1. Theory
1.1 Analytical Flow Model
The analytical secondary flow model used for this probabilistic study is a datamatched commercial turbofan engine that represents build of material hardware. The
one dimensional flow model is run via a graphical user interface (GUI). The flow model
calculates the internal engine cavity pressures and temperatures, internal cooling and
leakage fluid flow rates, and the axial load on the thrust bearings. The secondary flow
model is an analytical tool used to design the secondary flow system. The flow model
software is written in FORTRAN code and contains the mathematics required to accurately simulate the secondary flow system of a commercial turbofan engine. The tool
has many uses, and one of particular interest is that it allows system designers to predict
the effect of over or under machined parts on the secondary flow system. The flow
model is also used to validate the secondary flow system, verifying the system requirements are met. The results of the flow model are also used as input for other analyses
such as thermal analysis of the rotors, disks, blades and life estimates for bearings.
Secondary flow system requirements of interest for this analysis are the mass flow
rate through the pre-swirl nozzles of the high pressure turbine, the pressure supplied to
the blade for cooling, and rim cavity purge flow of the blade leading and trailing edges.
The flow model can solve a flow system for a steady state case, a transient case and a
statistical sensitivity or probabilistic analysis case. Only the steady state and probabilistic features will be discussed. The statistical probabilistic analysis used for this study is
the non linear Latin hypercube sampling method.
1.1.1
Flow Model Inputs
The flow and bearing load model is comprised of a series of “chambers” interconnected by various types of “restrictions” to the gas path. These graphical chambers and
resistor icons allow the user to model cavities and restrictions that make up an entire
engine. Chamber icons allow for the calculation of pressure and temperature of a
location, representing a large volume plenum where flow velocities are assumed to be
fully recovered and total pressure is equal to static pressure
1
Chamber pressures and temperatures can be either known or unknown and may be
input by a value or an equation, as in the case of data-matched locations. In order for the
model to solve, there must be at least one known chamber acting as a source and one
known chamber acting as a sink. The resistor icons model pressure losses and flows
between the chambers. There are 25 different types of restrictors available in the “GUI”,
the ones of most interest to the probabilistic analysis and ones pertinent to the TOBI
system’s design parameters are the flow parameter, orifice, labyrinth seal, vortex and
isentropic nozzle and each will be discussed in detail. The flow solver then calculates
the unknown chamber pressures and temperatures and restriction flow rates through
successive iterations until the flow rates are balanced.
The flow model used for this analysis is very detailed having hundreds of defined
chamber and resistor icons. The chambers and resistor icons are provided through an
input file that is generated using the flow model GUI. Every chamber and resistor has a
set of unknown states and governing equations that describe the local flow conditions
and are resolved by the flow solver. A coupled, nonlinear set of equations must be
solved to determine the flow in the flow network model. The flow solver uses a NewtonRaphson iterative method to solve the coupled equations.
1.1.2
Solution Technique
The flow network consists of the resistors and chambers with added interfaces between every resistor and chamber. These interfaces are utilized to improve the
modularity of the underlying solver and are not controlled by the user. A simple flow
network structure is shown in Figure 1 where squares represent the chambers and circles
represent the restrictions.
2
Figure 1 Example of flow network structure
Every chamber, resistor and interface has a set of unknown states and governing
equations that describe the local flow conditions and are resolved by the flow and
bearing load solver. The states for chambers are pressure and temperature. If there is no
initial guess input for each pressure or temperature, the value is set to equal the average
of the known pressures or temperatures.
The states for restrictors are mass flow rate,
temperature at the intended upstream boundary and temperature at the downstream
boundary. The states for interfaces are mass flow rate, temperature and pressure.
The specific governing equations for chambers are:
1. Conservation of mass. In steady flow, the sum of the interface mass flow rates
connected to the chamber is required to be zero:
Number ofint
Interfaces
numberof
erfaces
 m
i
i 1
0
(1)
 is the mass flow rate, or air flow.
Where m
For unsteady flow, appropriate time derivatives are added to account for a time rate
of change for the mass in the chamber.
2. Conservation of energy. In steady flow, the conservation of energy is given by:
Numberof
of Interfaces
number
int erfaces
 m
i 1
i
H (Ti up )  0
(2)
up
Where H is the stagnation enthalpy and Ti is the temperature taken from the upstream direction at interface i. In other words, if the interface is an inflow, this
temperature is equal to the interface temperature; however, at an outflow, this temperature is equal to the chamber temperature. For unsteady flow, additional terms are added
to account for a time rate of change of energy in the chamber.
Resistor governing equations:
1.
For most restrictors, the resistor mass flow rate is set up by a mass flow relation-
ship to the upstream and downstream pressures and temperatures that are taken from the
corresponding interfaces. However, for vortex resistors and fixed pressure ratio resistors,
a pressure ratio or pressure difference is set directly.
3
2.
Upstream temperature, which is based on the direction of flow, is set to the up-
stream interface temperature value.
3.
The downstream temperature is set by the resistor model and may include a tem-
perature set by the user.
Interface governing equations:
1.
The interface mass flow rate is set to the mass flow rate of the resistor it is at-
tached to.
2. The interface pressure
3. The
is set to the pressure of the chamber it is attached to.
interface temperature is set to the temperature from the upstream component,
for example the resistor or chamber which is upstream of the interface.
1.1.3
Restrictions and Chambers
The restrictors discussed here are the ones that will be varied in the input file created for the probabilistic study and consist of the flow parameter, orifice, labyrinth seal,
vortex and isentropic nozzle. Figure 2 shows schematic of locations varied in high
pressure turbine TOBI area.
4
Isentropic nozzle restrictors
Flow parameter
restrictors
Labyrinth seal
restrictors
Orifice restrictor
Figure 2 High pressure turbine cooling air and delivery system input variables locations
Ideally, the flow model restrictors and chambers include inputs that represent the
physical description for the hardware being modeled. The standard output contains the
input items as well as the calculated output values including flow area, temperature,
pressure, pressure ratio, flow rates and reference values if input. Basic discussions are
presented here for the restrictor inputs of interest to the probabilistic analysis and for
brevity are simplified. The way the flow model uses the inputted information is by
calculating a flow parameter and pressure ratio for each restrictor and chamber, which is
then used for the iterations solving the for the total sum of the mass flow rates.
1.1.3.1 Flow Parameter Restrictor
The flow parameter restrictor is used to model the cooling hole passages of the
blade’s leading edge, mid-body, trailing edge, the trailing edge platform overhang and
the TOBI. The input is a flow parameter versus pressure ratio curve and is derived from
5
flow measurements taken during manufacturing. This restrictor is used where the
relationship between flow parameter and pressure ratio is known.
  m T P
up
PR 
(3)
Pup
Pdown
An effective area, discharge coefficient (Cd), can also be calculated based on the cold
flow data. If no discharge coefficient is entered the flow model assumes an effective
area of 1.
ACd 
m T P 
m T P A
up measured
up
(4)
isentropic
Where A is the area.
1.1.3.2 Orifice Restrictor
An orifice restrictor is used to model the minidisk holes which feed the blades. This
restrictor is used where flow measurement or metering is done using a very short passage with a sharp edge on the upstream side and beveling downstream, or a square edge
with no beveling such as a drilled hole. The input includes a required flow area A and an
optional discharge coefficient Cd. The flow area can be input as a value, or an equation.
1.1.3.3 Labyrinth Seal Restrictor
The labyrinth seal restrictor is used to model the outer diameter and inner diameter
TOBI seals. Definitions of a typical labyrinth seal knife edge geometry required for the
flow model are illustrated in Figure 3.
6
Figure 3 Labyrinth seal knife edge geometry, inputs for flow model
Where:
c = Seal Clearance
b = Number of Teeth
pKE = Knife Edge Pitch
wKE = Knife Edge thickness
h = Land Step Height
rKE = knife Edge Leading Edge Radius (illustrated in figure for Up Flow)
Flow Direction = Up/Down
The flow area is calculated by the flow model using the supplied inputs.
1.1.3.4 Vortex Restrictor
The vortex restrictor is used to simulate fluid motion involving rotation about an axis. This simulation tool has no physical area restriction but can either increase or
decrease flows by imposing a pressure ratio on the adjacent chamber(s) with non-fixed
pressure(s). The flow model uses calculations for all the vortex motions that assume the
working fluid acts like a perfect gas and that the vortex is isentropic. These assumptions
are thought to provide a reasonable representation of a real vortex but if the vortex
pressure ratio is known it may be input directly.
7
1.1.3.5 Isentropic Nozzle
The isentropic nozzle restrictor is used to model the rim cavities. Input considerations include a required flow area, A, and an optional discharge coefficient, Cd. The flow
model assumes that the isentropic nozzle has an upstream area much larger than the
minimum flow area specified as input and uses the input flow area as the throat for this
restriction. The nozzle flow area must be input using the minimum passage area known
as the throat of the nozzle. The nozzle throat may have any cross-section shape; the flow
model will use the input flow area as the minimum flow area for the nozzle throat.
1.1.3.6 Chamber
The flow model assumes 1-D flow, constant enthalpy, velocity and density over the
area. Velocities are assumed normal to areas and no heat or work interactions with
surroundings. The air is assumed to be a perfect gas during standard flow model iteration loop, and it is assumed a real gas during flow model iteration loop in which
conservation of energy law is applied to each chamber using real fluid properties. The
chamber effectively acts as a mixing restriction with 0 or more control surfaces.
1.1.4
The Flow Network Solver
The coupled, nonlinear set of equations must be solved to determine the flow in the
flow and bearing load network model. The flow solver uses a Newton-Raphson iterative
method to solve the coupled equations. Specifically, the governing equations and unknown states for every chamber, resistor, and interface in the flow network can be
combined into a residual form,
R (U )  0
(3)
Where U is a vector containing all of the unknown states and R is the corresponding
vector containing all of the governing equations. A Newton-Rapshon method for solving
this nonlinear set of equations is found by linearizing the residuals about a current guess
for the solution,
8
R(U n  dU )  0
R(U n ) 
R
dU  0
U
(4)
Then, by solving for dU, an update for the state vector is given by,
U
n 1
R 1
U 
R(U n )
U
n
(5)
In practice, however, the full Newton update is not taken at every iteration especially early in the iterative process where the linearized update may result in non-physical
solutions. Thus, the Newton update is under-relaxed as follows,
U
n 1
U 1
U d
R(U n )
U
n
(6)
Where d is a relaxation factor and in the flow solver and is known as the drate. The
algorithm for determining the drate involves several parameters which may be modified
to control the convergence behavior of the flow solver. Two types of limiting of the
drate are used: (1) limiting based on the flow rate residual, and (2) limiting based on the
change in the states. For the flow rate residual limiting, the basic idea is to limit the
Newton update whenever the mass flow rate imbalance anywhere in the flow network is
large at the current iteration. For the state-based limiting, the basic idea is to limit the
changes in the state to guarantee that the states at the next iteration are physically
realistic. Thus, in both limiting procedures, as the solution converges, drate should
approach one, while initially drate is less than one.
1.2 Probabilistic Flow Model
The flow model’s sensitivity analysis permits the user to input variations of certain
parameters and then run a sensitivity study (linear or non linear) to obtain statistical
variations for other defined parameters. The secondary flow model, as a design feature,
has an option for varying input values and propagating them through the flow model.
Typically the flow model takes single input values per chambers and restrictors in the
form of flow areas, pressures and temperatures. The probabilistic flow analysis allows
9
the user to input variability in engine performance (day to day variability) and hardware
geometry (engine to engine variability) yielding the variability in mass flow rates and air
temperatures and pressures.
1.2.1
Probabilistic Flow Model Inputs and Outputs
The input file for the probabilistic study needs to identify the parameter (chamber or
restrictor) being varied the nominal value, the standard deviation, the distribution type
and the variance. The flow model probabilistic analysis generates regression output,
cumulative density functions, and probability density functions. After propagating input
variability through the model by means of an input file, it is possible to analyze the
model outcomes of means and deviations.
10