High-pressure stage gas turbine blades feature serpentine passages where rib turbulators are installed to enhance heat transfer between the relatively colder air bled off from the compressor and the hot internal walls. Most of the prior studies have been restricted to Reynolds number of 90,000 and several studies have been carried out to determine geometrically optimized parameters for achieving high levels of heat transfer in this range of Reynolds number. However, for land-based power generation gas turbines, the Reynolds numbers are significantly high and vary between 105 and 106. The present study is targeted toward these high Reynolds numbers where traditional rib turbulator shapes and prescribed optimum geometrical parameters have been investigated experimentally. A steady-state liquid crystal thermography technique is employed for measurement of detailed heat transfer coefficient. Five different rib configurations, viz., 45 deg, V-shaped, inverse V-shaped, W-shaped, and M-shaped have been investigated for Reynolds numbers ranging from 150,000 to 400,000. The ribs were installed on two opposite walls of a straight duct with an aspect ratio of unity. For very high Reynolds numbers, the heat transfer enhancement levels for different rib shapes varied between 1.4 and 1.7 and the thermal hydraulic performance was found to be less than unity.
Higher turbine inlet temperatures are required for high turbine efficiency. Hot gas path turbine components are subjected to elevated heat loads from hot gases exiting the combustor section. To protect the hot gas path components from hot gases, sophisticated internal and external cooling concepts are employed in turbine airfoils. Relatively colder air bled off from the compressor section is routed through complicated internal cooling passages to cool the hot internal walls of turbine airfoils. However, the coolant bled from the compressor reduces the turbine efficiency. Hence, it is imperative to develop cooling concepts, which have high heat transfer enhancement levels at relatively lower consumption of pumping power. Considerable research has been carried out to develop internal cooling concepts, which have high rates of heat transfer between the coolant and the blade internal walls.
Turbine airfoil internal cooling passages are equipped with rib turbulators for heat transfer enhancement. Han et al.  presented a detailed overview of the various geometric and flow parameters that influence ribbed channel heat transfer like rib spacing, their orientation, channel aspect ratio, rib shape, etc. First few studies were focused on rib turbulators placed orthogonal to flow, where the heat transfer enhancement mechanism was due to flow reattachment leading to very thin turbulent boundary layer . Heat transfer enhancement levels by rib turbulators underwent significant enhancement when ribs were placed at an angle of attack with the bulk flow. The heat transfer enhancement in such configurations was due to an increase in near-wall shear stress due to the generation of secondary flows and an increase in turbulent transport [3–6]. Different types of inclined ribs such as crossed ribs, V-shape, inverted V-shape, broken V-shaped rib, M-shaped, W-shaped, criss-cross pattern formed by 45-deg angled ribs, rib channel with grooves, rib dimpled compound channels, etc. have been studied in Refs. [7–15].
Heat transfer coefficient measurement techniques include steady-state copper blocks, transient liquid crystal and infrared thermography technique, steady-state liquid crystal thermography, etc. Leiss , Han et al. , and Han  have used thin foil heater steady-state technique for surface heat transfer coefficient measurement. Fan and Metzger , Wagner et al. , and Han et al.  have all used copper plates and presented regional averaged measurements inside stationary and rotating internal cooling channels. Steady-state techniques employing copper-based heating blocks provide regionally averaged heat transfer coefficient. However, for detailed knowledge of the local heat transfer coefficient, liquid crystal thermography is used. Ekkad and Han  were the first to present detailed Nusselt number distributions for a two-pass square channel using transient liquid crystal technique. Their study provided useful insight into the role of rib induced secondary flow on local heat transfer. Jenkins et al.  used transient liquid crystal technique to investigate the heat transfer distribution and pressure losses for a ribbed two-pass varying aspect ratio cooling channel with different divider wall-to-tip wall distance. Transient liquid crystal experiments provide useful insight into heat and fluid flow in complicated internal cooling ducts. These experiments are short duration and employ a low thermal conductivity material for the one-dimensional semi-infinite assumption. Liquid crystals can also be used in steady-state experiments as well. Tanda and Abram  carried out experiments using steady-state liquid crystal thermography on 45-deg angled turbulators for Reynolds number ranging from 9000 to 35,500 and reported heat transfer augmentation in the range of 2.7–1.8. Steady-state liquid crystal thermography technique was used by Tanda  to investigate the effects of pitch-to-height ratio of 45-deg ribs in a rectangular channel with an aspect ratio of five.
Based on our literature survey, we found that several studies have been carried out in the past to explore different rib shapes in terms of heat transfer enhancement and thermal hydraulic performance (THP) using different measurement techniques. However, most of these studies investigated heat transfer characteristics of rib roughened ducts upto Reynolds numbers of 90,000, catering toward aircraft engines. An experimental study by Rallabandi et al.  investigated heat transfer and frictional characteristics of 45-deg angled rib turbulators for Reynolds numbers ranging from 30,000 to 400,000. This was the first study to push the rib roughened duct heat transfer database to such high Reynolds numbers. The authors employed steady-state copper bar heat transfer measurement technique and reported regionally averaged Nusselt number ratio (Nu/Nu0).
The present study is focused on detailed measurement of heat transfer coefficient for different rib shapes at very high Reynolds numbers. The Reynolds numbers ranged from 150,000 to 400,000 and were chosen for land-based gas turbine engines. Detailed heat transfer coefficients were measured using steady-state liquid crystal thermography. Five different rib configurations, viz., 45-deg inline, M-shaped, W-shaped, V-shaped, and inverted V-shape, have been investigated. The rib turbulators were square in shape and the rib height to channel hydraulic diameter ratio (e/dh) was 0.1 and rib pitch to rib height ratio (p/e) was 10. The ribs were installed on two opposite walls of a straight duct with an aspect ratio of unity. Heat transfer enhancement levels (Nu/Nu0) and thermal hydraulic performance of different rib turbulator configurations have been reported and the heat transfer enhancement mechanism at high Reynolds numbers has been discussed.
Figure 1 presents a schematic of the experimental setup. Laboratory ambient air was drawn into the test section via a pressure blower (The New York Blower Company, wheel type 1710 ALUM, static pressure 20.1 “H2O”). The blower was driven by a 20 HP electric motor and the motor was controlled by a fine tune variable frequency drive. The blower was connected to a contraction section, which gradually converted the circular cross section to a square cross section. The square duct featured rib turbulators of different shapes. The duct was made out of plexiglass, which is a low thermal conductivity transparent material. The rib shapes were additively manufactured using ABS and had a thermal conductivity of 0.19 W/mK. Low thermal conductivity materials were used for assembling the test sections to minimize the heat loss during the forced convection experiments. Liquid crystal sheets were installed between the rib pitches for measurement of local wall temperatures. A wide-band liquid crystal sheet (R30C5W) manufactured by LCR Hallcrest was used in the present study. The assembly of liquid crystal sheet is shown in Fig. 2.
Flexible heater of size 101.6 mm × 304.8 mm and thickness of 0.127 mm, manufactured by Omega Engineering (KH-412/10-p) was glued on top of plexiglass enclosure. The power to the heater was provided by a fine control variac (Circuit specialist: TDGC-0.5KM). The variac was capable of supplying voltage up to 130 V at 60 Hz. The voltage and resistance across the heater were measured by a highly accurate Agilent digital Multimeter (34401A). To uniformly distribute the heat flux across the channel span, a thin brass sheet of thickness 0.635 mm was secured on top of the flexible heater. Liquid crystal sheets of thickness 0.0254 mm were then glued on top of the brass foil. Different layers of heater, brass foil, and liquid crystal sheets were compressed via uniformly spaced through bolts on the side walls. The side walls also had provisions for accommodating the extreme ends of ribs. Hence, compression force on the side walls through bolts also helped disburse the compression force along the ribs in the channel span. It was made sure that different layers of the heat transfer test section were in good contact with each other.
The liquid crystal color change was illuminated by diffused cool lighting kept at an angle of 45 deg with respect to the bulk flow direction. It was made sure that there were no reflections from the side walls and top wall on liquid crystal color change at the bottom end of the test section. The test section was covered with a black cover to block any ambient lighting. The liquid crystal color change was viewed from the opposite end through the transparent plexiglass. Hence, the data on the ribs could not be obtained due to optical blockage. The color change was captured by a GoPro Hero 4 camera at a spatial resolution of 1080 × 1920. The camera field of view was adjusted such that only the region of interest was captured.
Description of Test Configurations and Experimental Conditions.
Five different rib turbulator shapes were tested in the present study. Different rib configurations have been shown in Fig. 3. The channel hydraulic diameter was ten times the rib height. Hence, the channel blockage due to ribs was 20% since ribs were installed on opposite walls of the channel. Heat transfer coefficients were measured on only one wall given the symmetrical nature of the fluid domain. The total length of the test section was six times the channel hydraulic diameter. The rib pitch was ten times the rib height. The ribs were at an angle of attack of 45 deg with respect to the bulk flow direction. The heat transfer measurement pitch has been identified by red dashed region, which was at a distance of 3.75 times the channel hydraulic diameter. Although this length was not sufficient for proper flow development, the smooth converging duct helped in providing a smooth velocity profile at the entrance of the test section.
Steady-state liquid crystal thermography experiments were carried out at five Reynolds numbers ranging from 150,000 to 400,000. To calculate the normalized friction factor, the total pressure drop across the channel was measured.
Reynolds Number and Heat Transfer Coefficient Calculation Methodology
Reynolds Number Calculation.
Thermochromic Liquid Crystal Sheet Color Change Calibration.
The liquid crystal thermography technique works on the concept that the different colors can be correlated accurately to known wall temperatures. In this study, we used a Hue-based calibration technique to establish the relationship between Hue and wall temperature. Four T-type thermocouples were taped via aluminum tape on to the liquid crystal sheet. For calibration experiments, several different heat fluxes were provided to obtain multiple images of LC color change and wall temperatures. The LC color change captured by the CCD camera was transferred to the workstation. An in-house code was developed to convert the raw color change to Hue values. Hue value at every pixel was calculated from the red, green, and blue color content of the pixel .
Figure 4 shows the variation of Hue with respect to wall temperature. It should be noted that four different thermocouples yielded near-similar calibration curves and the color play was also obtained in the manufacturer specified band. The relationship between wall temperature and Hue will be used later for determination of local wall temperatures. Four identical hue-wall temperature curves indicate that the calibration was high quality, and the subsequent wall temperature calculation through this relationship was very accurate.
Heat Transfer Coefficient Calculations.
The LC sheet was assembled such that maximum heat transfer from heater through conduction only, occurred toward the airflow, in order to accurately capture the convective heat transfer rates. The thermal conductivity of brass foil (109 W/mK) was approximately 570 times higher than the plexiglass (0.19 W/mK)on the backside of the heater. Hence, it can be assumed that the heat losses were mostly in the direction of ambient air, i.e., the duct where coolant will flow in forced convection experiments. However, losses through the plexiglass on the back side of the foil heater have also been accounted for in the total heat loss.
After a certain Reynolds number was established through the duct, heat flux was increased such that every pixel in region of interest had some color change, so that it can be related to a known wall temperature Tw(x, y). The bulk fluid temperatures were measured at the inlet and exit of the test section and also calculated through overall energy balance. Due to the addition of heat into the fluid domain, bulk fluid temperature is expected to increase with increasing streamwise distance. However, the total heat addition into the fluid domain was negligible compared with the energy required to raise the fluid temperature by 0.1 °C considering very high heat capacity of the fluid flow. Hence, the bulk fluid temperature measured at the inlet of the rib pitch where the heat transfer coefficient has been calculated has been used as the reference fluid temperature in Eq. (3).
The thermal conductivity of air was calculated at the film temperature, Tf = 0.5(Tw + T∞). The Nusselt number was normalized with Dittus-Boelter correlation for Nusselt number in a developed turbulent flow in a circular tube.
The present paper reports heat transfer and pressure drop results at different Reynolds numbers. The quantities which have uncertainties are Reynolds number, friction factor, normalized friction factor, Nusselt number, Nusselt number ratio (Nu/Nu0), and thermal hydraulic performance. A sequential perturbation method prescribed by Moffat  was used for uncertainty calculation. The uncertainty for Reynolds number ranging from 170,000 to 370,000 varied between 3.7% and 0.68%. The uncertainty in calculation of Nusselt number ratio (Nu/Nu0) varied between 6% and 10%. The uncertainty in calculation of normalized friction factor (f/f0) varying between 8.5 and 10.5 was between 6.5% and 1.5%. The uncertainty in calculation of thermal hydraulic performance varying between 0.65 and 0.8 was between 3.9% and 6.8%.
Results and Discussion
Detailed Nusselt number ratio (Nu/Nu0) for different configurations have been presented and discussed in this section. Further, pitch-averaged Nusselt number ratio is presented with respect to Reynolds number. Normalized friction factor and thermal hydraulic performance have been presented with respect to Reynolds number.
Detailed Nusselt Number Ratio (Nu/Nu0).
Figure 5 shows the detailed Nusselt number ratio for 45-deg rib configuration. The heat transfer enhancement mechanism by ribs placed at an angle of attack with respect to bulk coolant flow is through the formation of thin thermal boundary layer regions and increase in near-wall shear due to the combined action of flow reattachment and increase in near-wall turbulent kinetic energy (TKE) due to rib-induced secondary flows. Further, the rib-induced secondary flows interact with duct sidewalls and result in the generation of a counterrotating vortex pair which increases turbulent transport between top and bottom walls subjected to heat loads in actual gas turbine blades. For the case of 45-deg angled rib turbulators, the secondary flows originate from the rib and duct sidewall intersection and travel along the rib. The reattachment of the flow separated due to ribs results in heat transfer enhancement as a result of a very thin thermal boundary layer. Also, the secondary flows enhance the near-wall shear due to a swirling action and increase the near-wall TKE.
For very high Reynolds number cases, the Nusselt number ratio levels were nearly similar to each other. In contrast to relatively lower Reynolds number studies on 45-deg angled rib turbulators , the higher Reynolds number heat transfer distribution was nonuniform within a rib pitch, with very high heat transfer at one corner and very low heat transfer upstream of the downstream rib. From detailed contours, it is apparent that the flow turning by 45-deg angle of attack was not very effective as has been observed in low Reynolds number studies (up to Reynolds number of 90,000) . Further, the deflected flow traveling along the rib, postinteraction with the duct sidewall, starts to travel along that wall, resulting in enhanced heat transfer. A parametric study of the effect of rib angle of attack is recommended for very high Reynolds numbers, as higher angle of attack appears to be more promising in terms of achieving higher heat transfer enhancement.
Figure 6 shows the detailed Nusselt number ratio for V-shaped rib turbulators. The heat transfer mechanism for V-shaped ribs is different compared with 45-deg ribs, as there are two 45-deg ribs joined at the channel centerline in V-shape ribs. The coolant interacts with the sharp corner at the channel centerline and secondary flows travel toward duct side wall. Since the ribs were installed on two opposite walls, two pairs of counterrotating vortices were present in V-shaped configurations, hence the turbulent transport mechanism in V-shaped ribs is stronger compared with 45-deg ribs. The individual strength of turbulent transport of one CRVP will be weaker when compared with the CRVP of 45-deg ribs. Also, the strength of secondary flows induced by V-shaped ribs will be weaker compared with that of 45-deg because of the difference in length of ribs. However, for V-shaped ribs, there are two CRVPs and two secondary flow streams compared with one CRVP and one secondary flow stream in 45-deg ribs. Hence, it is expected that overall heat transfer enhancement of V-shaped ribs will be higher than that of 45-deg ribs. From detailed contours, it can be seen that the heat transfer enhancement regions were aligned along the ribs. A large region of relatively lower heat transfer enhancement was also seen in V-shaped ribs, again indicating the conventional arrangement of p/e = 10, might be insufficient to allow the expected Nusselt number enhancement contour similar to low Reynolds number cases .
Figure 7 shows the detailed Nusselt number ratio for the inverse V-shaped ribs. In this case, secondary flows are generated from the corners near the duct sidewall and travel toward the channel centerline. Further, two pairs of counterrotating vortices will exist similar to V-shaped ribs. Inverse V-shaped ribs had lower heat transfer compared with the V-shaped ribs. Coolant momentum flux is expected to be higher along the channel centerline, hence the strength of rib induced secondary flows for the V-ribs is expected to be higher compared with the inverse V-shaped ribs. Low heat transfer regions were observed for inverse V-shaped ribs as well, just upstream of the downstream rib.
Figure 8 shows the detailed Nusselt number ratio for W-shaped ribs. In this case, four pairs of counterrotating vortices are observed due to the presence of two pairs of V-shaped ribs connected at the channel centerline and installed on two opposite walls. The heat transfer around the duct side wall was found to be higher, since the strength of secondary flows induced from the rib closest to the side wall was stronger than those present at channel centerline. Further, due to the higher proximity of the rib and side wall, the turbulence transport due to the CRVPs was stronger near the duct side walls. The dominant mechanism for heat transfer in V-shaped and W-shaped ribs is through the turbulence transport due to CRVPs compared with an increase in near-wall shear due to the interaction of secondary flows with the duct end wall.
For low Reynolds number cases, the effect of secondary flow induced enhancement in near-wall TKE of fluid is the dominant mechanism of heat transfer enhancement, where distinctly symmetric high and low heat transfer regions could be identified . For the inverse W-shaped ribs, referred as M-shaped ribs (Fig. 9), similar trends were observed as was between V-shaped and inverse V-shaped ribs.
Comparing all five configurations in the present study, some generalized comments can be made. 45-deg parallel rib configuration was different from other configurations, since the secondary flows induced to travel in only one direction, i.e., along the rib from one side wall to the other. However, V-shaped and W-shaped ribs had some geometrical similarities and inverse V- and M-shaped ribs had some geometrical similarities, in the way how secondary flows propagate within a rib pitch. For V- and W-shaped ribs, the secondary flows induced by the ribs travel toward channel side walls, and this mechanism has been proved to result in enhanced heat transfer because of strengthening of CRVP due to support from the channel side wall. Whereas in the case of inverse V- and M-ribs, the secondary flows originate from the channel side walls and travel toward channel centerline, leading to reduced strength of the CRVPs.
Regionally Averaged Nusselt Number Ratio (Nu/Nu0), Friction Factor (f/f0) and Thermal Hydraulic Performance (THP).
The variation of regionally averaged Nusselt number ratio (Nu/Nu0) with Reynolds number has been shown in Fig. 10. The rib pitch chosen for averaging the local Nusselt number has been shown in Fig. 3. The enhancement in Nusselt number ratio was found to be decreasing with increasing Reynolds number. This trend has been commonly seen in turbulent flows. With the increase in Reynolds number, the near-wall turbulent kinetic energy induced by secondary flows by ribs strengthens. However, this increase in near-wall TKE when normalized with the square of flow speed (TKE′) decreases with increasing Reynolds number. Hence, the potential for enhancement in Nusselt number ratio (Nu/Nu0) reduces. Interestingly, for very high Reynolds numbers, the enhancement levels varied only between 1.4 and 1.7 for all the rib configurations over the entire range of Reynolds number. Also, the dependence of Nusselt number ratio on Reynolds number was found to be weak, in contrast to studies on low Reynolds number range (<90,000) . As discussed above, V- and W-shaped ribs had the highest heat transfer followed by 45-deg parallel ribs. For the 45-deg parallel rib case, the strength of rib-induced secondary flows (due to longer rib length) balances the reduction in strength in turbulent transport due to less number of counterrotating vortex pairs. Inverse V- and M-shaped ribs had lowest heat transfer. M-shaped ribs had the lowest heat transfer because of low strength of rib-induced secondary flows and further lower strength of the generated CRVPs.
Figure 11 shows the variation of normalized friction factor with Reynolds number. An increasing trend was observed in friction factor with increasing Reynolds number, which is expected. The 45-deg rib turbulator had the lowest normalized friction factor, and the M-shaped rib had the highest frictional losses.
Detailed derivation of thermal hydraulic performance expression is presented in the Appendix. Variation of thermal hydraulic performance with Reynolds number has been shown in Fig. 12. Interestingly, for very high Reynolds number cases, the thermal hydraulic performance for all the investigated rib roughened channels was less than unity, which implies that the rib designs are not effective at elevated Reynolds numbers. Same rib shapes when investigated at Reynolds numbers less than 90,000, the thermal hydraulic performance was found to vary between 1.6 and 1.0 with a decreasing trend with increasing Reynolds number . If one has to bridge the thermal hydraulic performance trend between low Reynolds number studies and the present study's high Reynolds number cases, it can be concluded that THP shows an asymptotic trend with increasing Reynolds number with values less than unity at very high Reynolds numbers, and that the optimized rib turbulator shapes for low Reynolds number are not very effective for high Reynolds number cases. This is an important takeaway from the present study which provides caution to gas turbine hot gas path designers, particularly for the cases where rib designs for aircrafts are used in land-based power generation gas turbines.
Heat Transfer Roughness Function (G) and Friction Roughness Function (R) Variation With Roughness Reynolds Number (e+).
Heat transfer roughness function, friction roughness function, and roughness Reynolds number have been calculated based on the methodology described in Ref. . The equations for G, R, and e+ can be found in the “Nomenclature section.” These parameters (G, R, and e+) are based on law-of-wall for turbulent flows and are frequently used in the generalization of experimental data on different blockage ratio, the aspect ratio of channels. Figure 13 shows the variation of friction roughness function with roughness Reynolds number. Friction roughness Reynolds number relates the modified friction factor with channel aspect ratio and blockage ratio. Friction roughness function is inversely proportional to the square root of the modified friction factor . As shown in Fig. 11, the normalized friction factor (f/f0) increased monotonically with increasing Reynolds number. Further, the roughness Reynolds number (e+) is directly proportional to the Reynolds number (Re) and modified friction factor. Hence, the variation of R with e+ was found to have a monotonically decreasing trend with increasing roughness Reynolds number. 45-deg rib turbulators had the lowest frictional losses compared with other rib shapes, and hence the friction roughness function (R) of 45-deg ribs was highest and that of M-ribs was lowest. This parameter R is used in the determination of heat transfer roughness function G.
Conclusions and Recommendations
The present study was focused on measurement of detailed heat transfer coefficient at very high Reynolds numbers. A steady-state liquid crystal thermography technique was employed for calculation of heat transfer coefficient. The aim of the study was to evaluate the traditional rib designs at very high Reynolds number, which have been usually investigated for Reynolds numbers less than 90,000 in prior studies. For low Reynolds number studies, the optimum parameters (p/e = 10, e/dh = 0.1) defining the rib geometries for high levels of heat transfer were taken into consideration, and heat transfer enhancement levels were evaluated against the increase in frictional losses for Reynolds number ranging between 150,000 and 400,000. Five different rib configurations, viz., 45-deg, V, inverse V, W, and M shapes have been studied. From the detailed measurements of heat transfer coefficient, it was found that the general trends of heat transfer enhancement for high Reynolds number cases were similar to those observed for low Reynolds number studies; however, some differences were observed in local heat transfer enhancement characteristics. The overall normalized heat transfer coefficient (Nu/Nu0) showed weak dependence on Reynolds number in the investigated range and varied between 1.4 and 1.7. W-shaped ribs had the highest heat transfer and M-shaped ribs had the lowest heat transfer. The THP was found to be lower than unity for Reynolds numbers ranging from 150,000 to 400,000. Hence, gas turbine designers should use the optimized geometrical parameters for low range Reynolds number, in land-based power generation gas turbines with caution. Additionally, a parametric study is recommended to obtain optimized geometrical parameters defining rib roughened ducts for Reynolds numbers ranging between 105 and 106.
- e =
- f =
- h =
heat transfer coefficient
- p =
- u =
average coolant velocity in the duct
- G =
heat transfer roughness function,
- R =
friction roughness function,
- A =
area of the heated surface
- H =
height of the channel
- L =
length of the test section
- R =
resistance of flexible heater
- V =
voltage supplied by power source
- W =
width of the channel
modified friction factor, f + (H/W)(f − f0)
- dh =
channel hydraulic diameter
- f0 =
friction factor from modified Blasius correlation
- kair =
thermal conductivity of air
- ps =
- pt =
- qloss =
heat loss on the wall
- Tb =
- Tw =
- e+ =
roughness Reynolds number,
- Nu =
local Nusselt number
- Re =
- St =
Stanton number, Nu/(RePR)
- THP =
thermal hydraulic performance
- Nu0 =
Nusselt number (D-B correlation)
- Δp =
pressure drop across test section
Now, in heat exchanger design with heat transfer augmentation features with forced convection type cooling, following three situations can be considered depending upon the outcomes desired out of the design: (a) reduction in the heat transfer area (A) for the same heat exchange (Q) and pumping power (P), (b) increment in heat exchange (Q) for same heat transfer area (A) and pumping power (P), and (c) reduction in pumping power (P) for same heat exchange (Q) and heat transfer area (A). We are interested in determining an expression for PEC for the second scenario, where (A/As) = 1 and (P/Ps) = 1.
Another form of Stanton number is St = Nu/(Re Pr). Since, the Reynolds number was held constant between the smooth channel and the channel with heat transfer augmentation features, with working fluid being the same, Re = Res and Pr = Prs.
The above expression for THP signifies the net gain in heat transfer or same heat transfer area and pumping power.