Inverse approach for estimating boundary properties in a transient fin problem

Abstract

A solution methodology is proposed for an inverse estimation of boundary conditions from the knowledge of transient temperature data. A forward model based on prevalent time-dependent heat conduction fin equation is solved using a fully implicit finite volume method. First, the inverse model is formulated and accomplished for time-invariant heat flux at the fin base, and later extended to transient heat flux, base temperature and average heat transfer coefficient. Secondly, the Nusselt number is then replaced with Rayleigh number in the forward model to realistically estimate the base temperature, which varies with respect to time, based on in-house transient fin heat transfer experiments. This scenario further corroborates the validation of the proposed inverse approach. The experimental set-up consists of a mild steel $250 \times 150 \times 6\, \hbox {mm}^3$ fin mounted centrally on an aluminium base $250 \times 150 \times 8\, \hbox {mm}^3$ plate. The base is attached to an electrical heater and insulated with glass-wool to prevent heat loss to surroundings. Five calibrated K-type thermocouples are used to measure temperature along the fin. The functional form of the unknown parameters is not known beforehand; sensitivity studies are performed to determine suitability of the estimation and location of sensors for the inverse approach. Conjugate gradient method with adjoint equation is chosen as the inverse technique and the study is performed as a numerical optimization; subsequently, the estimates show satisfactory results.

Appendices

Nomenclature

$A_c$ :

area of cross section

d :

direction of descent

g :

acceleration due to gravity

h :

convective heat transfer coefficient

M :

number of sensors

k :

thermal conductivity of fin

$k_f$ :

thermal conductivity of fluid

L :

fin length

P(t):

unknown transient parameter

p :

fin perimeter

Nu :

Nusselt number

q :

heat flux

Ra :

Rayleigh number

T :

temperature

$T_b$ :

base temperature

t :

time, s

$T_{\infty }$ :

ambient temperature

x :

space coordinate

Y :

measured temperature

Greek symbols

$\alpha$ :

thermal diffusivity

$\beta$ :

coefficient of thermal expansion

$\beta _{k}$ :

step size

$\gamma$ :

coefficient of conjugation

$\epsilon$ :

stopping criterion/error

$\lambda$ :

Lagrange multiplier

$\nu$ :

kinematic viscosity

$\tau$ :

time step

Subscripts

exp :

experimental temperature

f :

final

i :

sensor index

m :

measured location

s :

simulated temperature

Superscripts

k :

iteration number