## 3 Results and Discussion  
### Effect of the Zigzag Ratio and Pr Number on the Nu Number  
The effect of changing the ratio \(\mathrm{e_{2}/e_{1}}\) on the average Nusselt number, \(\mathrm{Nu_{av}}\), while keeping the rib width and pitch fixed, is presented in Figure 6. Two cases are shown. The first case is a ribbed tube of equal rib height ratio, \(\mathrm{e_{2}/e_{1}}=1\), while the second case is a ribbed tube with a zigzag rib arrangement, \(\mathrm{e_{2}/e_{1}}=0.25\). The first rib height to tube diameter is \(\mathrm{e_{1}/d}=0.1\), the width of the ribs to the tube diameter is \(\mathrm{w/d}=0.05\), and the pitch between ribs to the diameter is \(\mathrm{p/d}=1\). Three Prandtl number (Pr) values (0.71, 3.42, and 7.0) were investigated. The presence of the ribs increases the Nu values compared with that of the smooth tubes, due to the disturbance of the viscous sublayer as a direct effect of the ribs' presence. The average Nusselt number increases almost linearly as the Reynolds number increases. The slope of such lines increases with the Pr number. Likewise, the higher the Pr number, the higher the average Nu number. The higher Pr number indicates that heat transfer is higher via fluid momentum than via conduction diffusion. The decrease in the \(\mathrm{e_{2}/e_{1}}\) value decreases the average Nusselt number at all values of the Re number and Pr numbers. At a Pr of 7.0, Re = 10,000 and 60,000, and the reductions in \(\mathrm{Nu_{av}}\) were only 3.4 and 1.6%, respectively, although the decrease in the rib height \(\mathrm{e_{2}}\) was by 75% compared with \(\mathrm{e_{1}}\).  
Figure 5: Verification of the present study of a ribbed tube at \(\mathrm{e/d}=0.1\), \(\mathrm{p/d}=1.0\), and \(\mathrm{w/d}=0.05\): (**a**) Nusselt number at Pr = 0.71 and (**b**) friction factor.  
Figure 6: Average Nusselt number (\(\mathrm{Nu_{av}}\)) at different values of Pr and two rib height ratio (\(\mathrm{e_{2}/e_{1}}\)) values.  
The detailed effect of changing \(\mathrm{e_{2}/e_{1}}\) on the average Nusselt number is better presented in Figure 7. Four values of \(\mathrm{e_{2}/e_{1}}\) (0.25, 0.5, 0.75, and 1) are plotted while keeping the values of \(\mathrm{e1/d}=0.1\), \(\mathrm{w/d}=0.05\), and \(\mathrm{p/d}=1\). Two Reynolds number values of 10,000 and 60,000 were selected to represent the minimum and maximum values, respectively, of the present study, while the Pr number was kept constant at 7.0. The average Nusselt number increases as \(\mathrm{e_{2}/e_{1}}\) increases. At a lower Reynolds number, the change in the average Nusselt number was nonlinear. The rate of change in the average Nusselt number decreases as \(\mathrm{e_{2}/e_{1}}\) increases. At a high Reynolds number, the change in the average Nusselt number is almost linear, with \(\mathrm{e_{2}/e_{1}}\).  
Figure 8 demonstrates the average Nusselt number between the ribs, Nu, plotted at the midspan between the ribs at three values of the Reynolds number (10,000, 30,000, and 60,000). Two values of \(\mathrm{e_{2}/e_{1}}\) (0.25 and 1) are presented, while the Pr number is 7.0. At a Re of 10,000, the Nusselt number between the ribs decreases slightly as the Reynolds number increases. Equal-height ribbed tubes, \(\mathrm{e_{2}/e_{1}}=1\), show almost identical Nu values between the ribs. When a zigzag arrangement of \(\mathrm{e_{2}/e_{1}}=0.25\) is used, a zigzag pattern of the Nu is clear, and its lower values are downstream of the shorter ribs. This zigzag behavior of the Nu decreases as the Reynolds number increases and almost vanishes at Re = 60,000. At this Re value, the Nu values are not affected by reducing the value of \(\mathrm{e_{2}}\) upstream of the test section.