat and work transfers, so you will often have: Remember that Q˙ and W 2 2 X X vin vout ˙ ˙ ˙ in (44) m ˙ steady hout + + gzout + Qout + Wout = m ˙ steady hin + + gzin + Q˙ in + W 2 2 Mass and volume flow rates: V˙ = A × v
(45)
m ˙ = ρ × A × v = ρV˙
(46)
3
Chapter 6 - THE SECOND LAW OF THERMODYNAMICS Reversible cycle (Carnot cycle - maximum achievable): QH TH = QL rev TL
(47)
Thermal efficiency (heat engine): ˙ desired W W QL TL = = =1− ≤1− ˙ required QH Q T QH H H
ηth =
(48)
Coefficient of performance of heat pumps (COPHP ): COPHP =
desired QH Q˙ H QH TH = = = ≤ ˙ required W QH − QL TH − TL W
(49)
Coefficient of performance of refrigerators (COPR ): COPR =
QL desired QL Q˙ L TL = = = ≤ ˙ required W QH − QL TH − TL W
(50)
Chapter 7 - ENTROPY Entropy: dS =
δQ Tsource
(51) rev
Inequality of Clausius: I
δQ ≤0 Tsource
(52)
Sgen ≥ 0
(53)
∆Stotal = ∆Ssystem + ∆Ssurroundings ≥ 0
(54)
Increase of entropy principle:
Insulated system (increase entropy): ∆Sinsulated ≥ 0
(55)
Second law (closed systems): Z ∆S = S2 − S1 = m (s2 − s1 ) =
Second law (open systems / control volumes): X δQk X X dSsystem = + Sgen + m ˙ in sin − m ˙ out sout + S˙ gen dt Tk out in
(59)
k
Isentropic (means adiabatic and reversible) =⇒ ∆S = 0 for ideal gases pV γ = k and p1−γ T γ = k and T V γ−1 = k Z 2 Qrev T dS =⇒ area under the curve on a T-s diagram 1