Dowding and Blackwell 1 derived sensitivity equations for general nonlinear heat conduction. What is surprising is that they chose to write these equations in dimensional form. One would expect that such a study must begin with writing the equations in nondimensional form and using Pi Theorem 2, p. 93 to find dimensionless groups of parameters on which the solution really depends. By failing to do this, the authors left undetected the fact that some of their sensitivity coefficients are linear dependent. In a practical calculation, this would unnecessarily increase the number of equations to solve.

The simple problem of steady one-dimensional heat conduction with temperature-dependent thermal conductivity used in 1 as a verification problem provides a good example to illustrate this point. Defining
$θ=T−TLTR−TL,z=xL,r=kk3$
(1)
one can re-write Eqs. (28)–(30) of 1 as
$ddz r dθdz=0$
(2)
$θ|z=0=0,θ|z=1=1$
(3)
$rθ=r1+r2−r1θ−θ1θ2−θ1,θ1⩽θ⩽θ2r2+1−r2θ−θ2θ3−θ2,θ2⩽θ⩽θ3$
(4)
Here
$r1=k1k3,r2=k2k3$
(5)
We see that the dependence of the solution on thermal conductivity changes is determined not by three parameters $k1,$$k2,$$k3,$ as in 1, but by only two parameters $r1,$$r2.$ Following 1, the corresponding sensitivity coefficients $θr1,$$θr2$ can be defined:
$θr1=r1 ∂θ∂r1,θr2=r2 ∂θ∂r2$
(6)
Applying the chain rule, one can relate $θr1,$$θr2$ to the dimensional sensitivities $Tk1,$$Tk2,$$Tk3$ defined in 1:
$Tk1=k1TR−TL ∂θ∂r1∂r1∂k1=TR−TLθr1$
(7)
$Tk2=k2TR−TL ∂θ∂r2∂r2∂k2=TR−TLθr2$
(8)
$Tk3=k3TR−TL∂θ∂r1∂r1∂k3+∂θ∂r2∂r2∂k3=−TR−TLθr1+θr2$
(9)
We can conclude that the sensitivities $Tk1,$$Tk2,$$Tk3$ are not independent but satisfy
$Tk1+Tk2+Tk3=0$
(10)
It is seen from Fig. 2 of 1 that Eq. (10) is indeed satisfied.

Department of Mechanical Engineering, California Institute of Technology, MC 104-44, Pasadena, CA 91125

1.
Dowding
,
K. J.
, and
Blackwell
,
B. F.
,
2001
. “
Sensitivity Analysis for Nonlinear Heat Conduction
,”
ASME J. Heat Transfer
,
123
(
1
), pp.
1
10
.
2.
Birkhoff, G., 1960, Hydrodynamics. A Study in Logic, Fact and Similitude, 2nd ed., Princeton University Press, Princeton.