# Numerical Study of Heat Transfer Enhancement for Laminar Nanofluids Flow
## 3 Results & Discussion
### Case 5: Water-Based Nanofluid in Straight Microtube with Constant Heat Flux Using SPH  
In case 5, all the different nanofluids Al\({}_{2}\)O\({}_{3}\)-water, TiO\({}_{2}\)-water, CuO-water, SiO\({}_{2}\)-water and ZnO-water are used to investigate the Nu in a straight microtube with a length of 0.3 m and an inner  
Figure 11: Proposed correlation vs computed results and experiments of Nu in a straight tube.  
Figure 10: Comparison of Nu for: (**a**) Al\({}_{2}\)O\({}_{3}\)-water, TiO\({}_{2}\)-water, CuO-water, SiO\({}_{2}\)-water and ZnO-water and at \(\phi_{b}\) = 4%; (**b**) Al\({}_{2}\)O\({}_{3}\)-water, Al\({}_{2}\)O\({}_{3}\)-EG and Al\({}_{2}\)O\({}_{3}\)-turbine oil at \(\phi_{b}\) = 4%.  
diameter (d\({}_{\rm{t}}\)) of 0.5 mm. Results from the numerical simulation were compared with [PERSON]'s [51] correlation (Equation (15)) for validation purpose. A good agreement can be seen in between the present computed results of Nu with the predicted values from [PERSON]'s correlation, as shown in Figure 12.  
\[\begin{array}{l}\text{Nu }=\ \left[\begin{array}{c}1.953\left(RePr\frac{D}{L} \right)^{\frac{1}{3}}RePr\frac{D}{L}\geq 33.3\\ 4.364+0.0722 RePr\frac{D}{L}RePr\frac{D}{L}<33.3\end{array}\right.\end{array} \tag{15}\]  
It may be noted from Figure 13a,b that similar to the straight tube results, the wall temperature as well as fluid mean temperature in the straight microtube also decreases with increase in the nanoparticle concentration. The decrease in the temperatures of the wall and the fluid will result in a better heat transfer.  
Figure 14a shows the effect of the different type of nanoparticles for a water-based nanofluid. The Al\({}_{2}\)O\({}_{3}\)-water nanofluid has a better heat transfer enhancement as compared to that of all the other nanoparticles due to the higher thermal conductivity. Figure 14b shows the Nu for different fluids at the same Re = 950. The turbine oil-based nanofluid has the higher Nu, however it is not the nanofluid with the higher enhancement.  
Figure 12: Comparison of Nu with literature: Al\({}_{2}\)O\({}_{3}\)-water nanofluid at \(\phi_{b}\) = 4%.  
Figure 13: Effect of nanoparticle concentration in tube flow: (**a**) axial development of wall temperatures; (**b**) fluid mean temperatures for Al\({}_{2}\)O\({}_{3}\)-water.  
The Nu enhancement was 6% for Al\({}_{2}\)O\({}_{3}\)-water compared to water, 12% for Al\({}_{2}\)O\({}_{3}\)-EG compared to ethylene glycol and 5% for Al\({}_{2}\)O\({}_{3}\)-turbine oil compared to turbine oil. Based on these results, it may be concluded that ethylene glycol-based nanofluids are the best choice for heat transfer enhancement at the same Re and \(\phi_{b}\).  
For the average Nu in a microtube with constant heat flux a new correlation (Equation (16)) is proposed by fitting the data obtained from the numerical simulations. The correlation covers a wide range of the process parameters (25 \(<\) Re \(<\) 1500, 0 \(<\phi_{b}\)\(<\) 10, 6 \(<\) Pr \(<\) 500) under the laminar flow condition.  
Figure 15 shows the comparison between the present Nu results along with the experiments from [PERSON] et al. [52] and the computed data from [PERSON] et al. [53] with the present proposed correlation (Equation [16]). The predicted values of Nu are in good agreement (variation of \(\pm\)5%) with the computed values and the previous experimental results.  
\[Nu\ =\ 0.4561 Re^{0.27}Pr^{0.30} \tag{16}\]