Modeling and Measurement of Thermal Process in Experimental

Borehole in Matlab&Simulink and Comsol Multiphysics

STEPAN OZANA, RADOVAN HAJOVSKY, MARTIN PIES, BLANKA FILIPOVA

Department of Cybernetics and Biomedical Engineering

VSB-Technical University of Ostrava

17. listopadu 15/2172

CZECH REPUBLIC

{stepan.ozana|radovan.hajovsky|martin.pies|blanka.filipova}@vsb.cz

Abstract: - The article deals with the measurement, modeling and simulation of thermal process within the

experimental borehole located at mining dump Hedvika. The model is based on Heat equation, solved in

Matlab and Comsol Multiphysics. The purpose of this model is to verify simulated data with the one retrieved

from sensoric network around and within the experimental borehole. Experiment that provided all the

necessary data has been running at Hedvika mining dump which is still one of the thermally active mining

dumps in the surroundings of Ostrava with subsurface fires in it. The temperature of the mining dump can

increase dramatically any time. The great benefit would be the knowledge of when and where fast temperature

changes occur.

Key-Words: - borehole, Matlab, Comsol Multiphysics, mining dump, modeling, simulation

1 Introduction The issue of mining dumps is very extensive.

The heaps are made from waste and tailings from

coal mines. Waste rocks can catch fire

spontaneously any time and mining dump starts to

burn. Temperatures can change immediately. Fast

change of temperature has a bad effect on the

environment, whether it is the fauna, flora, or the

surrounding buildings and humans. and

arise as a secondary product of combustion. It is

dangerous for living organisms due to exhaust

fumes and high temperature that can reach to the

buildings on the ground. It is one of the main

reasons why the presented data model is being

developed. Currently we are monitoring temperature

changes in the mining dump. It is a large network

made up of tens of sensors. The sensors measure

temperatures at depths of 3 and 6 meters.

Temperature distribution throughout the heap is

determined with using mathematical interpolated

methods based on the temperature measurements.

Current situation illustrated in Fig. 1 has been

changed by adding one more experimental borehole

with more additional equipment than the other

borehole have at these days. All of the data, as the

same as for the other sensors, is transferred to the

MySQL database by GPRS, then it is accessed and

processed by Matlab and Simulink environment.

The basic model (one-chamber) is solved in Matlab

only, then 3D model has been created with the use

of Comsol Multiphysics that allows to solve more

complex tasks by FEM (finite elements method) [6].

Fig. 1. Current sensoric network at Hedvika mining

dump [7].

2 Mathematical background The thermal process within the experimental

borehole can be analytically identified by use of

Heat equation which is then solved either in Matlab

(FDM, FEM) or Comsol Multiphysics (FEM) [1],

[3].

2.1 Heat equation The basic form of heat equation used in this

paper to analyze the dynamic model of thermal

process is according (1).

Advances in Systems Theory, Signal Processing and Computational Science

ISBN: 978-1-61804-115-9 84

(

) (1)

where stands for a short denotation of

and

2.2 Analysis of heat sources The environment of the experimental borehole

includes solid and leeks filled up with a liquid

phase. The liquid phase could consist of both liquid

and gas elements. Heat transfer is carried out by two

ways: convection and conduction. The first way is

bound for presence of media flow, the direction of

heat transfer is determined by the direction of this

flow and the magnitude of particular relative heat

flux is according (2).

Tcwq ttDK *** (2)

Darcy’s relative flow of liquid phase (for laminar

flow) is given by (3).

*

*

L

pKw DD

(3)

The relative heat flux for the heat transfer is

given by (4)

)(* TgradqV (4)

and partial differential equation for thermal field

includes the term (5)

*ca (5)

usually referred to as thermal conductivity

coefficient and other parameters refers to as

c…J/kg/deg…relative heat capacity

ct…J/kg/deg… relative heat capacity of liquid

component

qk…W/m2…relative heat flux (convection)

qv…W/m2…relative heat flux (conduction)

wD…m/s…Darcy’s flow

KD…m2 …permeability

L…m…length

T…K…temperature

…W/m/deg…relative heat conductivity

…N.s/m2…dynamic viscosity

…kg/m3…density

p…Pa…pressure difference

The equation for computing Darcy’s flow

includes permeability KD that depends on porosity

of the environment. Therefore besides relative

density, thermal capacity and thermal conductivity

the porosity has to be entered into the model.

The values of thermal conductivity coefficient

for common materials are rarely mentioned in the

literature resources, it is often determined as a result

of additional computational techniques [2], [4], [5].

3 Modeling of the thermal process in

experimental borehole Modeling of thermal process in experimental was

firstly carried out as one-chamber model and this

model then was extended into three-chamber model

that summarizes heat sources into three volumes.

3.1 Modeling of the basic one-chamber

model in Matlab The Matlab code shown below shows the

solution of basic one-chamber model, by use of

FDM (finite difference model). This solution was

used to determine basic parameters of the model and

mainly helped to turn the solution towards Comsol

Multiphysics.

Fig. 2 shows borehole cylindrical model (type of

single U-pipe) with placement of corresponding

variables: borehole parts, temperature variables and

four-variable functions, borehole essential

parameters (e. g. thermal conductivity λ or borehole

resistance RB), parameters of temperature initial

condition (IC), and temperature boundary conditions

(BC) based on geometrical configuration of this

borehole model (finite length circular cylinder).

Advances in Systems Theory, Signal Processing and Computational Science

ISBN: 978-1-61804-115-9 85

Fig. 2. Borehole cylindrical model (type of single U-pipe) [6].

3.2 Modeling of three-chamber model in

Comsol Multiphysics The main purpose of this experiment is to

measure heating, resp. cooling of a small area in

close surroundings of the experimental borehole by

use of temperature sensors Pt100. Inside there are

three interconnected chambers with cooling media.

The overall height of the apparatus is 4 meters,

having its top 1 meter under the ground, which

works out 5 meters of the depth altogether, as it can

be seen from Fig. 2. Around the borehole itself there

is a cross-cylindrical system of sensors for

temperature measurement, the distance between

sensors are 0.3 meter horizontally and they are

located in three levels vertically, corresponding to

the centers of particular chambers. It is supposed

that temperature changes caused by enforced

cooling happen at least 100 times faster than

temperature changes caused by burning of the

mining dump. Before the start of the experiment the

temperature in the borehole surroundings can be

considered as constant (and can be measured within

the borehole). By control intervention the

temperature in a close surroundings of the borehole

will be affected, ant the temperature trend lines can

be stored.

Fig. 3. Setting up the heat equation in Comsol

Multiphysics environment.

Advances in Systems Theory, Signal Processing and Computational Science

ISBN: 978-1-61804-115-9 86

Fig. 4. Transverse view of the experiment setup.

Fig. 5. Top view of the experiment setup.

Fig. 6. Detail of chamber outlets with Pt100

temperature sensor.

Fig. 7. Meshed geometry in Comsol Multiphysics.

Advances in Systems Theory, Signal Processing and Computational Science

ISBN: 978-1-61804-115-9 87

Fig. 8. 3D plot of temperature over the time in x-

direction

Setting of the partial differential equation

describing the model into Comsol Multiphysics can

be seen in Fig. 3. The experimental setup of the

apparatus that provides the data to be compared to

simulated results is illustrated in Fig. 4, Fig. 5 and

Fig. 6.

The created model is linear time invariant with

variant space parameters. Therefore, is not

necessary to deal with absolute values of

temperature, but it is enough to compute relative

temperature differences related to initial or steady

state. The top and sides of the modeled object will

have zero Dirichlet boundary condition that

determine the ambient temperature, or Neumann

condition computed from zero (relative) ambient

temperature, simulated temperature of cooled object

and heat transfer coefficient (iteration computation).

Meshed geometry ready to compute is illustrated

in Fig. 7. There can be many plots resulting from the

computation, such as the one in Fig. 8 representing

energy spread over x-axis and time.

The sequence of computation is as follows:

setting of zero initial condition

setting of zero Dirichlet boundary condition

input of step change of the power

is brought to the system

retrieving particular values of at given

points (corresponding to measured points)

calculation of from

4 Conclusion The paper presented main idea of modeling

experimental borehole. This model can be then

extended and applied for large areas of mining

dumps provided we have sufficient information

about crucial parameters of area of interest. The

verification between simulated and measured data is

just in the primal phase, but it has been proven that

the concept of the model and the methodology of

experiment are valid. The most crucial fact that has

to be explored in detail in future phases of the

project is heat sources represented by the

component . The Comsol Multiphysics appears to be the most

appropriate tool for solution of such complex model

due to the fact that it allows to model nonlinear

phenomena even in heterogeneous materials with

time and space variant coefficient, while the partial

differential equation(s) are already predefined in this

environment. Of course, it lets user easily define

1D, 2D or 3D plots with computed signals.

5 Acknowledgment The paper was supported by TACR TA01020282

- Enhancement of quality of environment with

respect to occurrence of endogenous fires in mine

dumps and industrial waste dumps, including its

modeling and spread prediction.

References:

[1] Phillips G. M., Taylor P. J.: Theory and

Applications of Numerical Analysis. Elsevier

Academic Press. London, 1996. ISBN 0-12-

553560-0.

[2] Filipova B., Hajovsky R. Using MATLAB for

modeling of thermal processes in a mining

dump. Recent advances in fluid mechanics and

heat & mass transfer. Proceedings of the 9th

IASME/WSEAS International Conference on

Heat Transfer, Thermal Engineering and

Environment. WSEAS Press, pp. 116-119,

2011. ISBN 978-1-61804-026-8.

[3] Stoer J., Bulirsch R. Introduction to Numerical

Analysis. Springer Science + Business Media,

New York, USA, 2002. ISBN 978-0-387-

95452-3.

[4] Vitasek E. Numericke metody. SNTL, Praha

1987.

[5] Lixin L., Revesz P. Interpolation methods for

spatio-temporal geographic data. Computers,

Environment and Urban Systems. Volume 28,

Issue 3, pp. 201 – 227, Elsevier, 2004.

[6] Vojcinak P., Vrtek M., Hajovsky R. Evaluation

and Monitoring of Effectiveness of Heat Pumps

via COP Parameter, In International

Conference on Circuits, Systems, Signals pp.

240-247, 2010. ISBN 978-960-474-226-4.

[7] Hajovsky R., Ozana S., Nevriva P. Remote

Sensor Net for Wireless Temperature and Gas

Measurement on Mining Dumps. In 7th WSEAS

International Conference on Remote Sensing

(REMOTE '11). Penang, Malaysia: WSEAS

Press, 2011. pp. 124-128. ISBN 978-1-61804-

039-8.

Advances in Systems Theory, Signal Processing and Computational Science

ISBN: 978-1-61804-115-9 88

Appendix clear all,close all

maxdel=10; %maximum of slices

dt=1; %integration step

maxtime=100; %number of integration steps

case "integer"

h=5; %length of the step in coordinates

q=1; %input heat source"

h2=h*h; %square of step in coordinates

%initial values:

i=1:1:maxdel;j=1:1:maxdel;k=1:1:maxdel;

Q(i,j,k)=0; %zero initial value

i=1:1:maxdel;j=1:1:maxdel;k=1:1:maxdel;

qt(i,j,k)=0; %zero initial value

n=1:1:maxtime;i=1:1:maxdel;

G(i,n)=0; %reset of regult matrix

%integration start

for n=1:1:maxtime; %cycle for integration steps

%second derivatives in the sides (going through

starting point)

j=1:1:maxdel;

k=1:1:maxdel;

d2Qdx2(1,j,k)=(Q(1,j,k)-2*Q(2,j,k)+Q(3,j,k))/h2;

%second derivative of the energy perpendicular to

x-axis

i=1:1:maxdel;k=1:1:maxdel;

d2Qdy2(i,1,k)=(Q(i,1,k)-2*Q(i,2,k)+Q(i,3,k))/h2;

%second derivative of the energy perpendicular to

y-axis

i=1:1:maxdel;j=1:1:maxdel;

d2Qdz2(i,j,1)=(Q(i,j,1)-2*Q(i,j,2)+Q(i,j,3))/h2;

%second derivative of the energy perpendicular to

z-axis

%second derivatives inside the objects (sides

excluded)

i=2:1:maxdel-1;j=1:1:maxdel;k=1:1:maxdel;

d2Qdx2(i,j,k)=(Q(i-1,j,k)-

2*Q(i,j,k)+Q(i+1,j,k))/h2; %second derivative of

the energy in x-axis

i=1:1:maxdel;j=2:1:maxdel-1;k=1:1:maxdel;

d2Qdy2(i,j,k)=(Q(i,j-1,k)-

2*Q(i,j,k)+Q(i,j+1,k))/h2; %second derivative of

the energy in y-axis

i=1:1:maxdel; j=1:1:maxdel;k=2:1:maxdel-1;

d2Qdz2(i,j,k)=(Q(i,j,k-1)-

2*Q(i,j,k)+Q(i,j,k+1))/h2; %second derivative of

the energy in z-axis

%second derivatives in the sides (going out of

starting point)

j=1:1:maxdel;k=1:1:maxdel;

d2Qdx2(maxdel,j,k)=(Q(maxdel-2,j,k)-

2*Q(maxdel-1,j,k)+Q(maxdel,j,k))/h2; %second

derivative of the energy in the side x=maxdel

i=1:1:maxdel;k=1:1:maxdel;

d2Qdy2(i,maxdel,k)=(Q(i,maxdel-2,k)-

2*Q(i,maxdel-1,k)+Q(i,maxdel,k))/h2; % second

derivative of the energy in the side y=maxdel

i=1:1:maxdel;j=1:1:maxdel;

d2Qdz2(i,j,maxdel)=(Q(i,j,maxdel-2)-

2*Q(i,j,maxdel-1)+Q(i,j,maxdel))/h2; % second

derivative of the energy in the side z=maxdel

%input heat source k=1:1:maxdel;

k=1:1:maxdel;

qt(1,1,k)=(1-exp(0.05*(-n+1)))-0.1*Q(1,1,k);

%time derivatives

i=1:1:maxdel;j=1:1:maxdel;k=1:1:maxdel;

dQ(i,j,k)=d2Qdx2(i,j,k)+d2Qdy2(i,j,k)+d2Qdz2(i,j,k

)+qt(i,j,k); %notation of derivatives in all points

%integration in time

for k=1:1:maxdel; for j=1:1:maxdel;

for i=1:1:maxdel;

Q(i,j,k)=Q(i,j,k)+dt*dQ(i,j,k); % Q(i,j,k)is a

vector depending on n index

G(i,n)=Q(i,3,2); %preparation for

plottingin x axisx

end

end

end

end

%3D plotting

figure(1),surface (G);

axis ([1 maxtime 1 maxdel 0 15]);

xlabel('time'),ylabel('x axis')

zlabel('energy'),grid

Advances in Systems Theory, Signal Processing and Computational Science

ISBN: 978-1-61804-115-9 89