Specific Heat, Volumetric Heat Capacity, and Thermal Diffusivity

How much energy do you need to change the temperature of a material by 1 degree? How fast does energy transfer from one side of a slab of a material to another? How fast does energy move away from a surface? These questions all rely on a material changing temperature as a function of time, whether it’s the time it takes to increase some ΔT or the time it takes one side of a material to heat up when heat is applied to the other side. While these questions can be answered in part by considering the material’s thermal conductivity, the specific heat of the material must also be considered. In fact, for any situation where temperature is changing as a function of time (i.e., any transient heat flow problem), the material’s specific heat enters the picture to fully describe the transient temperature changes.

The specific heat of a material is the amount of heat (Joules) per unit mass (g) or mole (mol) required to raise the temperature by one degree Celsius (or Kelvin). When conducting heat transfer analyses, typically we work with volumetric heat capacity, which is the specific heat per unit mass (J/g/K) multiplied by the mass density (g/m³). This volumetric heat capacity, C (J/m³/K), represents the amount of heat per unit volume required to raise the temperature by one degree Celsius. Where C enters into the heat transfer picture is when considering the time it takes for heat to flow across a thickness of a material or the ability of a material’s surface/interface to exchange heat with another material or its surroundings.

When considering the rate of heat transfer inside a material, we turn to the thermal diffusivity (D or a), which is defined as the ratio of the thermal conductivity to volumetric heat capacity of the material:

D = k/C (m²/s)
Thus, thermal diffusivity provides information about the competition between the ability to conduct heat in a material (k) vs the ability to store that heat (C). It is important to note that a high value of thermal diffusivity does not necessarily mean that heat is better dissipated, but rather that heat is more effectively dissipated than it is stored. Thus, for one to compare materials and assess their ability to dissipate heat, thermal conductivity and thermal resistances should be compared, so that any thermal diffusivity value or measurement must be multiplied by the materials heat capacity. To this end, technically, thermal conductivity measurements provide different information than thermal diffusivity measurements.

As thermal diffusivity is a property that is important during temperature transients, we can then relate the time it takes for heat to travel across some distance, or the time it takes for a material to reach steady state conditions, to the diffusivity, D. The time it takes to travel across a material of given thickness, d, can be approximated by the diffusion time.

t_diffusion> = d²/D
This diffusion time can then be used to approximate the time it takes for a material system to reach steady state. When the time scale of heating is much larger than the diffusion time, the temperature gradients in a material will progress to steady state, and thus the ratio of heating time to t_diffusion gives an indication of whether the material has reached steady state conditions. To this end, we define a non-dimensional number, the Fourier Number (Fo) given by:
Fo = t/t_diffusion = Dt/d²
where steady state conditions are achieved when Fo>>1. Under this condition of Fo>>1, the temperature gradients are not changing in time and the temperature changes across materials can be predicted by the thermal conductivity and thermal resistance.