THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS
345 E. 47th St., New York, N.Y. 10017
The Society shall not be responsible for statements or opinions advanced in
papers or discussion at meetings of the Society or of its Divisions or Sections,
or printed in its publications. Discussion is printed only if the paper is published in an ASME Journal. Papers are available from ASME for 15 months
after the meeting.
Copyright © 1993 by ASME
Printed in U.S.A.
PREDICTION OF THE TRANSIENT THERMODYNAMIC RESPONSE
OF A CLOSED-CYCLE REGENERATIVE GAS TURBINE
T. Korakianitis, J. I. Hochstein,* and D. Zou
Department of Mechanical Engineering
Washington University
St. Louis, Missouri
ABSTRACT
This paper presents instantaneous-response and transient-flow component models for the prediction of the transient response of gas turbine cycles. The component models are based on applications of the
principles of conservation of mass, energy, and momentum. The models
are coupled to simulate the system transient thermodynamic behavior, and used to predict the transient response of a closed-cycle regenerative Brayton cycle. Various system transients are simulated using: the instantaneous-response turbomachinery models coupled with
transient-flow heat-exchanger models; and transient-flow turbomachinery models coupled with transient-flow heat-exchanger models. The
component sizes are comparable to those under consideration for the
solar-powered Space Station (radial turbomachinery), but the models
can easily be expanded to other applications with axial turbomachinery. An iterative scheme based on the principle of conservation of
working-fluid mass in the system is used to compute the mass-flow
rate at the solar-receiver inlet during the transients. In the process
the mass-flow rate of every component at every time step is also computed. Representative results of different system models are compared
and discussed.
NOMENCLATURE
A
C
D
h
I
k
L
Nits
P
Q
r R
t
heat transfer area
specific heat capacity
heat capacity rate C
diameter of a component
heat-transfer coefficient
rotating-unit inertia
heat conductivity of the wall
length of a component
mass
number of heat transfer units
pressure
energy transferred as heat
radius
(universal gas constant)/(molecular weight)
time
Currently at Memphis State University, Memphis, TN 38152
T
U
temperature
absolute flow velocity
V
volume
W, VI/
work, power
heat-exchanger effectiveness
total-to-total polytropic efficiency
(emissivity) x (Stefan-Boltzmann constant)
a
angular velocity
fZ
Subscripts
1, 2, ... , 6 thermodynamic-cycle points
a, b
fluids in heat exchangers
input to alternator shaft
al
compressor
cv
control volume
cd
cold fluid
friction
fr
ht
hot fluid
element counter in difference equations
in
into component
middle of component
max
maximum
min
minimum
of
out of component
at constant pressure (c5 )
P
r
radiation
shaft power to alternator
sink (temperature)
sk
sl
eutectic salt (temperature)
T
total property
turbine
tp
impeller tip
at constant volume (c,)
w
wall
Cartesian directions
x, y
Superscript
counter in time-difference equations
n
II
primary and secondary coolant loops
non-dimensional quantity (non-dimensionalized
with values shown in Table 1)
Presented at the International Gas Turbine and Aeroengine Congress and Exposition
Cincinnati, Ohio May 24-27, 1993
This paper has been accepted for publication in the Transactions of the ASME
Discussion of it will be accepted at ASME Headquarters until September 30,1993
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/26/2015 Terms of Use: http://asme.org/terms
93-GT16
INTRODUCTION
Models for the steady-flow performance of various configurations
of gas-turbine engines have been presented in many previous studies
and with various intentions such as cycle optimization (for example,
Korakianitis and Wilson, 1992), or prediction of part-load performance
(for example, Korakianitis and Beier, 1992). The transient behavior
of turbomachinery components can be predicted with CPU-intensive
unsteady Euler and Navier-Stokes computer programs (for example,
Giles, 1990; Korakianitis, 1993). The time currently required for these
computational fluid dynamics (CFD) transient computations makes
them unsuitable for predicting the transient response of gas-turbine
components and power plants through speed, load, and mass transients.
waste heat
t
concentrator
ill
t
t
radiator
solar energy input
FC-75
coolant
T
gas
cooler
electrical
power output
receiver
Transient-flow models for components where the flow may be assumed incompressible (such as some heat exchanger passages) are easier to develop because of the constant-density assumption. Similarly
transient-flow models for ducts with compressible or incompressible
flow are easy to model by predicting with characteristics the time of
propagation of transients. Transient-flow models for axial and centrifugal compressor and turbine passages, where work is added to or extracted from the flow, are harder to develop. Non-CFD models suitable
for predicting the transient performance of compressors and turbines
fall into three main categories. The first category predicts the transient
performance by obtaining equations resembling transfer functions in
control theory using phenomenological analyses of component behavior (for example, Kuhlberg et al. 1969; Kalnitsky and Kwatny, 1981).
In the second category the unsteady conservation equations (mass,
momentum and energy) are written in a lumped-parameter approach
(or alternatively finite volumes are considered along a mid streamline)
through one or more components. Many of the models in the second
category use some assumption to minimize the effects of compressibility, or the effects of mass storage or mass depletion inside the component through the transient. For example, the system of equations
is solved assuming either incompressible fluid, or no storage of mass
in the component volumes through the transient, or another assumption that simplifies the transient for one or more of the flow properties
(for example, Adams et al., 1965; Corbett and Elder, 1974; Macdougal
and Elder, 1983). In the third category assumptions are made for the
delay of transport of perturbations from component inlet to exit (for
example, Ray and Bowman, 1976; Fink, Cumpsty and Greitzer, 1991).
All three categories may incorporate routines for predicting relations
between inlet and outlet properties from the component performance
maps.
--I alternator —
I
turbine
I
accumulator
comp essor
electronic
control unit
parasitic
load radiator
recuperator
Figure 1: Schematic of the solar-powered closed regenerative Brayton-cycle engine
SYSTEM DESCRIPTION
The important inputs to predict the transient performance of gas
turbines are the type and size of engine components, working fluid,
and inputs driving the transient. These will vary from application
to application. Simplified component and approximate temperatureentropy diagrams for the solar-powered closed regenerative Braytoncycle engine under consideration to provide electrical power to the
Space Station are shown in Figures 1 and 2. Approximate dimensions
and other information for the system are included in many NASA
and NASA-contractor reports such as the two reports by Rocketdyne
(1986, 1989). Some information used in the following is included in
the Appendix.
The purpose of this paper is to develop transient models of the second category (using the unsteady form of the conservation equations)
suitable for predicting the performance of gas-turbine components, and
to show how the models can be coupled to predict the transient performance of engines. The compressor and turbine models do not include assumptions (such as incompressible flow, constant density, no
mass storage through the transient, or no pressure variation through
the transient) that simplify the transient behavior for any of the flow
properties. The transient models for flow through radial impellers use
the unsteady forms of conservation of mass, angular momentum and
energy. These can easily be modified for flow through axial stages by
replacing the law of conservation of angular momentum with (the simpler equation of) the law of conservation of linear momentum through
the mid streamline of each blade row. As an application the models are used to predict the transient thermodynamic performance of
a regenerative gas-turbine engine that is approximately suitable for
providing power to the Space Station. Various combinations of component models are evaluated by predicting the transient performance
in constant-speed transients (suitable for alternator drives such as the
one for the Space Station), and variable-speed transients (suitable for
other drives for other applications).
The working fluid is a mixture of Helium and Xenon (He-Xe) gases.
Solar energy is collected by the concentrator, focused onto the phasechange eutectic salt (LiF—CaF2) in the receiver, and transferred to the
working fluid. Energy is rejected from the cycle through the gas cooler.
The gas cooler, in what is called the coolant loop, uses a liquid mixture
of 75% C2H3F30 and 25% H2O (called FC — 75) to reject energy via
the radiator to space. The centrifugal compressor, radial turbine, and
alternator rotate as a single unit at (approximately) constant speed.
The system rotates in space in and out of the shadow of the earth
with a period of approximately 90 minutes, implying transients for
the receiver and cooler. The receiver phase-change salt is used to
provide energy at almost-constant temperature to the cycle. These and
other system transients are handled by changing the mass-flow rate of
the working fluid through the components (by adding and subtracting
working fluid in the accumulator), and/or by shedding some of the
electrical power output in the parasitic-load radiator.
Power for the Space Station is partially provided by a photovoltaic
module, and partially by two regenerative Brayton cycles similar to
2
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/26/2015 Terms of Use: http://asme.org/terms
been obtained from similarly-sized turbocharger-type compressor and
turbine performance maps published by Heywood (1988). At designpoint operation tic = 0.916 and n t = 0.902. Wf r represents bearing
friction. The electrical power produced will be less than 1 ,i7, because
of the effects of alternator efficiency.
T [K]
1200 4 tur ine
1000
800
The transient radial compressor and turbine performance are modeled by considering the flow at three stations, at the inlet, middle and
outlet of the components. The whole component is considered as a
control volume for which the conservation of mass, angular momentum and energy are written as:
dm
recuperator high pressure side
600
2
We –Wt =
d(mc v T) m =
dt
dT,
dt
400
(2)
thin — ?hot
dt
d(rnflrl)
dt
Wc
±rill/r t — – —
P12
f2
n 2 r 2 dm + ornr 2 c1 C1 rht-2 24
dt
dt
(3)
(Mc p T)i n – (McpT) o t +W. – Wt
1
—[(titc,T) in – (Mc p T) ot
mcv
dm
+ViT` – cuTmdt
(4
)
where a component is either a compressor (with work W e going into
the control volume, and 1 ,17t = 0), or a turbine (with work Wt going
out of the control volume and W e = 0). The tip terms with "–" are
for compressors, and the terms with "+" are for turbines. The flow is
assumed going axially into the compressor impeller and going axially
out of the turbine impeller (no tangential momenta at these stations).
200
0
0.1
0.2
0.3
0.4
0.5
0.6
For axial components the unsteady form for the conservation of
angular momentum (Equation 3) is replaced with the unsteady form
for the conservation of linear momentum in each stator and rotor blade
row.
0.7
S [kJ/(kg K)]
Figure 2: Temperature-entropy diagram for the regenerative Brayton
cycle. The entropy value at station 1 is arbitrary
The equation of state for perfect gas is used for total (stagnation)
and static properties, the isentropic relation connecting total and static
properties, and the definition of total enthalpy (hr = h+U 2 12 T2 =
T + U 2 /(2 • c p )) are used at each station in the component. In order
to obtain closure of the models, the following assumptions connecting
properties at various locations and times are also used:
the one illustrated in Figures 1 and 2, each providing approximately
30 kW. The compressor-turbine-alternator units rotate at 32,000 rpm
(minimum 30,000 rpm, maximum 36,000 rpm). At the design point the
compressor and turbine pressure ratios are just below 2:1 (the turbine
pressure ratio is a little lower than the compressor pressure ratio due
to pressure losses), and the mass-flow rate is approximately 0.8 kg/s.
The compressor and turbine are small radial impellers.
=
mm(t) =
Although the following models are applied to the system illustrated
in Figure 1, the principles are general and they can be used in different
gas-turbine engines with different components.
172, =
pT,m(t)
COMPONENT MODELS
The working fluid is modeled as a perfect gas with constant c p =
520.4 J/(kg • K) and c„ = 312.2 J/(kg • K). It is assumed that the
compressor and turbine are adiabatic.
(PT2) 1R/(CP.n41
(TT5)
\.TT4/
Prl
p2, 5 )[R9t I CP]
k.PT4)
[TT i n (t) + TT,,,t(t)1
'
– [rhin(t)± th o t(t – At)]
2
PinVcv
(5)
TT m(t)
= PT,in(t)
'
[TT,i n (t)
licni(R'nen
1
TT m (t) IcPn`/R1
'
TT ,i n (t)
It is assumed that the efficiencies from the inlet to the middle of the
component are equal to the efficiencies from the inlet to the outlet of
the component. This is an approximation. (The efficiencies from the
inlet to the outlet of the component are read as functions of pressure
ratio, m and S/ from the non-dimensional performance maps). The
mass-flow rate through the inlet of the component at time t is assumed
equal to the average of the mass-flow rate through the inlet of the
component at the same time t and the mass-flow rate through the
outlet of the component at time t – At. This lagged assumption for the
mass-flow rate provides the required stability to the numerical scheme.
The steady state compressor and turbine performance are modeled
using the following equations:
TT1
1
PT,in(t) = PT,in(t) [
Compressor and Turbine
C
2
(1)
= fhtCp(TT4 TT5) thecp(TT2 TT1) Wfr
One additional word of caution is due here. This transient scheme
will predict an immediate response at the outlet of the compressor or
turbine due to a variation of properties at the inlet. This is correct only
if the time step is large enough for the disturbance to travel through the
where the efficiencies are read from compressor and turbine nondimensional performance maps typical for small radial impellers. The
compressor and turbine performance maps used in this study have
3
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/26/2015 Terms of Use: http://asme.org/terms
In the receiver the salt temperature (T ,t = Th t ) remains approximately constant during normal operation, so that Cra t is small. (In the
following we are using variations in salt temperature to study system
transients. Transients in sink temperature, or a combination of such
transients could also be used). In the recuperator Cht = Cad, so that
Cra t = 1. In the gas cooler Cra t has a value smaller than unity. The
values of c, Pitt, and C ra t for design-point operation of the receiver,
recuperator and gas cooler are given in the Appendix. The pressure
drops in the heat-exchanger passages are modeled as a function of
working-fluid properties, flow arrangement, and heat transfer characteristics using models presented in Kays and London (1984). Similarly
the pressure drops in other system ducts are modeled as a function of
the local properties in the ducts.
fluid a
Ta, i_i
Ta,i Ta,i+1
T w , 1 _ 1 Tw , 1 Ti.,,i +1
Tb,i—i
41.■sm
Tb,i
Tb,i+1
For transient performance the counter-flow and cross-counter-flow
heat exchangers are modeled by dividing the control volume into sections along the flow direction, with three elements in each section for
fluids a and b and the wall w, as illustrated in Figure 3. Fluid b may be
flowing in the counter-flow or cross-counter-flow direction with respect
to fluid a. It is assumed that the net energy transfer by convective
transport dominates the conduction by diffusion between adjoining
fluid elements. The transient continuity equation for fluids a and b
along the control volumes i can be written as:
fluid b
Figure 3: Heat-exchanger finite-difference scheme
impeller passage and reach the outlet, thus imposing a minimum timestep limit to the numerical scheme. The resultant mixture of algebraic
and differential equations is a "stiff" system of equations, so that its
integration in time imposes a maximum time-step limit. The time step
used in the numerical integration to predict engine transients must be
between these two limits.
dm a
at
am b
Equation 3 can be used to model: a constant-speed variable-work
component (elf I dt = 0 and clY17 I dt
0); a constant-work variablespeed component (dfl I dt 0 and cli;IT a I dt = 0); and a component
in which the inter-dependence of (W,S1) can be used to identify the
system transient.
dt
=
(7)
rha,i
=
The transient energy balances can be written as:
a(rnc.T)a
= (Ah) a (T. Ta) +
at
Heat Exchangers
a (Thc,T) a dx
(Ah) o (Ta — T.) + (Ah)b(Tb — T.)
w
(mc)waT =
a(mc„T)b
The steady-state performance of heat exchangers (receiver, regen— (Ah)b(T. Tb) + a (thcpT)o dx
ax
erator and gas cooler) is evaluated using the method of the number
at
of heat transfer units (Nt., Kays and London, 1984). Heat-exchanger
effectiveness (depending on heat-exchanger purpose) is defined by one
of:
At
By writing the partial differences as finite differences and letting superscript n represent time steps, Equations 8 are manipulated into:
Tn • — Tn7 1 1
Cht(Tht,in Tht,ot)
(rncb)a,s[ Cmdh(Tht,in Ted,ot)
The number of heat transfer units Nsu are defined as:
Cmin
maz
At
where T,a , is a suitably defined mean temperature difference (such as
the logarithmic mean temperature difference). The above are manipulated into:
Tht,in (1 — E Cit ) (Tht , in — Tcd,in)
Tht,ot =
Tca,in — (1 — E) (Tht,in Ted,in)
Ah
(Ah)b,i [
n
-
,
(9)
-
The pressure drops in the heat-exchanger passages during transient
flow are modeled assuming an equilibrium state at every time step
(similar to the corresponding pressure drops during steady flow). The
pressure distribution is assumed linear along the flow passages. The
equation of state for perfect gas is used to find the density in each
elemental volume from the local pressure and temperature, and the
mass of fluid in each elemental volume and at each time is found from
Ced
1 — exp[— b a (1 — Crat)]
1 — Crat exp[—Ntu (1 — Crat)]
C
and Crat =
(Ah)a,i [T;Z i —
(it can be shown analytically that the temperature variations in time
dominate the temporal changes in mass-flow rates). These equations
are rearranged to produce a matrix equation of the form [A] {T} = {B},
where {T} is the vector of unknown temperatures, which is solved using
a Gaussian elimination method with scaling and partial pivoting.
Q = AhAT„,, t = C ht(Tht,In Tht,ot) = C cd(T ed,of — cd ,irt) •
Ted,ot =
+(rncp )b,i [Tri+1 — Tri]
flow arrangement)
and
Cmin
At
Tri — T;7 1 ]
(mc.)b,i[ '
'
In general it is possible to express
f (Nt.,
w't
Tn-71.1
-F(Ah)b,i
Nta f (A, h, Cmin)•
Ns. =
= (Ah) a ,i [77„1 , i — Tan, i]
+(rncp ) a ,i [TZt_1 — 7T,i]
T n
Cmin(Tht,in Tcd,ot)
(77142,,,[
E=
"
Ccd(Tcd,ot Ted,in)
C
(8)
(6)
mi = PiVco •
The recuperator, receiver and gas-cooler transient models are de-
Cmaz
4
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/26/2015 Terms of Use: http://asme.org/terms
If' and " denote the primary and secondary loops respectively, Equations 11 for an arbitrary element i in the primary loop can be written
as (explicit formulation):
primary loop, fluid a
aT;
T`
1 [(Ah) o
( 7nevYai L
at j i
T"
(12)
+ (ii2 cP) so,i(TalT ,i)]
rani
V -
1
[(Ah)",,, i ( o
(mcY„ ,i
[ aT
secondary loop, fluid a
(T,„' -
-
41
- T:1
+ (a A r ).
.[(
T'a,i-1 T'a,i T'a,i+1
_ ( kit, ),
T'W7-2—,i- T'W,/ T'W,i+1
AY
_
-
\ x
_
w,i
Similarly, Equations 11 for an arbitrary element i in the secondary
loop can be written as (explicit formulation):
T"
w i T"w,i+1
w,i-1 T"
7
rani
T"
a,i T"a,•i +1
a,i-1 T"
at
( mc„1 ) ,
_ Tzi) ]
at
( mc1yz,,,
_ n, o +
rani
I. J i
Figure 4: Radiator finite-difference scheme
+ (aAr)w,i ( rsk
TV's)]
rA z y
veloped from Equations 9, with appropriate variations in constants to
account for design details. Some information about these components
is given in the Appendix. In the recuperator the values of c, and c„ for
the He-Xe mixture are used for both fluids a and b. In the gas-cooler
fluid a is the He-Xe mixture, and fluid b is liquid FC - 75, for which
cp ,b = co = cb = 1.0207 kJ/(kg • K) and pb = 1682.1 kg/m 3 (assumed
constant). In the receiver fluid a is the He-Xe mixture, and fluid b is
the LiF - CaF2 eutectic salt.
kA y y i
1
(mc) w ,i
-
Az w,2
\ AY
(13)
(T "
-
Steady-flow and transient pressure drops in the receiver and other
duct-type passages are modeled as described in the heat exchangers.
Rotor Inertia
Radiator
For constant-speed transients the inertia of the rotating unit is
not needed in the models. For variable speed operation the dynamic
equation of the unit is used to evaluate the instantaneous acceleration:
The radiator consists of two coolant loops, a primary and a secondary one, as shown in Figure 4. The steady-flow model for each
radiator loop is developed by considering differential slices along its
length. The heat rejected from each differential slice is evaluated by
assuming that: heat is radiated along the length of the radiator, but
not from its two ends; heat radiates according to the Stefan-Boltzmann
law; and the emissivity of the surface is constant. The liquid FC - 75
temperature at the radiator outlet is evaluated by integrating the resulting equation along the radiator length.
d (111 2 )
Tit 2
{3aDL1 } (-113)
.
mc, + Ts
at
az„
(712 w
= (Ah) a
M
D
Ta)+
a ( rhcpT) a dx
ax
aT„
ax
—
Wal
(14)
The above component models are coupled to form the system model.
The rate of change of energy in the system is equal to the rate of collected solar energy, minus the output power, minus the rate of waste
heat radiated to space. The transient formulations conserve energy,
and therefore at every instant the temperature-entropy diagram for
the cycle is a closed shape similar to the one shown in Figure 2. The
exact location of the diagram changes at different time instants, as
driven by transients in eutectic-salt and sink temperatures, or other
system transients. The new position of the diagram is found by iteratively evaluating the correct mass-flow rate through the components at
each instant. The control equation for the overall system is the equation of conservation of mass through the transient. The total mass of
working fluid and coolant in the system are assumed constant. The
resultant transients can also be manipulated by specifying the mass of
(11)
aTti,
-
r Wal
"' Wf r
(10) SYSTEM MODEL
= (Ah) a (To, - Tu,) crAr(TA Ti!)
-kAz—
-
K2
Wt
where for the compressor-alternator-turbine rotating unit I = 23.39 x
10 -6 kg • m 2 . The instantaneous acceleration SI is integrated in time
to give the rotational speed of the unit at the next time instant.
where a = (emissivity) x (Stefan-Boltzmann constant) = (0.9) x
(56.70 x 10 -9) = 51.03 x 10 -9 W/(m 2 .K 4 ). For the transient model it is
assumed that the sink temperature is a function of time, but constant
along the length of the panel, in addition to the above assumptions
for steady flow. The model is developed by dividing the radiator into
segments as shown in Figure 4. The energy balance for the fluid a and
the wall w become:
a(mc„T) a
— Wca =
- PV't -
4 = crri DdL = -thc p dT
To t =
= We
kAy—
ay
5
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/26/2015 Terms of Use: http://asme.org/terms
1.03
1.01 771
1.02
T* 1.00
1.01
0.99
T* 1.00
1.002
0.99
p4
1.02
1.000
1.01
p*, 1.00
0.998
0.99
1.02
1.02
1.01
1.01
1 1; 1.00
,
ris; 1.00
0.99
0.99
0.98
0.0
1.05
1500 2000
t [.s]
1.04
1.03
Figure 6: Instantaneous-response turbomachinery / transient heat exchangers. System response to sinusoidal variation in Ta i.
1.02
1.01
Table 1: Values at the beginning of transients
14; 1.00
1,1;
0.99
0.98
0.0
Variable
500
1000
1500
2000
2500
3000
3500
4000
Tek
t [S]
T4
P4
th3
12
Figure 5: Instantaneous-response turbomachinery / transient heat exchangers. System response to double step in Tg.
-
the working fluid in the system as a function of time through the period
of the transient (modeling the working-fluid mass in the accumulator
during the transient).
Value
186.0 K
1042.0 K
1012.2 K
354.8 kPa
0.835 kg/s
3351 rad/s
(32,000 rpm)
41.1 kW
perature Ts k constant. The recuperator, receiver and gas cooler are
divided into eight elements, while the radiator is divided into ten elements (for each fluid each wall section). Initially the models have been
run to an initial state of full convergence by specifying no changes
in the variable driving the transients. This operating point is identical (within numerical accuracy) to the corresponding point using the
steady-state models. At that operating point key parameters of the
cycle have the values shown in Table 1. The figures show the timeevolution of various parameters (nondimensionalized with their values
at the beginning of the transients, shown in Table 1).
The state of the system at all times is represented by the combination of {T,p, Till at all junction points between the components.
(During transients the mass flow rate is different at each flow station).
Transient iterations start from an initial steady state. Starting from
an imposed transient for properties at one or more junctions between
components, the He-Xe mass flow rate between the recuperator highpressure-side outlet and receiver inlet (rh3 at station 3 in Figure 2)
is iteratively increased and decreased using an under-relaxation technique until the conservation of total mass m in the system is satisfied,
ie the sum of the mass in all components is (within a user-specified
accuracy) equal to the specified total mass of the working fluid. The
under-relaxation factor is increased and decreased throughout the iterations by monitoring the effect of changes of m3 on total system
mass.
Figure 5 has been produced with a system model using the transient
heat exchanger models coupled to the instantaneous-response turbomachinery models (Equations 1). It shows select output of the system
response to a double-step transient of 20 K (about 2%) in salt temperature over 1080 seconds (18 minutes, about 20% of the 90 minute
orbital transient). The time step in the calculations is 1 s. As seen
from the figure, it takes over 18 minutes for the system to reach a new
steady state. T4 settles to its new value in about 800 s, but p 4 , th 3 and
W, require about 2000 s after the step to reach a new steady state. The
final steady state after 4000 s is identical (within numerical accuracy)
to the initial state at the beginning of the transient.
TRANSIENT PERFORMANCE
Many transients for the powering unit of the Space Station, including: 90 minute (orbital) sinusoidal transients in salt and sink temperature (Ti and TA); loss of coolant pumps; loss of some of the radiator
panels; and others, have been simulated using the above models. The
following system transients are illustrative of typical results. The cycle parameters and component sizes used in this paper are similar, but
not identical to those under consideration at the latest design iteration. The component sizes, pressures, temperatures, and power levels
for the powering unit of the Space Station have changed as the design
of the unit is evolving.
Figure 6 has also been produced with a system model using the
transient heat-exchanger models coupled to the instantaneous-response
turbomachinery models. It shows select output of the system response
to a sinusoidal transient in salt temperature of amplitude 5 K (0.47%)
over 628 seconds (about 11.6% of the 90 minute orbital transient).
The time step in the calculations is 1 s. As seen from the figure, the
relatively long thermal transient behavior of the heat exchangers causes
the transients to reach the various components at different phases. The
For illustration purposes the following transients are driven by
varying the eutectic salt temperature TA, while keeping the sink tern-
6
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/26/2015 Terms of Use: http://asme.org/terms
1.01 T;
1.05
T* 1.00
T; 1.00
0.99 1.003
0.95
1.00002
1.002
77 1.0001
1.001
1.0000
1.000
1.00004
0.999
\
1.00002
P:
0.998
p: 1.00000
JI
1.002
0.99998
1.001
1.00004
tit; 1.000
1.00002
0.999
fr4 1.00000
0.998
0.99998
1.01
1.0004
Cr 1.00
/
W: 1.0002 •
1.0000 /
0.00
0.99
4i7,* \
0.50
1.02
1.00
1.50
1.01
1 i; 1.00
0.99
2.00
,
t [s]
0.98
instantaneous-response turbomachinery models
0.0 500 1000 1500 2000 2500 3000 3500 4000
transient turbomachinery models
t
Figure 7: System response to sinusoidal variation in T3 1. Comparison
of system models with instantaneous-response turbomachinery (dashed
lines) and with transient turbomachinery models (solid lines)
[8]
Figure 8: System response to sinusoidal variation in T,1. Comparison
of constant and variable rotational speed models.
peaks in T4 are about 60 s later than the corresponding peaks in TEL ,
and they are of slightly lower amplitude. The sinusoidal variations in
P4, rhz and W, (after an initial system transient during the first two
or three cycles) follow a sinusoidal pattern similar to that for T4.
transient and instantaneous-response turbomachinery. In transients
with longer periods the slow thermal response of the heat exchangers
dominates the transient turbomachinery behavior.
Figure 7 indicates that the system responds at a frequency similar
to the driving frequency, and that the peaks in 7'4 are about 0.168 s
later than the peaks in Tat• (The delay between the peaks is a function
of the driving frequency). The 7'4 responses of the two system models
are identical. The sinusoidal patterns in T4, P4, ms and W., are similar, and there is little difference in p4 and ril3 between the two models.
Because the overall system takes about 2000 s to reach a new steady
state (Figure 5), once T4, p4, Th3 and W, have started increasing (first
cycle in Figure 7), they drop very little below their initial values within
the 2 s shown in Figure 7. (They drop below their values at the beginning of the transient at later times). The response of W, (and other
related temperatures and pressures at the inlet and exit of the compressor and turbine) have substantially different amplitudes between the
instantaneous-response and transient-turbomachinery models. In gas-
Figure 7 shows select output of the response of two different system models to a sinusoidal transient in salt temperature of amplitude
50 K (4.7%) over 0.628 seconds. The first system model (dashed lines)
has been produced with transient heat-exchanger models coupled to
the instantaneous-response turbomachinery models. The time step in
these calculations is 0.0025 s. The second system model (solid lines)
has been produced with transient heat exchanger models coupled to
the transient turbomachinery models (Equations 2, 3, 4 and 5). The
time step in these calculations is 0.000025 s. Using working-fluid properties and component sizes, it was estimated that acoustic disturbances
travel through the compressor passages in 0.000015 s, and through the
turbine passages in 0.000020 s. The short period of the transient has
been used in this case to show the differences in system output from the
7
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/26/2015 Terms of Use: http://asme.org/terms
Engineering Science, Vol. 16, No. 6, pp. 377-385.
Giles, M. B. (1990), "Stator/rotor interaction in a transonic turbine",
AIAA Journal of Propulsion and Power, Vol. 6, No. 5, Sep-Oct., pp. 621-627.
Fink, D. A., Cumpsty, N. A., and Greitzer, E. M. (1991), "Surge dynamics
in a free-spool centrifugal compressor system", Transactions of the ASME,
Journal of Turbomachinery (in print), ASME paper 91-GT-31.
Heywood, John B. (1988), "Internal combustion engine fundamentals",
McGraw-Hill.
Kays, W. M and London, A. L. (1984), "Compact heat exchangers",
McGraw-Hill.
Kalnitsky, K. C. and Kwatny, H. G. (1981), "A first principles model
for steam turbine control analysis", Transactions of the ASME, Journal of
Dynamic Systems, Measurement, and Control, Vol. 103, pp. 61-68.
Korakianitis, T. (1993), 'On the propagation of viscous wakes and potential flow in axial-turbine cascades", Transactions of the ASME, Journal
of Turbomachinery, Vol. 115, No. 1, pp. 118-127, Jan. (ASME paper 91-GT373).
Korakianitis, T. and Wilson, D. G. (1992), "Models for predicting the
performance of Brayton-cycle engines", Transactions of the ASME, Journal
of Engineering for Gas Turbines and Power (in print), ASME paper 92-GT361.
Korakianitis, T. and Beier, K. J. (1992), "Investigation of the part-load
turbine cycles with heat exchangers the long thermal-response times
of the heat exchangers dominates the turbomachinery response. It is
concluded that the transient turbomachinery models are required for
fast transients only. Such high-frequency transients may be introduced
by the electronic control systems (such as some of those required to
control the system by changing the distribution of mass between the
system and the accumulator). The transient turbomachinery models
are also required in simple cycles (those without heat exchangers).
Figure 8 has been produced with a system model using the transient
heat exchanger models coupled to the instantaneous response turbomachinery models. It shows select output of the system response to a
sinusoidal transient in salt temperature of amplitude 5 K (0.47%) over
628 seconds (about 11.6% of the 90 minute orbital transient). The
input to this transient is the same as that shown in Figure 6. The
time step in the calculations is 1 s. Two different transient system
models are shown. The first system model (dashed lines) simulates
constant-speed operation. The second system model (solid lines) simulates a system model with a power-speed law of the form W at = k11 3 .
The T4 responses of the two system models are identical. The sinusoidal variations in p4, rn3 and W, (after an initial system transient
during the first two or three cycles) follow a sinusoidal pattern similar to that for T4. The 1-2 variations in the variable-speed model are
0.25%. The amplitude of W, for the variable-speed model is less than
that for the constant-speed model because of the shape of the turbineefficiency contours (wider plateaus along constant-speed lines and narrower plateaus across the speed lines).
performance of two 1.12 MW regenerative marine gas turbines", Transactions
of the ASME, Journal of Engineering for Gas Turbines and Power (in print),
ASME paper 92 - GT - 86.
Kuhlberg, J. F., Sheppard, D. E., King, E. 0. and Baker, J. R. (1969).
"The dynamic simulation of turbine engine compressors", AIAA paper 69-
486.
Macdougal, I. and Elder, R. L. (1983), "Simulation of centrifugal compressor transient performance for process plant applications" Transactions of
the ASME, Journal of Engineering for Power, Vol. 105, pp. 885-890.
Ray, A. and Bowman, H. F. (1976), 'A nonlinear dynamic model of a oncethrough subcritical steam generator", Transactions of the ASME, Journal of
Dynamic Systems, Measurement, and Control, pp. 332-339.
Rocketdyne, (1986), "Space Station work package WP-04 power system
preliminary analysis and design document", Rocketdyne RI/RD85-320-2, December.
Rocketdyne, (1989), "Allied-Signal interim design review for the CBC /
PGS for the NASA Space Station Freedom", Rocketdyne report no. 41-9311,
November.
CONCLUSIONS
Steady-state and transient models suitable for predicting the behavior of gas-turbine components have been developed. The component models are based on the transient form of conservation of
mass, energy and momentum within each component. The component models have been coupled to simulate a closed-cycle regenerative
gas-turbine cycle with components similar to those under consideration for the powering unit of the Space Station. During transients the
cycle parameters of the overall system are iteratively evaluated. A converged solution for each time step is obtained by an iterative scheme
based on the principle of conservation of working-fluid mass in the
system. Various system transients have been studied. For illustration,
transients driven by step and sinusoidal variations in eutectic salt temperature (variations in energy input to the cycle), and under constant
and varying shaft speed, have been included. In gas-turbine cycles with
heat exchangers the long thermal-response times of the heat exchangers dominates the turbomachinery response. System models based
on instantaneous-response turbomachinery / transient heat-exchanger
component models can be used to predict low-frequency transients.
The transient turbomachinery models are required for higher-frequency
transients, such as those that may be introduced by the engine control system. The transient turbomachinery models are also required in
simple cycles (those without heat exchangers).
APPENDIX: DESCRIPTION OF COMPONENTS
(Note that the components of the Space Station are under development at the time of writing). The following information was used
to produce the results shown in this study. The receiver has 82 tubes
of Haynes 188 steel of outlet diameter 22.2 mm and active tube length
2.5 m. At design-point receiver operation c = 0.8730, Nt. = 2.0635,
and Cm , = 0.0000. The recuperator is a pure counterflow plate-fin unit
of length 0.37 m, width 0.32 m and height 0.49 m. The high-pressure
side has 62 sandwiches of fins, with 630 (0.15 mm thick) CRES 304L
fins per m, with height 3.2 mm. The low-pressure side has 63 sandwiches of fins, with 630 (0.15 mm thick) CRES 304L fins per m, with
height 3.9 mm. At design-point recuperator operation e = 0.9404,
Ntu = 15.7763, and Cm , = 1.0000. The gas cooler is an eight-pass
cross-counterflow plate-fin heat exchanger. The fin sandwiches are
rectangular offset, 2.3 mm high. The gas side has 470 (0.15 mm
thick) CRES 304L fins per m, with height 2.3 mm. The liquid side
has 790 (0.15 mm thick) CRES 304L fins per m, with height 1.9 mm.
At design-point gas-cooler operation e = 0.9430, Ntu = 6.4501, and
Cres t = 0.7428. The radiator has 8 aluminum 2 3 m x 8.0 m panels.
Key turbomachinery parameters are shown below.
ACKNOWLEDGEMENTS
Various aspects of this work were performed at Washington University between 1987 and 1991 with partial sponsorship from NASA
grant NAG-3-817 (technical officers: T. Peterson; and R. Slutz). In
addition to the authors, graduate students J. McTavish, A. Iqbal and
M. S. Ali contributed to the grant effort.
Variable
V,
REFERENCES
r e ap
re,m
)1,,in
Ac mt
.A 0 , 0,
Adams, J., Clark, D. R., Louis, J. R. and Spanbauer, J. P. (1965), "Mathematical modeling of once-through boiler dynamics", Transactions of the
IEEE, Power Apparatus and Systems, Vol. 84, No. 2, pp. 146-156.
Corbett, A. G. and Elder, R. L. (1974), 'Stability of an axial flow compressor with steady inlet conditions" I. Mech. E., Journal of Mechanical
Value
60 x 10 -6 m 3
0.065 m
0.037 m
3.60 x 10 -3 m 2
3.05 x 10 -3 m 2
2.50 x 10 -3 m 2
8
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/26/2015 Terms of Use: http://asme.org/terms
Variable
Vt
At,in
Value
100 x 10 -6 rn 3
0.095 m
0.053 m
4.00 x 10 -3 m 2
At,m
5.06 x 10 -3 m 2
A4,05
5.65 x 10 -3 m2
rt,tp
rt,m