Void fraction and effective thermal conductivity of binary particulate bed

Fusion Engineering and Design 88 (2013) 216–225

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Void fraction and effective thermal conductivity of binary particulate bed D. Mandal a,c,∗ , D. Sathiyamoorthy b , M. Vinjamur c a

Chemical Engineering Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India Powder Metallurgy Division, Bhabha Atomic Research Centre, Vashi, Sector 20, Navi Mumbai 4000705, India c Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India b

h i g h l i g h t s

g r a p h i c a l

 Variation of void fraction in binary particulate beds and its effect on effective thermal conductivity was investigated.  Variation of void fraction with volume fraction of smaller particles is ‘V-shaped’.  For any size ratio, two different volume fractions of particles can give the same void fraction other than the minimum.  It was found that the binary particulate bed has higher effective thermal conductivity than that of unary pebble bed.  The binary bed with less volume fraction of small particles has higher thermal conductivity than that of higher fractions.

From the experiments, it is observed that variation of void fraction with volume fraction of smaller particles is ‘V-shaped’ as shown in the Figure. That is, there exists a minimum void fraction and two different volume fractions of smaller pebbles can give same void fraction. Effective thermal conductivity of binary particulate bed is the maximum when its void fraction is the minimum. The binary bed with less volume fraction of small particles has higher conductivity than the binary bed of higher volume fraction of small particles and having same void fraction. This is due to the fact that with increase in volume fraction of small component, the number of small-to-small and large-to-small contact point increases resulting in higher resistance to heat transfers.

a r t i c l e

a b s t r a c t

i n f o

Article history: Received 2 January 2012 Received in revised form 8 October 2012 Accepted 8 February 2013 Available online 13 March 2013 Key words: Void fraction Particulate bed Coordination number Packed fluidized bed Effective thermal conductivity Lithium titanate

a b s t r a c t

In many industrial processes, solid particles of different sizes are mixed in different volume or mass fractions for various applications, primarily to reduce void volume or to increase the density of the mixture. A few of these processes include; production of high density ceramics, mortar, concrete, graphite, bricks and carbon blocks. Experiments were carried out with alumina and lithium titanate pebbles (size ≥ 1 mm) and particles (size < 1 mm) of different sizes and volume fractions in cylindrical and rectangular vessels to study the variation of void fraction with volume fraction of component pebbles and with large pebble to small pebble or pebble to particle size ratio. It was observed that the variation of void fraction with volume fraction of smaller pebbles or particles is ‘V-shaped’. Thus there exists a minimum void fraction and two different volume fractions of smaller pebbles or particles which can give same void fraction. The effect of void fraction on the effective thermal conductivity of binary bed of lithium titanate and alumina pebbles and particles of different sizes and volume fractions was investigated. From the experimental results it was found that the binary particulate bed has higher effective thermal conductivity that that of a unary particulate bed. Effective thermal conductivity of binary particulate bed is the maximum when its void fraction is the minimum. The binary particulate bed with less volume fraction of small particles has

∗ Corresponding author at: Chemical Engineering Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India. Tel.: +91 22 25593938; fax: +91 22 25505151. E-mail addresses: [email protected], [email protected], [email protected] (D. Mandal). 0920-3796/$ – see front matter. Crown Copyright © 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fusengdes.2013.02.033

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higher conductivity than that of the binary bed of higher volume fraction of small particles and having same void fraction. This is due to the fact that with increase in volume fraction of small pebbles or particles, the number of small-to-small and large-to-small pebbles and or particles contact point increases resulting in higher resistance to heat transfers. The experimental details and results are discussed in this paper. Crown Copyright © 2013 Published by Elsevier B.V. All rights reserved.

1. Introduction Void volume of a particulate bed is reduced when fine particles are mixed with relatively larger particles, because the former occupy the interstitial voids of the latter. By increasing the volume fraction of fine particles, void fraction is reduced to a minimum value and on further increase of volume fraction of smaller particles, the void fraction again increases. This interesting behavior of binary particle mixtures was initially observed by Westman and Hugill [1], Furnas [2] and Anderegg [3]. Void fraction in a binary particulate bed is always less than that of a unary bed. The void fraction (or packing density) in binary bed is a function of the (i) size ratio of particles, (ii) volume fraction of the small (or large) particles in the mixture, (iii) shape of the particles, and (iv) method of packing. Voidage of a particulate bed with different particle sizes can be less than that of binary particulate bed of same mass. The void fraction of a particulate bed also depends on volume fraction and size ratio between successive components. McGeary [4] reported that the minimum volume fraction of the small particles in a multi-particle system for closest packing is 0.27 and the closest packing voidage is 0.14. Void fraction in a particulate bed of different sizes pebbles and or particles was studied by Westman and Hugill [1], Furnas [2], Anderegg [3], McGeary [4], Fedors and Landel [5], Stovall et al. [6], Yu and Standish [7], Cumberland and Crawford [8] and others. Different mathematical models were proposed by different investigators to correlate voidage to size distribution of pebbles and or particles; these models were reviewed by Ouchiyama and Tanaka [9]. Yu and Standish [7] also reviewed the same. These reviews found that their experimental void fraction values differ widely from the estimated values using these models. Furnas [2] proposed an equation to find volume fraction of small pebbles or particles for the minimum voidage when the initial voidage of bed of large pebbles or particles is known. Zenz and Othmer [10] analyzed the experimental results of Furnas [2] and developed an equation to estimate the volume fraction of small pebbles or particles corresponding to the minimum void fraction of the bed. Effective thermal conductivity of compressed Li2 TiO3 pebble bed was studied by Tanigawa et al. [11]. Reimanna et al. [12] measure effective thermal conductivity of compressed beryllium pebble beds without flow of a gas through it. Both these studies showed that the effective thermal conductivity of particulate bed increases with increase in compressive load, which compacts the bed i.e., increases in bed density. Thus, decrease in void fraction increases the effective thermal conductivity of particulate bed. Okazaki et al. [13] developed a model to estimate the effective thermal conductivity (keff ) of a unary bed. Jaguaribe and Beasley [14] also predicted the same from their proposed model to estimate keff of unary bed with stagnant fluid. By analyzing the experimental data of Furnas [2] Zenz and Othmer [10] as well as Mandal [15] experimentally demonstrated that a ‘V’ shaped curve is obtained when the variation in bed voidage (ε) is plotted against volume fraction of small particulate solids in a binary particulate bed. That means, there exists a minimum void fraction and same void fraction for two different volume fractions of smaller pebbles or particles. The ‘V’ shape curve is distinct for small-to-large particle or pebble size ratio (dp /DP ). Till date no

investigator has investigated and compared the effective thermal conductivity of binary particulate bed at the same void fraction with two different volume fractions of smaller pebbles particles and also with that at the minimum void fraction. This paper reports a comparative study of the effective thermal conductivity of a binary particulate bed at the same void fraction for two different volume fractions of smaller pebbles or particles and also at the minimum void fraction. Experimental details and results are discussed in details in this paper. 2. Void fraction in binary particulate bed Consider a cylindrical vessel is filled with uniformly sized spherical pebbles of diameter DP and volume vP each. The bed volume is V and the number of pebbles required to fill the bed is N. Let the volume of pebbles in the bed is VP . Then, VP =

N 

vP,i = N vP = N

DP3 6

(1)

i=1

Now, if the bed is filled with smaller sized pebbles (or particles) of diameter dp and volume vp each instead of larger pebbles and suppose n such smaller pebbles (or particles) are required to fill the bed. The volume of the smaller pebbles (or particles) VP in the bed would be, Vp =

n 

vp,i = nvp = n

dp3 6

(2)

i=1

Assume εP and εp are the void fractions of bed when filled only with larger pebbles and with only smaller pebbles or particles respectively. Then, DP3 6

=

V (1 − εP ) N

(3)

=

V (1 − εp ) n

(4)

and, dP3 6

From Eqs. (3) and (4) we get Eq. (5) as shown below. R=



1 − εp dP = k1 DP 1 − εP

 (5)

 1/3

and R is the size ratio of smaller pebbles or where, k1 = Nn particles (dp ) to larger particles or pebbles (DP ). Eq. (5) may be considered valid when the bed is filled with well mixed smaller pebbles (or particles) and larger pebbles. The constant k1 in Eq. (5) may be considered as the function of the number of larger pebbles (N) to smaller pebbles or particles (n) in the binary particulate bed. That is, k1 is a function of n and N or the ratio of volume fractions of smaller pebbles or particles (xp ) or the volume fraction of pebbles (XP ). If εb is the void fraction of the binary particulate bed then, the volume occupied by particles, Vp may be expressed by Eq. (6). Vp = xp (1 − εb )V = n

dp3 6

(6)

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D. Mandal et al. / Fusion Engineering and Design 88 (2013) 216–225

Nomenclature At dp Db Dp Dt H k k1 keff ko kg mp mP n N Pb Po r R RO Rp RS t T Ta T¯ () uo

vp vP V Vp VP  q xp XP

cross-sectional area [m2 ] diameter of particles or smaller size pebbles [m] bed diameter [m] diameter of large particles or pebbles [m] diameter of cylindrical column [m] bed height [m] thermal conductivity of test specimen in TPS measurement [W m−1 K−1 ] constant in Eq. (5) [–] effective thermal conductivity [W m−1 K−1 ] effective thermal conductivity of pebble bed at zero gas flow-rate [W m−1 K−1 ] thermal conductivity of gas [W m−1 K−1 ] mass of small particles [kg] mass of large particles or pebbles [kg] number of particles or smaller pebbles [–] number of larger particles or pebbles [–] bed pressure ( stands for difference or drop) [N m−2 ] heat liberated from the sensor [J] radius of TPS sensor [m] small-to-large particle or pebble size ratio, dp /DP (suffix ‘max’ stands for its maximum value) [–] initial resistance of TPS sensor [] standard resistance in TPS measurement [] block resistance (suffix 1 and 2 in lieu of S, are for different resistances) [] time (s) temperature [K] ambient temperature [K] average temperature of the TPS element [K] operating gas velocity [m s−1 ] volume of one particle or smaller size pebble [m3 ] fluidization volume of one larger size particle or pebble [m3 ] volume of particles and or pebbles in binary particulate bed [m3 ] volume of particles and or smaller size pebbles in binary particulate bed [m3 ] volume of larger size particles or pebbles in binary particulate bed [m3 ] wave vector [m−1 ] volume fraction of particles or smaller size pebbles in mixed particulate bed [–] volume fraction of larger size particles pebbles in mixed particulate bed [–]

Greek alphabets Temperature coefficient of resistance [K−1 ] ˛ ˆ  difference or drop, e.g., P stands for pressure drop [N m−2 ] ε experimentally measured average void fraction in a particulate bed (suffix b, c are void fraction of binary particulate bed and that of a unary bed of bed of larger size pebbles, respectively) [–] εi internal porosity of pebbles [%] void fraction in unary bed of smaller size pebbles or εp particles [–] εP void fraction in unary bed of larger size particles or pebbles [–] that for different εpf void fraction in packed fluidized bed [–] minimum void fraction in pebble bed [–] εmin

εw

void fraction close to the wall of particulate bed in cylindrical column [–] viscosity of fluid (gas) [kg m−1 s−1 ] surface area of particles or pebbles [m2 kg−1 ] particle sphericity [–] average density of particulate bed [kg m−3 ] density of fluid (gas) [kg m−3 ] density of particles or smaller size pebbles [kg m−3 ] density of larger size particles or pebbles [kg m−3 ] a parameter defined in Eq. (21) [–] characteristic time of measurement in Eq. (21) [–]

 s

S av. g p P 

Similarly, the volume occupied by larger particles or pebbles, VP may be estimated using Eq. (7). VP = XP (1 − εb )V = N

DP3

(7)

6

εb may be estimated from Eqs. (6) and (7) as shown in Eq. (8), which is the same according to definition.



εb = 1 −

VP + Vp



(8)

V

Eq. (9) may be derived from Eqs. (5), (7) and (8). VP εb = 1 − V



1+

 R 3

(9)

k1

In Eq. (9), VP /V is a function of volume fraction of larger particles or pebbles (XP ). So, we can write, εb = f (XP , R)

(10)

From Eqs. (5)–(10) it may be concluded that εb is a function of volume fraction of components, XP (or xp ) and R. εb does not depend on the particle density. In the above expressions particles and pebbles have been considered perfectly spherical. For non spherical particles and pebbles sphericity must be considered as it may influence the volume fraction of particles and pebbles. 2.1. Determination of void fraction in binary particulate bed Packed fluidized bed may be considered as a binary particulate system. It is essential to evaluate void fraction in packed fluidized bed for our present studies. The value of void fraction (ε) in a cylindrical bed of cross-sectional area At , filled with larger pebbles of diameter DP , up to bed height H can be easily estimated, if the quantity of pebble, mP and density of pebbles, P are known. Consider, small particles of mass mp and particle density p are added in the interstitial voids of the bed of larger pebbles (called packing pebbles) and air is allowed to flow in the bed such that the small particles will fluidize in the interstices, while larger pebbles (i.e., packing pebbles) will remain stationary. It is a fluidized bed of small particles in the interstices of packed pebble bed, called Packed Fluidized Bed [16,17] and the void fraction in this packed fluidized bed (εpf ) may be estimated using Eq. (11). εpf = 1 −

 1  VP + Vp At H

(11)

where, VP and Vp are the volume of packing pebbles and fluidized particles inside the bed.In terms of measurable parameters Eq. (11) can be rewritten as, εpf

1 =1− At H



mp mP + P p



(12)

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Fig. 1. Hexagonal close packing arrangement of pebbles (a) without small particles in the interstices (b) with small particles in the interstices.

2.2. Coordination number The coordination number, or number of contact points per particle, is an important parameter in describing the geometrical arrangement of particles in a packing, and is widely used in the evaluation of properties related to the connectivity between particles, heat transfer [18], solid–solid reaction [19] etc. Like crystal lattice, the coordination number of a particle in binary particulate bed can be defined as the number of other particles that are in contact with the particle or pebble in a pebble bed. In a randomly packed binary particulate bed, the coordination number may not be the same for each particles or pebbles, but the overall mean coordination number is essentially a constant and independent of particle size and distribution. Assume pebbles of diameter DP are arranged in hexagonal closed packing as shown in Fig. 1. In one plane there will be six pebbles which are in contact with a pebble in the center (layer A). There will be three other pebbles above and three below a plane which are in contact to the central pebble (layer B). Hence, the coordination number of the central pebble is 6 + 3 + 3 = 12. Now, the bed is being filled with small particles of diameter dp , shown in dark red in Fig. 2. The particle size ratio R(= dp /DP ) plays an important role. It can be easily proved that for a hexagonally closed packed pebble bed, the maximum size of the particle that can be accommodated without disturbing the hexagonal close packing configuration of pebbles such that the maximum size ratio R = Rmax = 0.1547. That means that particles with R ≤ 0.1547 can be accommodated in the pebble bed. Consider one unit cell of hexagonal closed pack unit of packing pebbles in a cylindrical vessel. There is one pebble at the center and 12 (i.e., the coordination number) pebbles on the surrounding. In the unit cell, there will be 3 pebbles; with 1/3rd volume each of other cell will occupy to fill the space available at the top and bottom. So, the total number of pebbles in the unit cell will be 1 + 12 + 2 × 3x (1/3) = 15, as shown in Fig. 2. Consider that the above unit cell is being filled with fine particles of Rmax = 0.1547. There will be 6 particles on a plane (layer A, as shown in Fig. 2 around the central pebble. There will be one particle in between the three pebbles that are placed on the top and at the bottom (layer B, as shown in Fig. 2. Hence, the total number of fine particles in contact with the central pebble is 6 + 1 + 1 = 8. Hence, the minimum coordination number of fine particles around a pebble in packed fluidized bed is 8 and the number will increase as the R decrease from its maximum value, which is 0.1547.

Fig. 2. Coordination number in a unit cell of hexagonal closed packed pebble bed filled with small particles in the interstices. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)

Since, we are considering a single unit cell, 3 more fine particles may be accommodated at the top and bottom (layer B, as shown in Fig. 2) and 6 more particles may be accommodated at the periphery of layer B. Thus, total number of fine particles of the maximum size + 3 + 6 = 20. The that can be accommodated in the unit cell =8 + 3 √ unit cell considered has diameter 3DP and height 3DP . The void fraction, ε of the unit cell of packing pebbles and fine particles can be estimated using the following Eq. (13). ε=1− ε=1−

volume of solid volume of unit cell  6



15DP3 + 20dp3  2 √  3DP3 3DP 4

(13)



(14)

Replacing dp /DP by R, Eq. (14) can be written as, ε = 0.358 − 0.855 R3

(15)

Fig. 3 shows the variation of void fraction with particle size ratio in packed fluidized bed. When, R << 0.1547, the number of fine

Fig. 3. Variation of void fraction with the particle to pebble size ratio (R = dp /DP ) in binary particulate bed.

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Table 1 Particle size and physical properties of fluidized particles used in the study. Material

Particle or pebble size dp [␮m]

Density p [kg/m3 ]

Internal porosity εi [%]

Surface area s [m2 kg−1 ]

Sphericity s [–]

Li2 TiO3

231 550 1000 3000 5000 7000 10,000 1000 3000 5000 10,000

2847.810 2701.634 2634.170 3014.96 2887.65 2890.28 2923.48 3767.354 3662.991 3643.469 3624.805

15.43 18.65 22.11 12.45 14.44 15.59 15.83 3.354 6.077 6.657 7.056

198.305 211.506 234.506 177.90 200.23 222.57 244.90 31.233 19.566 10.815 5.232

0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.92 0.92 0.92 0.92

Al2 O3

particles will be more and void fraction will be less. Moreover, practically it is impossible to arrangement pebbles of size 3–10 mm in hexagonal close pack in a pebble bed of diameter 160 mm and bed height 300 mm. 2.3. Wall effect In a cylindrical pebble bed, the void fraction at the bulk and around the wall of the column is not the same. Void fraction around the wall will be slightly more than that at the bulk of the bed and this is referred as the wall effect. Furnas [2] was probably the first to study the wall effect. He proposed the following correlations for voidage close to the wall (εw ) and at the bulk (εb ) for a unary bed of stationary particle of size dp in cylindrical column of diameter Dt .



εw = 1 + 0.6

dp Dt





ε − 0.6

dp Dt

 εb = [ε + 0.3 (1 − ε)] 1 + 0.6



dp Dt

(16)



ε − 0.6

dp Dt

(17)

In Eqs. (16) and (17), ε is the experimentally measured average voidage. When the ratio of dp to Dt is less than 0.02, the wall effect is negligible. Void volume of a particulate bed is decreases when fine particles are added in the interstices of a unary bed of relatively large size pebbles. The fine or small particles will occupy the interstitial voids. By progressively increasing the volume fraction of small particles, the void fraction may be reduced to a minimum value. The details of the experimental works and results on void fraction in binary particulate bed are discussed below. 2.4. Void fraction and pressure drop When a fluid is allowed to flow through a pebble bed, the pressure of the fluid drops across beds due to the resistance caused by the presence of the particles. As the void fraction decreases the resistance caused by the particles increases and the pressure drop increases. The pressure drop across the bed is well represented by the semi-empirical correlation derived by Ergun [20] as shown below (Eq. (18)). Pb (1 − ε)2 uo (1 − ε) g u2o = 150 + 1.75 2 2 H ε3 ε3 dp s dp s

(18)

Ergun equation (Eq. (18)) explains how the bed pressure drop (Pb ) per unit bed height (H) of a stationary particulate bed varies with the void fraction (ε) and other bed properties viz. fluid viscosity (), fluid density (g ), particle size (dp ) and particle sphericity ( s ) and fluid velocity (uo ). In fact Pb /H increases with uo in general and as ε decreases Pb /H increases [21,22]. It may be noted

that the Ergun equation (Eq. (18)) is valid for the bed with structured packing as well as with random packing. Particles with a rough surface have void fractions a few percent larger than for smooth particles. With a distribution of particle sizes, the void fractions are lower than for uniform particles and Pb /H at any particular uo varies accordingly. Recently, Mandal et al. studied the variation of bed pressure drop [16] and void fraction [17] in packed fluidized bed. 3. Materials and methods 3.1. Materials Fabricated spherical Al2 O3 and Li2 TiO3 pebbles of size 1, 2, 3, 5 and 10 mm were used to study the void fraction in binary particulate bed and effect of pebble size ratio. Fabricated Li2 TiO3 pebbles of size 3 and 5 mm and Li2 TiO3 particles of size 231 ␮m were used to study the thermal conductivity of binary particulate bed. Li2 TiO3 particles and pebbles of different sizes used in the work were synthesized and fabricated by using solid state reaction process developed by Mandal et al. described in details in somewhere else [23,24]. The physical properties of these pebbles and particles are noted in Table 1. 3.2. Methods 3.2.1. Void fraction in binary particulate bed Binary pebble bed was prepared by arranging measured quantity of smaller pebbles and larger pebbles in layers. First, one layer of larger pebbles was arranged whose thickness was equal to diameter of the pebbles. A fraction of smaller pebbles were uniformly distributed in the interstitial void space of the larger size pebbles. Second layer of larger size was arranged and a fraction of fine particles was filled in the interstitial void and the same pattern was repeated till the desired bed height is reached. Larger pebbles were arranged randomly in the vessel but uniformly along the cross-section. Smaller pebbles were also distributed randomly, but uniformly to the extent possible, throughout the entire bed. No tapping, vibration etc. were used to compact the bed. Experiments were repeated by varying the volume fraction of smaller pebbles or particles and larger size pebbles or packing pebbles and different smaller pebble or particle to larger pebble or packing pebble size ratio R. Each experiment was repeated at least three times. Void fraction of binary particulate bed was measured by three methods, viz. (i) by water filling, (ii) by image analysis of still photographs, and (iii) by analyzing cross-sectional view of solid mold cut-out. The pebbles used to prepare the binary bed were kept under water overnight, separated from water by passing through a filter cloth and the surface of the pebbles was dried by blowing air. After arranging a binary pebble bed as discussed above, water was filled slowly in the interstitial voids till the water-level reached

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Fig. 4. Photographs of binary particulate bed for different smaller pebble to larger pebble size ratio (R = dp /DP ) of Li2 TiO3 particles (a) R = 0.15, (b) R = 0.26 and (c) R = 0.58.

Fig. 5. Binary phase of the photographs of a side of binary particulate bed of Li2 TiO3 pebbles, as shown in as shown in Fig. 4 for (a) R = 0.15, (b) R = 0.26, (c) R = 0.58.

upper surface of the bed. By measuring the quantity of the water required, the void fraction of the bed was estimated. This method was used for both cylindrical as well as rectangular bed. The second method by image analysis was used for rectangular beds. Photographs of a binary pebble bed were taken from four sides of the bed. Void fractions were estimated by analyzing photographs with the help of Image-J software. The photographs were converted to binary phase, and the void fractions at different crosssections were estimated. According to quantitative microscopy, the volume fraction is assumed to be equal to the average area fraction or line fraction of any plane. The estimated average values were considered as the void fraction of binary bed. In the third method, an instantly prepared solution of a polymeric powder and a liquid hardener was filled in the interstitial voids of a binary particulate bed. The polymer was allowed to solidify for 24 h at room temperature. The solid mass was cut in different sections using a diamond wheel. Each cut-out section was microscopically analyzed to measure the void fraction and the average value was taken as the void fraction of pebble bed. Fig. 4 shows some typical photographs of binary Li2 TiO3 pebble bed. The photographs were converted to binary phases using Image-J software to measure the average void fraction. Fig. 5 shows some typical projection images of pebbles in binary phase. It was observed that the value of the measured average void fraction by the three methods described above were close to each other.

3.2.2. Measurement of thermal conductivity of Li2 TiO3 pebbles Transient Plane Source (TPS) technique was used to measure the thermal conductivity of Li2 TiO3 pebbles at different temperatures. The TPS method utilizes a thin disk-shaped temperature dependent resistor simultaneously as the temperature sensor and as the heat source for the thermal conductivity measurements. The sensor is sandwiched between two specimen halves, as shown in Fig. 6. A direct current was passed through the sensor, sufficiently large to increase the sensor temperature by about 1–2 K. Due to the temperature increment, the resistance of the sensor will change and there

will be a corresponding detention in voltage drop over the sensor. By recording the voltage variation and the current variation over a certain time period from the onset of the heating current, it is possible to obtain precise information on the heat flow between the sensor and the test specimen. There will, however, be a small temperature drop, Ti over the electrically insulated Kapton layer. After a short initialization period this temperature drop will stay constant due to the liberation of constant power. The resistance of the sensor can then be expressed by the following Eq. (19).



R(t) = Ro 1 + ˛T ˆ ˆ T¯ () i + ˛

(19)

where, Ro is the resistance of the TPS element before the transient ˆ is the temperature coefficient of the resisrecording is initiated, ˛ tance and T¯ () is the average temperature increase of the TPS element assuming perfect thermal contact with the test specimen. Po

T¯ () =

2

D()

(20)

 3 rk In the Eq. (20) Po is the heat liberation from the sensor, k is the thermal conductivity of the test specimen, r is the radius of the TPS sensor and  is given by the Eq. (21). √ =

at = r



t

(21)

Fig. 6. Schematic arrangement of the sensor and samples for thermal conductivity measurement.

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Fig. 7. Schematic diagram of the (a) electrical circuit for TPS method and (b) TPS sensor.

where a is the thermal diffusivity of the test specimen, t is the measurement time and is the characteristic time of the measurement = r 2 /a and D() is a function of . Using a measurement time much longer than ≥ 2 the thermal conductivity can be calculated. Two Li2 TiO3 pellets of size 10 mm diameter and 15 mm height with density 2907 kg m−3 (85% of theoretical density of Li2 TiO3 were fabricated by compaction and sintering of Li2 TiO3 powder synthesized by the solid state reaction process developed by Mandal et al. [22–24]. The surfaces of Li2 TiO3 pellets were polished to have good thermal contact with the TPS sensor and to minimize contact resistance. The sensor is sandwiched between two identical samples as shown in Fig. 6. The schematic diagram of the electrical circuit of TPS method with TPS sensor is shown in Fig. 7. The Kapton sensor of radius 3.189 mm was used for room temperature measurement and Mica sensor of radius 3.189 mm was used for higher temperature. For high temperature measurement the samples were heated in high purity argon atmosphere. The samples were kept for 30 min at each set temperature before the measurement to avoid temperature drift inside the samples. A Keithley 2400 source meter supplies a constant voltage across the bridge. Before the measurement, the bridge is automatically balanced and as the resistance of the sensor increases the bridge becomes increasingly unbalanced. A Keithley 2000 digital voltmeter equipped with a scanner or multiplex card, records the unbalanced voltage. From these recorded voltages it is possible to determine the temperature changes of the sensor and consequently the thermal conductivity of the sample. In the Fig. 7, Rp is the standard resistance, which limits the current from a power supply; RS and R1 are two block resistances (ratio arm); DVM is the Digital Voltmeter; RO is the TPS element; and R2 adjustable resistance. A specially designed one side closed cylindrical sample holder having a small cut exactly at the middle to insert the sensor inside, was used for measurement of pebble samples in this present study. The experimental set up at room temperature is shown in Fig. 8. In order to make good thermal contact between the pebbles and the sensor, the sample holder was carefully filled with Li2 TiO3 slab.

3.2.3. Effective thermal conductivity of unary Li2 TiO3 particulate bed Unary bed was prepared by filling known quantity Li2 TiO3 pebbles of size 3 mm randomly in the sample holder as shown in Fig. 8. No tapping, vibration etc. were used to compact the bed. The same TPS technique was used to measure the effective thermal conductivity of unary particulate bed at room temperature. Experiments were repeated three times after rearranging the pebbles in the sample holder. The average keff value of the bed was noted. Similar experiments were conducted with 5 mm, 7 mm and 10 mm Li2 TiO3 pebbles and average keff of unary bed of each pebble size was measured.

Fig. 8. Experimental set up to measure thermal conductivity at room temperature. The inset shows the sample holder.

3.2.4. Effective thermal conductivity of binary Li2 TiO3 particulate bed Binary bed of 231 ␮m particles and 3 mm pebbles, containing 30 volume% 231 ␮m particles was prepared in the sample holder as discussed above for the measurement of void fraction in binary bed. Average keff of the binary bed was measured using TPS technique. Experiments were repeated with binary bed of 50 and 80 volume% of 550 ␮m particles in the binary mixture of 550 ␮m particles and 3 mm pebbles. Experiments were repeated with 231 ␮m Li2 TiO3 particles and 5 mm Li2 TiO3 pebbles and average keff of binary bed were measured for 30, 50 and 80 volume% 231 ␮m Li2 TiO3 particles in the binary mixture. 4. Results and discussions 4.1. Void fraction in binary pebble bed Experimental results on the binary bed void fractions with Al2 O3 pebbles are shown in Fig. 9 and that with Li2 TiO3 pebbles are shown in Fig. 10. Variation of void fractions of binary particulate bed (ε) are represented in terms of volume fraction of smaller pebbles or particles, xp . Figs. 9 and 10 show that ε decreases with xp to a minimum value, εmin and then increases for all R. Thus, there exists a minimum value εmin for each R. 4.1.1. Bed geometry The variation of void fraction with volume fraction of smaller pebbles or particles (xp ) for different, R for Al2 O3 pebbles in rectangular and cylindrical bed are shown in Fig. 9(a) and (b) respectively. Similarly, the variation of εb with xp for different R for Li2 TiO3 pebbles in rectangular and cylindrical bed are shown in Fig. 10(a) and (b) respectively. The effect of the shape of the container on the binary bed void fraction is found to be negligible. The void fraction in cylindrical bed was marginally higher than that in rectangular bed due to wall effect as discussed above. 4.1.2. Particle size ratio Figs. 9 and 10 also show the variation of εb with R for different xp . It was observed, that with increase in R, εb increases for any XP (or xp as xp = (1 − XP )), which is independent of bed geometry and pebble density as discussed in Section 2. 4.1.3. Other parameters εb is independent of particle properties viz. p , s and even the vessel geometry. However, for unary particulate bed ε may not be distributed uniformly in a cylindrical vessel. Near wall, it is more than that at the bulk of a cylindrical bed. This is called wall effect

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Fig. 9. Variation of void fraction in binary Al2 O3 pebbles in (a) rectangular and (b) cylindrical vessel.

Fig. 10. Variation of void fraction in binary Li2 TiO3 pebbles in (a) rectangular and (b) cylindrical vessel.

as discussed earlier and due this, the average ε in cylindrical vessel is marginally higher than that in a rectangular vessel for a unary pebble bed. For binary particulate bed viz., packed fluidized bed this is not true as the small fluidized particles remain distributed uniformly throughout the interstitial voids, so ε will be uniformly distributed even in cylindrical bed. 4.2. Effective thermal conductivity of unary bed Experimental results on the effective thermal conductivity of binary Li2 TiO3 particulate bed at room temperature (30 ◦ C) with stagnant air are shown in Table 2 and in Fig. 11. The effective thermal conductivity values shown in Table 2 are the average of three set of repeat experiments. The results show that the effective thermal conductivity of unary pebble bed is less than that of single pebble. This is due to the presence of significant amount of voids in the pebble bed and the low thermal conductivity of air (kg ) compared to that of Li2 TiO3 (kg /ks ≈ 1.44 × 10−2 at room temperature). From the Table 2 it is also observed that, in unary pebble bed, keff increases with increase in pebble size. As the particle size increases, the number of the particle-to-particle contact decreases and this reduces the resistance to heat transfer. So, the effective thermal conductivity increases. For 10 mm pebble, keff of unary bed is high

Fig. 11. Effective thermal conductivity of unary Li2 TiO3 particulate bed at 30 ◦ C without air flow. Dashed lines are shown to guide the eyes.

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Table 2 Thermal conductivity of unary Li2 TiO3 pebble bed at room temperature without flow of air. Particle size (dp ) or pebble size (DP ) [mm]

Bed density [kg m−3 ] Void fraction [–] Average keff [W m−1 K−1 ]

0.231

3

5

7

10

1295 0.439 0.192

1333 0.502 0.255

1389 0.494 0.273

1289 0.502 0.334

1348 0.485 0.605

compared to smaller pebble size. This is probably was due to the better contact of the pebble with the TPS sensor. However, the similar keff of unary 10 mm Li2 TiO3 pebble bed was observed as discussed in details above.

Table 3 Thermal conductivity of binary Li2 TiO3 particulate bed at room temperature and without flow of air. Volume% of particles in the binary particulate bed 30 volume%

4.3. Effective thermal conductivity of binary particulate bed Effective thermal conductivity of binary Li2 TiO3 particulate bed containing 30, 60 and 80 volume% 231 ␮m particles with 3 mm pebbles and 5 mm pebbles are shown in Table 3. Fig. 12(a) and (b) show the variation of keff and ε with the volume% of 231 ␮m Li2 TiO3 particles in binary bed with 3 mm and 5 mm Li2 TiO3

60 volume%

80 volume%

Binary particulate bed of 231 ␮m particles and 3 mm pebbles 1295 1433 Bed density [kg m−3 ] 0.423 0.323 Void fraction [–] −1 −1 0.332 0.357 Average keff [W m K ]

1389 0.423 0.298

Binary particulate bed of 550 ␮m particles and 3 mm pebbles 1296 1437 Bed density [kg m−3 ] 0.425 0.325 Void fraction [–] −1 −1 0.302 0.321 Average keff [W m K ]

1391 0.425 0.262

pebbles, respectively. Bed with 30 volume% of smaller particles has higher thermal conductivity than the bed with 80 volume% smaller particles even though the void fraction of the beds is same. Bed with 50 volume% smaller particles has the highest thermal conductivity for 3 and 5 mm larger pebbles. 50 volume% smaller particles has highest particle density (or lowest void fraction) and therefore has highest thermal conductivity. Compared to 80 volume% smaller particles, the bed with 30% has marginally higher density. As the numbers of smaller particles are less in the bed of 30 volume% smaller particles than that of the bed of 80 volume% smaller particles, the number of contact points between the particles is less. Hence, keff of the bed with low volume% of solid particles is higher than that of the bed with high volume% of smaller particles.

5. Conclusions

Fig. 12. Effective thermal conductivity of binary Li2 TiO3 particulate bed without air flow at 30 ◦ C, (a) binary bed of 231 ␮m particles and 3 mm pebbles (b) binary bed of 231 ␮m particles and 5 mm pebbles. Dashed lines are drawn to guide the eyes.

The variation of void fraction in binary particulate bed with volume fraction of smaller pebbles or particles has been studied. Void fraction of binary beds depends on the volume fraction of smaller particles or pebbles and the smaller particles (or pebbles) to larger pebble size ratio. For a particular size ratio, there is a minimum void fraction for any volume fraction of smaller (or larger) pebbles. Pressure drop of a flowing fluid through a particulate bed increases with decrease in void fraction. Moreover, other than the minimum void fraction, one may get same void fraction for two different volume fractions of smaller size pebbles. In a unary packed pebble bed, thermal conductivity is low owing to the poor conductivity of the air between the particles. The experimental results show that the binary particulate bed has higher effective thermal conductivity that that of unary pebble bed. Effective thermal conductivity of binary particulate bed is the maximum when its void fraction is the minimum. The binary bed with less volume fraction of small particles has higher conductivity than the binary bed of higher volume fraction of small particles and having same void fraction. This is due to the fact that with increase in volume fraction of small component, the number of small-to-small and large-to-small contact point increases resulting in higher resistance to heat transfers.

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