## 4 Results and Discussion
### Nusselt Number and Friction Factor  
Based on experimental and numerical results, Figures 9 and 10 show the variation of the shell-side Nusselt number \(Nu_{\mathrm{o}}\) and friction factor \(f_{\mathrm{o}}\) with the Reynolds number \(Re_{\mathrm{o}}\) for the base fluid and nanofluids at different concentrations, respectively. From Figure 9, it is found that \(Nu_{\mathrm{o}}\) increases with the increasing of \(Re_{\mathrm{o}}\). For a given \(Re_{\mathrm{o}}\), \(Nu_{\mathrm{o}}\) increases with an increasing in the MWCNTs concentration. Based on experimental data, the \(Nu_{\mathrm{o}}\) of the non-Newtonian nanofluids at weight fraction of 1.0 wt %, 0.5 wt %, and 0.2 wt % increases by 35%, 21% and 11% on average while compared with that of the base fluid at the same Reynolds number \(Re_{\mathrm{o}}\), respectively. It can be observed from Figure 10 that the experimental \(f_{\mathrm{o}}\) of the nanofluids are larger than that of the base fluid, and gradually increases with increasing in the weight fraction of nanoparticles and rapidly decreases with an increase in \(Re_{\mathrm{o}}\). Higher friction factor leads to greater pressure drop in shell side, which is the obvious drawback of nanofluid. The enhanced thermal conductivity and viscosity of nanofluid after adding MWCNTs are response for the increasing of \(Nu_{\mathrm{o}}\) and \(f_{\mathrm{o}}\), respectively.  
Figure 8: Ratio of shell side thermal resistance to overall thermal resistance at different shell-side Reynolds number.  
Both Figures 9 and 10 show that the simulation results fit very well with experimental data. The average deviations between experimental and numerical data are listed in Table 2. As can be seen in Table 2, the average deviations between numerical and experimental results are less than 6%, indicating that the single-phase flow model is able to simulate the heat transfer and flow characteristic of non-Newtonian base fluid and its nanofluids in shell side. However, with the increasing in weight fraction of nanoparticles, the values of \(Nu_{\mathrm{o}}\) and \(f_{\mathrm{o}}\) from simulation are less than those from experiment, indicating that the Brownian motion of nanoparticles and the interaction between nanoparticles and base fluid are other reasons for the enhancement of \(Nu_{\mathrm{o}}\) and \(f_{\mathrm{o}}\)[23], which are neglected in single-phase flow model. It can be predicted that the sing-phase flow model can effectively simulate the flow and heat transfer of nanofluids at low concentration because, in this case, the improvements of thermal properties are the main driver for the heat transfer enhancement, while, with the increasing of nanoparticles concentration, the Brownian motion of nanoparticles and the interaction between nanoparticles and base fluid play increasingly important role in the processes of flow and heat transfer.  
Figure 10: Friction factor based on experiment and simulation versus shell-side Reynolds number.  
Figure 9: Nusselt number based on experiment and simulation versus shell-side Reynolds number.  
The nanofluid tends to be two-phase flow, which should be simulated by multi-phase flow model for higher precision.  
Based on the results of numerical simulations, the local (Z = \(-500\) mm) temperature distributions of base fluid and nanofluids in helical baffled heat exchanger with elliptic tubes at \(V_{\text{f}}\) = 1.72 m\({}^{3}\cdot\)h\({}^{-1}\) are shown in Figure 11a-d. As shown in Figure 11, the temperature along the flow drops more slowly compared with nanofluids. As the concentrations of nanoparticles increase, lower temperature can be observed at the same position of heat exchanger due to the enhanced thermal conductivities after the addition of MWCNTs.