Waterjet first draft 9/23/04 from Prof. Carmichael notes.
9/17/06: modified to reflect w (V=>VA) and separate inlet
and outlet pressure loss (in addition to drag) to reflect paper
VA
velocity inlet
w
Vs
Vj
wake fraction
ship velocity
nozzle (outlet) velocity
h
Vj
VA := Vs⋅(1 − w)
(
)
T = m_dot⋅ VJ − VA
d
m_dot = mass_flow_rate
VA
p local = p atmos + ρ ⋅ g ⋅ d
at inlet centerline ...
1
1
2
2
p oin = p local + ⋅ρ ⋅VA = p atmos + ρ ⋅g⋅d + ⋅ρ ⋅VA
2
2
at this point ... total pressure (pitot tube)
pressure at inlet to pump ...
total pressure (pitot tube) ...
1
2
p op = p oin − ρ ⋅g⋅(d + h) = p atmos − ρ ⋅g⋅h + ⋅ρ ⋅VA
2
pressure at pump exit... total
pressure (pitot tube) ..
1
2
p oj = p atmos + ⋅ρ ⋅Vj
2
total pressure increase
across pump ...
1
1
2
2 1
2
2
p oj − p op = p atmos + ⋅ρ ⋅Vj − p atmos + ρ ⋅g⋅h − ⋅ρ ⋅VA = ⋅ρ ⋅ ⎛ Vj − VA ⎞ + ρ ⋅g⋅h
⎠
2
2
2 ⎝
p oj − p op
energy rise across the pump
per unit mass flow is ...
ρ
power absorbed by ideal pump is
therefore ...
ideal efficiency is then ...
ηi =
ηi =
PTi
ηD =
Ppi
PTi
Ppi
=
T⋅ VA
Ppi
=
=
⋅ ⎛ V − VA
2 ⎝ j
1
2⎞
2
Ppi = m_dot⋅
p oj − p op
ρ
⎠ + gh
1
2
2
= m_dot⋅ ⎡⎢ ⋅ ⎛ Vj − VA ⎞ + g h⎤⎥
⎝
⎠
⎣2
⎦
and quasi propulsive coefficient is ...
effective_power
power_delivered
(
=
PE
Ppi
=
R⋅ Vs
PPi
)
m_dot⋅ VA⋅ VJ − VA
1
2
2
m_dot⋅ ⎡⎢ ⋅ ⎛ Vj − VA ⎞ + g h⎤⎥
⎝
⎠
⎣2
⎦
=
=
(1 − t) ⋅ T⋅Vs
=
PPi
(
)
2 ⋅ VA⋅ VJ − VA
2
2
Vj − VA + 2g h
1 − t PTi
1 − t T⋅ VA
1 − t
⋅
=
⋅
=
⋅η
1 − w PPi
1 − w Ppi
1 − w
i
=
(
)
2 ⋅ VA⋅ VJ − VA
2
2
Vj − VA + 2g h
⎛ VJ
2⋅ ⎜
=
⎝ VA
2
⎞
− 1⎟
⎠
⎛ Vj ⎞
gh
⎜V ⎟ − 1 + 2 2
⎝ A⎠
V
⎛ VJ
if h = 0
2⋅ ⎜
⎞
⎝ VA
ηi =
− 1⎟
⎠ =
2
⎛ Vj ⎞
⎜ ⎟ −1
⎝ VA ⎠
same as propeller (we
developed following in
actuator disk
2
Vj
VA
+1
from actuator_disk.mcd using new variables to avoid duplication
Δv := VVj − VVA
Δv
VV := VVA +
2
η I :=
T⋅ VVA
T⋅ V
simplify → 2⋅
VVA
VVA + VVj
what are implications of
VVj = VVA ?
2
ηI =
1+
VVj
VVA
h cannot be negative (would be ducted prop, h limits efficiency
Real waterjet with losses
net thrust of waterjet
Tnet = T − Draginlet
conventional drag coefficient
Cd =
CD =
)
Drag
1
2
drag coefficient of inlet
(
T = m_dot⋅ Vj − VA
2
⋅ρ⋅v ⋅A
Drag
1
2
2
⋅ρ⋅v ⋅A
=
Drag
1
2
=
⋅ρ⋅VA⋅A⋅VA
Drag
1
2
⋅m_dot⋅VA
⎡⎛ Vj
⎞
1
1⎤
Tnet = T − Draginlet = m_dot⋅ Vj − VA − CD⋅ ⋅m_dot⋅ VA = m_dot⋅ VA⋅ ⎢⎜
− 1⎟ − CD⋅ ⎥
2
VA
2
(
net thrust
)
⎣⎝
⎠
⎞
1⎤
2 ⎡⎛ Vj
PT_net = Tnet⋅VA = m_dot⋅ VA ⋅ ⎢⎜
− 1⎟ − CD⋅ ⎥
2
⎣⎝ VA
⎠
⎦
net thrust power
delta p across pump must be increased to account for losses ...
we'll assume separate inlet and outlet losses
assume internal losses are ... ~ 1/2*ρ*v^2
and the pump pressure rise is ...
non-dimensional pressure loss coefficient is ....
and the real pump pressure rise is ...
Δp loss = Δp in_loss + Δp out_loss
Δp in_loss
Kin =
1
2
⋅ρ⋅VA
2
Kout =
Δp out_loss
1
2
⋅ρ⋅Vj
2
1
1
1
1
2
2
2
2
2
2
p oj − p op = ⋅ρ⋅ ⎛ Vj − VA ⎞ + ρ⋅g⋅h + Δp loss = ⋅ρ⋅ ⎛ Vj − VA ⎞ + ρ⋅g⋅h + Kin⋅ ⋅ρ⋅VA + Kout ⋅ ⋅ρ⋅Vj
⎠
⎠
2 ⎝
2 ⎝
2
2
2
⎡⎛ V 2
⎛ Vj ⎞ ⎤⎥
j ⎞
g⋅ h ⎢
p oj − p op = ⋅ρ⋅VA ⋅ ⎜
− 1 + 2⋅
+ Kin + Kout ⋅ ⎜ ⎟
⎢⎝ VA ⎟⎠
2
2
⎝ VA ⎠ ⎥⎦
VA
⎣
1
2
⎦
ideal pump power is ...
Ppi = m_dot⋅
define ηp such that
ηp =
p oj − p op
ρ
Ppi
2
⎡⎛ V ⎞ 2
⎛ Vj ⎞ ⎥⎤
g⋅ h
j
⎢
= m_dot⋅ ⋅ VA ⋅ ⎜
⎢ V ⎟ − 1 + 2⋅ 2 + Kin + Kout⋅ ⎜ V ⎟ ⎥
2
⎝
A ⎠
⎦
V
⎣⎝ A ⎠
1
2
Pp = actual_pump_power
Pp
⎡
⎛ Vj ⎞ ⎤⎥
m_dot p oj − p op
1 m_dot⋅ VA ⎢⎛ Vj ⎞
g⋅ h
Pp =
=
⋅
= ⋅
⋅ ⎜
⎢ V ⎟ − 1 + 2⋅ 2 + Kin + Kout⋅ ⎜ V ⎟ ⎥
2
ρ
ηp
ηp
ηp
⎝
A ⎠
⎦
V
⎣⎝ A ⎠
2
PPi
2
η real =
PT_net
Pp
2
2
⎡⎛ Vj
⎞
1⎤
− 1⎟ − CD⋅ ⎥
2
⎣⎝ VA
⎠
⎦
m_dot⋅ VA ⋅ ⎢⎜
=
⎡
⎡
⎛ Vj ⎞ ⎤⎥⎤⎥
g⋅ h
⎢
1 m_dot⋅ VA ⎢⎛ Vj ⎞
⋅
⋅ ⎜
−
1
+
2
⋅
+
K
+
K
⋅
⎟
out ⎜ V ⎟ ⎥⎥
⎢ in
⎢⎝ VA ⎠
2
2
ηp
⎣
⎝
A ⎠
⎦
VA
⎣
⎦
2
2
2
for a different form ... multiply numerator and
denominator by (V A/Vj)^2 ...
⎡⎛ Vj
⎞
1⎤
− 1⎟ − CD⋅ ⎥
2
⎣⎝ VA
⎠
⎦
2 ⋅ η p ⋅ ⎢⎜
η real =
2⋅ η p⋅
=
2
2
⎛ Vj ⎞
⎛ Vj ⎞
g⋅ h
⎜ V ⎟ − 1 + 2 ⋅ 2 + Kin + Kout⋅ ⎜ V ⎟
⎝
A ⎠
⎝
A ⎠
VA
and substitute μ for VA/Vj ...
η real =
VA
Vj
⎡⎛
⋅ ⎢⎜ 1 −
⎣⎝
VA ⎞
1 VA⎤
⎟ − CD⋅ ⋅ ⎥
Vj
2 Vj
⎠
⎦
2
2
⎛ VA ⎞
⎛ VA ⎞
g⋅ h
1−⎜
+ 2⋅
+ Kin⋅ ⎜
⎟
⎟ + Kout
2
⎝
Vj ⎠
⎝
Vj ⎠
Vj
1
2 ⋅ η p ⋅μ ⋅ ⎡⎢( 1 − μ )
− CD⋅ ⋅μ⎤⎥
2 ⎦
⎣
2
(
)
μ ⋅ Kin − 1 + 1 + Kout + 2 ⋅
g⋅ h
Vj
2
=
1
2 ⋅ η p ⋅μ ⋅ ⎡⎢( 1 − μ )
− CD⋅ ⋅μ⎥⎤
2 ⎦
⎣
2
(
)
1 + Kout − μ ⋅ 1 − Kin + 2 ⋅
g⋅ h
Vj
and the quasi propulsive coefficient
is then ..
⎡⎛ Vj
⎞
1⎤
1
2 ⋅ η p ⋅ ⎢⎜
− 1⎟ − CD⋅ ⎥
2⋅μ ⋅ ⎡⎢( 1 − μ )
− CD⋅ ⋅μ⎥⎤
V
2
1−t
1−t
2 ⎦
⎣⎝ A
⎠
⎦
⎣
ηD =
⋅η ⋅
=
⋅η ⋅
2
2
2
g⋅ h
1−w p
1−w p
1 + Kout − μ ⋅ 1 − Kin + 2 ⋅
⎛ Vj ⎞
⎛ Vj ⎞
g⋅ h
2
⎜ V ⎟ − 1 + 2 ⋅ 2 + Kin + Kout⋅ ⎜ V ⎟
Vj
⎝
A ⎠
⎝
A ⎠
VA
as from above ...
⎞
1⎤
2 ⎡⎛ Vj
net thrust power
PT_net = Tnet⋅V = m_dot⋅ VA ⋅ ⎢⎜
− 1⎟ − CD⋅ ⎥
2
⎣⎝ VA
⎠
⎦
(
)
2
first some comments to relate to previous lecture/notes version and Wärtsilä paper
with Kout = 0 (N.B. this just means lumping all the pressure losses into a factor of 1/2* ρ*VA^2 and
accounting for a drag increase due to the inlet ...
⎡⎛ Vj
ηD =
1−t
1−w
⎞
1⎤
− 1⎟ − CD⋅ ⎥
2
⎣⎝ VA
⎠
⎦
2⋅η p ⋅ ⎢⎜
⋅η p ⋅
this is the form previously
2
⎛ Vj ⎞
g⋅ h
⎜ ⎟ − 1 + 2⋅ 2
+ K
⎝ VA ⎠
VA
and ... with CD = 0 and assuming h = 0 (i.e. head loss is small compared to other terms ...
this is the form in the paper with
ηD =
1−t
1−w
⋅η p ⋅
2⋅μ ⋅ ( 1 − μ )
2
(
)
1 + Kout − μ ⋅ 1 − Kin
Kout = φ = nozzle_loss_coefficient
at this point, assuming K, C D, and ηp are constant, could differentiate wrt V j/VA (or
μ) and determine V j/VA for max propulsive coefficient, but minimum weight usually
pump background (Wislicenus)
example
Δp out_loss
1
2
Kin = ε = inlet_loss_coefficient
determines parameters.
Kout =
⋅ρ⋅Vj
Δp in_loss
Kin =
1
2
⋅ρ⋅VA
2
2