Systematic approach to estimate non-uniform heat generation rate in heat transfer problems using liquid crystal thermography and inverse methodology

ABSTRACT

This paper presents an inverse methodology to estimate the parameters of the non-uniform heat generation (function estimation) within a flat plate assembly using steady-state conjugate heat transfer experiments on the flat plate assembly. Steady-state laminar conjugate forced convection experiments on a flat plate assembly are conducted on a horizontal wind tunnel to estimate the parameters of the non-uniform heat generation within flat plate assembly using the inverse methodology. Bayesian inference based Metropolis Hastings–Markov Chain Monte Carlo (MH–MCMC) algorithm and experimental temperatures are employed in the inverse methodology. The experimental temperatures are measured at convenient locations of the flat plate assembly using liquid crystal thermography. In order to accomplish the retrieval, first, steady-state experiments on only the cork material are conducted to estimate the thermal conductivity of the cork material accurately for use in the estimation of the heat generation rate so that the additional error due to uncertainty in the thermal conductivity of the cork material does not affect our final goal of estimating heat generation rate. Following this, steady-state experiments on the cork setup (consisting of a non-uniform heat generation heater and two symmetric cork plates) are conducted to ascertain the nature of heat generation of the heater using measured temperatures and fundamental rate laws. The priors are generated using coupled artificial neural network (ANN) and Levenberg–Marquardt (LM) algorithm for Bayesian inference. Using the Bayesian inference with priors, the parameters of non-uniform heat generation are then estimated in terms of the mean, maximum a posteriori with standard deviation. Finally, the simulated heat powers and temperatures are estimated with retrieved parameters of the non-uniform heat generation. These compared very well with the measured heat powers and temperatures. Finally, a recipe for solving a practical problem, in which only measured temperatures are available, is provided.

KEYWORDS:

• Artificial neural networks
• Bayesian inference
• conjugate heat transfer
• function estimation
• Levenberg–Marquardt algorithm
• MH–MCMC
• non-uniform heat generation
• thermochromic liquid crystal

Disclosure statement

No potential conflict of interest was reported by the author(s).

Nomenclature

Gr	=	Grashof number
I	=	current, A
$k_b$	=	thermal conductivity of the bakelite, $W/(m.K)$
$k_c$	=	thermal conductivity of the cork, $W/(m.K)$
$k_h$	=	thermal conductivity of the heater (mica), $W/(m.K)$
N	=	normal
$P(\mathrm{\Psi })$	=	prior density function
$P(\mathrm{\Psi }/y)$	=	posterior probability density function
$P(y)$	=	normalizing constant
$P(y/\mathrm{\Psi })$	=	likelihood density function
$q_v$	=	volumetric heat generation, $W/m^3$
$R^2$	=	coefficient of determination
Re	=	Reynolds number
Ri	=	Richardson number
$s_1$	=	old solution
$s_2$	=	new solution
T	=	temperature, K
u	=	velocity in the x-direction, m/s
v	=	velocity in the y-direction, m/s
V	=	voltage, V
w	=	velocity in the z-direction, m/s

Greek letters

$\alpha$	=	thermal diffusivity of the fluid, $m^2/s$
$\vartheta$	=	kinematic viscosity of the fluid, $m^2/s$
$\rho$	=	density of the fluid, $m^2/s$
${\mu }_p$	=	mean of the prior density function
$\sigma$	=	uncertainty in the temperature measurement, °C
${\sigma }_p$	=	standard deviation of the prior
$\mathrm{\Psi }$	=	parameter vector
${\chi }^2$	=	chi squared

Subscripts

avg	=	average
cal	=	calculated
error	=	error
exp	=	experimental
meas	=	measured
sim	=	simulated
$\mathrm{\infty }$	=	ambient
LSR	=	least squares regression
ANN	=	artificial neural network
LM	=	Levenberg-Marquardt
LCT	=	liquid crystal thermography
MAP	=	maximum a posteriori
MH	=	Metropolis Hastings
MCMC	=	Markov Chain Monte Carlo
MSE	=	mean square error
PPDF	=	posterior probability density function
SD	=	standard deviation
TLC	=	thermochromic liquid crystals

Abbreviation

ANN	=	artificial neural network
LM	=	Levenberg-Marquardt
LCT	=	liquid crystal thermography
MAP	=	maximum a posteriori
MH	=	Metropolis Hastings
MCMC	=	Markov Chain Monte Carlo
MSE	=	mean square error
PPDF	=	posterior probability density function
SD	=	standard deviation
TLC	=	thermochromic liquid crystals