Dependable Dynamic Control Using Distributed Intelligent Agents

Paper number

IAC-04-A.4.04

Author

Dr. Peter Mendham, University of Dundee, United Kingdom

Year

2004

Abstract

The determination of in-depth, transient heat fluxes in high temperature environments is a critical issue requiring resolution in aerospace and defense sciences for survivability, energy management, and material evaluation. This paper presents recently developed integral relationships between heat flux q” ( W/ m ²) and temperature (K) or more specifically heating rate ( K/ s) in various one- and two-dimensional geometries for isotropic materials in semi-infinite regions. A unified theoretic approach has been developed for providing this highly important relationship in Cartesian, cylindrical and spherical geometries based on either Green’s functions or Fourier transforms; and, integral equation regularization.
For example, the simplest relationship (one-dimensional, constant property, no volumetric source, transient heat conduction in the half-space with trivial initial condition) is

q″( x, t)=

√

ρ  C  k  π

∫

t t o =0

∂  T  ∂  t o

( x, t o )

dt o √ t− t o

,    ( x, t)≥ 0,     (1)

and indicates that the local heat flux, q’’( x, t), x≥ 0 can be predicted using a single sensor that (a) measures temperature, T( x, t) (data would be filtered using a low-pass Gauss filter before differentiated) or (b) measures the heating/cooling rate, ∂ T/ ∂ t. Here, ρ=density, k=thermal conductivity, and C=heat capacity. Measurement of temperature leads to an ill-posed problem for predicting the local heat flux. When the temperature data are contaminated with white noise, the root-mean square error of the heat flux grows as √ M where M is the number of temperature samples. However, if one could measure the instantaneous heating rate, ∂ T/ ∂ t then the root-mean square error of the heat flux decreases as √ ( Ln( M)/ M) for sufficiently large M. Low-pass digital filtering takes advantage of the diffusion (low-frequencies are damped as x increases) process and should be used on the temperature data.
The two-dimensional, semi-infinite q _x ″− T integral relationship (heat flux normal to the surface) is

q x ″( x, y, t)=

k 2α π

∫

t t o =0

∫

∞ y o =−∞

e −  ( y− y o ) 2 4α ( t− t o )   t− t o

∂  T  ∂  t o

( x, y o , t o ) +

T( x, y o , t o )  2 ( t− t o )

/  |  |  \

1 −

( y− y o ) 2  2α( t− t o )

\  |  |  /

dy o dt o ,

x∈[0,∞)    y ∈ (−∞,∞),     t≥ 0,    (2)

when constant thermo-physical properties are assumed (here α= k/(ρ C)). In general, the constant property assumption is valid if the primitive properties ( k=thermal conductivity, C=heat capacity) are mathematically averaged in the measured temperature domain at the sensor site. The one-dimensional cylindrical coordinates ( r =radial) asymptotic q″− T integral relationship is

∼  q″

( r, t)=

k   ∼  T ( r, t) 2 r

+

√

ρ  C  k  π

∫

t t o =0

∂   ∼  T ∂  t o

( r, t o )

dt o √ t− t o

,     r> a,     t≥ 0,    (3)

where a=throat radius. This relationship has been implemented in nozzle throat studies to produce accurate findings. The thermo-environment consisted of a high-temperature combustion facility involving supersonic flow (M=2), nozzle exiting temperatures reaching 2700K and heat fluxes estimated at 1 kW/( cm ²) while the test ran for 5 seconds.
Additionally, we have investigated the effect of temperature dependent thermo-physical properties, use of local property averaging; and probe location uncertainty. These important issues will also be discussed in the paper. These new relationships are useful for material property evaluation and in-situ estimations of local heat flux for many practical aerospace applications.

Abstract document

IAC-04-A.4.04.pdf

Manuscript document

IAC-04-A.4.04.pdf (🔒 authorized access only).

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