Materials For Passive Solar Heating

• Introduction
• The Model
• The Selection
• Postscript
• Further reading
• Related case studies

Introduction

There are a number of schemes for capturing solar energy for home heating: solar cells, liquid filled heat exchangers, and solid heat reservoirs. The simplest of these is the heat-storing wall: a thick wall, the outer surface of which is heated by exposure to direct sunshine during the day, and from which heat is extracted at night by blowing air over its inner surface (Figure 23.1). The economy-minded householder will certainly wish that, if he is going to put half a meter or so of heat-storing material onto his walls, it should be cheap: the heat stored per dollar is what counts. A second essential of such a scheme is that the time-constant for heat flow through the wall be about 12 hours; then the wall first warms on the inner surface roughly 12 hours after the sun first warms the outer one, giving out at night what it took in during the day. We will suppose that, for architectural reasons, the wall must not be more than 1/2 m thick. What materials maximize the thermal energy captured by the wall while retaining a heat-diffusion time of up to 12 hours? Table 23.1 summarizes the requirements.

Figure 23.1 A heat-storing wall. The sun heats the wall during the day. The heat diffuses through the wall and is extracted during the night.

Table 23.1 The design requirements

The Model

At first sight, what we want here are materials with high heat capacity per unit cost, that is, materials with large values of

.

(M23.1)

where C_p is the specific heat capacity (J/kg.K) and C_m is the cost (£/kg) of the material. But this is misleading: we have ignored the requirement for a 12-hour time constant and a maximum wall thickness of 0.5 m. Instead, we proceed instead as follows.

The heat content, Q, per unit area of wall of thickness w, when heated through a temperature interval ΔT gives the objective function

,

(M23.2)

where ρC_p is the volumetric specific heat (the density ρ times the specific heat C_p). The thickness of the wall w is set by the 12-hour diffusion time, within the limit of the thickness constraint. If it is too thin, the heat it absorbs will diffuse out again in a few minutes; if it is too thick, very little heat will conduct through to the inside surface during the 12 hours of darkness. We must choose the thickness to give a 12-hour diffusion time.

This is a problem in transient heat flow. In any such transient problem, the thermal diffusion distance x in a time t is approximately

(M23.3)

where a is the thermal diffusivity (see Case Study Kiln Walls). Eliminating the free variable, w, in equation (M23.2) using equation (M23.3) gives

.

Using the fact that a = λ/ρC_p where λ is the thermal conductivity,

.

(M23.4)

The heat capacity of the wall is maximized by choosing material with a high value of

.

(M23.5)

This is the inverse of the index of Case Studies Kiln Walls and Sauna Walls. The restriction on maximum thickness w requires (from equation M23.3) that

.

(M23.6)

With w ≤ 0.5 m and t = 12 hours (4.3 × 10⁴s), we obtain the property limit

.

(M23.7)

The cost of the material of the wall, per unit area is

.

(M23.8)

Using equation (M23.3) for w gives

.

The cost of the wall is minimized by selecting materials with large values of

.

(M23.9)

As before, the restriction to a maximum thickness of 0.5 m imposes the property limit on a, given by equation (M23.7).

The Selection

Figure 23.2 shows a chart of thermal diffusivity, a, plotted against thermal conductivity, λ, with the index M₂ and the property limit on the diffusivity plotted on it. Figure 23.3 shows the second stage: it is a chart of thermal diffusivity, a, plotted against material cost per unit volume, C_mρ, with the index M₃ and the limit on diffusivity indicated. The two charts identify the group of materials listed in Table 23.2: they maximize M₂ and M₃ while meeting the constraint expressed by equation (M23.7). Solids are good; porous materials and foams (often used to insulate walls) are not.

Figure 23.2 A chart of thermal diffusivity, a, against thermal conductivity, λ. One possible line for the index M₂ is shown. The limit on a is shown.

Postscript

Old stone houses have thick walls — some approaching 0.5m in thickness. They are cool in the summer and — if adequately heated — warm in the winter, because the large heat capacity and 12 hour time constant evens out the difference in temperature between night and day. But — if the heating is internal — stone is not the best material for insulating walls; its thermal conductivity is too high; foams and cellular materials are much better (see Case Studies Isothermal Containers, Kiln Walls and Sauna Walls). The choice of stone here is driven by the objective of storing energy — it is the high heat capacity of stone that is attractive.

Figure 23.3(a) A chart of thermal diffusivity, a, against material cost per unit volume, C_mρ, with the index M₃ shown.

Table 23.2 Materials for heat-storing walls

Further Reading

Holman, JP (1981) 'Heat Transfer', 5th Edition, McGraw-Hill, NY, USA.

Related Case Studies

1. Insulation for Isothermal Containers

2. Energy-Efficient Kiln Walls

3. Materials for Sauna Walls

4. Materials for Storage Heaters