# Specific heat and thermal diffusivity

The specific heat capacity, cp [J/kg·K], of a material is the amount of energy per unit mass necessary to raise the temperature by one Kelvin. It therefore reflects the ability of the material to store thermal energy. For solid materials like paper, specific heat measurement is measured under constant pressure so that

(1)

where dQ is the amount of heat absorbed by mass m for temperature change dT.

Heating causes thermal expansion that would be prevented if the measurement occurred under constant volume. The constant volume specific heat cv would be larger than cp. For paper, changes in the moisture content are analogous to volume changes. The equivalent quantity to cp is the specific heat at constant vapour pressure. This includes the heat of evaporation. If the specific heat were determined at constant moisture content, the result would be smaller.

The saturation vapour pressure also changes with temperature. Special care would therefore be necessary to make certain that the measurement is made at constant vapour pressure. Alternatively, the moisture content should remain constant if specific heat at constant moisture content was necessary. The specific heat of oven-dry paper can be determined with a differential scanning calorimeter (DSC).

Specific heat can be combined with heat conduction (see Eq. 1 in Mechanisms of heat transfer in paper) to determine the temperature distribution through a paper sheet. The temperature distribution will evolve with time only if there is a different heat flux, q, through the two surfaces of the sheet. The rate of temperature evolution is controlled by thermal diffusivity. Materials with high thermal diffusivity react quickly to changes in external temperature. For example, thermal diffusivity controls the temperature of a paper sheet after a toner fusion nip.

The following diffusion equation governs the temperature distribution, T(z,t), in the thickness direction:

(2)

where α is thermal diffusivity [m2/s], defined by

(3)

where ρ is density.

Because of the ambiguity of paper thickness, thermal diffusivity is often expressed as α/ρ2. The normalisation is equivalent to changing thickness, z, to grammage in Eq. 2.