Page 177 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

169
TRANSIENT HEAT CONDUCTION ANALYSIS
with
∂T
−k
∂x = q(t) at x = 0 (6.67)
∂T
k = 0at x = l (6.68)
∂x
and
T = T o (x) at t = 0 (6.69)
where q(t) is the unknown heat flux and T o (x) is the initial temperature of the body.
The known temperature values at the sensor location are given as
T(t k ,x l ) = U k,l (6.70)

where k varies between 1 and the total number of measured data at the sensor location (l)
and t k indicates the corresponding time. Introducing a sensitivity coefficient Z k as
k,i
k
∗
T k,i = T ∗ + Z (q k − q ) (6.71)
k,i k,i k
where T k,i is the temperature at time t k and location i, T ∗ is the temperature calculated
k,i
∗
using q(k) = q(k) in Equation 6.67 and Z k are the sensitivity coefficients. Note that we
k,i
can write, using a Taylor series expansion,
∂T k,i
T k,i = T k,i + | q k =q (q k − q ) + ··· (6.72)
∗
∗
∗
k
∂q k k
The above equation shows that
Z k = ∂T k,i (6.73)
k,i
∂q k
In order to calculate the correct temperatures, the least squares error between the cal-
culated and measured temperature values needs to be minimized, that is,
I
2
(U k,i − T k,i ) = 0 (6.74)
i=1
where I is the number of sensors in the body. On substitution of Equation 6.71, into
Equation 6.74, and rearranging, we get

k
∗
I Z (U k,i − T )
∗ i=1 k,i k,i
q k = q + (6.75)
k
k
I (Z ) 2
i=1 k,i
If we assume only one sensor in the field, the above equation is reduced to
k
∗
Z (U k − T )
∗ k k
q k = q + (6.76)
k k 2
(Z )
k
In practice, the above equation is difficult to use in order to obtain a smooth heat flux
distribution. To arrive at such a smooth heat flux distribution, Beck (Beck 1968) suggested