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The energy cost of one firing-cycle of a large pottery kiln (Figure 21.1) is considerable. Part is the cost of the energy which is lost by conduction through the kiln walls; it is reduced by choosing a wall material with a low conductivity, and by making the wall thick. The rest is the cost of the energy used to raise the walls of the kiln and its contents to the operating temperature. It is reduced by choosing a wall material with a low heat capacity, and by making the wall thin. Is there a performance index which captures these apparently conflicting design goals? And if so, what is a good choice of material for kiln walls? The design requirements are listed in the Table 21.1.
Figure 21.1 A kiln. In a firing cycle, energy is lost both by conduction and in heating the structure of the kiln itself.
FUNCTION |
Thermal insulation for kiln walls |
OBJECTIVES |
Minimize energy consumed in firing cycle |
Minimize capital cost of insulating material |
|
CONSTRAINTS |
Maximum operating temperature = 1000 K |
Possible limit on kiln wall-thickness, for space reasons |
Table 21.1 The design requirements
When a kiln is fired, the temperature rises quickly from ambient, To, to the firing temperature, T, where it is held for the firing time t (Figure 21.1). The energy consumed in the firing time has, as we have said, two contributions. The first is the heat conducted out. Once a steady-state has been reached, the heat loss per unit area by conduction, Q1, is given by the first law of heat flow (Figure 21.1). Over the cycle time t (which we assume is long compared with the heat-up time) the heat loss is
 . |
(M21.1) |
Here λ is the thermal conductivity, dT/dx is the temperature gradient and w is the wall thickness.
The second contribution is the heat absorbed by the kiln wall itself. Per unit area, this is
 , |
(M21.2) |
where Cp is the specific heat and ρ is the density. The factor 2 enters because the average wall temperature is (T − To)/2. The total energy consumed per unit area of wall is the sum of these two heats:
 . |
(M21.3) |
A wall which is too thin loses much energy by conduction, but absorbs little energy in heating the wall itself. One which is too thick does the opposite. There is an optimum thickness, which we find by differentiating equation (M21.3) with respect to wall thickness w, giving:
 , |
(M21.4) |
where a = λ/Cpρ is the thermal diffusivity. The quantity (2at)1/2 has dimensions of length and is a measure of the distance heat can diffuse in time t. Equation M21.4 says that the most energy-efficient kiln wall is one that only starts to get really hot on the outside as the firing cycle approaches completion. That sounds as if it might lead to a very thick wall, so we must include a limit on wall thickness.
Substituting equation (M21.4) back into equation (M21.3) to eliminate w gives:
 . |
Q is minimized by choosing a material with a low value of the quantity (λCpr)1/2, that is, by maximizing
 . |
(M21.5) |
Now the limit on wall thickness. A given firing time, t, and wall thickness, w, defines, via equation (M21.4), an upper limit for the thermal diffusivity, a:
 . |
(M21.6) |
Selecting materials which maximize equation (M21.5) with the constraint (M21.6) minimizes the energy consumed per firing cycle.
Some candidates for the insulation could be very expensive. We therefore need a second index to optimize on cost. The cost of the insulation per unit area of wall is
 |
(M21.7) |
where Cm is the material cost per kg. Substituting for w from equation (M21.4) gives
 . |
(M21.8) |
The cost of the material is minimized by maximizing
 . |
(M21.9) |
And, finally, the material must be able to tolerate an operating temperature of 1000 K.
The neatest way to approach this problem is by a three-stage selection, starting with a chart of the thermal diffusivity (a compound-property)
 , |
plotted against thermal conductivity, λ, as in Figure 21.2. Contours of M1 are lines of slope 2. One has been positioned at M1 = 10-3. To this can be added lines of constant wall thickness, corresponding to fixed values of the thermal diffusivity, a (equation M21.6). The right-hand scale shows these limits, assuming a firing time of 6 hours; the horizontal broken line describes a thickness limit of 200 mm. We can now read-off the best materials for kiln walls to minimize energy, including the limit on wall thickness. Below the broken line, we seek materials which maximize M1; while meeting the constraint on w.
Figure 21.2 Thermal diffusivity, a, (a compound-property) plotted against thermal conductivity λ using the generic record subset. The selection line of slope 2 shows M1; materials above the line are the best choice, provided they lie within the thickness-limit (right-hand scale and horizontal broken line).
Figure 21.3 Thermal diffusivity, a, (a compound-property) plotted against cost per unit volume using the generic record subset. The selection line of slope -2 shows M2 ; materials below the line are the best choice, provided they lie within the thickness-limit (right-hand scale and horizontal broken line).
The second stage optimizes the cost of material for given firing conditions (equation M21.9). The line shows M2; once again limits on wall thickness can be added (right-hand scale and horizontal line).
The final stage is one for protection: it is a bar-chart of maximum operating temperature Tmax. The line limits the selection to the region
Tmax > 1000 K. |
Figure 21.4 Maximum service temperature plotted against material class. Only metals and ceramics can tolerate temperatures as high as 1000 K; metals are eliminated by their high thermal conductivities.
Table 21.2 lists the results. Porous ceramics, including firebrick, are the obvious choice. But the degree of porosity is important. The more porous (low density) firebricks lie highest under the dashed line on Figure 21.2 — they require the thickest wall. So it may pay to use a denser firebrick, to meet the requirements on wall thickness.
MATERIAL |
M1 = a1/2/λ (m2K/Ws1/2) |
COMMENT |
Porous Ceramics |
3 x 10-4 – 3 x 10-3 |
The obvious choice: the lower the density, the better the performance. |
(Fiberglass) |
10-2 |
Thermal properties comparable with polymer foams; usable to 500°C. |
Table 21.2 Materials for energy-efficient kilns
It is not generally appreciated that, in an efficiently-designed kiln, as much energy goes in heating up the kiln itself as is lost by thermal conduction to the outside environment. It is a mistake to make kiln walls too thick; a little is saved in reduced conduction-loss, but more is lost in the greater heat capacity of the kiln itself.
That, too is the reason that foams are good: they have a low thermal conductivity and a low heat capacity. Centrally heated houses in which the heat is turned off at night suffer a cycle like that of the kiln. Here (because Tmax is lower) the best choice is a polymeric foam, cork or fiberglass (which has thermal properties like those of foams). But as this case study shows — turning the heat off at night doesn't save you as much as you think, because you have to supply the heat capacity of the walls in the morning.
Holman, JP (1981) 'Heat Transfer' 5th Edition, McGraw-Hill, NY, USA.