Enhanced disk actuator modeling
Methods, systems, and apparatus, including computer programs encoded on computer storage media, for modeling turbine parameters. One of the methods includes obtaining, along multiple points of a blade of a turbine from a minimum radius rmin of the blade to a maximum radius rmax of the blade, lift coefficients C yi and drag coefficients C xi . At the multiple points of the blade from rmin to rmax, corresponding components of an upstream fluid flow velocity vector u h,Ri and u φ,Ri and components of a downstream fluid flow velocity u h,Li and u φ,Li are obtained. Averaged directions β i of the upstream and downstream fluid flow velocity vectors are computed using the components of the upstream fluid flow velocity vector u h,Ri and u φ,Ri and the components of the downstream fluid flow velocity u h,Li and u φ,Li . The total torque M of the turbine is computed including summing, from rmin to rmax, (C xi sin β i +C yi cos β i ).
1. A computer-implemented method comprising:
obtaining a plurality of parameters of a design of a blade wheel of an axial flow turbo machine, including lift coefficients and drag coefficients for the blade wheel for each of a plurality of angles of attack α i and for each of a plurality of cross sections r j of the blade wheel; from a minimum radius r min of the blade wheel to a maximum radius r max of the blade wheel;
obtaining mass jump conditions, momentum jump conditions, and energy jump conditions of a two-dimensional blade element model of the blade wheel, wherein the mass jump conditions, momentum jump conditions, and energy jump conditions each specify a relationship between upstream flow field parameters and downstream flow field parameters of the blade wheel;
computing upstream flow field parameters and downstream flow field parameters that satisfy the mass jump conditions, momentum jump conditions, and energy jump conditions of the two-dimensional blade element model;
computing momentum jump values across the swept area of the blade wheel using the computed upstream flow field parameters and the computed downstream flow field parameters that satisfy the mass jump conditions, momentum jump conditions, and energy jump conditions of the two-dimensional blade element model;
computing one or more turbine parameters for the axial flow turbo machine including computing an approximation of total torque for the design of the blade wheel by aggregating the momentum jump values across the swept area of the blade wheel; and
outputting the design of the blade wheel based on the computed one or more turbine parameters.
2. The method of claim 1 , wherein computing the momentum jump values from the computed upstream flow field parameters and the computed downstream flow field parameters comprises:
computing averaged directions β i of velocity components of the upstream flow field parameters and the downstream flow field parameters; and
computing each momentum jump values m i according to:
m i =q·σ·[C D sin β i +C L cos β i ], where q is an average dynamic pressure, C D are drag coefficients of the blade wheel, C L are lift coefficients of the blade wheel, and wherein σ is a solidity factor of the blade wheel.
3. The method of claim 1 , wherein computing the upstream flow field parameters and the downstream flow field parameters comprises computing upstream density ρ L , upstream pressure p L , upstream axial velocity (u h ) L , upstream azimuth velocity (u φ ) L , downstream density ρ R , downstream pressure p R , downstream axial velocity (u h ) R , and downstream azimuth velocity (u φ ) R , and further comprising:
computing an approximation of total drag X produced by the blade wheel according to:
X
=
-
2
π
∫
r
min
r
max
r
{
ρ
u
h
[
(
u
h
)
R
-
(
u
h
)
L
]
+
(
p
R
-
p
L
)
}
·
ⅆ
r
.
4. The method of claim 1 , wherein computing the upstream flow field parameters and the downstream flow field parameters comprises computing upstream density ρ L , upstream pressure p L , upstream axial velocity (u h ) L , upstream azimuth velocity (u φ ) L , downstream density ρ R , downstream pressure p R , downstream axial velocity (u h ) R , and downstream azimuth velocity (u φ ) R , and further comprising:
computing the approximation of the total torque M according to:
M
=
2
π
∫
r
min
r
max
r
2
ρ
u
h
[
(
u
φ
)
R
-
(
u
φ
)
L
]
·
ⅆ
r
.
5. The method of claim 4 , further comprising computing an approximation of total mechanical power W according to:
W=Ω·M,
wherein Ω is the rotational speed of the blade wheel.
6. The method of claim 4 , wherein computing the upstream flow field parameters and the downstream flow field parameters comprises computing upstream total enthalpy of the blade wheel H* L and downstream total enthalpy of the blade wheel H* R and further comprising:
computing an approximation of total mechanical power P* withdrawn from flow by the blade wheels in an incompressible fluid according to:
P
*
=
2
π
∫
r
min
r
max
r
ρ
u
h
(
H
R
*
-
H
L
*
)
·
ⅆ
r
.
7. The method of claim 1 , wherein the turbo machine is a wide angle ducted turbo machine, and further comprising numerically simulating the upstream flow field parameters and the downstream flow field parameters using representations of radial distributions of mass, momentum, and energy jumps across the swept area of the blade wheel derived with boundary-fitted coordinate transformation h(ξ, η) for coordinates ξ and η as h(ξ, η)=ξ, r(ξ, η)=r min (ξ)+η·[r max (ξ)−r min (ξ)].
8. A system comprising:
one or more computers and one or more storage devices storing instructions that are operable, when executed by the one or more computers, to cause the one or more computers to perform operations comprising:
obtaining a plurality of parameters of a design of a blade wheel of an axial flow turbo machine, including lift coefficients and drag coefficients for the blade wheel for each of a plurality of angles of attack α i and for each of a plurality of cross sections r j of the blade wheel from a minimum radius r min of the blade wheel to a maximum radius r max of the blade wheel;
obtaining mass jump conditions, momentum jump conditions, and energy jump conditions of a two-dimensional blade element model of the blade wheel, wherein the mass jump conditions, momentum jump conditions, and energy jump conditions each specify a relationship between upstream flow field parameters and downstream flow field parameters of the blade wheel;
computing upstream flow field parameters and downstream flow field parameters that satisfy the mass jump conditions, momentum jump conditions, and energy jump conditions of the two-dimensional blade element model;
computing momentum jump values across the swept area of the blade wheel using the computed upstream flow field parameters and the computed downstream flow field parameters that satisfy the mass jump conditions, momentum jump conditions, and energy jump conditions of the two-dimensional blade element model; and
computing one or more turbine parameters for the axial flow turbo machine including computing an approximation of total torque for the design of the blade wheel by aggregating the momentum jump values across the swept area of the blade wheel; and
outputting the design of the blade wheel based on the computed one or more turbine parameters.
9. The system of claim 8 , wherein computing the momentum jump values from the computed upstream flow field parameters and the computed downstream flow field parameters comprises:
computing averaged directions β i of velocity components of the upstream flow field parameters and the downstream flow field parameters; and
computing each momentum jump values m i according to:
m i =q·σ·[C D sin β i +C L cos β i ], where q is an average dynamic pressure, C D are drag coefficients of the blade wheel, C L are lift coefficients of the blade wheel, and wherein σ is a solidity factor of the blade wheel.
10. The system of claim 8 , wherein computing the upstream flow field parameters and the downstream flow field parameters comprises computing upstream density ρ L , upstream pressure p L , upstream axial velocity (u h ) L , upstream azimuth velocity (u φ ) L , downstream density ρ R , downstream pressure p R , downstream axial velocity (u h ) R , and downstream azimuth velocity (u φ ) R , and wherein the operations further comprise:
computing an approximation of total drag X produced by the blade wheel according to:
X
=
-
2
π
∫
r
min
r
max
r
{
ρ
u
h
[
(
u
h
)
R
-
(
u
h
)
L
]
+
(
p
R
-
p
L
)
}
·
ⅆ
r
.
11. The system of claim 8 , wherein computing the upstream flow field parameters and the downstream flow field parameters comprises computing upstream density ρ L , upstream pressure p L , upstream axial velocity (u h ) L , upstream azimuth velocity (u φ ) L , downstream density ρ R , downstream pressure p R , downstream axial velocity (u h ) R , and downstream azimuth velocity (u φ ) R , and wherein the operations further comprise:
computing the approximation of the total torque M according to:
M
=
2
π
∫
r
min
r
max
r
2
ρ
u
h
[
(
u
φ
)
R
-
(
u
φ
)
L
]
·
ⅆ
r
.
12. The system of claim 11 , wherein the operations further comprise computing an approximation of total mechanical power W according to:
W=Ω·M,
wherein Ω is the rotational speed of the blade wheel.
13. The system of claim 11 , wherein computing the upstream flow field parameters and the downstream flow field parameters comprises computing upstream total enthalpy of the blade wheel H* L and downstream total enthalpy of the blade wheel H* R and wherein the operations further comprise:
computing an approximation of total mechanical power P* withdrawn from flow by the blade wheels in an incompressible fluid according to:
P
*
=
2
π
∫
r
min
r
max
r
ρ
u
h
(
H
R
*
-
H
L
*
)
·
ⅆ
r
.
14. The system of claim 8 , wherein the turbo machine is a wide angle ducted turbo machine and wherein the operations further comprise:
numerically simulating the upstream flow field parameters and the downstream flow field parameters using representations of radial distributions of mass, momentum, and energy jumps across the swept area of the blade wheel derived with boundary-fitted coordinate transformation h(ξ, η) for coordinates ξ and η as h(ξ, η)=ξ, r(ξ, η)=r min (ξ)+η·[r max (ξ)−r min (ξ)].
15. A computer program product, encoded on one or more non-transitory computer storage media, comprising instructions that when executed by one or more computers cause the one or more computers to perform operations comprising:
obtaining a plurality of parameters of a design of a blade wheel of an axial flow turbo machine, including lift coefficients and drag coefficients for the blade wheel for each of a plurality of angles of attack α i and for each of a plurality of cross sections r j of the blade wheel from a minimum radius r min of the blade wheel to a maximum radius r max of the blade wheel;
obtaining mass jump conditions, momentum jump conditions, and energy jump conditions of a two-dimensional blade element model of the blade wheel, wherein the mass jump conditions, momentum jump conditions, and energy jump conditions each specify a relationship between upstream flow field parameters and downstream flow field parameters of the blade wheel;
computing upstream flow field parameters and downstream flow field parameters that satisfy the mass jump conditions, momentum jump conditions, and energy jump conditions of the two-dimensional blade element model;
computing momentum jump values across the swept area of the blade wheel using the computed upstream flow field parameters and the computed downstream flow field parameters that satisfy the mass jump conditions, momentum jump conditions, and energy jump conditions of the two-dimensional blade element model; and
computing one or more turbine parameters for the axial flow turbo machine including computing an approximation of total torque for the design of the blade wheel by aggregating the momentum jump values across the swept area of the blade wheel; and
outputting the design of the blade wheel based on the computed one or more turbine parameters.
16. The computer program product of claim 15 , wherein computing the momentum jump values from the computed upstream flow field parameters and the computed downstream flow field parameters comprises:
computing averaged directions β i , of velocity components of the upstream flow field parameters and the downstream flow field parameters; and
computing each momentum jump values m i according to:
m i =q·σ·[C D sin β i +C L cos β i ], where q is an average dynamic pressure C D are drag coefficients of the blade wheel, C L are lift coefficients of the blade wheel, and wherein σ is a solidity factor of the blade wheel.
17. The computer program product of claim 15 , wherein computing the upstream flow field parameters and the downstream flow field parameters comprises computing upstream density ρ L , upstream pressure p L , upstream axial velocity (u h ) L , upstream azimuth velocity (u φ ) L , downstream density ρ R , downstream pressure p R , downstream axial velocity (u h ) R , and downstream azimuth velocity (u φ ) R , and wherein the operations further comprise:
computing an approximation of total drag X produced by the blade wheel according to:
X
=
-
2
π
∫
r
min
r
max
r
{
ρ
u
h
[
(
u
h
)
R
-
(
u
h
)
L
]
+
(
p
R
-
p
L
)
}
·
ⅆ
r
.
18. The computer program product of claim 15 ,
wherein computing the upstream flow field parameters and the downstream flow field parameters comprises computing upstream density ρ L , upstream pressure p L , upstream axial velocity (u h ) L , upstream azimuth velocity (u φ ) L , downstream density ρ R , downstream pressure p R , downstream axial velocity (u h ) R , and downstream azimuth velocity (u φ ) R , and wherein the operations further comprise:
computing the approximation of the total torque M according to:
M
=
2
π
∫
r
min
r
max
r
2
ρ
u
h
[
(
u
φ
)
R
-
(
u
φ
)
L
]
·
ⅆ
r
.
19. The computer program product of claim 18 , wherein the operations further comprise computing an approximation of total mechanical power W according to:
W=Ω·M,
wherein Ω is the rotational speed of the blade wheel.
20. The computer program product of claim 18 , wherein computing the upstream flow field parameters and the downstream flow field parameters comprises computing upstream total enthalpy of the blade wheel H* L and downstream total enthalpy of the blade wheel H* R and wherein the operations further comprise:
computing total mechanical power P* withdrawn from flow by the blade wheels in an incompressible fluid according to:
P
*
=
2
π
∫
r
min
r
max
r
ρ
u
h
(
H
R
*
-
H
L
*
)
·
ⅆ
r
.
21. The computer program product of claim 15 , wherein the turbo machine is a wide angle ducted turbo machine and wherein the operations further comprise:
numerically simulating the upstream flow field parameters and the downstream flow field parameters using representations of radial distributions of mass, momentum, and energy jumps across the swept area of the blade wheel derived with boundary-fitted coordinate transformation h(ξ, η) for coordinates ξ and η as h(ξ, η)=ξ, r(ξ, η)=r min (ξ)+η·[r max (ξ)−r min (ξ)].