Volume 7, Issue 10, October – 2022 International Journal of Innovative Science and Research Technology

ISSN No:-2456-2165

A Rigorous Experimental Technique to Measure the

Thermal Diffusivity of Metals in Different 3D Forms
Dr. Ismail Abbas
Senior lecturer at MTC, Cairo

Abstract:- This present work is a continuation and other words, the total thermal energy stored in an object
validation of the results explained in a previous paper during its cooling curve is equal to the temperature at its CM
titled A Rigorous Experimental Technique for Measuring multiplied by the total number of free nodes in the grid (n).
the Thermal Diffusivity of Metals and goes further to
describe the notion of dimensionless time tD practical for This suggests the extension of the proposed experimental
solving the energy-density field distribution in 4D space. technique from cubic shaped objects to other regular shapes
Moreover, the notion of dimensionless time and statistical such as cylinders, hemispheres, pyramids, etc. by finding the
characteristic length of the 3D material object is cooling curve of the tested object at its CM and by relating its
introduced, defined and proved effective. exponent to the thermal diffusivity as explained in the
theoretical part.
We have carried out a preliminary experimental
investigation and a theoretical analysis on five 3D Recall that the so-called Cairo numerical technique
geometric objects of different shapes in aluminum and transforms continuous real time t into dimensionless discrete
steel and the results obtained for the thermal diffusivity time tD. tD is equal to N f where N is the number of iterations
are in good agreement with the thermal tables. performed on the transition matrix B through its chain and f
is a statistical factor.
I. INTRODUCTION
The dimensionless diffusive time is equal to the number of
This article is a generalization to non-cubic forms of the iterations N multiplied by a statistical factor f.
theory and experiment explained in a previous article entitled The transformation from real continuous time to the
A rigorous experimental technique for measuring the thermal dimensionless discrete time domain via the matrix B and vice
diffusivity of metals [1,2] and goes further by describing the versa requires the introduction of four parameters depending
notion of dimensionless time tD practical for solving energy on the geometric shape of the body and its thermal diffusivity.
density distribution in 4D space (x, y, z, t).
II. THEORY
In reference 1 we limited the proposed experimental
technique to experimental measurements of thermal Below is the general form of the partial differential
diffusivity in aluminum and steel in cubic shapes, while in the equation for the time evolution of the energy density U in 3D
present work we go further in other shapes. Regular shapes geometric space,
such as cylinders, hemispheres and pyramidal shapes have d / dt (partial) U (x,y,z,t) = D Nabla2 U (x,y,z,t) + S
been studied. (x,y,z,t) .. . . . . . . (1)

To be precise, the previous works [1,5,6] are based on In normal conventions. Equation (1) is subjected to
the numerical statistical method called Cairo technique which Dirichlet boundary conditions BC and arbitrary initial
predicts an exponential decay of the energy density in a conditions IC.
bounded medium and relates the exponent to the physical and
geometric properties of the object, under test. In fact, equation (1) characterizes the time evolution of
the energy density in real time t and in the 3D geometric space
Moreover, we assume that the general heat diffusion x,y,z where in the SI system (MKS) the unit of t is the second
PDE (Eq 1) cannot practically be solved numerically in real (s), that of x,y,z is meter (m) and that of thermal diffusivity is
time. Finite difference computation (FDM) methods of real- m^2/s.
time numerical solutions are extremely time-consuming and
prone to instability and inaccuracy, while the same in Our task is to show how to describe the solution in
dimensionless time tD are short, fast, and the stability and dimensionless time tD. In the proposed numerical method
accuracy are assured. called Cairo technique, this is done via B-Matrix strings
where the real time t is completely lost.
In the present experimental technique proposed to
measure the thermal diffusivity of metals in different 3D The notion of dimensionless time tD was recently
forms, we assume that, The spatio-temporal average of the introduced and described in signal processing theory [7].
energy density called the center of the energy density field
U(x,y,z,t) in the object under test coincides with its center of In the phenomena of diffusion in bounded objects, the
mass CM along the time evolution of its cooling curve. In dimensionless time is defined equal to f N where N is the

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Volume 7, Issue 10, October – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
whole number of operations or time step iterations carried out time domain. The statistical transition matrix B which
on the transition matrix B 1,2,3 . .. N and f is a scalar quantity contains all the information to solve Equation 1 in the time-
depending on the physical and geometric properties of the dependent 3D geometry of the cube in Figure 1 is specified
tested object. via a procedure similar to that followed in previous work
where the entries in the matrix B27X27 must be expressed in
The proposed experimental technique itself is not the following form [1,2,5,6],
complicated and can be summarized in the following five
consecutive steps, 27X27 B-Matrix inputs
i-Perform the experimental results of the temperature cooling Line1: RO 1/6-RO/6 0.0000 1/6- RO 1/6-RO/6 0.0000 0.0000
curve at the center of mass CM of the tested object and thus 0.0000 0.00001/6-RO/6 0.0000 0.0000 0.0000 0.0000 0.0000
find the real time - half-time decay value, i.e. T1/2, T1/4, T1/8 0.0000 0.0000 0.00000.0000 0.0000 0.00000.0000 0.0000
etc [1,2]. 0.0000 0.0000 0.0000 0.0000
Line 2: 1/6-RO/6 RO 1/6-RO/6 0.0000 1/6-RO/6 0.0000
ii- Calculate the statistical characteristic length of the tested 0.0000 0.0000 0.0000.0000 1/6-RO/6 0.0000 0.0000 0.0000
object Lc via the semi-imperial formula (2), [1,2] 0.0000 0.0000 0.0000 0.00000.00000.0000 0.0000 0.0000
Lc = {6*Volume of object V / Area of object A} . . . .(2) 0.0000 0.0000 0.0000 0.0000 0.0000
The statistical factor f emerges from another semi-imperial Line 3: 0.0000 1/6-RO/6 RO 0.0000 0.0000 1/6RO/6 0.0000
formula, 0.0000 0.00000.00000.0000 1/6-RO/6 0.0000 0.0000 0.0000
f = Pie /2. = 1.571 0.0000 0.0000 0.00000.0000 0.00000.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000
In fact, the characteristic length is of great importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\
in itself since the experimental temperature of the real-time Line 14: 0.0000 0.0000 0.0000 0.0000 1/6-RO/6 0.0000
cooling curve at the center of mass CM is described by, 0.0000 0.00000.00000.0000 1/6-RO/6 0.0000 1/6-RO/6 RO
T(t)=T(0).Exp (- D . f . t /Lc^2) . . . . . (3) 1/6-RO/6 0.0000 1/6-RO/60.00000.0000 0.0000 0.0000
0.0000 1/6-RO/6 0.0000 0.0000 0.0000 0.0000. . . . . .. . . . . .
Equation 3 is simply a consequence of defining the exponent . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .
of the cooling curve as the heat left per second dU/dt divided ..........
by the heat stored U. Line 25: 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.00000.00000.0000 0.0000 0.0000 0.0000 0.0000
Equations 2 and 3 suggest an important geometric 1/6-RO/6 0.0000 0.00000.0000 0.00000.0000 1/6-RO/6
physical rule, 0.0000 0.0000 RO 1/6-RO/6 0.0000
Line 26: 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Two 3D bodies of different shapes cannot have the same 0.0000 0.00000.00000.0000 0.0000 0.0000 0.0000 0.0000
volume to area ratio (V/A) unless both have exactly the 0.0000 1/6-RO/6 0.00000.0000 0.00000.0000 0.0000 1/6-
same volume and area. RO/6 0.0000 1/6-RO/6 RO 1/6-RO/6
It is simple to show that the half-time decay interval is given Line 27: 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
by, 0.0000 0.00000.00000.0000 0.0000 0.0000 0.0000 0.0000
T1/2= Log 2. Lc^2/D f . . . (4) 0.0000 0.0000 1/6-RO/60.0000 0.00000.0000 0.0000 0.0000
Obviously Log 2. = 0.693 1/6-RO/6 0.0000 1/6-RO/6 RO
with RO = 0.22 for steel and 0.13 for aluminum as shown in
In other words, the required thermal diffusivity will be given references 1 and 2.
by, In order not to worry too much about the details of the theory,
D= 0.693 *Lc^2 / (T1/2 * f ) . . . (5) let us present the following five illustrative experimental
applications with their experimental setups and experimental
Note that the statistical characteristic length Lc can be found results.
mathematically or experimentally as explained in references
1,2. III. EXPERIMENTAL SETUP AND
iv- Plot the experimental real-time cooling curve EXPERIMENTAL RESULTS
T(t)=T(0) .Exp(- D f t /Lc^2),
We move on to five different applications on different
Know the value of T1/2 and therefore calculate the 3D shapes, cubic and non-cubic, Al and steel where the tank
equivalent thermal diffusivity D using formula 5. cold water temperature is zero centigrade.
v- Also plot the dimensionless cooling time curve
proposed by the transition matrix B chains by choosing the The experimental setup is described in detail in
appropriate value of RO and compare their fit with the Reference 1 along with the composition of steel and
experimental results. aluminum used as the test material.

In this paper, we have arbitrarily chosen to apply the B In all five experiments, the hot water reservoir was
27X27 transition matrix as the transition to the dimensionless maintained at 76 C and the cold reservoir at 0 C.

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Volume 7, Issue 10, October – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
III(a)- Steel cube of 10 cm side Fig 1.

Fig 1. Steel cube with sides of 10 cm with holes and resistance thermometers.

The results of temperature T in centigrade at CM vs time in seconds is presented in Table III-a.

Table III-a , Cooling curve for steel cube 10 cm side length

t(sec) 0 30 60 90 120 150 180 210 240 300 360 420 480 540 600
T(c) 76 58 48 39 31 25.2 20.2 15.8 12.9 10.9 9.1 8.4 7.9 7.4 ---

We conclude from table III-a that T1/2 is close to 100 s.

Eq 2 yields Lc =10 cm for steel cube ie, equal to its side length
Finally , using Eq 5, ,then the value of the thermal diffusivity for steel equals
D= Lc^2. Log 2/ (T1/2.*Pie/2)
D=1 E-2*0.693 / (100*1.57) =44.2 E-6 m^2/s in good agreement with the thermal tables [8].

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Volume 7, Issue 10, October – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
III(b)-Aluminum cube of 10 cm side Fig 2.

Fig 2. Aluminum Cube of side length 10 cm.

The results of temperature T in centigrade at CM vs time in III(c). Aluminum cylinder of mass 2.61 Kg, a radius R of
seconds is presented in Table III-b. 14.8 and a length L of 14.8 cm.Fig.3

Table III-b , Cooling curve for Aluminum cube

10 cm side length
t(sec) 0 30 60 90 120 150 180 210 240 300 360
420 480 540 600
T(c) 76 45 33 26.5 23 20 17.6 15.5 13.8 11.9 10.
8.2 7.95 7.4 6.95

We conclude from table III-b that T1/2 is close to 45 s.

Equation 2 gives Lc = 10 cm for an aluminum cube, i.e. equal
to the length of its side.
Finally , using Eq 5, ,then the value of the thermal diffusivity
for Aluminum equals

D= Lc^2. Log 2/ (T1/2.*Pie/2)

D=1 E-2*0.693 /(45*1.57) =98 E-6 m^2/s
in good agreement with thermal tables[7]. Fig.3 Regular cylinder with circular base of radius R and
length L.

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Volume 7, Issue 10, October – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
The results of temperature T in centigrade at CM vs time Easy to calculate the surface area of the cylinder as
in seconds is presented in Table III-c nearly 600 cm^2 and

Table III-c , Cooling curve for Aluminum cylinder 14.0 Volume of the cylinder is nearly 1000 cm^3 which are
cm diameter and 6.5 cm length. the same as those of the preceding cube of 10 cm side length.
t(sec) 0 30 60 90 120 150 180 210 240 300 360 Cooling curve for Aluminum cylinder (Table III-c)14.0
420 480 540 600 cm diameter and 6.5 cm length is similar to that of the cube
T(c) 76 44 32 26. 23 20 17.2 15.6 13.8 11.6 (Table III-b) as Equation 3 predicts.
9.8 8.6 7.9 7.4 6.9
Obviously the calculated thermal diffusivity D is the same ,
We conclude from table III-c that T1/2 is close to 45 s D Alumium= 0.98 E-4 m^2/s.
close to that of the Aluminum cube as expected.

III(d). Aluminum pyramid with a mass of 6.5 Kg, a square base of 20 cm and a height of 19 cm.Fig.4

Fig.4 Regular pyramid with square base

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Volume 7, Issue 10, October – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
The results of temperature T in centigrade at CM vs time Easy to calculate the surface area of the pyramid as nearly
in seconds is presented in Table III-d. 1160 cm^2 and
Volume of the pyramid is nearly 2500 cm^3 .
Table III-d , Cooling curve for Aluminum pyramid with The characteristic length for the pyramid is 6 V/A=13.6 cm.
a square base of 20 cm and a height of 19 cm.Fig.4 Cooling curve for Aluminum pyramid (Table III-d) at its CM
t(sec) 0 30 60 90 120 150 180 210 240 300 is similar to that of an equivalent cube of side length 13.6 as
360 420 480 540 600 Equation 3 predicts.
T(c) 76 49 38.5 28 26 22.6 18.3 16.8 15.7 13.5
12.6 8.6 12 11.3 9.9 It is simple to calculate thermal diffusivity D from Eq. 5,
D= 0.693 *Lc^2 /(T1/2*f) = 132 E-6 m^2/s
We conclude from table III-d that T1/2 is close to 62 s which is slightly higher than that of thermal tables.

III(e). Aluminum half-sphere with a mass of 1.8 Kg and a diameter of 14.5 cm.Fig.5

Fig.5 Regular aluminum hemisphere with circular base

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Volume 7, Issue 10, October – 2022 International Journal of Innovative Science and Research Technology
ISSN No:-2456-2165
The results of temperature T in centigrade at CM vs time REFERENCES
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The dimensionless time tD inherent in the B-Matrix - 1119 ,DOI: 10.1109/27.533119
chain solution of time-dependent energy density scattering in
3D geometric objects has been shown to be consistent, stable,
fast, and accurate.

The theoretical and experimental results produced in

this article are consistent and suggest to introduce and
develop a generalized or unified theory to solve the problems
of energy density diffusion (thermal energy, electric potential
energy, sound kinetic energy, etc.) in 4D bounded media.

NB. All calculations in this article were produced

through the author's double precision algorithm to ensure
maximum accuracy, as followed by Ref. 10 for example

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