(1)
Work and Energy Relations
Wb = ∫ pdV
EQUATIONS FOR MAE 3524
W=
VI ∆t
e
ks 2
( x2 − x12 )
2
W=
spring
2
Wrot = 2π nτ
ΔKE = 1 m (V22 - V12 )
2
wcv =− ∫ νdp + (V12 − V22 ) / 2 + g ( z1 − z2 )
1
ΔPE = mg ( z2 - z1 )
Fspring = ks x
(2)
First Law Relations and Mass Conservation
∆Ε = ∆U + ∆KE + ∆PE = Q − W
dEcv 
= Qcv - Wcv + ∑ m i ( h + V 2 / 2 + gz ) - ∑ m e ( h + V 2 / 2 + gz )
i
e
dt
i
e
m = ρVA
(3)
dmcv
= ∑ m i - ∑ m e
dt
i
e
ν
2 δ q 
∆s =∫  
∫1
1
 T  int,rev
Q j
dScv
= ∑ m i si − ∑ m e se + ∑
+σ cv
dt
i
e
j Tj
2
δ Q 
 T  + σ
b

Q
dS
= ∑ j + σ
dt
j Tj
 δQ 
= -σ ≤ 0
T b
∫ 
Pressure, Temperature and Energy Relations
pabs = patm ± p gage
pV = nRT
pR = p / pc
p − patm = ρgL
= V ν
m
pV = mRT
R=R M
ν R =ν pc / RTc
TR = T / Tc
Pν =ZRT
k = Cp /C v
x = m g /(m f + m g )
y = (1 − x ) y f + xy g = y f + xy fg
y fg = y g − y f
h= u + Pv
∆u = ∫ c v dT
∆h = ∫ c p dT
Cp - Cv = R
∆h = ∆u + ν∆p
h (T,p ) ≅ h f (T )+ν f (T )[p - psat (T )]
ν(T,p ) ≅ ν f (T )
s(T,p) ≅ s f (T)
∆u = ∫ cdT
(5)
V
Second Law Relations
=
∆S
(4)
m =
u(T,p) ≅ u f (T)
Entropy Relations (and Isentropic Processes)
Tds = du + pd ν
T
ν
=
∆s cv ln 2 + R ln 2
T1
ν1
Tds= dh − νdp
T
p
=
∆s c p ln 2 − R ln 2
T1
p1
∆s = s o2 − s1o − R ln(p 2 / p1 )
T2 / T1 =( p2 / p1 )
( k −1)/ k
=( ν1 / ν 2 )
∆s = c ln(T2 / T1 )
k −1
(p 2 / p1 )s = p r 2 / p r1
( ν 2 / ν1 )s =νr 2 / ν r1
(6)
Device Efficiencies and Cycle Analyses
w act w isen
ηturbine =
ηth =
Wnet ,out
COPR =
Qin
ηth ,rev = 1-
TC
TH
=
ε th η=
II ,th
(7)
ηcompressor =
ηpump =
w isen w act
Qin
Wnet ,in
COPR,rev =
ηth
ηth ,rev
COPHP =
TC
TH - TC
=
ε R η=
II , R
COPHP,rev =
mi
m
yi =
ni
nj
j
mfi ai and a
∑=
p2
p1
m
ma
(10)
in
Rcond =
∆s =
ηvol
hˆ A
Q hm = c (h∞ − hs )
c p ,m
L
kA
j
i =1
i
 pv 

 pg T
EER =
Q cap
W
in
Tdp = Tsat ( pv )
 Btu 
W − hr 
L
1
1
1
= Rtot =
+
+
UA
hˆ1 A1 kA hˆ2 A2
Air-Cooled Condenser Analysis (Dry Coil Analysis)
Q cond ε c m c ,a c p ,a (Tcond − Tca ,i ) ε c = 1 − e − Ntuc
=
Tc=
Tca ,i +
a ,o
Q cond
m ca c pa
UAc =
1
1
1
+
ˆh A η hˆ A
r r
o a a
∑n a
=i 1
∑ [ ∆s ]
φ=
Heat Transfer Fundamentals
ˆ (T − T )

Q
=
hA
conv
s
∞
(11)
p2
p1
Compressor Analysis
η o , is
j
i i
∆si = si o,2 − si o,1 − R ln
m act
=
ρ sucωVdisp
mi ai
∑=
∑ya
h= ha + ω hg
m (h2, s − h1 )
=
W
j
j
M v pv
pv
pv
= 0.622
=
0.622
M a pa
pa
p − pv
hv ≅ hg
j
Ai
∑=
=i 1 =i 1
Psychrometric Relationships
ω=
= v
wturbine
TC  Q C 

=
TH  Q H  rev
COPHP
=
ε HP η=
II , HP
COPHP ,rev
Extensive Properties:
A
=
=i 1 =i 1
∆s = s2o − s1o − R ln
wpump
TH
TH - TC
COPR
COPR ,rev
pi = yi p
=
Intensive
Properties: a
(9)
BWR =
Mixtures of Ideal Gases
mfi =
(8)
Qout
Wnet ,in
Ntuc =
UAc
m a c p ,a
h=
h2 −
3
Rconv =
Q cond
m r
 T −T 
UAc ≅ −m a c p ,a ln 1 − ca ,o ca ,i 
 Tcond − Tca ,i 
1
ˆ
hA
i i
Mass and
Density
UNITS CONVERSION TABLE FOR MAE 3524
1 kg = 2.20 lb
m
3
3
1 g cm = 10 kg m
3
Length
1 lb = 0.454 kg
m
3
1 g cm = 62.4 lb
m
1 cm = 0.394 in
ft
3
Volume
1 cm = 0.061 in
3
1 m = 35.3 ft
1 L = 10
−3
m
Force
3
1 in = 16.4 cm
3
1 N = 1 kg ⋅ m s
3
1 gal = 0.134 ft
3
3
−3
1=
gal 3.79 × 10 m
2
1 lb
2
= 1.45 × 10
1=
Pa 1 N m
5
3
=
1 lb
32.2 lb ⋅ ft s
f
m
f
1 bar = 10 N m
1 atm = 1.01 bar
−4
lb
f
2
Energy
1 J = 1 N ⋅ m = 0.738 ft ⋅ lb
f
and
Specific =
1 kJ 737.56 ft ⋅ lb
f
Energy
1 kJ = 0.948 Btu
Energy
Transfer
Rate
3
1 ft = 0.0283 m
3
1 N = 0.224 lb
Pressure
3
3
1 L = 0.035 ft
in
2
1 lb
1 lb
f
2
= 4.45 N
f
f
2
in = 6894.8 Pa
2
in = 144 lb
1 atm = 14.7 lb
1 ft ⋅ lb
f
in
f
ft
f
1.36 J
=
=
1 Btu 778.17 ft ⋅ lb
f
1 Btu = 1.055 kJ
1 Btu lb = 2.326 kJ kg
m
1 kcal = 4.19 kJ
1 W = 1 J s = 3.413 Btu h
1 kW = 1.341 hp
1 Btu h = 0.293 W

1=
kJ kg ⋅ K 0.239 Btu lb m ⋅ R

=
⋅ K 1 Btu lb ⋅ R
1 kcal kg
m
Universal Gas Constant
2
2
1 kJ kg = 0.43 Btu lb
m
Specific
Heat
3
1 mile h = 1.61 km h
1 km h = 0.621 mile h
3
ft = 16.018 kg m
1 in = 2.54 cm
1 ft = 0.305 m
1 m = 3.28 ft
Velocity
3
1 lb
m
1 hp = 2545 Btu h
1 hp 550 ft ⋅ lb
=
f
s
1 hp = 0.746 kW
1 ton =12,000Btu/h=3.517kW

1 Btu
=
lb
R 4.19 kJ kg ⋅ K
m
8.314 kJ / kmol ⋅ K

o
R=
1545 ft ⋅ lb f / lbmol ⋅ R

o
1.986 Btu / lbmol ⋅ R
Standard Acceleration of Gravity
9.81 m / s 2
g=
2
32.174 ft / s
Standard Atmospheric Pressure
1.01bar
1 atm = 
2
14.7 lb f / in
Temperature Relations
T(o R ) = 1.8T(K )
o
T (=
C ) T ( K ) − 273
T=
(o F ) T (o R ) − 460