*3.5. Thermal Conductivity and Power Density Empirical Formulas of the Novel TEP*

In the present study, the purpose of deducing these empirical formulas, including thermal conductivity and power density of the TEP, is to utilize the intelligent dimensional analysis [22] to obtain these values under certain conditions and to enter several basic parameters that did not require instrument measurement. Comprehensive parameters in the empirical formulas were used to forecast the thermoelectric performances of the TEP in the present paper. According to the empirical formulas of the novel TEP derived from the experimental data, they were applicable for 2 wt.% titanium dioxide nanofluids between 25 and 100 ◦C with vacuum pressure between 400 and 760 torr and with an 80% filling ratio.

Originally, for the thermal conductivity formula of the novel TEP, we substituted the experimental parameters of *Kn f* , *ρn f* , *Ftp*, *Cn f* , and *μn f* into Equation (3). In order to facilitate a simple calculation, smaller changes of *ρn f* , *Cn f* , and *μn f* will be substituted into Equation (3) by mean values at different temperatures. As the product of (*Cn f* · *μn f*) was a negligible change as vacuum pressure increased, it was ignored because of no impact on the thermal conductivity of TEP and then it was assumed that *β* = 0. Therefore, Equation (5) can be briefly acquired as:

$$\mathbf{K}\_{tp} = 0.62a(50270.55T\_{tp})^\gamma (13.17P\_{tp})^\lambda \tag{5}$$

Equation (5) illustrated that *Ktp* was determined by the temperature and vacuum pressure of the novel TEP. The factor of vacuum pressure was temporarily not pondered. Setting λ = 0 and taking 400 torr as a benchmark revealed that *Ktp* changed along with temperature at 400 torr. Substituting values from Table 3 into Formula (5), we obtained Equation (6):

$$\mathbf{K}\_{tp} = 1.2042 T\_{tp}^{0.866} \tag{6}$$

**Table 3.** Parameters substituted into the empirical formulas of the novel TEP.


The parameters in Table 3 displayed that the changes in temperature and thermal conductivity of TEP are similar at different vacuum pressures. Consequently, λ value and polynomial function were estimated through the parameters of Table 3 based on 50 ◦C as the basis under different vacuum pressures. The vacuum pressure was then a function of λ. In summary, the final empirical formula of the novel TEP thermal conductivity is Equation (7):

$$\mathbf{K}\_{lp} = 1.2042 T\_{lp}^{0.866} (13.17 P\_{lp})^{\lambda} \lambda = -5.10769 \times 10^{-10} P\_{lp} ^{\;3} + 9.69923 \times 10^{-7} P\_{lp} ^{\;2} - 6.1349 \times 10^{-4} P\_{lp} + 0.1229 \quad \text{(7)}$$

For the power density formula of the novel TEP, the known parameters were substituted into Formula (4). Since the dimensionless parameters were regarded as having little effect on the power density of the novel TEP, the index was assumed to be zero, the same as the processes of deriving the empirical formula of TEP thermal conductivity. Subsequently, the derived results were exhibited in Equation (8), resulting from Equation (4):

$$\overline{P}\_{\rm tp} = 0.34 \alpha (209.19 T\_{tp})^{\lambda} (0.084 P\_{tp})^{\tau} \tag{8}$$

Without considering the factor of vacuum pressure first, *τ* was assumed to be zero when the vacuum pressure of 400 torr was utilized as a benchmark to find the functional relationship of the power density and temperature at this pressure, as shown in Equation (9):

$$
\overline{P}\_{\rm tp} = \left(2.167 \times 10^{-23}\right) T\_{\rm tp}^{-9.333} \tag{9}
$$

Finally, vacuum pressure was taken into consideration and 50 ◦C was employed as the benchmark to find the index for *τ* value. Equation (10) illustrates the power density empirical formula of the novel TEP:

$$\overline{P}\_{\rm lp} = \left(2.167 \times 10^{-22}\right) T\_{\rm lp}^{-9.333} (0.084 P\_{\rm lp})^{7} \tau = 9.1203 \times 10^{-9} P\_{\rm lp}^{-3} - 1.5595 \times 10^{-5} P\_{\rm lp}^{-2} + 8.4313 \times 10^{-3} P\_{\rm lp} - 1.461 \tag{10}$$

As mentioned above, many problems still exist in the present thermoelectric experiments that cause error rates for thermal conductivity and low and unstable current and power output. The calculation method of the error rate is shown in Formulae (11) and (12), where EK is the error value between the instrument value of Ktp,e measured by the equipment and the calculated value of Ktp,f calculated by the empirical formula of the thermal conductivity coefficient of the novel TEP. EP is the error value between the instrument value of *Ptp*,*<sup>e</sup>* measured by the equipment and the calculated value of *Ptp*, *<sup>f</sup>* calculated by the empirical formula of the power density of the novel TEP. Tables 4 and 5 illustrate the error rates of EK and EP. Initial estimations show that the calculated and measured values were similar. The derived empirical formulas were calculated based on the vacuum pressure of 400 torr and temperature of 50 ◦C, in which the error rates of the calculated value and the measured value will be lower, and the largest error rate of the thermal conductivity coefficient was 5.06%, indicating that the overall error rates were small. The power density will affect the stabilities of the current and voltage due to the oxidation of the aluminum electrode, and the longer the reaction time, the more the current will decay, and thus the error rate will also be affected. The maximum error rate of power density was 11.27%.

$$E\_K = \frac{K\_{tp, \varepsilon} - K\_{tp, f}}{K\_{tp, \varepsilon}} \times 100\% \tag{11}$$

$$E\_p = \frac{\overline{P}\_{tp, \mathcal{E}} - \overline{P}\_{tp, f}}{\overline{P}\_{tp, \mathcal{E}}} \times 100\% \tag{12}$$


**Table 4.** Comparison of measured and calculated thermal conductivities W/(m·K) of TEP and error rates under different vacuum pressures.

**Table 5.** Comparison of measured and calculated power densities μW/cm2 of TEP and error rates under different vacuum pressures.

