SlideShare a Scribd company logo
1 of 31
Download to read offline
Nonlinear estimation of a power law for the friction in a
pipeline
(IFAC MICNON 2018)
n m N
Authors: Lizeth Torres and Cristina Verde
ftorreso@iingen.unam.mx
http://lizeth-torres.info/
20th June 2018
Supported by Proyecto 280170, Convocatoria 2016-3, Fondo Sectorial CONACyT-Secretaría de Energía-Hidrocarburos.
Nonlinear estimation of a power law for the friction in a pipeline
Contents
1 Motivation
2 Contribution
3 Background
4 A power law for the total head losses
5 Identification approach
6 Design of the observers
7 Experimental results
Lizeth Torres | 20th June 2018 2 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Motivation
Contents
1 Motivation
2 Contribution
3 Background
4 A power law for the total head losses
5 Identification approach
6 Design of the observers
7 Experimental results
Lizeth Torres | 20th June 2018 3 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Motivation
Motivation of the research
Contributing to the proper management of distribution pipelines.
A proper management requires some tasks such as ...
The diagnosis and prognosis of the pipeline components.
The control for the adequate distribution of the fluids.
The solution of optimization problems.
⇓
Algorithms based on models formulated from physical laws that govern the
flow behavior throughout the pipelines.
Lizeth Torres | 20th June 2018 4 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Motivation
Disadvantage
Algorithms based on governing equations need to be frequently updated to
avoid being invalid because of the nonstop use and the passing of time.
Energy dissipation changes.
Affected Parameters
Roughness εa
.
Diameter φ.
a
It is not simple to measure it
Aging deterioration consequences
Corrosion
Tuberculation
Erosion
Mineral deposits
Lizeth Torres | 20th June 2018 5 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Motivation
Motivation
Main goal of this article
Proposing an algorithm to continuously estimate the energy dissipation in
a pipeline without measuring ε, φ and L.
Why?
To avoid miscalculations due to changes of the physical parameters that
could affect the performance of model-based algorithms.
Lizeth Torres | 20th June 2018 6 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Contribution
Contents
1 Motivation
2 Contribution
3 Background
4 A power law for the total head losses
5 Identification approach
6 Design of the observers
7 Experimental results
Lizeth Torres | 20th June 2018 7 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Contribution
Contribution
A power law to represent the energy dissipation mainly caused by
friction in a pipeline.
A simple method to estimate such a power law with the following
characteristics:
1 It is based on nonlinear state observers.
2 It only requires two pressure head recordings and a flow rate
measurement.
3 It performs the estimation in short-time.
Lizeth Torres | 20th June 2018 8 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Background
Contents
1 Motivation
2 Contribution
3 Background
4 A power law for the total head losses
5 Identification approach
6 Design of the observers
7 Experimental results
Lizeth Torres | 20th June 2018 9 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Background
The classic way to compute the energy dissipation
In pipelines, energy dissipation is called head loss and can be divided into
two main categories: major losses associated with energy loss per length of
pipe and minor losses associated with bends, fittings, valves, that act against
the fluid and reduce its energy.
∆Htot = ∆Hf + ∆Hm
∆Htot = f
L
φ
v2
2g
+
N
i=1
ki
v2
2g
= f
L
φ
+
N
i=1
ki
v2
2g
∆Htot =


f
L +
φ N
i=1 ki
f
φ



v2
2g
*Darcy-Weisbach equation
Lizeth Torres | 20th June 2018 10 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Background
Total head loss in pipeline
If Leq = L +
φ N
i=1 ki
f then
∆Htot = f
Leq
φ
v2
2g
Leq is known as equivalent length, which actually is the virtual length of a pipeline
with devices as if it were a horizontal pipeline without devices.
Note: If a device (e.g. a valve) in the pipeline is added or changed, the equivalent
length will change and consequently the total head loss.
The computation of f depends on the flow regime.
Lizeth Torres | 20th June 2018 11 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Background
Total head loss in pipeline
For laminar flow:
f = κ/Re
where κ depends on the pipeline geometry. For circular pipes κ = 64.
For turbulent flow (natural regime in pipelines): f can be accurately calculated by
using the Colebrook-White (CW) equation
1
f
= −2 log10
ε
3.71φ
+
2.51
Re f
Some disadvantages: ε is required, it is an implicit equation, it represents the
complete turbulent regimen of the Moody diagram! We don’t need that, we only
need to represent ∆Htot for the region where our pipelines work.
These disadvantages are reasons to propose new formulas to represent ∆Htot.
Lizeth Torres | 20th June 2018 12 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Background
Another reasons to propose new formulas for ∆Htot
An identifiable equation
To have an equation with empirical parameters that are easy to estimate
A differentiable equation
Some optimization and identification algorithms require the derivative of
the friction.
Lizeth Torres | 20th June 2018 13 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Background
Some proposed alternatives
1 Hazen William equation
∆Htot =
10.67
C1.852
LeqQ1.852
φ4.8704
2 Wood equation
∆Htot = (a(ε) + b(ε)Re−c(ε)
)
f
Q2Leq
2gφA2
r
3 Valiantzas equation
∆Htot = Leq
k0Q2
φ5.3
m
, k0 = 0.0126ε0.3
4 Quadratic (Prony) equation
∆Htot = aQ2
+ bQ
5 Rojas equations 1
∆Htot =
Leq
2gφA2
r
(θ1Q2
|Q| + θ2Q|Q|), ∆Htot =
Leq
2gφA2
r
θ1Q|Q| + θ2|Q|
1
Rojas Jorge et al. On-Line Head Loss Identification for Monitoring of Pipelines,
Safeprocess 2018, Poland.
Lizeth Torres | 20th June 2018 14 / 31
Nonlinear estimation of a power law for the friction in a pipeline | A power law for the total head losses
Contents
1 Motivation
2 Contribution
3 Background
4 A power law for the total head losses
5 Identification approach
6 Design of the observers
7 Experimental results
Lizeth Torres | 20th June 2018 15 / 31
Nonlinear estimation of a power law for the friction in a pipeline | A power law for the total head losses
Our proposition
∆Htot = ΩQ1+γ, where Ω =
α
β
.
α, β and γ are parameters to be estimated, which can be related to the
pipeline and fluid characteristics by using other equations for head losses.
For the DW equation: γ = 1, α =
f (Q)
2φArQγ−1
, β =
gAr
Leq
.
For the HW equation: Ω =
10.67Leq
C1.852φ4.8704
, γ = 0.852.
A general formula for friction losses
It involves other head loss equation.
Lizeth Torres | 20th June 2018 16 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Identification approach
Contents
1 Motivation
2 Contribution
3 Background
4 A power law for the total head losses
5 Identification approach
6 Design of the observers
7 Experimental results
Lizeth Torres | 20th June 2018 17 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Identification approach
Approach description
The least squares approach:
Steps
To obtain recordings (Time
Series Data) of Q and ∆Htot at
different operation points (from
the lowest to the highest
points).
To obtain the mean of each
recording: ¯Q, ¯∆Htot.
To fit ¯Q vs ∆Htot by using least
squares for instance.
0 200 400 600 800 1000 1200 1400
0
5
10
15
20
[m]
Htot
0 200 400 600 800 1000 1200 1400
[s]
2
4
6
8
10
12
[m3
/s]
10
-3
Q
2 3 4 5 6 7 8 9 10 11 12
Q [L]
0
2
4
6
8
10
12
14
16
18
20
Htot
[m]
Experimental Data
MATLAB fitted curve
Lizeth Torres | 20th June 2018 18 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Identification approach
Approach description
The proposed approach:
Steps
To induce steady-oscillatory flow in the pipeline provoking a
sinusoidal pressure at the upstream end.
To estimate β and γ by using a nonlinear observer.
To estimate α by using an algebraic equation or another nonlinear
observer.
∆Htot = ΩQ1+γ, where Ω =
α
β
.
Advantages
1 The time required for the estimation is shorter.
2 It is not necessary to set the pipeline at lowest and highest operation points.
Lizeth Torres | 20th June 2018 19 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Identification approach
OBSERVER 1
Mean
OBSERVER 2
Mean
totH
Q
Q
Q
totH
Q

 
 

tot in outH H H  
inH
outH
Q
d
dt
Lizeth Torres | 20th June 2018 20 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Design of the observers
Contents
1 Motivation
2 Contribution
3 Background
4 A power law for the total head losses
5 Identification approach
6 Design of the observers
7 Experimental results
Lizeth Torres | 20th June 2018 21 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Design of the observers
Design of the observers
Momentum equation:
˙Q =
gAr
Leq
∆Htot − Js(Q)
Js(Q) is the dissipation term that depends on the head losses formula used
(e.g. Js = f (Q)/2φArif DW equation is used). By substituting our proposed
power law in the momentum equation
˙Q = β∆Htot − αQ|Q|γ
If only positive flow is considered, then
˙Q = β∆Htot − αQ1+γ
Lizeth Torres | 20th June 2018 22 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Design of the observers
1st Observer: β and γ estimation
˙Q = β∆Htot − αQ1+γ
By x1 = Q, x2 = β, x3 = αQγ, x4 = γ, we get
˙x(t)=




0 u1 −y 0
0 0 0 0
0 0 0 0
0 0 0 0



 x +




0
0
x3x4u2/y
0




y = 1 0 0 0 x (1)
where u1 = ∆Htot and u2 = ˙Q is the derivative of the flow rate measurement.
˙x(t) = A(u, y)x + B(u, x)
y = Cx
Lizeth Torres | 20th June 2018 23 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Design of the observers
It has already been proven by Torres et al. 20122
that under excitation condition,
with u bounded and making A(u, y) bounded, if B(u, x) is globally Lipschitz in z
uniformly in u, one can obtain an estimation of the state for system with the
high-gain Kalman-like observer given by
˙ˆx = A(u, y)ˆx + B(u, ˆx) − Λ(λ)SCT
(u)(ˆy − y)
ˆy = Cˆx
˙S = λ(θS + [A(u, y) + dBλ(u, ˆx)]S + S[A(u, y) + dBλ(u, ˆx)]T
− SCT
CS),
with Λ(λ) =





λIN1
0
λ2
IN2
...
0 λq
INq





, dBλ =
1
λ
Λ−1
(λ)
∂B
∂x
Λ(λ), ˆx(0) ∈ RN
,
S(0) ≥ 0, which ensure
x(t) − ˆx(t) ≤ µe−σt
, µ > 0, ∀t ≥
1
λ
.
2
Torres, L., Besançon, G. and Georges, D. (2012). EKF-like observer with stability for a class
of nonlinear systems. IEEE Transactions on Automatic Control, 57(6), 1570-1574.
Lizeth Torres | 20th June 2018 24 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Design of the observers
2nd Observer: α estimation
˙Q = β∆Htot − αQ1+γ
By defining x1 = Q, x2 = α, we get system
˙x =
0 yˆγ|y|
0 0
x +
ˆβu1
0
,
y = 1 0 x(t) = Q, (2)
where ˆβ and ˆγ represent the parameters estimated in the previous step.
u1 = ∆Htot. To estimate the states of such a system, we used an exponential
proposed by Besançon19963:
3
Besançon, G., Bornard, G., and Hammouri, H. (1996). Observer synthesis for a class of
nonlinear control systems. European Journal of control, 2(3), 176-192.
Lizeth Torres | 20th June 2018 25 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Experimental results
Contents
1 Motivation
2 Contribution
3 Background
4 A power law for the total head losses
5 Identification approach
6 Design of the observers
7 Experimental results
Lizeth Torres | 20th June 2018 26 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Experimental results
Laboratory Pipeline Installations @ II-UNAM
Physical parameters
φ = 0.076 [m], L = 163.62 [m].
Lizeth Torres | 20th June 2018 27 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Experimental results
0 50 100 150 200 250 300 350 400 450
5
10
15
20
[m]
Δ Htot
0 50 100 150 200 250 300 350 400 450
[s]
6
8
10
12
[m3
/s]
×10-3
Q
Lizeth Torres | 20th June 2018 28 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Experimental results
0 50 100 150 200 250 300 350 400 450
[s]
-2
-1
0
1
2
3
4
×10-4
β
0 50 100 150 200 250 300 350 400 450
[s]
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
γ
0 50 100 150 200 250 300 350 400 450
[s]
0
0.5
1
1.5
2
2.5
3
α
Parameter average
¯β = 2.2941 × 10−4,
¯γ = 0.4403, ¯α = 2.5599.
Identified Momentum
Equation
˙Q = 2.56∆Htot − 0.4403Q1.44
Lizeth Torres | 20th June 2018 29 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Experimental results
Experimental Results
2 4 6 8 10 12
Q [L]
0
2
4
6
8
10
12
14
16
18
20
ΔH
tot
Experimental Data
Power Law
MATLAB fitted curve
3 4 5 6 7 8 9 10 11 12
Q [L] ×10-3
2
4
6
8
10
12
14
16
18
ΔH
tot
Steady State Points
Steady Oscillatory Flow
Lizeth Torres | 20th June 2018 30 / 31
Nonlinear estimation of a power law for the friction in a pipeline | Experimental results
qatlho’
Danke谢谢
Grazie
Спасибо
ขอบคุณ
9C4#5$Ì
‫ﺷﻜﺮا‬ Merçi
Gracias
நன்றி
Obrigado
Ευχαριστώ
감사합니다
ध यवाद
Terima kasih
Thank you
ありがとう
Tapadh leibh
ཐུགས་རྗེ་ཆེ་།
Go raibh maith agaibh
Xin cảm ơn
Questions? ftorreso@iingen.unam.mx
http://lizeth-torres.info
Lizeth Torres | 20th June 2018 31 / 31

More Related Content

Similar to Nonlinear estimation of a power law for the friction in a pipeline

IRJET - Experimental Analysis on Shell and Tube Heat Exchanger using ANSYS
IRJET -  	  Experimental Analysis on Shell and Tube Heat Exchanger using ANSYSIRJET -  	  Experimental Analysis on Shell and Tube Heat Exchanger using ANSYS
IRJET - Experimental Analysis on Shell and Tube Heat Exchanger using ANSYSIRJET Journal
 
work proposal presentation.pptx thesis new
work proposal presentation.pptx thesis newwork proposal presentation.pptx thesis new
work proposal presentation.pptx thesis newprabhasavita25
 
FORCED CONVECTIVE HEAT TRANSFER IN A LID-DRIVEN CAVITY
FORCED CONVECTIVE HEAT TRANSFER IN A LID-DRIVEN CAVITYFORCED CONVECTIVE HEAT TRANSFER IN A LID-DRIVEN CAVITY
FORCED CONVECTIVE HEAT TRANSFER IN A LID-DRIVEN CAVITYIRJET Journal
 
FLOW DISTRIBUTION NETWORK ANALYSIS FOR DISCHARGE SIDE OF CENTRIFUGAL PUMP
FLOW DISTRIBUTION NETWORK ANALYSIS FOR DISCHARGE SIDE OF CENTRIFUGAL PUMPFLOW DISTRIBUTION NETWORK ANALYSIS FOR DISCHARGE SIDE OF CENTRIFUGAL PUMP
FLOW DISTRIBUTION NETWORK ANALYSIS FOR DISCHARGE SIDE OF CENTRIFUGAL PUMPijiert bestjournal
 
Control Synthesis of Electro Hydraulic Drive Based on the Concept of Inverse ...
Control Synthesis of Electro Hydraulic Drive Based on the Concept of Inverse ...Control Synthesis of Electro Hydraulic Drive Based on the Concept of Inverse ...
Control Synthesis of Electro Hydraulic Drive Based on the Concept of Inverse ...ijtsrd
 
CFD analysis of flow through T-Junction of pipe
CFD analysis of flow through  T-Junction of pipeCFD analysis of flow through  T-Junction of pipe
CFD analysis of flow through T-Junction of pipeIRJET Journal
 
Updated Lagrangian SPH
Updated Lagrangian SPHUpdated Lagrangian SPH
Updated Lagrangian SPHpaulorrcampos1
 
A Spatio-temporal Optimization Model for the Analysis of Future Energy System...
A Spatio-temporal Optimization Model for the Analysis of Future Energy System...A Spatio-temporal Optimization Model for the Analysis of Future Energy System...
A Spatio-temporal Optimization Model for the Analysis of Future Energy System...IEA-ETSAP
 
Design and Implementation of Multiplier using Advanced Booth Multiplier and R...
Design and Implementation of Multiplier using Advanced Booth Multiplier and R...Design and Implementation of Multiplier using Advanced Booth Multiplier and R...
Design and Implementation of Multiplier using Advanced Booth Multiplier and R...IRJET Journal
 
IRJET- Enhancement of Heat Transfer Effectiveness of Plate-Pin Fin Heat S...
IRJET-  	  Enhancement of Heat Transfer Effectiveness of Plate-Pin Fin Heat S...IRJET-  	  Enhancement of Heat Transfer Effectiveness of Plate-Pin Fin Heat S...
IRJET- Enhancement of Heat Transfer Effectiveness of Plate-Pin Fin Heat S...IRJET Journal
 
AC-Based Differential Evolution Algorithm for Dynamic Transmission Expansion ...
AC-Based Differential Evolution Algorithm for Dynamic Transmission Expansion ...AC-Based Differential Evolution Algorithm for Dynamic Transmission Expansion ...
AC-Based Differential Evolution Algorithm for Dynamic Transmission Expansion ...TELKOMNIKA JOURNAL
 
Computational Investigation of Fluid Flow 90o Bend Pipe using Finite Volume A...
Computational Investigation of Fluid Flow 90o Bend Pipe using Finite Volume A...Computational Investigation of Fluid Flow 90o Bend Pipe using Finite Volume A...
Computational Investigation of Fluid Flow 90o Bend Pipe using Finite Volume A...IRJET Journal
 
Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...IAEME Publication
 
Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...IAEME Publication
 
Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...IAEME Publication
 
Overview of the FlexPlan project. Focus on EU regulatory analysis and TSO-DSO...
Overview of the FlexPlan project. Focus on EU regulatory analysis and TSO-DSO...Overview of the FlexPlan project. Focus on EU regulatory analysis and TSO-DSO...
Overview of the FlexPlan project. Focus on EU regulatory analysis and TSO-DSO...Leonardo ENERGY
 

Similar to Nonlinear estimation of a power law for the friction in a pipeline (20)

IRJET - Experimental Analysis on Shell and Tube Heat Exchanger using ANSYS
IRJET -  	  Experimental Analysis on Shell and Tube Heat Exchanger using ANSYSIRJET -  	  Experimental Analysis on Shell and Tube Heat Exchanger using ANSYS
IRJET - Experimental Analysis on Shell and Tube Heat Exchanger using ANSYS
 
work proposal presentation.pptx thesis new
work proposal presentation.pptx thesis newwork proposal presentation.pptx thesis new
work proposal presentation.pptx thesis new
 
FORCED CONVECTIVE HEAT TRANSFER IN A LID-DRIVEN CAVITY
FORCED CONVECTIVE HEAT TRANSFER IN A LID-DRIVEN CAVITYFORCED CONVECTIVE HEAT TRANSFER IN A LID-DRIVEN CAVITY
FORCED CONVECTIVE HEAT TRANSFER IN A LID-DRIVEN CAVITY
 
40220140507005
4022014050700540220140507005
40220140507005
 
40220140507005
4022014050700540220140507005
40220140507005
 
FluidDynamics.ppt
FluidDynamics.pptFluidDynamics.ppt
FluidDynamics.ppt
 
FLOW DISTRIBUTION NETWORK ANALYSIS FOR DISCHARGE SIDE OF CENTRIFUGAL PUMP
FLOW DISTRIBUTION NETWORK ANALYSIS FOR DISCHARGE SIDE OF CENTRIFUGAL PUMPFLOW DISTRIBUTION NETWORK ANALYSIS FOR DISCHARGE SIDE OF CENTRIFUGAL PUMP
FLOW DISTRIBUTION NETWORK ANALYSIS FOR DISCHARGE SIDE OF CENTRIFUGAL PUMP
 
Control Synthesis of Electro Hydraulic Drive Based on the Concept of Inverse ...
Control Synthesis of Electro Hydraulic Drive Based on the Concept of Inverse ...Control Synthesis of Electro Hydraulic Drive Based on the Concept of Inverse ...
Control Synthesis of Electro Hydraulic Drive Based on the Concept of Inverse ...
 
CFD analysis of flow through T-Junction of pipe
CFD analysis of flow through  T-Junction of pipeCFD analysis of flow through  T-Junction of pipe
CFD analysis of flow through T-Junction of pipe
 
Updated Lagrangian SPH
Updated Lagrangian SPHUpdated Lagrangian SPH
Updated Lagrangian SPH
 
A Spatio-temporal Optimization Model for the Analysis of Future Energy System...
A Spatio-temporal Optimization Model for the Analysis of Future Energy System...A Spatio-temporal Optimization Model for the Analysis of Future Energy System...
A Spatio-temporal Optimization Model for the Analysis of Future Energy System...
 
Design and Implementation of Multiplier using Advanced Booth Multiplier and R...
Design and Implementation of Multiplier using Advanced Booth Multiplier and R...Design and Implementation of Multiplier using Advanced Booth Multiplier and R...
Design and Implementation of Multiplier using Advanced Booth Multiplier and R...
 
IRJET- Enhancement of Heat Transfer Effectiveness of Plate-Pin Fin Heat S...
IRJET-  	  Enhancement of Heat Transfer Effectiveness of Plate-Pin Fin Heat S...IRJET-  	  Enhancement of Heat Transfer Effectiveness of Plate-Pin Fin Heat S...
IRJET- Enhancement of Heat Transfer Effectiveness of Plate-Pin Fin Heat S...
 
AC-Based Differential Evolution Algorithm for Dynamic Transmission Expansion ...
AC-Based Differential Evolution Algorithm for Dynamic Transmission Expansion ...AC-Based Differential Evolution Algorithm for Dynamic Transmission Expansion ...
AC-Based Differential Evolution Algorithm for Dynamic Transmission Expansion ...
 
module 1 PPT.pptx
module 1 PPT.pptxmodule 1 PPT.pptx
module 1 PPT.pptx
 
Computational Investigation of Fluid Flow 90o Bend Pipe using Finite Volume A...
Computational Investigation of Fluid Flow 90o Bend Pipe using Finite Volume A...Computational Investigation of Fluid Flow 90o Bend Pipe using Finite Volume A...
Computational Investigation of Fluid Flow 90o Bend Pipe using Finite Volume A...
 
Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...
 
Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...
 
Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...Effect of inertia weight functions of pso in optimization of water distributi...
Effect of inertia weight functions of pso in optimization of water distributi...
 
Overview of the FlexPlan project. Focus on EU regulatory analysis and TSO-DSO...
Overview of the FlexPlan project. Focus on EU regulatory analysis and TSO-DSO...Overview of the FlexPlan project. Focus on EU regulatory analysis and TSO-DSO...
Overview of the FlexPlan project. Focus on EU regulatory analysis and TSO-DSO...
 

Recently uploaded

PE 459 LECTURE 2- natural gas basic concepts and properties
PE 459 LECTURE 2- natural gas basic concepts and propertiesPE 459 LECTURE 2- natural gas basic concepts and properties
PE 459 LECTURE 2- natural gas basic concepts and propertiessarkmank1
 
Electromagnetic relays used for power system .pptx
Electromagnetic relays used for power system .pptxElectromagnetic relays used for power system .pptx
Electromagnetic relays used for power system .pptxNANDHAKUMARA10
 
Working Principle of Echo Sounder and Doppler Effect.pdf
Working Principle of Echo Sounder and Doppler Effect.pdfWorking Principle of Echo Sounder and Doppler Effect.pdf
Working Principle of Echo Sounder and Doppler Effect.pdfSkNahidulIslamShrabo
 
S1S2 B.Arch MGU - HOA1&2 Module 3 -Temple Architecture of Kerala.pptx
S1S2 B.Arch MGU - HOA1&2 Module 3 -Temple Architecture of Kerala.pptxS1S2 B.Arch MGU - HOA1&2 Module 3 -Temple Architecture of Kerala.pptx
S1S2 B.Arch MGU - HOA1&2 Module 3 -Temple Architecture of Kerala.pptxSCMS School of Architecture
 
Worksharing and 3D Modeling with Revit.pptx
Worksharing and 3D Modeling with Revit.pptxWorksharing and 3D Modeling with Revit.pptx
Worksharing and 3D Modeling with Revit.pptxMustafa Ahmed
 
Introduction to Geographic Information Systems
Introduction to Geographic Information SystemsIntroduction to Geographic Information Systems
Introduction to Geographic Information SystemsAnge Felix NSANZIYERA
 
Fundamentals of Internet of Things (IoT) Part-2
Fundamentals of Internet of Things (IoT) Part-2Fundamentals of Internet of Things (IoT) Part-2
Fundamentals of Internet of Things (IoT) Part-2ChandrakantDivate1
 
Augmented Reality (AR) with Augin Software.pptx
Augmented Reality (AR) with Augin Software.pptxAugmented Reality (AR) with Augin Software.pptx
Augmented Reality (AR) with Augin Software.pptxMustafa Ahmed
 
Danikor Product Catalog- Screw Feeder.pdf
Danikor Product Catalog- Screw Feeder.pdfDanikor Product Catalog- Screw Feeder.pdf
Danikor Product Catalog- Screw Feeder.pdfthietkevietthinh
 
Dr Mrs A A Miraje C Programming PPT.pptx
Dr Mrs A A Miraje C Programming PPT.pptxDr Mrs A A Miraje C Programming PPT.pptx
Dr Mrs A A Miraje C Programming PPT.pptxProfAAMiraje
 
1_Introduction + EAM Vocabulary + how to navigate in EAM.pdf
1_Introduction + EAM Vocabulary + how to navigate in EAM.pdf1_Introduction + EAM Vocabulary + how to navigate in EAM.pdf
1_Introduction + EAM Vocabulary + how to navigate in EAM.pdfAldoGarca30
 
Computer Graphics Introduction To Curves
Computer Graphics Introduction To CurvesComputer Graphics Introduction To Curves
Computer Graphics Introduction To CurvesChandrakantDivate1
 
Lect.1: Getting Started (CS771: Machine Learning by Prof. Purushottam Kar, II...
Lect.1: Getting Started (CS771: Machine Learning by Prof. Purushottam Kar, II...Lect.1: Getting Started (CS771: Machine Learning by Prof. Purushottam Kar, II...
Lect.1: Getting Started (CS771: Machine Learning by Prof. Purushottam Kar, II...singhalabhi53
 
Theory of Time 2024 (Universal Theory for Everything)
Theory of Time 2024 (Universal Theory for Everything)Theory of Time 2024 (Universal Theory for Everything)
Theory of Time 2024 (Universal Theory for Everything)Ramkumar k
 
Fundamentals of Structure in C Programming
Fundamentals of Structure in C ProgrammingFundamentals of Structure in C Programming
Fundamentals of Structure in C ProgrammingChandrakantDivate1
 
litvinenko_Henry_Intrusion_Hong-Kong_2024.pdf
litvinenko_Henry_Intrusion_Hong-Kong_2024.pdflitvinenko_Henry_Intrusion_Hong-Kong_2024.pdf
litvinenko_Henry_Intrusion_Hong-Kong_2024.pdfAlexander Litvinenko
 
Presentation on Slab, Beam, Column, and Foundation/Footing
Presentation on Slab,  Beam, Column, and Foundation/FootingPresentation on Slab,  Beam, Column, and Foundation/Footing
Presentation on Slab, Beam, Column, and Foundation/FootingEr. Suman Jyoti
 
8th International Conference on Soft Computing, Mathematics and Control (SMC ...
8th International Conference on Soft Computing, Mathematics and Control (SMC ...8th International Conference on Soft Computing, Mathematics and Control (SMC ...
8th International Conference on Soft Computing, Mathematics and Control (SMC ...josephjonse
 

Recently uploaded (20)

PE 459 LECTURE 2- natural gas basic concepts and properties
PE 459 LECTURE 2- natural gas basic concepts and propertiesPE 459 LECTURE 2- natural gas basic concepts and properties
PE 459 LECTURE 2- natural gas basic concepts and properties
 
Electromagnetic relays used for power system .pptx
Electromagnetic relays used for power system .pptxElectromagnetic relays used for power system .pptx
Electromagnetic relays used for power system .pptx
 
Working Principle of Echo Sounder and Doppler Effect.pdf
Working Principle of Echo Sounder and Doppler Effect.pdfWorking Principle of Echo Sounder and Doppler Effect.pdf
Working Principle of Echo Sounder and Doppler Effect.pdf
 
S1S2 B.Arch MGU - HOA1&2 Module 3 -Temple Architecture of Kerala.pptx
S1S2 B.Arch MGU - HOA1&2 Module 3 -Temple Architecture of Kerala.pptxS1S2 B.Arch MGU - HOA1&2 Module 3 -Temple Architecture of Kerala.pptx
S1S2 B.Arch MGU - HOA1&2 Module 3 -Temple Architecture of Kerala.pptx
 
Worksharing and 3D Modeling with Revit.pptx
Worksharing and 3D Modeling with Revit.pptxWorksharing and 3D Modeling with Revit.pptx
Worksharing and 3D Modeling with Revit.pptx
 
Introduction to Geographic Information Systems
Introduction to Geographic Information SystemsIntroduction to Geographic Information Systems
Introduction to Geographic Information Systems
 
Signal Processing and Linear System Analysis
Signal Processing and Linear System AnalysisSignal Processing and Linear System Analysis
Signal Processing and Linear System Analysis
 
Fundamentals of Internet of Things (IoT) Part-2
Fundamentals of Internet of Things (IoT) Part-2Fundamentals of Internet of Things (IoT) Part-2
Fundamentals of Internet of Things (IoT) Part-2
 
Augmented Reality (AR) with Augin Software.pptx
Augmented Reality (AR) with Augin Software.pptxAugmented Reality (AR) with Augin Software.pptx
Augmented Reality (AR) with Augin Software.pptx
 
Danikor Product Catalog- Screw Feeder.pdf
Danikor Product Catalog- Screw Feeder.pdfDanikor Product Catalog- Screw Feeder.pdf
Danikor Product Catalog- Screw Feeder.pdf
 
Dr Mrs A A Miraje C Programming PPT.pptx
Dr Mrs A A Miraje C Programming PPT.pptxDr Mrs A A Miraje C Programming PPT.pptx
Dr Mrs A A Miraje C Programming PPT.pptx
 
1_Introduction + EAM Vocabulary + how to navigate in EAM.pdf
1_Introduction + EAM Vocabulary + how to navigate in EAM.pdf1_Introduction + EAM Vocabulary + how to navigate in EAM.pdf
1_Introduction + EAM Vocabulary + how to navigate in EAM.pdf
 
Computer Graphics Introduction To Curves
Computer Graphics Introduction To CurvesComputer Graphics Introduction To Curves
Computer Graphics Introduction To Curves
 
Lect.1: Getting Started (CS771: Machine Learning by Prof. Purushottam Kar, II...
Lect.1: Getting Started (CS771: Machine Learning by Prof. Purushottam Kar, II...Lect.1: Getting Started (CS771: Machine Learning by Prof. Purushottam Kar, II...
Lect.1: Getting Started (CS771: Machine Learning by Prof. Purushottam Kar, II...
 
Theory of Time 2024 (Universal Theory for Everything)
Theory of Time 2024 (Universal Theory for Everything)Theory of Time 2024 (Universal Theory for Everything)
Theory of Time 2024 (Universal Theory for Everything)
 
Integrated Test Rig For HTFE-25 - Neometrix
Integrated Test Rig For HTFE-25 - NeometrixIntegrated Test Rig For HTFE-25 - Neometrix
Integrated Test Rig For HTFE-25 - Neometrix
 
Fundamentals of Structure in C Programming
Fundamentals of Structure in C ProgrammingFundamentals of Structure in C Programming
Fundamentals of Structure in C Programming
 
litvinenko_Henry_Intrusion_Hong-Kong_2024.pdf
litvinenko_Henry_Intrusion_Hong-Kong_2024.pdflitvinenko_Henry_Intrusion_Hong-Kong_2024.pdf
litvinenko_Henry_Intrusion_Hong-Kong_2024.pdf
 
Presentation on Slab, Beam, Column, and Foundation/Footing
Presentation on Slab,  Beam, Column, and Foundation/FootingPresentation on Slab,  Beam, Column, and Foundation/Footing
Presentation on Slab, Beam, Column, and Foundation/Footing
 
8th International Conference on Soft Computing, Mathematics and Control (SMC ...
8th International Conference on Soft Computing, Mathematics and Control (SMC ...8th International Conference on Soft Computing, Mathematics and Control (SMC ...
8th International Conference on Soft Computing, Mathematics and Control (SMC ...
 

Nonlinear estimation of a power law for the friction in a pipeline

  • 1. Nonlinear estimation of a power law for the friction in a pipeline (IFAC MICNON 2018) n m N Authors: Lizeth Torres and Cristina Verde ftorreso@iingen.unam.mx http://lizeth-torres.info/ 20th June 2018 Supported by Proyecto 280170, Convocatoria 2016-3, Fondo Sectorial CONACyT-Secretaría de Energía-Hidrocarburos.
  • 2. Nonlinear estimation of a power law for the friction in a pipeline Contents 1 Motivation 2 Contribution 3 Background 4 A power law for the total head losses 5 Identification approach 6 Design of the observers 7 Experimental results Lizeth Torres | 20th June 2018 2 / 31
  • 3. Nonlinear estimation of a power law for the friction in a pipeline | Motivation Contents 1 Motivation 2 Contribution 3 Background 4 A power law for the total head losses 5 Identification approach 6 Design of the observers 7 Experimental results Lizeth Torres | 20th June 2018 3 / 31
  • 4. Nonlinear estimation of a power law for the friction in a pipeline | Motivation Motivation of the research Contributing to the proper management of distribution pipelines. A proper management requires some tasks such as ... The diagnosis and prognosis of the pipeline components. The control for the adequate distribution of the fluids. The solution of optimization problems. ⇓ Algorithms based on models formulated from physical laws that govern the flow behavior throughout the pipelines. Lizeth Torres | 20th June 2018 4 / 31
  • 5. Nonlinear estimation of a power law for the friction in a pipeline | Motivation Disadvantage Algorithms based on governing equations need to be frequently updated to avoid being invalid because of the nonstop use and the passing of time. Energy dissipation changes. Affected Parameters Roughness εa . Diameter φ. a It is not simple to measure it Aging deterioration consequences Corrosion Tuberculation Erosion Mineral deposits Lizeth Torres | 20th June 2018 5 / 31
  • 6. Nonlinear estimation of a power law for the friction in a pipeline | Motivation Motivation Main goal of this article Proposing an algorithm to continuously estimate the energy dissipation in a pipeline without measuring ε, φ and L. Why? To avoid miscalculations due to changes of the physical parameters that could affect the performance of model-based algorithms. Lizeth Torres | 20th June 2018 6 / 31
  • 7. Nonlinear estimation of a power law for the friction in a pipeline | Contribution Contents 1 Motivation 2 Contribution 3 Background 4 A power law for the total head losses 5 Identification approach 6 Design of the observers 7 Experimental results Lizeth Torres | 20th June 2018 7 / 31
  • 8. Nonlinear estimation of a power law for the friction in a pipeline | Contribution Contribution A power law to represent the energy dissipation mainly caused by friction in a pipeline. A simple method to estimate such a power law with the following characteristics: 1 It is based on nonlinear state observers. 2 It only requires two pressure head recordings and a flow rate measurement. 3 It performs the estimation in short-time. Lizeth Torres | 20th June 2018 8 / 31
  • 9. Nonlinear estimation of a power law for the friction in a pipeline | Background Contents 1 Motivation 2 Contribution 3 Background 4 A power law for the total head losses 5 Identification approach 6 Design of the observers 7 Experimental results Lizeth Torres | 20th June 2018 9 / 31
  • 10. Nonlinear estimation of a power law for the friction in a pipeline | Background The classic way to compute the energy dissipation In pipelines, energy dissipation is called head loss and can be divided into two main categories: major losses associated with energy loss per length of pipe and minor losses associated with bends, fittings, valves, that act against the fluid and reduce its energy. ∆Htot = ∆Hf + ∆Hm ∆Htot = f L φ v2 2g + N i=1 ki v2 2g = f L φ + N i=1 ki v2 2g ∆Htot =   f L + φ N i=1 ki f φ    v2 2g *Darcy-Weisbach equation Lizeth Torres | 20th June 2018 10 / 31
  • 11. Nonlinear estimation of a power law for the friction in a pipeline | Background Total head loss in pipeline If Leq = L + φ N i=1 ki f then ∆Htot = f Leq φ v2 2g Leq is known as equivalent length, which actually is the virtual length of a pipeline with devices as if it were a horizontal pipeline without devices. Note: If a device (e.g. a valve) in the pipeline is added or changed, the equivalent length will change and consequently the total head loss. The computation of f depends on the flow regime. Lizeth Torres | 20th June 2018 11 / 31
  • 12. Nonlinear estimation of a power law for the friction in a pipeline | Background Total head loss in pipeline For laminar flow: f = κ/Re where κ depends on the pipeline geometry. For circular pipes κ = 64. For turbulent flow (natural regime in pipelines): f can be accurately calculated by using the Colebrook-White (CW) equation 1 f = −2 log10 ε 3.71φ + 2.51 Re f Some disadvantages: ε is required, it is an implicit equation, it represents the complete turbulent regimen of the Moody diagram! We don’t need that, we only need to represent ∆Htot for the region where our pipelines work. These disadvantages are reasons to propose new formulas to represent ∆Htot. Lizeth Torres | 20th June 2018 12 / 31
  • 13. Nonlinear estimation of a power law for the friction in a pipeline | Background Another reasons to propose new formulas for ∆Htot An identifiable equation To have an equation with empirical parameters that are easy to estimate A differentiable equation Some optimization and identification algorithms require the derivative of the friction. Lizeth Torres | 20th June 2018 13 / 31
  • 14. Nonlinear estimation of a power law for the friction in a pipeline | Background Some proposed alternatives 1 Hazen William equation ∆Htot = 10.67 C1.852 LeqQ1.852 φ4.8704 2 Wood equation ∆Htot = (a(ε) + b(ε)Re−c(ε) ) f Q2Leq 2gφA2 r 3 Valiantzas equation ∆Htot = Leq k0Q2 φ5.3 m , k0 = 0.0126ε0.3 4 Quadratic (Prony) equation ∆Htot = aQ2 + bQ 5 Rojas equations 1 ∆Htot = Leq 2gφA2 r (θ1Q2 |Q| + θ2Q|Q|), ∆Htot = Leq 2gφA2 r θ1Q|Q| + θ2|Q| 1 Rojas Jorge et al. On-Line Head Loss Identification for Monitoring of Pipelines, Safeprocess 2018, Poland. Lizeth Torres | 20th June 2018 14 / 31
  • 15. Nonlinear estimation of a power law for the friction in a pipeline | A power law for the total head losses Contents 1 Motivation 2 Contribution 3 Background 4 A power law for the total head losses 5 Identification approach 6 Design of the observers 7 Experimental results Lizeth Torres | 20th June 2018 15 / 31
  • 16. Nonlinear estimation of a power law for the friction in a pipeline | A power law for the total head losses Our proposition ∆Htot = ΩQ1+γ, where Ω = α β . α, β and γ are parameters to be estimated, which can be related to the pipeline and fluid characteristics by using other equations for head losses. For the DW equation: γ = 1, α = f (Q) 2φArQγ−1 , β = gAr Leq . For the HW equation: Ω = 10.67Leq C1.852φ4.8704 , γ = 0.852. A general formula for friction losses It involves other head loss equation. Lizeth Torres | 20th June 2018 16 / 31
  • 17. Nonlinear estimation of a power law for the friction in a pipeline | Identification approach Contents 1 Motivation 2 Contribution 3 Background 4 A power law for the total head losses 5 Identification approach 6 Design of the observers 7 Experimental results Lizeth Torres | 20th June 2018 17 / 31
  • 18. Nonlinear estimation of a power law for the friction in a pipeline | Identification approach Approach description The least squares approach: Steps To obtain recordings (Time Series Data) of Q and ∆Htot at different operation points (from the lowest to the highest points). To obtain the mean of each recording: ¯Q, ¯∆Htot. To fit ¯Q vs ∆Htot by using least squares for instance. 0 200 400 600 800 1000 1200 1400 0 5 10 15 20 [m] Htot 0 200 400 600 800 1000 1200 1400 [s] 2 4 6 8 10 12 [m3 /s] 10 -3 Q 2 3 4 5 6 7 8 9 10 11 12 Q [L] 0 2 4 6 8 10 12 14 16 18 20 Htot [m] Experimental Data MATLAB fitted curve Lizeth Torres | 20th June 2018 18 / 31
  • 19. Nonlinear estimation of a power law for the friction in a pipeline | Identification approach Approach description The proposed approach: Steps To induce steady-oscillatory flow in the pipeline provoking a sinusoidal pressure at the upstream end. To estimate β and γ by using a nonlinear observer. To estimate α by using an algebraic equation or another nonlinear observer. ∆Htot = ΩQ1+γ, where Ω = α β . Advantages 1 The time required for the estimation is shorter. 2 It is not necessary to set the pipeline at lowest and highest operation points. Lizeth Torres | 20th June 2018 19 / 31
  • 20. Nonlinear estimation of a power law for the friction in a pipeline | Identification approach OBSERVER 1 Mean OBSERVER 2 Mean totH Q Q Q totH Q       tot in outH H H   inH outH Q d dt Lizeth Torres | 20th June 2018 20 / 31
  • 21. Nonlinear estimation of a power law for the friction in a pipeline | Design of the observers Contents 1 Motivation 2 Contribution 3 Background 4 A power law for the total head losses 5 Identification approach 6 Design of the observers 7 Experimental results Lizeth Torres | 20th June 2018 21 / 31
  • 22. Nonlinear estimation of a power law for the friction in a pipeline | Design of the observers Design of the observers Momentum equation: ˙Q = gAr Leq ∆Htot − Js(Q) Js(Q) is the dissipation term that depends on the head losses formula used (e.g. Js = f (Q)/2φArif DW equation is used). By substituting our proposed power law in the momentum equation ˙Q = β∆Htot − αQ|Q|γ If only positive flow is considered, then ˙Q = β∆Htot − αQ1+γ Lizeth Torres | 20th June 2018 22 / 31
  • 23. Nonlinear estimation of a power law for the friction in a pipeline | Design of the observers 1st Observer: β and γ estimation ˙Q = β∆Htot − αQ1+γ By x1 = Q, x2 = β, x3 = αQγ, x4 = γ, we get ˙x(t)=     0 u1 −y 0 0 0 0 0 0 0 0 0 0 0 0 0     x +     0 0 x3x4u2/y 0     y = 1 0 0 0 x (1) where u1 = ∆Htot and u2 = ˙Q is the derivative of the flow rate measurement. ˙x(t) = A(u, y)x + B(u, x) y = Cx Lizeth Torres | 20th June 2018 23 / 31
  • 24. Nonlinear estimation of a power law for the friction in a pipeline | Design of the observers It has already been proven by Torres et al. 20122 that under excitation condition, with u bounded and making A(u, y) bounded, if B(u, x) is globally Lipschitz in z uniformly in u, one can obtain an estimation of the state for system with the high-gain Kalman-like observer given by ˙ˆx = A(u, y)ˆx + B(u, ˆx) − Λ(λ)SCT (u)(ˆy − y) ˆy = Cˆx ˙S = λ(θS + [A(u, y) + dBλ(u, ˆx)]S + S[A(u, y) + dBλ(u, ˆx)]T − SCT CS), with Λ(λ) =      λIN1 0 λ2 IN2 ... 0 λq INq      , dBλ = 1 λ Λ−1 (λ) ∂B ∂x Λ(λ), ˆx(0) ∈ RN , S(0) ≥ 0, which ensure x(t) − ˆx(t) ≤ µe−σt , µ > 0, ∀t ≥ 1 λ . 2 Torres, L., Besançon, G. and Georges, D. (2012). EKF-like observer with stability for a class of nonlinear systems. IEEE Transactions on Automatic Control, 57(6), 1570-1574. Lizeth Torres | 20th June 2018 24 / 31
  • 25. Nonlinear estimation of a power law for the friction in a pipeline | Design of the observers 2nd Observer: α estimation ˙Q = β∆Htot − αQ1+γ By defining x1 = Q, x2 = α, we get system ˙x = 0 yˆγ|y| 0 0 x + ˆβu1 0 , y = 1 0 x(t) = Q, (2) where ˆβ and ˆγ represent the parameters estimated in the previous step. u1 = ∆Htot. To estimate the states of such a system, we used an exponential proposed by Besançon19963: 3 Besançon, G., Bornard, G., and Hammouri, H. (1996). Observer synthesis for a class of nonlinear control systems. European Journal of control, 2(3), 176-192. Lizeth Torres | 20th June 2018 25 / 31
  • 26. Nonlinear estimation of a power law for the friction in a pipeline | Experimental results Contents 1 Motivation 2 Contribution 3 Background 4 A power law for the total head losses 5 Identification approach 6 Design of the observers 7 Experimental results Lizeth Torres | 20th June 2018 26 / 31
  • 27. Nonlinear estimation of a power law for the friction in a pipeline | Experimental results Laboratory Pipeline Installations @ II-UNAM Physical parameters φ = 0.076 [m], L = 163.62 [m]. Lizeth Torres | 20th June 2018 27 / 31
  • 28. Nonlinear estimation of a power law for the friction in a pipeline | Experimental results 0 50 100 150 200 250 300 350 400 450 5 10 15 20 [m] Δ Htot 0 50 100 150 200 250 300 350 400 450 [s] 6 8 10 12 [m3 /s] ×10-3 Q Lizeth Torres | 20th June 2018 28 / 31
  • 29. Nonlinear estimation of a power law for the friction in a pipeline | Experimental results 0 50 100 150 200 250 300 350 400 450 [s] -2 -1 0 1 2 3 4 ×10-4 β 0 50 100 150 200 250 300 350 400 450 [s] 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 γ 0 50 100 150 200 250 300 350 400 450 [s] 0 0.5 1 1.5 2 2.5 3 α Parameter average ¯β = 2.2941 × 10−4, ¯γ = 0.4403, ¯α = 2.5599. Identified Momentum Equation ˙Q = 2.56∆Htot − 0.4403Q1.44 Lizeth Torres | 20th June 2018 29 / 31
  • 30. Nonlinear estimation of a power law for the friction in a pipeline | Experimental results Experimental Results 2 4 6 8 10 12 Q [L] 0 2 4 6 8 10 12 14 16 18 20 ΔH tot Experimental Data Power Law MATLAB fitted curve 3 4 5 6 7 8 9 10 11 12 Q [L] ×10-3 2 4 6 8 10 12 14 16 18 ΔH tot Steady State Points Steady Oscillatory Flow Lizeth Torres | 20th June 2018 30 / 31
  • 31. Nonlinear estimation of a power law for the friction in a pipeline | Experimental results qatlho’ Danke谢谢 Grazie Спасибо ขอบคุณ 9C4#5$Ì ‫ﺷﻜﺮا‬ Merçi Gracias நன்றி Obrigado Ευχαριστώ 감사합니다 ध यवाद Terima kasih Thank you ありがとう Tapadh leibh ཐུགས་རྗེ་ཆེ་། Go raibh maith agaibh Xin cảm ơn Questions? ftorreso@iingen.unam.mx http://lizeth-torres.info Lizeth Torres | 20th June 2018 31 / 31