Thermal Capacitance

When time enters the equations, the ability of a material to store or release heat becomes a crucial parameter. This issue’s technical data column is devoted to a basic understanding of thermal capacitance. Thermal capacity (or heat capacity) is defined as:

C_th = V � � c_p [J/K]

where:
V =Volume (m³)
= Density (kg/m³)
c_p = Specific heat (J/kgK) at constant pressure (the usual situation)

However, some engineering exceptions exist; for example, when dealing with sealed enclosures. Fortunately, the difference between the heat capacity at constant volume and that at constant pressure (due to the expansion energy) is small, at least for solids. The essential point to keep in mind is that only the product c_p is important. Handbooks often quote only the individual values for and c_p, and you will notice that large differences exist between materials.

For example, let’s compare platinum and magnesium. Their densities are respectively 21,400 and 1740 kg/m³, more than a factor of twelve difference, but their thermal capacities are 2.9 � 10⁶ and 1.7 � 10⁶ J/m³K, much less than a factor of two. How come? Essentially, the specific heat is linked to the internal energy (such as rotational and vibrational energy) of a molecule. More correctly, the specific heat is the change in the internal energy per unit of temperature change. The more degrees of freedom the molecule has, the more energy it can store.

Material Cp (10+6J/m3K) Some heavy metal oxides/borides, etc 4.5 Water 4.2 Fe, Ni, Co, Cu, and alloys 3 – 4 Al, Ag, Mo, Au, Ti, and alloys 2.5 Ga, Cd, In, Sn, and Si 1.8 BN, Bi, and Na 1.2 Plastics, wood, glass, stone, and ceramics 1.5 – 2.5 Fluids 1.3 – 1.9 Porous insulators 0.01 – 1 Gases 0.001

Table 1. Orders of Magnitude for C_p

Now suppose we have a material with a high density, which means that the molecules are very close to each other. Certainly this reduces the ability to vibrate. In other words, the material has a low specific heat. The result is that the product c_p falls within a relatively narrow margin. This fact was already observed by Dulong and Petit, who stated in 1818 that the heat capacity should be constant, starting from first-order kinetic theory principles. Gases are a different story. ‘Free’ is ‘free’, and beyond a certain freedom it is only the density that counts. Table 1 provides some rules-of-thumb for certain classes of material.