aft arrow seminar week of 22 may, 2017 - prode aft fathom 7 and arrow 4 8 intro - additional...
TRANSCRIPT
-
AFT Arrow SeminarWeek of 22 May, 2017
-
Introduction
-
Overview of Seminar
AFT ARROWA1. Overview of AFT ArrowA2. Fundamental Equations of Compressible FlowA3. Demonstration Problem - Determining Delivery ConditionsA4. Understanding Solution Control OptionsA5. AFT Arrow Hands-On ModelingA6. Troubleshooting AFT Arrow Models A7. The Five Primary WindowsA8. Pipe and Junction Details A9. Special TopicsA10. Verification of SolutionsA11. Using Scenario ManagerA12. Customizing Arrow and Using Databases A13. Introduction to AFT Arrow ModulesA14. More AFT Arrow Hands-On Modeling
-
Intro -
About Applied Flow Technology
Applied Flow Technology (AFT), founded in 1993, is a world leader in providing high quality software to analyze flows, pressures and transients in systems with pipes, pumps and valves
Customers in 70+ countries Representatives in 32 locations around the world
1
-
Intro -
AFT Fathom 9
Models incompressible network pipe systems Liquid and low velocity gas systems
Models open and closed systems Models systems that are pressure, gravity or pump driven Models heat transfer and system energy balance Offers broad range of innovative reporting features
Printed output is of report quality Offers customizable component and property databases
Cost calculations Rheological data handling to support non-Newtonian fluids
2
-
Intro -
AFT Fathom Add-On Modules
XTS eXtended Time Simulation Simulate dynamic behavior of systems over time Models infinite and open and closed finite tanks of constant and
varying cross section Supports user defined time and event transients of pumps, valves
and other components GSC Goal Seek & Control
Automatically determines input variables that will yield specified output values
Extends Fathoms control simulation capabilities to include remote sensing
SSL Settling Slurry simulation Simulates settling slurry behavior Simulates pump performance degradation
3
-
Intro -
AFT Arrow 6
Models compressible network pipe systems High to low velocity gas systems High to low pressures
Models open and closed systems Accurately models
Real gases Heat transfer Highly compressible (sonic and near sonic) systems
Balances flow and energy throughout the system Offers broad range of innovative reporting features Offers customizable component and property databases Includes high accuracy steam/water properties to ASME
4
-
Intro -
AFT Arrow Add-On Module
GSC Goal Seek & Control Automatically determines input variables that will yield specified
output values Extends Arrows control simulation capabilities to include remote
sensing
5
-
Intro -
AFT Impulse 6
Models waterhammer/surge flow in pipe networks Models system transients caused by
Sudden valve closures Pump startups and shutdowns including pump inertia effects Relief valve cracking Events defined within the system (e.g. flow, pressure, etc.)
Includes modeling of Control and relief valves, vacuum breaker valves, pumps,
accumulators and surge tanks Includes a steady-state solver to determine initial conditions Calculates unbalanced transient forces
Forces can be graphed or exported as Force/Time data files Can also import AFT Fathom models
6
-
Intro -
AFT Impulse Add-On Module
SSL Settling Slurry simulation Simulates settling slurry behavior Simulates pump performance degradation
7
-
Intro -
AFT Mercury 7AFT Titan 4
Models and designs network pipe systems Combines a powerful hydraulic solver and flexible graphical
interface with an advanced optimization engine Automatically selects best pipe and component sizes to minimize
initial or life cycle cost, size or weight using IntelliFlow
Ability to apply multiple constraints to pipes and junctions Cost optimization may include;
non-recurring costs (materials and installation) recurring costs (energy and maintenance) including time varying
cost (energy costs varying with time) Offers customizable engineering and cost databases Includes powerful modeling and output capabilities
of AFT Fathom 7 and Arrow 48
-
Intro -
Additional Software Products
Chempak Property Database Property database of ~700 fluids Ability to define static pre-mixtures Dynamic mixing capability in Arrow
Chempak Viewer 2.0 & Chempak Add-in (for Excel) Viewer allows use of Chempak as a stand alone application Add-in makes all of the Chempak functions accessible within an
Excel spreadsheet SteamCalc 2.0
High accuracy ASME steam/water library for Windows and Excel
9
-
Intro -
Product Applications
AFT products are being successfully applied to a broad range of industrial systems: Power generation systems Chemical and petrochemical systems Oil and gas production, transportation, refining and delivery Automotive systems Aerospace systems Air conditioning and refrigeration systems Pulp and paper processing Fire suppression Water and Wastewater treatment plant design Mining processing and support systems Municipal water distribution
10
-
Intro -
AFT Flow Expert Package
Provides consulting services beyond typical technical support requests on the installation, upgrade assistance, and functionality of AFT software.
Access to a consulting engineer assigned as your primary point of contact.
Package Options: Blocks of 5 hours, 10 hours and 20 hours Typical ways to use your hours:
Receive online training on specific topics of your choice Request help on model results interpretation Get a second opinion of your assumptions, modeling choices
and reports
11
-
Intro -
AFT Flow Expert Package (2)
Additional ways to use your hours: Have an expert double check your modeling input and point out
common modeling mistakes or suggest better ways to model the desired behavior
Receive guidance in how to model pumps and pump-system interaction, relief valves and relief systems, surge suppression equipment, slurry pipelines, system transients, and anything having to do with flow in pipe systems
Discuss with an expert alternative solutions for hydraulic problems
Help launch AFT software within your company and reduce your learning curve
Help new hires get acquainted with AFT software
12
-
A1. Overview of AFT Arrow
-
Overview of Seminar
AFT ARROWA1. Overview of AFT ArrowA2. Fundamental Equations of Compressible FlowA3. Demonstration Problem - Determining Delivery ConditionsA4. Understanding Solution Control OptionsA5. AFT Arrow Hands-On ModelingA6. Troubleshooting AFT Arrow Models A7. The Five Primary WindowsA8. Pipe and Junction Details A9. Special TopicsA10. Verification of SolutionsA11. Using Scenario ManagerA12. Customizing Arrow and Using Databases A13. Introduction to AFT Arrow ModulesA14. More AFT Arrow Hands-On Modeling
-
Nomenclaturea sonic speed
A cross-sectional flow area of a pipe
C d discharge coefficientcp specific heat, constant pressure
cv specific heat, constant volume
D diameter of a pipe
e internal energy
f friction factor
F Force
F f Parameter in Section A2
F g Parameter in Section A2
F To Parameter in Section A2
F Parameter in Section A2
g gravitational constant
h internal convection coefficient
h enthalpy, static
ho enthalpy, stagnation
k thermal conductivity
K loss factor
L length of a pipem mass flow rate
M Mach Number
n constant
Nu Nusselt number
P pressure, static
Ph heated perimeter
Po pressure, stagnation
Pw wetted perimeter
Pr Prandtl number
q heat rate to a pipe
Q volumetric flow rate
q heat flux
r radius
r relaxation
R gas constant
Re Reynolds number
s fan speed
s entropy
T temperature, static
To temperature, stagnation
U overall heat transfer coefficient
v specific volume
-
NomenclatureV volume
V velocity
w work
x distance along pipe centerline
z elevation
Z compressibility factor
, , angle diameter ratio roughness specific heat ratio dynamic viscosity shear stress rotational velocity
Subscripts
1 location 1 in pipe
2 location 2 in pipe
i junction at which solution is sought
j junctions with pipes connecting to junction i
o stagnation infinity, far away, ambient
-
A1 -
AFT Arrow General Description
General purpose pipe network compressible flow analysis Drag-and-drop interface Calculates pressure drop, flow distribution and energy in pipe
networks Solves 5 equations for each pipe:
Continuity (Mass) Equation Momentum Equation Energy Equation Equation of State Mach Number
1
-
A1 -
AFT Arrow General Description (2)
Implements modified Newton-Raphson matrix method to solve network
Can model systems in any generalized configuration Open or closed systems Branching systems Looping systems
Can model sonic choking and heat transfer English and SI units supported
2
-
A1 -
Components That Can Be Modeled
Branching section (up to 25 pipes) Known pressure or flow boundaries Compressors and fans
Compressor/fan curves follow a polynomial equation Pressure and flow control valves Relief valves and check valves Heat exchanger pressure drop and heat transfer General fittings and components where the resistance curve
follows a polynomial relationship
3
-
A1 -
Engineering Limitations
No practical software limit to model size Flow is steady-state and one-dimensional No limit on number of fittings (i.e., additional losses) No limit on number of compressor/fans, control valves, etc. No limit on number of custom components, fluids or pipe
materials
4
-
A1 -
Arrow 6 Startup Window
5
-
A1 -
Primary Windows
The AFT Arrow modeling process flows through five Primary Windows Workspace Model Data Output Visual Report Graph Results
The Primary Windows offer a mixture and graphical and text-based features to assist in the modeling process
Tabbed Primary Windows allow for easier navigation Robust usage of dual monitors is supported
Can drag the Primary Window tabs into their own separate window
6
-
A1 -
Primary Window Process Flow
7
Graph Results
Visual Report
Workspace Output
Model Data
-
A1 -
Workspace
Multiple features available with Quick Access Panel Can pin Quick Access Panel to the Workspace or minimize with
thumbtack to allow for more Workspace area
8
These icons represent different components
This tool is used to draw new pipes
This tool will add annotation to the workspace
Minimize Quick Access Panel with thumbtack
Quick Access Panel
-
A1 -
Quick Access Panel Activate Modules
Ability to activate GSC, APS, and ANS Modules
9
-
A1 -
Workspace - Editing Features
Cut, copy, paste, delete, duplicate and undo features supported
Workspace can be sized to fit the model You can zoom out to see a larger area Objects can be selected as a group in several ways
Selecting the components by dragging the mouse over them Using the SHIFT key while clicking on the objects Using Select Flow Path on the Edit menu Using the Select Special tool on the Edit menu Using Groups / Select on the Edit menu Using the Select All feature
10
-
A1 -
Workspace Editing Features (2)
The Reference Flow Direction of a pipe can be changed The selected objects can be renumbered
Manually Renumber Automatic Renumber Wizard Renumber Increment
The Find tool will move the Workspace window to show a pipe or junction
11
-
A1 -
Workspace - Platform for Data Entry
All pipe and junction objects placed onto the Workspace are interactive
To open the Properties window for data entry, just double-click the graphical object Alternatively, you can select the object by clicking on it once and
then press the Enter key Or you can select the object by clicking on it once and then click
on the Open Pipe/Jct Window button on the Toolbar The Properties windows are the primary manner in which
component data is entered The Global Pipe Edit and Global Junction Edit window can
speed up data entry
12
-
A1 -
Workspace - Reporting
The Workspace image can be printed on printers and plotters Print Preview allows page customization
The image can be sized on the page A company logo and custom text can be added
13
-
A1 -
Model Data Window
Model Data is broken into three sections General Data Pipe Data Junction Data
Each section can be re-sized or collapsed allowing the user to focus on any of the sections
User can select all or portions of the Model Data Window content for printing Print format window allows customizing of content User can also select the font
14
-
A1 -
Model Data Window (2)
With a Workspace printout and the complete Model Data printout, the input can be printed in its entirety
Properties windows for data entry can be opened by double-clicking the far left column
15
-
A1 -
Output Window
The Output window is the primary vehicle for communicating the results of an analysis in text form
Output Window is broken into three sections General Results Pipe Results Junction Results
Each section can be re-sized or collapsed allowing the user to focus on any of the sections
Each section contains tabs to permit quick viewing of output by type
16
-
A1 -
Output Window (2)
User can select all or portions of the Output Window content for printing Print format window allows customizing of content User can also select the font
User can sort output according to any of the columns for quick review of data extreme maximums and minimums
Output Window content is specified by Output Control Window
17
-
A1 -
Visual Report Window
Visual Report allows user to display input and output results together with pipe system image
18
-
A1 -
Graph Results Window
The Graph Results Window allows creation of full-featured Windows graphs
19
-
A2. Fundamental Eqns. of Compressible Flow
-
Overview of Seminar
AFT ARROWA1. Overview of AFT ArrowA2. Fundamental Equations of Compressible FlowA3. Demonstration Problem - Determining Delivery ConditionsA4. Understanding Solution Control OptionsA5. AFT Arrow Hands-On ModelingA6. Troubleshooting AFT Arrow Models A7. The Five Primary WindowsA8. Pipe and Junction Details A9. Special TopicsA10. Verification of SolutionsA11. Using Scenario ManagerA12. Customizing Arrow and Using Databases A13. Introduction to AFT Arrow ModulesA14. More AFT Arrow Hands-On Modeling
-
A2 -
Introduction
AFT Arrow uses a modified Newton-Raphson Method to solve the flow distribution in a pipe network
This method is similar to that used in AFT Fathom, but more difficult to implement
There are no standard methods available to solve the full compressible flow equations for pipe networks
1
-
A2 -
Basic Laws of Incompressible Pipe Flow
Mass Conservation
Momentum Equation (Bernoulli, assuming incompressible)
The velocity (dynamic) pressure and static pressure can be combined into the total pressure, and the solution is then for total pressure Therefore, the momentum equation becomes
2
AVm =
lossPghVPghVP +++=++ 22
2212
11 21
21
lossoo PghPghP ++=+ 22,11,
-
A2 -
Law of Friction (Incompressible Flow)
Traditional method of friction loss calculation uses the Darcy-Weisbach friction factor, f
The friction factor is not a constant, but a function of the pipe wall characteristics and the Reynolds number
3
= 2
21 V
DLfPloss
-
A2 -
Law of Friction (Incompressible Flow) (2)
AFT Fathom uses the iterative Colebrook-White correlation for turbulent flow and the traditional laminar flow equation when laminar
Special friction models available for pulp and paper stock and crude oil
4
2
Re35.9log214.1
+=
fDf ( )4000Re >
Re64
=f ( )2300Re
-
A2 -
Modified Form for Law of Friction
Basic law (incompressible flow)
Substituting mass flow rate definition
Defining new term, where R is a pipe resistance
Bernoullis equation then becomes
5
= 2
21 V
DLfPloss
=
2
21
Am
DLfPloss
2mRPloss =
= 22
1AD
LfR
222,11, mRghPghP oo ++=+
-
A2 -
Balancing Mass at Branches
Applying law of mass conservation to a branching section
Substituting yields the following equation to be solved for every branch, i, (incompressible flow)
where sgn = 1 depending on flow direction
6
=
=n
jijm
10
( )( ) ( )=
=
++
n
j ij
ijiojoijiojo R
hhgPPhhgPP
1
5.0
,,,, 0sgn
-
A2 -
Balancing Mass at Branches (2)
The objective is to find all of the P values that satisfy the above equation applied to every branch
We will then have a solution for two unknowns: pressure at all junctions mass flow rate in all pipes
7
( )( ) ( )=
=
++
n
j ij
ijiojoijiojo R
hhgPPhhgPP
1
5.0
,,,, 0sgn
-
A2 -
Solving the Equations
We need to solve as many equations as there are flow splits All of the equations are non-linear AFT Fathom uses the Newton-Raphson Method to solve the
system of equations Newton-Raphson is an iterative method used to solve for roots of
equations
8
-
A2 -
Solving the Equations (2)
Initially the pipe flow rates are not known so an error, F, exists at each branch (incompressible flow)
The objective is to use the Newton-Raphson Method to drive all of the F errors to zero (within some tolerance)
9
( )( ) ( )=
=
++
n
ji
ij
ijiojoijiojo FR
hhgPPhhgPP
1
5.0
,,,,sgn
-
A2 -
The Newton-Raphson Method
The procedure for applying Newton-Raphson to a single equation is as follows:
1) Take a guess at the solution to function F2) Calculate an improved guess using the following equation:
3) Substitute the improved guess back into the above equation until the change in x is small
10
( )( )i
iii xF
xFxx'1
=+
x
F(x)
xi
F(xi)
xi+1
-F'(xi)
-
A2 -
Solving the System (Incompressible Flow) When applied to a system of equations with P as the
unknown, Newton-Raphson looks as follows
where P is the vector of pressures and JF is the Jacobian matrix of error function derivatives - both of a size, n, which is the number of branches (i.e., equations in the system)
=
no
n
o
n
o
n
nooo
nooo
F
PF
PF
PF
PF
PF
PF
PF
PF
PF
J
,2,1,
,
2
2,
2
1,
2
,
1
2,
1
1,
1
11
= FJPP Foldonewo1
,,
-
A2 -
Derivative Terms in Jacobian
The diagonal derivative terms in the Jacobian can be calculated analytically (incompressible flow)
The off-diagonal terms can also be calculated analytically (incompressible flow)
12
( )( ) ( )=
++=
n
j ij
ijiojoijiojoi R
hhgPPhhgPPF
1
5.0
,,,,sgn
( )( )=
+
=
n
jijiojo
iji
i hhgPPRP
F1
5.0
,,5.05.0
( )( ) 5.0,,5.05.0
+
=
ijiojoiji
i hhgPPRP
F
-
A2 -
Solving the Matrix (Incompressible Flow)
Rather than inverting the Jacobian matrix, it is usually faster to solve a linear system of equations as follows
We need to solve for the values in vector, z, that satisfy the above
13
= zPP oldonewo ,,
= FJz F1
= FzJ F
-
A2 -
Solving the Matrix (Incompressible Flow) (2) Use Gaussian Elimination to solve for z
By multiple substitutions, we progressively eliminate terms in JFand eventually obtain the identity matrix, where all terms are zero except the diagonal, which is unity
We then have the solution for z, which can be substituted back into the original equation at the top to improve our guess for all of the pressures in the pressure vector
14
= zPP oldonewo ,,
-
A2 -
Test Problem #1 (Incompressible Flow)P = 175 psiah = 0 feet
P = 200 psiah = 0 feet 1 2
3
4
P = 160 psiah = 0 feet
h = 0 feet
pipe 1 pipe 2
pipe 3pipe f L (ft) D (in) Fluid
1 0.0219 100 4 Water @ 70F2 0.0156 75 4 Water @ 70F3 0.0180 125 6 Water @ 70F
Jct P (psia)1 2002 1753 160In this test problem, pipe
resistances can be calculated based on known friction factor (shown in the table) 15
-
A2 -
Test Problem #1 (Incompressible Flow) (2) To start the solution, we
need to guess P4, so guess 180 psia
Note: All pressures here are stagnation 16
( )[ ]=
=
n
j ij
ijij R
PPPPF
1
5.0
sgn
( )[ ] ( )[ ] ( )[ ]5.0
43
4343
5.0
42
4242
5.0
41
4141 sgnsgnsgn
+
+
=
RPP
PPR
PPPP
RPP
PPF
1489.277=F5.0
435.043
5.0425.0
42
5.0415.0
41
5.05.05.0'
+
+
= PPR
PPR
PPR
F
6139.18' =F( )( )old
oldoldnew PF
PFPP'
=
-
A2 -
Test Problem #1 (Incompressible Flow) (3)
17
Pj=4 Mpipe=1 Mpipe=2 Mpipe=3 F F'Iteration1234567
180.0000 115.3464 -78.9048 -313.5906 -277.1489 -18.6139165.1106 152.3476 110.9691 -158.5207 104.7960 -23.3027169.6078 142.1903 81.9409 -217.3503 6.7810 -21.2485169.9269 141.4418 79.4792 -220.9305 -0.0095 -21.3129169.9265 141.4429 79.4827 -220.9255 0.0000 -21.3128169.9265 141.4429 79.4827 -220.9255 0.0000 -21.3128169.9265 141.4429 79.4827 -220.9255 0.0000 -21.3128
P (psia), M (lbm/s)
-
A2 -
Whats Special About Compressible Flow? Compressible flow is defined as fluid flow where density
changes are significant Changing density has several important ramifications
Velocity changes in a pipe Velocity change is generally non-linear
Density depends on temperature so that flow is coupled to energy equation
Accelerating flow is limited to sonic velocity, thus sonic choking can become a dominant characteristic of the system
Sonic choking may occur in multiple locations
18
)( AVm =
-
A2 -
Whats Special About Compressible Flow? (2) All governing equations are strongly coupled
An accurate solution must address all aspects of the gas flow Pipe networks introduce an order of magnitude complexity
into compressible flow analysis
19
-
A2 -
Possible Methods of Analysis
Use incompressible flow methods Inherently inaccurate Large safety margins required Engineer is never sure of analysis results Sonic choking glossed over
Use simplified compressible flow pressure drop correlations Crane manual isothermal flow equation (Eqn. 1-6) , Weymouth,
Panhandle, etc. Thermal and real gas effects are ignored Cannot extend method to pipe networks Large safety margins required Engineer is never sure of analysis results Sonic choking glossed over
20
-
A2 -
Possible Methods of Analysis (2)
Use iterative spreadsheet or in-house software Usually based on simplified correlations Usually assumes ideal gas behavior and ideal energy process
(adiabatic or isothermal) Time consuming to use and difficult to interpret results Often developed by non-specialists in compressible flow
21
-
A2 -
Basic Problems With Traditional Methods
Engineer is never sure of analysis results Gases frequently are not ideal Gases frequently are not isothermal or adiabatic
In many cases engineers believe their analysis is better than it really is
Sheer quantity of important variables means that important data can be easily overlooked
Low quality analysis leads to higher costs and reduced safety Over-design costs more during construction & over the life-cycle Safe operation of design is jeopardized if analysis is not properly
performed Parallel flow pipe networks cannot be properly
analyzed22
-
A2 -
AFT Arrow Approach to Compressible Flow Solve all governing equations simultaneously Include all thermal and real gas effects Balance mass and energy throughout the network
Implement special flow and energy balance iterative methods Offer several solution methods to increase flexibility Encapsulate powerful solution method in an easy-to-use
graphical Windows interface
23
-
A2 -
Governing Equations of Compressible Flow Equations for each pipe
[1] Mass:
[2] Momentum:
[3] Energy:
[4] Equation of State:
[5] Mach Number:
d dVV
+ = 0
P Z RT=
MVZRT
=
dP VfD
dx VdV+ + =12
02 gdz+
m d h V q+
=
12
2 + gz
24
-
A2 -
Governing Equations of Compressible Flow (2) Equations for each junction
25
[6] Balance Mass:
[7] Balance Energy:
mijj
n
=
=1
0
m h Vij ij ijj
n
+
=
=
1
202
1
0
1
,
1
===
npipes
j
ijc
ngases
c
m
[8+] Balance Species:
-
A2 -
Stagnation vs. Static Properties
The static properties are the true thermodynamic properties: pressure, temperature, density, enthalpy, etc.
The stagnation properties are those that combine the thermodynamic properties with the fluid dynamic effects
Classic example is a pitot tube that is normal to the flow or pointed directly into flow
Static Pressure Stagnation Pressure
26
-
A2 -
Stagnation vs. Static Properties (2)
Compressible flow calculations are greatly aided by using stagnation properties Effects of flow area changes are inherently taken care of
In incompressible flow, the stagnation pressure is the static plus dynamic pressure
27
2
21 VPPo +=
-
A2 -
Stagnation vs. Static Properties (3)
In compressible flow the relationship is more complicated It can be shown that the incompressible case is a simplification
while the compressible case is the true equation
28
+= 2
211 M
TTo
( )12
211
+=
MPPo
2
2Vhho +=
-
A2 -
AFT Arrows Solution Methods
AFT Arrow offers six solution methods altogether Two methods are "lumped" methods
Lumped Adiabatic Lumped Isothermal These are not as accurate but solve much faster Cannot model heat transfer Cannot model elevation changes (usually this is not very
important)
29
-
A2 -
AFT Arrows Solution Methods (2)
Four methods are marching methods Length March Mach Number March Two hybrid methods based on these two These four methods are highly accurate but have longer run
times Can accurately model heat transfer, elevation changes and
rotating systems
30
-
A2 -
AFT Arrows Solution Methods (3)
The two lumped methods use traditional handbook methods to solve a single pipe
Mach Number March method is a marching method that is optimized for sonic and near sonic systems
Length March method is a marching method that works well at all velocities, but is not as accurate or reliable as Mach Number March method at near-sonic conditions
For many systems all six methods will work fine and will give similar results
31
-
A2 -
These methods have closed form solutions and can be found in textbooks
Adiabatic flow equation and integrated solution
Isothermal flow equation and integrated solution
20 24
22
12
11
1 dMMM
MdxDfL M
M
+
=
+
+
++
=
21
22
22
21
22
21
211
211
ln2
1111
M
M
M
M
MMDfL
( ) 20 4
22
1
1 dMM
MdxDfTL M
M
=
=2
1
22
21
22
21
ln
1
M
M
M
M
M
DfLT
Lumped Adiabatic & Isothermal - Single Pipe
32
-
A2 -
Length March: Single Pipe
Using substitution and calculus, the following equation can be derived:*
Integration of the above yields:
In this approach, each distance step, x2, allows calculation of a new P0,2 .
The solution is obtained by marching down each pipe until x2 = L.
33
+++=
2 sin2 ZRT
dxgdZ
dZDfdx
TdTM
PdP
o
o
o
o
( ) ( )
+++= 1212
1
2
1
2
1,
2,2
1,2,sinlnlnln
2exp
TZRxxg
Dxxf
ZZ
TTMPP
o
ooo
* See AFT Arrow Help System topic Length March Method for complete derivation
Flow
1 2
-
A2 -
Mach Number March: Single Pipe
Using substitution and calculus, the following equation can be derived:*
where:
34
Flow
1 20sin2
2=
+ gf
o
oT ZRT
dxgFDfdxFdF
ZdZ
TdTF
M
dMo
F f = 2
22
12
11
M
MM
+
FTo =( )
2
22
12
111
M
MM
++
=( )
2
22
12
111
M
MM
+
F F g = 2
2
12
112
M
M
+
* See AFT Arrow Help System Topic Mach Number March Method for complete derivation
-
A2 -
Mach Number March: Single Pipe (2)
Integration of the above yields:*
By selecting a target Mach number M2 for each step, one can march down a pipe to calculate the corresponding x2 distance.
35
+
+
+=sin
lnlnlnln12
12
21
22
121
2
gf
o
oTo
TZRgF
DfF
FZZ
T
TF
M
M
xx
Flow
1 2
* See AFT Arrow Help System Topic Mach Number March Method for complete derivation
-
A2 -
Tying Things Together - Solving Networks To solve the network, energy and mass flow must balance at
each branching section AFT Arrow employs a modified Newton-Raphson method to
solve the mass flow balance Solves the non-linear flow equations using matrix techniques Similar to incompressible network solution method
Energy equation is linear and is solved by multiple substitution and iteration method
36
-
A2 -
Compressible Flow Solution Difficulties
Not possible to analytically calculate the derivative terms Pressure drop also depends on energy transfer Velocity and friction factors vary along pipe Negative pressures cannot be allowed Sonic choking Result is that AFT Arrow has to iterate much more than AFT
Fathom, leading to significantly longer run times
37
-
A2 -
Taking Care of Details
Sonic choking is accounted for by constant checking of Mach Numbers
Flow can choke at: A flow restriction (e.g., orifice) A flow expansion (area increase) The end of a pipe as it exits the system (endpoint choking)
Sonic choking is heavily influenced by friction effects and thermal effects AFT Arrow accounts for these effects when it solves the
equations simultaneously When flow is choked, special iterative techniques are
employed to converge on flow and pressure solutions
38
-
A2 -
Convergence
When the change in pressures, flow rate and temperature decrease to some small amount, the calculation is converged
Different criteria can be applied for identifying convergence Percentage change in result Absolute change in result
We will cover convergence in a later section
39
-
A2 -
Flow Rate and Enthalpy Updates
After the pressure solution is obtained a new flow solution and enthalpy are calculated
The new flows and enthalpies are then compared against the old flows and enthalpies
If the flow or enthalpy change too much it is updated and the pressure solution repeated
This whole procedure is repeated until flow, enthalpy and pressure updates are small
If mixing is modeled, a concentration update is also performed
40
-
A2 -
Solver Flow Chart
RecalculateConcentrations *
Start
RecalculateEnthalpies
Solve Junction Pressures
Yes
> Max Iterations ?Converged ? EndYesNo
No
> Max Iterations ?Converged ?No No
Yes
Return
Recalculate Mass Flow Rates
> Max Iterations ?Converged ? EndYesNo
No
Yes
Update Flow Losses and Compressors
RecalculateEnthalpies
RecalculateConcentrations *
> Max Iterations ?Converged ? EndYesNo
No
Yes Yes
* A concentration balance is performed only if dynamic mixing is modeled. If not, the Solver passes through this block.
41
-
A2 -
Known Flow Vs. Known Pressure Junctions At all system boundaries AFT Arrow must solve for either flow
or pressure User cannot specify both flow and pressure at the same point
because there would be nothing for AFT Arrow to solve Either the flow rate calculation or the pressure calculation
must be available to AFT Arrow
42
-
A3. Demo. Problem - Delivery System
-
Overview of Seminar
AFT ARROWA1. Overview of AFT ArrowA2. Fundamental Equations of Compressible FlowA3. Demonstration Problem - Determining Delivery ConditionsA4. Understanding Solution Control OptionsA5. AFT Arrow Hands-On ModelingA6. Troubleshooting AFT Arrow Models A7. The Five Primary WindowsA8. Pipe and Junction Details A9. Special TopicsA10. Verification of SolutionsA11. Using Scenario ManagerA12. Customizing Arrow and Using Databases A13. Introduction to AFT Arrow ModulesA14. More AFT Arrow Hands-On Modeling
-
A3 -
Pipes
AFT Arrow uses two system constructs: pipes and junctions Pipes are conduits for steady-state, compressible, one-
dimensional fluid flow The mass flow rate through the entire length of the pipe is
always constant Volumetric flow rate is not constant!
Pipes have constant diameters but the fluid velocity is not constant
Each pipe must be connected to a junction on each end
1
-
A3 -
Pipes (2)
A pipe differs from a junction in that it has a reference positive flow direction To say a pipe has a flow rate of 1 lbm/sec is meaningless unless
the flow direction is specified. In cases where there is uncertainty about flow direction, you
do not need to specify the actual flow direction in a pipe AFT Arrow sorts out the true physical flow directions of the
system you define However, the pipe orientation is critical when using pressure-
dependent junctions like pumps and control valves
2
-
A3 -
Junctions
Junctions are connector points for pipes Junctions are elements at which flow and energy balances
are made Some junction types can only connect to one pipe while
others can connect with up to twenty five AFT Arrow provides a total of twenty standard junction types
3
-
A3 -
Junctions (2)
In addition to balancing flow and energy, junctions also influence the flow or pressure behavior of the system A tank junction applies a constant pressure at a location, and the
flow at a tank is free to adjust in whatever manner is consistent with the governing equations
An assigned flow junction applies a known flow rate at its location, allowing the pressure to adjust to that level dictated by the governing equations
The twenty standard junction types allow you to specify special kinds of irrecoverable pressure losses or fluid behavior
Junctions communicate with each other through the pipes connecting them
4
-
A3 -
Creating Objects
Pipe and junction objects are created using the Workspace Toolbox New pipes and junctions can also be derived from previous ones
by duplication Pipes are drawn on the Workspace Junctions are dragged from the Toolbox
5
-
A3 -
Creating Objects (2)
Pipe and junctions have default numbers assigned Users can reassign numbers Pipes numbers are displayed near the pipe center preceded by a
"P" Junction numbers are displayed over the junction icons
preceded by a "J" Pipes also have a direction arrow displayed with the number to
indicate the positive flow direction
6
-
A3 -
Moving Objects
The objects on the Workspace can be moved individually or as groups
To move an object, select it, drag it within the Workspace, and drop it in the desired location When an object is dragged off the existing Workspace area, the
Workspace is expanded accordingly The pipe object can be stretched by grabbing the handles at
the pipe endpoints and moving an endpoint to a new location
7
-
A3 -
Moving Objects (2)
To prevent accidental movement of objects, lock the objects on the Workspace The Lock feature is accessed from the Edit menu or the lock
button on the Toolbar. To group multiple objects for movement or other operations,
hold down the SHIFT key when selecting the objects Objects can also be selected by using the Selection Tool on
the Workspace Toolbar Click on the Workspace and drag the mouse to draw a box
around the objects Holding down the SHIFT key while drawing multiple boxes
permits multiple sets of grouped selections
8
-
A3 -
Connecting Pipes and Junctions
Pipes and junction objects can be placed anywhere on the Workspace
Remember that connectivity ONLY exists between junctions and pipes There are no junctions that connect to junctions, and no pipes
that connect to pipes The model connectivity you establish on the Workspace
remains only as long as you maintain the graphical objects in their current visual relationship to each other
The most certain way to maintain the connectivity of your model is to Lock the objects to the Workspace so they cannot be moved
9
-
A3 -
Connecting Pipes and Junctions (2)
To establish a connection between a junction and a pipe, the following three steps are required:
1) Graphically connect the objects on the Workspace (the pipe endpoint must terminate within the boundaries of a junction icon)
2) Enter data for the pipes through the Pipe Property window or globally
3) Enter data for the junctions through the Junction Property window or globally
10
-
A3 -
Editing Objects
The objects you place on the Workspace can be edited with the editing commands from the Edit menu or the Workspace Toolbar
Objects can be cut, copied, pasted, duplicated, and deleted These operations can be performed on individual objects or
on groups One level of undo is available for each editing operation
through the Edit menu
11
-
A3 -
Lay Out the Model
Open branching system Need to find the delivery conditions at J5, J6 and J7 Model looks as below
12
-
A3 -
Using the Checklist
The Checklist tracks the status of your model Communicates what items must be completed before you can
run the model
13
You can open the Checklist box from the Toolbar, View menu, or Quick Access Panel
-
A3 -
Using the Checklist (2)
The first item is always checked off because AFT Arrow assigns default Solution Control parameters The default Solution Control parameters work satisfactorily in
most cases The fourth item is disabled because no costs are applied by
default The fifth item may not be visible or may be disabled
depending on GSC module usage
14
-
A3 -
Checklist Quick Access Panel
Checklist status is available from Status Light on the Quick Access Panel
15
Status Light
-
A3 -
Using the Object Status Feature
Each pipe or junction object requires some minimum input data
Until each object has the required input, it is "undefined" The Show Object Status feature checks the required data for
each object and reports to the user which objects are and are not defined Undefined object numbers change color (to red by default) Right clicking on an object will display a listing of the input,
output, and undefined items for that object
16
-
A3 -
Using the Object Status Feature (2)
Show Object Status is toggled on and off from the Workspace Toolbar (flood light) or the View Menu
Show Object Status should be used selectively because it slows down the Workspace graphics if left in the ON state For large models, users should turn it ON only when needed
17
-
A3 -
Using Undefined Objects Window
Opened from the View menu, undefined pipes and junctions are displayed in lists
Click on a pipe or junction to see undefined properties
18
-
A3 -
Solution Control Windows
Solution Control Window is opened from the Analysis Menu or by clicking the Solution Control in the Checklist area of the Quick Access Panel.
This window gives user control of how the Solver behaves The default parameters are sufficient for the majority of
analyses
19
-
A3 -
Output Control Window
Output Control Window is opened from the Tools Menu or by clicking the Output Control icon on the Toolbar
Users can modify and keep Output Control formats for future use
20
-
A3 -
Output Control Window (2)
Output Control offers users control over the following items: The pipe and junction output parameters to be included in the
output The engineering units in which the output parameters will be
expressed The order in which the output parameters will appear The title appearing on the output report Reference information to keep with model Special summary reports The minimum number of significant digits to appear in the output
parameters Where to direct the output once it has been obtained and
formatting
21
-
A3 -
System Properties Window
System Properties Window is opened from the Analysis Menu or by clicking the Systems Properties on the Checklist in the Quick Access Panel
This window allows the user to select a fluid (or fluids) for use in the model With Chempak, static mixtures can be created The AFT Standard fluid database is customizable ASME Steam properties can be used
Other options include: Change the gravity level, atmospheric pressure, transition
Reynolds numbers, and STP conditions A rotational velocity can be specified here for modeling
turbomachinery22
-
A3 -
Cost Setting Window
Cost Calculations are enabled on the Analysis Menu Pump Energy Only or Full Cost Calculations can be calculated
Cost Settings Window is opened from the Analysis Menu or clicking the Specify Cost Settings on the Checklist in the Quick Access Panel when Cost Calculations are being calculated
Various costs can be calculated such as material, installation, and operation/energy
23
-
A3 -
Entering Pipe and Junction Data
Data for pipes and junctions are entered into Properties Windows
Properties Windows are opened either by double-clicking or single-click then pressing enter for the pipe or junction of interest Properties windows may also be opened by double clicking an
object within the Model Data and Output windows Data can also be entered through Global Edit Windows
24
-
A3 -
Input Data For Pipes
All pipes must have data for Length Diameter Roughness Heat transfer model
In addition, each pipe must have at least two connecting junctions
25
-
A3 -
Input Data For Junctions
All junctions must have Elevation data
Connecting pipes are assumed to travel linearly between junctions Sufficient number of connecting pipes
Number of connecting pipes is different for each junction type
There are twenty different junctions
26
-
A3 -
Data For Bend Junctions
Use standard elbow for all bend Bend junction K factors may depend on diameter Diameter is picked up from upstream pipe
All Bend junctions must have two connecting pipes
27
-
A3 -
Data For Tanks
Can have 1-25 connecting pipes For open systems do not select balance energy option Tank junctions maintain a constant stagnation pressure
Also maintain constant stagnation temperature for flow into system
28
-
A3 -
Data For Branches
Branches can have from 2-25 pipes No additional data is needed for branches
29
-
A3 -
Data For Assigned Flow
Assigned Flow junctions connect to one pipe only Allows you to define an inflow or outflow Requires fluid temperature, but only uses temperature for
inflows
30
-
A3 -
Inspecting Objects
The data in a pipe or junction can be reviewed quickly using the inspection feature
Inspecting is done by pressing down the right mouse button on the graphical pipe or junction
Inspecting is much quicker than opening the Properties Window Using the inspection window also does not clear the output
results as opening a Properties window can
31
-
A3 -
Inspecting Objects Quick Access Panel
Pipe and Junction input/output data can be viewed in Quick Access Panel Click the Property tab on Quick Access Panel Select a pipe or a junction on Workspace
32
Property Tab
-
A3 -
Model Data Window
The Model Data window is useful for reviewing the text input for the model All data can be printed out for documentation
Model Data can be accessed from the Model Data Primary Window tab or from the Window menu
Use the Model Data window to do a quick sanity check of the input Incorrect units or a typo become more obvious in Model Data
Double-clicking the far left column of the tables opens the appropriate Properties Window
33
-
A3 -
Running Models - Solution Progress Window When a model is complete, the Run command is enabled The model can be run by choosing Run from the Analysis
Menu or clicking the appropriate toolbar icon When a model is running, the Solution Progress Window
displays The Solution Progress Window shows the status of the
Solver's progress towards convergence
34
-
A3 -
Running Models - Solution Progress Window (2) The Solution Progress Window allows you to Cancel or Pause
the run so that Solution Control parameters can be modified Modifying Solution Control parameters during runtime may help
for difficult models When the solution converges, you are notified When you select View Output, you are immediately taken to
the Output Window
35
-
A3 -
Output Window
The Output Window displays text output for your model and is accessed from the Primary Window tabs or Window menu
The Output Control Window allows you to customize the content of the output
Each section can be re-sized or collapsed allowing the user to focus on any of the sections
Each section may have multiple tabs to quickly view data by type
Print Format allows you to select the content of the printed report
36
-
A3 -
Output Window (2)
Transfer Results to Initial Guesses saves the current output results as the initial conditions Transfer Results to Initial Guesses may be accessed from the
Edit menu or the Output Toolbar (push pin) Warnings are placed into the General Results section
When warnings exist the text color is changed to red Sort allows you to sort the Output according to one the
columns Double-clicking the column header allows you to change the
units for that column
37
-
A3 -
Graph Results
Graphs are created with the Graph Results Window This window is one of the Primary Window tabs Graph Results can also be accessed from the Window menu
Various parameters can be graphed by clicking on the Select Graph Data button in the Graph Results window
The graph can be printed, copied to the clipboard, or saved to a file
The graph x-y data can be exported to a file or copied to the clipboard
38
-
A3 -
Visual Report
Visual Report allows you to see the results superimposed on the Workspace graphic This is one of the Primary Window tabs Visual Report can also be accessed from the Window menu
The Visual Report Control allows you to select the type of results you want to see
You can print the image at full size or fit it to a single page with Print Special
Text locations are automatically saved with the model
39
-
A3 -
Input for Demo 1
40
All pipes are Steel - ANSI, standard schedule (STD), standard friction dataGN2, Redlich-Kwong and Generalized Enthalpy, SolutionDefault Solution Control All elevations are zero
US
-
A3 -
Output for Demo 1
41
US
-
A3 -
Input for Demo 1
42
SI
All pipes are Steel - ANSI, standard schedule (STD), standard friction dataGN2, Redlich-Kwong and Generalized Enthalpy, SolutionDefault Solution Control All elevations are zero
-
A3 -
Output for Demo 1
43
SI
-
A4. Understanding Solution Control Options
-
Overview of Seminar
AFT ARROWA1. Overview of AFT ArrowA2. Fundamental Equations of Compressible FlowA3. Demonstration Problem - Determining Delivery ConditionsA4. Understanding Solution Control OptionsA5. AFT Arrow Hands-On ModelingA6. Troubleshooting AFT Arrow Models A7. The Five Primary WindowsA8. Pipe and Junction Details A9. Special TopicsA10. Verification of SolutionsA11. Using Scenario ManagerA12. Customizing Arrow and Using Databases A13. Introduction to AFT Arrow ModulesA14. More AFT Arrow Hands-On Modeling
-
A4 -
Solution Control Window Summary
The Solution Control Window is opened from the Analysis Menu
Solution Control is one of the Checklist items Solution Control is required for every model
AFT Arrow provides robust Solution Control defaults Parameters that can be modified include solution method,
solution step size, tolerance, relaxation and maximum iterations
You can also keep track of the iteration history
1
-
A4 -
Solution Control Window Summary (2)
2
-
A4 -
Solution Control Window Summary (3)
3
-
A4 -
How To Use Solution Control
In general, the defaults provided by AFT Arrow are sufficient to guide a model to convergence Care should be taken to review output to ensure operating
conditions are consistent with solution method It is recommended you avoid changing the Solution Control
parameters unless you understand how to use them or it is recommended by AFT or a more experienced user The danger is that it is possible to modify the Solution Control
parameters in such a way that the model will converge on the wrong answer
We will cover different convergence strategies later in the seminar
4
-
A4 -
Solution Methods
The Length March method and Mach Number March method have been discussed previously
Arrow 6 offers two methods that are hybrids of the two basic methods Length March with Mach Number Limits Mach Number March with Length Limits
Arrow 6 also offers two lumped methods Lumped Adiabatic Lumped Isothermal
5
-
A4 -
Length March Method
The Length March method takes solution steps over equal length steps The user can specify steps either as number per pipe, or by
absolute length The Length March method solution for a single pipe looks as
follows: Note equal length steps
6
-
A4 -
Mach Number March Method
The Mach Number March method takes solution steps over equal Mach number steps This allows the solver to follow rapidly accelerating flow in pipes
The Mach Number March method solution for a single pipe looks as follows: Note how the distance steps shorten towards the end of the pipe
1 2 3 4 5 6 7 8 9 10 11 Note equal Mach number increments
7
-
A4 -
Hybrid Solution Methods
The two hybrid methods combine the best of the two basic solution methods They prevent length steps that are excessively large, as well as
Mach number increments that are excessively large The hybrid methods switch between the two basic methods
dynamically
8
-
A4 -
Hybrid Solution Methods (2)
9
Note how equal length increments are used until Point 6, then equal Mach number increments are used
-
A4 -
Default Solution Method
The default solution method is the Length March method (with two increments per pipe) with Mach Number increments limits
Note that this method uses two pipe sections on every pipe in the model, no matter how long
Using more sections per pipe causes longer run times
10
-
A4 -
Tolerance Summary
There are three tolerance inputs for the three variables (four if dynamic mixing is modeled) Pressure (at all junctions) Mass Flow Rate (in all pipes) Enthalpy (at all junctions) Concentration (at all junctions if dynamic mixing)
Each tolerance has four criteria to choose from Absolute Relative Either Absolute or Relative Both Absolute and Relative
11
-
A4 -
Tolerances and Convergence
When solution iterations are performed, the values of all junction pressures, enthalpies and pipe flow rates progress from the initial guesses (which are incorrect) to the true results (which satisfy the governing equations)
The solution method needs to have a criteria to decide when the results are good enough so it can stop iterating The tolerance values are the criteria the solution method
compares against to decide to stop iterating
12
-
A4 -
Tolerances and Convergence (2)
The best way to determine whether results are converged is to compare the results of the current iteration to those of the previous iteration If the results do not change appreciably then the true results
have been obtained Each iteration AFT Arrow does this check and when the
change in results for all the pipes and junctions is less than the specified tolerance, it considers the results converged
13
-
A4 -
The relative tolerance approach does the comparison of current vs. previous on a relative change (i.e., percentage change) basis
Relative tolerance is the AFT Arrow default because it is the
most robust AFT Arrow uses 0.0001 (i.e., 0.01%) as the
criteria for both pressure and flow
Relative Tolerance
( ) P P P
j new j old
j new
, ,
,
< TOL If (For All Junctions) Then
Convergence = True Else
Convergence = False End If
14
-
A4 -
This method is especially good for systems with highly different flow rates because each flow rate must converge to a percentage value only
One drawback of this method is if systems have zero or near zero flow rates
Relative Tolerance (2)
15
( ) P P P
j new j old
j new
, ,
,
< TOL If (For All Junctions) Then
Convergence = True Else
Convergence = False End If
-
A4 -
The absolute tolerance approach does the comparison of current vs. previous on an absolute change basis (i.e., number of psi's)
Absolute tolerance has units associated with it
Absolute Tolerance
If (For All Junctions) Then
Convergence = True Else
Convergence = False
End If
P P j new j old , , < TOL
16
-
A4 -
This method is good for systems with flows that are all of a similar magnitude
Typically, both tolerance settings will give (and should give) the same answer Usually relative tolerance is more efficient and reliable
Absolute Tolerance (2)
If (For All Junctions) Then
Convergence = True Else
Convergence = False
End If
P P j new j old , , < TOL
17
-
A4 -
Tolerance Application:
Note that this convergence and tolerance is for pressure
(psia) -------------(lbm/s)------------- (lbm/s) (lbm/s/psia) --- (psia) P4 M1 M2 M3 F F' REL CHNG ABS CHNG 180.0000 115.3464 -78.9048 -313.5906 -277.1489 -18.6139 --- --- 165.1106 152.3476 110.9691 -158.5207 104.7960 -23.3027 9.0178E-02 1.4889E+01 169.6078 142.1903 81.9409 -217.3503 6.7810 -21.2485 2.6515E-02 4.4972E+00 169.9269 141.4418 79.4792 -220.9305 -0.0095 -21.3129 1.8780E-03 3.1913E-01 169.9265 141.4429 79.4827 -220.9255 0.0000 -21.3128 2.6127E-06 4.4398E-04 169.9265 141.4429 79.4827 -220.9255 0.0000 -21.3128 6.8205E-12 1.1590E-09 169.9265 141.4429 79.4827 -220.9255 0.0000 -21.3128 0.0000E+00 0.0000E+00
1 2 3 4 5 6 7
Iter #
18
-
A4 -
Solver Flow Chart
Recalculate Concentrations *
Start
Recalculate Enthalpies
Solve Junction Pressures
Yes
> Max Iterations ? Converged ? End Yes No
No
> Max Iterations ? Converged ? No No
Yes
Return
Recalculate Mass Flow Rates
> Max Iterations ? Converged ? End Yes No
No
Yes
Update Flow Losses and Compressors
Recalculate Enthalpies
Recalculate Concentrations *
> Max Iterations ? Converged ? End Yes No
No
Yes Yes
* A concentration balance is performed only if dynamic mixing is modeled. If not, the Solver passes through this block.
19
-
A4 -
Relaxation Overview
Relaxation slows the amount of flow rate change allowed by the solution Relaxation is like a damping factor that smoothens the convergence
process while also slowing the process Relaxation is applied to the flow rate and pressure update for all pipes, i
Relaxation is always greater than zero and less than or equal to one Relaxation of 1 is the same as no relaxation Relaxation of 0 would never update the flow rates
A relaxation of 1 is fastest Arrow will automatically reduce flow relaxation if dictated by the solution
progress Flow relaxation less than 0.01 is almost never required If you relaxation values -
For flow, typical settings for highly non-linear models are 0.1 or 0.05 For pressure, never use anything other than 0.5 or 1
m m m m ( ) , , , , r i new i new i old i old = + p r p p p i new p i new i old i old , , , , ( ) = +
20
-
A4 -
Relaxation Application
Calculate the new flow rates for several values of relaxation Relaxation Old Flow Rate (lbm/s)
Ideal New Flow Rate (lbm/s)
New Flow Rate (lbm/s)
1 10 20 20 0.5 10 20 15 0.2 10 20 12 0.1 10 20 11
0.05 10 20 10.5 0 10 20 10
21
-
A4 -
The Solution Progress Window (which displays while the Solver is running) communicates the maximum out of tolerance junction pressure, junction enthalpy and pipe mass flow rate
Junction pressures are solved first and the pipe flow rates and junction enthalpies are updated
Completing the Picture on Tolerance
22
-
A4 -
Using Transfer Results to Initial Feature
The Output Window has a feature called Transfer Results to Initial
This features takes the current results and transfers them to the initial guess for each pipe and junction
This makes the model run much faster in the future Transfer Results to Initial can be always enabled with the
Output Control window
23
-
A4 -
Maximum Global Iterations
The Maximum Iterations parameters restricts the total number of iterations for the Solver to calculate
The Maximum Global Iterations can be as high or as low as you want - it has no effect on the behavior of the Solver
The purpose of this parameter is to keep the Solver from searching forever for a solution it cannot obtain
Most models will converge within 50000 iterations, which is the default
24
-
A4 -
Maximum Local Iterations
Maximum Local Iterations should usually be left at 500 It can be changed to as high as 2000 in certain cases
Local Iterations are those iterations Arrow performs when it marches down a specific pipe
When marching down a pipe Arrow will iterate on all of the local solution parameters to make them consistent with each other for the step size
In certain cases Arrow cannot drive all of the local solution parameters to convergence within the desired local tolerance
25
-
A4 -
Maximum Local Iterations (2)
If no limit is set on this process, Arrow will get stuck in an infinite loop To make sure this does not happen, Arrow checks the local
iteration loop counter against the Maximum Local Iterations set in Solution Control
When the local counter reaches the limit, Arrow will warn the user and offer several options
26
-
A4 -
Local Iteration Control
If a local iteration problem occurs on the final iteration, AFT Arrow keeps a record of the problem and automatically retries the iteration (up to 3 times by default)
If the local iteration problem happens and it is not the final iteration, AFT Arrow ignores it It will not affect the solution because it is not the final iteration This happens frequently inside the Solver and the
user will never see it
27
-
A4 -
Local Iteration Control (2)
28
If a local iteration problem happens three times in a row, AFT Arrow displays convergence and then adds a warning to the output IF the value is more than 50 (by default) times greater than the tolerance
If the Error Value is far above the Tolerance, the answer is not reliable If it is only slightly out of
tolerance, this is usually not a problem
-
A5. AFT Arrow Example Models
-
Overview of Seminar
AFT ARROWA1. Overview of AFT ArrowA2. Fundamental Equations of Compressible FlowA3. Demonstration Problem - Determining Delivery ConditionsA4. Understanding Solution Control OptionsA5. AFT Arrow Hands-On ModelingA6. Troubleshooting AFT Arrow Models A7. The Five Primary WindowsA8. Pipe and Junction Details A9. Special TopicsA10. Verification of SolutionsA11. Using Scenario ManagerA12. Customizing Arrow and Using Databases A13. Introduction to AFT Arrow ModulesA14. More AFT Arrow Hands-On Modeling
-
A5 -
Introduction to Scenario Manager
The Scenario Manager allows you to keep variants of a model all with the same model
The types of changes that can be made are very broad Junctions can be turned on and off to evaluate different
operating conditions Pipe and junction data can be varied to parametrically evaluate
competing designs You can build an existing system as your base model then add to
the system to evaluate expansion possibilities on the existing system
You can easily evaluate different working fluids by setting them up as different children scenarios
1
-
A5 -
Introduction to Scenario Manager (2)
Scenarios are created, manipulated, and loaded using the Scenario Manager window
The Scenario Manager can be opened from the Tools menu in the Workspace window, the Scenario Manager button on the toolbar, or Quick Access Panel
2
From Quick Access Panel
-
A5 -
Build Model of Nitrogen Transfer System
Hands-on problem #1 (TEST1.ARO - "Length March 2 Segments" Scenario) - Find flow rate through pipe
SolutionControl NotDefaultLengthMarch,2segmentsGN2IdealGasReferenceEnthalpy
Po =225psiaTo =100FEl =0
Po =400psiaTo =100FEl =0
1
2
pipe1
US
3
L = 200 ftSteel - ANSI3 in., STD (Sch. 40)Adiabatic
-
A5 -
Modify Test Model #1 for Real Gases
Modify model to use real gas features, Redlich-Kwong and Generalized Enthalpy ("Real Gas" Scenario) - Find flow rate through pipe
Po =225psiaTo =100FEl =0
Po =400psiaTo =100FEl =0
1
2
pipe1L = 200 ftSteel - ANSI3 in., STD (Sch. 40)Adiabatic
SolutionControlLengthMarch,2segmentsGN2RedlichKwongGeneralizedEnthalpy
US
4
-
A5 -
Modify Test Model #1 for More Sections
Change number of pipe sections ("Length March 10 Segments" Scenario) - Find flow rate through pipe
Po =225psiaTo =100FEl =0
Po =400psiaTo =100FEl =0
1
2
pipe1
SolutionControlLengthMarch,10segmentsGN2RedlichKwongGeneralizedEnthalpy
US
5
L = 200 ftSteel - ANSI3 in., STD (Sch. 40)Adiabatic
-
A5 -
Modify Test Model #1 Solution Method
Change solution method to Mach March method ("Mach March Step 0.01" Scenario) - Find flow rate through pipe
Po =225psiaTo =100FEl =0
Po =400psiaTo =100FEl =0
1
2
pipe1
SolutionControlMachMarch,.01incrementsGN2RedlichKwongGeneralizedEnthalpy
US
6
L = 200 ftSteel - ANSI3 in., STD (Sch. 40)Adiabatic
-
A5 -
Modify Test Model #1 Solution Method
Change solution method to Arrow defaults (Length March to Mach March transition) - Find flow rate through pipe
Po =225psiaTo =100FEl =0
Po =400psiaTo =100FEl =0
1
2
pipe1
SolutionControlLengthMarchwithMachNumberLimits,2segments&0.01incrementsGN2RedlichKwongGeneralizedEnthalpy
US
7
L = 200 ftSteel - ANSI3 in., STD (Sch. 40)Adiabatic
-
A5 -
Modify Test Model #1 - Add Heat Transfer
Change model so that ambient temperature is 20F, no insulation and external convection coefficient is 100 Btu/Hr-ft2-R ("Heat Transfer" Scenario) Use Default Length March with Mach Number Limits
How does this affect the flow rate? What was the flow rate prediction error from the original
calculation?Eqn.State Enthalpy SolutionMethod Increment FlowrateIdeal Reference LengthMarch 2 20.20RK General LengthMarch 2 20.29RK General LengthMarch 10 20.73RK General MachMarch 0.01 20.74
Heat/RK General MachMarch 0.01 21.26RK General Default 0.01 20.74
US
8
-
A5 -
Graph Results
Use Graph Results to see how velocity changes along the pipe for the Heat Transfer scenario
US
9
-
A5 -
Size Helium Storage Tank
It is required to supply two systems with at least 2.5 lbm/s of helium (each) at a minimum 100 psia stagnation pressure
A single line runs from your tank for 1000 feet and then splits into two lines each 500 feet long. These two lines supply the two demand points. All pipe is steel - ANSI, 4 inch, schedule 40.
The lines are well insulated and thus heat transfer can be neglected to ambient
Elevation changes can be neglected
10
US
-
A5 -
Size Helium Storage Tank (2)
The system must function on hot and cold days, with the design ambient temperatures at 30 F for cold days and 100 F for hot days
The helium supply vessel will be outside, and thus will always be at the ambient temperature
What minimum (stagnation) pressure must the storage tank be designed to guarantee adequate supply year round?
TEST2.ARO (Hot and Cold Scenarios)
11
US
-
A5 -
Model Control Valve
Model flow control valve (TEST3.ARO, Base Scenario) Steam flow Use steam properties from AFT Standard Use real gas and real enthalpy models Inlet (stagnation) pressure is 250 psia at 425 F Exit (stagnation) pressure is 150 psia (assume 425F - this
temperature at the discharge will not be used) Pipe is horizontal Pipe is uninsulated 450 feet long and Standard Schedule (40)
steel - ANSI
12
US
-
A5 -
Model Control Valve (2)
Model flow control valve (TEST3.ARO, Base Scenario): contd Ambient temperature is 60F External heat transfer coefficient is 10 Btu/hr-ft2-F (same as
Btu/hr-ft2-R) Minimum pressure drop is 20 psid across valve Required flow rate is 5 lbm/s Assume valve is in the middle
Find minimum pipe size if valve is in middle of pipe Do the results change if the valve is not in the middle?
13
US
-
A5 -
Model Control Valve (3)
Fluids in the AFT Standard database do not have saturation line data It is not possible to evaluate condensation Chempak fluids and the ASME Steam data do have saturation
line data Use steam data from the Chempak database to evaluate
whether condensation will occur. Does it? TEST3.ARO - "Chempak - No Insulation" Scenario
14
US
-
A5 -
Model Control Valve (4)
To prevent condensation one can add insulation. The selected insulation has a thermal conductivity of 0.1 Btu/hr-ft-R
Insulation thickness of 0.25, 0.5 and 1.0 inch are being considered. Will they work? If so, which is best?
Is the control valve pressure drop still acceptable? TEST3.ARO - "Insulation XX inch" Scenarios
15
US
-
A5 -
Answers to Problems
TEST2 T-hot needs 265 psia (stagnation) T-cold needs 250 psia (stagnation) Requirement is thus 265 psia
TEST3, Use 3 inch pipes, which gives a stagnation pressure drop of 26.30 psid Results change if valve is not in middle (less pressure drop
available as valve is moved towards pipe inlet)
16
US
-
A5 -
Answers to Problems (2)
TEST3 with Chempak Condensation begins in pipe 1 before the control valve The previous sizing calculation is thus invalid
TEST3 with insulation Insulation of 0.5 or 1 inch prevents condensation. The 0.25 inch
does not. With 0.5 inch insulation the control valve pressure drop is 22.29
psid using 3 inch pipe With 1 inch insulation the control valve pressure drop is 21.67
psid using 3 inch pipe Use 0.5 inch insulation
17
US
-
A5 -
Build Model of Nitrogen Transfer System
Hands-on problem #1 (TEST1 (SI).ARO - "Length March 2 Segments" Scenario) - Find flow rate through pipe
SolutionControl NotDefaultLengthMarch,2segmentsGN2IdealGasReferenceEnthalpy
Po =1600kPaTo =40CEl =0meters
Po =2800kPaTo =40CEl =0meters
1
2
pipe1
L=60metersSteel ANSI3in.,Sch.40.(7.8cmID)Adiabatic
18
SI
-
A5 -
Modify Test Model #1 for Real Gases
Modify model to use real gas features, Redlich-Kwong and Generalized Enthalpy ("Real Gas" Scenario) - Find flow rate through pipe
Po =1600kPaTo =40CEl =0meters
Po =2800kPaTo =40CEl =0meters
1
2
pipe1
L=60metersSteel ANSI3in.,Sch.40(7.8cmID)Adiabatic
SolutionControlLengthMarch,2segmentsGN2RedlichKwongGeneralizedEnthalpy 19
SI
-
A5 -
Modify Test Model #1 for More Sections
Change number of pipe sections ("Length March 10 Segments" Scenario) - Find flow rate through pipe
Po =1600kPaTo =40CEl =0meters
Po =2800kPaTo =40CEl =0meters
1
2
pipe1
L=60metersSteel ANSI3in.,Sch.40.(7.8cmID)Adiabatic
SolutionControlLengthMarch,10segmentsGN2RedlichKwongGeneralizedEnthalpy 20
SI
-
A5 -
Modify Test Model #1 Solution Method
Change solution method to Mach March method ("Mach March Step 0.01" Scenario) - Find flow rate through pipe
Po =1600kPaTo =40CEl =0meters
Po =2800kPaTo =40CEl =0meters
1
2
pipe1
L=60metersSteel ANSI3in.,Sch.40.(7.8cmID)Adiabatic
SolutionControlMachMarch,.01incrementsGN2RedlichKwongGeneralizedEnthalpy 21
SI
-
A5 -
Modify Test Model #1 Solution Method
Change solution method to Arrow defaults (Length March to Mach March transition) - Find flow rate through pipe
Po =1600kPaTo =40CEl =0meters
Po =2800kPaTo =40CEl =0meters
1
2
pipe1
L=60metersSteel ANSI3in.,Sch.40.(7.8cmID)Adiabatic
SolutionControlLengthMarchwithMachNumberLimits,2segments&0.01incrementsGN2RedlichKwongGeneralizedEnthalpy
22
SI
-
A5 -
Modify Test Model #1 - Add Heat Transfer
Change model so that ambient temperature is -7.0 C, no insulation and external convection coefficient is 570 W/m2-K ("Heat Transfer" Scenario) Use Default Length March with Mach Number Limits
How does this affect the flow rate? What was the flow rate prediction error from the original
calculation?Eqn.State Enthalpy SolutionMethod Increment FlowrateIdeal Reference LengthMarch 2 9.290RK General LengthMarch 2 9.332RK General LengthMarch 10 9.522RK General MachMarch 0.01 9.526Heat/RK General MachMarch 0.01 9.771RK General Default 0.01 9.526
23
SI
-
A5 -
Graph Results
Use Graph Results to see how velocity changes along the pipe for the Heat Transfer scenario
24
SI
-
A5 -
Size Helium Storage Tank
It is required to supply two systems with at least 1.1 kg/s of helium (each) at a minimum 700 kPa stagnation pressure
A single line runs from your tank for 300 meters and then splits into two lines each 150 meters long. These two lines supply the two demand points. All pipe is steel - ANSI, 4 inch (10.2 cm ID), schedule 40.
The lines are well insulated and thus heat transfer can be neglected to ambient
Elevation changes can be neglected
25
SI
-
A5 -
Size Helium Storage Tank (2)
The system must function on hot and cold days, with the design ambient temperatures at -1.0 C for cold days and 40 C for hot days
The helium supply vessel will be outside, and thus will always be at the ambient temperature
What minimum (stagnation) pressure must the storage tank be designed to guarantee adequate supply year round?
TEST2 (SI).ARO (Hot and Cold Scenarios)
26
SI
-
A5 -
Model Control Valve
Model flow control valve (TEST3 (SI).ARO, Base Scenario) Steam flow Use steam properties from AFT Standard Use real gas and real enthalpy models Inlet (stagnation) pressure is 1725 kPa at 220 C Exit (stagnation) pressure is 1050 kPa (assume 220C - this
temperature at the discharge will not be used) Pipe is horizontal Pipe is uninsulated 140 meters long and Standard Schedule (40)
steel - ANSI
27
SI
-
A5 -
Model Control Valve (2)
Model flow control valve (TEST3 (SI).ARO, Base Scenario) contd Ambient temperature is 16C External heat transfer coefficient is 57 W/m2-K Minimum pressure drop is 140 kPa across valve Required flow rate is 2.25 kg/s Assume valve is in the middle
Find minimum pipe size if valve is in middle of pipe Do the results change if the valve is not in the middle?
28
SI
-
A5 -
Model Control Valve (3)
Fluids in the AFT Standard database do not have saturation line data It is not possible to evaluate condensation Chempak fluids and the ASME Steam data do have saturation
line data Use steam data from the Chempak database to evaluate
whether condensation will occur. Does it? TEST3 (SI).ARO - "Chempak - No Insulation" Scenario
29
SI
-
A5 -
Model Control Valve (4)
To prevent condensation one can add insulation. The selected insulation has a thermal conductivity of 0.2 W/m-K
Insulation thickness of 0.5, 1.25 and 2.5 cm are being considered. Will they work? If so, which is best?
Is the control valve pressure drop still acceptable? TEST3 (SI).ARO - "Insulation XX cm" Scenarios
30
SI
-
A5 -
Answers to Problems
TEST2 (SI) T-hot needs 1778 kPa (stagnation) T-cold needs 1675 kPa (stagnation) Requirement is thus 1778 kPa
TEST3 (SI), Use 3 inch (7.8 cm ID) pipes, which gives a stagnation pressure drop of 168.2 kPa Results change if valve is not in middle (less pressure drop
available as valve is moved towards pipe inlet)
31
SI
-
A5 -
Answers to Problems (2)
TEST3 (SI) with Chempak Condensation begins in pipe 1 before the control valve The previous sizing calculation is thus invalid
TEST3 (SI) with insulation Insulation of 1.25 or 2.5 cm prevents condensation. The 0.5 cm
does not. With 1.25 cm insulation the control valve pressure drop is 141.7
kPa using 3 inch pipe With 2.5 cm insulation the control valve pressure drop is 135.4
kPa using 3 inch pipe Use 1.25 cm insulation
32
SI
-
A6. Troubleshooting AFT Arrow Models
-
Overview of Seminar
AFT ARROWA1. Overview of AFT ArrowA2. Fundamental Equations of Compressible FlowA3. Demonstration Problem - Determining Delivery ConditionsA4. Understanding Solution Control OptionsA5. AFT Arrow Hands-On ModelingA6. Troubleshooting AFT Arrow Models A7. The Five Primary WindowsA8. Pipe and Junction Details A9. Special TopicsA10. Verification of SolutionsA11. Using Scenario ManagerA12. Customizing Arrow and Using Databases A13. Introduction to AFT Arrow ModulesA14. More AFT Arrow Hands-On Modeling
-
A6 -
Getting the Right Results
There are a number of modeling problems AFT sees frequently
This section offers numerous strategies and suggestions for approaching modeling problems
1
-
A6 -
0
2
4
6
8
10
12
0 50 100 150 200
Flow (ft3/sec)
Pres
sure
(psi
d)
0
2
4
6
8
10
12
0 50 100 150 200
Flow (ft3/sec)
Pres
sure
(psi
d)
Compressor/Fan Curves
AFT Arrow allows you to use compressor/fan curves up to fourth order
Quadratic compressor/fan curves always reach a zero head value
Third or fourth order compressor/fan curves may not reach the zero head axis
This is a problem because the compressor/fan curve may start increasing This will cause problems for
the Solver 2BAD!
Chart1
10
9.95
9.8
9.55
9.2
8.75
8.2
7.55
6.8
5.95
5
3.95
2.8
1.8
1.2
1
1.2
1.8
2.8
4.2
6
Flow (ft3/sec)
Pressure (psid)
Sheet8
Pointx (feet)Mass Flow (lbm/sec)Velocity (feet/sec)Mach #P Stag. (psia)P Static (psia)T Stag. (deg. F)T Static (deg. F)H Stag. (Btu/lbm)H Static (Btu/lbm)fRho Static (lbm/ft3)
102.10648343.8560.22404110096.6032540.317530.781310.699308.3370.01935960.23449500.2240411
210.05072.10648359.0380.23404195.982492.4316540.316529.919310.699308.1240.01935950.22457980.1005070.2340410.959824
318.842.10648374.1990.24404192.305788.6005540.316529.021310.699307.9020.01935950.21548060.18840.2440410.923057
426.56472.10648389.3390.25404188.929585.0696540.316528.087310.699307.6710.01935950.20710110.2656470.2540410.889295
533.38472.10648404.4580.26404185.819881.8049540.316527.118310.699307.4320.01935950.19935950.3338470.2640410.858198
639.43132.10648419.5550.27404182.947578.7771540.316526.113310.699307.1830.01935940.19218610.3943130.2740410.829475
744.81292.10648434.6280.28404180.287575.9612540.316525.073310.699306.9260.01935940.1855210.4481290.2840410.802875
849.61992.10648449.6770.29404177.818273.3357540.316523.998310.699306.660.01935940.17931210.4961990.2940410.778182
953.92772.10648464.7020.30404175.520870.8816540.316522.888310.699306.3860.01935930.17351470.5392770.3040410.755208
1057.82.10648479.7010.31404173.37968.5827540.316521.743310.699306.1030.01935930.16808930.5780.3140410.73379
1161.29082.10648494.6740.32404171.378566.4246540.316520.564310.699305.8120.01935920.16300160.6129080.3240410.713785
1264.44592.10648509.6190.33404169.506564.3946540.316519.351310.699305.5120.01935920.15822120.6444590.3340410.695065
1367.30472.10648524.5370.34404167.75262.4815540.317518.105310.699305.2040.01935910.15372140.6730470.3440410.67752
1469.90092.10648539.4260.35404166.10560.6755540.317516.825310.699304.8880.01935910.14947840.6990090.3540410.66105
1572.26372.10648554.2860.36404164.556758.9677540.318515.511310.699304.5630.01935910.1454710.7226370.3640410.645567
1674.41832.10648569.1170.37404163.099357.3503540.318514.165310.699304.230.0193590.14168030.7441830.3740410.630993
1776.38682.10648583.9160.38404161.725855.8162540.319512.786310.699303.8890.0193590.13808940.7638680.3840410.617258
1878.18842.10648598.6840.39404160.429854.359540.32511.374310.699303.5410.01935890.13468320.7818840.3940410.604298
1979.83992.10648613.4190.40404159.205752.9731540.321509.93310.699303.1840.01935880.13144780.7983990.4040410.592057
2081.35612.10648628.1220.41404158.048251.6532540.322508.455310.699302.8190.01935880.12837090.8135610.4140410.580482
2182.75022.10648642.7910.42404156.952750.3948540.323506.948310.699302.4470.01935870.12544130.8275020.4240410.569527
2284.03352.10648657.4260.43404155.91549.1935540.324505.409310.699302.0670.01935870.12264890.8403350.4340410.55915
2385.21652.10648672.0260.44404154.931348.0456540.326503.84310.699301.6790.01935860.11998430.8521650.4440410.549313
2486.30812.10648686.5910.45404153.998146.9475540.328502.24310.699301.2840.01935860.11743910.8630810.4540410.539981
2587.31662.10648701.1190.46404153.112145.8959540.329500.61310.699300.8820.01935850.11500560.8731660.4640410.531121
2688.24922.10648715.610.47404152.270544.888540.331498.95310.699300.4720.01935840.11267670.8824920.4740410.522705
2789.11242.10648730.0640.48404151.470543.921540.333497.26310.699300.0540.01935840.11044590.8911240.4840410.514705
2889.91212.10648744.480.49404150.709742.9925540.335495.541310.699299.630.01935830.10830730.8991210.4940410.507097
2990.65352.10648758.8570.50404149.985842.1001540.338493.793310.699299.1980.01935820.10625530.9065350.5040410.499858
3091.34142.10648773.1940.51404149.296741.2419540.34492.017310.699298.760.01935810.1042850.9134140.5140410.492967
3191.982.10648787.4920.52404148.640640.4157540.343490.212310.699298.3140.01935810.10239160.91980.5240410.486406
3292.57332.10648801.7490.53404148.015539.6199540.346488.379310.699297.8610.0193580.10057090.9257330.5340410.480155
3393.12482.10648815.9650.54404147.4238.8527540.349486.519310.699297.4020.01935790.09881870.9312480.5440410.4742
3493.63762.10648830.1390.55404146.852438.1127540.352484.631310.699296.9360.01935780.09713150.9363760.5540410.468524
3594.11482.10648844.2710.56404146.311337.3983540.355482.717310.699296.4640.01935770.09550560.9411480.5640410.463113
3694.55882.10648858.3590.57404145.795436.7082540.359480.776310.699295.9850.01935770.0939380.9455880.5740410.457954
3794.97222.10648872.4050.58404145.303536.0413540.363478.809310.699295.4990.01935760.09242570.9497220.5840410.453035
3895.35722.10648886.4060.59404144.834435.3963540.367476.816310.699295.0070.01935750.09096570.9535720.5940410.448344
3995.71572.10648900.3630.60404144.387134.7721540.371474.798310.699294.5090.01935740.08955560.9571570.6040410.443871
4096.04972.10648914.2750.61404143.960634.1677540.375472.755310.699294.0050.01935730.08819290.9604970.6140410.439606
4196.36092.10648928.1420.62404143.553933.5823540.38470.687310.699293.4950.01935720.08687530.9636090.6240410.435539
4296.65082.10648941.9620.63404143.166233.0148540.384468.595310.699292.9790.01935710.08560060.9665080.6340410.431662
4396.92092.10648955.7370.64404142.796732.4645540.389466.479310.699292.4570.0193570.0843670.9692090.6440410.427967
4497.17252.10648969.4640.65404142.444631.9306540.395464.339310.699291.9290.01935690.08317240.9717250.6540410.424446
4597.40682.10648983.1430.66404142.109131.4123540.4462.177310.699291.3960.01935680.08201510.9740680.6640410.421091
4697.62492.10648996.7750.67404141.789730.909540.406459.991310.699290.8570.01935670.08089350.9762490.6740410.417897
4797.8282.106481010.3590.68404141.485630.4199540.412457.784310.699290.3120.01935660.07980590.978280.6840410.414856
4898.0172.106481023.8930.69404141.196329.9446540.418455.554310.699289.7620.01935650.0787510.980170.6940410.411963
4998.19272.106481037.3790.70404140.921329.4824540.425453.303310.699289.2070.01935630.07772720.9819270.7040410.409213
5098.35612.106481050.8150.71404140.6629.0327540.431451.03310.699288.6470.01935620.07673340.9835610.7140410.4066
5198.50782.106481064.2010.72404140.411828.5951540.438448.737310.699288.0810.01935610.07576820.9850780.7240410.404118
5298.64872.106481077.5370.73404140.176428.1691540.446446.423310.699287.5110.0193560.07483050.9864870.7340410.401764
5398.77932.106481090.8210.74404139.953327.7541540.453444.09310.699286.9360.01935590.07391920.9877930.7440410.399533
5498.90032.106481104.0550.75404139.742127.3498540.461441.736310.699286.3560.01935570.07303310.9890030.7540410.397421
5599.01232.106481117.2370.76404139.542426.9558540.469439.364310.699285.7710.01935560.07217140.9901230.7640410.395424
5699.11592.106481130.3670.77404139.353826.5716540.477436.972310.699285.1820.01935550.07133310.9911590.7740410.393538
5799.21142.106481143.4450.78404139.17626.1969540.486434.562310.699284.5880.01935540.07051720.9921140.7840410.39176
5899.29952.106481156.4710.79404139.008625.8313540.495432.134310.699283.9890.01935520.0697230.9929950.7940410.390086
5999.38042.106481169.4440.80404138.851325.4746540.504429.689310.699283.3870.01935510.06894950.9938040.8040410.388513
6099.45482.106481182.3630.81404138.703925.1263540.514427.226310.699282.780.01935490.06819610.9945480.8140410.387039
6199.52292.106481195.2290.82404138.56624.7861540.524424.746310.699282.1690.01935480.0674620.9952290.8240410.38566
6299.5852.106481208.0410.83404138.437424.4539540.534422.25310.699281.5540.01935