# Mechanisms of heat transfer in paper

Heat can transfer through paper thickness by conduction, convection and radiation. Thermal conduction is the diffusion-like transfer of energy. The basic mechanism is the exchange of energy, in gases and liquids, say, in molecular collisions from more energetic particles to less energetic ones by random collisions. In solids, conduction can result from the interaction of atoms in the form of lattice vibrations called phonons 1.

The following rate equation governs heat conduction in one dimension:

(1)

where q is the rate of energy transfer across unit area,
λ thermal conductivity, and
T temperature.

The minus sign indicates that heat flows in the direction of decreasing temperature.

Dry paper is a poor thermal conductor. This is because the air-filled pores have low conductivity as such, and that of the three-dimensional fibre network reflects the low thermal conductivity of fibres and the tortuous, disordered structure.

The heat conductivity problem in paper is related to that of, say, water vapour diffusion: the former takes place in the fibres, and the second in the pore space, but mathematically they are close.

At interfaces between two bodies, the rate of heat transfer decreases due to thermal contact resistance that causes the temperature drop shown in Figure 1. Thermal contact resistance is due to surface roughness that reduces the microscopic contact area. Pressure reduces contact resistance. For most materials, 700 kPa is sufficient to remove contact resistance completely 2. When paper is pressed against a hot cylinder, the apparent contact resistance may be less sensitive to pressure because of convective and radiative heat transfer. Figure 1. Temperature distribution at the interface of two materials showing perfect contact (a) and the temperature drop at the interface in the case of a finite contact resistance (b) 2.

Convection is a mode where moving fluid carries heat. A good example of convection is a liquid moving on a heated plate. The equation for convective heat transfer is the following:

(2)

where h is the convection heat transfer coefficient, and
Ts, Tinf the surface and fluid temperatures, respectively.

The coefficient h depends on the fluid and surface properties, flow conditions, geometry, etc. Convective heat transfer is a much more effective mechanism of heat transfer than conduction.

For paper, the evaporation and re-condensation of water vapour is an effective mechanism of convective heat transfer (/LINK: 03.09.07 Moisture and fluid transport/). The evaporated molecules take energy from warm areas and release it to cooler areas in condensation. Because of the porous nature of paper, convection is important if there is enough moisture available. The diffusion of water vapour will then significantly increase the heat transfer rate above the pure conduction along fibres.

Because of the effects of water vapour, measurement of the thermal conductivity of paper gives an apparent thermal conductivity that combines conduction and convection 3,

(3)

where λ is the “true” thermal conductivity without convection,
l the evaporation enthalpy of adsorbed water,
εb the effective diffusion resistance coefficient,
Mw the mole mass of water,
R the gas constant,
Dwa the water vapour diffusion coefficient, and
pw the vapour pressure.

Figure 2 shows the true thermal conductivity, λ, and apparent conductivity, λa, calculated as a function of moisture content at two different temperatures. At low moisture contents, λa increases dramatically because of water vapour convection.

At high moisture contents, the vapour flux decreases from collisions between water molecules, and λa declines. Wet paper has no room for water vapour convection. Then λa is entirely due to actual thermal conduction. Figure 2. Calculated apparent thermal conductivity, λa, and ordinary thermal conductivity, λ, of newsprint as a function of moisture content at two temperatures 3.

Any material at a finite temperature emits thermal blackbody radiation. Photons emitted in electronic transitions carry the energy. Their wavelength distribution depends on the material and its temperature. The maximum thermal radiation shifts to shorter wavelengths as temperature increases. At room temperature, thermal radiation is in the IR range. It becomes visible when temperature increases above 800 K. The net heat flux is the following:

(4)

where ε is the emissivity of the material,
σ the Stefan-Boltzmann constant, and
Ts, Tsur the temperatures of the surface and its surroundings, respectively.

Unlike conduction and convection, thermal radiation does not require any intervening medium for heat transfer. The transfer of thermal energy by radiation is most effective in a vacuum.

At all relevant temperatures, heat transfer by radiation is negligible inside paper. In some applications such as drying of coatings and toner fusion in electrophotography, paper is heated with IR radiation. IR radiation can penetrate paper effectively and provide uniform heating of the contained water. In paper cooling, thermal radiation needs to be considered even at normal temperatures 4.