Vol 23, No. 1;Jan 2016
MODELING AND SIMULATION OF TRANSIENT HEAT DIFFUSION IN RUBBERIZED CONCRETE SLAB BY FINITE DIFFERENCE EXPLICIT METHOD Asmarah Kanwal*, Omair Inderyas*, Dr. Siraj-ul-Islam**, Sidrah Kanwal***, S. Farasat Ali**** *Sarhad University of Science & Information Technology Peshawar. PO Box. Landi Akhun Ahmad, Hayatabad Link, Ring Road, Peshawar, Pakistan. Tel: +92-91-5230931 E-mail:
[email protected] The research is financed by: Sarhad University of Science & IT Peshawar, Pakistan. ABSTRACT The paper suggests a mathematical model and simulation of two-dimensional transient heat diffusion through a rubberized concrete slab with 0%, 5%, 10% and 15% of rubber aggregate mixed with concrete to increase thermal resistance of construction material. The purpose of research is to use numerical techniques based on their efficiency and stability for the solution of practical problems occur in the field of civil engineering regarding thermal properties and effect of proposed materials for construction. Finite difference (FD) explicit scheme is the simplest choice for analysis of transient heat diffusion in such problems with constant boundary and initial condition generated for manual testing of the slab in hot box design. The modeling and simulation of such problems provide mathematical and computational approximations to the problem. It was observed that the comparison of sixth diffusion level shows that 15% rubberized concrete has highest thermal resistance than the 10%, 5% and concrete without the rubber aggregate. Also, the numerical simulation interprets the effect of rubber aggregate on heat diffusion in each case depending on chemical and physical properties of the slab. KEYWORDS Rubberized Concrete, Finite Difference Explicit Method and Numerical Simulation. INTRODUCTION: Several type of materials are used to improve the quality of concrete in terms of its physical, chemical and thermal properties. Rubber is the heat insulator by its properties so the recycled chip rubber obtained
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Vol 23, No. 1;Jan 2016
from scrap tire is used to increase the thermal resistance of concrete see with percentage of 5%, 10% and 15% aggregate in the mixture [1]. To test the samples hot box design experiment is conducted in which the temperature in measured through the thermometers placed on both sides of the specimen.
Figure 1: Hot Box Design
Figure 2: Hot Box Image For the whole process, each slab is tested in 180 minutes in which the flanking loss is avoided for the purpose of minimum error. The modern computing techniques is the best solution in such problems to reduce the time factor and human efforts. Computational mathematics and numerical techniques provides a suitable approximation to the engineering problems in case of heat diffusion, wave propagations, earthquake vibrations and flows etc. For this purpose the numerical finite difference, finite element and finite volume schemes are used with MATLAB simulation accordingly for steady state and transient heat diffusion in different types of
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boundary and initial conditions. To follow on with the problem we first define the equation of model and the conditions accordingly to the experimental work. MATHEMATICAL MODEL OF THE PROBLEM The model equation we use is the transient heat equation written as ππ’π = π β 2 π’π (1) ππ‘ In equation (1) π’π = π’π (π₯, π¦, π‘) In which π₯ πππ π¦ represents the Cartesian coordinates as the width and length of the slab. π β€ π₯ β€ π, π β€ π¦ β€ π According to [2], each slab has the dimensions of 3ββΓ7ββ ππ πππππ’π πππ πππ‘π π = (2) ππ ππ Where ππ is thermal conductivity, ππ is density and ππ is the specific heat of slab. For the boundary and initial conditions we refer [2]. The slab is provided homogeneous heat with 80Β°C where the other side is at room temperature of 30Β°C.the slab is initially at temperature of 30Β°πΆ then we let boundary conditions are π’π (π, π¦, π‘) = 80Β°πΆ
(3)
π’π (π, π¦, π‘) = π’π (π₯, π, π‘) = π’π (π₯, π, π‘) = 30Β°πΆ
(4)
METHODOLOGY For finite difference explicit method we first discretize the equation (1) on the domain i-e the slab of length L=7ββ=177mm and thickness is W=3ββ=76mm. The time derivative is chosen to be discretized by forward difference while the spatial derivatives are discretized by the central difference as ππ’π π’π (π, π, π + 1) β π’π (π, π, π) = ππ‘ βπ‘
(5)
π 2 π’π π’π (π + 1, π, π) β 2π’π (π, π, π) + π’π (π β 1, π, π) = ππ₯ 2 βπ₯ 2 2 π π’π π’π (π, π + 1, π) β 2π’π (π, π, π) + π’π (π, π β 1, π) = ππ¦ 2 βπ¦ 2 Then equation (1) implies π’π (π,π,π+1)β π’π (π,π,π) βπ‘
π’π (π+1,π,π)β 2π’π (π,π,π)+π’π (πβ1,π,π)
= π[
βπ₯ 2
+
(6) (7)
π’π (π,π+1,π)β 2π’π (π,π,π)+π’π (π,πβ1,π) βπ¦ 2
]
(8)
Where βπ₯ πππ βπ¦ indicates the nodes spacing in both spatial directions, and there are now two indices for space,π πππ π πππ π₯π πππ π¦π respectively. k represents the time level. The equation (8) can be written as π’π (π, π, π + 1) β π’π (π, π, π) =
πβπ‘
βπ₯ 2 πβπ‘
βπ¦ 2
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[π’π (π + 1, π, π) β 2π’π (π, π, π) + π’π (π β 1, π, π)] +
[π’π (π, π + 1, π) β 2π’π (π, π, π) + π’π (π, π β 1, π)] (9)
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If we let πΌ =
πβπ‘ βπ₯ 2
and π½ =
πβπ‘ βπ¦ 2
then equation (9) implies
π’π (π, π, π + 1) = (1 β 2πΌ β 2π½)π’π (π, π, π) + πΌ[π’π (π + 1, π, π) + π’π (π β 1, π, π)] + π½[π’π (π, π + 1, π) + π’π (π, π β 1, π)] (10) The above equation is the finite difference scheme for transient heat diffusion in a rectangular region and the stability region of finite difference explicit scheme is 2πβπ‘ β€1 (11) min(βπ₯ 2 , βπ¦ 2 ) RESULTS AND DICUSSIONS For simulation we have used MATLAB and generated codes accordingly to the following tables for each slab For n=7 and for 180 minutes Table No. 1: Slab With 0% Rubber Aggregate Slump size
Density
Specific heat
π
mm
g/ππ
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0.0024971
J/kg.K
Thermal conductivity W/m.K
0.90157
1.3260
Then the heat diffusion is shown in the figures below;
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heat diffusion
70 60 50 40 30 200 150 100 50 0 y=177 mm
0
10
20
30
40
50
70
60
80
x=76 mm
Figure 3: Heat Distribution in Slab with 0% Rubber Aggregate
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80 75 70
heat diffusion
65 60 55 50 45 40 35 30
0
20
40
60
80 100 y=177 mm
120
140
160
180
Figure 4: Y-axis-Heat Diffusion View Table No. 2: Slab With 5% Rubber Aggregate Slump size
Density
Specific heat
mm
g/πππ
J/kg.K
22
0.002321
0.91655
Thermal conductivity W/mK 1.2848
80
heat diffusion
70 60 50 40 30 200 150 100 50 0 y=177 mm
0
10
20
30
40
50
60
70
80
x=76 mm
Figure 5: Heat Diffusion in slab with 5% Rubber Aggregate
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80 75 70
heat diffusion
65 60 55 50 45 40 35 30
0
20
40
60
80 100 y=177 mm
120
140
160
180
Figure 6: Y-axis and Heat Diffusion View Table No. 3: Slab With 10% Rubber Aggregate Slump size
Density
Specific heat
mm
g/πππ
J/kg.K
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0.002201
0.9316
Thermal conductivity W/mK 1.1892
80
heat diffusion
70 60 50 40 30 200 150 100 50 0 y=177 mm
0
10
20
30
40
50
60
70
80
x=76 mm
Figure 7: Heat Diffusion in slab with 10% Rubber Aggregate
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Vol 23, No. 1;Jan 2016
80 75 70
heat diffusion
65 60 55 50 45 40 35 30
0
20
40
60
80 100 y=177 mm
120
140
160
180
Figure 8: Y-axis and Heat Diffusion View Table No. 4: Slab With 15% Rubber Aggregate Slump size
Density
Specific heat
mm
g/πππ
J/kg.K
15
0.002032
0.94644
Thermal conductivity W/mK 1.0780
80
heat diffusion
70 60 50 40 30 200 150 100 50 0 y=177 mm
0
10
20
30
40
50
60
70
80
x=76 mm
Figure 9: Heat Diffusion in slab with 15% Rubber Aggregate
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Vol 23, No. 1;Jan 2016
80 75 70
heat diffusion
65 60 55 50 45 40 35 30
0
20
40
60
80 100 y=177 mm
120
140
160
180
Figure 10: Y-axis and Heat Diffusion View Before conclusion compared the values at any time level in each case. The following graph shows difference of heat diffusion at any time level. The green curve represents 15% rubberized slab, magenta curve represents 10% rubberizes concrete, blue represents 5% rubberized concrete and red represents 0% of rubberized concrete results of heat flow within the slab.
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comparison of heat diffusion in each case
70 65 60 55 50 45 40 35 30
0
20
40
60
80
100
120
140
160
180
y=177mm
Figure 11: Difference of Heat Flow in 0%, 5%, 10% and 15% Rubberize Concrete
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CONCLUSIONS: The comparison of heat flow within the slab of 0%, 5%, 10% and 15% of rubber aggregate was carried out shown in figure 11, it was observed that the comparison of sixth diffusion level shows that 15% rubberized concrete has highest thermal resistance than the 10%, 5% and concrete without the rubber aggregate. The mathematical approximations using finite difference explicit method may not be the best approximation to the problem but is a simplest method to make the conclusion about thermal resistance of concrete based on its chemical and physical properties. NOTATION β2 is the sum of 2nd order partial derivatives with respect to spatial variables . ALGORITHM clear; W = width in mm along x-axis L = length in mm along y-axis Tend = final time maxk =maximum time level dt = Tend/maxk; n = node points defining the initial conditions u(1:n+1,1:n+1) =initial conditions dx = W/n; dy = L/n; h1= dx; h2 = dy; b1 = dt/ (h1*h1); b2=dt/ (h2*h2); cond = thermal conductivity of slab spheat = specific heat of slab rho = density of slab a = cond/(spheat*rho); alpha = a*b1; beta=a*b2; for i = 1:n+1 x(i) =(i-1)*h1; y(i) =(i-1)*h2; end for k=1:maxk+1
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time(k) = (k-1)*dt; for j=1:n+1 set of boundary conditions u(1,j) =; u(n+1,j) =; u(i,1) =; u(i,n+1) =; end end finite difference explicit method for k=1:span:maxk for j = 2:n for i = 2:n u(i,j)=(1-2*alpha-2*beta)*u(i,j)... +alpha*(u(i-1,j)+u(i+1,j))... +beta*(u(i,j-1)+u(i,j+1)); end end end % temperature versus space at the final time mesh(x,y,u') REFERENCES 1.
2.
3.
4. 5. 6. 7.
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S.F.A.Shah, A.Naseer, S.A.A.Shah, M.Ashraf, Evaluation of Thermal and Mechanical Properties of Concrete Containing Rubber Aggregate, Arabian Journal For Science and Engineering ISSN 1319-8025 Arab J Sci Eng DOI 10.1007/s13369-014-1294-1. Syed Farasat Ali Shah, Evaluation of Thermal and Mechanical Properties of Concrete Containing Rubber Aggregate, unpublished, University of Engineering and Technology, Peshawar, Pakistan, 2012. Matthew R. Halla, Khalid B. Najima, Christina J. Hopfeb, Transient thermal behavior of crumb rubber-modified concrete and implications for thermal response and energy efficiency in buildings , M.R. Hall et al. / Applied Thermal Engineering 33 (2012) 77-85. Won Young Yang, Wenwu Cao, Tae-Sang Chung, John Morris, Applies numerical methods with MATLAB, John Wiley & Sons INC, Publications. th Richard l.Burden, J.Douglas Faires, numerical analysis, Brooks/Cole, Cengage Learning, 9 edition. MD Azree Othuman Mydin. Modeling of Transient Heat Transfer in Foamed Concrete Slab. Journal of Engineering Science and Technology Vol. 8, No. 3 (2013) 326 β 343. Two dimensional heat equation with FD. Modeling Earth Systems. USC GEOL55.
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