Modeling thermal contact resistance at the finger-object interface

Figures & data

Figure 1. (a) Schematic of heat transfer at finger-object interface (inspired by work of Ho [1]). The close-up schematic and corresponding plot show that surface roughness leads to reduced heat transfer area, thus reduced effective heat flux q”, and a temperature jump at the interface ∆T_C that decreases with time, t (F indicates the applied force, $A_a$ indicates the apparent area, and $A_r$ indicates the real contact area, which is a sum of the areas of the microscopic contacts, $A_{ri}$); a three dimensional optical scan of a replica of a male finger surface is also included (see Supplemental Material for full-size image); (b) and (c) schematics of microscale contact geometry during (b) plastic deformation of contacting surfaces with random surface roughness assumed in the CMY model ($a$ indicates circular contact radius, $\sigma$ mean surface height, $tan\theta$ mean surface asperity slope) and (c) elastic deformation of one-dimensional wavy surface in contact with a stiff and flat substrate assumed to derive the functional form of the PM model ($\bar{p}$ indicates mean pressure and $p^{\ast }$ pressure at which gaps fully collapse on such surface).

Table 1. Thermal properties of skin and BaF₂ [27,31] as well as aluminum, marble, and wood [12,38] (in the case of the latter upper value of thermal conductivity was from [38] was used). Since values in italic were not reported, the typical range reported in literature was used instead [37,39–41]. The reported uncertainty range corresponds to 68% confidence interval.

Material	Skin	BaF2	Aluminum	Marble	Wood
Thermal Conductivity, $k$, Wm^−1K^−1	0.37	11.7	238	2.9	0.23–0.4
Density, $\rho$, kgm^−3	1000	4890	2700	2700	675
Specific Heat, $c_p$, Jkg^−1K^−1,	3770	410	917	881	3156
Thermal Diffusivity, $\alpha$, 10^6m^2s^−1	0.098	5.9	0.96	1.2	0.1
RMS roughness, $\sigma$, µm	21.7 ± 2	0.008	1–6	5–30	5–30
Microhardness, $H$, kPa	122.5 ± 5.8	-	-	-	-
Effective Shear Modulus, G, kPa	0.41 ± 0.1	-	-	-	-
Surface Asperity Slope, Δa, rad	0.3 ± 0.03	0.009	0.01–0.3	0.01–0.3	0.01–0.3

Figure 2. (a) Comparison between thermal contact resistance of finger and barium fluoride crystal interface calculated using CMY model (Equation (5)) and PM model (Equation (6)) and (b) comparison between experimentally measured (adapted from Ho and Jones [32]) and analytically predicted difference between finger temperature at the beginning and end of the 10 s contact period for contact pressure in the range of 0.78–11 kPa; the shaded areas correspond to possible range of properties indicated in Table 1 and specified in text, (c) temporal evolution of the finger temperature during the contact calculated using average properties in reported in Table 1 and closed-form expression in Equation (12) and CMY model (dashed lines) or PM model (solid lines) contact resistance models (arrows are used as visual guides to highlight the difference between the two), and (d) lower and upper limits of the values in (c) along with Ho and Jones experimental data [32].

Figure 3. Comparison between theoretical predictions of thermal contact resistance by CMY and PM models including experimental data on finger contact with (a) aluminum, (b) marble, and (c) wood surfaces from Maamir et al. [17]. The shaded areas correspond to possible range of properties indicated in Table 1 and specified in text.

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