Ireneusz Szczygieł Institute of Thermal Technology Silesian University of Technology

Ireneusz SzczygiełInstitute of Thermal Technology

Silesian University of TechnologyGliwice, Poland

Identification of Boundary Velocity Basing on Internal Temperature

Measurements – sensitivity analysis

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Outline

Introduction Direct problem formulation Inverse problem formulation Discussion on sensitivity coefficient field Results of numerical tests Final remarks

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Basic assumptions

•2D, Cartesian geometry 2D, Cartesian geometry •Steady stateSteady state•Potential or laminar flowPotential or laminar flow•Isoparametric fluidIsoparametric fluid

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Direct problem formulation

temperature field descriptiontemperature field description


xux

y

u y

velocity field description – potential flowvelocity field description – potential flow

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velocity field description – incompressible flowvelocity field description – incompressible flow

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Boundary conditions inflow profileinflow profile

0

n

ui

iTT

symmetry wallssymmetry walls 0nT

0n

heating pipes surfaceheating pipes surface pTT

0n

outlet profileoutlet profile 0nT

const

flow direction symmetry walls

heating pipes

inflow surface outflow surface

0; yix uuu

0;0 yxx unuuu

0;0 yx uu

0;0 yx uu

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Inverse problem formulation

definition of sensitivity coefficients:definition of sensitivity coefficients:

iT u

TZ

Estimate inflow velocity knowing the value of internal temperature

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Evaluation of sensitivity coefficients

Tuy

cTux

cqyTk

yxTk

x yxv

iu

TyTxATT Zu

ycZu

xcS

yZ

kyx

Zk

x

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Evaluation of sensitivity coefficient

boundary conditionsboundary conditions

iu

iTT

0nT

pTT

inflow profileinflow profile

symmetry wallssymmetry walls

heating pipes surfaceheating pipes surface

0TZ

0nZT

0TZ

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Evaluation of sensitivity coefficient

source term Ssource term SAA

yT

uu

xT

uu

kcS

i

y

i

xA

soso

yTZ

xTZ

kcS uyuxA

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Evaluation of sensitivity coefficients – potential flow

02 iu

02 Z

boundary conditionsboundary conditions

0

n

ui

0n iu

01

nZ

0

nZ

xZ

Zux

iuZ

yZ

Zuy

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Evaluation of sensitivity coefficients – NS flow

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source term Ssource term Suxux

yuZ

xuZS x

uyx

uxux

Evaluation of sensitivity coefficients – NS flow

yu

Zxu

ZS yuy

yuxuy

source term Ssource term Suyuy

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Evaluation of sensitivity coefficients – NS flowboundary conditionsboundary conditions

inflow profileinflow profile

symmetry wallssymmetry walls

heating pipes surfaceheating pipes surface

outlet profileoutlet profile

0; yix uuu

0;0 yxx unuuu

0;0 yx uu

0;0 yx uu

0;1 uyux ZZ

0;0 uyuxux ZnZZu

0;0 uyux ZZ

0;0 uyux ZZ

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Examplary iterative inverse procedure –potential flow

• calculations of the Zcalculations of the Z fieldfield•assumption of the inflow velocity valueassumption of the inflow velocity value•evaluation of the potential and the velocity fieldevaluation of the potential and the velocity field•evaluation of the temperature sensitivity coefficient evaluation of the temperature sensitivity coefficient distributiondistribution•calculation of the new boundary velocitycalculation of the new boundary velocity

•convergence test for the inflow velocityconvergence test for the inflow velocity

Tii ZTYuu /** where Y stands for the measurementswhere Y stands for the measurements resultsresults

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Examplary iterative inverse procedure –NS flow

•assumption of the inflow velocity valueassumption of the inflow velocity value•evaluation of the velocity fieldevaluation of the velocity field•evaluation of the evaluation of the velocityvelocity sensitivity coefficient sensitivity coefficient distributiondistribution Z Zuxux

•evaluation of the evaluation of the velocityvelocity sensitivity coefficient sensitivity coefficient distributiondistribution Z Zuyuy

•evaluation of the temperature sensitivity coefficient evaluation of the temperature sensitivity coefficient distributiondistribution•calculation of the new boundary velocitycalculation of the new boundary velocity•convergence test for the inflow velocityconvergence test for the inflow velocity


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Examplary iterative inverse procedure –NS flow

•assumption of the inflow velocity valueassumption of the inflow velocity value•evaluation of the velocity fieldevaluation of the velocity field•evaluation of the evaluation of the velocityvelocity sensitivity coefficient sensitivity coefficient distributiondistribution Z Zuxux

•evaluation of the evaluation of the velocityvelocity sensitivity coefficient sensitivity coefficient distributiondistribution Z Zuyuy

•evaluation of the temperature sensitivity coefficient evaluation of the temperature sensitivity coefficient distributiondistribution•calculation of the new boundary velocitycalculation of the new boundary velocity•convergence test for the inflow velocityconvergence test for the inflow velocity


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Results of numerical tests Simple channel, Re=500Simple channel, Re=500

ZZux ux distribution distribution

ZZuuyy distribution distribution

Velocity distribution Velocity distribution

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Results of numerical tests Potential flowPotential flow, Re=20, Re=20

Temperature

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Results of numerical tests Potential flowPotential flow, Re=20, Re=20

Temperature sensitivity coefficients

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Results of numerical tests NSNS flow flow, Re=200, Re=200

Temperature

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Velocity magnitude

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Sensitivity coefficients Zux

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Sensitivity coefficients Zuy

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Sensitivity coefficients ZT

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Results of numerical tests Estimation of nodal quantity, NS flow, Re=200Estimation of nodal quantity, NS flow, Re=200

Temperature

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Velocity

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Sensitivity coefficient Zux

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Sensitivity coefficient ZT

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Results of numerical tests Estimation of mass velocity, NS flow, Re=50Estimation of mass velocity, NS flow, Re=50

Velocity

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• the shape of the sensitivity coefficient distribution is a the shape of the sensitivity coefficient distribution is a function of inlet velocity value. It means, the that the function of inlet velocity value. It means, the that the optimal location for the measurement sensors varies with optimal location for the measurement sensors varies with the inlet velocity value;the inlet velocity value;•the region of maximum values of the sensitivity coefficient the region of maximum values of the sensitivity coefficient is shifted from the inlet surface toward the flow direction: is shifted from the inlet surface toward the flow direction: bringing the measurement sensors close to the inflow bringing the measurement sensors close to the inflow boundary can result in worse estimation of the inflow boundary can result in worse estimation of the inflow velocity;velocity;• the application of the inverse algorithm is possible on in the application of the inverse algorithm is possible on in the regions with nonuniform temperature distribution;the regions with nonuniform temperature distribution;•inverse procedure for the NS flow is much more time inverse procedure for the NS flow is much more time consuming than the procedure for the potential flowconsuming than the procedure for the potential flow;;

Final remarks

Documents

Ireneusz Szczygieł Institute of Thermal Technology Silesian University of Technology