Calculations of Temperature, Conductive Heat Flux, and 

Heat Wave Velocities Due to Radiant Heating of 

Opaque Materials 

by Arthur Cohen 

ARL-TR-5804 November 2011 

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Army Research Laboratory 

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ARL-TR-5804 November 2011 

Calculations of Temperature, Conductive Heat Flux, and 

Heat Wave Velocities Due to Radiant Heating of 

Opaque Materials 

Arthur Cohen 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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Calculations of Temperature, Conductive Heat Flux, and Heat Wave Velocities 

Due to Radiant Heating of Opaque Materials 

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Arthur Cohen 

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14. ABSTRACT 

The analytic solutions of the one-dimensional Fourier conductive heat flux law and corresponding transient heat transfer equation 

have been used to calculate temperature, conductive heat flux, and their trajectories due to radiant heating of opaque materials. 

Heat wave trajectories (and velocities) are defined by the value of a constant temperature or constant conductive heat flux that 

propagates through the materials. For constant values of zero, the analytic solutions lead to infinite heat wave velocities. For nonzero 

constant values, the numerical solutions at ultra-short transient times result in other nonphysical velocities (i.e., >3E(10) cm/s). These 

calculations demonstrate that there are limitations to the validity of the Fourier heat flux law and corresponding heat transfer 

equation, although they have been used successfully to solve engineering and scientific heat conduction problems for over 180 years. 

15. SUBJECT TERMS

radiant heating, opaque materials, Fourier heat transfer, temperature/conductive heat flux, infinite heat wave velocity 

calculations 

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Arthur Cohen 

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iii

Contents 

List of Figures iv

List of Tables iv

Acknowledgments v

1. Introduction 1

2. Nomenclature 1

3. Mathematical Formulation 2

4. Conclusion 7

5. References 8 

Distribution List 9 
iv

List of Figures 

Figure 1. Conductive flux vs. distance. ..........................................................................................3

Figure 2. Temperature vs. distance. ................................................................................................3

Figure 3. Temperature trajectory distance vs. time. ........................................................................5

Figure 4. Temperature wave velocity vs. time. ...............................................................................6

Figure 5. Flux wave trajectory distance vs. time. ..........................................................................6

Figure 6. Flux wave velocity vs. time............................................................................................6

List of Tables 

Table 1. Nomenclature. ...................................................................................................................1
v

Acknowledgments 

The suggestions of Dr. Gene Cooper for clarifying the mathematical presentation and the help of 

Dr. Anthony Kotlar with the Mathcad calculations are greatly appreciated. 
vi

INTENTIONALLY LEFT BLANK. 
1 

1. Introduction 

The objective of this report is to document recent one-dimensional (1-D) calculations of 

temperature, conductive heat flux, and heat wave velocities for an opaque, semi-infinite cylinder 

due to radiant heating. The Fourier conductive heat flux law is assumed to be valid. An 

idealized model is used for the calculations. Radiant flux enters the planar surface. It is 

uniform and steady with beam and cylinder diameters equal. The initial temperature is 

uniform, and thermal properties are constant. Heat wave trajectories (and velocities) are 

defined by the value of a constant temperature or constant conductive heat flux that propagates 

through the opaque material. For the constant values of zero, the analytic solutions (1) of the 

1-D Fourier conductive heat flux law and transient heat transfer equation (derived by applying 

energy conservation to the Fourier heat flux law) lead to infinite heat wave velocities. For the 

constant values >0, the numerical solutions at ultra-short transient times result in other 

nonphysical velocities (i.e., >3E(10) cm/s). 

These calculations suggest that there are limitations to the validity of the Fourier heat flux law 

and heat transfer equation, although they have been used successfully to solve engineering and 

scientific heat conduction problems for over 180 years.

2. Nomenclature 

Table 1 shows the units and values of the thermophysical properties and F that were used in all 

calculations. The thermophysical values chosen were those for RDX at T  288 K (2). 

Table 1. Nomenclature. 

Symbols Quantity Units Values 

t radiant flux time s — 

d distance from the planar surface cm — 

V temperature increment, i.e., V (d,0) = 0 K — 

f conductive heat flux cal/cm2-s — 

λ thermal conductivity cal/cm-s-K 2.5 × 10–4

ρ density g/cm3 1.76 

c heat capacity cal/g-K 0.3 

α thermal diffusivity = λ/ρc cm2/s 4.7 × 10–4

F radiant flux cal/cm2-s 100 
2 

3. Mathematical Formulation 

For the simplified model with opaque (surface absorption) materials, the Fourier heat flux law 

is 

f(d,t) = –(1/ ) V(d,t) d

  



(1)

with boundary condition 

 f(d,t) = F; d = 0, t > 0 . (2)

The Fourier heat transfer equation is 

2

V(d,t) = V(d,t) 2 t d

 



  (3) 

with boundary condition 

V(d,t) = –F/ ; d = 0, t > 0

d

 

 . (4) 

Using equations 1 and 3, Carslaw and Jaeger (1) show that the Fourier flux transfer equation is 

2

2 f(d,t) = f(d,t) ; d > 0 , t > 0

t d

 



  (5) 

with the boundary condition 

f(d,t) = F ; d = 0 . (6) 

The solution for f(d,t) (1) is 

d f(d,t) = F erfc 2 t

         

. (7)

The solution for V(d,t) , obtained by integration of equation 1, is (1)

2 F t –d d d V(d,t) = exp erfc 4t 2 2 t

                            

. (8) 

Figures 1 and 2 are calculated values of f (cal/cm2-s) and V (K) vs. d (cm) at t (s) = 0.001, 0.01, 

0.05, and 0.1.
3 

Figure 1. Conductive flux vs. distance. 

Figure 2. Temperature vs. distance.

Average heat wave velocities for conductive flux at f = 10 and 60 and for temperature at V = 50 

and 150 can be obtained from the distances between intersections of their corresponding lines 

with those calculated for different values of t. Distances vary inversely with the values of both 

f(d) and V(d), which suggests that the largest heat wave velocities are for f = 0 and V = 0. To 

obtain heat wave velocities at points on trajectories, let x = trajectory distance (cm) from the 

planar surface. 

Values of x at given values of t can be obtained from numerical solutions of equations 7 and 8. 

For f = 0 and V = 0, the numerical solutions (3) are unstable. To calculate these trajectories, 
4 

x let z = and substitute into equations 7 and 8. 2 t  

f (z) = F erfc(z)  . (7a) 

  2 2 F t z z V(z, t) 2 e 1 z e erfc(z) 

        

  . (8a) 

The trajectories (x 2 z t)     can be calculated for given values of t from solutions of 

equations 7a and 8a for z. 

When f(z) = 0, then erfc(z) = 0. The solution to equation 7a is z = ∞. 

When V(z,t) = 0, then 

2 z z e erfc(z) 1   . (9)

Using L’Hopital’s rule, when z = ∞, then 

 2 z z e erfc(z) 1   .

 (10)

The solution to equation 8a is z = ∞. 

For V > 0 and f > 0, the trajectories (x 2 z t)     can be calculated from the values of x 

corresponding to given values of t from numerical solutions of equations 7 and 8 or the 

corresponding z from equations 7a and 8a. 

Corresponding velocities d

x

dt (fv for flux and Vv for temperature) can be calculated by 

numerical differentiation. Calculating trajectories with z = ∞ is a problem. The heat wave 

velocities can also be obtained analytically from the calculated trajectories using g(x,t) = 

constant ; 

d 1 x – (x, t) dt t g(x, t)

x

g

        

  

. 

When f(x,t) = g(x,t), then 

x fv = z

2t t

    . (11) 

When f(x,t) = 0, then z = ∞. 

When x = 0, then z < ∞ , i.e., 0 < x ≤ ∞ . 

When x = ∞ and t = ∞, then z = 1 < ∞, i.e. , 0 ≤ t < ∞. 
5 

Using equation 11 with 0 < x ≤ ∞ and 0 ≤ t < ∞, 

 fv = ∞. (11a) 

When V(x,t) = g(x,t), then 

2

2

x –

4 t

z

e z

t t Vv =

x z e erfc(z) erfc

2 t

   

     

       

. (12) 

When V(x,t) = 0, then z = ∞. Using equation 10, 2 z z e erfc(z) 1   .

Vv = z

t

  . (13) 

When 0 < x ≤ ∞ and 0 ≤ t < ∞, then 

 Vv = ∞, (13a) 

Equations 11a and 13a show that when 0 < x ≤ ∞ and 0 ≤ t < ∞, and f(z) = V(z,t) = 0 , then 

Vv = fv = ∞. This is consistent with the results shown in figures 1 and 2. 

Figures 3 and 4 show the calculated values of x(cm) and vV(cm/s) vs. t(s) for constant values of 

V. The x ratios (RxV) and vf ratios (RvV) are shown on the right y-axis. In figure 4, the 

vertical line indicates that for V = 1E(–19) ; vV > 3E10 at t < –23 1.37 10 . 

Figure 3. Temperature trajectory distance vs. time. 
6 

Figure 4. Temperature wave velocity vs. time. 

Figures 5 and 6 correspond to figures 3 and 4 for the calculated values of x and vf vs. time for 

constant values of f/F. The x ratios (Rxf) and vf ratios (Rvf) are shown on the right y-axis. In 

figure 6, the vertical lines indicate that for f/F = 1E(–10) and 1E(–16), vf > 3E(10) at t < 

1.1 10–23 and 1.8 respectively. 

Figure 5. Flux wave trajectory distance vs. time. 

Figure 6. Flux wave velocity vs. time. 
7 

4. Conclusion 

The calculations of nonphysical heat wave velocities for temperature and heat flux using analytic 

solutions of the Fourier heat flux law and the corresponding transient heat transfer equation 

indicate that there are limitations to their validity. This problem has been well noted in the heat 

transfer literature (4). The inability of the Fourier heat transfer (parabolic) equation to predict 

measurements of heat propagation velocities in solids near zero absolute temperatures and for 

pulsed laser processing of solid surfaces involving ultra-short transient heating times is probably 

due to the Fourier flux law being constitutive and not valid under these extreme conditions. The 

Fourier equations are based on the linear dependence of heat flux and temperature gradient. The 

details of the mechanism for thermal energy transfer (molecular collisions in the absence of net 

mass motion) are still under investigation. Consideration of phonon interactions as a molecular 

gas (5) has recently been used to describe the thermal energy transfer mechanism for conductive 

heat flux. 

Non-Fourier-type conductive heat flux laws in solids have been proposed and include both time 

and temperature gradient dependence (4). These laws combined with energy conservation can 

lead to hyperbolic type transient heat transfer equations, which avoid the Fourier problem of 

nonphysical heat propagation velocity predictions and give better agreement with measurements 

under extreme conditions. Under normal conditions, these time-dependent conductive heat flux 

laws transit into the Fourier temperature gradient flux law. 

Temperature measurements (using different techniques) of the rate at which equilibrium is 

established when solids (meat) at different (near room) temperatures are placed in contact have 

been used to compare heat transfer velocities with predictions of the Fourier and non-Fourier 

type conductive heat transfer equations in solids (6). The conclusions as to which type equation 

is in better agreement with the data are in disagreement. It is of interest to determine the lower 

time limits at which Fourier’s heat transfer equation remains valid. It may be possible to obtain 

estimates from the published results of recent pulsed radiant heating experiments on opaque and 

semi-transparent solids. 
8 

5. References 

1. Carslaw, H.; Jaeger, J. Conduction of Heat in Solids, 2nd ed. Oxford Science 

Publications: London, 1959. 

2. Miller, M. S. Thermophysical Properties of Cyclotrimethylenetrinitramine. Journal of 

Thermophysics and Heat Transfer 1994, 8, 803–805. 

3. Parametric Technology Corporation. Mathcad 2001; Needham, MA, 2001. 

4. Joseph, D. D.; Preziosi, L. Heat Waves. Rev. Modern Physics 1989, 61 (1), 41–73. 

5. Cao, B-Y; Guo, Z-Y. Journal of Applied Physics 2007, 102, 053503. 

6. Herwig, H.; Beckert, K. Journal of Heat Transfer May 2000, 122, 363–365.
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