# Viscosity Prediction of Different Ethylene Glycol/Water Based Nanofluids Using a RBF Neural Network
## 4 Results and Discussion
As shown in Table 2, the values of the four evaluation criteria are \(RMSE=0.04630\), \(MAPE=2.09967\%\), \(SSE=0.42454\), and \(R^{2}=0.99927\) for the training samples, and \(RMSE=0.04443\), \(MAPE=2.43228\%\), \(SSE=0.03553\), and \(R^{2}=0.99904\) for the testing samples, which preliminarily indicates that the RBF neural network has a good ability to predict the viscosity ratio of ethylene glycol/water based nanofluids. To further evaluate the prediction performance of the RBF neural network for nanofluid viscosity, the following typical viscosity models which consider the effects of nanoparticle Brownian motion and aggregation are selected for analysis. Batchelor model [32]:  
\[\mu_{nf}=(1+2.5\phi_{p}+6.25\phi_{p}^{2})\mu_{f} \tag{9}\]  
Chen model [33]:  
\[\mu_{nf}=[1-\frac{\phi_{p}}{0.605}(\frac{r_{a}}{r_{p}})^{1.2}]^{-1.5125}\mu_{f} \tag{10}\]  
where \(r_{p}\) and \(r_{a}\) are the radius of nanoparticle and nanoparticle aggregation, respectively. Figure 5: Scatter plots of (**a**) training and (**b**) testing \(\mu_{nf}/\mu_{f}\) for the RBF predicted results and experimental data. Figure 6 and Table 4 respectively compare the prediction performances of the different models for the total viscosity data. It is easily seen that the RBF neural network has a better prediction accuracy than the above two typical models. The main reason is that the Batchelor model and [PERSON] model cannot fully quantitatively describe the relationship between the nanofluid viscosity ratio and the various factors including the nanoparticle properties, temperature, and base fluid. Moreover, Tables 5-7 respectively present the comparisons between the predicted viscosity ratio of the RBF neural network and the corresponding experimental data of [PERSON] et al. [12], [PERSON] et al. [13], and [PERSON] et al. [16]. It can be seen that there is good agreement between the RBF predicted and the experimental viscosity ratio of the different ethylene glycol/water based nanofluids. At the temperature range of 20-40 \({}^{\circ}\)C, the maximum and minimum prediction errors of the RBF neural network are respectively 5.788% and 0.434% for the experimental data of [PERSON] et al. [12]. For the viscosity ratio of the SiO\({}_{2}\)-ethylene glycol/water (50:50 by volume ratio) nanofluid provided by [PERSON] et al. [13], the RBF neural network can accurately predict the viscosity ratio with an average error of 1.772% at the nanoparticle volume fraction of 0.1%. Moreover, the comparisons shown in Table 7 further illustrate that the developed RBF neural network has high accuracy (average error: 2.097%) for predicting the viscosity ratio of CuO-ethylene glycol/water (60:40 by weight ratio) nanofluids. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Evaluation Criteria** & **RBF Neural Network** & **Batchelor Model** & **Chen Model** \\ \hline \(RMSE\) & 0.04615 & 0.82200 & 0.77129 \\ \(MAPE\) (\%) & 2.12738 & 24.2349 & 21.79539 \\ \(SSE\) & 0.46007 & 145.94871 & 128.49640 \\ \(R^{2}\) & 0.99925 & 0.76244 & 0.79085 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance evaluation of different modes for the total viscosity data. Figure 6: Prediction relative errors of different models for the total viscosity data. \begin{table}
\begin{tabular}{c c c c c} \hline \(T\) (\({}^{\circ}\)C) & \(\phi_{p}\) (\%) & Experiment (P) & RBF Prediction (Q) & \(\frac{1P-Q1}{P}\times\)**100\%** \\ \hline
28.65 & 0.1 & 1.082 & 1.099 & 1.549 \\
38.18 & 0.1 & 1.076 & 1.073 & 0.258 \\
47.91 & 0.1 & 1.064 & 1.063 & 0.067 \\
55.81 & 0.1 & 1.100 & 1.098 & 0.202 \\
61.28 & 0.1 & 1.132 & 1.078 & 4.746 \\
28.45 & 0.1 & 1.128 & 1.098 & 2.703 \\
36.55 & 0.1 & 1.119 & 1.082 & 3.323 \\
45.07 & 0.1 & 1.054 & 1.056 & 0.196 \\
50.95 & 0.1 & 1.091 & 1.078 & 1.243 \\
58.85 & 0.1 & 1.134 & 1.095 & 3.432 \\ \hline \end{tabular}
\end{table} 
Table 6: Comparisons of the RBF predicted viscosity ratio of SiO\({}_{2}\)-ethylene glycol/water (50:50 by volume ratio) nanofluids with the experimental data of [PERSON] et al. [13].