Why does a piece of room temperature metal feel colder to the touch than paper wood or cloth?

Photo by Ilse Orsel on Unsplash

On a cold winter morning, when you touch a piece of metal and a piece of wood, the metal feels a lot colder than the wood, though both are at equal temperatures. This observation is usually attributed to metals having higher thermal conductivity than wood. They extract more heat from your hand than wood in a given time. Therefore, you perceive the metal as being colder than the wood. However, thermal conductivity is not the only property that plays a significant role here. In this post, in addition to thermal conductivity, I will the discuss roles of other thermal properties of a material that govern physics behind this trivial but interesting observation.

Let’s do a thought experiment.

You have a slab of SAE 304 stainless steel (which is the most commonly used stainless steel) and a slab of teak wood, which are at 15 °C each, on a cold winter morning. Imagine that one of your hands is made of potassium (K) and the other is made of cobalt (Co), which have nearly equal thermal conductivities (k): k_K = 102 W/m·K and k_Co = 100 W/m·K [2]. Both of your hands are at 37 °C each (which is the normal human body temperature). Now, touch the slab of SAE 304 stainless steel (SS slab) and the slab of teak wood by the potassium hand, one by one. Repeat the same with the cobalt hand.

When you touch the slabs, the degree of ‘hotness’ or ‘coldness’ you feel could be quantified in terms of the temperature at the interface between your hand and the slab. Table 1 shows the interface temperatures (Tᵢ) for all the combinations:

The interface temperatures for the combinations of K-teak wood and Co-teak wood are close to each other. So, both the hands would feel the wooden slab to be nearly equally cold. In contrast, the temperatures for the combinations of K-SS and Co-SS are significantly different — your cobalt hand would feel the SS slab to be significantly hotter than the potassium hand. This should not happen because thermal conductivities of potassium and cobalt are nearly equal, and thus, they should extract an equal amount of heat in a given time from the SS slab, making the SS slab equally cold to both hands. What is the reason behind this discrepancy?

Under steady conditions, thermal conductivity is the only property that governs heat conduction. It essentially measures the heat transfer rate by conduction. Under steady conditions, when placed in contact with a thermal reservoir, a good thermal conductor such as copper, extracts or loses more heat (depending on the temperature difference) per unit time than a bad conductor such as wood. However, under transient conditions, in addition to thermal conductivity, thermal diffusivity also governs heat conduction. Thermal diffusivity (α) is defined as,

where ρ and c, respectively represent the density and specific heat capacity of a material [2]. The product ρc can be considered as volumetric heat capacity, which describes how quickly the temperature of a material changes when it is heated. Hence, thermal diffusivity, which takes into account the effects of thermal conductivity and volumetric heat capacity, is essentially a measure of how quickly a material achieves a steady state heat transfer after an initial transient heat conduction period [5]. In other words, thermal diffusivity is associated with the speed of propagation of heat by conduction in a material where the temperature is changing with time [2]. Now, let’s try to address the discrepancy observed in Table 1.

Touching the SS slab and the teak wood slab with your hands involves transient heat conduction. A combination of thermal conductivity and thermal diffusivity plays a significant role here. This combination is known as thermal effusivity (e):

Thermal effusivity, in broad terms, is a measure of a material’s ability to exchange heat with its surroundings [2]. If two semi-infinite solids at temperatures T₁ and T₂ (T₁≠T₂) are suddenly placed in perfect thermal contact, the interface temperature quickly reaches a steady value,

where e₁ and e₂ are the thermal effusivities of those solids [2]. The SS slab, the teak wood slab, and your hand could be considered semi-infinite solids. [A semi-infinite solid is an idealized body that has a single plane surface and extends to infinity in all directions except one [6]. A slab can be considered a semi-infinite solid as long as (a) we are interested in temperature variation in the region close to one of the surfaces, and (b) the observation time is so short that the other surface does not have a significant effect on the region of interest. When you touch the SS slab or the teak wood slab, the times involved are of the order of few seconds, and thus, the slabs and your hand could be considered semi-infinite solids.]

The temperatures in Table 1 were calculated using Eq. 3. This equation can also be used to calculate the interface temperature when you touch the SS slab and the teak wood slab with your normal hand. Table 2 shows the interface temperatures for different combinations (the temperatures from Table 1 are also shown for comparison), and Table 3 shows the thermal effusivities of the involved materials required to calculate those interface temperatures.

In conclusion, under transient conditions, thermal effusivity governs heat conduction, and thus, our feelings of ‘hotness’ or ‘coldness’ of a material.

What if you stay in touch with the SS slab or the teak wood slab a little longer? Would it affect the interface temperature? Figures 1 and 2 show representative temperature profiles at different times within 5 mm from the interface of the combinations of K-SS and K-teak wood from Table 2 (the temperature profiles are not obtained empirically).

I encourage readers to guess temperature profiles for the other combinations from Table 2. Be careful about the slopes of the curves!

Food for thought

1. It seems that, for short time intervals, the temperature profiles inside materials change with time, whereas the interface temperatures do not (Figs. 1 and 2). Would the interface temperature differ from the initial interface temperature once steady conditions are achieved?
2. You have a high fever. Your body temperature is 39 °C. A friend with normal body temperature touches your forehead and diagnoses that you don’t have a fever. What could his hand be made of 😄? Choose the correct options from the following: (a) glass (b) water (liquid) (c) aluminum (d) diamond (more than one option could be right)

Thanks for reading! Constructive criticism and feedback are always welcome!

References:

1. Predicting Human Thermal Comfort in Automobiles
2. Augustin Salazar, On thermal diffusivity, European Journal of Physics, 2003.
3. Varlamov, A. A., Aslamazov, L. G., The wonders of physics, World Scientific.
4. Database of thermal properties of materials
5. Thermal diffusivity and effusivity
6. Transient heat conduction in a semi-infinite solid with surface convection
7. Yoshida, A., Kagata K., Yamada T., Measurement of Thermal Effusivity of Human Skin Using the Photoacoustic Method, Int J Thermophys, 2010.
8. List of thermal conductivities

I would like to thank Dr. T. G. Vignesh for proofreading the article.