Experimental observation of heat transparency
Lunwu Zeng,a) and Runxia Song
Jiangsu Key Laboratory of Intelligent Agricultural Equipment, College of Engineering, Nanjing
Agricultural University, Nanjing, 210031, China
1. The relations between effective heat conductivity and filling ratio
The spherical core is composed of pure iron powder and pure copper powder. Theoretically, the effective
heat conductivity of the mixture is k eff  k Cu f  k Fe (1  f ) , where k eff , k Cu and k Fe are the heat
conductivity of effective material, copper and iron, respectively, f is the filling ratio of copper. But the
effective heat conductivity in experiments is smaller than the effective heat conductivity in theory; this is
because the copper powder and iron powder mixed with air. The relations between effective heat
conductivity and filling ratio are shown in Fig. S1, the black square is the theoretical value and the red
circle is the experimental value. By adjusting the proportion of copper and iron powder, we can obtain the
effective heat conductivity expected.
FIG. S1. The relations between effective heat conductivity and filling ratio
2. Time-dependent sine temperature curves in the holes of sample II
The four horizontal lines are A2B2C2D2, E2F2G2H2, I2J2K2L2 and M2N2O2P2; the time-dependent
temperatures and temperature gradients can be measured by the computer. The average temperature
gradients of two neighboring holes are 2.26C/cm in Fig. S2a, 2.28C/cm in Fig. S2b, 2.27C/cm
1
in Fig. S2c and 2.26C/cm in Fig. S2d, correspondingly.
2
FIG. S2. Time-dependent sine temperature curves in the holes of sample II. (a) Time-dependent
temperatures in the holes of the segment A2B2C2D2, the average temperature gradient of two neighboring
holes is 2.26C/cm . (b) Time-dependent temperatures in the holes of the segment E2F2G2H2, the
average temperature gradient of two neighboring holes is 2.28C/cm . (c) Time-dependent temperatures
in the holes of the segment I2J2K2L2, the average temperature gradient of two neighboring holes is
2.27C/cm . (d) Time-dependent temperatures in the holes of the segment M2N2O2P2, the average
temperature gradient of two neighboring holes is 2.26C/cm
3. Heat diffusion theory
FIG. S3. Schematic of heat energy diffusion.
Heat diffusion is a common phenomenon in the natural world, and heat energy will diffuse from the high
temperature region to the low temperature region when the temperature gradient exists. Heat diffusion can
be described by Fourier’s law, for simplicity, we discussed one-dimensional heat diffusion, Fig. S3 shows
3
that the heat energy is diffused along an aluminum bar, then the Fourier’s law can be written as
dQ
dT
 kA
,
dt
dx
（S1）
where k is the heat conductivity, A is the cross section area, Q is the heat quantity, t is the time,
dT / dx is the temperature gradient. Differential two sides of the Eq. (S1), we obtain
dQ
d 2T
d
 kA 2 dx ,
dt
dx
（S2）
According to the law of energy conservation, the heat balance equation is
 cAdx
dT
dQ
d 2T
d
 kA 2 dx ,
dt
dt
dx
（S3）
where c ,  are the specific heat and mass density, respectively, so
dT
d 2T
D 2 ,
dt
dx
（S4）
where D  k / c is the heat diffusion coefficient. In order to obtain the solution of Eq. (S4), we
designed boundary conditions: (1) the temperature in the endpoint of low temperature ( x   ) is T0 , (2)
the temperature in the endpoint of high temperature ( x  0 ) is
T  T0  Tm sin t ,
（S5）
where Tm is the amplitude of temperature,
 is angle frequency, the temperature in the endpoint of high
temperature is time-dependent sine function. According to the boundary conditions, the solution of Eq. (S4)
can be written as
T  T0   Tm e
n

n
x
2D
sin( nt 
n
x) ,
2D
（S6）
Experimentally and theoretically, when the heat energy is diffused about 7.0cm along the aluminum bar
from the heat source, the high order harmonics will rapidly attenuate, only fundamental harmonic exists,
i.e., n  1 , so
4
T  T0  Tm e


2D
x
sin( t 

2D
x) ，
（S7）
when x  0 , T  T0  Tm sin t ; when x   , T  T0 , the boundary conditions we designed are the
solutions of Eq. (S7). Set the heat source at x  7.0cm , the endpoint of high temperature at the origin of
coordinates x  0 , and the endpoint of low temperature at x   , then Eq. (S7) is alike the equation of
wave motion, the “wave velocity” of heat diffusion is v 
2 D , and the “wavelength” is
  2 2 D /  .
The shape of the time-dependent sine temperature curve is alike “wave motion curve”, the “wavelength”,
“wave velocity” and “phase difference” are
  2 kT / c , v  2 k / Tc and   2x /  ,
respectively, we take k  220.00W /( mK ) , c  880.0 J /( kgK ) ,
we obtain
  2700.0kg / m 3 , T  180.0s ,
  0.4576m , v  0.00254m / s . The distance between two neighboring holes is
x  0.02m , the phase difference   0.0874 Rad. Figs. 5a-5d show that the phase of green curve is
ahead of the phase of red curve. It should be pointed out that the “wave” here is only used to describe the
shape of time-dependent temperature curve rather than a wave in physical conception31. The self-made heat
diffusion device is an artificial periodical heating device, the period is 180.0s , i.e., the heating frequency
is 1 / 180.0Hz , it is quite different from the frequency of lattice wave motion, the latter is much largerS1.
S1M.
Maldovan, Nature 503, 209 (2013).
5