DETERMINATION OF THE EFFECTIVE THERMAL CONDUCTIVITY OF PACKED BEDS OF MAGNESIUM-MANGANESE-OXIDE By Faezeh Masoomi A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mechanical Engineering – Master of Science 2021 ABSTRACT DETERMINATION OF THE EFFECTIVE THERMAL CONDUCTIVITY OF PACKED BEDS OF MAGNESIUM-MANGANESE-OXIDE By Faezeh Masoomi Magnesium-manganese-oxide redox materials have significant potential for thermochemical en- ergy storage; however, the material high-temperature physical properties, in particular, its thermal conductivity are critical for engineering storage devices. In this thesis, the effective thermal conductivity of packed beds of magnesium-manganese-oxide pellets is measured in the temperature range of 300 °C to 1300 °C with a 100 °C span using the transient hot-probe method. The transient hot-probe method is a well-established technique to determine the effective thermal conductivity of materials by measuring the transient temperature response of a 1-D heat source with constant heat generation. In this thesis, a thin platinum wire covered by a thin alumina sheet is used as both 1-D heat source, and the temperature measurement device. A constant current is passed through the platinum wire emulating 1-D heat generation. Si- multaneously, the electrical resistance of the wire is measured versus time by measuring the voltage across the platinum wire. The wire temperature is calculated based on the wire’s temperature- dependent resistance using a calibration curve established prior to the experiment. The effective thermal conductivity is calculated by curve-fitting a conductivity-dependent model to the wire temperature. The effective thermal conductivity ranges from 0.46 to 1.64 (𝑊𝑚 −1 𝐾 −1 ), and increases signifi- cantly with the temperature. The increase in thermal conductivity with temperature is primarily attributed to thermal radiation. The experimental results are compared to a theoretical dual-porosity model, showing good agreement. ACKNOWLEDGEMENTS First and foremost, I am incredibly grateful to my advisor, Dr. Joerg Petrasch, for his invaluable advice, continuous support, and patience during my master’s study. I would also like to thank Dr. Klausner and Dr. Randhir for their supervision and support. I want to thank all my colleagues for their kind help and support which made my study at Michigan State University a wonderful time. Finally, I would like to express my gratitude to my parents. Without their tremendous understanding and encouragement in the past few years, it would be impossible to complete my study. iii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . xi CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Energy storage Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Thermal conductivity measurement techniques . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Steady state methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Transient Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 CHAPTER 2 MATHEMATICAL MODELING . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Conduction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Radiation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Thermal conductivity of magnesium-manganese-oxide with zero porosity . . . . . 9 2.2.1 Thermal conductivity of magnesium-oxide . . . . . . . . . . . . . . . . . . 10 2.2.2 Thermal conductivity of manganese-oxide . . . . . . . . . . . . . . . . . . 10 2.2.3 Thermal conductivity of magnesium-manganese-oxide . . . . . . . . . . . 10 2.3 Effective thermal conductivity of a single pellet . . . . . . . . . . . . . . . . . . . 12 2.4 Effective thermal conductivity of the packed bed of magnesium-manganese-oxide pellets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 CHAPTER 3 TRANSIENT HOT-PROBE METHOD . . . . . . . . . . . . . . . . . . . . 23 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 CHAPTER 4 EXPERIMENTAL METHODS . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.1 Hot Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.2 Test Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4 Test Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 CHAPTER 5 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . 37 5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.3 Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 CHAPTER 6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 iv BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 v LIST OF TABLES Table 1.1: Existing technologies for energy storage . . . . . . . . . . . . . . . . . . . . . . 2 Table 2.1: Polynomial fitting coefficients to estimate magnesium-oxide thermal conductivity 10 Table 2.2: Exponential fitting coefficients to estimate manganese-oxide thermal conductivity 11 Table 2.3: Estimated values of MgMnO2 ’s thermal conductivity based on a mean geo- metrical model, and its components, magnesium-oxide and manganese-oxide . . 13 Table 2.4: Polynomial fitting coefficients to estimate air thermal conductivity . . . . . . . . 15 Table 2.5: Estimated values of a single pellet effective thermal conductivity and its com- ponents, conduction, and radiation. The radiation component is calculated for 𝑀pellet = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Table 2.6: Radiation component of the packed bed effective thermal conductivity for different extinction coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Table 2.7: The comparison between different conduction models and experimental results of the packed bed effective thermal conductivity . . . . . . . . . . . . . . . . . . 18 Table 2.8: Effective thermal conductivity of the packed bed of magnesium-manganese- oxide pellets and its components, conduction, and radiation. . . . . . . . . . . . 21 Table 4.1: Magnesium-manganese-oxide pellet’s physical properties . . . . . . . . . . . . . 30 Table 4.2: Test setup dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Table 5.1: The electrical resistance of the platinum wire at various temperatures measured in the calibration process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Table 5.2: Calibration curve coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Table 5.3: Experiment 1- The effective thermal conductivity of the packed bed of magnesium- manganese oxide pellets measured in Experiment 1 . . . . . . . . . . . . . . . . 41 Table 5.4: Experiment 2- The effective thermal conductivity of the packed bed of magnesium- manganese oxide pellets measured in Experiment 2 . . . . . . . . . . . . . . . . 41 Table 5.5: Experiment 3- The effective thermal conductivity of the packed bed of magnesium- manganese oxide pellets measured in Experiment 3 . . . . . . . . . . . . . . . . 45 vi Table 5.6: The effective thermal conductivity of the packed bed of magnesium-manganese- oxide pellets for different 𝑡start - Experiment 1 - 𝑇 = 700°C . . . . . . . . . . . . 46 Table 5.7: The effective thermal conductivity of the packed bed of magnesium-manganese- oxide pellets for different 𝑡start - Experiment 1 - 𝑇 = 800°C . . . . . . . . . . . . 47 vii LIST OF FIGURES Figure 2.1: Theoretical modeling of the effective thermal conductivity of the packed bed of magnesium-manganese oxide pellets based on a dual-porosity method: Step 1 - Estimating the magnesium-manganese-oxide thermal conductivity based on its components, magnesium-oxide, and manganese-oxide, Step 2 - Estimating the effective thermal conductivity of a single pellet, step 3 - Esti- mating the effective thermal conductivity of the packed bed of magnesium- manganese-oxide pellet. Green blocks have known values from the literature, and the gray blocks are estimated through modeling. . . . . . . . . . . . . . . . 6 Figure 2.2: Thermal conductivity of magnesium-oxide: A 6th order polynomial is fitted through the empirical data [1], giving the magnesium oxide thermal conduc- tivity as a function of temperature. . . . . . . . . . . . . . . . . . . . . . . . . 11 Figure 2.3: Thermal conductivity of manganese-oxide: An exponential curve is fitted through the empirical data, giving the manganese-oxide thermal conductivity as a function of temperature. No empirical data is found for temperatures over 800 °C, so the thermal conductivity in this region is estimated by extrapolation. [1] 12 Figure 2.4: Estimated thermal conductivity of magnesium-manganese oxide based on a geometrical mean model versus temperature . . . . . . . . . . . . . . . . . . . 14 Figure 2.5: Effective thermal conductivity of a single pellet versus temperature: As shown, the radiation component does not affect the effective thermal conduc- tivity of the pellet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Figure 2.6: The conduction component of the packed bed’s effective thermal conductivity calculated based on different models . . . . . . . . . . . . . . . . . . . . . . . 19 Figure 2.7: Effective thermal conductivity of the packed bed of magnesium-manganese- oxide pellets versus temperature for different extinction coefficients (𝑀bed = 6). 20 Figure 2.8: Effective thermal conductivity of the packed bed of magnesium-manganese- oxide pellet versus temperature, and its components, conduction, and radiation (for 𝑀bed = 6). As shown in the figure, the radiation component grows rapidly with temperature and is the dominant mode in higher temperatures. . . . . . . . 22 Figure 3.1: A schematic of the external medium: the external medium is modeled as an infinite hollow cylinder with constant heat flux at the inner boundary . . . . . . 25 viii Figure 4.1: Magnesium-manganese-oxide pellets: These pellets are made by solid-state mixing of the magnesium- and manganese-oxide followed by a heat-treatment process. The mixed powder is added the mixed powder to an automatic pellet press. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Figure 4.2: A picture of the hot probe: The primary platinum wire with a diameter of 0.5 mm is covered by a thin alumina sheet (OD = 1.7 mm), and used both as the 1-D heat generation source and the temperature measurement device. Two secondary platinum wires with the same purity are welded across the center for voltage measurement. As shown, both primary and secondary wires are extended out of the test container. . . . . . . . . . . . . . . . . . . . . . . . . . 31 Figure 4.3: A schematic of test apparatus: A constant current generated by a DC power supply passed through the primary platinum wire. The voltage across the shunt and the secondary platinum wire is measured to calculate the current and the electrical resistance of the primary wire. A B-type thermocouple is inserted inside the packed bed to measure the bed temperature. Insulation materials are inserted outside of the insulation bricks to prevent heat loss from the ends. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Figure 4.4: Test container in the tube furnace: The thermocouple and voltage measure- ment wires are connected to a LabJack T7 pro. Two compressed air tubes were installed outside the tube furnace to promote the cooling process. . . . . . 33 Figure 4.5: A schematic of the calibration procedure for each measurement. . . . . . . . . . 35 Figure 4.6: A schematic of the experimental procedure . . . . . . . . . . . . . . . . . . . . 36 Figure 5.1: Calibration Process: The experimental data are shown with black dots. A second-order polynomial curve is fitted through the experimental data, dis- played by the dashed line. The experimental resistance data are compared to the calculated values based on the electrical resistivity of platinum from the literature (shown in green dots), offering excellent agreement. . . . . . . . . . . 38 Figure 5.2: A schematic of data analysis steps to calculate the effective thermal conduc- tivity of packed bed of magnesium-manganese-oxide pellets . . . . . . . . . . . 39 Figure 5.3: A sample plot of the platinum’s wire electrical resistance and the calculated wire’s temperature versus time. Experiment 3 - 𝑇 = 300°C. . . . . . . . . . . . 40 Figure 5.4: Temperature rise of the platinum wire versus ln(𝑡). Experiment 3 - 𝑇 = 300°C . 42 Figure 5.5: Temperature rise of the platinum wire versus ln(𝑡). Experiment 3 - 𝑇 = 600°C . 43 Figure 5.6: Temperature rise of the platinum wire versus ln(𝑡). Experiment 3 - 𝑇 = 1000°C 44 ix Figure 5.7: Temperature rise of the platinum wire versus ln(𝑡). Experiment 3 - 𝑇 = 1400°C 44 Figure 5.8: The temperature rise of the platinum wire versus ln(𝑡) - Experiment 1 - 𝑇 = 700°C with an 𝑅 2 of 0.9991, the highest achieved 𝑅 2 in the experiment set. 46 Figure 5.9: The temperature rise of the platinum wire versus ln(𝑡) - Experiment 1 - 𝑇 = 800°C with an 𝑅 2 of 0.9940, one the lowest achieved 𝑅 2 in the experiment set. 47 Figure 5.10: The temperature rise of the platinum wire versus ln 𝑡 - Experiment 1 - 𝑇 = 700°C 48 Figure 5.11: The temperature rise of the platinum wire versus ln 𝑡 - Experiment 1 - 𝑇 = 800°C 48 Figure 5.12: The effect of 𝑡start on calculation of the effective thermal conductivity of the packed bed at 𝑇 = 700 °C and 𝑇 = 800 °C - Experiment 1 . . . . . . . . . . . . 49 Figure 5.13: The effective thermal conductivity of the packed bed of magnesium-manganese- oxide pellets measured by a modified transient hot-probe method: Experi- ments are done in three sets to ensure the repeatability of measurements. The experimental results are compared to the theoretical modeling, and the result shows the best agreement when 𝑀bed = 6. . . . . . . . . . . . . . . . . . . . . 50 x KEY TO SYMBOLS AND ABBREVIATIONS 𝑎𝑤 Coefficient in Woodside and Messmer model 𝑎1 Coefficient in the calibration formula 𝑎2 Coefficient in the calibration formula 𝑎3 Coefficient in the calibration formula 𝑎4 Coefficient in the calibration formula 𝐴 Coefficient in Equation 3.9 𝐴𝑖 Coefficient in the polynomial fit of magnesium-oxide thermal conductivity 𝐵 Coefficient in Equation 3.9 𝐵0 scaled coefficient of B 𝐵𝑖 Coefficient in the polynomial fit of air thermal conductivity 𝐵𝑖 Biot number 𝑐𝑤 Coefficient in Woodside and Messmer model 𝐶 Coefficient in Equation 3.9   J 𝐶𝑝 Specific heat capacity in constant pressure, kg°C 𝑑 The smallest distance between the probe and the boundary, (m) 𝑑pellet Manganese-magnesium-oxide pellet diameter, (m) 𝑑pore Average diameter of pores in the pellet, (𝑚) 𝑑𝑤 Coefficient in Woodside and Messmer model 𝐷 Coefficient in Equation 3.9 𝐹𝑜 Fourier number 𝐼 Electrical current, (A) 𝐽0 Bessel function of the first kind or order zero 𝐽1 Bessel function of the first kind of order one   W 𝑘 Thermal conductivity m°C   W 𝑘 air Air thermal conductivity, m°C   W 𝑘 cond Conduction component of the effective thermal conductivity, m°C xi   W 𝑘 cond−bed Conduction component of the packed bed effective thermal conductivity, m°C   W 𝑘 cond−pellet Conduction component of the pellet effective thermal conductivity, m°C   W 𝑘 eff Effective thermal conductivity of packed bed measured in the experiment, m°C   W 𝑘 eff−model Effective thermal conductivity of packed bed calculated from modeling, m°C   W 𝑘 MgO Thermal conductivity of magnesium-oxide, m°C   W 𝑘 MnO Thermal conductivity of manganese-oxide, m°C   W 𝑘 MgMnO2 Thermal conductivity of magnesium-manganese-oxide without porosity, m°C   W 𝑘 pellet Effective thermal conductivity of a single pellet, m°C   W 𝑘 rad Radiation component of the effective thermal conductivity m°C   W 𝑘 rad−bed Radiation component of the packed bed thermal conductivity m°C   W 𝑘 rad−pellet Radiation component of the pellet thermal conductivity, m°C   W 𝑘 solid Thermal conductivity of solid, m°C 𝐾 Coefficient in Zehner and Schlunder model 𝑚 The linear line slope in Δ𝑇𝑚 versus ln(𝑡) plot 𝑚 probe Mass of the probe (kg) 𝑀 Coefficient in the extinction coefficient MgO Magnesium-oxide MgMnO2 Magnesium-manganese-oxide MnO Manganese-oxide 𝑀bed Extinction coefficient of the packed bed 𝑀pellet Extinction coefficient of the pellet 𝑛 Refractive index 𝑁 Coefficient in Zehner and Schlunder model xii 𝑝 Coefficient in the exponential fit of manganese-oxide 𝑞 Coefficient in the exponential fit of manganese-oxide   W 𝑄 Power of 1-D heater per meter, m 𝑟 Radial direction in cylindrical coordinate system 𝑅2 R-squared ! m2 𝑅𝑐 Contact resistance, W°C 𝑅shunt Electrical resistance of shunt, (Ω) 𝑅wire Electrical resistance of platinum wire, (Ω)   J 𝑆𝑚 Heat capacity of the material, °C   J 𝑆𝑠 Heat capacity of the probe, °C 𝑡 Time, (s) 𝑡0 Initial time, (s) 𝑡max The upper limit on time in the transient hot-probe method suggested by Vos [2], (s) 𝑡min The lower limit on time in the transient hot-probe method suggested by Vos [2]. (s) 𝑡start The lower limit on time based on maximizing 𝑅 2 , (s) 𝑇 Temperature, (°C) 𝑇𝑚 Temperature of the heating probe, (°C) 𝑉shunt The voltage measurement across the shunt, (V) 𝑉wire The voltage measurement across the platinum wire, (V) 𝑌0 Bessel function of the second kind of order zero 𝑌1 Bessel function of the second kind of order one 𝑧 z-direction in cylindrical coordinate ! m 2 𝛼 Thermal diffusivity, s 𝛽 Extinction coefficient 𝛽bed Extinction coefficient of the packed bed xiii 𝛽pellet Extinction coefficient of the pellet 𝛾 Euler’s constant 𝜎 Stefan-Boltzman constant 𝜙 Porosity 𝜙bed Packed bed overall porosity 𝜙pellet Pellet porosity Φ The volume fraction of magnesium-oxide in the magnesium-manganese-oxide mixture Ω Inertia contrast xiv CHAPTER 1 INTRODUCTION 1.1 Energy storage Technologies Renewable energy resources are a subject of interest worldwide due to their advantages over fossil fuel, the primary energy resource in today’s energy market. Renewable energy resources, such as solar, wind, hydro, and biomass, are sustainable and abundant. Unlike fossil fuels, which take millions of years to be formed, renewable resources do not deplete. Renewable energy resources are better for the environment. They have no greenhouse gas emissions, the leading cause of global warming, they also produce less waste, and they do not contaminate the environment. Furthermore, renewable energy resources can bring economic benefits since they have low maintenance costs and can generate electricity locally, lowering the electricity transfer costs, especially for remote areas. Despite these advantages, renewable energies have difficulties finding their potential role in today’s energy market. One of the drawbacks is the unpredictable nature of renewable energy resources like solar and wind. The uncertainty associated with these resources requires higher electrical grid flexibility, causing an increase in electricity prices. Electrical Storage Units (ESU) can be a solution to this problem. They can store energy when renewable energy resource is available and release it to the electrical grid when demand is high. Some of these technologies are introduced in Table 1.1. Table 1.1 summarizes relevant technologies for energy storage. Thermochemical storage is one of the existing technologies for renewable energy storage. It uses reversible chemical reactions to store thermal energy in the form of chemical compounds. In recent years many reactor concepts and materials have been introduced to design a thermochemical unit [3, 4]. Randhir et al. [5] introduced magnesium-manganese-oxide as a promising material to be used in thermochemical storage applications. Magnesium-manganese-oxide is a low-cost, abundant, non-toxic material that uses air as the reacting gas, eliminating the need for a gas storage system. 1 Water is being pumped to a reservoir at a higher level during Pumped storage hydropower off-peak. The energy is stored in the form of potential energy (PSH) and can be released by allowing the water to flow back to the lower reservoir and rotating a turbine during peak time. During peak-off, the air is compressed into the tanks using Compressed air energy stor- a rotary compressor. The compressed air can be released to age (CAES) a gas turbine to generate electricity from high-pressure air during peak time. Batteries use chemical reactions to store and release energy. Batteries Batteries can be used in distribution, off-grid, and short-term storage. It uses a reversible chemical reaction to store energy. The Thermochemical storage reversible chemical reaction can occur in high or low temper- atures based on the nature of the chemical reaction. It uses hydrogen as the energy carrier to store electricity, for Chemical hydrogen storage example, through electrolysis. Mechanical devices that store electricity in the form of rota- Flywheels tional energy. Slowing down these devices can provide bursts of energy to the electrical grid. Energy is stored in the form of electrostatic between two con- Supercapacitors ductive plates. This technology is used for short terms, pro- ducing a short burst of power. In this technology, electricity is stored in a magnetic field Superconducting magnetic with a direct current going into a super-conducting coil. The energy storage (SMES) resistance of superconducting material is almost zero, so the energy can be stored for a long time. Table 1.1: Existing technologies for energy storage This material stores energy by undergoing a reversible chemical reaction at higher temperatures as follows: 𝐶 − 𝐶2 MgMnO2+𝐶 ⇒ MgMnO2+𝐶 + 1 O2 (1.1) 1 2 2 In which 𝐶1 and 𝐶2 denote the excess oxygen in the material. Randhir et al. [5] investigated the reactive stability of magnesium-manganese-oxide at high tem- peratures (1000 < 𝑇 < 1500 °C) for three different molar ratios (Mn/Mg = 2/3, 1/1, and 2/1), and shown than the material’s released and absorbed oxygen remains unchanged over 20 cycles, cycling between 1200 and 1500 °C with an oxygen partial pressure of 0.2 atm. Based on these results, 2 they concluded this material has excellent reactivity stability at higher temperatures. They also measured the energy density of the material in temperatures 1000 to 1500 °C using a combination of acid-solution calorimetry and drop calorimetry. They obtained the volumetric energy density (chemical, phase change, and sensible) of Mn/Mg, for 2/3, 1/1, and 2/1 molar ratios cycles 1596,   MJ 1626, and 1654 , respectively. m3 King et al. [6] investigated different methods to enhance thermochemical energy storage density of magnesium-manganese-oxide. They varied the manganese to magnesium molar ratios, and they found that the maximum energy density is at the 1/1 molar ratio. They also observed lowering the oxygen partial pressure below atmospheric during thermal reduction enhances the thermochemical energy storage density. However, to design a thermochemical storage unit, the magnesium-manganese-oxide’s thermo- physical properties need to be investigated. Randhir et al. [7] determined the oxidation kinetics of porous magnesium-manganese-oxide pellets at high temperatures. The current thesis investigates the effective thermal conductivity of a packed bed of magnesium-manganese oxide pellets. In the following sections, different measurement techniques will be briefly introduced. 1.2 Thermal conductivity measurement techniques Experimental approaches for measuring the effective thermal conductivity fall into two general categories: transient and steady-state [8]. In the following sections these methods will be discussed. 1.2.1 Steady state methods In this category of methods, a heat source in contact with the sample affects a temperature gradient within the material at steady-state condition; the thermal conductivity can be calculated based on the Fourier law. The steady-state methods are analytically simple. However, there are significant drawbacks to these methods. Achieving high accuracies with steady-state methods is challenging and requires rather 3 complex experimental setup due to temperature sensor uncertainties and parasitic heat loss [9]. Also, The long wait time to develop the steady-state condition makes it a less appealing candidate for experiments with multiple measurements. 1.2.2 Transient Methods The transient methods use temperature gradients generated by short-time heating of a 1-D heat source embedded inside the medium to measure the effective thermal conductivity. The transient methods fall into different categories based on various contact forms, for example, transient hot strip, transient hot plane, or transient hot disk. The transient methods are used extensively due to simplicity and the short wait time [8, 10, 11]. These methods have been proven to be the best for investigating the thermal conductivity of liquids and gases. However, it has been popular with solids as well, for example, in applications like geometrical features [12], nuclear pebble beds [13],and porous materials [14]. The commercial hot probe, an instrument for in situ measurements, is designed based on the transient hot-wire method. However, recent works put more limitations on transient probe methods. A study at the University of Plymouth has shown that commercial thermal probe measurements can only   𝑊 be reliable for materials with thermal conductivity above 0.07 [15]. 𝑚°𝐶 Based on the simplified analytical solution of the heat conduction equation, the temperature of a 1-D heat source with constant heat generation must increase linearly with logarithmic time after a short period. The thermal conductivity of the material is calculated based on the slope of this linear function. However, this curve can deviate from the linear shape due to boundary conditions, in- homogeneity, or moisture content. Marmoret et al. [16] demonstrated that estimating the effective thermal conductivity from the slope of the sigmoid curve will result in inaccurate estimations. 4 CHAPTER 2 MATHEMATICAL MODELING 2.1 Introduction Different models have been introduced to estimate the effective thermal conductivity of porous media [17]. These models are introduced based on various parameters, such as porosity, texture, and porous structure. Different modes of heat transfer can contribute to the effective thermal conductivity of porous materials. For high-temperature applications, radiation will play a significant role in the effective thermal conductivity and needs to be considered in the modeling. For our application, the overall effective thermal conductivity is assumed to have two major contributing factors: conduction and radiation. The summation of these two components will make up the total effective thermal conductivity. 𝑘 eff−model = 𝑘 cond + 𝑘 rad (2.1) Multiple models need to be used at different modeling stages to accurately estimate the effective thermal conductivity of the packed bed of magnesium-manganese-oxide pellets. As a well-accepted approximation, a packed bed of pellets can be considered as a 1-D homogeneous porous system [18]. Still, the magnesium-manganese-oxides pellets have an internal porous structure, so their conductivity needs to be calculated prior to the packed bed. Also, since there is no experimental data, the bulk material thermal conductivity of magnesium-manganese-oxide needs to be estimated from its components. The bulk material thermal conductivity can be modeled using a dual-porosity model [19], and a mixing rule as follows: Step 1: Calculating the thermal conductivity of magnesium-manganese-oxide based on its com- ponents, magnesium-oxide and manganese-oxide via a mixing rule. Step 2: Calculating the effective thermal conductivity of a single pellet, considering the pellet porosity and manganese-magnesium-oxide thermal conductivity from step 1, as the small scale. 5 Step 3: Calculating the effective thermal conductivity of the packed bed of magnesium-manganese- oxide pellets, considering the bed porosity and the effective thermal conductivity of a single pellet from step 2, as the large scale. Figure 2.1: Theoretical modeling of the effective thermal conductivity of the packed bed of magnesium-manganese oxide pellets based on a dual-porosity method: Step 1 - Estimating the magnesium-manganese-oxide thermal conductivity based on its components, magnesium-oxide, and manganese-oxide, Step 2 - Estimating the effective thermal conductivity of a single pellet, step 3 - Estimating the effective thermal conductivity of the packed bed of magnesium-manganese-oxide pellet. Green blocks have known values from the literature, and the gray blocks are estimated through modeling. 2.1.1 Conduction Models Conduction is the dominant mode of heat transfer in porous materials at low to moderate tempera- tures. The porous structure can significantly affect the heat transfer due to conduction. Numerous models have been suggested to predict the effective thermal conductivity due to conduction. These models come at different levels of complexity. In general, there is a trade-off between accuracy and 6 simplicity. In the following part, a few of these models will be introduced: 1- Parallel conduction model: The maximum value of the effective thermal conductivity of a two-phase system can be achieved when the solid and fluid phases are in parallel with the direction of heat flow. In this case, the effective thermal conductivity is: 𝑘 cond = 𝜙𝑘 𝑓 + (1 − 𝜙)𝑘 𝑠 (2.2) In which 𝑘 𝑠 is the thermal conductivity of solid, 𝑘 𝑓 is the thermal conductivity of the fluid, and 𝜙 is the porosity. 2- Series conduction model: The minimum value of effective thermal conductivty of a two-phase system can be achieved when the solid and fluid phases are normal to the direction of heat flow. In this case, the effective thermal conductivity can be found as follows: 𝑘𝑠𝑘 𝑓 𝑘 cond = (2.3) 𝜙𝑘 𝑠 + (1 − 𝜙)𝑘 𝑓 3- Geometric mean model: This model considers the random distribution of the solid and fluid phases and estimates the effective thermal conductivity based on the weighted geometric mean of the conductivity of phases as follows: 𝜙 1−𝜙 𝑘 cond = 𝑘 𝑓 𝑘 𝑠 (2.4) 4-Maxwell model: Maxwell used the potential theory to obtain the exact solution for thermal conductivity of randomly distributed and non-interacting homogeneous solid spheres as follows [20]: 2𝜙𝑘 𝑓 + (3 − 2𝜙)𝑘 𝑠 𝑘 cond = 𝑘 𝑓 (2.5) (3 − 𝜙)𝑘 𝑓 + 𝜙𝑘 𝑠 This model is only applicable when the porosity is large. 5- Krupiczka model: Krupiczka derived a general model for granular materials based on the numerical solution of the effective thermal conductivity of a packed bed of long cylinders with porosity of 0.215 and spheres in a cubic lattice with porosity of 0.476. Taking into account the effect of porosity, he approximated these two solutions by the following general correlation: !!   0.28−0.757 log(𝜙)−0.057 log 𝑘 𝑠 𝑘𝑠 𝑘𝑓 𝑘 cond = 𝑘 𝑓 (2.6) 𝑘𝑓 7 This correlation is valid for 0.215 < 𝜙 < 0.476. 6- Zehner and Schlunder model: This model is based on a one-dimensional heat flow through a packed bed of spherical particles. The effective thermal conductivity is estimated as follows: 1/2  (1 − 𝐾)𝑁 ! 2(1 − 𝜙) 1 𝑁 + 1 𝑁 − 1 𝑘 cond = 𝑘 𝑓 1 − (1 − 𝜙) 1/2 + ln ( )− − (2.7) 1 − 𝐾𝑁 (1 − 𝐾 𝑁) 2 𝐾𝑁 2 1 − 𝐾𝑁 in which 𝑘𝑓 𝐾= (2.8) 𝑘𝑠 and 1 − 𝜙 10/9   𝑁 = 1.25 (2.9) 𝜙 7- Woodside and Messmer model: This model is suggested based on a combination of the series and parallel distribution as follows: 𝑎𝑤 𝑘 𝑠 𝑘 𝑓 𝑘𝑠 = + 𝑐𝑤 𝑘 𝑓 (2.10) 𝑘 𝑠 (1 − 𝑑 𝑤 ) + 𝑑𝑘 𝑓 in which, 𝑐 𝑤 = 𝜙 − 0.03, (2.11) 𝑎𝑤 = 1 − 𝑐𝑤 , (2.12) 1−𝜙 𝑑𝑤 = (2.13) 𝑎𝑤 2.1.2 Radiation Model The radiation component of the effective thermal conductivity can be calculated based on the Rosseland diffusion approximation [21, 22], as follows: 16𝑛2 𝜎𝑇 3 𝑘 rad = (2.14) 3𝛽 In which 𝑛 is the refractive index of the medium, 𝜎 is the Stefan-Boltzmann constant, and 𝛽 is the extinction coefficient. Extinction coefficients incorporates the specific radiation characteristics of 8 the porous material in the radiation model. Hsu and Howel [23] proposed the following model to predict the extinction coefficient of ceramic materials: (1 − 𝜙) 𝛽=𝑀 (2.15) 𝑑pore In which 𝑀 is a constant coefficient. They used a guarded hot-plate test apparatus to determine the effective thermal conductivity and radiative properties of partially stabilized zirconia (PS ZrO2 ). The constant 𝑀 was set to 3 by Hsu and Howell. Hendricks and Howell [21] determined absosrption and scattering coefficient as well as scattering phase function of PS ZrO2 and oxide-bonded (OB) SiC porous materials, and noted that the extinxtion correlates betterm when 𝑀 is set to 4.4 for PS ZrO2 , and 4.8 for OB SiC [24]. Zaversky et al. [25] used this correlation with 𝑀 = 4.8 for their investigation of radiative heat transfer in SiC foams as solar absorbers. It can be concluded that 𝑀 is a material-specific parameter, and the exact value needs to be investigated for each material. Determination of the extinct coefficient of the packed bed of magnesium-manganese-oxide pellets is out of the scope of this thesis, so the radiation model is investigated for different extinction coefficients (𝑀 = 4 − 10) to find the best match with our experimental results. 2.2 Thermal conductivity of magnesium-manganese-oxide with zero poros- ity Since there is no empirical data, the thermal conductivity of magnesium-manganese-oxide is estimated based on its components, magnesium-oxide, and manganese-oxide using a mixing rule. In this step, we will consider zero porosity for the mixture of magnesium-oxide and manganese- oxide, and we will refer to it as "MgMnO2 " throughout the thesis. The thermal conductivity of manganese-oxide and magnesium-oxide is estimated as a function of temperature by curve-fitting to the experimental data, which is discussed in the following sections. 9 𝐴0 52.849 𝐴1 -0.1641 𝐴2 2.7056E-4 𝐴3 -2.4997E-7 𝐴4 1.2753E-10 𝐴5 -3.2666E-14 𝐴6 3.2900E-18 Table 2.1: Polynomial fitting coefficients to estimate magnesium-oxide thermal conductivity 2.2.1 Thermal conductivity of magnesium-oxide In Figure 2.2, the empirical data for thermal conductivity of magnesium-oxide [1] are shown with black dots. A 6-th order polynomial is fitted through the experimental data (shown by the dashed line). The thermal conductivity of magnesium-oxide can be calculated as a function of temperature as follows: Õ 6 𝑘 MgO (𝑇) = 𝐴𝑖 𝑇 𝑖 (2.16) 𝑛=0 The coefficient of this polynomial are given in Table 2.4. 2.2.2 Thermal conductivity of manganese-oxide Similar to magnesium-oxide, the empirical data for manganese-oxide is obtained from Reference [1]. Since the empirical data is limited to temperatures below 800 °C, the thermal conductivity is extrapolated for temperatures higher than 800 °C. An exponential curve is fitted through the empirical data to estimate the thermal conductivity of manganese-oxide, as follows: 𝑘 MnO (𝑇) = 𝑝 exp (𝑞) (2.17) Coefficients of this exponential fit are given in Table 2.4. 2.2.3 Thermal conductivity of magnesium-manganese-oxide After calculating the thermal conductivity of magnesium-oxide and manganese-oxide in 100 °C intervals, the thermal conductivity of MgMnO2 is estimated using the geometric mean model, 10 Figure 2.2: Thermal conductivity of magnesium-oxide: A 6th order polynomial is fitted through the empirical data [1], giving the magnesium oxide thermal conductivity as a function of temperature. 𝑝 4.1893 𝑞 -6.734E-4 Table 2.2: Exponential fitting coefficients to estimate manganese-oxide thermal conductivity assuming the random distribution of magnesium-oxide and manganese-oxide powders in the solid mixing process. The geometric model is used as follows: Φ 𝑘 (1−Φ) 𝑘 MgMnO2 = 𝑘 MgO MnO (2.18) In which Φ is the volume fraction of magnesium-oxide in the magnesium-manganese-oxide mix- ture, equal to 0.460117. The calculated magnesium-manganese-oxide thermal conductivity versus temperature is shown in Figure 2.4. In this step, the porosity is assumed to be zero, and is replaced by the volume fraction of com- 11 Figure 2.3: Thermal conductivity of manganese-oxide: An exponential curve is fitted through the empirical data, giving the manganese-oxide thermal conductivity as a function of temperature. No empirical data is found for temperatures over 800 °C, so the thermal conductivity in this region is estimated by extrapolation. [1] ponents. Since there is no void space in the mixed material, the only contributing mode of heat transfer is conduction. 2.3 Effective thermal conductivity of a single pellet After estimating the thermal conductivity of MgMnO2 , the effective thermal conductivity of a single porous pellet can be found as follows: 𝑘 pellet = 𝑘 cond−pellet + 𝑘 rad−pellet (2.19) Considering the random structure of the porous, the geometrical mean model is used for the conduction mode in the pellet as well. 𝜙pellet (1−𝜙pellet ) 𝑘 cond = 𝑘 air 𝑘 MgMnO (2.20) 2 12 𝑇 (°C) 𝑘 MgO 𝑘 MnO 𝑘 MgMnO2 0 52.6885 4.1893 13.4260 50 45.1292 4.0506 12.2776 100 38.7461 3.9165 11.2395 150 33.3789 3.7868 10.3054 200 28.8841 3.6614 9.4683 250 25.1337 3.5402 8.7215 300 22.0144 3.4230 8.0579 350 19.4262 3.3096 7.4703 400 17.2818 3.2001 6.9515 450 15.5051 2.8926 5.7340 500 14.0308 2.9917 6.0905 550 12.8029 2.8926 5.7340 600 11.7743 2.7968 5.4179 650 10.9058 2.7042 5.1360 700 10.1653 2.6147 4.8830 750 9.5271 2.5281 4.6541 800 8.9709 2.4444 4.4455 850 8.4815 2.3635 4.2542 900 8.0479 2.2852 4.0779 950 7.6629 2.2096 3.9152 1000 7.3223 2.1364 3.7650 1050 7.0242 2.0657 3.6272 1100 6.7692 1.9973 3.5018 1150 6.5591 1.9312 3.3891 1200 6.3970 1.8672 3.2900 1250 6.2867 1.8054 3.2050 1300 6.2324 1.7456 3.1347 1350 6.2384 1.6878 3.0796 1400 6.3089 1.6319 3.0398 1450 6.4474 1.5779 3.0150 1500 6.6570 1.5257 3.0045 Table 2.3: Estimated values of MgMnO2 ’s thermal conductivity based on a mean geometrical model, and its components, magnesium-oxide and manganese-oxide 13 Figure 2.4: Estimated thermal conductivity of magnesium-manganese oxide based on a geometrical mean model versus temperature In this equation, the thermal conductivity of MgMnO2 is calculated from the previous step, the porosity of the pellet (𝜙pellet ) is equal to 0.264, and the thermal conductivity of air can be found by the following equation: Õ2 𝑘 air = 𝐵𝑖 𝑇 𝑖 (2.21) 𝑛=0 The radiation component is modeled using Rosseland diffusion approximation, as follows: 16𝑛2 𝜎𝑇 3 𝑘 rad = (2.22) 3𝛽pellet In which: 1 − 𝜙pellet 𝛽pellet = 𝑀pellet (2.23) 𝑑pore−pellet The radiation component is calculated with 𝑀pellet = 4 to maximize the effect of radiation. As shown in Figure 4.5, even with the lowest coefficient, which will result in higher radiation effects, the 14 Figure 2.5: Effective thermal conductivity of a single pellet versus temperature: As shown, the radiation component does not affect the effective thermal conductivity of the pellet. 𝐵0 2.4674E-2 𝐵1 6.8601E-5 𝐵2 1.2242E-8 Table 2.4: Polynomial fitting coefficients to estimate air thermal conductivity radiation component has no role in the pellet’s effective thermal conductivity due to the microscale size of porous (< 0.14%). 2.4 Effective thermal conductivity of the packed bed of magnesium-manganese- oxide pellets After finding the pellet’s effective thermal conductivity, the same procedure is followed to estimate the effective thermal conductivity of the packed bed of magnesium-manganese-oxide pellets. In this case, the solid phase is the pellet, and the porosity of the bed is 0.369. The effective thermal conductivity of the packed bed is estimated considering conduction and radiation 15 𝑇 (°C) 𝑘 MgMnO2 𝑘 air 𝑘 cond−pellet 𝑘 rad−pellet 𝑘 eff−pellet 0 13.4260 0.0247 2.5452 0.0000 2.5452 50 12.2775 0.0281 2.4657 0.0000 2.4657 100 11.2395 0.0314 2.3800 0.0000 2.3801 150 10.3054 0.0347 2.2921 0.0000 2.2921 200 9.4683 0.0379 2.2045 0.0000 2.2045 250 8.7215 0.0411 2.1194 0.0000 2.1195 300 8.0579 0.0442 2.0382 0.0001 2.0382 350 7.4703 0.0472 1.9618 0.0001 1.9619 400 6.9515 0.0502 1.8908 0.0001 1.8909 450 6.4939 0.0531 1.8254 0.0001 1.8255 500 6.0905 0.0559 1.7654 0.0001 1.7656 550 5.7340 0.0587 1.7106 0.0002 1.7108 600 5.4179 0.0614 1.6605 0.0002 1.6607 650 5.1360 0.0641 1.6144 0.0002 1.6147 700 4.8830 0.0667 1.5719 0.0003 1.5722 750 4.6541 0.0692 1.5324 0.0003 1.5328 800 4.4455 0.0717 1.4954 0.0004 1.4958 850 4.2542 0.0741 1.4605 0.0004 1.4609 900 4.0779 0.0765 1.4275 0.0005 1.4280 950 3.9152 0.0788 1.3962 0.0006 1.3967 1000 3.7650 0.0810 1.3666 0.0006 1.3673 1050 3.6272 0.0832 1.3390 0.0007 1.3397 1100 3.5018 0.0853 1.3134 0.0008 1.3142 1150 3.3891 0.0874 1.2902 0.0009 1.2911 1200 3.2900 0.0894 1.2699 0.0010 1.2709 1250 3.2050 0.0913 1.2527 0.0011 1.2538 1300 3.1347 0.0932 1.2390 0.0012 1.2402 1350 3.0796 0.0950 1.2292 0.0013 1.2305 1400 3.0398 0.0967 1.2233 0.0014 1.2248 1450 3.0150 0.0984 1.2216 0.0016 1.2231 1500 3.0045 0.1000 1.2237 0.0017 1.2254 Table 2.5: Estimated values of a single pellet effective thermal conductivity and its components, conduction, and radiation. The radiation component is calculated for 𝑀pellet = 4 16 components, as follows: 𝑘 bed = 𝑘 cond−bed + 𝑘 rad−bed (2.24) The radiation component is calculated as follows: 16𝑛2 𝜎𝑇 3 𝑘 rad−bed = (2.25) 3𝛽bed In which: 1 − 𝜙bed 𝛽bed = 𝑀bed (2.26) 𝑑pellet The radiation component of the packed bed effective thermal conductivity is calculated for different extinction coefficients (𝑀bed = 4 − 10), and the results are shown in Table 2.6. As shown, the radiation component is strongly dependent on the extinction coefficient in higher temperatures. Unlike the pellet, the radiation component of the packed is growing rapidly and becomes prominent in higher temperatures. The conduction component is calculated for different models and the results are shown in Figure 2.6. In order to find the appropriate model for the conduction, the theoretical modeling results are compared with our experimental results, which will be discussed in the following chapters. The comparison is made at 𝑇 = 310°C, the lowest temperature that the measurements are done. The lowest temperature is chosen to minimize the contribution of the radiation component to the packed bed effective thermal conductivity. Since the radiation component has a minor role, the conduction component is the dominant term in the packed bed effective thermal conductivity. The results of different modelings and experimental measurements are shown in Table 2.7. The assumptions of parallel and series models are not realistic for the packed bed structure, and the predicted thermal conductivity is not close to the experimental results. However, these values are calculated to have an upper and lower limit for the packed bed. The geometric model assumptions are applicable to the packed bed. However, this model overpredicts the packed bed’s effective thermal conductivity. The Krupiczka model is based on long cylindrical particles, which is close to the packed bed structure. The porosity of the packed bed is within the suggested range of porosity by the model, and it predicts the most comparative values to our experimental results. Due to 17 𝑇 (°C) 𝑘 rad−bed (𝑀bed = 4) 𝑘 rad−bed (𝑀bed = 6) 𝑘 rad−bed (𝑀bed = 8) 0 0.0092 0.0062 0.0046 50 0.0153 0.0102 0.0077 100 0.0236 0.0158 0.0118 150 0.0345 0.0230 0.0172 200 0.0482 0.0321 0.0241 250 0.0651 0.0434 0.0326 300 0.0857 0.0571 0.0428 350 0.1101 0.0734 0.0550 400 0.1388 0.0925 0.0694 450 0.1721 0.1147 0.0860 500 0.2103 0.1402 0.1051 550 0.2538 0.1692 0.1269 600 0.3029 0.2019 0.1515 650 0.3580 0.2387 0.1790 700 0.4194 0.2796 0.2097 750 0.4874 0.3250 0.2437 800 0.5624 0.3750 0.2812 850 0.6448 0.4299 0.3223 900 0.7348 0.4899 0.3674 950 0.8328 0.5552 0.4164 1000 0.9392 0.6261 0.4696 1050 1.0543 0.7029 0.5271 1100 1.1784 0.7856 0.5892 1150 1.3119 0.8746 0.6559 1200 1.4551 0.9701 0.7275 1250 1.6083 1.0722 0.8042 1300 1.7720 1.1813 0.8860 1350 1.9464 1.2976 0.9732 1400 2.1319 1.4212 1.0659 1450 2.3288 1.5523 1.1644 1500 2.5375 1.6917 1.2687 Table 2.6: Radiation component of the packed bed effective thermal conductivity for different extinction coefficients 𝑇 (°C) Parallel Series Geometric Krupiczka Exp 1 Exp 2 Exp 3 310 5.4835 0.0873 1.3284 0.3150 0.4644 0.4722 0.4879 Table 2.7: The comparison between different conduction models and experimental results of the packed bed effective thermal conductivity 18 Figure 2.6: The conduction component of the packed bed’s effective thermal conductivity calculated based on different models these reasons, this model is chosen for the conduction component, even though it under-predicts the packed bed’s effective thermal conductivity. This can be due to neglecting the radiation components or the assumption that we made to model the thermal conductivity of MgMnO2 . This model is given as follows: !! 𝑘 pellet   0.28−0.757 log (𝜙pellet )−0.057 log 𝑘 pellet 𝑘 air 𝑘 cond−bed = 𝑘 air (2.27) 𝑘 air The effective thermal conductivity of the packed is calculated for different radiation components (𝑀bed = 4 − 10), and the results are shown in Figure 2.7. The rate of increase in the effective thermal conductivity of the packed in our experimental results (Table 5.3, 5.4, and 5.5) best matches the theoretical modeling when 𝑀bed = 6. The radiation component is calculated for 𝑀bed , is shown in Table 2.8, along with the conduction component and the overall effective thermal conductivity of the packed bed. These results are plotted in Figure 2.8, as it shows the conduction component is the dominant mode for temperatures below 900 °C. This component stays relatively constant in 19 Figure 2.7: Effective thermal conductivity of the packed bed of magnesium-manganese-oxide pellets versus temperature for different extinction coefficients (𝑀bed = 6). higher temperatures while the radiation component overgrows, so we can conclude that the rate of increase in the packed bed’s effective thermal conductivity is attributed to the radiation component. 20 𝑇 (°C) 𝑘 pellet 𝑘 air 𝑘 cond−bed 𝑘 rad−bed 𝑘 eff−bed 0 2.5452 0.0247 0.2426 0.0046 0.2472 50 2.4657 0.0281 0.2596 0.0077 0.2672 100 2.3801 0.0314 0.2742 0.0118 0.2860 150 2.2921 0.0347 0.2868 0.0172 0.3040 200 2.2045 0.0379 0.2976 0.0241 0.3217 250 2.1195 0.0411 0.3070 0.0326 0.3396 300 2.0383 0.0442 0.3152 0.0428 0.3580 350 1.9619 0.0472 0.3223 0.0550 0.3774 400 1.8909 0.0502 0.3287 0.0694 0.3980 450 1.8255 0.0531 0.3343 0.0860 0.4203 500 1.7656 0.0559 0.3393 0.1051 0.4445 550 1.7108 0.0587 0.3439 0.1269 0.4708 600 1.6607 0.0614 0.3481 0.1515 0.4996 650 1.6147 0.0641 0.3520 0.1790 0.5310 700 1.5722 0.0667 0.3555 0.2097 0.5652 750 1.5328 0.0692 0.3587 0.2437 0.6024 800 1.4958 0.0717 0.3616 0.2812 0.6428 850 1.4609 0.0741 0.3642 0.3224 0.6866 900 1.4280 0.0765 0.3665 0.3674 0.7339 950 1.3967 0.0788 0.3686 0.4164 0.7850 1000 1.3673 0.0810 0.3704 0.4696 0.8400 1050 1.3397 0.0832 0.3721 0.5271 0.8992 1100 1.3142 0.0853 0.3736 0.5892 0.9628 1150 1.2911 0.0873 0.3752 0.6559 1.0311 1200 1.2709 0.0894 0.3768 0.7275 1.1043 1250 1.2538 0.0913 0.3786 0.8042 1.1828 1300 1.2402 0.0932 0.3182 0.8860 1.2042 1350 1.2305 0.0950 0.3830 0.9732 1.3562 1400 1.2248 0.0967 0.3858 1.0659 1.4517 1450 1.2231 0.0984 0.3890 1.1644 1.5534 1500 1.2254 0.1000 0.3304 1.2687 1.5991 Table 2.8: Effective thermal conductivity of the packed bed of magnesium-manganese-oxide pellets and its components, conduction, and radiation. 21 Figure 2.8: Effective thermal conductivity of the packed bed of magnesium-manganese-oxide pellet versus temperature, and its components, conduction, and radiation (for 𝑀bed = 6). As shown in the figure, the radiation component grows rapidly with temperature and is the dominant mode in higher temperatures. 22 CHAPTER 3 TRANSIENT HOT-PROBE METHOD 3.1 Introduction The transient hot-wire method is one of the most common and well-established methods for measuring thermal conductivity [26]. This method uses the rate of temperature change of a 1-D heat source embedded in the material to determine the effective thermal conductivity. This method has been used widely due to its simplicity and adopted for measuring the effective thermal conductivity of different materials [8]. Van der Held et al. [27, 28] developed an apparatus to determine the thermal conductivity of liquids using transient methods. Blackwell [29] recognized inadequacies in the mathematical analysis of the transient hot-wire method in some applications, as follows: 1- The assumption of line-source of heat, which is not generally valid, e.g., in geophysical applica- tions. 2- Neglecting thermal contact resistance between the hot probe and the external medium. 3- Semi-empirical estimation of the minimum probe length to ensure radial flow conditions. He revised the theory and the experimental method accordingly and proposed a modified transient method to determine the thermal conductivity of insulating materials. He also derived a condition for the minimum probe length to radius ratio to avoid significant errors due to axial heat losses [30]. Vos [2] discussed a few aspects of the transient hot-wire method, for example, the necessary con- ditions to hold the linear relationship between the temperature rise of hot wire and ln(𝑡). He also defined an upper and a lower limit for the linear section. The temperature rise and ln(𝑡) can deviate from a linear due to different reasons. One of the reasons of deviation is the inhomogeneity of the material. In this case, long-duration measurements are necessary to reflect the material’s average thermal conductivity. Jaeger [31] improved the hot-probe technique by measuring the temperature rise at a defined dis- 23 tance from the linear heat source. Vries [32] extended this theory to the case of a homogeneous probe of finite conductivity with a line source embedded in its center. 3.2 Theory The three-dimensional, homogeneous differential equation of heat conduction in the cylindrical coordinate system assuming constant properties is given by: 𝜕 2𝑇 1 𝜕𝑇 1 𝜕 2𝑇 𝜕 2𝑇 1 𝜕𝑇 + + + = (3.1) 𝜕𝑟 2 𝑟 𝜕𝑟 𝑟 2 𝜕𝜙2 𝜕𝑧 2 𝛼 𝜕𝑡 If the temperature is independent of 𝑧 and 𝜙 direction, the differential equation can be simplified to the transient 1-D heat conduction differential equation as follows: 𝜕 2𝑇 1 𝜕𝑇 1 𝜕𝑇 + = (3.2) 𝜕𝑟 2 𝑟 𝜕𝑟 𝛼 𝜕𝑡 This equation can be solved for different boundary and initial conditions. The external medium can be modeled as a 1-D infinite hollow cylinder with an inner diameter of 2𝑅 𝑠 , in which 𝑅 𝑠 is the outer diameter of the hot probe. The temperature of the external medium can be found by solving the transient, 1-D heat conduction equation considering the following boundary conditions. 1- Constant heat flux at 𝑟 = 𝑅 𝑠 (due to the constant heat generation in the hot probe). 2- Infinite hollow cylinder (The external medium’s radius is much larger than the hot probe). The general solution for the temperature distribution is given by [16]: ˆ 𝑡 !  𝑟 2 + 𝑅 2𝑠  𝑄 1 𝑟 𝑅𝑠 𝑇 (𝑟, 𝑡) = exp 𝐼 𝑑𝑡 (3.3) 4𝜋𝑘 𝑡0 𝑡 − 𝑡 0 4𝛼(𝑡 − 𝑡0 ) 0 2𝛼(𝑡 − 𝑡0 ) In which 𝛼 is the thermal diffusivity, and 𝐼0 is the modified Bessel function of zero order. Knowing the heat flux and the transient temperature at any point of the medium, the thermal conductivity can be calculated by solving Equation 3.3. However, this equation has a complex form, and further simplification can be done. Marmoret et al.[16] used the following general solution to the transient, 1-D heat conduction 24 Figure 3.1: A schematic of the external medium: the external medium is modeled as an infinite hollow cylinder with constant heat flux at the inner boundary equation to estimate the thermal conductivity as follows: ˆ 𝑘𝑇 8Ω2 ∞ 1 − exp (−𝐹𝑜.𝑥 2 ) = 𝐺 (𝐹𝑜, Ω, 𝐵𝑖) = 𝑑𝑥 (3.4) 𝑄 𝜋3 0 𝑥 3 Δ(𝑥) In which, " ! #2 " ! #2 𝑥2 𝑥2 Δ(𝑥) = 𝑥𝐽0 (𝑥) − 2Ω − 𝐽 (𝑥) + 𝑥𝑌0 (𝑥) − 2Ω − 𝑌 (𝑥) (3.5) 𝐵𝑖 1 𝐵𝑖 1 In this equation, 𝐽0 and 𝐽1 are the Bessel functions of the first kind of order 0 and 1, 𝑌0 and 𝑌1 are Bessel functions of the second kind of order 0 and 1. This solution is expressed in the form of three non-dimensionalized numbers: Biot number (𝐵𝑖), Fourier number (𝐹𝑜), and Interia contrast (Ω). The contact resistance of the air conductance (𝑅𝑐 ) is considered in the equation as well. These three dimensionless parameters are introduced in the following paragraphs: Biot number (𝐵𝑖): In general, the Biot number represents the ratio between thermal resistance by conduction and thermal resistance by convection at the body’s surface. The Biot number smaller than 0.1 implies that the heat conduction inside the body is much faster than the heat convection outside, so the temperature gradient is negligible inside the body. In the hot probe application, the Biot number can be calculated based on thermal conductivity (𝑘), probe volume (𝑉𝑠 ), probe surface 25 (𝐴𝑠 ), and probe length (𝐼 𝑠 ), as follows: 𝑉𝑠 /𝐴𝑠 𝑘 𝐴𝑠 𝑅𝑠 𝐵𝑖 = = (3.6) 1 𝑘.𝑅𝑐 𝐻 𝐴𝑠 In which 𝐻 is the conductance, the inverse of the contact resistance (𝑅𝑐 ). Fourier number (𝐹𝑜): The Fourier number determines the ratio of the heat conducted through the medium and the heat stored in the material and is defined as follows: 𝛼.𝑡 𝐹𝑜 = (3.7) 𝑅 2𝑠 Inertia contrast (Ω): Inertia contrast represents the ratio between the heat capacity of the material (𝑆 𝑚 ) and the heat capacity of the probe (𝑆 𝑠 ), which is equal to: 𝑆𝑠 𝜋𝑅 2𝑠 𝜌 𝑠 𝐶𝑠 Ω= = (3.8) 𝑆𝑚 𝑆𝑚 Knowing the temperature profile at the external surface of the hot probe, the three non-dimensionalized numbers, 𝐵𝑖, 𝐹𝑜, and Ω, can be estimated by taking the inverse of the integral using optimization methods. Subsequently, the contact resistance (𝑅𝑐 ), thermal conductivity (𝑘), and thermal diffu- sivity (𝛼) can be calculated. Marmoret et al. [16] demonstrated that this method accurately predicts the properties of glass wool. However, the complex form of the general solution is a drawback for this analysis method. Blackman [29] derived the approximated solution of the hot probe temperature at small- and large- time. The large-time solution gives the temperature of the probe as a function of time, as follows: 1 Δ𝑇𝑚 (𝑡) ≈ 𝐴 ln (𝑡) + 𝐵 + {𝐶 ln (𝑡) + 𝐷} (3.9) 𝑡 26 In which Δ𝑇𝑚 is the temperature rise of the probe, 𝑡 is time, and 𝐴, 𝐵, 𝐶, and 𝐷 are constant coefficients, which can be derived from the following equations: 𝑄 𝐴= (3.10) 4𝜋𝑘 ! 4𝛼 2𝑘 𝑅𝑐 𝐵 = ln −𝛾+ (3.11) 2 𝑅𝑠 𝑅𝑠 !" !# 𝑟2 𝑚 probe 𝑐 𝑝 𝛼 𝐶= 1− (3.12) 2𝛼 𝜋𝑅 2𝑠 𝑘      4𝛼 2𝑘  ! ! ln − 𝛾 + 𝑚 𝑐 𝛼 probe 𝑝   𝑟 2  4𝛼 𝑟2 𝑅𝑠 𝐻 𝐷= ln 2 − 𝛾 + 1 − (3.13)  2  2𝛼  𝑅 𝑠 𝜋𝑅 𝑠 𝐿𝑘        In which 𝑄 is the probe power per length, 𝑘 is the effective thermal conductivity, 𝛼 is the thermal diffusivity, 𝑅 𝑠 is the probe radius, 𝛾 is Euler’s constant equal to 0.5772, 𝑚 probe is the mass of the probe, and 𝐶 𝑝 is the specific heat capacity of the probe. In large-time, terms related to 𝐶 and 𝐷 become insignificant, so Equation 3.9 can be simplified as follows: 𝑄 Δ𝑇𝑚 = ln (𝑡) + 𝐵0 (3.14) 4𝜋𝑘 In which, 𝑄 𝐵0 = 𝐵 (3.15) 4𝜋𝑘 So the thermal conductivity can be derived as follows:   𝑄 𝑡max 𝑘= ln (3.16) 4𝜋 (𝑇 (𝑡max ) − 𝑇 (𝑡min )) 𝑡min As mentioned before, Blackman’s analysis is based on the long-time assumption, so a lower limit on time needs to be considered for this analysis. This lower limit (𝑡min ) is associated with heating the 1-D hot probe evenly and heat flux propagation in the external medium. In materials with a non-homogeneous structure, the minimum time to reach the equilibrium state is higher than homogeneous materials, to neglect the spatial effect of the external medium. An upper limit on time (𝑡max ) is needed as well. Since the radius of the external medium is finite 27 (even though it is much larger than the hot probe radius), the heat flux will eventually reach the outer boundary and starts leaking out. Since the hot probe method is developed based on the heat conduction of an infinite cylinder, this equation does not hold up when the heat reaches the boundary. Vos [2] suggested the following lower and upper limits for the time: 50𝑅 2𝑠 𝑡min = 4𝛼 0.6𝑑 𝑡 max = 4𝛼 However, the ASTM standard [33] suggests considering the most linear part of the (Δ𝑇𝑚 − ln 𝑡) plot to calculate the thermal conductivity whenever the curve deviates from linearity. 28 CHAPTER 4 EXPERIMENTAL METHODS 4.1 Material Figure 4.5 shows a picture of magnesium-manganese-oxide pelletst. The magnesium-manganese- oxide pellets are synthesized by a solid-state mixing method followed by heat treatment. First, the magnesium-oxide powder is mixed with manganese-oxide using a cement mixer. Cylindrical zir- conia milling media is added to assist in the mixing process. Mixing takes about 3 to 4 hours. The mixed powder is pelletized using an automatic pellet press (TDP 5 Desktop Tablet Press from LFA machines). When the mixing process is completed , the pressed pellets are heat treated in a box furnace. Heat treatment is performed at 1500 °C for 24 hours followed by 4 hours at 1000 °C. The physical properties of magnesium-manganese-oxides are listed in Table 4.1. Figure 4.1: Magnesium-manganese-oxide pellets: These pellets are made by solid-state mixing of the magnesium- and manganese-oxide followed by a heat-treatment process. The mixed powder is added the mixed powder to an automatic pellet press. 29 Diameter (mm) 3.7 Height (mm) 3 porosity 0.264 Porous diameter (mm) 0.0003 Table 4.1: Magnesium-manganese-oxide pellet’s physical properties 4.2 Experimental Setup The experimental setup and hot probe are designed based on ASTM C1113 standard [34]. In a commercial heating probe, the temperature measurement device is packed with the 1-D heat source. But in this thesis, we used the platinum wire both as the heat generation source and the temperature measurement device as it is suggested ASTM C1113 [34]. In the following section, the design of the hot probe is explained. 4.2.1 Hot Probe A 0.5 mm diameter platinum wire with 99.95% purity (SigmaAldrich), covered with a thin alumina sheet, is used both as the hot probe and the temperature measurement device. Heat is generated by passing a constant current through the platinum wire, and the platinum wire’s temperature is calcu- lated by measuring the wire’s electrical resistance. The thin alumina sheet insulates the platinum’s wire from to packed bed to prevent the electrical current leakage since magnesium-manganese- oxide pellets are electrically conductive at higher temperatures. Platinum is chosen as the wire material because of its advantageous properties. First, the platinum’s melting temperature is very high (1768 °C), making it a perfect candidate for high-temperature experiments; second, platinum is chemically stable and does not oxidize in higher temperatures; and third, The electrical resistance of platinum is known to have high accuracy and can be represented as a second-order polynomial of temperature. So, by knowing the wire’s electrical resistance, the temperature can be calculated with high accuracy. Two secondary platinum wires with the same purity are welded on both sides of the platinum wire’s 30 center with a distance of 7.5 cm for voltage measurements. It is crucial to have the same material for the secondary wires as the primary wire to avoid measurement errors due to the Seebeck effect. The wire resistance can be calculated by measuring the voltage across the secondary wires and dividing by the current passed through the primary wire. Even though this small portion is used for the measurements, both primary and secondary wires are extended out of the furnace. The primary platinum wire (the heat generation source) is extended to prevent melting and oxidation, and the secondary wires are continued to avoid the Seebeck effect at elevated temperatures. Figure 4.2: A picture of the hot probe: The primary platinum wire with a diameter of 0.5 mm is covered by a thin alumina sheet (OD = 1.7 mm), and used both as the 1-D heat generation source and the temperature measurement device. Two secondary platinum wires with the same purity are welded across the center for voltage measurement. As shown, both primary and secondary wires are extended out of the test container. 4.2.2 Test Apparatus The experimental setup is designed based on ASTM C1113 standard [34]. Figure 4.3 shows a schematic of the test apparatus. The bed space is made by placing two low thermal conductive bricks with a distance of 15 cm. A small hole is placed at the center of one of the bricks to insert the hot probe. On the other brick, a sector with an angle of 45° is removed to pack the pellets. After placing the hot probe at the center of the bricks, alumina paste is applied to secure the hot probe’s location. After drying the paste, the hot probe and the insulation bricks are slid into the test container (An alumina tube 31 Figure 4.3: A schematic of test apparatus: A constant current generated by a DC power supply passed through the primary platinum wire. The voltage across the shunt and the secondary platinum wire is measured to calculate the current and the electrical resistance of the primary wire. A B-type thermocouple is inserted inside the packed bed to measure the bed temperature. Insulation materials are inserted outside of the insulation bricks to prevent heat loss from the ends. with an inner diameter of 2 inches). The insulation material is packed outside of the insulation brick without the empty sector to secure the bricks’ location. Then the packing of the pellets is achieved by pouring and shaking. A B-type thermocouple is inserted inside the packed bed during the packing process. After packing the bed, the insulation material is placed outside the second insulation brick to prevent heat loss from the container’s end. Finally, the test container placed in a tube furnace (Sentrotech, Model: STT-1600 °C-12) [35], to perform the calibration and experiment process. The physical dimensions of the test setup are shown in Table 4.2. After placing the test container in the tube furnace, the primary platinum wire is connected to 32 Bed length (cm) 15 Container ID (in) 2 container length (cm) 46 platinum wire length (cm) 7.5 platinum wire diameter (mm) 0.5 Probe OD (mm) 1.7 Bed porosity 0.369 Table 4.2: Test setup dimension Figure 4.4: Test container in the tube furnace: The thermocouple and voltage measurement wires are connected to a LabJack T7 pro. Two compressed air tubes were installed outside the tube furnace to promote the cooling process. a DC power supply, with a shunt placed in series downstream of the platinum wire. The voltage across the shunt and the platinum wire is measured by the data acquisition system. The current in the circuit is calculated by measuring the voltage across the shunt, as follows: 𝑉 𝐼 = shunt (4.1) 𝑅shunt 33 And respectively, the resistance of the platinum wire is calculated as follows: 𝑉 𝑅wire = wire (4.2) 𝐼 Kipling [36] is used as the data acquisition software. The LabJack T7 pro’s precision is limited by the recording frequency. For our experiment, the precision is far more important than frequency, so the precision is set to the highest degree, and subsequently, the recording frequency is set to 2 Hz. This frequency is fast enough for our application. 4.3 Calibration Calibration is conducted per C1113 standard [34]. The purpose of calibration is to find the electrical resistance of the platinum wire as a function of temperature by fitting a second-order polynomial to the experimental electrical resistance measurements. Calibration measurements are performed in 100 °C intervals, starting from 300 °C and ending at 1300 °C. The following procedure is followed for each measurement: The furnace is ramped up from the initial temperature with a rate of 10 °C per minute. Once the desired temperature is reached, the furnace is dwelled for 4 hours to stabilize temperature throughout the packed bed. After stabilizing the temperature, a 0.05𝐴 current is passed through the platinum wire. This current is low enough to ensure the wire’s temperature is not rising significantly with respect to the packed bed. The voltage across the wire and the calibration shunt (100 Ω) is recorded for 30 s. The wire’s electrical resistance is calculated and plotted versus the packed bed’s temperature, measured by the B-type thermocouple. Once the wire resistance is calculated, the setup is ready to move to the subsequent measurement. After conducting the calibration procedure for all temperatures, a second-order polynomial is fitted through the experimental measurements to obtain the calibration curve. 34 Figure 4.5: A schematic of the calibration procedure for each measurement. 4.4 Test Method Similar to the calibration procedure, the furnace is ramped up from the initial temperature to reach the desired temperature with a rate of 10 °C per minute. Once the desired temperature is reached, the furnace is dwelled for 4 hours to stabilize the temperature of the test section. After keeping the furnace at a constant temperature for four hours, a constant current (3.8 to 4.2 𝐴), is passed through the platinum wire for 300 seconds. Unlike the calibration, this current is high enough to raise the platinum wire’s temperature with respect to the packed bed. During this period, the bed’s temperature, the voltage across the shunt, and the voltage across the platinum wire are recorded, and the electrical resistance of the platinum wire is calculated based on the voltage measurements. This procedure is repeated from 300 to 1300 °C in 100 °C intervals for three experiments. An additional measurement at 1400 °C is done in experiment 3 as the final measurement. 35 Figure 4.6: A schematic of the experimental procedure 36 CHAPTER 5 RESULTS AND DISCUSSION 5.1 Calibration The platinum wire’s electrical resistance at different temperatures is shown in Table 5.1. A second-order polynomial curve is fitted through the experimental data. The inverse of this curve predicts the platinum wire’s temperature as a function of electrical resistance as follows: p 𝑇𝑚 = 𝑎 1 + 𝑎 2 𝑎 3 + 𝑎 4 𝑅 (5.1) The values of 𝑎 1 to 𝑎 4 are shown in Table 5.2. The polynomial fitting has an 𝑅 2 of 0.999978, showing the high accuracy of the calibration process. The measured electrical resistance values are compared to the calculated resistance based on the resistivity data from the literature [37], showing excellent agreement (Figure 5.1). The calibration process was repeated three times at different stages of the experiment, resulting in similar calibration curves. 𝑇 (°𝐶) 𝑅(Ω) 314.26 0.082391 411.69 0.095537 510.28 0.108342 608.07 0.120451 704.28 0.132164 801.65 0.143497 900.21 0.154493 1000.92 0.165161 1095.94 0.175056 1191.87 0.184905 1288.40 0.193665 Table 5.1: The electrical resistance of the platinum wire at various temperatures measured in the calibration process 37 𝑎1 3353.6170 𝑎2 -8.8508E-12 𝑎3 1.6495E+29 𝑎4 -5.7281E+29 Table 5.2: Calibration curve coefficients Figure 5.1: Calibration Process: The experimental data are shown with black dots. A second-order polynomial curve is fitted through the experimental data, displayed by the dashed line. The experimental resistance data are compared to the calculated values based on the electrical resistivity of platinum from the literature (shown in green dots), offering excellent agreement. 5.2 Experimental results Once the data for each measurement is collected, the following steps are taken for data analysis. First, the wire’s temperature is calculated based on the wire’s electrical resistance using the inverse of the calibration curve. A sample figure of this procedure is shown in Figure 5.4. After calculating 38 Figure 5.2: A schematic of data analysis steps to calculate the effective thermal conductivity of packed bed of magnesium-manganese-oxide pellets the wire’s temperature, the temperature rise can be derived by subtracting the temperature at 𝑡 = 0 from wire’s temperature. Then temperature rise of the platinum wire is plotted versus ln(𝑡), and a straight line is fitted through the most linear segment of the curve. This segment was chosen by maximizing 𝑅 2 as suggested by ASTM standard [33]. Samples of these plots are shown in Figures 5.4 - 5.7. After fitting the straight line, the effective thermal conductivity of the packed bed is calculated as follows: 𝑄 𝑘 eff = (5.2) 4𝜋𝑚 In which 𝑚 is the slope of the linear curve, and 𝑄 is the power of the platinum wire per length, which can be calculated as follows: 𝑉 𝐼 𝑄 = wire (5.3) 𝑙 wire 39 And 𝑙 wire is the length of the platinum wire. This procedure is repeated for all of the measurements in three experiments. The results are shown Figure 5.3: A sample plot of the platinum’s wire electrical resistance and the calculated wire’s temperature versus time. Experiment 3 - 𝑇 = 300°C. in Tables 5.3 - 5.5. The cells with green color have an 𝑅 2 higher than 0.9998, in which the noise level is low and have the highest level of accuracy. Two measurement in experiment 1 have an 𝑅 2 lower than 0.999, which are ruled out due to the relatively high level of noise. As shown in the Table 5.5, Experiment 3 has the highest 𝑅 2 values and 𝑅 2 and 𝑡start tend to decrease with increasing temperature. The lower 𝑡start in higher temperatures is consistent with the lower limit introduced by Vos [2]. As the effective thermal conductivity increases, less time is need for the material to reach thermochemical equilibrium. 5.3 Uncertainty Analysis In this section, the effect of 𝑡start on the effective thermal conductivity of the packed bed is discussed. As mentioned before, for all of the measurements, 𝑡start is chosen by maximizing 𝑅 2 , as suggested by ASTM standard [34], rather than the lower limit offered by Vos [2]. This is due to the 40 𝑇 (°C) 𝑘 eff 𝑡 start 𝑅2 312.5 0.4644 81.5 0.9990 411.6 0.5296 55.0 0.9991 509.8 0.6097 55.0 0.9965 603.8 0.7240 55.0 0.9986 704.1 0.8152 55.0 0.9991 800.1 0.8470 20.5 0.9942 898.9 0.9534 45.0 0.9974 999.0 1.0021 16.5 0.9885 1092.7 1.2033 40.5 0.9974 1190.0 1.3410 25.0 0.9961 1287.5 1.3970 33.5 0.9814 Table 5.3: Experiment 1- The effective thermal conductivity of the packed bed of magnesium-manganese oxide pellets measured in Experiment 1 𝑇 (°C) 𝑘 eff 𝑡 start 𝑅2 312.6 0.4722 73.7 0.9986 409.8 0.5536 66.8 0.9987 509.8 0.6403 60.4 0.9986 606.7 0.7483 54.6 0.9985 704.1 0.8152 33.6 0.9974 800.8 0.9415 60.5 0.9998 898.9 0.9995 54.8 0.9996 999.7 1.1113 45.0 0.9986 1097.3 1.2726 40.0 0.9995 1190.0 1.3911 33.5 0.9992 1295.2 1.4543 15.02 0.9967 Table 5.4: Experiment 2- The effective thermal conductivity of the packed bed of magnesium-manganese oxide pellets measured in Experiment 2 41 Figure 5.4: Temperature rise of the platinum wire versus ln(𝑡). Experiment 3 - 𝑇 = 300°C non-homogeneous structure of the packed bed; as non-homogeneous materials need a longer time to reach the equilibrium state. Two measurements from experiment one are chosen to assess the effect of 𝑡start on the effective thermal conductivity of the packed. The first measurement is 𝑇 = 700°C, with a maximum 𝑅 2 of 0.9991, the highest achieved 𝑅 2 in the set, and the second one is 𝑇 = 800°C with an 𝑅 2 of 0.991, showing relatively high levels of noise in the measurement. The plots for these measurements are shown in Figures 5.8 - 5.11. The effective thermal conductivity is calculated by choosing different ln (𝑡start ) for these two measurements, and results are shown in Tables 5.6, and 5.7. For a better comparison, the results are plotted in Figure 5.12. The red dot indicates the highest 𝑅 2 achieved in the measurement. As shown in Figure 5.12, the effective thermal conductivity increases rapidly with increasing 𝑡 start before the red dot. The effective thermal conductivity still continues to increase after the red dot, but with a significantly lower rate. The error between the highest calculated thermal conductivity 42 Figure 5.5: Temperature rise of the platinum wire versus ln(𝑡). Experiment 3 - 𝑇 = 600°C and the thermal conductivity with the highest 𝑅 2 is three percent for 𝑇 = 700 °C, and seven percent for 𝑇 = 800 °C. The seven percent is chosen as the uncertainty level for all measurements. Although the effective thermal conductivity converged for both cases, the noisier measurement achieved the highest 𝑅 2 at a lower 𝑡start compared to the previous and subsequent temperature 5.3. Choosing the lower 𝑡start results in underpredicting the effective thermal conductivity. So, it can be concluded that the noise level of measurement is crucial for accurate results. Due to this fact, the results of two measurements with 𝑅 2 s below 0.99, are discarded. The results of three experiments, beside the mathematical modeling with 𝑀bed = 6 is shown in Figure. 43 Figure 5.6: Temperature rise of the platinum wire versus ln(𝑡). Experiment 3 - 𝑇 = 1000°C Figure 5.7: Temperature rise of the platinum wire versus ln(𝑡). Experiment 3 - 𝑇 = 1400°C 44 𝑇 (°C) 𝑘 eff 𝑡 start 𝑅2 310.0 0.4879 122 0.9997 410.3 0.5796 90.1 0.9996 508.7 0.6403 81.5 0.9996 606.6 0.7589 67.0 0.9994 704.3 0.8707 60.5 0.9995 799.7 0.9534 49.5 0.9993 898.7 0.9988 33.5 0.9993 999.6 1.1302 30.0 0.9988 1097.6 1.1934 25.0 0.9995 1195.9 1.3663 22.5 0.9988 1294.6 1.5172 18.5 0.9978 1394.0 1.6430 11.5 0.9970 Table 5.5: Experiment 3- The effective thermal conductivity of the packed bed of magnesium-manganese oxide pellets measured in Experiment 3 45 ln (𝑡start ) 𝑡start (𝑠) 𝑘 eff 𝑅2 0.6935 2.0 0.6394 0.9608 1.0986 3.0 0.6556 0.9657 1.5041 4.5 0.6769 0.9729 2.0149 7.5 0.7098 0.9836 2.5257 12.5 0.7447 0.9922 3.0204 20.5 0.7747 0.9966 3.5116 33.5 0.7996 0.9986 4.0073 55.0 0.8188 0.9991 4.5034 90.5 0.8307 0.9986 5.0006 148.5 0.8406 0.9962 5.5013 245.0 0.8461 0.9550 Table 5.6: The effective thermal conductivity of the packed bed of magnesium-manganese-oxide pellets for different 𝑡start - Experiment 1 - 𝑇 = 700°C Figure 5.8: The temperature rise of the platinum wire versus ln(𝑡) - Experiment 1 - 𝑇 = 700°C with an 𝑅 2 of 0.9991, the highest achieved 𝑅 2 in the experiment set. 46 Figure 5.9: The temperature rise of the platinum wire versus ln(𝑡) - Experiment 1 - 𝑇 = 800°C with an 𝑅 2 of 0.9940, one the lowest achieved 𝑅 2 in the experiment set. ln (𝑡start ) 𝑡start (𝑠) 𝑘 eff 𝑅2 0.6932 2.0 0.6974 0.9605 1.0986 3.0 0.7163 0.9659 1.5041 4.5 0.7410 0.9737 2.0149 7.5 0.7777 0.9842 2.5257 12.5 0.8135 0.9907 3.0204 20.5 0.8470 0.9942 3.5116 33.5 0.8691 0.9938 4.0073 55.0 0.8813 0.9900 4.5034 90.5 0.8952 0.9805 5.0006 148.5 0.9061 0.9492 5.5013 245.0 0.8733 0.4359 Table 5.7: The effective thermal conductivity of the packed bed of magnesium-manganese-oxide pellets for different 𝑡start - Experiment 1 - 𝑇 = 800°C 47 Figure 5.10: The temperature rise of the platinum wire versus ln 𝑡 - Experiment 1 - 𝑇 = 700°C Figure 5.11: The temperature rise of the platinum wire versus ln 𝑡 - Experiment 1 - 𝑇 = 800°C 48 Figure 5.12: The effect of 𝑡start on calculation of the effective thermal conductivity of the packed bed at 𝑇 = 700 °C and 𝑇 = 800 °C - Experiment 1 49 Figure 5.13: The effective thermal conductivity of the packed bed of magnesium-manganese-oxide pellets measured by a modified transient hot-probe method: Experiments are done in three sets to ensure the repeatability of measurements. The experimental results are compared to the theoretical modeling, and the result shows the best agreement when 𝑀bed = 6. 50 CHAPTER 6 CONCLUSION In this thesis, a modified transient hot-wire method is used to determine the effective thermal conductivity of the packed bed of magnesium-manganese-oxide pellets at various temperatures (300-1400 °C with 100 °C intervals). The modified transient hot-wire is a simple and effective method to measure effective thermal conductivity at higher temperatures. This method predicts the thermal conductivity based on the linear segment of the temperature rise versus ln(𝑡). The experimental results show the effective thermal conductivity of the packed bed is strongly depen-   𝑊 dent on the temperature, increasing from 0.45 to 1.64 in temperatures 300 to 1400 °C. 𝑚°𝐶 This increase in the effective thermal conductivity is primarily attributed to radiation. Experiments are conducted in three sets to ensure repeatability, and the results of the three experiments were consistent with each other. Even though this method is mathematically simple, it is highly sensitive to the data noise level. Experiments with a higher level of noise (𝑅 2 < 0.998) can result in underpredicting the effective thermal conductivity, so having the highest level of accuracy in measurements is crucial for this technique. The uncertainty is calculated by choosing different portions of the Δ𝑇 − ln(𝑡) curve to estimate the effective thermal conductivity. The experimental results are compared to a dual-porosity model, estimating the effective thermal conductivity at small-scale (magnesium-manganese-oxide pellet) and large-scale (packed bed). The theoretical model consists of two terms, conduction and radiation. Krupiczka model is used for the conduction component of the packed bed. The radiation component is estimated based on the Rosseland diffusion approximation, using different extinction coefficients. The rate of increase in the effective thermal conductivity of the packed bed is best matched when 𝑀bed = 6, which is a coefficient in the extinction coefficient offered by Hsu and Howel [23]. The dual-porosity model predicts lower values of the packed bed’s effective thermal conductivity compared to exper- imental results. The underprediction can be due to several model assumptions. For example, the 51 thermal conductivity of magnesium-manganese-oxide is estimated by a mixing model, while we know magnesium-oxide and manganese-oxide chemically bond and form new compounds, such as perovskites, during the solid mixing process. Furthermore, the thermal conductivity of manganese- oxide has to be extrapolated for temperatures above 800 °C due to the lack of experimental data, which may introduce errors for higher temperatures. 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