*2.1. Empirical Formula Derivation of the Novel TEP Thermoelectric Performance*

Ordinarily, most of the engineering related to the heat-flow physical mechanics can be analyzed via the motion equations and underlying theories, but there are still many deductions that should be experimentally inspected in order to acquire realistic findings, because the deductions derived from the fundamental theories and motion equations may only be employed for the basic estimations. Units and scales are a manual conception with underlying relevance in the physical world, in which it is a more official way of signifying that kind thought. Accordingly, the dimensional analysis does not frequently render a whole exploration, yet it supplies the beneficial procedures toward an intact comprehension for exploiting the helpful results that are not petty and not distinct. Dimensional analysis is a variable skill and manner that may be accustomed to clarifying and demonstrating the conjunctions between physical quantities, and makes it possible to gather the consequences of estimations and tests in a concise and widespread formula, which can apply forecasts expeditiously. Dimensional analysis is actually adopted to assemble the consequences of experiments in simple and clear modus, so that we can achieve the ordinary fitting from a little number of examinations at a model scale. The present study obtains the empirical formulas of the novel TEP thermoelectric performance, which are derived from the experimental data and the dimensional analysis of factors in Vashy-Buckingham Pi (Π) Theorem. The dimensional analysis with intelligent experiment [22] was introduced, and the application of the empirical formula was searched for the electric charge density output of the novel TEP and waste heat development in the present work. The major ideology of the dimensional analysis with intelligent experiment is that the relationship can continuously be expressed as conjunctions between these Π-dimensional groups. The present study aims to illustrate that the dimensional analysis is a formidable means and covers extended, evident, and new achievements. The analysis procedure is performed to find out all the variables of the novel TEP thermoelectric performance through the repeating variable method resulting from the basic dimensional qualifications. For effective thermal conductivity and power generation empirical formulas of the novel TEP, the dimensional analysis procedure [22] was as follows:

$$K\_{tp} = \text{Function}\left\{ K\_{nf}, \mathcal{C}\_{nf}, \mathcal{F}\_{tp}, \ T\_{tp}, \mathcal{P}\_{tp}, \mu\_{nf}, \rho\_{nf} \right\} \tag{1}$$

$$\overline{P}\_{tp} = \text{Function}\left\{ V\_{n\nu} K\_{nf\nu} F\_{tp\nu} \; \rho\_{nf\prime} \; V\_{\epsilon\nu} T\_{tp\prime} \overline{P}\_{nf\prime} \; \mathbb{C}\_{nf\prime} \; P\_{tp} \right\} \tag{2}$$

Definitions for the correlated variables of thermoelectric values include the thermal conductivity of the novel TEP, *Ktp*, the thermal conductivity of nanofluid, *Kn f* , the nanofluid specific heat, *Cn f* , the nanofluid viscosity, *μn f* , the nanofluid density, *ρn f* , the temperature of novel TEP, *Ttp*, the filling amount of novel TEP, *Ftp*, the pressure of novel TEP, *Ptp*, the zeta potential of nanofluid, *Vn*, the electric charge density output of the novel TEP with nanofluid as electrolyte, *Ptp*, the standard electric potential of the novel TEP electrode, *Ve*, and the electric charge density output of the copper-aluminum battery cell with nanofluid as electrolyte, *Pn f* . We determined the relevant physical quantities and expressed the variables as the basic physical quantities, then selected the required variables to represent

each Π term and repeated variables. We then calculated through repeated variables and multiplied each Π term to find the dimensionless parameters of each Π term. The obtained Π term is expressed as a functional relationship. Equation (1) reveals the *Ktp* function, which was defined via the other seven variables, four of which were independent physical quantities, namely, mass (M), length (L), time (T), and temperature (Θ). *Ptp* was decided by the other nine variables, five of which were independent physical quantities consisting of M, L, T, Θ, and voltage (V). These can be employed as in Equation (2). Expressions of all variables were adopted through the M, L, T, Θ, and V, as follows: they are *Ktp* = MLT−3θ−<sup>1</sup> , *Kn f* = MLT−3θ−<sup>1</sup> , *Cn f* = L2T<sup>−</sup>2θ−<sup>1</sup> , *μn f* = ML<sup>−</sup>1T<sup>−</sup><sup>1</sup> , *ρn f* = ML−<sup>3</sup> , *Pn f* = MT−<sup>3</sup> , *Ttp* = [θ], *Ftp* = L3 , *Ptp* = ML<sup>−</sup>1T<sup>−</sup><sup>2</sup> , *Vn* = [V],

and *Ptp* = MT−<sup>3</sup> . Regarding the effective thermal conductivity of the novel TEP, there are four dimensionless Π numbers for the *Ktp*. This study chose four repeating variables, *Kn f* , *ρn f* , *Ftp*, and *Cn f* for extrapolations. These four repeating variables are multiplied through other non-repeating variables to gain the dimensionless Π parameters. The four Π groups are shown in Equation (3). For the electric charge density of the novel TEP, *Ptp*, five dimensionless Π number groups are determined separately and five repeated variables (*Pn f* , *Ve*, *Ftp*, *ρn f* , and *Cn f*) are chosen. The analysis procedure is the same as the empirical formula of Equation (3). Equation (2) displayed the electric charge density output functional equation. The five Π groups are exhibited in Equation (4). The known attributes of the novel TEP acquired from the thermoelectric experiment and experimental data are substituted into Equations (3) and (4) to obtain the indeterminate values of α, β, γ, λ, and τ. After simplification, the thermoelectric empirical formula of the novel TEP are derived in the present study.

$$\frac{\mathbf{K}\_{lp}}{\mathbf{K}\_{\mathrm{nf}}} = a \left\{ \frac{\mu\_{\mathrm{nf}} \cdot \mathrm{F}\_{\mathrm{tp}}^{\frac{4}{3}} \cdot \mathrm{C}\_{\mathrm{nf}}}{\mathrm{K}\_{\mathrm{nf}}} \right\}^{\frac{\alpha}{\beta}} \left\{ \frac{\mathrm{T}\_{tp} \cdot \rho\_{\mathrm{nf}}^{2} \cdot \mathrm{F}\_{\mathrm{tp}}^{\frac{2}{3}} \cdot \mathrm{C}\_{\mathrm{nf}}^{3}}{\mathrm{K}\_{\mathrm{nf}} c^{2}} \right\}^{\gamma} \left\{ \frac{\mathrm{P}\_{\mathrm{tp}} \cdot \rho\_{\mathrm{nf}} \cdot \mathrm{F}\_{\mathrm{tp}}^{\frac{2}{3}} \cdot \mathrm{C}\_{\mathrm{nf}}^{2}}{\mathrm{K}\_{\mathrm{nf}} c^{2}} \right\}^{\lambda} \tag{3}$$

$$\frac{\overline{\mathbf{P}}\_{\rm tp}}{\overline{\mathbf{P}}\_{\rm nf}} = \alpha \left\{ \frac{\mathbf{V}\_{\rm n}}{\mathbf{V}\_{\rm t}} \right\}^{\beta} \left\{ \frac{\mu\_{\rm nf}}{\overline{\mathbf{P}}\_{\rm nf}^{\frac{1}{3}} \cdot \mathbf{F}\_{\rm tp}^{\frac{1}{3}} \cdot \boldsymbol{\rho}\_{\rm nf}^{\frac{2}{3}}} \right\}^{\gamma} \left\{ \frac{\mathbf{T}\_{\rm tp} \cdot \boldsymbol{\rho}\_{\rm nf}^{\frac{2}{3}} \cdot \mathbf{C}\_{\rm nf}}{\overline{\mathbf{P}}\_{\rm nf}^{\frac{2}{3}}} \right\}^{\lambda} \left\{ \frac{\mathbf{P}\_{\rm tp} \cdot \mathbf{F}\_{\rm tp}^{\frac{1}{3}}}{\overline{\mathbf{P}}\_{\rm nf}^{\frac{2}{3}} \cdot \boldsymbol{\rho}\_{\rm nf}^{\frac{1}{3}}} \right\}^{\mathsf{T}} \tag{4}$$

This research reports the intelligent dimensional analysis methods [22] to be used to find the effective thermal conductivity values and the electric charge density output values of the novel TEP. Consequently, inevitable errors certainly exist among the real values owing to the artificial operation, the restriction of preciseness of the experimental instrument, the measured data during experiment, and the values deriving from experimental data. It is essential to premeditate the trial errors so as to find the experimental reliance before resolving the experimental results based on this, where the notion of the error propagation is employed to appraise the experimental errors and basic functionary relations. Many items of effective thermal conductivities and electric charge density outputs are applied separately to survey the effective thermal conductivity values and the electric charge density output values of the novel TEP during the thermoelectric experiments. The effective thermal conductivity values and electric charge density output values respectively pertain to derived variables consisting of *Ktp*, *Kn f* , *Cn f* , *μn f* , *ρn f* , *Ttp*, *Ftp*, *Ptp*, *Vn*, *Ptp*, *Ve*, and *Pn f* which are measured with experimental apparatus. The error of experimental apparatus is propagated to the consequence during deduction, and thereby transforms the errors of effective thermal conductivities and electric charge density output values. The experimental error is indicated with a relative error, and the maximum relative errors of effective thermal conductivities and electric charge density outputs are within ±5.5% and ±10%, respectively. The experimental results are exploited in dimensional analysis to derive the empirical formulas of the novel TEP.
