Cellular Ceramics / 4

352 Part 4 Properties

ture difference for which the material experienced a 10 % reduction in Young’s modulus [15] and the damage parameter DE [17] were introduced. The damage parameter is defined as

where E is the elastic modulus after thermal shock, and E0 is the elastic modulus in the as-received state.

4.3.5

Volumetric Convective Heat Transfer

In some applications cellular ceramics are employed to heat or to cool a fluid which flows through the open pores of the material. Such applications are heat exchangers, solar receivers, electric air heaters with a cellular ceramic as a heating element, and air-cooling systems for gas turbines or combustion chambers. In all cases good convective heat transfer from the solid to the fluid or vice versa is needed. In some cases the problem of a fluid flowing through a cellular body is handled with homogenous approaches, considering fluid and solid as one medium with averaged effective quantities. Whenever a heterogeneous approach is used, the temperatures of fluid and solid are different. In this case the convective heat transfer coefficient a [W m–2 K–1] describes the capability of a surface to transfer heat to a fluid which flows along the surface. It is defined by the equation which describes the typical volumetric heat-transfer problem:

area of the cellular ceramic, and TS [K] and TF [K] the wall and fluid temperatures, respectively. Unlike many two-dimensional heat-transfer problems here the quantities qv and Av are volumetric properties.

Investigations of the convective heat transfer of open-celled foams revealed significant deviations (up to 100 %) from semi-empirical correlations for packed beds [22]. Therefore, for a detailed calculation of any heat-transfer problem in a cellular ceramic, it is necessary to determine the heat-transfer coefficient experimentally.

One way of measuring the heat-transfer coefficient is described in the following. It is an experiment based on the method by Younis and Viskanta [22]. The experimental setup is shown in Fig. 6. An air stream of alternating temperature flows through the porous sample. The air temperature can be regarded as a temperature wave T(t,x), where x denotes the direction of air flow. This temperature wave is generated by an electrically driven heating element. The amplitude and the frequency of the temperature wave can be controlled. The tube through which the air flows is isolated, and the air temperature is measured at the inlet and outlet of the air flow through the sample. The sample induces a phase shift Du and an attenuation

4.3 Thermal Properties 353

Figure 6 Experimental setup for measuring convective heattransfer properties of cellular materials.

of the temperature wave. From these, the product of the specific surface area Av and the heat-transfer coefficient a can be calculated [23].

With the assumptions of a uniform temperature across any cross section of the tube T = T(t,x), a constant heat transfer coefficient a and a negligible heat conductivity of the air, the energy transport equations of the air and the wall are:

where the subscript F denotes the fluid and the subscript S the solid (ceramic), Po the porosity of the cellular ceramic, and cpF and cpS the corresponding specific heat capacities.

These equations are coupled differential equations. It could be shown by numerical simulations [23] that the analytical solution of this problem, which can achieved by neglecting the thermal conductivity k of the wall, is valid in many cases with high porosity and thus low thermal conductivity. The general analytical solution is a dampened wave function. So by measuring the phase shift or the dampening the product of the heat-transfer coefficient and the specific area Av can be calculated. As for most heat-transfer problems only the knowledge of the product Av is essential, this is not a major drawback of this measurement technique.

As an example, results of an investigation on ceramic foam materials are presented [24]. The foam materials consist of cordierite, clay-bound silicon carbide (CBSiC), and sintered silicon carbide (S-SiC). The results are presented as volumetric heat transfer coefficient aAv as a function of fluid velocity. Figure 7 shows the results for various 20 ppi materials with a porosity of 0.76. The coefficients increase with increasing fluid velocity. The differences between materials of the same cell geome-

354 Part 4 Properties

try but different material are small. In Fig. 8 the volumetric heat-transfer coefficients of CB-SiC foams with various cell diameters are compared. Heat transfer increases significantly with increasing ppi value, mainly due to the larger heat-transfer surface of materials with smaller cells. In the next section, the relationship between cell size and heat-transfer surface is quantified in more detail.

Figure 7 Volumetric heat transfer coefficients of various 20 ppi ceramic foam materials (porosity 76 %).

Figure 8 Volumetric heat-transfer coefficients of clay-bound silicon carbide foams with various cell diameters

10, 20, 45 ppi; porosity: 76 %).

4.3.5.1

Nusselt/Reynold Correlations and Comparison with Theoretical Data

To describe the heat-transfer properties of foams independent of their cell dimensions, similarity theory can be employed. Here, heat transfer is described by the Nusselt number Nu:

where dh is the hydraulic diameter, and k the thermal conductivity of the fluid. For a regular channel geometry, for example, rectangular channels, the specific area Av can be easily calculated geometrically. For these kinds of materials many Nu–Re–Pr correlations can be found in publications such as Kays and London [25].

4.3 Thermal Properties 355

To calculate Nusselt/Reynolds correlations from the volumetric heat-transfer data, assumptions and calculations concerning the pore level fluid velocity UPORE, the hydraulic diameter dh, and the specific surface area AV [m2 m–3] must be made. The calculation of the specific surface was performed with:

This empirical equation is the result of a numerical investigation of various micrographs [26]. The fluid velocity in the pore UPORE = U0/P0 is calculated from the porosity P0 and the fluid velocity outside the cellular body U0.

To calculate the hydraulic diameter dh in a ceramic foam, various approaches have been published. Buck used a simple model in which the foam is considered to be a mesh made of cylindrical wires [27]. With this model, the length s of a characteristic cylindrical element and the hydraulic diameter dh can be calculated:

A further possibility to calculate dh arises if the pore of the foam is considered to be a polyhedron made of solid struts, as was done by Gibson and Ashby [8]. In this case the cell of the foam consists of 24 vertices, linked by 6 faces consisting of 4 edges and 8 faces consisting of 8 edges, which results in a total of 36 edges (Fig. 9). Heat transfer can be regarded as a flow around the struts.

With the help of this model the diameter l [m] of the struts can be calculated:

Figure 9 Tetrakaidecahedron as a model for foams investigated (after Gibson and Ashby [8]).

356 Part 4 Properties

Thus, the geometry of the pore can be derived from the macroscopic quantities porosity and cell density. As examples these quantities were calculated for some ceramic foams (Table 4). Additionally, the quantities derived from Buck’s model and the reciprocal specific surface area, which also can be regarded as a characteristic length, are listed.

Table 4 Macroscopic and pore-level geometric data of various ceramic foams.

For the Nusselt/Reynold plots in the further course of the text, the strut diameter t as calculated by Gibson and Ashby was chosen as hydraulic diameter dh.

From the point of view of similarity theory the Nusselt number should be independent of the cell dimensions. However, the Nusselt/Reynold plots in Fig. 10 show that for the materials investigated this is not the case. The materials with larger cell dimensions (10 ppi) show significantly higher Nusselt numbers. In contrast, no significant differences are observed between the 20 and 45 ppi materials.

To derive Nusselt/Reynold relations, the experimental data were fitted with a potential function

The parameters a and b, derived from heat-transfer measurements, are included in Table 4 for all materials investigated.

By means of the parameters derived from the measurements the Nusselt number of a foam can be calculated for a range of Reynold numbers of 5–150. An approach to explicitly determine the functional dependence of the Nusselt number on the cell density can now be performed by means of a modified fit. Three materials with a porosity of 0.76 were taken to perform a fit Nu = aReb for cell densities of 10, 20, and

358 Part 4 Properties

son for this deviation is that a significant volume of the packed bed does not play a role in fluid flow due to inhomogeneities of the cell size. Smaller cell sizes induce higher pressure losses and consequently lower or vanishing flow velocities. The fact that in particular materials with smaller cell sizes show lower Nusselt numbers is an argument that in ceramic foams similar mechanisms take place.

In the following a comparison is presented between the data experimentally acquired for the foam materials mentioned above and a theoretical model describing heat transfer in ceramic foam materials. Before employing a certain model, the heat-transfer mechanism must be considered. If the flow through the cellular body is considered as a flow through a network of struts similar to the sketch in Fig. 9, the mechanism of heat transfer can be regarded as cross-flow over a network of cylinders. For this simple case the following relations can be found [29]:

q

The equations are valid for a range of Reynold numbers of 10 < Re < 107 and for a range of Prandtl numbers of 0.6 < Pr <1000. The experiments were carried out at temperatures between 40 and 60 C and at flow speeds of 0.6–3 m s–1. Prandtl and Reynold numbers were Pr » 0.7 and 5 < Re <160, that is, within the range of validity of Gnielinski’s equations.

The comparison in Fig. 12 shows that the values derived from the model of Gnielinski are in good accordance with the values experimentally derived for 10 ppi foams. For higher cell densities (20 and 45 ppi) the experimental data deviate significantly from the model. This leads to the assumption that, similar to packed beds, flow vanishes in parts of the volume of the material. This effect is considered in Eq. (2).

Figure 12 Comparison of experimentally derived Nusselt/ Reynold correlations with theoretical data for cross-flow over a cylinder (after Gnielinski).