Comprehensive Equation Sheet for M E 320
Author: John M. Cimbala, Penn State University. Latest revision: 03 February 2015
Notation for this equation sheet: V = volume, V = velocity, v = y-component of velocity, ν = kinematic viscosity
kg
kg
kg
m
Patm = 101.3 kPa rair = 1.204 3 r water = 998.0 3 r mercury ≅ 13, 600 3
2
m
m
m
s
•
Constants: g = 9.807
•
 N ⋅ s 2   Pa ⋅ m 2   kW ⋅ s   m3   2π rad 
Conversions: 
 
 
 
 T (K) ≅ T
 
 kg ⋅ m   N   1000 N ⋅ m   1000 L   rotation 
•
/ H2 O
Specific gravity: SG = ρρ
•
Simple shear flow and viscosity: u ( y ) = Vy / h for flow sandwiched between two infinite flat plates; τ = µ
•
Surface tension: ∆Pdroplet = Pinside − Poutside = 2
•
Pgage Pabsolute − Patm Pvacuum
= Patm − Pabsolute Pv ≅ 2.0 kPa for water at T ≅ 20o C
Gage, vacuum, and vapor pressure: =
•

 dP
ρ gh
= − ρ g Pbelow = Pabove + ρ g ∆z ∆P =
ρg
Hydrostatics: ∇P =
dz
•
Buoyant force: FB = r f gVsubmerged
•
Forces on submerged, plane surfaces (see sketch):
( C ) + 273.15
o
= cP − cv k = cP / cv kair = 1.40 Rair ≅ 300
Ideal gas: P = ρ RT R
yC +
F=
PC A Pavg A y=
( P0 + ρ ghC ) A ==
P
ss
R
∆Pbubble = Pinside − Poutside = 4
ss
R
m2
s2 ⋅ K
Capillary tube: h =
du
dy
2s s
cos φ
ρ gR
I xx ,C
 yC + P0 / ( ρ g sin θ )  A
π R4
ab3
. Circle of radius R: I xx ,C =
4
12


 
G=
Rigid body acceleration: ∇P= ρρ
( g − a ) . Use modified gravity vector


G in place of g everywhere.
Rectangle a wide and b tall: I xx ,C =
•
•
Rigid body rotation (vertical axis, cylinder radius R):
zsurface =
h0 −
ω2
(R
4g
2
P0 +
− 2r 2 ) P =
rω 2
2
r 2 − r gz
•





 
∂V
∂V
∂V Db ∂b
DV ∂V

= +u
+v
+w
=
+ V ⋅∇ b
Acceleration field and material derivative: a ( x, y, z , t ) =
∂t
∂x
∂y
∂z Dt ∂t
Dt
•
  
  ∂w ∂v    ∂u ∂w    ∂v ∂u  
− i + 
−
Vorticity vector: z = ∇ × V = curl V = 
 j +  ∂x − ∂y  k
 ∂y ∂z   ∂z ∂x 


•
Equation for a 2-D streamline:
•


 
∂u
∂v
∂w
=
ε xx
=
, ε yy =
, ε zz
Rates of motion and deformation of fluid particles: V = ui + vj + wk
∂x
∂y
∂z
(
( )

w=
•
)


 
dy 
v
= where in Cartesian coordinates, V = ui + vj + wk

dx along a streamline u
1  ∂w ∂v   1  ∂u ∂w   1  ∂v ∂u  
1  ∂u ∂v 
1  ∂w ∂u 
1  ∂v ∂w 
+  , ε yz = +
− i +  −


 j +  −  k ε xy = +  , ε zx =
2  ∂y ∂x 
2  ∂x ∂z 
2  ∂z ∂y 
2  ∂y ∂z 
2  ∂z ∂x 
2  ∂x ∂y 
Volumetric strain rate in Cartesian coordinates:
1 DV 1 dV
∂u ∂v ∂w
=
= ε xx + ε yy + ε zz =
+
+
V Dt V dt
∂x ∂y ∂z
•

1  ∂u ∂v 
∂u
+ 


2  ∂y ∂x 
∂x

 ε xx ε xy ε xz  

  1  ∂v ∂u 
∂v
+ 
Strain rate tensor in Cartesian coordinates: ε ij =

 ε yx ε yy ε yz  =
 2  ∂x ∂y 
∂y
ε

 zx ε zy ε zz  
 1  ∂w ∂u  1  ∂w ∂v 
 2  ∂x + ∂z  2  ∂y + ∂z 



•
Reynolds transport theorem (RTT): For a fixed or non-fixed control volume, where B = some flow property, and b = B/m,
dBsys d
=
dt
dt
•
∫
d d



r bdV + ∫ r bVr ⋅ ndA where Vr= V − VCS is the relative velocity
CV
CS
dmCV
=
dt
Conservation of mass for a CV:
∑ m − ∑ m
in
For steady-state, steady flow (SSSF) problems:
out
Conservation of energy for a CV: Q net in + Wshaft,=
net in
P
V2
1
+ gz h= u +
m = ρVavg Ac a =
where e =u +
2
ρ
Ac
Bernoulli equation:
•
Momentum equation for a CV:
•
out
in
out




d
V2
V2


e
d
m
h
gz
m
h
+
+
+
−
+
+ gz 
ρ
a
a
V



∑
∑
∫
CV
dt
2
2
out

avg in 
avg
 V
∫Ac  Vavg

3

a = 2.0 fully developed laminar pipe flow
 dAc
a ≈ 1.05 fully developed turbulent pipe flow

P1 V12
P2 V2 2
+
+ z1 =
+
+ z2 along a streamline from 1 to 2.
g 2g
g 2g
ρρ
•
•
1
=
m ρρ
=
V
Vavg Ac at an inlet or outlet, where Vavg =
Vnormal dAc
Ac ∫
∑ m = ∑ m . For incompressible SSSF: ∑V = ∑V .
in
•
1  ∂u ∂w  
+

2  ∂z ∂x  

1  ∂v ∂w  
 +

2  ∂z ∂y 

∂w


∂z

dddddddd
d
 − ∑ β mV

β mV
gravity + ∑ Fpressure + ∑ Fviscous + ∑ Fother =
∫ rVdV + ∑
dt CV
out
in
∑F = ∑F
P1
V12
P2
V2
+ αα
+ z1 + hpump, u=
+ 2 2 + z2 + hturbine, e + hL , where 1 = inlet, 2 = outlet, and the
1
r1 g
r2 g
2g
2g
hpumpWpump shaft
W
useful pump head and extracted turbine head are hpump, u =
and hturbine, e = turbine shaft


mg
h turbine mg
Head form of energy equation:
 P V2

 P V2

+
+ z −
+
+ z
Turbomachinery: Net head = H = hpump,u = 
g 2g
 ρρ
out  g 2 g
in
= ρV gH , Brake horsepower = Wshaft = bhp
= mgH
Water horsepower = Wpump,u
To match a pump (available head) to a piping system (required head), match H available = H required
Pump and turbine performance parameters:
bhp
V
gH
CQ = Capacity coefficient =
CH = Head coefficient = 2 2 CP = Power coefficient =
3
rw 3 D 5
ωD
ω D
=
hpump
Wpump,u ρV gH CQ CH
= = =
bhp
bhp
CP
function of CQ h
=
turbine
CP
bhp
bhp
= = =


Wturbine,e rV gH CQ CH
function of CP
3
2
2
3
5
VB ωB  DB  H B  ωB   DB  bhp B ρ B  ωB   DB 
=
=
Pump and turbine affinity laws:  =


 


 

VA ωA  DA  H A  ωA   DA  bhp A ρ A  ωA   DA 
•
Grade lines: Energy Grade Line = EGL =
•
Re
Pipe flows:=
ρVD VD
=
hL , total
=
µ
ν
m = ρVA , Minor head loss = hL = K L
P
P V2
=
+z
+
+ z , Hydraulic Grade Line = HGL
ρg
ρ g 2g
∑h
V2
du 
, Fully developed laminar pipe flow, α = 2 f = 64 / Re . Near wall, τ wall = µ

2g
dy  wall
e /D
2.51
≈ −2.0 log10 
+

f
 3.7 Re f
1
Fully developed turbulent pipe flow, α ≅ 1.05
•
Hydraulic diameter: Dh =
•
Continuity equation:
•
•
•

 (Colebrook eq., Moody chart)

4 Ac
, where Ac is the cross-sectional area and p is the wetted perimeter
p
 

∂ρ 
+ ∇ ⋅ ρV = 0 . If incompressible, ∇ ⋅ V = 0
∂t
( )
∂u ∂v ∂w
1 ∂ ( rur ) 1 ∂uθ ∂u z
+
+
=
0 , and in cylindrical coordinates,
+
+
=
0
∂x ∂y ∂z
r ∂r
r ∂θ
∂z
In Cartesian coordinates,
•
8τ w
e 
L V2

=
f
= fnc  Re,  ,
D 2g
D
ρV 2


+ ∑ hL , minor , Major head loss = hL = f
L , major
∂y
∂y
∂ψ
∂ψ
1 ∂ψ
uθ = −
; Cylindrical planar (r-θ plane): ur =
∂x
∂r
r ∂θ


  


 ∂V

DV
= 
+ V ⋅∇ V  = −∇P + g + µ∇ 2V
Navier-Stokes equation: (for incompressible, Newtonian flow) ρρρ
Dt
 ∂t



Creeping flow: For Re << 1, ∇P ≈ µ∇ 2V .
 
 
0
Potential flow (Irrotational flow): Since ∇ × V = 0 , then V = ∇φ , where φ is the velocity potential function. ∇ 2φ =
Stream function: Cartesian (x-y plane): u =
v= −
(
)
∂ 2φ ∂ 2φ ∂ 2φ
1 ∂  ∂φ  1 ∂ 2φ ∂ 2φ
+ 2 + 2 = 0 . Cylindrical coordinates:
r
0
=
∇ 2φ
+
=
+
2
r ∂r  ∂r  r 2 ∂θ 2 ∂z 2
∂x
∂y
∂z
Cartesian coordinates: ∇ 2φ=
0 as well. Superposition of both φ and y is valid for potential flow.
If flow is also 2-D, then ∇ 2ψ =
•
Re x
Boundary layers: =
ρUx Ux
=
, where x is along the body. U(x) is the outer flow (just outside the boundary layer).
µ
ν
For steady flow, continuity:
•
Flat plate boundary layer:
If laminar, ( Re x < 5 × 105 ),
∂u ∂v
∂P
dU
∂u
∂u
∂ 2u
+
=
0 ; x-momentum: u
≈0
+v
= U
+ ν 2 ; y-momentum:
∂y
∂x ∂y
dx
∂x
∂y
∂y
δ
x
=
4.91
δ*
Re x
x
If turbulent and smooth, ( 5 × 105 < Re x < 107 ),
•
Drag and Lift on bodies: CD =
FD
1
2
ρV 2 A
1.72
=
Re x
δ
CL =
x
≈
δ*
0.38
( Re x )
FL
1
2
2τ w
0.664
C=
C=
=
f
D
ρU 2
Re x
C f ,x
=
ρV 2 A
1/ 5
x
≈
0.048
( Re x )
1/ 5
C f ,x ≈
1.33
Re x
0.059
( Re x )
1/ 5
C=
C=
f
D
0.074
Re x1/ 5
, where A = projected frontal area or planform area. CD includes skin friction
and pressure drag. For bodies without ground effect, required power = W = FDV . For vehicles in ground effect, the required power
=
WV + 1 rV 3C A , where µ rolling = coefficient of rolling resistance, and W is the vehicle weight.
to the wheels
= W µ
rolling
2
D
•
Isentropic compressible flow for air (k = 1.4):
T0
ρ
= 1 + 0.2Ma 2 0=
T
ρ
•
T2
Normal shock equations for air (k = 1.4): =
T1
− 0.4
( 2 + 0.4Ma ) 2.8Ma
5.76Ma
2
1
(1 + 0.2Ma )
2 2.5
2
1
2
1
P0
=
P
(1 + 0.2Ma )
2 3.5
m max =
0.6847P0 A*
P2 ( 2.8Ma1 − 0.4 ) ρ 2
2.4Ma12
=
=
P1
2.4
ρ1 ( 2 + 0.4Ma12 )
2
RT0
•
Moody Chart: