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Heat Transfer Monitoring in Solids By Means of Finite

Element Analysis Software

J. Hernndez Wong*, V. Suarez, J. Guarachi, A. Caldern, A. G. Jurez, J. B. Rojas-Trigos and E. Marn

Instituto Politcnico Nacional, CICATA Unidad Legaria, Mxico D.F., C. P. 11500, Mxico.

Abstract: We study the radial heat transfer in a homogeneous and isotropic substance with a heat linear source in its axial axis. For this, we used hot wire photothermal technique in order to obtain the temperature distribution as a function of radial

distance and time exposure. Also, the solution of the transient heat transport equation for this problem was obtained with

appropriate boundary conditions, by means of finite element technique. The comparison of the experimental and simulated

results shows a good agree, which demonstrate the utility of this methodology in the investigation of the thermal response of

substances, in the radial configuration.

Keywords: Heat Transfer; Hot Wire; Finite element.

1. Introduction

Heat transfer is the area which describes the energy

transport between material bodies due to a difference in

temperature, and its development and applications are of

fundamental importance in many branches of engineering

since provides economical and efficient solutions for critical

problems encountered in many engineering items of equip-

ment. Among the parameters that determine the thermal

behavior of a material, the thermal conductivity is especially

important because it represents the ability of a material to

transfer heat, and it is one of the physical quantities whose

measurement is very difficult and it requires high precision

in the determination of the parameters involved in its calcu-

lations [1, 2].

The hot wire technique is an absolute, non-steady state and

direct method which is considered an effective and accurate

procedure to determining the thermal conductivity of a va-

riety of materials, including ceramics, fluids, food and

polymers [3-6]. However, this technique is based in a con-

ventional mathematical model which is an approximation of

the physical reality in the experimental setup because the

complexity of the mathematical problem has been an obsta-

cle to obtain a more realistic theoretical model [7, 8]. For-

tunately, nowadays the development of the advanced nu-

merical methods and computing systems allow the applica-

tion of high level software for obtain an approximate solu-

tion to a complex mathematical problem with a boundary

conditions congruent with the physical reality. In particular,

Comsol Multiphysics is a powerful Finite Element (FEM)

Partial Differential Equation (PDE) solution engine [9]

useful to obtain a numerical solution in complex problems.

* E-mail corresponding author: hjoel@tetraa.com.mx , hjoel@lycos.com

Telephone: +52-(55)-57 29 60 00

In this work, the Comsol Multiphysics software is use to

determinate the numerical solution of a transient temperature

distribution in a sample measured by the hot wire technique

configuration.

2. Experimental

The hot wire technique is based in a needle with a heater

and temperature sensor inside. A rectangular linear power

density is passed through the heater and the temperature of

the probe is monitored over a specific period of time. An

analysis of the transient temperature response is used to

investigate the thermal response of the sample.

In the conventional mathematical model the hot wire is

assumed to be an ideal infinitely thin and long heat source

which is immersed in an infinite surrounding material. By

means of the solution of the heat conduction equation, in

cylindrical coordinates, it can be obtained the temperature

distribution divided in two stages [7]:

The heating stage: htt ,0

t

rEi

k

QrtThs

44),(

2

0

The cooling stage: ,htt

)(444),(

22

0

h

cstt

rEi

t

rEi

k

QrtT

(1)

(2)

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Here, r is the radial distance from the lineal source, k is the

thermal conductivity, is the thermal diffusivity, Q0 is the

linear heat source, th is the heating time, and Ei is the expo-

nential integral [10]. By means of the fitting the expression

of Ths (heating stage) or Tcs (cooling stage) to the experi-

mental data, it can be obtained the thermal parameters of the

sample. The black curve in Figure 1 shows the graph of Ths,

and, Tcs of the theoretical model, in function of t for r = a

(fixed). The radius of the needle probe is a = 0.6 mm.

3. Simulation Process

It was apply Comsol Multiphysics software with the

purpose to solve the heat transport equations for the config-

uration of hot wire technique with boundary conditions

corresponding to the physical reality.

In the software Graphic user interface (GUI), it was built a

2D hollow cylinder of radius a and b, which represent the

sample-probe arrangement. The hollow cylinder gives cer-

tain advantages in the numerical calculation process, one of

these advantages is to reduce complexity in the model and

the other is to simplify the boundary regions, see Figure 1.

Figure 1. 2D Geometry.

Once the geometry is defined, the physical process that

takes place in the experiment is defined in the heat transfer

module of the Comsol software. The heat transfer module

use the heat diffusion equation (HDE), given by:

(3)

In the simulation setup, only the Laplacian term and the

time dependent contribution are needed, therefore the HDE

is reduced to:

(4)

The boundary conditions and initial values proposed are

given as follow:

For r = a, the heat flux is represented by a rectangular, or

boxcar function, Figure 2:

elswere;0

300;1

)(

st

tQ

Figure 2. Rectangular function.

For r = b, the boundary condition is given by

(6)

where Tamb = 294.15 K. And, initial conditions take the val-

ues:

4. Results and Discussion

The material used in the simulation set up was the glyc-

erine (CAS 56-81-5). In table 1 is described the thermo-

physical properties of glycerine.

Table 1. Thermophysical properties of glycerin

Property Magnitude Unit

1261 kg/m3

k 0.285 W/mK

c 2470 J/kg-K

The time vector it was intoduced for the transient

response, in the close interval [0,60s], with increments of

0.1 seconds.

The resultant surface, with a rainbow representation, gives

the radial heat distribution in the sample, Figure 3.

ptranspp WQTkTuCt

TC

Tkt

TCp

,),( ambTtbrT

0

,15.294

t

KTini (7)

(5)

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Figure 3. Temperature Field evolution in time

Neumann Boundary condition.

Figure 4 shows the temperature distribution in time at

different radial distance, the first curve (Blue curve) repre-

sent the one needle hot wire experiment and has a good

correlation with the conventional mathematical model.

In the other hand, the figure 4 shows the heat diffusion in

the entire sample and it is observing that with the increment

of radial distance, the amplitude decay drastically.

Figure 4. Temperature vs time at different radial distance.

Figure 5 show the theoretical (red curve) and the simu-

lated (black curve) results for the temperature difference

versus time. It is observed a difference between these re-

sults, which is better appreciated in Figure 6, where the

differences between the curves are represented.

Figure 5. Comparison of conventional method and

Comsol Simulation.

Figure 6. Differential Temperature between the theoretical

and simulated results.

In this form, the use of numerical simulation software to

obtain the solutions of the PDE involved in this problem, is

an important tool for investigate the heat transfer in sub-

stances that can be used like complement in the application

of the hot wire technique.

5. Conclusion

It was obtained that the finite element simulation result

has a similar shape with that obtained by the conventional

theoretical model; however, there is a significant difference

between them that can be important to have in account in the

use of this results in the analysis of the experimental data. In

addition, the final element simulation analysis provides a

substantial advantage in this problem, being able to vary the

different parameters of the experiment, such as the radial

distance, the heat flux, the initial temperature, among others,

and thereby optimize the results.

Acknowledgments.

The authors would like to thank to Consejo Nacional de

Ciencia y Tecnologa (CONACyT) of Mxico, Secretaria de

Investigacin y Posgrado (SIP) from Instituto Politcnico

Nacional of Mxico, for its support to this work.

___________________________________________

References.

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[5] M. Khayet and J. M. Ortz Zarate. (2005). Int. J. Thermophys. 26: 637-646.

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