In the last years, many technological advances have emerged in the turbo machinery industry, mainly in the area of power generation using gas turbines . The main target in this science field consists on designing and building more efficient machines with a higher life-time by means of applied research. However, in order to achieve this, it is necessary that the gas turbine operates at high compression pressure ratios as well as high turbine inlet temperatures (TIT), but these operating conditions generate thermal consequences or degradations in the gas turbine components that are exposed to the high temperatures, like blades and vanes of the first stage. For this reason, it is necessary to have an internal cooling system in gas turbines to avoid the reduction of the useful life of their hot components, since the useful life of turbine blades is reduced to half with every 10 – 15 ºC rise in metal temperature . Nowadays basic methods exist, which improve the gas turbines operating conditions, having as a result improvements of the external cooling , where the use of micro-jets with smaller diameters enhanced the overall heat transfer coefficient, or internal cooling where square ribbed channels are employed to study the thermal behaviour of the flow inside the channel , turbulence promoters with different geometries to study the temperature distribution in the gas turbine blades . Also, it is possible using serpentine passages inside the turbine blade to improve internal convective cooling , ribs in the internal surface of the cooling passages where the rib-to-rib pitch and angle of attack that yield a maximum heat transfer and maximum thermal performance are determined  or ribs as turbulence promoters to increase the rate of heat transfer . To increase the heat transfer with minimal friction in compact heat exchangers, the internal surfaces are ribbed with protuberances that have convex and concave forms . To study the heat transfer characteristics of laminar flow in parallel-plate dimpled channels are used  or square-channel fitted with baffles . However, a common way to increase the internal cooling efficiency in gas turbines is to add ribs, this method offers a better mixed fluid near to the hot internal surface of the channel thus increasing the thermal heat transfer. The present study shows a numerical analysis of the first stage blades in a gas turbine with internal cooling system (model MS7001E) applying the conjugated heat transfer. This method considers the direct coupling of fluid flow and solid body using the same mesh distribution and numerical principles for both domains. This coupling is achieved by using boundary conditions called double-side wall.
2. Mathematical formulation
The numerical analysis of a gas turbine at first stage blade with internal cooling system considers the solution of the conjugate heat transfer in steady state between the hot combustion gases flowing around the blade and the coolant flowing inside the cooling channels of gas turbine blades. The following assumptions are made to model the conjugate heat transfer problem:
Compressible and turbulent flow
Rotational frame with relative velocity formulation
Fluid is considered as an ideal gas.
2.1. Governing equations
The applied governing equations are the 3D Reynolds-averaged Navier-Stokes equations , which were solved by commercial Computational Fluid Dynamics software . The governing equations solved by the model are:
The energy equation for the fluid domain is given by
Since the behaviour of the flow in the gas turbine is very chaotic, it is necessary to incorporate a turbulence model in the numerical analysis to determine the Reynolds stresses. The Standard k-ε model was used, which relates the Reynolds stresses to the mean velocity by the Boussinesq hypothesis :
The eddy turbulent viscosity, µ
In these equations, G
Equation of state
The density variation in both fluids, the hot combustion gases and cooling air, is assumed according to the gas ideal law:
where R is the gas constant. This equation of state provides the linkage between the energy equation on one side, and the mass conservation and moment equations on the other. This linkage emerges from the density variations.
2.2. Computational model and grid
The computational model and mesh were generated in the pre-processor GAMBIT . In order to avoid many simplifications in the computational model, such as the use of boundary conditions at surfaces and outlets of the cooling channels, the computational model was generated using the total blade geometry, which includes the plenum, gap in the tip, gaps between the seal and the plenum and the ribbed cooling channels. Figure 1 shows the rotor blade geometry at the first stage of the gas turbine MS7001E.
In this Figure the internal cooling system can be seen. This system has 13 cylindrical channels inside the blade. The ribs were placed on the inner surface of the cooling channels in order to increase the heat transfer from the solid body to the cooling air. Also, in Figure 1 can be seen that the ribs were only placed in the middle blade zone inside the cooling channels; because in this zone the largest temperature gradient is present , causing failure such as cracks in the blade structure .
Three different geometries of rib configurations are studied, which are square, triangular and semi-circular forms. The ribs are placed perpendicularly to the air flow. Figure 2 shows a sketch of the different rib parameters used in the square cross-section ribs. These parameters are used for the other two geometries (triangular and semi-circular). Moreover, these configurations are applied for both arrangements (full and half-ribs). Figure 3 shows the form and parameters of the ribs cross-section.
Due to complexity of the geometry, different computational grid sizes are required. These grids are not uniform in all directions and were structured by mixing different types of cells (hexahedral and prismatic elements). For the grid used in the blade computational model with smooth cooling channels, the mesh density is high in the near-wall region of the blade body. The first wall-adjacent cells height in the vicinity of regions corresponding to the leading and trailing edge as well as in the suction and pressure side of the blade is 0.0012 mm, while in the region corresponding to the internal cooling channels, the wall-adjacent cell height is 0.0035 mm. This is developed in order to get a better solution into the boundary layer, obtaining y
For the blade models with ribbed cooling channels, the same grid distribution was used at the region of hot combustion gases and the external blade surface. The height of the first wall-adjacent cells in the vicinity of regions corresponding to ribbed cooling channels was of 0.002 mm, obtaining
For the models with half-ribs, it was used the same grid distribution showed in the Figure 5, having a small variation in the ribs domains, defining one half of the domain as fluid and the other half as solid. Figure 6 shows the grid used in the internal cooling channels with half-ribs having an aspect ratio of
2.3. Boundary conditions and thermal properties
The boundary conditions used for the computational model with smooth cooling channels are defined according to approximate values of the gas turbine operating conditions in steady state. Figure 7 shows the boundary conditions used in the computational model.
In the hot combustion gases inlet, the operational conditions are: mass flow of 2.47 kg/s, static pressure of 508700 Pa and total temperature profile is a function of the radial coordinate. This total temperature profile is described by the next equation :
This total temperature profile is imposed in order to match the oxidation mark of hot combustion gases over the airfoil of the blade, which has been operating until the end of its useful life , and because this profile is similar to the one obtained in the radial edge direction of the exit of the transition piece. Figure 8 shows the oxidation marks on the blade.
The units for the independent variables of Equation 12 are: (m) for the rotational radius and (K) for the total temperature. The turbulence quantities at the inlet of the model are defined using a turbulence intensity of 5% and 0.006 m for turbulent length scale.
A mass flow of 0.0048 kg/s of air, a static pressure of 897300 Pa, and a total temperature of 853.15 K were specified for each inlet zones of the thirteen cooling channels. Also, turbulence parameters are defined using a turbulence intensity of 5% and a hydraulic diameter of 0.004 m for these boundaries. At the left and right sides of the plenum inlets of cooling air were adjusted with a mass flow of 0.152 kg/s and 0.025 kg/s, respectively. As well, a turbulence intensity of 5% and turbulent length scale of 0.005 m was set. In the remaining inlet section of the cooling plenum, boundary conditions were adjusted to the same conditions used at the inlets of the cooling channels for the parameters of pressure and temperature. At the exit of the gas-air mixture a static pressure of 473170 Pa, and a backflow total temperature of 1226 K were specified.
For the solid surfaces the conditions were imposed as no-slip condition, while for the thermal condition were imposed as coupled. With these considerations it is possible to solve simultaneously the solid-fluid interfaces. At the interfaces, the temperature and heat flux could be continuous. These conditions are developed by the use of the boundary conditions denominated as two-side wall, which can be expressed as:
Rotational periodic boundary conditions are defined for the suction and pressure side of the computational model and a nominal angular velocity vector were prescribed.
The flow and heat transfer analysis were performed under the assumption that the fluid behaviour is compressible and viscous. For the case of the air properties, these are temperature dependant, while the thermo-physical properties of the solid domain were assumed to behave like Inconel 738LC alloy. The thermo-physical properties of the fluid and solid domains are showed in Table 1  and Table 2 , respectively.
2.4. Numerical method
Fluid flow and turbulent heat transfer analysis of the first stage gas turbine blade (MS7001E) with different ribs configurations placed in the internal cooling channels were realized using commercial Computational Fluid Dynamic software (FLUENT®). This code allows to solve the Reynolds averaged Navier-Stokes and the transport equations of the turbulent quantities for the compressible viscous flow. This CFD code solves the equations using the finite volume technique  to discretize the governing equations inside the computational domains. The Standard
The governing equations were solved simultaneously by the approach of the pressure-based solver. The pressure-based approach is recommended in the literature  to be used for flows with moderate compressibility, since it offers a better convergence. Due to the fact that the governing equations are non-linear and coupled, several iterations were needed to reach a converged solution. The Gauss-Seidel linear algorithm was used to solve the set of algebraic equations obtained by the discretization in FLUENT®. The convergence was reached when the residuals of the velocity components in the Reynolds averaged Navier-Stokes equations, continuity and turbulent quantities were smaller than 10-5, while for the energy conservation equation the residuals were smaller than 10-6.
Six computational equipments were used to solve the model. Each computer has a 3 GHz processor and 2 GB in RAM. These equipments were connected in a scheme of parallel processing. Figure 9 shows a sketch of the parallel processing equipment using a basic LAN topology.
3. Results and discussions
In the first part, a comparison between the results obtained experimentally by Kwak  and numerically for the external flow is presented. Also, a comparison between results obtained from semi-empirical correlations derived from the law of the wall  and the numerical results of the internal flow are presented. In the second part the effects on internal flow through the internal cooling channels are presented. Finally, the temperature distribution inside the blade body and the surface temperature distribution in the blade body with and without ribs are showed.
3.1. Comparison with experimental and semi-empirical correlation data
In order to validate the external flow around the blade a qualitative comparison between the pressure distribution obtained numerically and experimentally , has been performed. In  the pressure distribution on the gas turbine blade of GE-E3 was measured. Figure 10 shows the comparison between the numerical and experimental pressure distribution at the middle of the blade. Several turbulence models were used. The turbulence models used in the comparison were Standard
In Figure 10 can be seen that the pressure distribution, p
For comparison purposes of the internal flow in the cooling channels, the pressure drop given by the section with square full-ribs having an aspect ratio of P/e = 10 for the central cooling channel was determined. This pressure drop was calculated by the friction factor for ribbed tubes, using semi-empirical correlations derived from the law of wall. Nikuradse  developed a friction factor correlation to be used in geometries with sand-grain roughness. His results were excellent for a wide range of roughness sizes. This correlation is expressed by Equation (15).
The term e+ = eu*/v is the roughness Reynolds number and D
where A is the cross section area and P
This equation is valid in the range e+ > 35. By solving Equations (15) and (17) the friction factor can be found from the geometrical parameters of the internal structure of the ribbed channels. Equation (18) shows the result obtained.
Equation (18) is valid for channels with ribs placed 90º to the flow direction and an aspect ratio of
|Correlation Eq. (18)||Numeric|
The pressure drop calculated numerically presents a good approximation, having an absolute difference of 3.25%.
3.2. Effects of the internal flow through internal cooling channels
In order to determine the effects generated by the ribs, a line along the central cooling channel was created. This centerline is dimensionless with a range from 0 to 1, where y is the dimensionless distance of the axial length of the internal cooling channel, measured from the base of the blade until the outlet of the cooling channel. This centerline was used to obtain data of the flow parameters, such as temperature, velocity, Mach number and pressure loss.
Figure 11 shows the temperature distributions of the coolant flow along the centerline of the central cooling channels. Figures 11a and 11c show the results for different types of full-ribs, and Figures 11b, and 11d show the results for the half-ribs studies, having a ratio of
The temperature contours along the central cooling channel at a longitudinal plane are shown in Figure 12. Figures 12a, 12c and 12e show the results for different types of full-ribs, and Figures 12b, 12d and 12f show results for the half-ribs, both models have a ratio of P/e = 10. The fluid temperature increases while it goes along the channel for all cases (arrow indicates the direction of flow). The surface temperature of the channels is higher in the cases with half-ribs. Thus, the flow is heated at surface near the ribs. For the cases with full-ribs, the surface temperature is lower, showing a more uniform temperature distribution near to the wall.
The triangular ribs configuration presents the lowest temperatures, because this configuration offers the best cooling design inside the blade body. These effects are similar to the configurations with ribs whose
Figure 13 shows the comparison between velocity magnitude and Mach number distributions obtained for the cases of blades with smooth and ribbed channels.
In the ribbed channel corresponding to the configuration of square full-ribs with an aspect ratio of
Temperature distribution at the centerline inside of the central cooling channel
In order to have a better description of the effects caused by the acceleration and deceleration of the flow mentioned above, Figure 14 shows the contours of the axial velocity through the central cooling channel at a longitudinal plane. Figures 14a, 14c and 14e show the results for the different types of full-ribs. Figures 14b, 14d and 14f show the results for the half-ribs, both models have a ratio of
On the other hand, the square and semi-circular ribs produce recirculation zones as well as stagnation points over the upstream and downstream rib surfaces, respectively. The triangular ribs only produce recirculation zones in the downstream surfaces. These effects are similar to configurations with a ratio of
The local static pressure is presented in terms of the normalized pressure difference, calculated by the equation (19)
In Figures 15 and 16 can be observed that the slope of pressure drop in the smooth inlet section decreases due to a gradual reduction of channel cross-section. This area reduction is localized in the joint between the plenum and the blade. After this section, the pressure increases while the channel distance increases to the ribbed section. This is produced by the stagnation point when the flow shocks with the first ribs. In the ribbed section, the slope of the pressure drop becomes unstable, presenting periodical increments and decrements due to the cross-section variation, producing accelerations and decelerations of the flow. At the smooth outlet section, the slope of the pressure drop is relatively higher than that in the smooth inlet section. This is due to the increase of the flow velocity at this zone due to the rotational force, ejecting the flow inside the hot gases stream in the tip of the blade.
3.3. Temperature distribution inside the blade body
Figure 17 shows the temperature contours at a transversal plane of the blade body right in the middle of the blade for the cases with smooth and full-rib channels with a
As can be seen in Figure 17, the maximum temperature decreases, approximately about 10 to 20 degrees and is reached close to the internal surfaces for the cases of blade with ribbed channels (Figures 17b, 17c and 17d), noticing that the cooling zone covers the major part of the internal cooling channels, propagating to the leading and trailing edge. In the cases of the blade with smooth channels, it is only present a smaller cooling zone at the three central cooling channels (Figure 17a).
Models with square and triangular cross-section full-ribs show a similar temperature distribution and major heat dissipation compared with the semi-circular full-ribs.
Mazur  performed an analysis of a gas turbine bucket failure made of Inconel 738LC super alloy. This bucket operated for 24,000 hours. Mazur et al.  found that the maximum stresses are present in the blade cooling channels, producing cracks. Figure 18 shows that kind of cracks. These start in the coating of the cooling channels, propagating and following intergranular trajectories, reaching a depth up to 0.4 mm.
In this way, the effect generated by increasing the internal cooling zone produces an increment in the useful life of the blade, since the useful life of gas turbine blades is reduced to half with every 10-15 °C rise in metal temperature . On the other hand, the use of ribs increases the heat transfer, generating an increase in thermal gradients at internal surface of cooling channels. In the leading edge another interesting effect is presented. There is a minor penetration of the blade body temperature (Figures 17b to 17d) caused by the use of the ribs. However, it cannot be adequate due to the fact that the thermal gradients at the leading edge are increasing. Due to these thermal effects, these zones must be taken into account to be protected by means of the film cooling method.
Figure 19 shows the blade profile right in the middle along the blade height. In this section, a perpendicular line to the blade chord is created to obtain the temperature distribution inside the solid body as well as in the cooling air through the central cooling channel. The distance is a dimensionless parameter, taking values between 0 and 1, starting in the suction side and ending in the pressure side, respectively.
In Figures 20a and 20b the temperature distributions for the cases with full and half-ribs are presented, respectively. Both models have an aspect ratio between pitch and height of the ribs (
Figures 21a and 21b show the temperature distributions for the cases of cooling channels with full and half-ribs with an aspect ratio of P/e = 20. Figure 21a shows that the temperature distributions in the solid body are similar for the three ribbed cases, having the lowest temperature in the square ribs model. With these configurations a higher penetration of the cooling blade using any rib geometry is achieved. These configurations present a similar behaviour, in comparison with the results presented in Figure 20a. These cases present a temperature reduction up to 22 K in regions close to the channel surface and up to 10 K in the pressure and suction sides. In Figure 21b can be observed a uniform behaviour of the temperature distribution for the three types of ribs. This behaviour is basically the same for all the cases. However, the blade cooling is affected due to the temperature distributions obtained for all the cases with different tendency to be similar for the smooth case, having a smaller improvement on the blade temperature when compared with the temperature profiles shown in Figure 20b.
The temperature distributions of cooling air presented in Figures 20 and 21 have a symmetrical parabolic behaviour due to mixing flow. This is generated by placing ribs, while the profile related with the smooth channel has an asymmetrical behaviour. In this case the profile presents a tendency to attach to the pressure side due to the blade rotational force.
The effect of having a symmetric profile improves the heat transfer from the internal surface of the cooling channels to the air, due to the turbulence generated in the flow which is increased because of the ribs removing a high quantity of heat.
3.4. Surface temperature distribution in the body blade
Figure 22 shows a comparison between the surface temperature distribution on the pressure side and the suction side of the blade for the cases of models with smooth and ribbed cooling channels. The temperature distributions of the blade with ribbed internal cooling channels correspond to the configuration with square full-ribs with a ratio P/e = 10. As can be seen in Figure 22, internal cooling generated by ribs has an effect on the blade surface temperature, since it presented a substantial decrease in surface temperature on the pressure and suction sides of the blade. Also, it is observed a reduction of the spot of maximum temperature on the leading and trailing edges of the blade generated by the parabolic temperature profile at the inlet of the hot combustion gases. Another effect which can be seen is the stain of cooling at the root of the blade on the suction side, which is generated by the flow of air entering the vane platform.
In the present work, a numerical study was performed, with the aim to assess the effect generated by the ribs in the temperature distributions inside the blade body as well as the pressure drop through the cooling channels with different types of ribs. The main conclusions are:
The validation of the numerical model by comparing the internal and external flow with experimental  and semi-empirical [23, 24] data was developed. The pressure drop in the internal flow obtained through the numerical solution and semi-empirical [23, 24] data, offers a close enough agreement, with an absolute difference of 3.25%.
The higher and smaller temperatures of the internal flow are presented for the configurations of full-ribs in the ribbed section, reaching temperatures from 937 K to 741 K, respectively, while the half-ribs configurations offer a smaller difference of temperature in this section. This range is between temperature of 927 K and 791 K. The configurations of full-ribs have a larger contact area than configurations of half-ribs. Due to this reason, the configurations of full-ribs remove more thermal energy from the blade body.
The configuration with full-ribs with P/e = 20, offers the best cooling with any rib type. This could be due to the fact that the flow has a high recirculation zone between the ribs, generating a hydrodynamic perturbation to provoke a separation of the boundary layer.
The acceleration and deceleration effects, which are presented in the ribbed section, play an important role in the flow behaviour of the compressible fluid, since the high velocity of the flow shows a strong influence on the variations of temperature in the flow field. The highest Mach number and velocity are obtained with the ribbed channel, whose values are 1.45 and 823 m/s, respectively, while that smooth channel presents a continuous acceleration of the flow along the channels.
The ratio between the required inlet pressure in the cooling channels and the outlet pressure increases from 4 to 4.5 times approximately for the cases with full-ribs with aspect ratios (P/e) of 10 and 20, respectively. For the half-ribs, this ratio is between 3 to 4 times, approximately. These values are higher than the values obtained in smooth cooling channels.
The ribbed cooling channels present different pressure drops, ordered from higher to lower pressure drops, they are triangular, square and semi-circular ribs, respectively. The triangular ribs offer the highest cooling effects of the analyzed cases; however, this configuration presents the highest pressure drop when compared with any other case.
The turbulence promoters allow to obtain a maximum temperature decrease, approximately about 10 to 20 degrees, close to the internal surfaces of the blade body. This allows reducing damages by fatigue and thermal stresses.
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