*2.4. Heat Transfer Enhancement*

The considered test structure was analyzed, including the surrounding air environment. Furthermore, platinum resistors' separation distance *d* = 100 nm in the benchmark structure is comparable to the surrounded air mean free path length Λ ≈ 65 nm. Therefore, the heat flow can be approximated by the following equation [41]:

$$q\_{\rm air} = \frac{k\_{\rm air}}{\langle d + a\Lambda \rangle} (T\_{\rm Pt,A} - T\_{\rm Pt,B}) \tag{23}$$

where *a* is used to describe the interaction between the gas molecules and the solid walls [42], typically *a* = 1, and <·> stands for ensemble average. The final equivalent air conductance between the mentioned resistors was estimated by using the following simplified formula (compare with value in Table 2):

$$k\_{\rm air,qw,PtA-PtB} = k\_{\rm air} \frac{d}{\langle d + a\Lambda \rangle} \approx 0.961 \, k\_{\rm air} \approx 0.961 \cdot 0.0263 \, \text{W/(mK)} \tag{24}$$

The photon tunneling and the radiative heat transfer between platinum resistor parallel surfaces were also analyzed. The photon tunneling was neglected due to the small amount of heat flux transport (*Sz*)*evamescant* max = 2.2 · <sup>10</sup>−19W/m2 (calculated for vacuum environment) in comparison to the main heat flux Pt-SiO2 *q*Pt\_SiO2,cond<sup>≈</sup>19.3 MW/m<sup>2</sup> and the heat conduction through *q*Pt\_Pt,cond ≈ 0.917 *q*Pt-SiO2,cond >> (*Sz*)*evamescant* max in the air for the steady state [43,44]:

$$\left(\left(S\_z\right)\_{\text{max}}^{\text{cumuresant}}\right) \approx \frac{k\_B^2 \cdot \left(T\_{PtA} + 273.15\,\text{K}\right)}{24\hbar b^2} \left[\text{W/m}^2\right] \tag{25}$$

where *b* is an inter-atomic distance *b* ≈ 39.12 nm for Pt, *k*B means the Boltzmann constant, and - is the reduced Planck constant. Moreover, the radiative heat transfer (*q*SB rad) was also neglected, due to the insignificant emitted energy from warmer (*TPtA*) to colder platinum resistor (*TPtB*) parallel surfaces *q*Pt-SiO2, cond / *q*SB rad ≈ 8.3·1025:

$$\begin{aligned} q\_{SB\,md} & \approx 5.6693 \,\frac{\text{W}}{\text{m}^2 \left(\frac{\text{K}}{100}\right)} \cdot \left(\sqrt{2} - 1\right) \cdot \left\{ \left(\frac{T\_{PA} + 273.15 \text{K}}{100}\right)^4 - \left(\frac{T\_{PB} + 273.15 \text{K}}{100}\right)^4 \right\} \\ & \text{and } 72 \le q\_{SB\,md} \le 146 \,\text{W}/\text{m}^2 \text{ for } T\_{PA} \approx 59 \text{K and } 4.7 \text{K} \le T\_{PB} \le 31 \text{K} \end{aligned} \tag{26}$$

Moreover, considering investigation presented above, some additional changes in the proposed model are needed. Thus, for the air area between resistors' surfaces and for contact areas between platinum and silicon dioxide, a fractional order of the temperature-rise function space derivative has been employed, according to the theory described in [31,45]. This theory, based on the Grünwald–Letnikov definition of the fractional derivative, allows us to establish the following formula reflecting the temperature-rise function's improvement for the central difference of the FDM scheme [31,45]:

$$\begin{split} \, \_{GL}D\_{0,\mathbf{s}}^{a\_{\mathbf{s}}}T(\mathbf{s}) &= \frac{1}{(\Lambda s)^{\mathbf{d}}} \cdot \sum\_{k=0}^{\text{round}(a\_{\mathbf{s}},0)} (-1)^{k} \frac{\Gamma(a\_{\mathbf{s}}+1)}{\Gamma(k+1)\cdot\Gamma(a\_{\mathbf{s}}-k+1)} T(\mathbf{s}-k\cdot\Delta\mathbf{s} + \frac{a\_{\mathbf{s}}\cdot\Delta\mathbf{s}}{2}), \\ \text{for } a\_{\mathbf{s}} \in \mathbb{R}\_{+}, \text{ } \Delta\mathbf{s} \to \mathbf{0} \end{split} \tag{27}$$

where Δ*s* is the mesh nodes distance, α is investigated fractional order, Γ is the special gamma function, and *round*(*<sup>m</sup>*,*<sup>n</sup>*) is the function rounding the value *m* to *n* digits. Considering fractional order of derivative, we needed approximating investigated function values in points between nodes of a discretization mesh. The mentioned approximation depends on function values in neighboring mesh points. Thus, the right-hand-side part of Equation (27), being an investigated approximation, can be reflected by using the following equations [45]:

1 (<sup>Δ</sup>*s*)<sup>α</sup>*<sup>x</sup>* · ) 2 *k*=0 (−<sup>1</sup>)*<sup>k</sup>* Γ (<sup>α</sup>*x*+<sup>1</sup>) Γ (*k*+<sup>1</sup>)·<sup>Γ</sup> (<sup>α</sup>*x*<sup>−</sup>*k*+<sup>1</sup>)*<sup>T</sup>* **x** − *k* · Δ**x** + <sup>α</sup>*x*·Δ**<sup>x</sup>** 2 , *t* · 1 = = ( α*x*2 −<sup>1</sup>)·*<sup>T</sup>* (*x*+2·Δ*x*,*y*,*<sup>t</sup>*)+(<sup>2</sup>−α*x*<sup>2</sup> <sup>−</sup>α*x*·( α*x*2 −<sup>1</sup>))·*<sup>T</sup>* (*x*+Δ*x*,*y*,*<sup>t</sup>*) (<sup>Δ</sup>*x*)<sup>α</sup>*<sup>x</sup>* + + <sup>α</sup>*x*·(<sup>α</sup>*x*−<sup>1</sup>) 2 ·( α*x*2 <sup>−</sup><sup>1</sup>)−α*x*·(<sup>2</sup>−α*x*2 ) ·*T* (*<sup>x</sup>*,*y*,*<sup>t</sup>*)+(<sup>2</sup>−α*x*<sup>2</sup> )· <sup>α</sup>*x*·(<sup>α</sup>*x*−<sup>1</sup>) 2 ·*T* (*<sup>x</sup>*−Δ*x*,*y*,*<sup>t</sup>*) (<sup>Δ</sup>*x*)<sup>α</sup>*<sup>x</sup>* +( α*x*2 −<sup>1</sup>)·*<sup>T</sup>* (*<sup>x</sup>*,*y*+2·Δ*x*,*<sup>t</sup>*)+(<sup>2</sup>−α*x*<sup>2</sup> <sup>−</sup>α*x*·( α*x*2 −<sup>1</sup>))·*<sup>T</sup>* (*<sup>x</sup>*,*y*+Δ*x*,*<sup>t</sup>*) (<sup>Δ</sup>*x*)<sup>α</sup>*<sup>x</sup>* + + <sup>α</sup>*x*·(<sup>α</sup>*x*−<sup>1</sup>) 2 ·( α*x*2 <sup>−</sup><sup>1</sup>)−α*x*·(<sup>2</sup>−α*x*2 ) ·*T* (*<sup>x</sup>*,*y*,*<sup>t</sup>*)+(<sup>2</sup>−α*x*<sup>2</sup> )· <sup>α</sup>*x*·(<sup>α</sup>*x*−<sup>1</sup>) 2 ·*T* (*<sup>x</sup>*,*y*−Δ*x*,*<sup>t</sup>*) (<sup>Δ</sup>*x*)<sup>α</sup>*<sup>x</sup>* for α*x* ∈ (2, 2.5), Δ*x* → 0 (28)

**Table 2.** Considered material parameters' values [46].


\* See Section 2.4.

The formula above replaces the classic approximation of the Laplacian described by Equation (9).
