Abstract
In the present work, the effects of radii ratios of curved rib on heat transfer performance, friction factor, thermal performance factor and exergy loss in roughened duct are investigated. The curved ribs are formed in concave and convex shapes, with and without cut. Reynolds number values are varied in the range of 5000 to 30,000, which were most suitable for roughened duct. The radii ratios of curved rib are studied with values of (2, 3 and 4), and the effects are discussed and compared with the transverse straight rib (R00). The results show that the heat transfer enhances by using a convex broken rib at different radii ratios. The curved rib with a radius ratio of 2 offers the best heat transfer characteristics and exergy loss compared with the other cases. Convex broken rib (case D) shows the highest thermal performance factor compared with the other investigated cases. Case D shows a good improvement in heat transfer with lower pumping power consumption compared with the other investigations.
Introduction
Turbine inlet temperature is considered one of the important factors to improve the thermal efficiency and power output of modern gas turbines by increasing the value of this temperature. This temperature already exceeded the allowable temperature of the component material. So, it must be finding a new effective cooling technique to protect turbine blades from excessive thermal damages. The film cooling technique is usually used on external blade surface by providing an air protective layer to eliminate the direct contact between the blade surface and hot combustion gases. Another cooling technique is known as internal cooling such as, impingement, rib turbulated and pin–fin techniques. In internal cooling, the flow fluid is subjected to separation and reattachment and creating secondary flows near the wall. This phenomenon causes reduction in the pressure inside the cooling channel. It is desirable to increase the heat transfer coefficient with a limited pressure drop through channel. The configuration of the rib turbulated and pin–fin has a significant effect on pressure drop and heat transfer characteristics. This rib breaks the laminar sublayer, creates flow separation, increases the reattachment length, reduces the thermal resistance and enhances the heat transfer.
More researchers conducted experimental studies on the heat transfer characteristics (Islam et al. 2002; Chen et al. 2000; Graham et al. 2004; Fu et al. 2005a). They studied the effects of geometric configurations and flow conditions on heat transfer in ribbed ducts. A comparison between V and broken V ribs in internal cooling gas turbine was performed (Kumar and Amano 2015). The comparison results showed that the performance of the broken V ribs was higher than the V ribs in twopass cooling channels. The relation between the heat transfer performance and rib height was investigated (Liou et al. 2003). Their results showed that the ratio of rib height to channel hydraulic diameter must be about 5–10% to produce high heat transfer performance. The best ratio between rib pitch and rib height was investigated (Bailey and Bunker 2003; Han 1984; Park et al. 1992). From their study, they showed that the best ratios were 7 and 15. The effects of the rotation on the Nusselt number and friction factor for angled ribbed duct with different aspect ratios were investigated (Fu et al. 2005b; Han et al. 1989; Huh et al. 2011). They showed that the higher Nusselt number ratios and the lower pressure drop can be obtained for duct with low aspect ratios. To enhance the thermal performance of ribbed duct with a higher aspect ratio under an effect of the rotation, Vshaped and Wshaped ribs were used (Wright et al. 2004; Maurer et al. 2007).
Several numerical studies investigated the performance of flow and heat transfer characteristics in ribbed cooling channels by different turbulence models such as k–ω SST model (Lin et al. 2001; Shih et al. 2001). Another simulation study investigated the flow and heat transfer in twopass rectangular channel with 45° rib turbulators using chimera RANS method (AlQahtani et al. 2002; Su et al. 2003) which it used a nearwall secondmoment turbulence closure. The effects of rib configurations on flow and heat transfer were investigated by using k–ε turbulence model (Bharath et al. 2017; Kızılırmak et al. 2017; Menni et al. 2017, 2018). Another study used each of an explicit algebraic stress model (EASM) for compute the Reynolds turbulent stresses and a simple eddy and diffusivity model (SED) to compute the turbulent heat fluxes (Jia et al. 2002).
According to the previous review, it is observed that there is not enough work involving curved ribs for promoting turbulence of air flowing through the ribbed duct to enhance heat transfer rate from hot surface to cooling medium inside the gas turbine. On the other hand, there is lack of studies focused on the best curvature radii to enhance the turbulence. The curved wall ribs have a higher surface area than that for transverse rib. The high surface area leads to increase the heat transfer rate from the surface to the cross air and also to increase the turbulence levels. Finally, no available work investigates the exergy losses for curved ribs and to choose the better curvature radii from point view of exergy analysis. Thus, the main objectives of this analysis are to:

Simulate 3D of roughness duct with curved ribs at various Reynolds number of the air flow.

Evaluate the effect of curvature radii and curvature direction on heat transfer characteristics and fluid flow of roughened duct.

Evaluate the exergy losses rate produced from all shapes.
Geometry and Computational Domain
In the present work, the analysis will be conducted on a rectangular cross section roughened duct. The computational domain dimensions are considered such as the experiment (Tanda 2004).
In this study, the reference case is considered as a duct with ribs 90° called (R00) as shown in Fig. 1.
The reference case is divided to three parts, the unheated entrance length (245 mm), heated wall (280 mm) and unheated exit length (115 mm). The transverse ribs are on the one wall of rectangular duct; the rib’s angle is a right angle based on the flow direction.
The cases under test are transverse convex rib, transverse convex broken rib, transverse concave rib and transverse concave broken rib as shown in Fig. 2. The dimensions of theses configurations are presented in Table 1.
The heated wall consists of seven ribs while the other sides are smooth. The rectangular cross section roughened duct is adopted as a flow domain for the predictions of the heat transfer, flow friction characteristics and exergy loss. The simulation results are compared with experimental results of Tanda (2004). A multiple block structured grid is used in this domain. To obtain grid independence solution, the number of cells was changed from 1.3 to 1.7 million cells for rib 90° (Elwekeel et al. 2014). The commercial grid generator ICEMCFD was used to create a multiple block, hexahedral, structure mesh for this geometry. The interrib region was provided with high density mesh to observe the heat transfer variation as well as to catch reattachment point. The mesh was refined toward the wall to obtain appropriate y^{+} value less than 1.
CFD Simulations and Data Reduction
The computational domain is simulated using the commercial ANSYS CFX (xxx xxx), which is a fully implicit multigrid coupling N–S solver based on finite volume method. So it can ensure the conservation as well as the accuracy of numerical method. The equations were discretized using a secondorder accurate upwind finite volume scheme. The convergence criterion is that the root means square residuals of both pressure and velocity components in three directions and was set to less than 10^{−5}. The simulations were executed on a high performance parallel computer cluster equipped with 20 compute node with IntelCPUs and 200 cores. The parameters of interest for this work were heat transfer coefficient, heated wall temperature, pressure distributed, friction factor, and exergy loss.
The local heat transfer coefficient is (Elwekeel et al. 2014):
The averaged heat transfer coefficient can be obtained by
The friction factor can be defined as the following
where \( \Delta p \) is static pressure drop across the length (L) of heated wall. The friction factor ratio can be written by:
\( f_{o} \) is friction factor in smooth duct, it can be calculated by modified Blasius equation as following:
Thermal performance factor (Tanda 2004) can be defined as
The local Nusselt number depending on hydraulic diameter is written as,
The Nusselt number \( \left( {{\text{Nu}}_{\text{o}} } \right) \) at nonswirling flow and for cooling the fluid can be calculated by Dittus–Boelter correlation (Su et al. 2003);
Exergy balance equations on roughened duct as following
\( \dot{E}_{\text{heat}} \) is the exergy transfer rate due to heat transfer. Q is the heat transfer rate from any heat source at temperature T and contact with ambient temperature T_{o}.
Turbulence Models and Boundary Conditions
Three turbulence models (standard k–ε turbulence model, ωRS turbulence model, and SST turbulence model) are validated with experimental data. In the simulation, the air is modeled as an ideal gas and a continuous fluid. Simulation performs at Reynolds number values from 5000 to 30,000. Air temperature at inlet is 300 K, the reference pressure and outlet pressure are 1 atm and 0 atm, respectively. The heat flux of 1100 W/m^{2} was applied at heated wall keeping all other walls smooth and insulated. In the experimental study, the ribs made of Plexiglas and glued onto the heated plate, so the seven ribs are considered as an adiabatic, which the main objective from these ribs is to generate turbulence in the air path to increase the heat transfer from the heated wall to the air (Tanda 2004). The turbulence intensity of 5% was assigned at the mainstream inlet. The flow exit of computational domain was assumed to be a constant atmospheric pressure. The nonslip boundary condition was assigned at all the walls.
Results and Discussion
In this study, the computational model for air path was simulated and compared with experimental results of Tanda (2004). Figure 3 presents the validation of three turbulence models, Reynolds number and average heat transfer coefficient over heated wall. The SST turbulence model has a good agreement with the experimental results and that is according to simulation results of Elwekeel et al. (2014). The SST turbulence model has significant advantages for heat transfer prediction and flow separations. Because it works by solving a turbulence frequencybased model (k–ω) at the wall and (k–ε) in the bulk flow. The combination function ensures a smooth transition between the two models. Table 2 shows the grid independence study; in this study, five grids of 0.3, 0.5, 0.75, 1.3, and 1.7 million cells at Reynolds number of 8900 were investigated. From this table, it is clear that the error is almost constant for grids of 1.3 and 1.7 million cells. So, in this study the grid size of 1.7 million cells was chosen.
In this work, the numerical simulations were conducted on four rib shapes with three curvature radii. The effects of these novel configurations on heat transfer flow, friction factor and exergy analysis were investigated and were compared with the experimental results.
Heat Transfer and Fluid Flow Characteristics
The stream velocity in X–Y direction is presented in Fig. 4. In this figure, the streamlines collide with the ribs, reflect the direction and spread on spanwise plate in the convex ribs. From this figure, it can be seen that the curved ribs create more turbulence than traditional case. In case A, the ribs are more convexity, which creates a circulation zones at upstream and downstream of the rib.
When the convexity of the ribs approaches from the flat shape, the flow divides into both sides and then the flow merges together behind the ribs as shown in case C and creates lower pressure at downstream. In the cases of concave shape (cases G, H and I), the streamlines have similar direction of rib 90° while the streamlines of concave rib collide with a large area of the ribs that is enhanced the mixing in sublayers. The fluid flows through broken areas of the ribs (cases J, K and L) resulting in small eddies in the upstream and downstream regions. But in convex broken ribs (cases D, E and F) have a large recirculation downstream region.
Figure 5 shows the temperature distribution over heated wall. It is clear that the convex rib and concave rib have a lower temperature on heated wall compared to case of R00. Also, it can be seen that the curved ribs enhance the wall temperature. This is because the curved shape increases the surface area and creates more turbulences.
A convex rib case A clearly shows an improvement in wall temperature compared to other cases B and C. This can be explained by generating a large turbulences that can be attributed to increased the convexity of rib shape.
In three cases of convex broken rib (cases D, E and F), the wall temperature is decreased than that of extend convex rib and R00. Because the broken rib increases the collision force between the flow and the rib leads to enhance the turbulence and decrease the thermal resistance.
Cases H and I with concave rib have a low wall temperature at upstream and a high wall temperature at downstream of trailing plate, but in case G the wall temperature is more enhancement than the other cases H and I. The fluid is accelerated to flow through the concave broken rib to create a small eddies behind the rib; the broken area enhances the mixing in sublayer leading to diffuse more heat from the wall. From this figure, it can be noted that the convex broken rib (case D) gives the lower wall temperature. The averaged wall temperature of case D is decreased by 26.23 K than that for R00.
Averaged heat transfer coefficient with Re number is presented in Figs. 6, 7 and 8 for radii ratios of 2, 3 and 4, respectively. Obviously, the averaged heat transfer coefficients increase with Re number for all cases. Increasing of Re number increases the turbulent intensity, and this leads to increase in turbulent dissipation rate and turbulence kinetic energy and then the averaged heat transfer coefficient increases. From Fig. 6, it can be seen that the heat transfer coefficient values of case D are the highest over Re range while R00 shows the lowest heat transfer coefficient values. The rib shape of case D creates more vortices, and the number of reattachment points over the heated wall increases as shown in Fig. 4. This leads to increase the heat transfer rate compared to the other studies cases.
Figure 7 shows the relation between averaged heat transfer coefficient for radius ratio of 3 and Re numbers. The relation between these parameters is directly proportional. Case E has the highest heat transfer coefficient, but the lower heat transfer coefficient can be achieved by case B. The same trend can be observed for Cases F and C as shown in Fig. 8 for radius ratio of 4.
From Figs. 6, 7 and 8, it can be concluded that the heat transfer enhances by convex broken rib. The lower heat transfer values are obtained by rib 90° (R00). At Re of 30,000, the heat transfer coefficients of Case D, Case E and Case F are higher than that for rib 90° (R00) by 52.34%, 48.25% and 45.79%, respectively. This is due to the fact that, as curvature radius increases, the number of reattachment points over the heated wall reduces resulting in a decrease in heat transfer. Also from these figures, it can be seen a good agreement of simulation results of R00 with experiment results of Tanda (2004). It can be found that all cases are better than traditional case (case M), but the heat transfer coefficient varies depending on ribs shape and radius ratio because the curved ribs enhance the turbulence near the wall and thus increase heat transfer.
The friction factor ratios as a function of Re number are shown in Figs. 9, 10 and 11. It is observed that the friction factor ratio increases with increasing Re number due to increase the wall shear stress.
It is also observed that the friction factor ratios of concave shape ribs have the highest values than those other cases. The dissipation of vortices accompanying with the concave shape rib causes an additional loss in the kinetic energy leading to increase in the friction factor. By increasing the radii of convex rib, the friction factor ratio of convex broken ribs increases compared with the friction factor ratio of convex ribs. This is because the high radius ratio of broken ribs helps the fluid layers to create more shear stress. The different values of friction factor for the different cases depended on the rib shape and therefore amount of kinetic energy losses of these ribs. The flow friction phenomenon over heated wall is presented by the contour map of pressure. The contour maps of pressure for convex and concave ribs at Re number of 5000 demonstrate in Fig. 12. The major aerodynamic loss is a secondary flow as shown in the curved ribs, that the vortices are stronger than that of transverse case (see Fig. 4). Also the shape of curvature contributes in pressure loss. In curved ribs the fluid separates and creates more vortices at downstream. From this figure, the flow in concave ribs (G, I and H) slows down, which leads to an increase in pressure drop.
With comparison of all study cases, it can be concluded that the highest friction factor value can be achieved by case G, the value increased by 14.9 times than that of R00 at Re of 30,000. The case A has a lowest friction factor value compared with curved ribs while it increases by 80% than that reference case (R00).
Figure 13 shows the thermal performance factor for all study cases. The thermal performance factor decreases with increasing Re numbers that because of the increase of friction factor ratio is higher than enhancement of Nusselt number ratio over Re ranges. From this figure, it can be seen that the case D has the highest heat transfer coefficient, so the highest thermal performance factor is achieved by case D.
Figure 14 presents the contour maps of turbulence ratio between 3rd and 5th rib for study cases at Re number of 5000. It can be seen that the concave ribs have the highest turbulence ratio while the lowest turbulence ratio can be achieved by convex ribs. The turbulence ratio of convex ribs is approach to turbulence ratio of R00. This explains the mentioned trends of friction factors ratio as shown in Figs. 9, 10 and 11.
Exergy Losses Analysis
The exergy losses are a difference between maximum useful energy and actual energy. The exergy loss as function of rib shapes, Re number and radius ratio will be presented in Fig. 15.
At low Re numbers (5000 and 8500), the exergetic losses of curved ribs with radii ratios of 2, 3 and 4 are approaching to reference case (R00). While at high Re number, the concave ribs (G, H and I) with different radii ratios show a highest values of exergetic losses than those of other cases. That is because the exergy loss accompanying by friction for these cases is highest values. So, it can conclude that although the curved ribs enhance of heat transfer but it has high energy destroyed at high Re number.
Comparisons with Other Studies
Figure 16 presents the globally Nu number ratio versus the friction factor ratio for cases (D, E and F) and other rib studies (Han and Park 1988; Han et al. 1991, 1993).
From this figure, it can be concluded that the case D gives the best globally Nusselt number ratio compared with the other studies. The value of Nusselt number ratio for case D reaches 3.68 higher than all comparison cases at the same friction factor ratio. Note that in these references the augmentations of heat transfer ratio changed according to rib arrangement, rib dimensions, heat flux and Reynolds number. The periodic ribs placed on the twoopposite channel at Re of 30,000 in Han and Park (1988). The heat transfer ratio presented in Han et al. (1991) at different rib angle with Re of 80,000. The wedge continuous rib and wedge broken rib tested at Re = 80,000 in Han et al. (1993).
Conclusions
In this study, CFD simulations were used to investigate the effects of curved ribs with different radii on the performance of roughened duct. Heat transfer characteristics, friction factor and exergy loss were investigated at different Reynolds numbers. Four different configurations such as transverse convex rib, transverse convex broken rib, transverse concave rib and transverse concave broken rib were investigated. Three radii ratios of 2, 3 and 4 were used and were compared with the transverse straight rib (R00). The simulation results showed that:

The lowest heated wall temperature can be achieved by convex broken rib (case D), the average temperature of heated wall decreases by 26.23 K than that for reference case (R00).

The heat transfer coefficient increases by using the curved rib compared with straight transverse rib (R00).

The convex broken with radius ratio of 2 has highest heat transfer coefficient while R00 has lowest values over study range of Re.

The curved rib has higher values of friction factor ratios than that of R00. The concave rib has highest friction factor ratio.

The exergetic losses of curved rib are close at low Re, while at high Re the curved rib has high exergy losses. This means the good heat transfer and exergy loss can be achieved by curved heated wall at low Re number.

Convex broken rib (case D) shows a higher enhancement in heat transfer with less pumping power consumption compared with the other rib studies.
Abbreviations
 Q :

Heat transfer rate (W)
 q :

Heat flux (W/m^{2})
 \( \dot{m} \) :

Mass flow rate Kg/s
 \( \dot{W} \) :

Work kW
 \( \dot{E} \) :

Exergy rate kW
 \( I \) :

Specific enthalpy (kJ/kg)
 \( \dot{\sigma } \) :

Exergy destruction rate (irreversibility) kW
 A:

Area (m^{2})
 D:

Diameter (m)
 E :

Energy kW
 FR:

Friction factor ratio (−)
 h:

Heat transfer coefficient (W/m^{2} K)
 I:

Enthalpy (J/kg)
 L:

Heated wall length (m)
 p :

Static pressure (Pa)
 R :

Radius ()
 R ^{*} :

Radii ratio (R/D_{h}) ()
 Re :

Reynolds number, ud/ν ()
 s:

Specific entropy kJ/kg
 T:

Temperature (K)
 TPF:

Thermal performance factor
 u :

Velocity (m/s)
 HTC:

Heat transfer coefficient
 R00:

Rib 90°
 \( \rho \) :

Density (kg/m^{3})
 ∆:

Difference
 av:

Average
 b :

Bulk temperature
 h :

Hydraulic
 w :

Wall temperature
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Elwekeel, F.N.M., Abdala, A.M.M. & Zheng, Q. Effects of Curved Ribs on Heat Transfer, Friction and Exergy Loss in Rectangular Cooling Channels by CFD. Iran J Sci Technol Trans Mech Eng (2020). https://doi.org/10.1007/s40997020003763
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Keywords
 Heat transfer
 Exergy loss
 Curved rib
 Internal cooling
 CFD