A NEW EXPERIMENTAL METHOD FOR MEASURING THERMAL CONDUCTIVITY AND THERMAL PARTITION COEFFICIENT OF BUILDING MATERIALS A. Bodalal1†, and F. Haghighat2 1
Mechanical Engineering Department, University of Garyounis, Benghazi-Libya 2 Environmental and Architectural Engineering Department, University of Concordia, Montreal-Canada
ABSTRACT In this paper, a new experimental method for measuring thermal conductivity and thermal partition coefficient for building materials is presented. The uniqueness of this method is that the aforementioned physical properties are measured for the building materials under the same conditions as in real life. This method is based on monitoring the temperature difference across the sample and the solution of the transient conduction equation. The proposed method and designed apparatus were tested using some building materials (i.e. sheet metal, glass windows, concrete bricks, concrete bricks with plaster and marble) and the preliminary tests showed promising results.
KEYWORDS thermal conductivity, partition coefficient, thermal comfort, building materials
INTRODUCTION Nowadays, a wide variety of building materials form the building envelope such as concrete bricks, wooden boards, metal curtains etc. The physical properties of these building materials play a pivotal role in the energy conservation of the building as well its thermal load. With growing awareness of the problems associated with indoor air quality and the energy consumed in the building, there is now a strong impetus for manufacturing and selecting building materials which are friendly to the environment as well as thermally efficient. Success in both manufacturing and choosing environmentally conscious building materials will be greatly facilitated by the availability of mathematical models for predicting the thermal performance of buildings made of these materials. Among the key input parameters of any heat transferred based model designed for predicting the building thermal performance are the thermal conductivity (k) and thermal partition coefficient (ke). The accuracy of the prediction model is directly affected by the accuracy of the previously stated input parameters. This research is aimed at developing a new experimental approach to determine simultaneously the thermal diffusivity (α) and the thermal partition coefficient (ke). The uniqueness of this method is that the test facility is designed to simulate the real case scenario where the building materials usually exist. In other words, the material under investigation is bounded by two layers of air which mimic the indoor and outdoor environment.
THEORY Basically, this method is based on solving the transient one-dimensional conduction equation by using Laplace-Carson Transformation Technique. An experimental arrangement will be described later in this research in which the heat transfer rate through a test specimen is observed by measuring temperatures on its two sides. Therefore, from the transient one-dimensional conduction equation we have [11]:
∂ 2T 1 = ∂x 2 α †
∂T ∂τ
Corresponding Author: E-mail address:
[email protected]
(1 )
Where;
T is the temperature of the test specimen x is the distance in the test specimen
τ
is the time
From the boundary conditions we can write the following:
T = k e t1 T = ket 2
The first boundary condition is The second boundary condition is
at τ = 0,
The initial condition is
at x = 0 at x = l 0≤χ ≤l
(2 − a) (2 − b)
T =0
(2 − c)
Also, from the heat balance around the inner and outer air test chambers we get;
αA
∂T ) x= 0 dτ ∂x αA τ ∂T t 2 = t 0 − ( ) ∫ ( ) x=l dτ v2 0 ∂x t1 = (
v1
τ
)∫ (
(3)
0
(4)
Where
T
t1 t2 to ke v1 and v2 A α l
is the temperature of the test specimen (variable with time τ) is the outer test chamber air temperature (variable with time τ) is the inner test chamber air temperature (variable with time τ) is the initial air temperature of the inner test chamber (at τ = 0) is the partition coefficient between the specimen surface and the air are the volume of inner test chamber and outer chamber respectively is the specimen surface area is the specimen thermal diffusivity is the specimen thickness
The governing equation and its boundary and initial equations are then solved analytically using laplace transformation , where S, s1,and s2 are a transformation of T, t1, and t2 respectively. Hence, the transformed equations can be written as: ∞
∫
S ( x, p ) = l{t ( x,τ )} = p t ( x∞, τ ) exp{− pτ }dτ
(5)
s1 ( p ) = l{t1 (τ )} = p ∫ t1 (τ ) exp{ − p τ } d τ
(6 )
s 2 ( p ) = l{t 2 (τ )} = p ∫ t 2 (τ ) exp{ − p τ }d τ
(7 )
0
0 ∞
0
For instance, upon transforming equation (1) it will take the following form:
α
d 2S = p {S − t ( x , 0 )} d x2
(8 )
And equations (3) and (4) can be transformed as follows: s1 ( p ) = (
α A v1 p
)(
dS ) dx
X =0
(9 )
α A
dS )( ) X =l (10 ) v2 p dx Upon imposing the initial condition ( Eq. 2-c) the solution of the ordinary differential equation (Eq. (8) will take the form: s2 ( p ) = t0 − (
S = λ 1 sinh{ x ( p / α ) 0 . 5 } + λ 2 cosh{ x ( p / α ) 0 . 5
(11 )
Also by applying the boundary conditions (Eqs. 2-a and 2-b) on equations (9) and (10) we get:
s1 ( q ) =
β 1t 0 ( β 1 β 2 / q 2 − 1) q sin( q ) + ( β 1 + β 2 ) cos( q )
(12 )
s2 (q ) =
{ β 1 cos( q ) − q sin( q ) } t 0 ( β 1 β 2 / q 2 − 1) q sin( q ) + ( β 1 + β 2 ) cos( q )
(13 )
Where;
q = l( p / α ) 0.5 , β1 =
ke Al ke Al , β2 = v1 v2
(14)
In case where the air volume of the inner chamber (v2) became equal to the air volume of outer chamber (v1), (which has been imposed experimentally), the value of (β1+β2) and (β1β2) can be replaced with (2β) and (β2) respectively. The inverse transformation of Eqs. (12 ) and (13) according to the generalized Heaviside Theorem [12] yields the relationships for measuring quantities t1 and t2 −1 − q 2ατ (t 2 − t1 ) ∞ = ∑i =1 4β {qi2 + β (2 + β )} Exp( i2 ) t0 l
(15)
where qi are the positive roots of the characteristic equation
tan( q ) =
(q
2 β q 2 − β i
i 2
( 16 )
)
It is worth mentioning here that equation (15) converges rapidly for small time intervals (τ). Therefore, if the first value of the series is considered we end-up with:
( t 2 − t1 ) − q 12 α τ 4β exp{ } = 2 t0 q1 + β ( 2 + β ) l2
(17 )
Taking the natural logarithms for both sides of equation (17) gives:
ln
t 2 − t1 q2 α 4β }− 1 2 τ = ln { 2 t0 q1 + β ( 2 + β ) l
(18 )
Equation (18) is a straight line equation with a slope ( r) equal to:
r =
−
q 12 α l2
And (u) represents the intercept of the axis
u = ln {
4β } q 12 + β ( 2 + β )
(19 ) ln (
t 2 − t1 ) and is given by: t0
( 20 )
On solving Eqs. (16),(19) and (20), the auxiliary parameters, q1 and the quantities α and β can be computed from experimentally determined values r and u. the thermal partition coefficient ke can then be determined by Eq.(14) where β1 =β2 =β .
DESCRIPTION OF THE TEST FACILITY The developed apparatus consists of two chambers. Figure (1) gives a general view of the test facility. The inner chamber, which constructed from the material under investigation, is placed inside the outer chamber. Sets of thermocouples were distributed in selected positions in both chambers. Five thermocouples were designated for the outer chamber while four thermocouples were designated for the inner chamber. A detailed description of each chamber is given below: (i) The outer chamber: The size of the outer chamber is (51.5 x 56.5 x 96) cm .The outermost frame of the outer frame is fabricated from wood. Figure (2) gives a cross section of the wall of the outer chamber. As shown, the total wall thickness of the outer room is approximately equal to 12 cm and consists of two main frames (outer wooden frame and inner Styrofoam layer) sandwiching a layer of highly insulating materials, to minimize the heat loss from the outer chamber walls. The outer room is fixed on a portable trailer for easier movement of the apparatus and is also fitted with a very tightly sealed door for loading and unloading of the test specimen (inner chamber). As we can see, special attention was paid to the thermal insulation of the outer chamber since the boundary conditions (namely Eq. 3) assumes that the heat loss from the outer chamber wall is zero. (ii) The inner chamber: The inner chamber or, as we may call it, the ‘test specimen’ is fabricated from the material which is required to measure its thermal conductivity. For instance, figures 3 and 4 show two specimens made of steel sheet and glass windows respectively. The test specimen (inner chamber) is fitted with two ports one is used for charging the inner chamber with hot air by mains of hot air blower (see Fig. 7) while the other port is used for inserting the wires of the thermocouples which are used for measuring the temperature of the inner air chamber.
Figure (1): General view of the test facility
Figure (2): Cross section of the wall of the outer chamber
Figure (3): Specimen ‘inner chamber’ made from steel sheet
Figure (4): Specimen ‘inner chamber’ made from glass windows
EVALUATION OF THE TEST FACILITY A set of tests, aimed at checking some of the assumptions stated earlier as well as refining the experimental procedure, were conducted in the beginning of this research. The main points of concern were the insulation efficiency of the outer chamber, the repeatability of the test data and the homogeneity of the two chambers air temperatures readings. The remainder of this section is devoted for the discussion of the abovementioned concerns:
(a) Calibration of the Thermocouples Nine (9) thermocouples were employed in this apparatus. All thermocouples were calibrated using a reference thermocouple and the precision was found to be within ± 0.5 C0. (b) Evaluation of the insulation efficiency of the outer chamber The efficiency of the outer chamber thermal insulation is evaluated by raising the chamber air temperature to about (50 0C) and monitoring its temperature reading as a function of time. As shown in figure (5) the outer chamber air temperature did show any significant change within the period of 50 minutes, which represent the maximum test duration for specimen. This stability in the chamber air temperature means that the outer chamber insulation is very efficient.
Figure (5): Evaluation of the outer chamber insulation
TEST MATERIALS AND SPECIMEN PREPARATION As mentioned earlier, the materials investigated in this research are: ( a ) Concrete block without plastering ( b ) Concrete block plastered from both sides ( c ) Marble ( d ) Glass windows (both 5 mm and 10 mm thickness) ( e ) 3 mm thickness steel sheet (for calibration and validation) It is to be mentioned here that in this research work, the steel sheet and the window glass samples were testes first. This has been done on the basis that the thermal conductivities for these materials are well documented in the literature [5-7]. This will allow for comparison between these literature values and the results obtained from this method. Therefore, the accuracy of the developed method can be investigated. Samples were built in the same way as that would be made in practice. For example the concrete block has been castled from a mix which is prepared by one the concrete block local factories. Also the plastering for the plastered concrete block samples was made by one of the plastering workers who was hired from the local building market. All of the samples were made to form the inner room for the testing apparatus. Table (1) shows the details of the samples (inner room) prepared and their dimensions. It is to be mentioned here that the differences observed in the samples sizes is due to the fact that the measuring method assumes equal volumes of the inner and the outer room air. As the samples have different thickness, it was necessary to produce samples with different outer dimensions which will produce equal air volumes (v1 = v2). Figure (6) a, b, and c shows three of the fabricated samples which have been considered in this study. They are the concrete block without plastering, the concrete block with plastering from both internal and external surfaces, and the imported marble. Figure (7) shows the adopted air charging method.
Table (1) Detailed dimensions of the specimens (inner chamber) 3 Materials Tested Thickness of the Air volume (m ) specimen (mm) Glass Window (5 mm) 0.5 14079 Glass Window (10 1.0 12312 mm) concrete block with-out 2.2 6624.576 plastering concrete block with 3.2 6178.176 plastering steel sheet (mm) 0.3 6664 Marble 3 4284
b:
Inner surface area of the specimen (m2) 3686 3384 2356.56 2308.56 2296 1816
Figure (6) a: Concrete block without plastering
Figure (6) plastering
Concrete
block
with
Figure (6) c: Marble sample
Figure (7) : the heater connected to the test specimen
TESTING PROCEDURE The test is performed according to the following sequence: (1) For compatibility of the initial conditions (at T = 0, τ = 0), the sample was put in a freezer for 24 hours to make sure that the samples is completely reached zero degree centigrade. (2) In the day of the experiment the thermocouples were re-calibrated (six thermocouples) for confirmation. (3) As mentioned previously each sample is provided with two openings for air charging by the air blowing tool prepared particularly for this purpose (see Figure 8). This air blowing tool consists of a blowing fan, a heating coil and a thermal switch. One opening of this tool is fitted to the inner room as shown for a predetermined period of time which will change all of the inner room air while the other inner room opening is lift open. (4) After air charging of the inner room (the sample), both of the inner room openings are closed by a piece of cork. Two thermocouples were fitted in the inner room through the cork pieces from each opening of the sample. Average of the two thermocouples reading was considered for more accuracy. (5) After charging and fitting of the two thermocouples the sample is instantly fitted in the outer room and the apparatus is firmly closed after fitting the other four thermocouples in suitable predetermined places in the air surrounding the inner room.
(6) The temperatures are measured from the fitted six thermocouples at certain pre-defined time intervals and then the results are tabulated for further data sorting and analysis. The mean value of the inner room two thermocouples readings and the mean of the four outer room thermocouples is then determined and also fitted in the same previous table. (7) It is to be emphasized that the time spent from the start of the air charging process until the closing of the outer room ranges between 4 to 5 minutes, and the time to start temperature reading (zero time) is calculated from closing of the outer room.
SOLUTION PROCEDURE t −t The following procedure was used: ln ( 2 1 ) (1) from the straight line part of the plot vs τ determined from experiment, the quantities r t0 and u (the slope and the negative intercept) were calculated by the least-square regression method. (2) the root q of the characteristic equation [Eq.(16)] was first computed by using the Bisection method [10] for selected values of β . Note that until now the correct values of β and q were unknown. (3) from Eq. (20) q1 values were calculated and plotted against β for a given u. (4) the values q1 and β corresponding to the intercept between Eq. (15) and Eq.(20) for the experimentally determined u are the correct q1 and β values. (5) finally, the thermal diffusivity α and thermal partition coefficient ke were calculated using Eqs. (19) and (14) respectively.
RESULTS AND DISCUSSIONS The temperature readings of the thermocouples in the two chambers are recorded, averaged and tabulated as shown in tables 2 to 7. Also the variation of the ln (t2 -t1 /t0) with time for the tested materials are given in figures 8 to 13. As we can see, the linearity of ln (t2 -t1 /t0) with time supports the assumption stated earlier regarding the truncation of the series solution and considering only its first term. This also reflects the accuracy of the measurement of temperature in the two chambers. Since the solution procedure is based upon the accuracy of the values of the slope and the intercept from the ln (t2 -t1 /t0) axis, it may be deduced that the closer the line in the graph is to a straight line, the more accurate and reliable are its readings. In that context, due attention was paid to the straightness of the slope. As is noticed from the figures below, the accuracy index of the result, in general, is above 0.95 (i.e. 95 %). Table (8) summarizes the results of the thermal diffusivity, partition coefficient, and the thermal conductivity for the tested materials. As a validation of the solution procedure and also to the test facility, the measured thermal conductivity from this study for steel sheet and glass window [41.633(w/m.c0) and 1.034 (w/m.c0)] are compared with the results listed in the literature from previous studies [11]. The results of the validations showed that the differences between the results from this study and the results from the previous studies for steel sheet and glass window were 3.1 % and 5.7 % respectively. This difference might be due to two reasons. Firstly, in this study the value of the thermal partition coefficient (ke) is considered while it was ignored in the previous studies. Secondly, the tested specimen in this study is bounded by two layers of air, which mimic the indoor and outdoor environment air, whereas in the majority of the previous studies, the specimen was placed between two plates and the temperature drop between them was measured. The method used in this study simulates the actual conditions faced by the materials and its results are presumably more correct. This study has ventured into the calculation of two important thermal properties of building materials (i.e. thermal conductivity and thermal partition coefficient). The results of this method, though promising are still preliminary in nature and are meant to stimulate further research in this method of calculation to refine the test procedure and to generalize its use.
Table (2): Readings of steel sheet sample Outer chamber air temperature (Co) τ τ min hr t1 ' t1 ' t1 ' t1 ( average) 0 0 33 30 31 31.33333 5 0.083333 33.5 34 36 34.5 10 0.166667 35 35 37 35.66667 15 0.25 36 36 36.5 36.16667 20
0.333333
36
36
36
36
25 30 35
0.416667 0.5 0.583333
37 37 36.5
37 36 35
36 37 36
36.66667 36.66667 35.83333
Inner chamber air temperature (Co) t2 ' t2 ' t2 ( average) 47 48 47.5 45 46 45.5 43 44 43.5 41 41 41 38. 39 38.75 5 38 39 38.5 38 38 38 37 37 37
Table (3): Readings of flat glass window 5mm thick sample Outer chamber air temperature (Co) Inner chamber air temperature (Co) τ τ t1 ' t1 ' t1 ' t1 t2 ' t2 ' t2 ( min ) ( hr ) ( average) averag e 0 0 15 14 16 15 25 25 25 5 0.083333 15 14 16 15 25 25 25 10 0.166667 15 14.5 16.5 15.33333 24.5 24.5 24.5 15 0.25 15.5 14.5 17 15.66667 24.5 24.5 24.5 20 0.333333 15.5 14.5 17 15.66667 24 24 24 25 0.416667 16 14.5 17 15.83333 24 24 24 30 0.5 16 15 17.5 16.16667 23.5 23 23.25 35 0.583333 16 15 17.5 16.16667 23.5 23 23.25
Figure(8): The Variation of ln (t2 -t1 /t0) with the time τ for steel sheet
ln (t2 -t1 /t0) -0.43592 -0.82098 -1.16049 -1.64334 -2.20727 -2.61274 -2.93119 -3.06473
ln (t2 -t1 /t0) -0.91629 -0.91629 -1.0033 -1.04034 -1.09861 -1.11881 -1.26113 -1.26113
Figure(9):The Variation of ln (t2 -t1 /t0) with the time τ for window glass 5 mm thick
Table (4): Readings of flat glass window 10 mm thick sample Outer chamber air temperature Inner chamber air temperature (Co) τ τ (Co) min ( hr ) t1 ' t1 ' t1 ' t1 t2 ' t2 ' t2 ( average) ( average) 0 0 11 12 14 12.33333 25 25 25 5 0.083333 11 12 14 12.33333 25 25 25 10 0.166667 11.5 12.5 15 13 24.5 24.5 24.5 15 0.25 11.5 12.5 15 13 24.5 24.5 24.5 20 0.333333 12 12.5 15.5 13.33333 24 24 24 25 0.416667 12 13 15.5 13.5 24 24 24 30 0.5 12 13 16 13.66667 23.5 23 23.25 35 0.583333 13 13.5 16 14.16667 23.5 23 23.25
ln (t2 -t1 /t0) -0.6799 -0.6799 -0.77653 -0.77653 -0.85175 -0.8675 -0.95885 -1.01243
Table (5): Readings of Concrete block with plaster sample Outer chamber air Inner chamber air temperature (Co) temperature (Co) τ τ t2 ' t1 ' t1 ' t1 ' t1 t2 ' t2 ( average ( min ) ( hr ) ( average ) 0 0 15 14 12 13.66667 25 25.5 25.25 14. 5 0.083333 16 5 12 14.16667 25 25 25 10 0.166667 16 15 12 14.33333 24.5 25 24.75 15 0.25 16 14 13 14.33333 24.5 24.5 24.5 20 0.333333 16 14 13 14.33333 24 24.5 24.25 25 0.416667 16 14 13 14.33333 24 24 24
Figure(10):The Variation of ln (t2 -t1 /t0) with the time τ for window glass 10 mm thick
ln (t2 -t1 /t0) -0.95163 -1.01857 -1.05779 -1.08208 -1.10698 -1.13251
Figure(11):The Variation of ln (t2 -t1 /t0) with the time τ for concrete block with plastering
Table (6): Readings of Concrete block without plaster sample Outer chamber air temperature Inner chamber air o (Co) τ τ temperature (C ) min ( hr ) t1 ' t1 ' t1 ' t1 t2 ' t2 ' t2 average average 0 0 16 13 10.5 13.16667 20 20 20 5 0.083333 17 13 11 13.66667 20 19.5 19.75 10 0.166667 18 14 11 14.33333 19.5 19.5 19.5 15 0.25 18.5 14 12 14.83333 19 19 19 20 0.333333 19 15 12 15.33333 19 19 19 25 0.416667 19 15 12.5 15.5 18.5 19 18.75 30 0.5 19 15 12.5 15.5 18 18.5 18.25 Table (7): Readings of marble sample Outer chamber air temperature (Co) τ τ t1 ' t1 ' t1 ' t1 min ( hr ) ( average ) 0 0 11 13 14 12.66667 5 0.083333 11 13 14.5 12.83333 10 0.166667 11.5 13 14.5 13 15 0.25 11.5 13 14.5 13 20 0.333333 11.5 13.5 14.5 13.16667 25 0.416667 11.5 13.5 15 13.33333
Inner chamber air temperature (Co) t2 ' t2 ' t2 averag e 29 29 29 29 29 29 28.5 29 28.75 28.5 28.5 28.5 28.5 28.5 28.5 28.5 28.5 28.5
ln (t2 -t1 /t0) -1.29706 -1.41332 -1.57665 -1.79176 -1.91959 -2.04022 -2.20727
ln (t2 -t1 /t0) -0.60799 -0.61825 -0.64436 -0.66036 -0.67117 -0.6821
Figure(12):The Variation of ln (t2 -t1 /t0) with the time (τ) for concrete block without plastering Table (8): Results summary Material tested
5mm thick glass window 10mm thick glass window Concrete block without plaster Concrete block with plaster Marble Steel sheet
Figure(13):The Variation of ln (t2 -t1 /t0) with the time (τ) for marble
Thermal diffusivity (m2/s), "α"
Thermal partition coefficient (ke)
4.5774688x10-7
48.12
-7
46.736
1.68904243x10-5
15.332
1.596982428x10-6
6.774
4.85771322x10
1.484390038x10 -5 1.12831x10
-6
3.381 28.584
Thermal conductivity coefficient (w/m.co) 1.034 1.1017 2.542 1.916 3.087 41.633
CONCLUSIONS AND RECOMMENDATIONS The following conclusions can be drawn from this study: 1. A novel approach for simultaneous measurements of the thermal conductivity and thermal partition coefficient of building materials is developed. 2. The uniqueness of this method is that the measured properties are estimated for the building materials under the same conditions as in real life whereas in the previous studies, the tested material was placed between two plates and the temperature drop between them was measured. These are not the actual conditions faced by the building material and hence a more accurate result can be expected to be obtained from the method developed here. 3. This method is based on monitoring the temperature difference across the sample and the solution of the transient conduction equation. 4. The proposed method and designed apparatus were tested using some building materials (i.e. sheet metal, glass windows, concrete bricks, concrete bricks with plaster and marble) and the preliminary tests showed promising results.
ACKNOWLEDGMENT The authors are very thankful to the Garyounis University Research and Consultations Center for its financial and logistical support. In addition, we are equally grateful for the assistance rendered by our colleague Mr. I. El-Badri.
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