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=TTLTRTL,z=xL,r=kk3
(1)
one can re-write Eqs. (28)–(30) of 1 as  
ddzrdθdz=0
(2)
 
θ|z=0=0,θ|z=1=1
(3)
 
rθ=r1+r2r1θθ1θ2θ1,θ1θθ2r2+1r2θθ2θ3θ2,θ2θθ3
(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=r1θr1,θr2=r2θr2
(6)
Applying the chain rule, one can relate θr1,θr2 to the dimensional sensitivities Tk1,Tk2,Tk3 defined in 1:  
Tk1=k1TRTLθr1r1k1=TRTLθr1
(7)
 
Tk2=k2TRTLθr2r2k2=TRTLθr2
(8)
 
Tk3=k3TRTLθr1r1k3+θr2r2k3=TRTLθr1+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.