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Equationsheet - equation sheet for practice material
Introduktion til vindenergi (46000)
Danmarks Tekniske Universitet
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Section 1 - Integral Relations
- Hydrostatic pressure equation:
∆p = ρg∆z
- Hydrostatic force on a panel:
F/A panel = (p atm + ρgh CG )
where the force acts on centre of pressure:
x CP = −
I xy sin θ
h CG A ,
y CP = −
I xx sin θ
h CG A
- Mass conservation for a CV:
∫
CV
∂ρ
∂t dV
∮
CS
ρ(
~ V · ~n)dS = 0
which reduces to m ̇ in
= m ̇ out for steady flow, where
m ̇ = ρ(
~ V · ~n)A.
- Linear momentum conservation for steady,
one-dimensional flow:
∑
~ F CV =
[∑
i
m ̇ i
~ V i
]
out
−
[∑
i
m ̇ i
~ V i
]
in
- Angular momentum conservation for steady,
one-dimensional flow:
∑
~ M 0 =
[∑
i
(~r i ×
~ V i ) ̇m i
]
out
−
[∑
i
(~r i ×
~ V i ) ̇m i
]
in
- Energy conservation for a control volume:
̇ Q −
̇ W shaft
−
̇ W viscous =
∂
∂t
∫
CV
(u +
V
2
2
- gz)ρdV
∮
CS
(h +
V
2
2
- gz)ρ(
~ V · ~n)dS
- Steady flow energy equation (SFEE):
(
p
ρg
V
2
2 g
- z
)
in
=
(
p
ρg
V
2
2 g
- z
)
out
- h friction − h pump + h turbine
- Bernoulli equation:
p 1
ρ 1
V
2
1
2
- gz 1 =
p 2
ρ 2
V
2
2
2
- gz 2
Frictionless, incompressible flow with no work or heat
transfer. Holds for steady flow along a single streamline.
- Frictional (major) losses:
∆p L
= f L
D ρ V
2
2
, f = 8
τ w
ρV
2
- Minor losses: Relates losses in valves, fittings, elbows, etc.
Defined in terms of resistance coefficients or loss factor,
K L
=
h L
V
2 /(2g)
, h L
= head loss
- Total head loss:
h L,major + h L,minor =
(
f L
D
∑
K L
)
V
2
2 g
Section 2 - Turbomachinery
- Flow rate:
̇ V = m/ρ ̇ = (
~ V · ~n)A
- Pump head: net energy imparted to the flow
H =
(
p
ρg
V
2
2 g
- z
)
out
−
(
p
ρg
V
2
2 g
- z
)
in
- Useful pump power (water horsepower):
̇ W water = ρg
̇ V H
- Total mechanical power (shaft horsepower):
bhp = ωT shaft
- Pump efficiency:
η pump =
̇ W water
̇ W shaft
=
ρg
̇ V H
ωT shaft
- Pump steady operating point is where the available head
matches the required:
H required =
p 2 − p 1
ρg
αV
2
2
− αV
2
1
2 g
- (z 2 − z 1 ) + h L
where α = 1 for turbulent flows.
- Net positive suction head:
N P SH =
(
p
ρg
V
2
2 g
)
inlet
−
p v
ρg
where p v is the vapour pressure.
- Euler pump equation:
H =
η pump
g
(ωr 2 V 2 ,t − ωr 1 V 1 ,t )
=
η pump
2 g
[(
V
2
2
− V
2
1
)
(
ω
2
r
2
2
− ω
2
r
2
1
)
−
(
V
2
2 ,r
− V
2
1 ,r
)]
- Dimensionless pump coefficients:
Head coefficient: C H =
gH
ω
2 D
2
Flow coefficient: C Q =
̇ V
ωD
3
Power coeffieicnt: C P =
bhp
ρω
3 D
5
Pump efficiency: η pump =
C Q C H
C P
=
ρ
̇ V gH
bhp
- Pump specific speed:
N sp =
C
1 / 2
Q
C
3 / 4
H
=
ω
̇ V
1 / 2
(gH)
3 / 4
N sp,U S = ( ̇
n, rpm)( ̇V , gpm)
1 / 2
(H, ft)
3 / 4
N sp,Eur = ( ̇
n, Hz)( ̇V , m
3 /s)
1 / 2
(H, m)
3 / 4
Conversion factors:
N sp N sp,U S
N sp,Eur
N sp 1 2734 1/2π
N sp,U S 3× 10
− 4 1 5× 10
− 5
N sp,Eur
2 π 17180 1
- Pump/turbine affinity laws:
̇ V B
̇ V A
=
ω B
ω A
(
D B
D A
) 3
H B
H A
=
(
ω B
ω A
) 2
(
D B
D A
) 2
bhp B
bhp A
=
ρ B
ρ A
(
ω B
ω A
) 3
(
D B
D A
) 5
- Euler turbine equation:
̇ W shaft
= ρω
̇ V (r 2
V 2 ,t
− r 1
V 1 ,t
)
- Impulse (Pelton) turbine shaft power:
̇ W shaft = ρωr
̇ V (V jet − rω)(1 − cos β)
where max efficiency is achieved when β = 160
◦ -
◦ and
V jet = 2ωr.
- Reaction (Francis, Kaplan) turbine power:
̇ W water = ρg
̇ V H
̇ W shaft
= T shaft
ω
η turbine
=
̇ W shaft
̇ W water
=
bhp
ρgH
̇ V
- Turbine scaling laws:
Head coefficient: C H
=
gH
ω
2 D
2
Flow coefficient: C Q =
̇ V
ωD
3
Power coefficient: C P =
bhp
ρω
3 D
5
Turbine efficiency: η turbine =
C P
C Q C H
- Turbine specific speed:
N st =
C
1 / 2
P
C
5 / 4
H
=
ω(bhp)
1 / 2
ρ
1 / 2 (gH)
5 / 4
N st,U S
= ( ̇
n, rpm)(bhp, hp)
1 / 2
(H, ft)
5 / 4
N st,SI
= ( ̇
n, rpm)( ̇V , m
3 /s)
1 / 2
(H, m)
3 / 4
Section 3 - Differential Analysis
- Integral form of the continuity equation:
∫
CV
∂ρ
∂t dV
∮
CS
ρ(
~ V · ~n) dA = 0
- Gauss’ divergence theorem:
∫
CV
~ ∇ · G ij
dV =
∮
CS
G ij
· ~n dA
where G ij is a tensor of arbitrary rank.
- Differential continuity equation:
∂ρ
∂t
~ ∇ · (ρ~V ) = 0
- Differential continuity equation in Cartesian coordinates:
∂ρ
∂t
∂(ρu)
∂x
∂(ρv)
∂y
∂(ρw)
∂z
= 0
- Differential continuity equation in cylindrical coordinates:
∂ρ
∂t
- 1
r
∂(rρu r )
∂r
- 1
r
∂(ρu θ
)
∂θ
∂(ρu z )
∂z
= 0
- Material derivative:
DG ij
Dt
=
∂G ij
∂t
~ V ·
~ ∇G ij
where G ij
is a tensor of arbitrary rank.
- Constitutive relation for Newtonian fluids:
τ ij
= 2μǫ ij
where τ ij is the shear stress tensor and the strain rate
tensor is ǫ ij = 1
2
(
∂u i
∂x j
∂u j
∂x i
)
null- Incompressible Navier-Stokes equation in vector form:
ρ D ~
V
Dt
= −∇p + ρ~g + μ ~∇
2 ~ V
- Incompressible Navier-Stokes equation in Cartesian coordinates:
x-component: ρ
(
∂u
∂t
- u ∂u
∂x
- v ∂u
∂y
- w ∂u
∂z
)
= −
∂p
∂x
- ρg x + μ
(
∂
2 u
∂x
2
∂
2 u
∂y
2
∂
2 u
∂z
2
)
y-component: ρ
(
∂v
∂t
- u ∂v
∂x
- v ∂v
∂y
- w ∂v
∂z
)
= −
∂p
∂y
- ρg y + μ
(
∂
2 v
∂x
2
∂
2 v
∂y
2
∂
2 v
∂z
2
)
z-component: ρ
(
∂w
∂t
- u ∂w
∂x
- v ∂w
∂y
- w ∂w
∂z
)
= −
∂p
∂z
- ρg z + μ
(
∂
2 w
∂x
2
∂
2 w
∂y
2
∂
2 w
∂z
2
)
- Incompressible Navier-Stokes equation in cylindrical coordinates:
r-component: ρ
(
∂u r
∂t
- u r
∂u r
∂r
u θ
r
∂u r
∂θ
−
u
2
θ
r
- u z
∂u r
∂z
)
= −
∂p
∂r
- ρg r +
μ
[
1
r
∂
∂r
(
r ∂u
r
∂r
)
−
u r
r
2
1
r
2
∂
2 u r
∂θ
2
−
2
r
2
∂u θ
∂θ
∂
2 u r
∂z
2
]
θ-component: ρ
(
∂u θ
∂t
- u r
∂u θ
∂r
u θ
r
∂u θ
∂θ
u θ u r
r
- u z
∂u θ
∂z
)
= −
1
r
∂p
∂θ
- ρg θ +
μ
[
1
r
∂
∂r
(
r ∂u
θ
∂r
)
−
u θ
r
2
1
r
2
∂
2 u θ
∂θ
2
2
r
2
∂u r
∂θ
∂
2 u θ
∂z
2
]
z-component: ρ
(
∂u z
∂t
- u r
∂u z
∂r
u θ
r
∂u z
∂θ
- u z
∂u z
∂z
)
= −
∂p
∂z
- ρg z +
μ
[
1
r
∂
∂r
(
r ∂u
z
∂r
)
1
r
2
∂
2 u z
∂θ
2
∂
2 u z
∂z
2
]
- Pressure and velocity distribution for Couette flow with a wall speed of V and gap h:
p = p 0 − ρgz
u =
V
h y
- Pressure and velocity distribution for Couette flow with an applied pressure gradient:
p = p 0 +
(
∂p
∂x
)
x − ρgz
u =
V y
h
1
2 μ
(
∂p
∂x
)
(y
2 − hy)
- Pressure and velocity distribution for a laminar film flowing down a vertical wall by gravity:
p = p atm
w =
ρg
2 μ x
(x − 2 h)
- Pressure and velocity distribution for Poiseuille flow:
p = p 0
u x =
1
4 μ
dp
dx
(r
2
− R
2
)
- Poiseuille max and average velocity and shear stress:
u max =
−R
2
4 μ
dp
dx
u avg =
−R
2
8 μ
dp
dx
τ rx =
R
2
dp
dx
- Friction drag of flat plates:
Laminar boundary layer: C f =
- 33
√
Re L
, Re L . 5 × 10
5
Turbulent boundary layer: C f =
- 074
Re
1 / 5
L
, Re L & 10
7
Transitional boundary layer: C f
=
- 074
Re
1 / 5
L
−
1742
Re L
, 5 × 10
5 . Re L
. 10
7
- Wing aspect ratio: (AR = b
2 /A) where b is the wingspan and A is the planform area (=bc
for rectangular wing).
Section 6 - Compressible Flows
- Total or stagnation quantities:
Enthalpy: h 0 = h +
V
2
2
, Pressure: p 0 = p + ρ V
2
2
, Temperature: T 0 = T +
V
2
2 C
####### p
####### p
####### p
- Isentropic flow relations for ideal gases with constant specific heats:
p 0
p
=
(
####### T
0
T
)
k
k − 1 ,
####### ρ
0
ρ
=
(
T 0
T
)
1
k − 1
where k = C p /C v is the specific heat ratio.
- Speed of sound:
c =
√ (
∂p
∂ρ
)
s
︸ ︷︷ ︸
General definition
=
√
k
(
∂p
∂ρ
)
T
︸ ︷︷ ︸
Constant k
=
√
kRT
︸ ︷︷ ︸
Ideal gas
where R = R/M , R = 8314 J/kmol·K is the universal gas constant and M is the molar
mass.
Mach number: V /c
Area change for 1D isentropic flow:
dA
A
= −
dV
V
(1 − M a
2 )
- Isentropic flow relations:
T 0
T
= 1 +
(
k − 1
2
)
####### M a
####### M a
2
,
p 0
p
=
[
1 +
(
k − 1
2
)
M a
2
]
k
k − 1 ,
ρ 0
ρ
=
[
1 +
(
k − 1
2
)
M a
2
]
1
k − 1
These are tabulated in Table A-13 for k = 1.
- Critical properties for choked isentropic flow in a nozzle:
T
∗
T 0
=
2
k + 1
,
p
∗
p 0
=
(
2
k + 1
)
k
k − 1 ,
ρ
∗
ρ 0
=
(
2
k + 1
)
1
k − 1
- Mass flow rate for isentropic flow in a nozzle:
m ̇ =
p 0 M aA
√
k/(RT 0 )
[
1 + (k − 1)
M a
2
2
] (k+1)/[2(k−1)]
, m ̇ max = A
∗ p 0
√
k
RT 0
(
2
k + 1
) (k+1)/[2(k−1)]
- Flow area relative to throat in a converging isentropic flow nozzle:
A
A
∗
=
1
M a
[(
2
k + 1
) (
1 +
k − 1
2
M a
2
)]
k + 1
2(k − 1)
- Mach number based on throat speed of sound: M a
∗ =
V
c
∗
= M a
√
T
T
∗
- Normal shock relations:
T 01 = T 02 , M a 2 =
√
(k − 1)M a
2
1
- 2
2 kM a
2
1
− k + 1
p 2
p 1
= 1 +
kM a 1
2
1 + kM a
2
2
= 2
kM a
2
1
− k + 1
k + 1
,
ρ 2
ρ 1
=
p 2
####### /p
####### /p
####### /p
1
T 2
####### /T
1
=
(k + 1)M a
2
1
2 + (k + 1)M a
2
1
=
V 1
V 2
T 2
T 1
= 2 +
M a
2
1
(k − 1)
2 + M a
2
2
(k − 1)
,
p 02
p 01
=
M a 1
M a 2
[
1 + M a
2
2
(k − 1)/ 2
1 + M a
2
1
(k − 1)/ 2
] (k−1)/[2(k−1)]
p 02
p 1
= (1 +
kM a
2
1
)[1 + M a
2
2
(k − 1)/2]
k/(k−1)
1 + kM a
2
2
These are tabulated in Table A-14 for k = 1.
- Oblique shock components:
M a 1 , n = M a 1 sin β, M a 2 , n = M a 2 sin(β − θ)
where β is the shock angle relative to oncoming flow and θ is the deflection angle relative
to oncoming flow.
- Relationship between M a, θ, and β in oblique shocks:
tan θ = 2 cot
β(M a
2
1
sin
2 β − 1)
M a
2
1
(k + cos 2β) + 2
This is plotted in Figure 1.
M a p/p A/A∗ - Table A-13: One-dimensional isentropic compressible flow functions for an ideal gas with k = 1. - - T /T - ρ/ρ - 0 9-01 9-01 9-01 5+ 0 1+00 1+00 1+00 ∞ - 0 9-01 9-01 9-01 2+ - 0 9-01 9-01 9-01 2+ - 0 8-01 9-01 9-01 1+ - 0 8-01 9-01 8-01 1+ - 0 7-01 9-01 8-01 1+ - 0 7-01 9-01 7-01 1+ - 0 6-01 8-01 7-01 1+ - 0 5-01 8-01 6-01 1+ - 1 5-01 8-01 6-01 1+ - 1 4-01 8-01 5-01 1+ - 1 4-01 7-01 5-01 1+ - 1 3-01 7-01 4-01 1+ - 1 3-01 7-01 4-01 1+ - 1 2-01 6-01 3-01 1+ - 1 2-01 6-01 3-01 1+ - 1 2-01 6-01 3-01 1+ - 1 1-01 6-01 2-01 1+ - 1 1-01 5-01 2-01 1+ - 2 1-01 5-01 2-01 1+ - 2 1-01 5-01 2-01 1+ - 2 9-02 5-01 1-01 2+ - 2 7-02 4-01 1-01 2+ - 2 6-02 4-01 1-01 2+ - 2 5-02 4-01 1-01 2+ - 2 5-02 4-01 1-01 2+ - 2 4-02 4-01 1-01 3+ - 2 3-02 3-01 9-02 3+ - 2 3-02 3-01 8-02 3+ - 3 2-02 3-01 7-02 4+ - 3 2-02 3-01 6-02 4+ - 3 2-02 3-01 6-02 5+ - 3 1-02 3-01 5-02 5+ - 3 1-02 3-01 5-02 6+ - 3 1-02 2-01 4-02 6+ - 3 1-02 2-01 4-02 7+ - 3 9-03 2-01 3-02 8+ - 3 8-03 2-01 3-02 8+ - 3 7-03 2-01 3-02 9+ - 4 6-03 2-01 2-02 1+ - 4 5-03 2-01 2-02 1+ - 4 5-03 2-01 2-02 1+ - 4 4-03 2-01 2-02 1+ - 4 3-03 2-01 1-02 1+ - 4 3-03 1-01 1-02 1+ - 4 3-03 1-01 1-02 1+ - 4 2-03 1-01 1-02 1+ - 4 2-03 1-01 1-02 2+ - 4 2-03 1-01 1-02 2+ - 5 1-03 1-01 1-02 2+ - Table A-14: One-dimensional normal shock functions for an ideal gas with k = 1. - M a - M a - p - /p - p - /p - p - /p - T - /T - ρ - /ρ - 1 1+00 1+00 1+00 1+00 1+00 1+ - 1 9-01 1+00 9-01 2+00 1+00 1+ - 1 8-01 1+00 9-01 2+00 1+00 1+ - 1 7-01 1+00 9-01 2+00 1+00 1+ - 1 7-01 2+00 9-01 3+00 1+00 1+ - 1 7-01 2+00 9-01 3+00 1+00 1+ - 1 6-01 2+00 8-01 3+00 1+00 2+ - 1 6-01 3+00 8-01 4+00 1+00 2+ - 1 6-01 3+00 8-01 4+00 1+00 2+ - 1 5-01 4+00 7-01 5+00 1+00 2+ - 2 5-01 4+00 7-01 5+00 1+00 2+ - 2 5-01 4+00 6-01 6+00 1+00 2+ - 2 5-01 5+00 6-01 6+00 1+00 2+ - 2 5-01 6+00 5-01 7+00 1+00 3+ - 2 5-01 6+00 5-01 7+00 2+00 3+ - 2 5-01 7+00 4-01 8+00 2+00 3+ - 2 5-01 7+00 4-01 9+00 2+00 3+ - 2 4-01 8+00 4-01 9+00 2+00 3+ - 2 4-01 8+00 3-01 1+01 2+00 3+ - 2 4-01 9+00 3-01 1+01 2+00 3+ - 3 4-01 1+01 3-01 1+01 2+00 3+ - 3 4-01 1+01 3-01 1+01 2+00 3+ - 3 4-01 1+01 2-01 1+01 2+00 4+ - 3 4-01 1+01 2-01 1+01 3+00 4+ - 3 4-01 1+01 2-01 1+01 3+00 4+ - 3 4-01 1+01 2-01 1+01 3+00 4+ - 3 4-01 1+01 1-01 1+01 3+00 4+ - 3 4-01 1+01 1-01 1+01 3+00 4+ - 3 4-01 1+01 1-01 1+01 3+00 4+ - 3 4-01 1+01 1-01 2+01 3+00 4+ - 4 4-01 1+01 1-01 2+01 4+00 4+ - 4 4-01 1+01 1-01 2+01 4+00 4+ - 4 4-01 2+01 1-01 2+01 4+00 4+ - 4 4-01 2+01 1-01 2+01 4+00 4+ - 4 4-01 2+01 9-02 2+01 4+00 4+ - 4 4-01 2+01 9-02 2+01 4+00 4+ - 4 4-01 2+01 8-02 2+01 5+00 4+ - 4 4-01 2+01 7-02 2+01 5+00 4+ - 4 4-01 2+01 7-02 3+01 5+00 4+ - 4 4-01 2+01 6-02 3+01 5+00 4+ - 5 4-01 2+01 6-02 3+01 5+00 5+
Equationsheet - equation sheet for practice material
Kursus: Introduktion til vindenergi (46000)
Universitet: Danmarks Tekniske Universitet
