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Wärme - und Stoffübertragung

, Volume 27, Issue 8, pp 505–513 | Cite as

Numerical solutions for general inverse heat conduction problem

  • J. Taler
Article

Abstract

A family of numerical methods for determining the space-and time-variable heat transfer coefficient, based on experimentally acquired interior temperature-time data, is presented. Newton-type methods are utilized to compute simultaneously the unknown heat transfer coefficient components. To reduce the influence of random errors in the measurement data on the estimated heat transfer coefficients, the noisy data are smoothed using least squares approximation by cubic splines. Three test examples using experimental and random simulated data are used to illustrate the computation efficiency and generality of the present methods.

Keywords

Heat Transfer Heat Conduction Heat Transfer Coefficient Noisy Data Computation Efficiency 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

Bi

Biot number,h rin/k

c

specific heat

Ei

distance of thei-th temperature sensor from thei-th boundary segment

F

number of future time steps in analysis interval

Fo

Fourier number

gi

cubic spline

g″i

second derivative ofgi(t) att

h

heat transfer coefficient

H

number of time steps in analysis interval

J

number of data points in time domain

k

thermal conductivity

K

total number of iterations

l

knot coordinate

L

thickness of vessel, also the number of knots in cubic spline approximation

M

total number of iterations in the secant method

n

unit outward normal to boundary surface

N

number of heat transfer coefficients

N1

number of grid points

q

heat flux

q

rate of power generation per unit-volume

r

radial coordinate

r

space variable vector

ri

position of thei-th temperature sensor

s

coordinate along boundary surface

S

boundary

Si

smoothing parameter

t

time variable

tF

time at the end of the time analysis interval

tk

time at the end of the time interval

T

temperature

TF

temperature computed at timetF

Tk

temperature computed at timetk

w

weighting factor

y

experimental data

yi

measured temperature at thei-th location

z

axial coordinate

Greek symbols

α

thermal diffusivity

β

angle (see Fig. 4)

δ

small positive scalar

δy

weighting factor

Δr

space increment inr direction

ΔFo

differential Fourier number,α(Tk)Δtk/E i 2

Δt

time step

ΔtF

future time interval for computing heat transfer coefficient or heat flux,tF−tF−1

Δtk

time interval,tk−tk−1=H Δt

Δz

space increment inz direction

εi, j

random error

εh

convergence tolerance

θ1

smoothing factor

ξ1

coordinate of thel-th knot

ϱ

density

σ

sample variance

ϕ

angle coordinate

ω

relaxation factor

gradient operator

if

partial derivative with respect to thei-th variable

Subscripts

f

fluid

F

denotes quantities defined at timetF

i

index of heat transfer coefficients and temperature sensor locations

in

at inside surface

j

index of data points used in piecewise cubic approximation

k

denotes quantities defined timetk

o

at outside surface

0

initial

Superscripts

(k)

iteration number

(m)

iteration number, Eqs. (8)–(9)

T

transpose of matrix or column vector

-

mean

*

dimensionless

Numerische Lösungen des allgemeinen inversen Wärmeleitungsproblems

Zusammenfassung

Es wird eine Reihe von numerischen Verfahren zur Bestimmung von orts- und zeitabhängigen Wärmeübergangs-koeffizienten, auf der Basis der im Körperinnern gemessenen Temperaturen, dargestellt. Zur Berechnung der Komponenten des Wärmeübergangskoeffizienten werden verschiedene Varianten des Verfahrens von Newton angewandt. Der Einfluß der zufälligen Fehler in gemessenen Temperaturen auf die zu bestimmenden Wärmeübergangsk oeffizienten wird durch die Ausgleichung der zeitlichen Temperaturverläufe mit kubischen Splines reduziert. Drei Beispiele, in welchen die gemessenen und stochastisch simulierten Temperaturverläufe verwendet wurden, zeigen, daß die entwickelten Methoden sehr effizient und allgemein sind.

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References

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Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • J. Taler
    • 1
  1. 1.Institute for Industrial Apparatus and Power EngineeringTechnical University of CracowKraków

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