i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 7 ( 2 0 1 2 ) 1 4 6 0 3 e1 4 6 1 1
Available online at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/he
Hydrogen tube vehicle for supersonic transport: 4. Hydrogen
propeller
Arnold R. Miller*
Supersonic Institute, 200 Violet St, Suite 100, Golden, CO 80401, USA
article info
abstract
Article history:
A hydrogen propeller is the method of propulsion of a conceptual supersonic vehicle that
Received 6 April 2012
operates within a hydrogen-filled tube at cruise speed of 1 km/s. Because Mach number
Received in revised form
governs formation of shock waves at the blade tips, the high sonic speed of hydrogen
19 June 2012
allows a rotational frequency 3.85 times faster than the same propeller operating in air
Accepted 19 June 2012
immediately outside the tube. RankineeFroude propulsive efficiency and e for a given
Available online 11 August 2012
vehicle Mach numberepropeller pitch and helix angle are invariant with respect to the
atmosphere. To achieve constant efficiency at a given thrust and for adequate acceleration,
Keywords:
the low density of hydrogen requires some combination of higher frequency, more blades,
Hydrogen
or larger diameter. The hydrogen propeller conceptual design employs 14 contra-rotating
Propeller
blades, 4.11 m diameter, and rotational frequency of 40.4 s1 at translational velocity of
RankineeFroude theory
970 m/s.
Supersonic transport
Copyright ª 2012, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights
Tube vehicle
1.
Introduction
Tube vehicles, for example, subway trains, provide benefits
that include operation independent of the weather, shortestdistance travel, avoidance of surface traffic, and groundlevel noise abatement. A large literature exists on the aerodynamics of vehicles operating in air-filled tubes [1e6].
Because of flow restriction in the gap between tube and
vehicle, a tube vehicle experiences higher drag than the same
vehicle operating in an open atmosphere [4]; to lower drag,
evacuated tubes with air pressure of 9 kPa have been
proposed [7].
One factor in realizing the potential of a tube vehicle is to
maximize tube cross-sectional area while minimizing infrastructure cost. Such a method has been proposed as
“conjoined tubes” [8]. Because a practical tube system must
allow parallel or bidirectional transit, the idea is to join
a bundle of circular-section tubes to form a single, fluted tube
reserved.
with grooves or waists along its length. Fig. 1 illustrates the
idea for three conjoined tubes.
Another new idea is to use hydrogen as the tube atmosphere
[9e11]. The conceptual supersonic hydrogen vehicle operates
within an enclosed, hydrogen-filled tube at cruise speed of 1 km/
s. To prevent leakage of air into the tube, hydrogen pressure is
minimally above outside air pressure, and the tube serves as
a phase separator rather than a pressure vessel. Within the tube,
a vehicle levitates above a guideway on a magnetic field or gas
film and uses propeller propulsion; it breathes fuel for fuelcell
power from the tube, stores liquid oxygen onboard, and collects
the product water as it operates [9].
Parameters associated with sonic speed and aerodynamic
drag can be very different inside the tube than outside. The
hydrogen vehicle has two Mach numbers: one with respect to
hydrogen inside the tube and one with respect to air outside. A
speed of Mach 0.74 inside (970 m/s) corresponds to Mach 2.8 for
a land vehicle immediately outside the tube and to Mach 3.3
* Corresponding author. Tel.: þ1 303 296 4218; fax: þ1 303 296 4219.
E-mail address: arnold.miller@vehicleprojects.com.
0360-3199/$ e see front matter Copyright ª 2012, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijhydene.2012.06.079
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Fig. 1 e Schematic of conjoined tubes: Three circularsection tubes joined along their lengths form a fluted tube
having three waists. Shown are the propeller disk of one
vehicle and the possible positions of two others (dashed).
for the same body hypothetically at 11,000 m, a typical cruise
altitude of transport airplanes. Thus, the vehicle can be
strongly supersonic outside while subsonic inside. For laminar
flow over a flat plate, skin-friction drag varies inversely with
the square root of the Reynolds number [11,12]. Assuming the
same speed and same characteristic length [4,5], the Reynolds
number for the hydrogen vehicle is half the number for the
land vehicle but twice that of the body at altitude.
A gas-filled tube, vis-à-vis an evacuated tube, offers the
potential efficiency, simplicity, and low infrastructure cost of
propeller propulsion. Propeller propulsion has been studied
for air-filled tube vehicles [3]. However, a hydrogen propeller
has not been studied or built, and no propeller-driven vehicle
in air could achieve a speed of 1 km/s. The high sonic speed
and low density of hydrogen make the design of a hydrogen
propeller distinct from that of air propellers; the linear nature
of a tube constrains multiple propellers to be contra-rotating
pairs or tandem pairs, one at either end of the vehicle. In
this paper, we investigate contra-rotating, tandem propellers
driving a hydrogen tube vehicle and propose a conceptual
propeller design based on the results.
2.
Thrust and power
We assume the main features of RankineeFroude momentum
theory [13,14] but apply it to contra-rotating, tandem propellers in a tube. A propeller is modeled as an ideal actuator disk,
equivalent to a propeller with an infinite number of blades [14],
which exhibits no resistance to gas flow, transfers all
mechanical energy to the gas, and experiences a uniform
distribution of gas pressure and velocity over its surface. As an
ideal actuator disk, a contra-rotating propeller has twice the
area of an actuator disk representing a single propeller. RankineeFroude theory assumes isentropic, incompressible, irrotational flow in an ideal gas. By the assumption of isentropic
flow, the theory ignores presence of the vehicle in the propeller
flows. In the case of a tube vehicle, by continuity, propeller
flow accelerates as it enters the gap between vehicle and tube,
but it equally decelerates as it exits. Thus, by the isentropic
assumption, we also ignore presence of the vehicle in the tube.
Flow velocity Vf through front disk Df (see Fig. 2) is derived
from the Bernoulli equation. To avoid violating the Bernoulli
assumption of adiabatic flow, the equation is separately
applied before the disk and after the disk. It can thus be shown
[13] that Vf equals the mean
1
Vf ¼ ðV þ Vs Þ
2
(1)
where V is flow velocity far ahead of the disk (i.e., vehicle
velocity, or initial inflow velocity, within the vehicle-fixed
frame of reference) and Vs is the fully-developed slipstream
velocity far behind the disk. The analysis based on the
Bernoulli equation is not changed by the fact that the flows
occur within a tube e as long as they are isentropic and
incompressible e and Eq. (1) applies equally to an actuator
disk within a finite-diameter, infinite-length tube as to one in
an open atmosphere. The tube may have any cross-sectional
geometry; e.g., it may be circular-section or fluted.
Following the RankineeFroude theory for a single disk, the
thrust and power of the front disk derives from Newton’s
second law. In unit time increment, mass flow mf ¼ rAVf,
where r is gas density and A is the area of the disk. Substitution of Eq. (1) into this expression gives
1
mf ¼ rAðV þ Vs Þ
2
(2)
for the mass flow through the front disk. Since thrust
Tf ¼ mf(VsV), substitution of Eq. (2) for mf gives
1 Tf ¼ rA Vs2 V2 ;
2
Vs V
(3)
as the thrust provided by the front disk of the tandem pair. In
unit time, power is the kinetic energy to change V to Vs, that is,
1
1
Pf ¼ mf Vs2 mf V2 . Substituting Eq. (2) into this expression
2
2
yields
Fig. 2 e Propulsion in a tube: Front disk Df and rear disk Dr are identical up to location. Possible streamline velocities are vehicle
velocity V, velocity through front disk Vf, and slipstream velocity from front disk Vs. Mean inflow velocity to rear disk, V0 (not
shown), is the weighted mean of V and Vs. The tube has unbounded length.
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1 Pf ¼ rA Vs2 V2 ðVs þ VÞ;
4
Vs V
(4)
as the slipstream power required by the front disk of the
tandem pair.
Thrust Tr and power Pr for the rear disk Dr differ from the
front disk because the initial inflow velocity V0 may be greater
than V, and its final slipstream velocity Vs0 may be greater than
Vs. Possible velocities of streamlines entering the rear disk are
illustrated in Fig. 2. Because the vehicle length is large, before
the rear disk is reached, we assume (a) the slipstream from the
front disk achieves its final velocity Vs and (b) complete mixing
of the flows occurs. By continuity, if a streamline with velocity
Vs misses the rear disk, a streamline of velocity V must replace
it. Hence, the initial inflow velocity to the rear disk is the
weighted mean
V0 ¼ ð1 wÞV þ wVs ;
0w1
(5)
where w is the probability that a streamline in the slipstream
from the front disk will intersect the rear disk. Weight
w¼
A
At
(6)
where A is the area of the rear actuator disk and At is the
cross-sectional area of the tube. We assume that Vs0 =Vs ¼ V0 =V
and hence the final slipstream velocity from the rear disk is
Vs0 ¼
V0
Vs
V
(7)
This assumption is equivalent to the standard Rankinee
Froude assumption of an inflow factor a [13,15], defined as
Vs ¼ V(1þ2a), a 0, where a is a constant1 positive real
number. If a is the same for the front and rear disks, we have
Vs0 ¼ V0 ð1 þ 2aÞ and Vs0 =Vs ¼ V0 (1þ2a)/V(1þ2a) ¼ V0 /V, which is
equivalent to (7).
Substitution of V0 for V and substitution of Vs0 for Vs into
Eqs. (3) and (4) give
ð1 wÞV þ wVs 2
1 ;
Tr ¼ rA Vs2 V2
V
2
0 w 1;
0 < V Vs
(8)
3
1 ð1 wÞV þ wVs
; 0 w 1; 0 < V Vs
Pr ¼ rA Vs2 V2 ðVs þVÞ
V
4
(9)
for the thrust and power, respectively, for rear disk Dr.
1
The standard assumption is that a is constant for all V.
However, because of the large speed range and the cubic dependency of slipstream power on V, a constant a is not realistic for
the hydrogen vehicle. For example, taking h ¼ 0.90, the Rankinee
Froude expression [13] for propulsive efficiency h ¼ 1/(1 þ a) yields
constant a ¼ 0.111. With an actuator disk area A ¼ Re ¼ 81.2 m2, as
developed in the section Calculations for Design, Eq. (4) calculates
1.7 GW for slipstream power at 970 m/s; even if we reduce A to that
of the airplane, 13.27 m2, the power is 270 MW. Rather than being
a constant, the inflow factor must be a decreasing function of V,
which amounts to Vs being a nonlinear function g of V, with
monotone decreasing slope. Derivation of Eq. (7) is valid because it
only assumes that each tandem propeller has the same function g
and same argument V.
3.
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Propulsive efficiency
The RankineeFroude propulsive efficiency h establishes an
upper bound on efficiency of propeller propulsion. Extension
of the definition to tandem propellers gives the following
equation for propulsive efficiency
h¼
V Tf þ Tr
Pf þ Pr
(10)
where Tf and Tr are the thrust provided by the front and rear
disks, respectively, and Pf and Pr are the slipstream power
required by the front and rear disks, respectively. Substitution
into Eq. (10) of Eqs. (3), (4), (8) and (9) gives
h¼
2V V3 þ V½ð1 wÞV þ wVs 2
;
Vs þ V V3 þ ½ð1 wÞV þ wVs 3
0 w 1;
0 < V Vs
(11)
as the propulsive efficiency for tandem disks in a tube; V can
validly assume the value zero in (11), but V and Vs cannot
simultaneously be zero. By inspection, because h is a continuous function of Vs and w, with V fixed, h converges to its
maximum value of unity as Vs / V and w ¼ 0.
Proposition: Efficiency h as described by (11) is a monotone
decreasing function of w (with fixed V and Vs), and independently, h is a monotone decreasing function of Vs (fixed V and
w).
Proof: For brevity, using the V0 notation as defined by Eq. (5),
Eq. (11) may be written as
h ¼ 41 $42 ¼
2V V3 þ VV02
$
Vs þ V V3 þ V03
(12)
where 41 and 42 are positive real numbers for all values of the
independent variables. It is obvious that 41 is a monotone
decreasing function of Vs (fixed V and w), and it is not false
that 41 is a monotone decreasing function of w (fixed V and
Vs). For factor 42, consider that V0 is a weighted mean and
hence V V0 Vs. It follows that, for fixed V, a non-negative
0
real number c exists such that V ¼ cV0 , c 1. Therefore, the VV 2
term in (12) becomes
VV02 ¼ ðcV0 ÞV02 ¼ cV03 ; c 1
(13)
and the second factor becomes
42 ¼
V3 þ cV03
; c1
V3 þ V03
(14)
Consider the continued inequality 0 V V0 Vs, with V
constant: With Vs fixed, by the properties of a weighted mean,
c decreases monotonically as w increases; with w fixed, c
decreases monotonically as Vs increases. Hence, noting that
the numerator of 42 is a monotone increasing function of c, it
follows that 42 is a monotone decreasing function of w, and 42
is a monotone decreasing function of Vs. Because positive real
numbers 41 and 42 each obey the conditions of the Proposition, their product h does also.
Corollary: With V and w fixed, h converges monotonically
to unity as Vs converges to V.
Proof: Let V and w be fixed. By the Proposition, if Vs
decreases monotonically, h increases monotonically. Since Vs
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is bounded below by V, and h is bounded above by unity, it
follows that h/1 monotonically as Vs/V.
Efficiency as defined by Eq. (11) is not explicitly a function
of r or A; however, for efficiency to increase and thrust to
remain constant, either r or A (or both) must also increase. It
is sufficient to show this by considering only the front disk:
Because efficiency is increased by the convergence Vs/V,
the factor ðVs2 V2 Þ in Eq. (3) simultaneously converges
monotonically to zero; hence, for thrust to remain constant,
the product rA must increase as efficiency increases. Likewise, for given thrust, speed, and density, efficiency is an
increasing function of A, and for given thrust, speed, and
area, it is an increasing function of r.
Fig. 3 compares h from Eq. (11), as a function of Vs, with
the standard RankineeFroude propulsive efficiency h0 ¼ 2V/
(VsþV) for a single disk [13]. With vehicle velocity fixed at
V ¼ 970 m/s and weight w ¼ 0.3, a reasonable value of w for
a fluted tube, Vs is varied from 970 m/s to 1020 m/s.
Because w > 0, the efficiency of the tandem disks is lower
than that of a single disk; however, the maximum absolute
difference in efficiency, over the range examined, is less
than 1%.
Because propulsive efficiency is maximized when Vs is
near V, we can derive an approximation from the limit of Eq.
(11) as Vs/V. Define
DV ¼ Vs V;
Vs V
(15)
then, substitution into Eq. (11) of Vs as thus defined gives, after
dropping terms containing DV2 and DV3, an expression
for lim h from which we establish the approximation
DV/0
2V
V þ wDV
hy
;
2V þ DV V þ 3= wDV
2
0 w 1;
(16)
which is valid when DV is small. With V ¼ 970 m/s and w ¼ 0.3,
the relative error is less than 0.25% for values of Vs up to
1120 m/s.
4.
Pitch and frequency
Because a propeller rotates simultaneously to translating,
a fixed point B on a blade traces out a helix as the vehicle
advances. Pitch p is the distance the point moves forward in
one revolution. Letting r be the orthogonal distance of B from
the rotational axis, the helix angle, or pitch angle, q(r) is the
angle between the helix and any edge of the right sections of
the cylinder, with radius r, on which the helix lies; the coarser
the helix, the larger the angle. Helix angle is given as [14]
qðrÞ ¼ arctan
p
2pr
(17)
Because pitch by definition is
p¼
V
f
(18)
where V is vehicle speed and f is propeller rotational
frequency, we have
qðrÞ ¼ arctan
V
2prf
(19)
for the helix angle. Ignoring angle of attack, if a blade’s nosetail line is twisted along its length according to Eq. (19), the
blade will strike the atmosphere by its edge, and most of its
drag will be skin-friction drag.
For a given propeller diameter, rotational frequency f is
limited by the formation of shock waves at the blade tips. The
tips experience the highest flow velocity on the vehicle. Tip
velocity Vtip is the vector sum Vtip ¼ VþVrot, where V is
translational velocity and Vrot is tangential velocity of the
rotating tip, and hence, tip speed is Vtip ¼ ðjVj2 þ jVrot j2 Þ1=2 .
Since tangential speed is Vrot ¼ pdf, where d is propeller
diameter, tip speed is
Vtip ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
V2 þ ðpdf Þ
(20)
Because formation of shock waves is governed by Mach
number, assuming the same blade shape, the acceptable
Mach number of the tips in hydrogen is the same as the
acceptable Mach number in air or any ideal gas. Consider two
gases: (a) hydrogen or (b) any ideal gas (referred to as the
“second gas”). Let fi be the rotational frequency and Vi be the
speed in the second gas. If the vehicles have the same Mach
number in their respective gases, then V ¼ sVi, where s is the
sonic ratio. Suppose the propeller frequencies are likewise
related by f ¼ sfi. Then, from (20), the ratio of tip speeds is also
in the ratio of s:
Vtip
¼
Vtip; i
Fig. 3 e Propulsive efficiency: Efficiency h versus
slipstream velocity for a RankineeFroude single disk and
tandem disks. Initial inflow velocity is fixed at 970 m/s.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi
ðsVi Þ2 þ pdsfi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi ¼ s
Vi2 þ pdfi
(21)
where Vtip, i is the tip speed in the second gas. Eq. (21) shows
that the Mach numbers of the blade tips in the two gases are
identical, and for a given propeller diameter, blade shape, and
vehicle Mach number, a hydrogen propeller may rotate faster
than a propeller in air or any ideal gas by the sonic ratio s.
Denoting the sonic ratio as s0 when the air propeller is operating in the standard atmosphere at zero altitude, we have
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s0 ¼ 3.85. If the comparison is made with an air propeller
operating at higher altitude than zero, then s > s0.
With given vehicle Mach number and propeller diameter,
the pitch and helix angle are invariant with respect to the
identity of the atmosphere. Let pi be the pitch of a propeller in
the second gas. Because V ¼ sVi and f ¼ sfi, the pitch p of the
hydrogen propeller is
V sVi Vi
p¼ ¼
¼
¼ pi
sfi
fi
f
Calculations for design
Based on the theoretical results above and recommendations
for contra-rotating propeller design [16], we will develop
a conceptual design by extrapolating the propeller of an
airplane to operation in a hydrogen atmosphere. The
Bombardier Dash 8 Q400 [9] uses the Dowty R408 propeller
[17], a six-bladed, carbon-fiber composite propeller with
scimitar-shaped blades. We assume that the hub diameter,
blade shape, and propeller diameter of the hydrogen propeller
are the same as the air propeller and that it uses two contrarotating rows of blades. To be determined are the rotational
frequency and the number of blades. The blade will be twisted
to satisfy Eq. (19).
To accommodate the potential for higher rotational
frequency, as well as adding more blades and two rows of
contra-rotating blades, consider a transformation that
converts the geometric rotor area Ra of the air propeller to an
equivalent, hypothetical rotor area Re of the hydrogen
propeller. The transformation is defined as
Re ¼ ðFCNÞRa
Vehicle
Q400 airplane
Hydrogen
(22)
Hence, pitch of the hydrogen propeller is the same as the pitch
in any ideal gas; likewise, the helix angle is invariant.
5.
Table 1 e Frequency and tip speed.a
(23)
where F is the ratio of thrusts obtainable (with fixed flow
velocity) by increasing rotational frequency of the hydrogen
rotors versus the air propeller; C is the ratio of thrusts
obtainable by increasing the number of blade-rows from one
to two (contra-rotating); and N is the ratio of thrusts by
increasing the mean number of blades from six to (n1 þ n2)/2,
where n1 is the number of blades in the first row, n2 is the
number in the second, and N ¼ (n1 þ n2)/12. For a given flow
velocity, each factor increases mass flow through the rotor.
Analogous to airplane takeoff, rotor frequency at startup is
higher than at cruise. Table 1 lists the Q400’s known
frequencies at takeoff and cruise and computes the corresponding tip Mach numbers using Eq. (20). The table also
displays the frequencies of the hydrogen vehicle at V ¼ 0 and
V ¼ 970 m/s and the corresponding tip Mach numbers
matching those of the airplane. At V ¼ 0, the hydrogen
propeller can rotate at 65.5 s1 compared to 17.0 s1 for the air
propeller; at cruise, it can rotate at 40.4 s1 compared to
14.17 s1. Hence, the ratio of thrusts is F ¼ 3.85 at startup and
F ¼ 2.85 at cruise.
The thrust provided by the two rotors could, in principle, be
equal and thence C ¼ 2. However, because of gearbox requirements [16], the thrust provided by the two rotors may not be
Speed
m/s
Frequency
s1
Tip
Speedb
m/s
Tip
Speed
Mach
0
185
0
970
17.0
14.17
65.5
40.4
219.8
260.3
846.9
1102
0.646
0.841c
0.646
0.841
a Propeller diameter 4.11 m for both vehicles.
b From Eq. (20).
c At 7620 m altitude in standard atmosphere.
equal. As an empirical model for the effect on thrust of adding
a second, contra-rotating row of blades, we note that the front
row of the contra-rotating propellers of the Tupolev Tu-95
(n1 ¼ n2 ¼ 4) absorbs 54.4% of the power for the pair and the
rear row absorbs 45.6% [18]. Therefore, we let C ¼ 100/54.4 ¼ 1.84.
What remains to be determined is the ratio N.
To determine ratio N that provides adequate startup thrust,
set V ¼ 0 in Eq. (3), as well as in the analogous equation for
thrust of the airplane propeller, and assume that ideal actuator disk area is proportional to physical rotor geometric area.
Then, the ratio of thrusts yields
2
Re
T0 ra0 Vsa
¼
Ra Ta0 r VsH
(24)
where T0 is the maximum static thrust of the hydrogen
vehicle, Ta0 is the maximum static thrust of the airplane, ra0 is
the density of air at zero altitude in the standard atmosphere,
Vsa is the slipstream velocity of the airplane at maximum
static thrust, and VsH is the slipstream velocity of the
hydrogen vehicle at maximum static thrust. Assuming at
V ¼ 0 that Vsa ¼ VsH, and using transformation (23), Eq. (24)
reduces to
FCN ¼
T0 ra0
Ta0 r
(25)
Static thrust of the hydrogen vehicle is computed from
Newton’s second law. Because the vehicle must reach 970 m/s
in a reasonable elapsed time, its mass must be similar to an
airplane’s. We estimate a mass from the 74-passenger Q400
airplane [9]. Based on the weight breakdown of a similar-sized
airplane [19], the Q400 without its wing and tail should weigh
approximately 87% of its total takeoff mass of 29.3 t. Assuming
the mass, in total, of all other components and fluids of the
two vehicles is equal, the mass at V ¼ 0 of the hydrogen
vehicle is taken as 26 t. If the acceleration is a decreasing
linear function of time t, starting from initial value a0 ¼ 2.4 m/
s2, it can be shown2 that 970 m/s can be attained in 13.5 min,
which is deemed a satisfactory elapsed time. Static thrust of
the hydrogen vehicle is therefore T0 ¼ (26 103 kg)(2.4 m/
s2) ¼ 62.4 kN, or 31.2 kN per propeller.
2
Let the linear acceleration function be a0(1t/teq), where teq is
the timeto reach equilibrium
speed of 970 m/s. Integration gives
VðtÞ ¼ a0 t 1= t2 =teq . Letting t ¼ teq, this equation yields
2
teq ¼ 808 s.
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Table 2 e Data for Q400 Airplane.
Parameter
Value
a
Altitude at cruise, m
Density of air at cruise,b kg/m3
Density of air at zero altitude
in Std. atmosphere,c kg/m3
Mass at takeoff,a t
Power, equivalent, at cruise,d MW
Power, shaft, normal takeoff,e MW
Power, shaft, maximum at cruise, MW
Propeller diameter,a m
Propeller geometric disk area, m2
Propeller rotational frequency
at takeoff,f s1 (rpm)
Propeller rotational frequency
at cruise,f s1 (rpm)
Speed of vehicle at cruise,g m/s (Mach)
7620
0.548
1.2256
29.3
7.30
6.83
5.89
4.11
13.27
17.0 (1020)
Symbol
ra
ra0
Pa
2Pmax
Ra
14.17 (850)
185 (0.60)
Va
a Ref [21].
b Ref [13], Appendix 2; by linear interpolation.
c Ref [13], Appendix 2.
d Equivalent power (shaft power plus exhaust power): maximum
shaft power at cruise divided by the observed ratio 0.806 of shaft
power to equivalent power [22].
e 2 3.415 MW [21].
f 1020 rpm at takeoff and 850 rpm at cruise [23].
g Cruise speed of 667 km/h [21].
We will determine that 14 blades are adequate at cruise by
showing that the hydrogen propeller’s power loading, power
per unit area, falls within the propfan loading limits
proposed by Strack et al. [16], namely, 0.31e0.41 MW/m2.
With the parameters F, C, and N discussed above and the
geometrical rotor area Ra ¼ 13.27 given in Table 2, we have
the equivalent rotor area Re ¼ 81.2 m2. Hence, the recommended power loadings give an acceptable power range of
27e37 MW per contra-rotating propeller for the hydrogen
vehicle. The range of power estimates determined in the
Appendix, 19e38 MW, falls almost entirely within or below
this range, which supports a propeller design having 14
blades.
The result of the analysis is a hydrogen propeller conceptual design with rotational frequency of 40.4 s1 at cruise, 14
contra-rotating blades, hub diameter of 0.375 m, and outer
diameter of 4.11 m. The analysis of Strack et al. [16] found that
contra-rotating propeller efficiency is maximized when rotor
spacing is minimized. The minimum spacing with the R408
blade is 0.37 m, occurring when the blades are feathered; we
use a rotor spacing of 0.43 m. From Eq. (18), the pitch of the
hydrogen propeller is 24.0 m, which compares to 13.1 m for the
Q400 airplane propeller. Fig. 4 is a CAD drawing of the
hydrogen propeller conceptual design, and Table 4 collects
relevant design parameters for the hydrogen vehicle and
airplane.
The static thrust for the airplane is estimated by a model of
the maximum static thrust of a propeller as a function of its
maximum shaft power [20]
2=3
Ta0 ¼ 0:90ð2ra0 Ra Þ1=3 Pmax
(26)
where Pmax is its maximum shaft power per rotor. From the
data in Table 2, with Pmax ¼ 3.415 MW, Eq. (26) yields
Ta0 ¼ 65.15 kN.
With F ¼ 3.85, C ¼ 1.84, the two thrust values just
computed, and hydrogen density from Table 3, we solve (25)
for N to give N ¼ 1.005. Since n1 ¼ 7 and n2 ¼ 6 correspond to
N ¼ 1.083, a 13-bladed propeller may be adequate. However, if
the true static thrust of the airplane is lower than calculated
by Eq. (26), the error would make the value of N calculated via
(25) too small. With 14 blades, (25) calculates that the true
thrust Ta0 could be as low as 39.3 kN. Propulsive efficiency at
cruise will be higher for 14 blades than 13 because h is an
increasing function of A (see Propulsive Efficiency) and likewise an increasing function of Re. Hence, for a design margin
and increased efficiency, the conceptual design uses 14
blades, seven in each row.
Table 3 e Data for Hydrogen Vehicle.
Parameter
a
3
Density of hydrogen, kg/m
Mass at startup, t
Speed of vehicle at cruise, m/s (Mach)b
3
Value
Symbol
0.0824
26
970 (0.74)
r
V
a At tube pressure 101.3 10 Pa and temperature 298e300 K [24].
b For 1310 m/s speed of sound [24].
Fig. 4 e Hydrogen propeller conceptual design. Fourteen
contra-rotating blades of 4.11 m diameter are shown with
a nose-tail helix angle q(r) designed for a cruise speed of
970 m/s at rotational frequency of 40.4 sL1. As per Eq. (19),
q(0.375) [ 84.4 at the blade root, and q(2.055) [ 61.7 at the
tips. Rotor spacing is 0.43 m, and the person is 1.7 m tall.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 7 ( 2 0 1 2 ) 1 4 6 0 3 e1 4 6 1 1
Table 4 e Comparison of Parameters.
Parameter
Mass at startup, t
Number of blades per propeller
Number of propellers
Number of passengers
Pitch at cruise, m
Propeller diameter, m
Propeller frequency
at startup, s1
Propeller frequency
at cruise, s1
Vehicle speed at cruise,
m/s (Mach)
a
b
c
d
6.
Q400 airplane Hydrogen vehicle
29.3a
6
2 (lateral)b
74b
13.1
4.11
17.0
26
14
2 (tandem)
74c
24.0
4.11
65.5
14.17
40.4
185 (0.60)d
970 (0.74)
At takeoff [21].
Ref [21].
Ref [9].
At 7620 m altitude [21].
Discussion of results
For a given vehicle Mach number, the propeller pitch and helix
angle are invariant with respect to the atmosphere. These
parameters, the same in hydrogen as in air or any ideal gas,
are invariant because both vehicle velocity and rotational
frequency are multiplied by the sonic ratio, which therefore
cancels. This notwithstanding, the design pitch of the
hydrogen propeller, at 24.0 m, is nearly twice the 13.1 m pitch
of the Q400 airplane propeller model. Higher pitch ultimately
obtains because the cruise speed V of the hydrogen vehicle in
Eq. (18) is much higher than that of the given airplane e 970 m/
s versus 185 m/s at 7620 m altitude e likewise, higher Mach
number has the effect of lowering acceptable propeller
frequency f to avoid shock waves.
For constant thrust, propulsive efficiency is an increasing
function of the product rA, and therefore high efficiency in the
low density of hydrogen requires a relatively large actuator
disk area. Our argument is based on the monotone convergence Vs/V, as described in the Corollary, and on thrust Eq.
(3). (For a different argument e but from the thrust equation e
that propeller efficiency increases as A increases, see Clancy
[15]). When statistical weight w ¼ 0, the rear disk is equivalent
to the front disk, and the efficiency equation reduces to
h0 ¼ 2V/(VsþV), the standard RankineeFroude propulsive
efficiency for a single disk [13]. Efficiency is given exactly by
approximation (16) in the limit as DV/0, and the approximation agrees well with Eq. (11) for values of Vs reasonably
near V. Efficiency in any case is slightly lower than a Rankinee
Froude single disk because tandem disks increase the inflow
velocity and slipstream kinetic energy for the rear disk. The
Proposition and Corollary apply to tandem propellers in
hydrogen, air, or any ideal gas.
Because Mach number governs formation of shock waves,
the sonic ratio s allows a hydrogen propeller rotational
frequency 3.85 times faster than the same propeller, at the
same translational speed, operating in air immediately
outside the tube. The frequency of 40.4 s1 is designed for the
cruise speed of 970 m/s, and as per Eq. (20), a higher frequency
14609
of 65.5 s1 is feasible at V ¼ 0. The ratio F of cruise design
frequencies of the hydrogen vehicle and airplane, 40.4 s1/
14.17 s1 ¼ 2.85, is below the sonic ratio s ¼ 4.23 for air at an
altitude of 7620 m because the vehicle Mach numbers, as
required by Eq. (21), are not the same, with the hydrogen
vehicle’s being higher.
The conceptual design of a hydrogen propeller is based on
transforming the rotor area of an air propeller to a hypothetically larger rotor area that is generated by increasing the
rotational frequency, the number of blade-rows, or the number
of blades per row. Although the geometrical area of the two
rotors is the same, each of these three factors e F, C, and
N e increases thrust by increasing mass flow at a given flow
velocity, and the transformed area behaves as though it were
a larger geometrical area. Fourteen contra-rotating blades
rotating at 65.5 s1 provide sufficient static thrust and at
40.4 s1 provide a cruise power-loading within recommended
limits for propfans.
7.
Conclusions
To achieve satisfactory propeller static thrust, power loading,
and efficiency, the low density of hydrogen requires some
combination of higher frequency, more blades, or larger
diameter. This conceptual design uses a standard airplane
propeller diameter but uses more blades rotating at higher
frequency. Tandem propellers are necessary for the tube
vehicle to generate sufficient thrust.
High speed of sound and low density of hydrogen play
key roles in hydrogen propeller characteristics. With such
a high cruise speed, propeller thrust to achieve practical
vehicle acceleration may be the limiting factor in passenger
carrying-capacity of a hydrogen tube vehicle. The vehicle
could consist of several passenger cars in a train [9];
however, achievable acceleration may limit the number of
cars or, alternatively, the maximum speed. At the maximum
speed of 1 km/s, passenger carrying-capacity could be
augmented by multiple discrete vehicles operating in
parallel in a fluted tube. High sonic speed allows a 3.85-fold
higher rotational frequency than an air propeller immediately outside the tube, and high frequency partially offsets
the need for an awkwardly large-diameter propeller. A
higher rotational frequency, however, requires superior
dynamic balance and greater strength.
This analysis discovers characteristics of a hydrogen
propeller e potential high rotational frequency and invariance
of pitch and helix angle e that suggest it can be developed as
a practical device. As the next step of development, we
propose that the work justifies implementing experiments on
a physical prototype.
Acknowledgements
I thank my research assistant, Valerie A. Traina, for assistance
with the literature references. Thanks to the technical staff of
Vehicle Projects Inc for reviewing the manuscript in a series of
review sessions and to Dan Lassiter for confirming the
14610
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 7 ( 2 0 1 2 ) 1 4 6 0 3 e1 4 6 1 1
derivations of Eqs. (8), (9), (11) and (16). This research was
supported in part by Vehicle Projects Inc.
f
fi
mf
N
Appendix
We estimate the power of the hydrogen vehicle by considering
the Q400 fuselage as a model of the hydrogen-vehicle fuselage.
When an airplane is minimized for total drag, parasitic drag
equals induced drag [19], and half its cruise power Pa is due to
induced drag. Hence, Pa0 ¼ ½ Pa, where Pa0 is the power with
zero angle of attack, i.e., the zero-lift condition. Start by eliminating induced drag at cruise speed Va in air by instantaneously reducing the effective angle of attack to zero, and then
transfer the airplane to an open hydrogen atmosphere with
zero lift and new speed V. Since, in any case, the airplane’s
power is directly proportional to the product rV3, we have
PH0 ¼ Pa0
rV3
1 rV3
¼ Pa
ra Va3 2 ra Va3
(A.1)
for the power of the transferred airplane. From the parameters
in Tables 2 and 3, Eq. (A.1) yields PH0 ¼ 79 MW for the airplane
at zero lift in an open hydrogen atmosphere at 970 m/s.
Consider the airplane as constructed from a complete,
enclosed fuselage body to which wings and tail have been
added. Let Sfus be the surface area of this enclosed body, and
let S be the exposed area of the entire airplane (minus
propeller blades). From a commercial CAD model of the Q400,
we estimate [25] Sfus/S ¼ 0.48. Assume that the hypothetical
power required by the fuselage body itself, Pfus, is proportional
to this ratio of surface areas, and hence
Sfus
Pfus yPH0
S
(A.2)
Eq. (A.2) computes Pfus ¼ 38 MW for the airplane fuselage as
a model of the hydrogen-vehicle fuselage in the open
hydrogen atmosphere at 970 m/s, which corresponds to
19 MW per contra-rotating propeller.
This estimate omits power to overcome increased drag due
to operation within a tube and to provide vehicle levitation;
however, these adjustments should be smaller than the basic
power estimate itself. The power due to operation within
a tube can be made as small as desired by increasing tube
cross-sectional area At; if levitation were magnetic, based on
the known power of 1.7 kW/t for the Transrapid train [26],
power consumption should be only 44 kW. Hence, we estimate that the hydrogen vehicle power P, per contra-rotating
propeller, falls within the range 19 < P 38 MW at 970 m/s.
Nomenclature
A
At
a0
B
C
c
F
area of ideal actuator disk of hydrogen vehicle, m2
cross-sectional area of hydrogen tube, m2
initial acceleration of vehicle, m s2
fixed point on a propeller blade
ratio of thrust for two contra-rotating rotors versus
one rotor
real constant
n1
n2
P
Pa
Pa0
Pf
Pfus
PH0
Pmax
Pr
p
pi
Ra
Re
r
S
Sfus
s
s0
Ta0
Tf
Tr
T0
t
teq
V
V
V0
DV
Va
Vf
Vi
Vrot
Vrot
Vs
Vs’
Vsa
VsH
Vtip
Vtip
Vtip, i
w
ratio of thrust obtainable by increasing rotational
frequency of hydrogen rotor
propeller rotational frequency in hydrogen, s1
propeller rotational frequency in any ideal gas, s1
mass flow through front actuator disk, kg/s
ratio of thrust due to increasing the number of
blades in rotors
number of blades in first row
number of blades in second row
power per propeller of the hydrogen vehicle at cruise
(equilibrium), MW
power of airplane at cruise, MW
power of airplane with zero-lift, zero angle of attack
power required by the front actuator disk, W
power attributable to airplane fuselage
power of an airplane at zero lift in hydrogen
maximum shaft power of airplane
power required by the rear actuator disk, W
pitch of propeller
pitch of propeller in an ideal gas
geometric rotor area of airplane, m2
transformed, equivalent rotor area in hydrogen
vehicle, m2
radius, m
area measure of vehicle; surface area of entire
airplane
surface area attributable to an airplane’s fuselage
ratio of speed of sound in hydrogen to speed of
sound in a specified gas, sonic ratio
ratio of sonic speeds when air is the standard
atmosphere at zero altitude
maximum static thrust of airplane, N
thrust provided to the vehicle by the front actuator
disk, N
thrust provided to the vehicle by the rear actuator
disk, N
maximum static thrust of hydrogen vehicle, N
time
time to reach equilibrium speed
speed of vehicle, m/s; initial inflow velocity for front
actuator disk, m/s
velocity of vehicle, m/s
initial flow velocity for rear actuator disk, m/s
velocity increment imparted by actuator disk, m/s
speed of airplane at cruise, m/s
flow velocity through front actuator disk, m/s
speed of vehicle in any ideal gas, m/s
tangential speed of rotating propeller blade tips, m/s
tangential velocity of rotating propeller blade tips, m/s
final slipstream velocity from front actuator disk, m/s
final slipstream velocity from rear actuator disk, m/s
slipstream velocity of airplane at maximum static
thrust, m/s
slipstream velocity of hydrogen vehicle at maximum
static thrust, m/s
propeller blade-tip speed, m/s
propeller blade-tip velocity, m/s
propeller tip velocity in any ideal gas, m/s
probability (statistical weight) that a streamline from
front disk will intersect rear disk
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 7 ( 2 0 1 2 ) 1 4 6 0 3 e1 4 6 1 1
Greek letters
a
RankineeFroude inflow factor, dimensionless
front actuator disk
Df
rear actuator disk
Dr
d
propeller diameter, m
h
propulsive efficiency of tandem actuator disks in
a tube, dimensionless
RankineeFroude propulsive efficiency (w ¼ 0),
h0
dimensionless
q
helix (pitch) angle, r
density of hydrogen in the tube, kg m3
ra
density of air at cruise altitude of airplane, kg m3
ra0
density of air at sea level, kg m3
41
positive real factor of h
positive real factor of h
42
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