Declaration

Title: MS1 - Conduction Heat Transfer in Porous Media Name: Taahir Bhaiyat Student Number: 782500 Group Number: 19 Date: 17 October 2016

Declaration

UNIVERSITY OF THE WITWATERSRAND, JOHANNESBURG SCHOOL OF MECHANICAL, INDUSTRIAL AND AERONAUTICAL ENGINEERING Name: Taahir Bhaiyat

Student Number: 782500

Course code: MECN3007

Course Name: Mechanical Engineering Laboratory II

Submission Date : 17 October 2016

Project Title: Conduction Heat Transfer In Porous Media

I hereby declare the following: o I am aware that plagiarism (the use of someone else’s work without their permission and / or without acknowledging the original source) is wrong; o I confirm that the work submitted herewith for assessment in the above course is our my unaided work except where I have explicitly indicated otherwise; o This task has not been submitted before, either individually or jointly, for any course requirement, examination or degree at this or any other tertiary educational institution; o I have followed the required conventions in referencing the thoughts and ideas of others; o I understand that the University of the Witwatersrand may take disciplinary action against me if it can be shown that this task is not my own unaided work or that I have failed to acknowledge the sources of the ideas or words in my writing in this task Signature: Date: 17 October 2016

Abstract A porous 60mm x 100mm x 100mm sample with porosity of 0.4 was enclosed in an insulated material and the top surface of the sample was exposed to a heat source, such that the heat conduction length was 60mm. The purpose of this study was to investigate the effect of heat flux on the temperature gradient across the sample, and to see what effect the inclusion of pores had on the effective thermal conductivity of the sample. Physical interpretations of these phenomena were also of interest. Data was obtained for power inputs ranging from 14W to 20W. The temperature difference across the sample was seen to increase linearly with the power input, in accordance with Fourier’s Law for one dimensional steady state heat conduction, and this was reasoned to be due to the simultaneous action of different parts of excess heat between adjacent molecules. It was also observed that the thermal conductivity of homogeneous dense steel was reduced from 10 W /m∙ ℃ to 3.60 W /m∙ ℃ when pores were present. Thus the inclusion of pores had an insulating effect on the sample, and this was seen to be even more pronounced with larger porosities. The reason behind this insulating effect was attributed to the increased heat conduction length caused by inclusions in the porous sample. Several models for predicting the effective thermal conductivity of the sample such as the Maxwell, Rayleigh, and EMT models were considered, and these were critically analysed for their accuracy in predicting the effective thermal conductivity.

1

Contents Declaration ...............................................................................................................................................................................1 Abstract ....................................................................................................................................................................................1 List of Figures ..........................................................................................................................................................................2 List of Tables ...........................................................................................................................................................................2 Nomenclature ...........................................................................................................................................................................2 1

Introduction ......................................................................................................................................................................3 1.1

Background and Motivation ...................................................................................................................................3

1.2

Literature Review ...................................................................................................................................................3

1.2.1

Conduction Heat Transfer ..................................................................................................................................3

1.2.2

Effect of Porosity................................................................................................................................................3

1.3 2

Objectives ...............................................................................................................................................................4

Experimentation ...............................................................................................................................................................4 2.1

Apparatus ................................................................................................................................................................4

2.2

Procedure and Precautions ......................................................................................................................................5

2.3

Uncertainty Analysis ..............................................................................................................................................5

2.4

Observations and Data Processing ..........................................................................................................................5

3

Results and Discussion .....................................................................................................................................................5

4

Conclusion .......................................................................................................................................................................8

5

References ........................................................................................................................................................................9

List of Figures Figure 1: Schematic of test facility...........................................................................................................................................4 Figure 2: Relationship between temperature difference and power input ................................................................................6 Figure 3: Change in thermal conductivity with mean temperature ..........................................................................................6 Figure 4: Effective thermal conductivity models for non-conducting pores (𝑘𝑑 ≈ 0) ............................................................7 Figure 5: Heat flux vectors for a dispersion of spheres in a continuous medium (a) with 𝑘𝑐 > 𝑘𝑑; (b) with 𝑘𝑐 < 𝑘𝑑 ...........8 Figure 6: Distortion of local temperature field and constant heat flow lines obtained from numerical simulations with constant wall temperature at 𝑥 = 0 and 𝑥 = 𝐿 for 𝜙 = 0.15 .................................................................................................8

List of Tables Table 1: Experimental data ......................................................................................................................................................5

Nomenclature Symbol 𝑞𝑥′′ 𝑘 𝑘𝑒 𝑘𝑑 𝑘𝑐 𝑑𝑇/𝑑𝑥 𝐿 Δ𝑇

Meaning Heat flux Thermal conductivity Effective Thermal conductivity Thermal conductivity of dispersed phase Thermal conductivity of continuous phase Temperature gradient in xdirection Heat conduction length Temperature difference

Units 𝑊/𝑚2 𝑊/𝑚 ∙ ℃ 𝑊/𝑚 ∙ ℃ 𝑊/𝑚 ∙ ℃

Symbol 𝑇1 𝑇2 𝑇𝑚𝑒𝑎𝑛 𝑄

Meaning Temperature at 𝑥 = 0 Temperature at 𝑥 = 𝐿 Mean temprerature Power input

Units ℃ ℃ ℃ 𝑊

𝑊/𝑚 ∙ ℃

𝑉

Voltage

𝑉

𝐾/𝑚

𝐼

Current

𝐴

𝑚 ℃

𝜙

Volume fraction of dispersed phase

-

2

1

Introduction

1.1

Background and Motivation

Nearly every energy conversion and production device makes use of energy in the form of heat [1].For example, power plants make use of boilers, condensers, evaporators, pre-heaters, intercoolers and cooling towers in order to convert heat energy into mechanical work. The blades of a gas turbine are specially designed using alloys and Thermal Barrier Coatings (TBC), as well as intricate internal cooling passages, in order to withstand extremely high operating temperatures. Electrical and mechanical systems such as transformers and gearboxes dissipate a large proportion of energy in the form of heat. These are but a few of numerous examples where the understanding of heat transfer is important in analysing the performance and safety of the systems at hand.

1.2

Literature Review

1.2.1

Conduction Heat Transfer

From our everyday experience, we know that heat moves from a place of high temperature to that of low temperature. For example, the end of a steel spoon partly submerged in a hot cup of tea eventually becomes warmer. This concept of diffusive heat transfer is termed conduction. It was formalised by Joseph Fourier in 1822, where he used experiments and dimensional homogeneity to arrive at a quantitative relationship for the heat energy transferred between two ends of a solid which are at different temperatures [2]. In one dimension, Fourier’s Law is expressed as: 𝑑𝑇 (1) 𝑞𝑥′′ = −𝑘 𝑑𝑥 𝑑𝑇 where𝑞𝑥′′ is the heat flux in the x-direction (𝑊/𝑚2 ), is the temperature gradient (𝐾/𝑚), and 𝑘 is the thermal 𝑑𝑥 conductivity (𝑊/𝑚 ∙ ℃ ) - an intrinsic, bulk property which depends on the thermodynamic state of the material [3].The negative sign in equation 1 stresses the second law of thermodynamics (i.e. heat flows in the direction of decreasing temperature). It must be noted that equation 1 is considered valid for homogeneous, isotropic materials under steady state conditions with zero internal heat generation [4]. The thermal conductivity is a macroscopic representation of the molecular interactions which occur during conduction; i.e. electron migration in metals and phonon contribution in non-metals [5, 6]. Thus its numerical value provides an indication of how fast heat is conducted through a material. For monoatomic gases, kinetic theory can be used to predict 𝑘 analytically. In general, however, 𝑘 must be determined experimentally. Values of 𝑘 have been documented for a wide variety of materials in the literature [5-8]. For constant pressure and small linear temperature variation, 𝑘 can be considered approximately constant [9]. Under these conditions, it can be shown that 𝐿 (2) 𝑘 = −𝑞𝑥′′ Δ𝑇 where, Δ𝑇 = 𝑇2 − 𝑇1 is the temperature difference between the two ends of the sample and 𝐿 is the heat conduction length (see Figure 1).

1.2.2

Effect of Porosity

In engineering practice, a large number of applications involve the use of heterogeneous materials such as alloys, ceramics, and metallic foams. Such materials consist of one or more phases dispersed within a continuous medium. If the thermal conductivity of the dispersed phase (𝑘𝑑 ) is smaller than the thermal conductivity of the continuous phase (𝑘𝑐 ) then this is referred to as an ‘internal porosity’ material. In contrast, an ‘external porosity’ material is one in which 𝑘𝑑 > 𝑘𝑐 [10]. The porosity of a material is defined as the ratio of the volume of the dispersed phase to the volume of the combination of the dispersed and continuous phases [6]. Determining the heat flux through a porous medium is somewhat a difficult task because (1) Fourier’s law only applies to homogeneous materials and (2) the pore structure is often complicated. As a result, a porous medium is treated as a homogeneous material with an effective thermal conductivity 𝑘𝑒 (which combines the thermal conductivities of the coexisting phases) and then Fourier’s Law is applied. Logically then, 𝑘𝑒 must be a function of the independent thermal conductivities of the constituent phases, the porosity of the material and the distribution of the dispersed phase [11]. There is no universally applicable model for predicting 𝑘𝑒 ; various models continue to appear in the heat transfer literature [12]. These models are usually specific to the structure of the pores and are limited to within certain porosity bounds. Maxwell [13] was the first to propose a model for the effective thermal conductivity, where he considered spherical particles dispersed in a continuous matrix (and spaced out) such that the thermal interactions between the spheres could be ignored. Maxwell’s expression is as follows: 𝑘𝑒 3𝜙(1 − 𝜆) (3) =1− 𝑘𝑐 2 + 𝜆 + 𝜙(1 − 𝜆) 𝑘 where 𝜆 = 𝑑 and 𝜙 is the porosity. Because thermal interaction between pores was neglected, Maxwell’s model is only 𝑘𝑐

valid for low porosities (under about 25%). Many researchers built on Maxwell’s model to include various effects such as the size, shape, geometry, distribution and interfacial thermal resistance of the pores [14, 15]. Some models are completely 3

theoretical while others may be semi- or fully empirical. Models like Eucken [16] accounted for multiple constituent phases, and other effects such as pressure variations and unsteady heat conduction were studied by Zhao et al. [17, 18] and Monde and Mitsutake [19] respectively. Numerical models were also proposed to model specific geometries and phenomena which are difficult to model using classical mathematics [20-23]. The finite element method [24] is frequently used to verify the results obtained by other classical methods. One model that is widely covered in the literature is the Rayleigh model [25], which considers spheres arranged in a cubic array within a continuous medium and also takes into account the thermal interaction between adjacent spheres. Thus Rayleigh’s model is applicable to higher porosity materials (compared to Maxwell’s model). Also, it can easily be extended to pores with cylindrical geometry. For cylindrical pores and heat transfer direction perpendicular to the axis of the cylinders, Rayleigh’s model is expressed as: 𝑘𝑒 2𝜙 (4) = 1+ 𝑘𝑐 𝐶1 − 𝜙 + 𝐶2 (0.30584𝜙 4 + 0.013363𝜙 8 ) where 𝐶1 =

𝑘𝑑 +𝑘𝑐 𝑘𝑑 −𝑘𝑐

and 𝐶2 =

𝑘𝑑 −𝑘𝑐 𝑘𝑑 +𝑘𝑐

. Another model is the Effective Medium Theory (EMT) [26], which considers a random

distribution such that neither of the phases is necessarily continuous nor dispersed. This is a good medium approximation which can be used to determine the boundary between internal and external porosity materials, but it must be used with caution as it is limited to low porosity bounds. At higher porosities the model predicts an underestimation of 𝑘𝑒 . The EMT model is expressed as: (5) 𝑘𝑒 = 0.25 ((3𝜙 − 1)𝑘𝑑 + [3(1 − 𝜙) − 1]𝑘𝑐 + √[(3𝜙 − 1)𝑘𝑑 + (3{1 − 𝜙} − 1)𝑘𝑐 ]2 + 8𝑘𝑑 𝑘𝑐 ) The series and parallel models, which consider the constituent phases in series and parallel distributions respectively, provide the upper and lower bounds in which all other models must lie within [10]. The series and parallel models are given by: 1 Series: (6) 𝑘𝑒 = (1 − 𝜙)/𝑘𝑐 + 𝜙/𝑘𝑑 Parallel:

1.3

𝑘𝑒 = (1 − 𝜙)𝑘𝑐 + 𝜙𝑘𝑑

(7)

Objectives 1. 2. 3.

Obtain experimentally and explain physically the relationship between heat flux and temperature difference of a given material based on Fourier’s Law Obtain experimentally and explain physically the relationship between effective thermal conductivity and porosity for a two phase porous medium Compare the experimental data obtained with the literature and suggest the material comprising the continuous phase

2

Experimentation

2.1

Apparatus

Figure 1: Schematic of test facility The present study utilised a porous sample block of porosity 𝜙 = 0.4. The dispersed phase consisted of staggered cylinder pores (of conductivity 𝑘𝑑 = 0) placed inside an insulating material (Perspex with wall thickness = 40mm and thick Teflon top surface) of inner dimensions 100mm (width) x 100mm (length) x 60mm (height along the x-axis) as shown in Figure 1. The substrates on the upper and lower sides of the sample were copper (thermal conductivity: 398 𝑊/𝑚 ∙ ℃).The heating 4

pad, controlled by a FLUKE 52II DC power supply was used to provide a constant heat input. The two T-type foil thermocouples of thickness 5𝜇𝑚 (one attached on either side of the sample) connected to an EDISON DM664 voltmeter were used to measure the temperatures 𝑇1 and 𝑇3 .

2.2

Procedure and Precautions

The experiment was set up as shown in Figure 1. The DC power supply was first set to produce approximately 10W by adjusting the voltage 𝑉 and current 𝐼. The temperatures 𝑇1 and 𝑇3 were recorded after sufficient time (20 minutes) for the system to reach thermal equilibrium. (NB: it was assumed that the thermal resistance due to the copper substrates was negligible, thus 𝑇2 = 𝑇3 ). This was repeated for 12W, 14W, 16W, 18W and 20W. Due to time constraints, the experiment was split into two groups. Group 1 measured data for 10W, 12W, and 18W while group 2 measured data for 14W, 16W, and 20W; each group using a separate apparatus. The compiled data is presented in section 2.4. It was important that the power supplied was not too high, in order to prevent inducing large temperatures, as this may have increased the effects of radiation and hence obscured the results.

2.3

Uncertainty Analysis

Using the method outlined in Holman [27], if 𝑦 = 𝑓(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) then the propagated uncertainty in the variable 𝑦 is given by: 2 2 2 𝜕𝑦 𝜕𝑦 𝜕𝑦 Δ𝑦 = √( Δ𝑥1 ) + ( Δ𝑥2 ) + ⋯ + ( Δ𝑥𝑛 ) 𝜕𝑥1 𝜕𝑥2 𝜕𝑥𝑛

(8)

where Δ𝑥𝑖 is the absolute uncertainty in the measured variable 𝑥𝑖 . Applying this to equation 2 we get Δ𝑘𝑒 Δ𝑞𝑥′′ 2 Δ𝑇1 2 Δ𝑇2 2 = √( ) +( ) +( ) 𝑘𝑒 𝑞𝑥 ′′ 𝑇1 − 𝑇2 𝑇1 − 𝑇2

(9)

Thus the maximum uncertainty in the present measurement of effective thermal conductivity can be estimated by equation 9. The absolute errors associated with the thermocouples are assumed to be Δ𝑇1 = Δ𝑇2 = 0.2°𝐶 and for the heat flux, it is estimated that Δ𝑞𝑥′′ = 5 𝑊/𝑚2 .

2.4

Observations and Data Processing 𝑇 +𝑇

The data captured in the experiment is presented in Table 1. Here, 𝑇𝑚𝑒𝑎𝑛 = 1 3 and Δ𝑇 = 𝑇1 − 𝑇3 . The effective thermal 2 conductivity 𝑘𝑒 was calculated by substituting 𝑞𝑥′′, Δ𝑇 and 𝐿 = 0.06 𝑚 into equation 2. The uncertainty Δ𝑘𝑒 was obtained using equation 9 and the absolute uncertainties which are recorded in the column headings of Table 1. Table 1: Experimental data

1

Q (𝑾) 10

𝑽×𝑰 (𝑾) 9.6

𝒒′′ 𝒙 (± 𝟓 𝑾/𝒎𝟐 ) 960

𝑻𝟏 [±𝟎. 𝟐°𝑪] 20

𝑻𝟑 [±𝟎. 𝟐°𝑪] 37.9

𝑻𝒎𝒆𝒂𝒏 [±𝟎. 𝟐°𝑪] 28.9

∆𝑻 [±𝟎. 𝟐°𝑪] 17.9

𝒌𝒆 [𝑾/𝒎 ∙ ℃)] 3.22

𝚫𝒌𝒆 [𝑾/𝒎 ∙ ℃)] 0.05

2

12

11.2

1120

25.6

50.7

38.2

25.1

2.68

0.03

3

14

13

1300

20.6

40.5

30.6

19.9

3.92

0.06

4

16

16.8

1680

25.8

49.1

37.5

13.3

4.33

0.05

5

18

18.6

1860

32.2

61.5

46.9

29.3

3.81

0.04

6

20

19.2

1920

30.8

62.6

46.7

31.8

3.62

0.03

3.60

0.04

NO

Average

3

Results and Discussion

The present study aims to investigate and physically explain (1) the relationship between the heat flux and the temperature difference across a conductor and (2) the relationship between the porosity and the effective thermal conductivity of a porous medium. Figure 2 shows the experimentally determined temperature difference across the sample (8th column in Table 1) plotted against the power input (3rd column in Table 1), together with a least squares fit. The linear relationship between these two parameters means that as the power input is increased, the temperature difference between the two ends of the conductor increases proportionally. This observation is in accordance with Fourier’s law for one-dimensional, steady state conduction (equation 1) which dictates that the heat flux and temperature gradient are directly proportional.To explain this result physically, consider two adjacent molecules of the sample block in Figure 1; molecule 𝑎 located at 𝑥 = 0 and molecule 𝑏 located a small distance away 𝑥 = 𝛿 but still in contact with 𝑎. If 𝑎 is in excess of heat, that is, if the temperature of 𝑎 is 5

𝑇 + 𝜏 and that of 𝑏 is 𝑇, then by the second law of thermodynamics, a portion of the excess heat from 𝑎, associated with the temperature difference 𝜏, must be communicated to 𝑏. Now suppose that the excess of heat is doubled, or, which is the same thing as if 𝑎 possessed a temperature 𝑇 + 2𝜏, and 𝑏 possessed a temperature 𝑇. In this case, the exceeding heat will be composed of two equal parts corresponding to the two halves of the whole difference of temperature 2𝜏; each of these parts would have its proper effect as if it alone existed [2]. Therefore, the quantity of heat communicated from 𝑎 to 𝑏 would now be twice as great compared to when the temperature difference was only 𝜏. This simultaneous action of the different parts of exceeding heat is what constitutes the thermal communication of heat, and the reasoning can be extended to the sum of molecules encompassing the entire sample. Indeed, it can be reasoned that if the temperature difference was double, triple, quadruple, etc. then the heat communicated will also be double, triple, quadruple, etc. thus confirming that the temperature difference (Δ𝑇) must be proportional to the heat transfer rate (𝑄).

Figure 2: Relationship between temperature difference and power input Figure 3 shows the relationship between the effective thermal conductivity (with error bars) and the mean temperature. The average percentage uncertainty in 𝑘𝑒 is 1.1% hence it is small enough to be neglected. The trend line shows a slight increase in the thermal conductivity with temperature even though the temperatures are within the valid range. In the valid range, electrons in the metal lattice do not possess enough energy for collisions to be sufficiently frequent such that the thermal conductivity of the material is affected. Therefore, the thermal conductivity is supposed to be constant in this range [28, 29]. One reason for the discrepancy could be due to the fact that different apparatus was used by the two groups in carrying out the experiment. Another could be that during the experiment, the power input was increased and decreased while trying to get the correct 𝑉 × 𝐼 product, instead of progressively increasing the power input. The experiment must be conducted over a wider range of temperatures, with consequently greater number of data points in order to obtain more accurate results in this respect.

Figure 3: Change in thermal conductivity with mean temperature 6

Figure 4 shows the relationship between effective thermal conductivity and porosity using different effective thermal conductivity models (equations 3-7) with𝑘𝑑 ≈ 0. Using Figure 4, if we choose the Maxwell model, a porosity of 𝜙 = 0.4 gives the ratio 𝑘𝑒 /𝑘𝑐 = 0.5 and, from Table 1, the average effective thermal conductivity is 𝑘𝑒 = 3.60 𝑊/𝑚 ∙ ℃. Therefore the estimated average thermal conductivity of the continuous phase using the Maxwell model is 𝑘𝑐 = 7.20 𝑊/𝑚 ∙ ℃. If we instead choose the Rayleigh model, we find that 𝑘𝑒 /𝑘𝑐 = 0.4 (when 𝜙 = 0.4). By applying the same procedure as before we have 𝑘𝑐 = 9 𝑊/𝑚 ∙ ℃. Similarly, using the Effective Medium Theory (EMT) model will yield a slightly lower 𝑘𝑒 /𝑘𝑐 ratio and hence a slightly higher approximation for 𝑘𝑐 at 𝜙 = 0.4. Using data from the literature [6, 8], the accepted value for the thermal conductivity of dense steel (without pores) is approximately 10 𝑊/𝑚 ∙ ℃. Thus the validity of the experiment is considered to be within acceptable limits, and the EMT model is expected to give the best model for the effective thermal conductivity. Furthermore, it can now be confirmed that the Maxwell model is less accurate for large porosity (above 0.25) and that since the Rayleigh model accounts for thermal interactions between pores, it gives a better approximation than the Maxwell model as anticipated in section 1.2.2.

Figure 4: Effective thermal conductivity models for non-conducting pores (𝑘𝑑 ≈ 0) Note that the effective thermal conductivity for a porous material is lower than that of the same material without pores, and that all of the models presented in the literature predict that the effective thermal conductivity of a porous material decreases as the volume fraction of the pores increases regardless of the nature of the pores. For this reason, porous materials are used extensively as insulators [6]. The reason behind the drop in thermal conductivity due to porosity is not fully understood to date. But accepted explanations involving “optimal heat transfer pathways” and “thermal stretching” have been proposed to provide a physical basis for Maxwellian models [10, 30]. Consider a porous sample having spherical inclusions subjected to a constant temperature difference. Heat, much like current, takes the path of least resistance [13]. Thus if 𝑘𝑑 < 𝑘𝑐 the heat flow tries to avoid the dispersed phase as much as possible. Conversely, if 𝑘𝑐 < 𝑘𝑑 the heat flow tries to stay within the dispersed phase as much as possible (see Figure 5). This in effect demands that the heat flux vector travels a further distance than if the inclusions were not present. Put another way, the presence of inclusions causes a local temperature distortion near the interface of the inclusion and the continuous phase. This in turn elongates the constant heat-flow lines as shown in Figure 6, thereby increasing the heat conduction length. Fourier’s Law thus suggests that the rate of heat transfer will be reduced. Since heat flux is proportional to the temperature gradient, the temperature difference across the sample must therefore reduce. This implies an increasing “insulator” effect as the porosity increases, regardless of the nature of the pores. The preceding discussion refers to a microscopic interpretation of heat transfer phenomena through porous media. Macroscopically, it is understood that there is reduced effective heat flow area due to the volume fraction of the dispersed phase. However, little work has been done in linking the two levels of interpretation [30]. 7

4

Conclusion

The aims of the experiment were to study the relationship between (1) heat flux and temperature gradient and (2) porosity and the effective thermal conductivity. Experimental observations and physical interpretations are summarised below: 1. The heat transfer rate across the sample block was found to be linearly proportional to the temperature difference across the sample. This is in accordance with Fourier’s Law for one dimensional, steady state heat conduction with zero internal heat generation. The reason behind this observation was attributed to the simultaneous action of different parts of excess heat between adjacent molecules. 2. The effective thermal conductivity of a sample was seen to be reduced when the porosity of the sample was increased (i.e. the inclusion of pores has an insulating effect). The experiment confirmed this result by using a sample with non conducting staggered cylinders dispersed in a continuous phase of dense steel. The thermal conductivity of dense steel (without pores) was reduced from 10 𝑊/𝑚𝑜 𝐶 to 3.60 𝑊/𝑚𝑜 𝐶 due to a volume fraction of inclusions 𝜙 = 0.4. On a microscopic level, the physical mechanism for this result was a consequence of increased heat conduction length due to temperature distortions near the interface of the inclusions and the continuous phase. 3. The uncertainty in the measured effective thermal conductivity was in effect negligible, and the data correlated well with theoretical models such as Rayleigh’s model and the EMT model. The porosity was seen to be too high for the Maxwell model to make accurate predictions, and this is in line with the literature. The EMT model was also seen to have the best prediction for the effective thermal conductivity at the given porosity.

Figure 5: Heat flux vectors for a dispersion of spheres in a continuous medium (a) with 𝑘𝑐 > 𝑘𝑑 ; (b) with 𝑘𝑐 < 𝑘𝑑 Extracted from [10]

Figure 6: Distortion of local temperature field and constant heat flow lines obtained from numerical simulations with constant wall temperature at 𝑥 = 0 and 𝑥 = 𝐿 for 𝜙 = 0.15 Extracted from [30]

8

5

References [1] Moran, M. J., Shapiro, H. N., Boettner, D. D., Bailey, M. B., Principles of Engineering Thermodynamics, 7 ed., JohnWiley& Sons, Inc, 2012. [2] Fourier, J., 1822, The Analytical Theory of Heat, English translation by Freeman(1878); republication by Dover, New York (1955) [3] Zhao, C.Y., Lu, T.J., Hodson, H.P., Jackson, J.D., The temperature dependence of effective thermal conductivity of open-celled steel alloy foams, Materials Science and Engineering A 367 (2004) 123–131 [4] Rajput, R. K., Engineering Thermodynamics, Laxmi Publications (P) Ltd, New Delhi, 2007, ISBN: 978-0-76378272-63678 [5] Janna, W. S., Engineering Heat Transfer, CRC Press LLC, ISBN 0-8493-2126-3 [6] Incropera, F. P., Bergman, T. L., Lavine, A. S., Dewitt, D. P., Fundamentals of Heat and Mass Transfer, 7th ed., Hoboken, NJ: John Wiley & Sons, Inc, 2011. [7] Touloukian, Y. S., and C. Y. Ho, Eds., ThermophysicalProperties of Matter, The TPRC Data Series (13 volumeson thermophysical properties: thermal conductivity,specific heat, thermal radiative, thermal diffusivity,and thermal linear expansion), Plenum Press, New York,1970 through 1977. [8] Holman, J. P., Heat Transfer, Seventh ed., McGraw-Hill, Inc, 1990, pp. 2-3. [9] Lienhard V, J., Lienhard IV, J., A Heat Transfer Textbook, Phlogiston Press, Cambridge, MA, U.S.A, 2012 [10] Carson, J. K., Lovatt, S. J., Tanner, D. J., Cleland, A. C., Thermal conductivity bounds for isotropic, porous materials, International Journal of Heat and Mass Transfer 48 (2005) 2150-2158. [11] Woodside, W., Messmer, J. H., Thermal Conductivity of Porous Media. I. Unconsolidated Sands, Journal of Applied Physics, Vol 32, No. 9, 1961. [12] Pietrak, K., Wisniewski, T. S., A review for effective thermal conductivity of composite materials, Journal of Power Technologies, 95(1) (2015) 14-24. [13] Maxwell, J. C., A treatise on electricity and magnetism, vol. I, 3rd Ed, Oxford University Press, 1904. [14] Hasselman, D. P. H., Johnson, L. F., Effective thermal conductivity of composites with interfacial thermal barrier resistance, J. Compos. Mater. 21 (6) (1987) 508. [15] Hamilton, R. L., Crosser, O. K., Thermal conductivity of heterogeneous two component systems, Ind. Eng. Chem. Fundamen, 1 (3) (1962) 187. [16] Eucken, A., The thermal conductivity ceramic, resistant materials - your calculation of the thermal conductivity of the components, VDI Research Bulletin 353, Supplement to, research in the Ggebiete engineering, output B, Volume 3, 1932 [17] Zhao, J.-J., Duan, Y.-Y., Wang, X.-D., Wang, B.-X., Experimental and analytical analyses of the thermal conductivities and high-temperature characteristics of silica aerogels based on microstructures, J. Phys. D: Appl. Phys. 46 (2013) 12, 015304. [18] Zhao, J.-J., Duan, Y.-Y., Wang, X.-D., Wang, B.-X., Effects of solid–gas coupling and pore and particle microstructures on the effective gaseous thermal conductivity in aerogels, J Nanopart Res 14 (2012) 1024. [19] Monde, M., Mitsutake, Y., A new estimation method of thermal diffusivity using analytical inverse solution for one-dimensional heat conduction, International Journal of Heat and Mass Transfer 44 (2001) 3169–3177. [20] Ganapathy, K. Singh, P. E. Phelan, R. Prasher, An effective unit cell approach to compute the thermal conductivity of composites with cylindrical particles, J. Heat Transfer 127 (2005) 553. [21] Devpura, A., Phelan, P. E., Prasher, R. S., Size effects on the thermal conductivity of polymers laden with highly conductive filler particles, Microscale Thermophysical Engineering 5 (2001) 177. [22] Devpura, A., Phelan, P. E., Prasher, R. S., Percolation theory applied to the analysis of thermal interface materials in flip-chip technology, in: Thermomechanical Phenomena in Electronic Systems - Proceedings of the Intersociety Conference, Vol. 1, Las Vegas, Nevada, 200, p. 21. [23] Zhao, J.-J., Duan, Y.-Y., Wang, X.-D., Wang, B.-X., A 3-d numerical heat transfer model for silica aerogels based on the porous secondary nanoparticle aggregate structure, Journal of Non-Crystalline Solids 358 (2012) 1287–1297. [24] Carson, J. K., Lovatt, S. J., Tanner, D. J., Cleland, A. C., An analysis of the influence of material structure on the effective thermal conductivity of porous materials using finite element simulations, Int. J. Refrig. 26 (2003) 873– 880. [25] Strutt (Lord Rayleigh), J., On the influence of obstacles arranged in rectangular order upon the properties of a medium, Phil. Mag.,vol. 34, pp. 481, 1892 [26] Gong, L., Wang, Y., Cheng, X., Zhang, R., Zhang, H., A novel effective medium theory for modelling the thermal conductivity of porous materials, International Journal of Heat and Mass Transfer 68 (2014) 295–298 [27] Holman, J.P, Experimental Methods for Engineers, McGraw-Hill, 2001, ISBN 978-0-07-352930-1 [28] Zhao, C.Y., Lu, T.J., Hodson, H.P., Jackson, J., D., The temperature dependence of effective thermal conductivity of open-celled steel alloy foams, Materials Science and Engineering A 367 (2004) 123–131 [29] Leibowitz, L., Blomquist , R. A., Thermal Conductivity and Thermal Expansion of Stainless Steels DO and HT9 , Argonne National Laboratory, Chemical Technology Division, 9700 S. Cass Avenue, Argonne IL 60439 USA [30] Yang, X., Lu., T., Kim, T., Thermal stretching in two-phase porous media: Physical basis for Maxwell model, Theoretical & Applied Mechanics Letters 3, 021011 (2013) 9

Laboratory Risk Assessment Checklist Campus: University of Witwatersrand, Johannesburg

the

Faculty/School/Unit: School Mechanical, Industrial Aeronautical Engineering

of and

Laboratory Type: Heat Transfer in Porous Media Assessment Date: 23 September 2016 Assessors Name: Taahir Bhaiyat

Assessors Signature: Signature:

Approved by:

Main Laboratory Functions: 1. Obtain experimentally and explain physically the relationship between heat flux and temperature difference of a given material based on Fourier’s Law 2. Obtain experimentally and explain physically the relationship between effective thermal conductivity and porosity for a two phase porous medium 3. Compare the experimental data obtained with the literature and suggest the material comprising the continuous phase

Part 1 - Hazard Identification The table below will assist you in identifying hazards when completing the section entitled “These Hazard Exist” A Could people be injured or made sick by things such as:

B

What could go wrong?

         

     

What if equipment is misused? What might people do that they should not How could someone be killed? How could people be injured? What may make people ill? Are there any special emergency procedures required?

D

How might these injuries happen to people?

     

Broken bones Eye damage Hearing problems Strains or sprains Cuts or abrasions Bruises

   C     

Noise Light Radiation Toxicity Infection High or low temperatures Electricity Moving or falling objects (or people) Flammable or explosive materials Malenals under tension or pressure (compressed gas or liquid; springs) Any other energy sources or stresses Bio-hazardous material Laser Can workplace practices cause injury or sickness? Are there heavy or awkward lifting jobs? Can people work in a comfortable posture? If the work is repetitive, can people take breaks? Are people properly trained? Do people follow correct work practices?

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  

E  

Are there adequate facilities for the work being performed? Are universal safety precautions for biohazards followed? Is there poor housekeeping? Look out for clutter, torn or slippery flooring, sharp objects sticking out, obstacles etc

    

Burns Lung problems including inhalation injury/infection Skin contact Poisoning Needle-stick injury

Imagine that a child was to enter your work area

F

What are the special hazards?



What occurs only occasionally - e.g. during maintenance and other irregular work?

Of what, would you warn them to be extra careful? What would do to reduce the harm to them?

These hazards exist: (Please √) Physical:

Chemical:

Biological:

Mechanical/Ergonomic:

Psycho-social:

Noise

Liquids

Posture

Worry

Vibration UV X-ray Laser Heat and cold Electricity Extremes of pressure Heavy weights Sharps Needles Physical activity, exertion

Dusts Fumes Fibres Mists Vapours Gases Compressed gases Acids

Human blood and saliva Insects Mites Moulds Yeasts Fungi Bacteria Viruses

Movement Repetitive actions Illumination and visibility

Work pressure Monotony Unsocial hours Shift work

Mercury

Animals: Rats Mice Rabbits

Sheep brain Pig heart Toads

Major Equipment: Autoclave

Fume hood

Major sports equipment

Bio-safety cabinet

Vacuum

Compressed air

Trolleys/mechanical aids

Radioactive

Sharps

Minor Equipment: Bunsen burner

Water bath

Microscope

Materials:

Waste Generated: Biological Gloves

Chemical Plastic apron

Carcinogenic

ACU National - Laboratory Risk Assessment Checklist

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Waste disposed of by: General waste Disinfection

Autoclaving Flushing sink

Freezing Fume hood and water

Medical waste bin Sharps bin

Incineration Glass bin

Fire Extinguisher

To help me prepare for an emergency I may need: First Aid training Evacuation procedures

CPR training

Safety Signs

MSDS

Spill kit

Australian Standards

Personal Protective Equipment

Emergency contact numbers could be needed for (List extension numbers below) Security First aiders Floor Warden Emergency phone numbers are posted where? On notice board in laboratories and by all telephones (Yes/No)

Part 2 - How to Asses Risk – Explanatory Notes

ASSESSMENT OF RISK

Risk Score:

CONSEQUENCES: How severely it hurts someone (if it happens)? LIKELIHOOD How likely is it to happen?

Almost certain expected in most circumstances Likely – will probably occur in most circumstances Possible – might occur at some time Unlikely – could occur at some time Rare may occur, only in exceptional circumstances

Insignificant (no injuries)

Minor (first aid treatment only; spillage contained at site)

Moderate (medical treatment; spillage contained but with outside help)

Major (extensive injuries; loss of production)

Catastrophic (death; toxic release of chemicals)

3 High

3 High

4 Acute

4 Acute

4 Acute

2 Moderate

3 High

3 High

4 Acute

4 Acute

1 Low

2 Moderate

3 High

4 Acute

4 Acute

1 Low

1 Low

2 Moderate

3 High

4 Acute

1 Low

1 Low

2 Moderate

3 High

3 High

1. To use the matrix, first find the CONSEQUENCES column that best describes the risk. Then follow the LIKELIHOOD row to find the description that best suits the likelihood that the consequence will occur.

ACU National - Laboratory Risk Assessment Checklist

Score and statement

Action

4 A: Acute

ACT NOW – Urgent do something about the risks immediately. Requires immediate attention. Senior management decision is required urgently. Follow management instructions e.g. policy/guidelines.

3 H: High 2 M: Moderate 1 L: Low

OK for now. Record and review if any equipment/ people/ materials/ work processes or procedures change.

Note:

3

The risk level is given in the box where the row and column meet. 2. When considering the likelihood of injury or disease, the number of people exposed, the extent of the exposure to the hazard and the likelihood that exposure will result in harm, all to be taken into account. 3. The estimate of likelihood will also depend on the effectiveness of the control in place. It is important to indicate what assumptions are being made about the controls in place.

ACUTE or HIGH Risk must be reported to the School’s Senior Management (HOS and/or Dean) and require detailed treatment plans to reduce the risk, where possible, to MODERATE or LOW.

Adapted from Standards Australia Risk Management AS/NZS 4360: 2004 Risk Control Emphasis is on controlling hazard at source. For instance, for those risks that are assessed as “High”, steps should be taken immediately to minimise risk of injury. Use the “hierarchy of controls” as listed below to determine the type of control measures that should be implemented:

Order No. Firstly

Control Eliminate

Secondly

Substitute

Thirdly

Isolation

Fourthly

Engineering

Fifthly

Administrative

Sixthly

Personal Protective Equipment

Example Disposing of unwanted chemicals and out-of-service hazardous equipment, prompt repair of damaged equipment. Using water-based instead of a solvent-based paint, using chemicals of lower concentration. barricades around trenches, fume cupboards, bio-safety cabinets. Ensure proper machine guarding, ventilation and extraction systems Appropriate training to all staff, provision of adequate warning signs Use of gloves, glasses, ear muffs, aprons, safety footwear, dust masks, etc.

Part 3 – Completion of Laboratory Risk Assessment Now that you have identified the hazards and using the information above, complete the following Risk Assessment Form. Once the risk assessment has been completed, copies should be provided to:

Copies: 1. Laboratory/Academic Supervisor or their Representative 2. Summary/ Report - Campus OHS Committee Chair (for

tabling at next OHS

Committee Meeting)

3. Manager, Campus Operations 4. Head of School

ACU National - Laboratory Risk Assessment Checklist

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RISK ASSESSMENT HAZARD IDENTIFICATION No. What harm can happen to people or equipment?

Risk Score* (See Matrix)

List any control measures already implemented

RISK CONTROL

Describe what can be done reduce the harm?

REVIEW

Whom Responsible

By when

Are the Controls Effective?

Date Finalised

(see Risk Controls )

(See page 3 for list of possible hazards)

1

High powered DC supply could possess and electrical and fire hazard to both the operator and other major equipment.

2

The lab instructor ensures that the maximum power output of the DC supply is not exceeded and that the experiment is performed within a safe range.

Only authorised people should operate the equipment or it should be used under the supervision of a senior person

Lab coordinator

co-

N/A

YES

23/09/16

2

Burns due surfaces

hot

2

The outer surfaces insulate the hot inner surfaces.

Lab coordinator

co-

N/A

YES

23/09/16

3

Electrical wires could posses a tripping hazard as well as a potential fire hazard should they short out Equipment could potentially fall off the work surface.

2

The wires are well insulated and positioned away from the operator.

Lab coordinator

co-

N/A

YES

23/09/16

1

PPE is a requirement for any person that is present in the laboratory

Warn people about being careful when handling the equipment. Fire extinguishers should be available at all times in the event of a fire. People should be made aware of their surroundings Secure the equipment and ensure that they are not placed in such a way that it could fall.

Lab coordinator

co-

N/A

YES

23/09/16

4

to

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