A
simple model of an industrial turbocharger has been
developed to illustrate key concepts for modeling
systems involving more than one turbomachinery component.
These concepts include the calculation of net torque,
the calculation of the shaft speed that balances
torque, and shaft speed transients based on transient
equations of motion (namely, T = I*dw/dt).
In
the case of a turbocharger, a turbine provides the
torque to drive a compressor. There is no gear box
in this system, though representations of gearing,
gear losses, bearing losses, etc. do not represent
significant modeling challenges if the data (gear
ratios, torque coefficients, etc.) is readily available.
Similarly, starter motors and loads (e.g., generators)
can be modeled as well.
The
concepts and modeling methods developed are
applicable to other systems involving multiple, linked
turbomachines including:
- brayton
cycles, including jet engines
- rankine
cycles
- liquid
rocket turbopumps
The
turbocharger system, including the engine definition
and component map information for both turbine and
compressor, were provided by SoftInWay.
System
Description
The
figure below represents the system schematic.
Air
at ambient pressure and 20°C enters the compressor
at point 1, and is discharged at point 2 (nominally
3.5:1 pressure ratio), the engine inlet. The nominal
(design point) flow rate into the compressor is 10.47
kg/s, and the nominal shaft speed is 16000 rpm. The
engine is modeled as a source of hot air (with combustion
products neglected for simplicity), with a constant
flow rate of 0.52 kg/s.
The
engine representation is very simple: it adds 5.93MW
of energy to the air. The nominal flow rate through
the turbine, from point 3 to 4, is the sum of the
flows through the compressor and engine: 10.99 kg/s.
The
nominal exhaust pressure of the turbine is 1.9MPa.
The exhaust system resistance (from turbine outlet
to ambient) is estimated to be equivalent to a K-factor
loss of about 16.8 at the dynamic head corresponding
to the turbine exhaust. (This exhaust system resistance
value will be varied parametrically later to test
sensitivity).
The
compressor is a centrifugal compressor, with an inlet
meanline diameter of 230mm, a rotor outer diameter
of 474mm, and a stator outer diameter of 676mm. The
turbine is a radial design, with a stator inlet diameter
of 709mm, rotor inlet diameter of 541mm, and a meanline
outlet diameter of 252mm.
Basic
Model Description
The
model was developed using Thermal Desktop® and
FloCAD®.
The lump identifiers shown below correspond to the
state points defined above, except that an additional
state (#5) has been added as the engine exhaust source,
and another (#6) as the turbine exhaust point. The
temperature and pressure of the engine mass source
(#5) has been set to be equal to that of point #2,
the compressor outlet. The exhaust to the atmosphere
(#6) is the same state as the inlet (#1).
The
compressor was modeled using the performance map
information (flow and efficiency versus pressure
ratio) provided by SoftInWay (as outputs from their
turbomachinery design software AxSTREAM). FloCAD-produced
plots of this information are provided below. The
basis for compressor performance are total temperatures
and pressures at both the inlet and outlet (total-total).
Performance Map Input
for Compressor
Similarly,
the performance of the turbine is plotted below.
The basis for the turbine is total-static, which
was defined as part of the TURBINE device information.
The
compressor inlet pressure was specified, though the
compressor outlet, turbine inlet and turbine outlet
pressures are to be calculated. A small duct (representing
a negligible resistance) is used to interconnect
the compressor outlet with the turbine inlet (at
which point warm engine exhaust joins the flow).
A K-factor based LOSS path was used at the turbine
outlet, exhausting into a FLUINT plenum representing
ambient. The K-factor for this loss was set to a
symbol and register named ExhaustK, which was initially
equal to 16.8.
Since
the flow rate from the engine was specified, an MFRSET
(set mass flow rate) path was used to inject the
flow rate into lump #3. The inlet to the engine,
a plenum, was set with variable temperature and pressure
to match the compressor outlet state, and the corresponding
power was added to the engine outlet (turbine inlet)
using the QL source term on that lump (#3).
All
major variations (inlet conditions, flow rates, etc.)
were supplied as Thermal Desktop symbols, which facilitates
changing them parametrically, or solving for them
if they are unknown. Two sets of symbols, as depicted
in Thermal Desktop’s Symbol Manager under different
tabs of the same form, are shown below.
Thermal Desktop
Symbols
Some
of these symbols (e.g., the transient event duration,
the shaft inertia, etc.) are defined in later subsections.
Others (e.g., resistive torque terms) are available
for demonstration purposes, but have been zeroed.
Others (e.g., compressor outlet pressure) represent
nominal design points used for initial conditions
and to check answers.
Using
the Case Set Manager, a steady state solution is
invoked (see the case “steady”)
within the drawing. This case runs a simple steady
state at constant rpm, one of the chief products
being the prediction of the torque imbalance, as
denoted by the register NetTorq, defined as the negative
of the sum of the hydraulic torques for the compressor
and turbine:
NetTorq
= - (flow.torq1+ flow.torq3)
Negation
is used because the TORQ values produced by SINDA/FLUINT
use the sign convention of “positive for energy
into the fluid.” Therefore, NetTorq uses the
negative of the sum of torques such that it corresponds
to the direction of positive speed (net power generated
by the fluid system): a larger value of NetTorq drives
a higher shaft speed at equilibrium.
The
calculated net torque is predictably small (about
11 N-m) given that 16000 rpm is the design point:
this is basically a confirmation of the design point,
taking into account numerical uncertainties and approximations.
At the design point, the turbine inlet temperature
and pressure (total) are 935K and 3.43 bar. The turbine
outlet static pressure is 1.91 bar. The turbine flow
rate is 10.9 kg/s.
Solving
for RPM at Zero Net Torque
In
the above example, shaft speed is constant and the
net torque is predicted. Often, the balance point
is required: what shaft speed will result in equal
but opposite compressor and turbine torques?
In
SINDA/FLUINT, the Solver module can be used to find
a traditional input (speed) given a traditional output
(net torque), in a manner similar to the Excel goal
seeking capability. This functionality is demonstrated
by the case “RPM
solver” in the Thermal Desktop Case Set Manager.
Unfortunately,
setting NetTorq to have a goal of zero is too simplistic,
since it is difficult for the Solver to assign an
uncertainty to zero.
While
control constants (e.g., AERRO) are available (see
below), an alternative is used in this model: the
GOAL is set to a small but non-zero number, 0.1,
and the OBJECT (objective) is set to NetTorq. Furthermore,
a constraint is added to provide the Solver with
further guidance with respect to allowable tolerance
(TorqTol = 3 N-m):
-TorqTol <= NetTorq <= TorqTol
The
only design variable is RPM, a register used to define
the speeds of the turbine and compressor.
Using
this set-up, the Solver repeats about 15-25 steady
state solutions, iteratively determining that the
balance point is 16073 rpm.
[Other
simpler approaches work well, but require an understanding
of advanced Solver controls. For example, setting
AERRO=0.1 (basically, allowing an absolute tolerance
of 0.1 N-m rather than the small default of 0.0001
N-m) and setting RDERRO=0.002 (taking a smaller relative perturbation in shaft speed when testing derivatives:
0.2% rather than the large default of 2%) allows
the user to set GOAL=0.0 without any constraints.
This alternative method is used in the “FKsweep” set,
as described next.]
The
above set-up, like the “steady” run
that preceded it, results in a single operational
point. More information is gained by making more
variations. Therefore, a parametric sweep was made
of the exhaust system resistance (holding engine
injection flows and powers constant). This case is
presented as “FKsweep” in the case set
manager, and consists of making repeated Solver calls
in a Fortran DO loop in OPERATIONS, with each Solver
call itself invoking several steady state solutions
as needed to find the shaft speed that balances torque.
The overall runtime, however, remains at just a few
seconds. Key results are plotted below as a function
of exhaust system K-factor.
Shaft
Speed Transient Example
To
illustrate the solution of a combined mechanical
and thermohydraulic set of equations, an artificial
transient is run by perturbing the shaft speed from
its equilibrium value (just above 16000 rpm) to 14000
rpm … the lowest value for which turbine and
compressor data are available. Initially, this lower
speed will cause a net positive value of torque (NetTorq).
The shaft will then be allowed to speed back up to
its design point.
The
inertia of the shaft and rotors are, unfortunately,
completely unknown. Therefore, for demonstration
purpose a mass of 50kg was assumed, and the compressor
and turbine and shaft systems were estimated as solid
cylinders 500 mm in diameter (RadEff = 0.25m).
This
case is presented as “transient” in
the Case Set Manager. The variable omega (the shaft
speed in units of radians per second) is added, and
will be co-solved by a first order ODE (ordinary
differential equation) per the following equation:
NetTorq
= Ishaft * d(omega)/dt
Where
Ishaft is the estimated inertia:
Ishaft
= 0.5*Mass*RadEff**2
These
inputs are placed in a call to DIFFEQN1 in FLOGIC
2 (which is executed at the completion of each thermohydraulic
solution time step), along with conversions from
omega back to the register RPM, the shaft speed,
which as the name implies has units of rpm:
if
(NSOL .EQ. 0) return
call
DIFFEQ1 (1, omega, Ishaft, 0.0, NetTorq)
rpm = omega*2.*pie/60.
The
first line prevents the ODE from solving in a steady
state that is invoked before the transient begins,
in order to set initial conditions. The first argument
in DIFFEQ1 is the ODE equation ID (#1), and the fourth
argument is a resistive term (currently zero). The
value of “pie” is pi.
An
event duration of 360 seconds (6 minutes) proves
enough for the shaft speed to return to its equilibrium
value, as shown in the responses below:
Download
Sample Problem
This
sample problem and it full documentation are available
for download This
sample is a
single Thermal Desktop® drawing containing four
pre-defined cases. Associated EZXY® plots are
also provided. Execution of the model requires SINDA/FLUINT,
Thermal Desktop, and FloCAD to be installed (fully
functional demo versions are available).
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