Estimation of effective thermal conductivity of packed bed with internal heat generation

Fusion Engineering and Design 152 (2020) 111458

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Estimation of effective thermal conductivity of packed bed with internal heat generation

T

D. Mandala,*, P.A. Dabhadea,*, N. Kulkarnib a b

Alkali Material & Metal Division, Bhabha Atomic Research Centre, Trombay, Mumbai, 400085, India Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumbai, 400019, India

G R A P H I C A L A B S T R A C T

The following figure shows the effect of induction heating temperature on the effective thermal conductivity of packed bed filled with 10 mm lithium titanate pebbles.

A R T I C LE I N FO

A B S T R A C T

Keywords: Non-uniform internal heat generation Test blanket module Fusion rector Effective thermal conductivity Finite difference method

Test Blanket Module (TBM) of the fusion reactor has the vital role of removing heat generated from the fusion reactor along with the heat generated internally due to tritium breeding reaction. In the initial startup phase of fusion reactor, neutron flux has a climbing ramp nature and heat gets generated in discrete locations of TBM due to non-uniform nature of neutron irradiation. Thus, there is need to establish mathematical method to estimate Effective Thermal Conductivity (ETC) of packed bed with randomly scattered internal heat sources. Novel Induction Heating technique was used to simulate above mentioned conditions in packed bed of Li2TiO3 pebbles. An input-output FDM (Finite Difference Method) based model has been developed to estimate ETC of packed beds with non-uniform and discrete Internal Heat Generation (IHG). The inputs to the FDM based model are experimental results along with various operating and process parameters of the packed bed system under consideration. Also, an empirical correlation has been proposed to yield ETC of above mentioned packed bed systems valid for bed voidage of 0.44-0.47. The experimental and analytical methods used for this study are detailed in this paper.

⁎

Corresponding authors at: Alkali Material & Metal Division, Bhabha Atomic Research Centre, Trombay, Mumbai, 400085, India. E-mail address: [email protected] (D. Mandal).

https://doi.org/10.1016/j.fusengdes.2020.111458 Received 3 June 2019; Received in revised form 9 December 2019; Accepted 7 January 2020 0920-3796/ © 2020 Elsevier B.V. All rights reserved.

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Nomenclature

TIH ρg Cpg V˙ inlet Apeb H αg m˙ g dp VIHS μ g Tout Ting

Symbol description i j k ks

kg q˙ V Dbed σ

Generic element of regular rectangular mesh present inside bed Element located adjacent to element i inside bed Thermocouple index Thermal conductivity of solid phase (Li2TiO3 pebble/steel sphere) Thermal conductivity of gas phase (air) Rate of heat generation per unit volume of steel sphere Bed Diameter (m) Stefan Boltzmann constant

1. Introduction

Induction Heating Temperature Density of gas phase Specific heat of gas phase Inlet volumetric flowrate of gas phase Surface area of Li2TiO3 pebble/steel sphere Height of bed Void fraction of bed Mass flowrate of gas Diameter of Li2TiO3 pebble/steel sphere Volume of steel sphere (internal heat source) Viscosity of gas phase Outlet temperature of gas Inlet temperature of gas

lithium density followed lithium orthosilicate, lithium meta-titanate and lithium meta-zirconate but the lithium vaporization rate and moisture sensitivity of lithium oxide and lithium orthosilicate is also high [4–7] making them vulnerable to deterioration with time. Lithium meta-titanate is chemically inert, has good tritium release behaviour and low activation characteristics making it the most promising LCTB material [5]. From the above discussion it is well understood that Li2 TiO3 pebble bed plays an important role and the heat transfer characteristics of the bed is a key parameter for the application in TBM and ITER. Effective Thermal Conductivity (ETC) of packed bed is a well-studied concept and plenty of literature is available on this subject [11–25]. Also, heat transfer characteristics of packed bed systems are well established both for external heating conditions and for uniform internal heat generation [26–39]. Different LCTB materials have been

Lithium-based ceramic materials enriched in Li6 isotope are prime candidates for application as tritium breeders in blankets of fusion reactors [1–10]. Lithium ceramics can withstand long time irradiation at high temperatures and high temperature gradients [1–3]. Pebble bed of Lithium based Ceramic Tritium Breeder (LCTB) material is considered for the Test Blanket Module (TBM) of ITER (acronym of International Thermonuclear Experimental Reactor) [4,5]. The choice of lithium ceramic breeder material depends on varies factors viz., thermo-mechanical stability, chemical stability, activation level and tritium retention capability among others. The candidate LCTB materials are lithium oxide (Li2O), lithium orthosilicate (Li 4 SiO4 ), lithium metatitanate (Li2 TiO3 ), lithium meta-zirconate (Li2 ZrO3) lithium aluminate (LiAlO2 ) etc. Among LCTB materials, lithium oxide has the highest

Fig. 1. Schematic diagram of experimental setup used to study packed bed with internal heat generation. 2

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studied with stagnant gas flows [13–15]. Mandal et. Al. [23] conducted experimental effective thermal conductivity study of Li2 TiO3 pebbles of diameters ranging from 1−10 mm under flowing gas medium under external heating conditions and observed significant enhancement in flowing gas configuration as compared to stagnant gas configuration. The test blanket module (TBM) of fusion reactor has to remove heat under external heating arising from high temperature fusion reaction and also the heat generated from in-situ tritium breeding reaction which can be termed as internal heat generation. Also, based on the neutronics analysis of ITER [6], it can be observed that in the initial startup sequence of fusion reactor, the neutron spectrum has climbing ramp nature. This implies that with increase in neutron flux, the fraction of TBM undergoing tritium breeding reaction will rise from 0 to nearly 100 %. The neutron irradiation in TBM is non uniform in the initial powering up phase of fusion reactor.Hence there is a need to study effective thermal conductivity of packed bed with randomly scattered internal heat sources to simulate the non-uniform internal heat generation due to tritium breeding reactions arising from discrete pebbles in TBM during the initial startup phase of fusion reactor. With the help of novel induction heating technique, this type of packed bed systems was simulated using steel spheres as the randomly scattered internal heat sources. Mathematical model for estimation of effective thermal conductivity along with experimental methods used are detailed in the following sections.

Table 1 Location of thermocouples in experimental packed bed. Thermocouple index (k)

r (m)

θ (o)

Z (m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.0037 0.0279 0.0522 0.0764 0.0037 0.0279 0.0522 0.0764 0.0037 0.0279 0.0522 0.0764 0.0037 0.0279 0.0522 0.0764

45 45 45 45 135 135 135 135 225 225 225 225 315 315 315 315

0.295 0.295 0.295 0.295 0.195 0.195 0.195 0.195 0.95 0.95 0.95 0.95 0.15 0.15 0.15 0.15

given composite bed. In this experiment, the sizes taken are 5 mm, 7 mm and 10 mm steel spheres/ pebble diameter. The diameter of column in 162.74 mm and the maximum height of packing is 330 mm. The snap of induction heating furnace along with test section can be seen in Fig. 2. For temperature measurement, 16 thermocouples were inserted at various axial, radial and circumferential coordinates as shown in Table 1.

2. Experimental details 2.1.1. Induction heating system The induction heating furnace has 15 kW capacity and its power rating is 3 Phase with 440 V AC. The maximum attainable temperature inside the bed is 1100°C and the maximum rated working temperature is 900°C. Due to presence of alumina wall, the maximum surface wall temperature is not more than 70°C. The OD of induction coil was 250.637 mm and thickness of coil was 10 mm.

2.1. Experimental setup The schematic of experimental setup is shown in Fig. 1. The test section can be reference bed viz. packed bed completely composed of internal heat sources (here, steel spheres) or the bed can be composite bed of steel spheres and Li2TiO3 pebbles mixed randomly in a predefined manner. The Li2TiO3 pebbles were fabricated using the extrusion and spherodization technique using Li2 TiO3 powder synthesized by solid State reaction process developed by Mandal et al. [4,5]. Particle density was estimated from the true density of pebbles using Helium Pycnometer. The wall of test section is made of alumina so that it can withstand high temperatures inside the induction furnace and also can act as an insulator so that only internal heat generation is present in the test section. The steel spheres and Li2TiO3 pebbles have same sizes for a

2.1.2. Pressure measurement system As was adopted in our previous work on heat transfer study of packed bed with external heating [12], the pressure measurement system has 8 DPTs which measure pressure at different axial locations of packed bed.

Fig. 2. Snap of Induction Heating Furnace along with packed bed test section used in experimental setup. 3

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thermocouples) were installed at different bed height to measure the temperature at different radial as well as at different axial locations of the pebble bed.

2.1.3. Flow measurement and control One on-line gas mass-flow-meter with inbuilt controller was installed to measure and control gas flow-rate. The mass-flow-meter was calibrated for air at 6.0 kg/cm2 (g) inlets and 5.0 kg/cm2 (g) discharge pressure. To ensure air supply at 6.0 kg/cm2 (g) one PRV was installed between air receiver and mass-flow-meter as shown in Fig. 2. The massflow-meter was connected to a PLC based SCADA system to control and measure of flow and for storage of data.

2.1.5. Data acquisition and storage system In the experimental work the bed wall temperature and air flow rate were controlled parameters while pressures and bed temperatures were measured parameters. Two PLC based controller modules were installed (one in use and other as redundant).Ellipse software was used for data acquisition, display, and storage of all the process and controlled parameters of heating system, flow measurement and control system and temperature measurement system.

2.1.4. Temperature measurement To measure temperature at different radial locations four thermocouples were placed in 8 mm diameter thermo-well with 3 mm diameter cut-out to well place the thermocouple sensors. Four such thermo-wells containing four thermocouples each (i.e. total 16

Fig. 3. Algorithm for ETC estimation of packed bed with internal heat generation using FDM based Model. 4

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2.2. Experimental method

4⎛ αg = 1 − ⎝ ⎜

The Experimental methods for reference bed systems and composite bed systems are discussed in following subsections.

mp ρp

⎞⎞ ⎠⎠ ⎟

2 πDbed Hbed

(2)

3. FDM based model The aim of FDM based Model is to estimate the effective thermal conductivity of the ceramic component of the composite packed bed. The ceramic component comprises of the bed domain not occupied by internal heat sources. Finite Difference Method (FDM) was used to solve the solid phase and gas phase constitutive energy equations for the unary or composite packed bed systems. Domain discretization was done using regular rectangular cartesian mesh. The algorithm of the model is shown in Fig. 3. The inputs to the model are the experimental readings for reference bed systems (completely filled with internal heat sources) and composite packed bed systems (internal heat sources mixed with Li2TiO3 pebbles) along with the process operating parameters like inlet gas flowrate q˙ INLET, Induction Heating temperature TIH and other bed parameters like pebble diameter dp, bed voidage α g and mass fraction of internal heat sources χIHS for composite bed systems of Li2TiO3 pebbles and steel spheres of equal sizes. The output of the model is the effective thermal conductivity of composite packed bed under consideration and the total error associated with this value.

mref

⎜

mIHS ρIHS

The steel spheres used have the same diameter as that of Li2TiO3 pebbles for any given bed configuration. The experiments were again repeated for 10 mm, 7 mm and 5 mm steel spheres/Li2TiO3 pebble mixed bed under induction heating temperature of 150 °C, 300 °C, 450 °C and 600 °C and inlet gas flowrates of 5−20 m3/h.

2.2.1. Experiments on reference bed The test section was completely filled with steel spheres in such a way that the bed voidage lies in the range 0.44-0.47 so that this bed can be virtually simulated with our random close pack generation algorithm. The bed voidage was calculated using Eq. 1.

4 ⎛ ρ IHS ⎞ IHS αg = 1 − ⎝2 ⎠ πDbed Hbed

( ) + ⎛⎝

⎟

(1)

ref mIHS is the total mass of steel spheres used to fill the bed and is noted before pouring into column. The experiments were done for 10 mm, 7 mm and 5 mm steel spheres under induction heating temperature of 150 °C, 300 °C, 450 °C and 600 °C and inlet gas flowrates of 5−20 m3/h.

2.2.2. Experiments on composite bed The test section was filled with mixture of steel spheres and Li2TiO3 pebbles based on the chosen internal heat source mass fraction χIHS which is the ratio of mass of steel spheres to the sum of steel spheres mass and Li2TiO3 pebbles mass.. The Li2TiO3 pebbles are first weighed in a container and accordingly the steel spheres were collected in another container as per the chosen IHS mass fraction. Four values of χIHS were chosen for the experimental runs namely 10 %, 20 %, 30 % and 50 %.The solid steel spheres are first poured in the container of Li2TiO3 pebbles and the composite mixture then poured in the empty container and this process was repeated several times to obtain proper random bed configuration.. The height of bed was so adjusted that the bed voidage lies in the range of 0.44-0.47. The bed voidage was calculated using Eq. 2.

3.1. Allocation of internal heat sources For composite random packed bed systems, the actual coordinates of the internal heat sources inside the bed are not known, hence there is a need to optimally allocate the internal heat sources such that the temperature profile obtained using FDM based model should

Fig. 4. Zonal division of Test Section (Binary Packed Bed) based on thermocouple locations. 5

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denotes the ceiling function. α and β are fixed for given bed allocation / configuration and change in them brings change in configuration. mIHS is the total mass of steel spheres inside the composite bed, this value is noted before mixing the steel spheres with the Li2TiO3 pebbles. Once α and β are known, the number of internal heat sources in each zone is found using Eqs 5–8. 16 Zone1 NIHS =

∑

NIHS (k )

(5)

k = 13

12

Fig. 5. Thermal Resistance model for solid to gas phase heat transfer.

Zone2 NIHS =

∑ NIHS (k )

(6)

k=9

correspond to the experimental temperature profile. Eqs. 3–8 collectively define the allocation scheme.

8 Zone3 NIHS =

∑ NIHS (k )

(7)

k=5

⎛ Texpt (k ) ⎞

β

4

NIHS (k ) = α ⎜ ⎟ ⎝ Tref (k ) ⎠

Zone 4 NIHS =

(3)

∑ NIHS (k )

(8)

k=1 16

6 ⎞ ⎛ mIHS ⎞ ⎤ ⎟ 3 ⎟⎜ ⎥ ρ πd ⎣ ⎝ IHS ⎠ ⎝ IHS ⎠ ⎦

⎡ total = ⎢⎛ ∑ NIHS (k ) = NIHS

Fig. 4 shows the 4 zones of packed bed divided based on the axial Zone i is locations of thermowells. Each zone is divided into 4 sectors. NIHS the number of internal heat sources in zone i (1 ≤ i ≤ 4) . The internal heat sources are then divided in 4 equal group. Each group is then randomly allocated to each sector belonging to the respective zone.

⎜

k=1

(4)

NIHS (k ) is the number of internal heat sources associated with thermocouple index k. Texpt (k) is the experimental temperature reading of kth thermocouple of composite packed bed and Tref (k ) is experimental temperature reading of kth thermocouple of reference bed (packed bed completely filled with internal heat sources, here steel spheres) under the same operating conditions of that of composite packed bed viz, inlet gas flowrate, induction heating temperature, pebble diameter and bed voidage. α and β are constants such that Eq.4 is satisfied, here []

3.2. Constitutive energy balance equations The FDM based model uses Eqs. 9–16 to obtain the temperature profile of gas phase and solid phase. The bulk phase equations are given by Eqs. 9–10.

Table 2 FDM gridsize optimization and evaluation of average error associated with domain discretization. Sr no.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Bedwall Temperature [oC]

Pebble Diameter [mm]

Inlet Gas Flowrate [m3/hr]

Optimum gridsize [-]

ε FDM

200 200 200 200 400 400 400 400 600 600 600 600 200 200 200 200 400 400 400 400 600 600 600 600 200 200 200 200 400 400 400 400 600 600 600 600

10 10 10 10 10 10 10 10 10 10 10 10 7 7 7 7 7 7 7 7 7 7 7 7 5 5 5 5 5 5 5 5 5 5 5 5

5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20

75 × 75 × 119 75 × 75 × 119 75 × 75 × 119 75 × 75 × 119 75 × 75 × 119 75 × 75 × 119 75 × 75 × 119 75 × 75 × 119 75 × 75 × 119 75 × 75 × 119 75 × 75 × 119 75 × 75 × 119 83 × 83 × 105 83 × 83 × 105 83 × 83 × 105 83 × 83 × 105 83 × 83 × 105 83 × 83 × 105 83 × 83 × 105 83 × 83 × 105 83 × 83 × 105 83 × 83 × 105 83 × 83 × 105 83 × 83 × 105 91 × 91 × 103 91 × 91 × 103 91 × 91 × 103 91 × 91 × 103 91 × 91 × 103 91 × 91 × 103 91 × 91 × 103 91 × 91 × 103 91 × 91 × 103 91 × 91 × 103 91 × 91 × 103 91 × 91 × 103

0.0742 0.0823 0.0611 0.0931 0.0735 0.0881 0.0924 0.0756 0.0666 0.0649 0.0741 0.0698 0.0997 0.0546 0.0687 0.0946 0.0754 0.0961 0.0559 0.0878 0.0784 0.0818 0.0988 0.1053 0.1021 0.0775 0.0717 0.0648 0.0846 0.1019 0.0842 0.0646 0.0778 0.1137 0.0983 0.1076

6

[−]

¯

εFDM [-]

0.0763

0.0831

0.0874

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Fig. 6. Temperature profile of packed bed composed of 10 mm Li2TiO3 pebbles under external heating with bedwall temperature of 200°C and inlet gas flowrate of 5 m3/hr used for FDM gridsize optimization (75 × 75 × 119 optimal grid) (a): Random Close Pack generated to simulate experimental bed under external heating (green: Li2TiO3) (b): XZ cutplane of 3-D temperature profile (c) YZ cutplane of 3-D temperature profile. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

Fig. 8. Evaluation of Error associated with ETC calculated with FDM based model using previously reported ETC values for external heating case.

Fig. 7. Gridsize optimization (75 × 75 × 119 optimal grid) of FDM model for 10 mm pebble bed under external heating with bedwall temperature of 200°C and inlet gas flowrate of 5 m3/hr. Grid Independence of FDM based Model can be inferred.

k g ∇ 2 T − ρg Cpg ug

(10)

At bed wall:

For bulk region of solid phase:

kS ∇ 2 T + q V̇ = 0

∂T =0 ∂z

T = Tambient

(9)

q˙ V term is the heat generation by induction heating in the internal heat source (here,steel sphere) and is defined in subsection 3.2.1. q˙ V is zero for the Li2TiO3 pebbles. For bulk region of gas phase:

(11)

The boundary condition is based on the fact that the bed wall is ceramic in nature and acts as insulator, hence the wall temperature is close to the ambient temperature as shown in Eq. 11. FDM based model assumes plug flow of gas phase and therefore ug is calculated as shown 7

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Table 3 Induction Heating Power Profile optimization and evaluation of average error associated. Sr no.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Induction Heating Temperature [oC]

Steel Sphere Diameter [mm]

Inlet Gas Flowrate [m3/hr]

εFDM + εIHP [-]

ε FDM

150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600 150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600 150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20

0.1088 0.1134 0.0924 0.1179 0.1159 0.1021 0.0923 0.0972 0.0937 0.0955 0.0972 0.1025 0.0915 0.1171 0.1183 0.1047 0.1047 0.1001 0.117 0.1011 0.0933 0.1134 0.1017 0.0973 0.1021 0.0929 0.094 0.1183 0.1187 0.1073 0.0918 0.097 0.1006 0.1146 0.0905 0.0913 0.0951 0.1095 0.112 0.1094 0.1035 0.1064 0.0989 0.1123 0.0957 0.1106 0.0955 0.1011

0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874

[−]

¯ εIHP

_as =

* ε IHP

Apeb Vpeb

=

πdp2 (πdp3/3)

0.0325 0.0371 0.0161 0.0416 0.0396 0.0258 0.016 0.0209 0.0174 0.0192 0.0209 0.0262 0.0152 0.0408 0.042 0.0284 0.0216 0.017 0.0339 0.018 0.0102 0.0303 0.0186 0.0142 0.019 0.0098 0.0109 0.0352 0.0356 0.0242 0.0087 0.0139 0.0132 0.0272 0.0031 0.0039 0.0077 0.0221 0.0246 0.022 0.0161 0.019 0.0115 0.0249 0.0083 0.0232 0.0081 0.0137 0.0210292

max PIHS =

Ptotal ref NIHS

(16) max PIHS

is the total number of steel spheres in the reference bed and is the maximum power transmitted to a given steel sphere/internal max heating source. Value of PIHS is used as it is for composite packed bed system having same induction heating temperature, pebble diameter and bed voidage as that of reference bed. For 10 mm steel spheres: 2 Ptotal = −0.021TIH (°C ) + 30.127 TIH (°C ) − 1278.5

(17)

For 7 mm steel spheres: 2 Ptotal = −0.015TIH (°C ) + 28.218TIH (°C ) − 644.23

(18)

For 5 mm steel spheres: 2 Ptotal = −0.031TIH (°C ) + 41.826TIH (°C ) − 1690.2

(19)

Furthermore, the induction heating is non uniform inside the packed bed given that its power decreases near the ends, therefore not max . To acevery internal heat source can generate power equal to PIHS count for this non uniformity, an empirical profile was generated to minimize the error associated with internal heat generation term.

q˙ V =

max δ (rc , z c ) PIHS VIHS

(20)

δ (rc , z c ) = a4 (z c ) rc4 + a2 (z c ) rc2 + a0 (z c )

(21)

Where: 2

a4 (z ) = 13 + 0.3e

z − 0.5H ⎞ −⎛ c ⎝ 0.5 ⎠

(22) 2

z − 0.5H ⎞ −⎛ c ⎝ 0.5 ⎠

(23) 2

a0 (z ) = 0.5 + 0.48e

z − 0.5H ⎞ −⎛ c ⎝ 0.2 ⎠

(24)

Here, (rc , z c ) are the radial and axial coordinates of the steel sphere centroid. It is assumed that power generated inside steel sphere is uniform throughout its volume since dimensions of steel sphere are very small when compared to that of bed.

(12)

The constitutive equations at gas solid interface are given in Eqs. 13–15. The calculation of overall heat transfer coefficient U gs is explained in subsection 3.2.2. Gas-Solid Interface: Solid Phase:

3.2.2. Thermal resistance models Heat transfer at interface is either between solid phase and solid phase or between solid phase and gas phase. Eq. 25 gives the overall heat transfer coefficient between two elements i and j both belonging to different solid spheres.

(13)

Gas Phase:

∂T k g ∇ 2 T − ρg Cpg ug + U gs_asΔTsg = 0 ∂z

(15)

ref NIHS

in Eq. 12.

kS ∇ 2 T + q V̇ − U gs_a s ΔTsg = 0

6 dp

3.2.1. Power generation term The internal heat generation term depends on the total power transmitted by the induction heating furnace which in turn is a function of induction heating temperature. The spatial variation of Power generation term was optimized after performing experimental runs with packed beds completely filled with mono sized steel spheres and comparing them with resultant temperature profile from FDM based model. Empirical Eqs. 16–24 are the optimized equations that result in minimal error when used in FDM based model to obtain the temperature profile of a given reference bed. Ptotal is the total power transmitted by induction heating furnace which was observed to depend on steel sphere diameter and induction heating furnace.

[−]

a2 (z ) = −30 + 0.5e

V˙ ug = inlet αg Abed

=

dij dij 1 = + U s1 − s2 2ks1 2ks2 (14)

(25)

The gas–solid thermal resistance model used to obtain overall heat 8

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Fig. 9. a) Optimized Contour Plot for Spatial Variation of δ(rC, zc) inside Test Section. (b) Optimized Total power transmitted Ptotal inside Test Section as a function of Induction Heating Temperature.

transfer coefficient is shown in Fig. 5. Eqs 26–28 define the equations used to obtain overall heat transfer coefficient U gs for gas solid heat transfer.

Here, (dij⊥)2 is the cross sectional area of heat transfer between element i and j. On rearranging, we obtain the effective thermal conductivity as shown in Eq. 30.

(1 − αg ) dij αg dij 1 = + U gs ks dij αg hg + k g + kr

k eff =

(26)

1 2

hgs =

kr =

2 3⎞

∑iεC ∑jεBed (dij⊥ )2

Total error associated with the effective thermal conductivity value is sum of four errors namely FDM discretization error εFDM , Induction Heating profile error εIHP , IHS (Internal Heat Source) allocation error εallocation and ETC model error εETC , each of which are defined as follows. The error estimation model was such that the four errors are independent of each other.

Here, dij is the distance between centroids of elements i and j. Convection heat transfer coefficient hgs is evaluated using Whitakers correlation [33]. The radiation effect though not very significant at temperatures below 600 °C is taken into account by using kr [38].

3.4.1. FDM discretization error εFDM The FDM discretization error depends on the grid size used for Finite Difference Method. The error decreases with increase in the number of discrete cells. The optimal grid size is the one which leads to acceptable error at moderate computational time. The optimal number of discrete cells increases with decrease in pebble size to adequately capture the solid - solid and gas – solid interface phenomenon as seen in Table 2. The optimal grid size for dp equal to 10 mm is 75 × 75 × 119 which results in average error of 7.63 %. For dp equal to 7 mm, the optimal grid size is 83 × 83 × 105, resulting in average error of 8.31 % and for dp equal to 5 mm, the optimal grid size is 91 × 91 × 103, resulting in average error of 8.74 %.

3.3. ETC estimation model The effective thermal conductivity (ETC) of ceramic component of composite packed was evaluated using overall energy balance on the ceramic domain. At steady state heat gained from induction heat generation in steel spheres is completely carried away by gas as per adiabatic wall assumption as shown in Eq. 29. The left hand side term is the net energy output from every element in the ceramic domain C and right hand term is the energy gained by gas phase between inlet and outlet of packed bed.

16

keff ∑

∑

iεC jεBed

dij

=

m˙ g Cpg (Tgout

− Tg

(30)

3.4. Error associated with ETC value obtained from FDM based model

(28)

ΔTij

dij||

(27)

4σdp T 3 (K )

(dij⊥ )2

Δ Tij

1 3⎞

⎛ ⎛ ρg ug dpeb ⎞ kg ⎛ ⎛ ρg ug dpeb ⎞ ⎛ Cpg μg ⎞ + 0.06 ⎜ ⎜2 + ⎜0.4 ⎜ ⎟ ⎟ ⎟⎜ dpeb ⎜ μg ⎟ ⎟ ⎝ k g ⎠ ⎟ ⎜ ⎝ μg ⎟⎠ ⎠ ⎠ ⎝ ⎝ ⎠ ⎝

3

m˙ g Cpg (Tgout − Tg in)

εFDM =

in)

(29) 9

1 ∑ 16 k = 1

2

⎛ Texpt (k ) − TFDM (k ) ⎞ ⎜ ⎟ Texpt ⎝ ⎠

(31)

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Fig. 10. Temperature profile of reference bed (completely filled with internal heat sources) composed of 10 mm steel spheres under internal heat generation with induction temperature of 300°C and inlet gas flowrate of 5 m3/hr used for FDM gridsize optimization (75 × 75 × 119 optimal grid) (a): Random Close Pack generated to simulate experimental bed under internal heat generation (red: steel spheres) (b): XZ cutplane of 3-D temperature profile (c) YZ cutplane of 3-D temperature profile. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

method for sample bed shown in Fig. 6. The grid independence of FDM based model can also be inferred from Fig. 7. The experimental temperature values used in Fig. 7 were obtained earlier to calculate effective thermal conductivity of packed bed composed of Li2TiO3 pebbles under gas flow and external heating conditions in our previous work [23]. 3.4.2. ETC model error εETC ETC model error is the error associated with the methodology used to obtain effective thermal conductivity value using Eqs. 29–30. This error is calculated by comparing the ETC values obtained by FDM based model with the reported effective thermal conductivity values by Mandal et. Al [23] for packed beds with external heating systems as shown in Fig. 8. The ETC value obtained by Eq. 30 includes FDM discretization error and ETC model error for external heating systems hence upon subtracting the FDM discretization error from net error, ETC model error can be evaluated. The εETC is equal to 1.387 % for 10 mm pebbles, 3.789 % for 7 mm pebbles and 7.316 % for 5 mm pebbles. The increase in εETC value on decrease in pebble diameter is based on the fact that the overall energy balance of Eq. 29 is more accurate for packed pebble beds with larger pebble sizes owing to reduction in solid to solid contact.

Fig. 11. Induction Heating Profile optimization (75 × 75 × 119 optimal grid) of FDM model for packed bed composed of 10 mm steel sphere bed under internal heat generation with induction temperature of 300°C and inlet gas flowrate of 5 m3/hr.

Eq.31 shows the calculation method for FDM discretization error. Fig. 6 shows the application of FDM based model to obtain temperature profile of sample Li2TiO3 pebble bed under external heating with pebble diameter of 10 mm, external bed wall temperature of 200 °C and inlet gas flowrate of 5 m3/h. Fig. 7 shows the grid size optimization

3.4.3. Induction heating profile error εIHP Induction Heating Profile Error εIHP is the error associated with the empirical equations used to optimize internal heat generation term. Table 3 lists the error associated with optimized empirical Eqs. 10

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Fig. 12. Composite Bed Configurations based on different internal heat source allocations, used for optimization of allocation scheme.

allocation method used to allocate internal heat sources inside packed bed. This error is evaluated by applying FDM based model to obtain temperature profile of composite packed beds and comparing them with the experimental temperature readings obtained after applying induction heating on composite beds. Fig. 12 shows 4 configurations based on different IHS (Internal Heat Source) allocation for composite packed bed with 30 % IHS mass fraction, 300 °C induction heating temperature, 5 m3/ h inlet gas flowrate and 10 mm pebble/steel sphere diameter. Fig. 13 shows the temperature profile for above mentioned packed bed system obtained using FDM based model with optimal grid size established in Table 2 using configuration 2 of Fig. 12. Fig. 14 shows that configuration 2 is the optimum allocation among the 4

17–19 and Eqs. 21–24. The optimized induction power profile represented by these equations is shown in Fig. 9. εIHP is evaluated by comparing the experimental temperature readings of reference beds viz. packed beds completely filled with internal heat sources and those obtained by FDM based model. The sample temperature profile for reference bed with 10 mm steel sphere diameter, 5 m3/h inlet gas flowrate and induction heating temperature of 300 °C is shown in Fig. 10. εIHP for reference packed bed system mentioned in Fig. 10 is calculated as shown in Fig. 11.

3.4.4. Allocation error εallocation Allocation error εallocation is error associated with the random 11

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Fig. 13. Temperature Profile obtained using FDM based model for Optimal Bed Configuration (conFig. 2) with Induction Heating Temperature of 300°C, Li2TiO3 Pebble /Steel sphere diameter of 10 mm, inlet gas flowrate of 5 m3/hr and Internal Heat Source mass fraction of 30 %. (a) XZ cut plane of 3D temperature profile (b) YZ cut plane of 3D temperature profile.

in Tables 4–6 for pebble sizes of 10 mm, 7 mm and 5 mm respectively. The keff value for composite Li2TiO3 pebble/steel sphere packed bed with pebble size of 10 mm lies in the range of 0.4614–0.6795 W/mK for 10 % internal heat source (IHS) mass fraction, 0.5747– 0.9045 W/mK for 20 % IHS mass fraction, 0.6668–1.0031 Wm K for 30 % IHS mass fraction and 0.7941–1.1983 W/mK for 50 % IHS mass fraction. For 7 mm pebble bed, keff lies in the range of 0.3724– 0.6002 W/mK for 10 % IHS mass fraction, 0.5089– 0.7668 W/mK for 20 % IHS mass fraction, 0.5935– 0.8245 W/mK for 30 % IHS mass fraction and 0.7006–1.0279 W/mK for 50 % IHS mass fraction. Lastly for 5 mm pebble bed, keff lies in the range of 0.3244– 0.4637 W/mK for 10 % IHS mass fraction, 0.4326– 0.6534 W/mK for 20 % IHS mass fraction, 0.4849– 0.7180 W/mK for 30 % IHS mass fraction and 0.5701– 0.9060 W/mK for 50 % IHS mass fraction. The effect of different process parameters on keff value is discussed below. The average total error is 21.717 % which is majorly due to discretization and allocation errors. Fig. 14. Comparison of experimental Temperature readings and Temperature values based on FDM Model to optimize Internal Heat Source allocation (Configuration 2 of Fig. 9 is the optimal one for the process parameters mentioned in Fig. 10).

4.1. Effect of Pebble diameter Fig. 15 shows the effect of pebble size on effective thermal conductivity values for 5 m3/h inlet gas flowrate and 30 % IHS mass fraction. The keff increases with increase in pebble diameter dp . This can be due to the fact that with increase in dp , solid-solid and as-solid heat transfer increases due to decrease in number of contact points and also, with increase in pebble diameter, the heat transfer through radiation also increases. As seen from section 4.6, the first term in the proposed correlation indicates that packing factor has an important influence on the effective thermal conductivity. For a given bed voidage, the effec-

configurations shown in Fig.12 with an error of 16.86 % which corresponds to allocation error of 7.13 % after deducting FDM discretization error and IHP (Induction Heating Power) profile error. 4. Results & discussions The results obtained after performing experiments and using FDM based model to estimate effective thermal conductivity of composite packed beds for different operating and process parameters is detailed

tive thermal conductivity decreases with increase in

( ) viz. the ratio Dbed dp

of bed diameter to pebble diameter which is directly related to the 12

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Table 4 Estimated Effective Thermal Conductivity data for packed composite bed of 10 mm Li2TiO3 pebbles and Steel spheres under Internal Heat Generation based on FDM model. Sr no.

Dp [mm]

1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 10 15 10 16 10 17 10 18 10 19 10 20 10 21 10 22 10 23 10 24 10 25 10 26 10 27 10 28 10 29 10 30 10 31 10 32 10 33 10 34 10 10 35 36 10 37 10 38 10 39 10 40 10 41 10 42 10 43 10 44 10 45 10 46 10 47 10 48 10 49 10 50 10 51 10 52 10 53 10 54 10 55 10 56 10 57 10 58 10 59 10 60 10 61 10 62 10 63 10 64 10 Average Error

3 [m/hr]

TIH o [C]

χIHS [%]

FDM based ETC [W/mK]

* εIHP [−]

* εFDM [−]

* εalloc [−]

* εETC [−]

* εTOTAL [−]

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600 150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600 150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600 150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600

10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50

0.6241 0.8096 0.9438 1.0677 0.5393 0.7072 0.7965 0.9182 0.4918 0.6149 0.7703 0.832 0.4614 0.5747 0.6668 0.7941 0.6348 0.8158 0.9569 1.1218 0.5789 0.7256 0.8381 0.9209 0.5029 0.6545 0.7741 0.8658 0.472 0.6163 0.6831 0.8275 0.6424 0.8843 0.9799 1.1587 0.6155 0.7347 0.829 0.9346 0.5124 0.7078 0.7789 0.9373 0.5108 0.6351 0.7175 0.8811 0.6795 0.9045 1.0031 1.1983 0.6264 0.8017 0.9031 1.1245 0.5823 0.7364 0.8489 0.9879 0.5312 0.6576 0.8021 0.9139

0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021

0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763 0.0763

0.0722 0.0709 0.0712 0.0677 0.0727 0.0728 0.0713 0.0799 0.0761 0.0695 0.0803 0.0733 0.0688 0.0835 0.0819 0.0738 0.0756 0.0747 0.0652 0.0675 0.0718 0.0658 0.0811 0.0649 0.0656 0.0643 0.0657 0.0709 0.0678 0.0831 0.0708 0.0646 0.0826 0.0845 0.071 0.0628 0.0665 0.0702 0.0749 0.0666 0.0751 0.0778 0.0655 0.0629 0.0674 0.068 0.0706 0.0727 0.0621 0.0666 0.08 0.0607 0.0832 0.0783 0.0722 0.0745 0.0659 0.0715 0.0841 0.0737 0.073 0.0658 0.0722 0.0756

0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139 0.0139

0.1834 0.1821 0.1824 0.1789 0.1839 0.184 0.1825 0.1911 0.1873 0.1807 0.1915 0.1845 0.18 0.1947 0.1931 0.185 0.1868 0.1859 0.1764 0.1787 0.183 0.177 0.1923 0.1761 0.1768 0.1755 0.1769 0.1821 0.179 0.1943 0.182 0.1758 0.1938 0.1957 0.1822 0.174 0.1777 0.1814 0.1861 0.1778 0.1863 0.189 0.1767 0.1741 0.1786 0.1792 0.1818 0.1839 0.1733 0.1778 0.1912 0.1719 0.1944 0.1895 0.1834 0.1857 0.1771 0.1827 0.1953 0.1849 0.1842 0.177 0.1834 0.1868 0.1833

q˙ INLET

conductivity values for 10 mm pebble diameter and 30 % IHS mass fraction. It can be observed that with increase in inlet gas flowrate, the keff also increases. This increase can be attributed to increase in heat transfer by convection with increase in inlet gas flowrate.

number of point to point contact in solid phase of packed bed. 4.2. Effect of Inlet gas flowrate Fig.16 shows the effect of inlet gas flowrate on effective thermal 13

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Table 5 Estimated Effective Thermal Conductivity data for packed composite bed of 7 mm Li2TiO3 pebbles and Steel spheres under Internal Heat Generation based on FDM model. Sr no.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Average Error

Dp [mm]

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

3 [m/hr]

TIH o [C]

χIHS [%]

FDM based ETC [W/mK]

* εIHP [-]

* εFDM [-]

* εalloc [-]

* εETC [-]

* εTOTAL [-]

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600 150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600 150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600 150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600

10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50

0.5021 0.7144 0.7234 0.8901 0.4675 0.5745 0.6615 0.7966 0.4161 0.5337 0.642 0.7026 0.3724 0.5089 0.5935 0.7006 0.5139 0.7172 0.7428 0.9118 0.4751 0.6178 0.7181 0.8851 0.4678 0.5431 0.6521 0.7195 0.4456 0.5282 0.5985 0.7142 0.5618 0.7383 0.7576 0.9633 0.4971 0.6338 0.7625 0.9072 0.4715 0.5551 0.6635 0.7718 0.4617 0.5296 0.6088 0.7278 0.6002 0.7668 0.8245 1.0279 0.5175 0.6252 0.7726 0.9145 0.4793 0.5639 0.6738 0.7795 0.468 0.5586 0.655 0.7323

0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021

0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831 0.0831

0.077 0.0699 0.0692 0.0847 0.0609 0.0821 0.0828 0.0799 0.0625 0.0665 0.0684 0.077 0.0634 0.078 0.0627 0.0763 0.0724 0.0795 0.0779 0.0826 0.0823 0.0684 0.0775 0.0649 0.0608 0.0786 0.0725 0.072 0.0781 0.0637 0.0765 0.073 0.0708 0.0806 0.0621 0.0633 0.0615 0.07 0.0732 0.0704 0.0708 0.0604 0.0846 0.0642 0.0613 0.0784 0.0667 0.0706 0.0693 0.065 0.0722 0.0685 0.082 0.0767 0.0648 0.0692 0.0735 0.0775 0.0767 0.0645 0.085 0.0643 0.0608 0.074

0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379 0.0379

0.219 0.2119 0.2112 0.2267 0.2029 0.2241 0.2248 0.2219 0.2045 0.2085 0.2104 0.219 0.2054 0.22 0.2047 0.2183 0.2144 0.2215 0.2199 0.2246 0.2243 0.2104 0.2195 0.2069 0.2028 0.2206 0.2145 0.214 0.2201 0.2057 0.2185 0.215 0.2128 0.2226 0.2041 0.2053 0.2035 0.212 0.2152 0.2124 0.2128 0.2024 0.2266 0.2062 0.2033 0.2204 0.2087 0.2126 0.2113 0.207 0.2142 0.2105 0.224 0.2187 0.2068 0.2112 0.2155 0.2195 0.2187 0.2065 0.227 0.2063 0.2028 0.216 0.2138

q˙ INLET

mass fraction. It can be seen that keff decreases with increase in induction heating temperature. This is because the decrease in thermal conductivity of ceramic Li2TiO3 pebbles and steel spheres more than compensates the increase in thermal conductivity of air with increase in

4.3. Effect of induction heating temperature Fig. 17 shows the effect of induction heating temperature on effective thermal conductivity for 5 m3/h inlet gas flowrate and 30 % IHS

14

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Table 6 Estimated Effective Thermal Conductivity data for packed composite bed of 5 mm Li2TiO3 pebbles and Steel spheres under Internal Heat Generation based on FDM model. Sr no.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Average Error

Dp [mm]

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

q˙ INLET

TIH

3 [m/hr]

o [C]

χIHS [%]

FDM based ETC [W/mK]

* εIHP [-]

* εFDM [-]

* εalloc [-]

* εETC [-]

* εTOTAL [-]

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600 150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600 150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600 150 150 150 150 300 300 300 300 450 450 450 450 600 600 600 600

10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50

0.4152 0.5856 0.6222 0.7309 0.3914 0.4789 0.5833 0.673 0.3535 0.441 0.5088 0.6068 0.3244 0.4326 0.4849 0.5701 0.4489 0.5889 0.6567 0.7961 0.3936 0.5269 0.6234 0.7062 0.3597 0.4503 0.5524 0.634 0.3329 0.4329 0.5125 0.6093 0.4596 0.6371 0.6887 0.8468 0.428 0.5683 0.6263 0.7091 0.3939 0.4766 0.6007 0.6632 0.3696 0.4404 0.5402 0.6352 0.4637 0.6534 0.718 0.906 0.4523 0.588 0.6435 0.7371 0.4018 0.5036 0.6058 0.7169 0.3902 0.4565 0.5564 0.652

0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021 0.021

0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874 0.0874

0.0836 0.0704 0.0846 0.0675 0.0721 0.063 0.0747 0.0657 0.0845 0.0639 0.0814 0.0761 0.0696 0.0746 0.0663 0.0673 0.0643 0.0698 0.0808 0.0801 0.066 0.0822 0.0607 0.0722 0.0752 0.0754 0.0815 0.0801 0.0715 0.0694 0.0648 0.0707 0.0775 0.0767 0.0754 0.0666 0.073 0.0624 0.0805 0.0804 0.0845 0.0778 0.0725 0.0718 0.0838 0.083 0.0737 0.0632 0.0615 0.077 0.0611 0.0618 0.0843 0.0762 0.08 0.0713 0.0764 0.0757 0.0673 0.0627 0.0826 0.0744 0.0646 0.0642

0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073 0.073

0.265 0.2518 0.266 0.2489 0.2535 0.2444 0.2561 0.2471 0.2659 0.2453 0.2628 0.2575 0.251 0.256 0.2477 0.2487 0.2457 0.2512 0.2622 0.2615 0.2474 0.2636 0.2421 0.2536 0.2566 0.2568 0.2629 0.2615 0.2529 0.2508 0.2462 0.2521 0.2589 0.2581 0.2568 0.248 0.2544 0.2438 0.2619 0.2618 0.2659 0.2592 0.2539 0.2532 0.2652 0.2644 0.2551 0.2446 0.2429 0.2584 0.2425 0.2432 0.2657 0.2576 0.2614 0.2527 0.2578 0.2571 0.2487 0.2441 0.264 0.2558 0.246 0.2456 0.2544

induction heating temperature. Table 7 shows the effect of temperature on thermal conductivity of lithium titanate pebbles, solid steel spheres and air.

4.4. Effect of internal heat source (IHS) mass fraction Fig. 18 shows the effect of increase in IHS mass fraction on keff for 10 mm pebble diameter and 5 m3/h inlet gas flowrate. The value of keff increases with increase in IHS mass fraction which can be due to the 15

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Table 7 Variation of thermal conductivity of lithium titanate and steel with temperature. Temperature [oC]

Thermal Conductivity of Li2TiO3 [11] [W/mK]

Thermal Conductivity of Solid Steel [40] [W/mK]

Thermal Conductivity of air [23] [W/mK]

30 50 150 300 450 600

3.561 3.528 3.085 2.634 2.506 2.379

39.618 39.272 37.922 37.052 36.182 35.312

0.0258 0.0271 0.0337 0.0429 0.0512 0.0585

Fig. 15. Effect of Pebble diameter on Effective Thermal Conductivity of Packed Bed with Internal Heat Generation.

Fig. 18. Effect of Internal Heat Source Mass Fraction on Effective Thermal Conductivity of Packed Bed with Internal Heat Generation.

Fig. 16. Effect of Inlet Gas Flowrate on Effective Thermal Conductivity of Packed Bed with Internal Heat Generation.

Fig. 19. Evaluation of error associated with effective thermal conductivity values obtained from proposed empirical correlation when compared to values obtained from FDM based model.

increase in heat flux input into the ceramic domain at given induction heating temperature along with reduction in point to point solid contact in the ceramic domain due to reduction in number of lithium titanate pebbles. The steady state heat flux input (hence, output at steady state) per unit volume of ceramic domain increases from 7 kW/m3 to 60 kW/ m3 for 10 mm pebble diameter, 5 m3/h inlet gas flowrate and 150 °C induction heating temperature when the IHS mass fraction increases

Fig. 17. Effect of Induction Heating Temperature on Effective Thermal Conductivity of Packed Bed with Internal Heat Generation.

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1.6 times on increase of IHS mass fraction from 10 % to 50 %. An Internal Heat Source (IHS) allocation scheme was developed and optimized. The induction heating power profile was mathematically modelled and was optimized to give an error of 2.1 %. An empirical correlation was proposed to yield value of effective thermal conductivity after giving the process and operating parameters as input with an accuracy of 75–83 %. Similar experiments using helium instead of air are also underway to get closer to the actual TBM conditions.

from 10 % to 50 % as per the FDM based model. Similar observations were made for different bed operating parameters. Although at steady state, the temperature of ceramic domain increases with increase in heat flux input to the domain when IHS mass fraction increases and hence the effective thermal conductivity should have decreased as seen from section 4.3. Hence, from obtained results, it may be inferred that this decreasing effect is more than compensated by the increase in heat flux input and reduction in point to point solid contact in ceramic domain. This is due to the maintenance of a constant induction heating temperature and the consequent temperature increase in ceramic domain has a minor effect on its effective thermal conductivity. Hence, the heat removal characteristics of ceramic domain improves with increase in IHS mass fraction.

Author contribution We the authors mentioned above have contributed in the preparation of this paper. We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. We understand that the Corresponding Author is the sole contact for the Editorial process (including EVISE and direct communications with the office). He is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author and which has been configured to accept email from [email protected]; dmandal10@ gmail.com

4.5. Effect of bed voidage The random close pack generation algorithm gives model beds with bed voidage of 0.44-0.47. Hence the experimental beds are filled such that the bed voidage lies within this range. In our previous work, we have observed that with increase in bed voidage, the effective thermal conductivity decreases for packed beds under external heating. The same trend can be expected for the internal heat generation case as the thermal conductivity of gas phase is much less as compared to that of solid phase as seen from the first term of proposed correlation in section 4.6. Also wall effect in packed beds significantly increases the bed voidage near the bed walls, hence effective thermal conductivity of ceramic domain of composite bed decreases in the near wall region of packed bed. 4.6. Proposed correlation The data reported in Tables 4–6 was analyzed to obtain an empirical correlation to obtain the value of keff with the knowledge of bed process parameters and operating parameters. Eq. 32 gives an empirical correlation to obtain effective thermal conductivity of packed bed with internal heat generation. The proposed correlation is valid for bed voidage in the range of 0.44-0.47. Fig. 19 compares the values of keff obtained from Eq. 32 with the ones estimated using FDM based model and the error is around 4.1 % which is within acceptable limits. −αg

k eff (TIH )

D = 3.493 ⎜⎛ bed ⎞⎟ k g (Tamb) ⎝ dp ⎠

Acknowledgements ⎜

⎛ kp (TIH ) ⎞ ⎟ ⎝ kp (Tamb ) ⎠

0.48

The authors hereby certified that the article entitled, ‘Estimation of Effective Thermal Conductivity of Packed Bed with Internal Heat Generation’ is submitted to the journal of Fusion Engineering & Design for the interest of its readers. The authors are happy to share their findings for the benefit of researches working in the similar fields.

⎛ ⎜1 ⎝

⎛ 0.35 0.0375 ⎛ kIHS (TIH ) ⎞ Rep + 44.95 ⎜χIHS ⎝ kIHS (Tamb ) ⎠ ⎝ ⎜

Declaration of Competing Interest

The authors are thankful to Shri B.K. Chougule, Shri M C Jadeja, Shri C. A Shinde and Shri S.Sarang for their constant supervision and assistance in the experimental studies.The authors are also thankful to Shri N. Ghuge, Shri S Satre and Shri C. P. Shringi for their expertise on handling of induction heating furnace and for constantly monitoring safety aspects of the experiments.

⎟

−0.16

⎛ k g (TIH ) ⎞ ⎟ ⎝ k g (Tamb ) ⎠

⎜

0.48

⎞⎞ ⎟⎟ ⎠⎠

(32)

From Eq. 32, it can be observed that keff decreases with increase in

( ) viz. with increase in number of solid to solid contact points. Also for given ( ), k decreases with increase in void fraction of bed α . Dbed dp

Dbed dp

eff

Appendix A. Supplementary data

g

Tamb is the ambient temperature and TIH is the induction heating temperature.

Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.fusengdes.2020. 111458.

5. Conclusions

References

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