Cellular Ceramics / 4

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4.3

Thermal Properties

Thomas Fend, Dimosthenis Trimis, Robert Pitz-Paal, Bernhard Hoffschmidt, and Oliver Reutter

4.3.1

Introduction

Many cellular ceramic materials are used in high-temperature applications such as insulating materials, heat exchangers, solar receivers, and porous burners. Consequently, knowledge of their thermal properties is essential for the design engineer. Some quantities, such as the heat capacity, may be easily derived from the solid and gaseous components of the cellular ceramic; for others, such as the thermal conductivity, this derivation is complex and temperature-dependent. In addition, similar to the mechanical properties, the thermal properties strongly depend on the chemical composition and the manufacturing process. This emphasizes the need for measurements. As the literature provides little data on the thermal properties of cellular ceramics, this chapter describes the most important measurement techniques for heat conduction and heat transfer and provides some data. Knowledge of these thermal properties is also essential when solving heat-transfer and flow problems numerically with heterogeneous models.

4.3.2

Thermal Conductivity

Fourier’s law describes heat transport in a medium and includes a definition of the thermal conductivity k [W m–1 K–1]. According to this law, heat transport is proportional to thermal conductivity and temperature gradient:

where q [W m–2] denotes the heat flux density (heat per unit time and unit area), and T [K] the temperature.

In one dimension this equation has the form:

which can be integrated for a homogeneous one-dimensional case:

Cellular Ceramics: Structure, Manufacturing, Properties and Applications.

Michael Scheffler, Paolo Colombo (Eds.)

Copyright 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-31320-6

_

where one can imagine Q [W] as the heat per unit time transferred through a rod of length l [m] and a cross sectional area A [m2] linking two reservoirs at constant temperatures T1 and T2.

In this stationary case at each point of the rod the temperature is constant. For the transient case the heat flux in media is described by the following equation:

defining – because of its similarity to the equation of diffusion – the quantity thermal diffusivity k [m2/s]

These definitions are made for condensed or gaseous media, in which heat transport can be imagined as coupled vibrations of lattices or collisions of molecules or electrons. Cellular ceramics generally consist of a solid framework which is filled with a fluid. The fluid may flow through the solid framework in the case of open porosity. Consequently, thermal conductivity in a cellular solid is not a homogenous material property. It is a combination of several different mechanisms taking place at the pore-size level. Besides heat conduction through the solid walls or struts, there may be convective heat transfer in open or closed cells, heat conduction in the fluid, and radiative heat transfer. If all mechanisms are regarded separately, calculations of heat-transfer problems become quite complicated. Consequently, an “effective thermal conductivity” keff has been introduced to simplify the calculations. The cellular material is considered as a “black box” and “local thermal equilibrium” is assumed. Then, the solution of the equations mentioned is still possible:

An enormous number of studies have been published on the theoretical prediction of the effective thermal conductivity of cellular materials. Good reviews of these studies are given by Kaviany [1] and Hsu [2] for porous media in general. Tsotsas and Martin [3] reviewed theoretical approaches and experimental methods to determine the effective thermal conductivity of packed beds. Taylor [4], Mottram and Taylor [5], and Hale [6] give very general and comprehensive reviews on the physical properties of composites which can also be applied to cellular ceramics. Collishaw [7] gives a more specific review on thermal conductivity of cellular materials. Ceramic foams have not been in the focus of these approaches; therefore, experimental

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data on the thermal conductivity of these materials are rare. Some experimental investigations have been performed by the authors and are reported below.

To roughly quantify the contributions of the single mechanisms, one has to consider that the thermal conductivity of ceramic materials is high compared to air, the most frequently used fluid. Thermal conductivity data of several (dense) ceramic materials and for air are given in Table 1. Consequently, thermal conduction by the air can be neglected in most cases of cellular ceramics as long as the porosity is not too high.

Table 1 Approximate thermal conductivities k of some dense ceramic materials at room temperature compared to the heat conductivity of air.

The contribution of natural convection to heat transfer in closed cells can be estimated by calculating the Grashof number [8]

which becomes 1000 or greater if convection heat transfer is of importance (g = acceleration due to gravity 9.81 m s–2, b = volume coefficient of expansion for the gas, DTC = temperature difference across one cell, l = cell size, = density of the gas, l = dynamic viscosity of the gas). The equation includes the cell size l as a main parameter of influence. For a particular fluid, a critical cell size can be calculated. For all cell sizes larger than this critical cell size, natural convection is of importance. By using data appropriate to air at 105 Pa (T = 300 K, b = 1/T, DTC = 10 K, = 1 kg m–3, l = 2 10–5 N s m–2), Gibson and Ashby calculate a critical cell size of 10 mm, which is larger than in most common cellular ceramics; thus, the contribution of natural convection to heat transfer can also be neglected [8].

This leaves three contributions, which are denoted as kR, kS, kC, representing radiative heat transport, heat transport through the solid, and convective heat transport by means of a flow through the open cells of the material (forced convection), respectively:

4.3 Thermal Properties 345

Note that forced convection only occurs if a flow of the fluid arises due to external forces or internal chemical reactions. Examples of the first case are exhaust catalysts and heat exchangers. Examples of the second case are porous burners and regeneration of diesel particle filters by burning off soot particles in the pores of the filter.

Closer examination of the contribution of forced convection kC reveals firstly that it is strongly dependent on the main direction of flow, that is, in the presence of flow through the cellular ceramic the effective thermal conductivity becomes anisotropic. Secondly, the contribution of forced convection is dependent on the geometry of the cells. In a foam, mixing of the fluid in directions perpendicular to the main flow direction leads to additional heat transfer. Imagining a tube filled with a cellular material, as illustrated in Fig. 1 and assuming cylindrical symmetry two effective conductivities describe the heat-transport properties of the material.

Figure 1 Anisotropic heat-transport properties of a cellular material inserted in a tube and subjected to flow.

This consideration leads to the two equations

The contributions kR and kS remain isotropic if an isotropic material is assumed. They are frequently denoted as keff,0, the effective thermal conductivity without flow

4.3.2.1

Experimental Methods to Determine the Effective Thermal Conductivity without Flow

Besides the approaches mentioned to predict the effective thermal conductivity of cellular ceramics theoretically, there are also several experimental methods to determine this quantity. For the case without forced convection, which excludes flow through the open pores of the material, methods can be employed which are also well known for dense ceramic materials. In the case of porous materials, in contrast to dense materials, sample dimensions should be chosen that are large compared to

346 Part 4 Properties

the pore dimensions. This is frequently not the case for the well known laser-flash method, for which sample dimensions of approximately 10 mm are required. It is the case for the transient plane-source technique and the hot-plate method, which are both presented here in more detail.

The transient plane-source technique, also known as the hot-disk method, uses a thin nickel double spiral as a heat source, placed between two identical material samples. Simultaneously it is used as a sensor for temperature measurement (Fig. 2). Depending on pore size and sample dimensions, sensor sizes of 10 to 50 mm are available. A defined quantity of electrical power is delivered in a defined time period to the hot disk, and the curve of the increasing temperature is recorded. The thermal conductivity of the sample material now influences the slope of the curve. A numerical fit of the temperature curve yields thermal conductivity k and thermal diffusivity k of the sample material. Specific heat capacity cp is then given with the density by

A detailed description of the method is given by Gustaffson [9].

Figure 2 Transient plane-source technique to measure thermal conductivity of porous materials.

In the hot-plate method, a temperature gradient inside the test body generates a one-dimensional heat flow in the area of measurement. Then, the effective thermal conductivity without flow keff,0 [W m–1 K–1] can be calculated from the heat flow density q [W m–2], the sample thickness s [m], and the measured temperature difference DT [K]:

Temperatures at the boundaries of the samples can be varied, so that the variation of the thermal conductivity as a function of temperature (in this case the average temperature) can be determined. For this purpose, the temperature difference should be kept small.

The measurement includes heat flow through the material as well as radiative heat transport. The measurement principle is illustrated in Fig. 3. Typical sample dimensions are a diameter of 200 mm and a thickness of 70 mm. The minimum sample thickness has to be large enough to suppress direct radiation from the hot to the cold plate and therefore depends on the extinction coefficient of the sample.

4.3 Thermal Properties 347

Figure 3 Photograph and sketch of an installation to measure thermal conductivity without flow by the hot-plate method (apparatus built at the Institute of Fluid Dynamics, Friedrich Alexander University of Erlangen-Nuremberg).

Figure 4 Thermal conductivities without flow of various ceramic foam materials with porosity of 0.76 determined by the hotplate method [10].

Figure 4 shows thermal conductivity data of various ceramic foam materials [10]. As expected, thermal conductivity increases significantly with temperature. The reason for this is the increasing contribution of radiation. The radiative contribution can be quantified by considering that the heat flow density transferred by radiation from a body of temperature T1 to a body of temperature T0 increases in accordance with the Stefan–Boltzmann equation

where r denotes the Stefan–Boltzmann constant, and e the emissivity of the cell material ranging between 1 for an ideal absorber and 0 for an ideal reflector. Now, imagining a cellular ceramic between these two bodies, radiation will then be attenuated in the material following an exponential law with the extinction coefficient k [m–1] and the thickness of the material x [m]

for the radiative contribution to the effective thermal conductivity. However, the contribution of heat conduction through the solid decreases with increasing temperature for ceramic materials. These two dependences explain the course of the curves in Fig. 4, which were all fitted with a second-order polynomial function.

4.3.2.2

Method to Determine the Effective Thermal Conductivity with Flow

In the above discussion on effective thermal conductivity with flow through the open cells of a ceramic material (see Fig. 1) the anisotropy of this quantity due to forced convection was mentioned. In the literature, this effect is often called dispersion. Assuming a steady flow in a tube filled with a cellular body one can imagine that heat transport is enhanced by mixing of the fluid in directions perpendicular to the tube axis. These dispersion effects depend on geometrical properties of the cells and on the flow conditions in such a way that they play an increasing role with increasing flow velocity. In a complex cell geometry, as in ceramic foams, it is difficult to derive the influence of the cell geometry on dispersion from fluid mechanics on the pore-size level. Therefore, there is a need to characterize this quantity as an effective, integrated property. For cellular ceramics a simple model derived from a packed bed leads to the following equations describing the influence of flow on the effective thermal conductivity [3, 11]. This procedure is legitimate as long as the macroscopic scales needed to describe the physical phenomenon are large in comparison to the microscopic scales describing the structure [3].

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Table 2 Radial and axial mixing coefficients of various ceramic foam materials.

* Cell density of ceramic foams (pores per inch).

4.3.3

Specific Heat Capacity

The specific heat capacity cp [J kg–1 K–1] characterizes the capability of a material to store energy in the form of heat. In contrast to the thermal conductivity, the specific heat of a cellular ceramic cpCC consisting of the solid material s and the fluid f can be directly derived from the specific heat capacity of the components cpS and cpF as the sum of their weight fractions [8]:

In most cases the mass fraction of the gas is so small that the specific heat capacity of the cellular ceramic is equal to that of the solid of which it is made. In the case of known chemical compositions, cpS can be estimated with the well-known rule of Dulong and Petit [14]

which involves Boltzmann’s constant k, the mass number of the atoms involved lAM, and the atomic mass unit mH; f denotes the number of degrees of freedom of each particle. For most solids f » 6. The rule yields a good approximation and can be seen as an upper limit for “high temperatures”. For most solids this limit is already reached at room temperature. For pure silicon carbide (SiC) and pure cordierite (2 MgO Al2O3 SiO2) the rule yields values of 1244 and 1151 J kg–1 K–1, respectively, which are close to those derived experimentally.

4.3.4

Thermal Shock

Thermal shock resistance is a quantity of interest to nearly every designer employing cellular or noncellular ceramics in high-temperature applications. Thermal shock occurs when changes in temperature lead to thermal stress due to the different amounts of thermal expansion in a piece of material. Due to the poor fracture toughness KIC of most ceramic materials, thermal shock resistance of cellular ceramics is

4.3 Thermal Properties 351

low compared to metallic materials. Thermal shock resistance of a material increases with increasing strength rC, and decreases with increasing elastic modulus E and thermal expansion al. Additionally, if heat flow is regarded, the thermal conductivity k influences the thermal shock behavior. Experiments in which heated foams were quenched in water, oil, or cold air jets have shown the dependence of shock resistance on cell size, density, and material [15–19]. The thermal shock resistance rises with increasing cell size and is only weakly dependent on density. Thermal shock leads to the propagation of pre-existing cracks. The main source of stress is the temperature difference across the bulk and not across the individual struts, as could be shown by Orenstein, Green, and Vedula [15–17]. When cyclic thermal shocks are applied to ceramic foams there is more damage if higher temperatures or faster cooling rates are used [18]. By special processing, Colombo et al. were able to obtain materials with much higher shock resistances [19].

Usually, two quantities R1 and R2 characterize the thermal shock behavior. They are frequently called “hard” and “soft” thermal shock parameters (m denotes Poisson’s ratio):

The hard parameter R1 gives an approximate maximum allowable temperature difference for a rapid superficial thermal shock. The soft parameter includes thermal conductivity and considers a long-term effect of the temperature gradient on the material. Heat flux into the material is taken into account. Table 3 includes the thermal shock parameters of selected dense ceramic materials. To describe the thermal shock behavior of ceramic foams the parameters DT10 describing the tempera-

Table 3 Quantities influencing the thermal shock behavior for a number of dense ceramic materials [20, 21].

* HIP = hot isostatically pressed.