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Research Papers

# A Three-Dimensional Conjugate Approach for Analyzing a Double-Walled Effusion-Cooled Turbine BladePUBLIC ACCESS

[+] Author and Article Information

Department of Engineering Science,
University of Oxford,
Oxford OX1 3PJ, UK

Alexander V. Murray

Department of Engineering Science,
University of Oxford,
Oxford OX1 3PJ, UK
e-mail: alexander.murray@eng.ox.ac.uk

Peter T. Ireland

Department of Engineering Science,
University of Oxford,
Oxford OX1 3PJ, UK
e-mail: peter.ireland@eng.ox.ac.uk

Eduardo Romero

Rolls-Royce Plc.,
Bristol BS34 7QE, UK
e-mail: eduardo.romero@rolls-royce.com

1Corresponding author.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received July 20, 2018; final manuscript received August 29, 2018; published online October 17, 2018. Editor: Kenneth Hall.

J. Turbomach 141(1), 011002 (Oct 17, 2018) (10 pages) Paper No: TURBO-18-1168; doi: 10.1115/1.4041379 History: Received July 20, 2018; Revised August 29, 2018

## Abstract

A double-wall cooling scheme combined with effusion cooling offers a practical approximation to transpiration cooling which in turn presents the potential for very high cooling effectiveness. The use of the conventional conjugate computational fluid dynamics (CFD) for the double-wall blade can be computationally expensive and this approach is therefore less than ideal in cases where only the preliminary results are required. This paper presents a computationally efficient numerical approach for analyzing a double-wall effusion cooled gas turbine blade. An existing correlation from the literature was modified and used to represent the two-dimensional distribution of film cooling effectiveness. The internal heat transfer coefficient was calculated from a validated conjugate analysis of a wall element representing an element of the aerofoil wall and the conduction through the blade solved using a finite element code in ANSYS. The numerical procedure developed has permitted a rapid evaluation of the critical parameters including film cooling effectiveness, blade temperature distribution (and hence metal effectiveness), as well as coolant mass flow consumption. Good agreement was found between the results from this study and that from literature. This paper shows that a straightforward numerical approach that combines an existing correlation for film cooling from the literature with a conjugate analysis of a small wall element can be used to quickly predict the blade temperature distribution and other crucial blade performance parameters.

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## Introduction

The desire to build a gas turbine with both high efficiency and specific power output has led to the use of ever-increasing turbine entry temperatures [1]. The continuous increase in turbine entry temperatures results in an extremely harsh environment for turbine blades and other critical hot stage components. Film cooling combined with a multipass system has been the conventional cooling method for turbine aerofoils to date. However, the desire to attain much higher engine efficiency and at the same time reduce the amount of cooling air requirements has led to the need to research and implement advanced cooling techniques such as transpiration cooling or those that closely approximate transpiration, for instance, porous multiwall cooling schemes like double-wall cooling. This paper presents an efficient analysis procedure for the class of double skin cooling systems for advanced engines shown schematically in Fig. 1.

## Literature Review

A literature review of earlier conjugate heat transfer (CHT) simulations and film cooling effectiveness applicable to effusion cooling is presented here.

###### Three-Dimensional Conjugate Simulation Approaches.

A fully coupled CHT simulation allows both the solution of the fluid flow and heat transfer to be determined in just one code without the need to carry out interpolation of boundary conditions between codes. However, carrying out a coupled CHT of a double-walled effusion cooled blade presents a challenge not only because of the very many small-diameter cooling holes that need to be modeled, but also owing to inadequate discretization and turbulence modeling included in the computational fluid dynamics (CFD). Even though numerical simulation approaches like direct numerical and large-eddy simulations offer improved accuracy in solving complex turbulent flows, in the case of practical cooling systems, they remain time-consuming and computationally costly [2] and thus not ideal especially in some cases for example where preliminary results are desired to enable redesign and optimization. To avoid the use of such computationally expensive numerical simulation methods, several authors have developed simplified numerical approaches for studying effusion cooled systems.

Laschet et al. [35] presented a 3D conjugate approach for modeling both the fluid flow and heat transfer using a homogenization method—an approach which assumes that a given multilayered model consists of a periodic repetition of a unit volume cell. This approach was used to predict the aerothermal behavior of flat and curved multilayered plates as well as the aerothermal behavior of transpiration cooled plates. A CHT flow solver that depends on an implicit finite volume method in conjunction with multiblock approach was used. The domain was divided into separate blocks and the full, compressible, 3D Navier–Stokes equations were solved in each of the fluid blocks. The simulation time required to run the homogenous model was very small and the authors suggested possible future application of this technique for analysis of complex cooled gas turbine components. The limitation of this approach is that it is restricted to homogeneous models and may not be possible to be used in the analysis of a heterogeneous system where a unit volume cell cannot be simply reproduced to represent the whole domain.

Zecchi et al. [6] developed a quick decoupled conjugate simulation tool for preliminary design stage analysis of an uncooled turbine vane. The program inputs were the hot-side and cold-side heat transfer coefficients (HTC) (from correlations) and the fluid temperature. Their simulation tool permitted not only the evaluation of metal temperature distribution but also coolant mass flow distribution on the vane. The authors compared the results from this simplified analysis approach with the experimental results, of the same vane from Hylton et al. [7], and a satisfactory match was found.

Heidmann et al. [8] developed a CHT approach to study a film cooled turbine vane. This method was later applied to a gas turbine blade by Kassab et al. [9]. In the latter method, fluid convection was simulated using the Glenn-HT [10] general multiblock heat transfer code and conduction solved using a boundary element method that was coupled directly to the fluid flow solver. The authors noted that the use of the boundary element method saved computational time as no volumetric grid within the solid was required.

Mendez et al. [11] offered a simplified approach for modeling and simulating effusion cooling using a flat plate. In this approach, the number of rows was assumed to be infinitely large and a small, finite calculation domain was extracted from the plate. The calculation domain was chosen so that it contained a single perforated hole. Periodic boundaries were specified to reproduce infinite plate geometry. The results obtained from their simplified approach were found to match the existing experimental results well.

Amaral et al. [12] evaluated temperature of an internally cooled gas turbine blade by employing a decoupled CHT, which involved use of three different solvers; (1) Navier–Stoke solver to evaluate nonadiabatic external flow and heat transfer, (2) finite element analysis to obtain heat and stress solution within the solid domain and (3) a 1D aerothermal model, in the cooling channels, that used empirical heat transfer and friction formulae available in the open literature. An iterative procedure was used to stabilize the input/output data from the solvers. The authors validated the results from this method against experimental test results. Even though the empirical formulae were used in the 1D solver, the obtained results were satisfactory. It was noted that this method offered an acceptable a trade-off between accuracy and computational cost.

Bonini et al. [13] and Andreini et al. [14] presented a simplified decoupled 3D CHT procedure for evaluating the performance of gas turbine blades. In their methodology, the internal cooling system was modeled using an in-house 1D thermo-fluid network solver, external heat loads, and pressure distribution evaluated using three-dimensional CFD. Film cooling effectiveness was calculated using correlations for shaped and cylindrical holes developed by Colban et al. [15] and L'Ecuyer and Soechting [16], respectively. The effect of multiple rows of films was accounted for using the superposition approach proposed by Sellers [17]. Heat conduction through the metal of the blade was calculated using three-dimensional ANSYS steady-state thermal module. Their CHT entailed an iterative procedure of guessing a reasonable first metal temperature, running simulations, updating the metal temperature, and repeating the process until the solution converged. The predictions from their work demonstrated that decoupled CHT produced results comparable to those from experiments and satisfactory agreement between the two was demonstrated.

The decoupled CHT presented by Bonini et al. [13] and Andreini et al. [14] was validated by Andrei et al. [18] using both an internally cooled and an internally and film cooled gas turbine vane. The authors compared both the metal temperature and adiabatic effectiveness distribution results against those from experiments (carried out at the NASA Lewis Research Centre (Cleveland, OH) by Hylton et al. [7]) and a fully 3D coupled CHT CFD analysis and a good match was noted between the results.

###### Methodology for Predicting Film Cooling Effectiveness.

Several authors have developed film effectiveness correlations, including Colban et al. [15] who developed a correlation specifically for both laid-back and regularly shaped cooling holes, Baldauf et al. [19] for a row of cylindrical holes and L'Ecuyer and Soechting [16] for a row of cylindrical cooling holes. Display Formula

(1)$ϵf=1−∏i=1n1−ϵfix$

Baldauf et al. [20] carried out a comprehensive analysis of laterally averaged film cooling effectiveness. Their correlation included the influence of a full set of parameters and the impact of film cooling performance demonstrated agreement with the measured results. However, these researchers only considered a single row of cooling holes.

Sellers [17] presented a simple way of predicting the film cooling effectiveness for many rows of cooling holes from correlations (or data) from single rows. Sellers [17] approach replaces the free stream temperature for downstream rows with the adiabatic wall temperature calculated from upstream films. Sellers [17] superposition approach can be generalized to any number of rows as shown experimentally by Murray et al. [21], see Eq. (1). There are many reviews of film cooling effectiveness in the literature (for example, Goldstein [22]) but a few deal with effusion cooling applied to turbine aerofoils or predict the two-dimensional distribution of effectiveness downstream of the cooling hole.

The present work aims to extend the decoupled 3D CHT approach presented by Bonini et al. [13] and Andreini et al. [14] to a double-walled effusion cooled gas turbine blade so as to develop a computationally efficient conjugate approach that can be used to produce an assessment of the performance of the complex double-walled blade. The Goldstein [22] correlation was employed for film cooling effectiveness for a single hole and film superposition using Sellers [17] to represent the 2D distribution of film cooling effectiveness of an array of cooling holes. The suitability of this approach was confirmed for multiple rows of films by Murray et al. [21]. The internal heat transfer coefficient was evaluated from conjugate analysis of a unit wall element and the conduction through the blade was simulated using the finite element code in ANSYS steady-state thermal module. Figure 2 shows a concept double-wall turbine blade with a cooling design for the leading edge and trailing edge (TE).

###### Test Geometry in the Present Study.

A practical double-wall blade, as shown with more details in Fig. 2, comprises of three main regions with different cooling designs. (1) Leading edge cooling is achieved through a combination of impingement jets from the inner skin and showerhead cooling holes on the outer skin. (2) Midchord region, which is cooled both internally (through impingement jets from the inner skin and convective heat transfer in the array of pedestals) and externally by an almost-continuous film cover from the staggered array of effusion cooling holes. (3) Early and late TE cooling is achieved using pin-fin bank and TE slots, respectively. The present study focuses on the double-wall midchord region of the blade.

Pedestal height, size, spacing as well as impingement and effusion cooling holes size and spacing are set to match that of the desired wall element. Each of the wall elements studied by Murray et al. [23] has distinct aerothermal characteristics (such as convective efficiency, discharge coefficient, and mass flow rate). A sample of CFD results of a unit cell showing the flow velocity distribution is as shown in Fig. 3. To demonstrate how the numerical analysis approach developed herein works, one of the best-performing wall elements named geometry 3 (from Murray et al. [23]) was used. The entire span of the midchord region of the aerofoil is made up of 88 of these wall elements.

To simplify the analysis, geometry 3 wall element (whose geometrical parameters and dimensions are as shown in Fig. 4 and Table 1, respectively) was used in the whole midchord region. However, owing to the varying flow structure and heat load around a turbine blade, it will be essential in the near future of the double-wall blade design to consider employing different wall elements around the blade. For instance, employing a wall element with the low-mass flow in the transonic region of the suction surface where coolant ejection onto the surface is undesirable because of the aerodynamic losses. Cylindrical cooling holes with 1 mm diameter and inclined at an angle of $30$ deg to the external surface of the blade were modeled in a staggered pattern on the blade surface, as shown in Fig. 5. This design, with equal spanwise and streamwise pitches of 10 diameters, resulted in a total of 12 staggered rows of cooling holes; 7 on the suction side and 5 on the pressure surface.

###### Wall Block Analysis.

The researchers in Ref. [23] carried out an elaborate conjugate CFD while performing aerothermal and thermomechanical analysis of seven different geometries of wall blocks (wall elements). Each of the elements comprises a small domain with square sides equal to half the streamwise pitch between the effusion holes. In this wall element, the external and internal skins are connected via a bank of pedestals, see Fig. 4. To carry out numerical analysis in the present study, the internal heat transfer coefficient of the unit wall element from conjugate CFD was correlated using a power law, Eq. (2). Therefore, for a given Re and with known values of constants A and B it is possible to evaluate the Nu and hence internal heat transfer coefficient for a specific wall element geometry. Display Formula

(2)$Nu=AReB$

where A = 0.07 and B = 0.80.

The wall element analysis also provided an effective discharge coefficient which was subsequently used to predict the coolant flow rate through the effusion holes on the aerofoil, using Eq. (3). Ideal coolant mass flow rate, $m˙ideal$ is calculated assuming an isentropic one-dimension expansion from the coolant total pressure inside the blade to the static pressure local to the exit of the cooling hole. In the present work, the discharge coefficient was correlated from the conjugate analysis of Murray et al. [23] for average aerofoil cross-flow but later work has correlated $Cd$ with blowing ratio Display Formula

(3)$m˙c,actual=Cdm˙c,ideal$

To evaluate the internal heat transfer coefficient in the blade, $hi,blade$, heat balance was carried out between the block and the blade resulting in Eq. (4). Where $Ablade$ is the internal surface area which the wall element occupies on the blade and $ηc$ is the convective efficiency of the wall element (Fig. 6 gives a graph of $ηc$ variation with Re for the wall element used in this study).

###### Numerical Simulation.

As aforementioned, the conjugate model in the present study adopted a decoupled approach, which modeled the blade in three separate steps; internal heat load modeling, film cooling (external heat load) modeling and a conjugate simulation of the whole domain. In practice, the outer skin of the aerofoil, which is effusion cooled, is connected to the internal impingement plate via a bank of pedestals forming a double-skin system, as shown in Fig. 1. In our approach, we predict the blade temperature in the aerofoil wall (modeled as 1 mm thick) but account for impingement cooling and the pedestals by increasing the internal heat transfer coefficient Display Formula

(4)$hi,blade=m˙c,actualCp,cηcAblade$

The internal and external heat load modeling are described in detail in the Wall Block Analysis and Film Cooling Modelling sections respectively.

###### Film Cooling Modeling.

A MATLAB2 script was written to compute both the adiabatic wall temperature and the film cooling effectiveness at every grid point on the blade. The Mainstream Velocity and Density, Adiabatic Wall Temperature and Film Cooling Effectivenes sections describe how each of the inputs necessary for the film cooling evaluation was obtained.

###### Mainstream Velocity and Density.

The mainstream Mach number and velocity were evaluated from the local static pressure assuming an isentropic flow. The blade profile used is similar to the one used by Gurram et al. [24] for blade trailing edge studies but was scaled to an engine representative size. The engine-representative conditions used by Colladay [25], summarized in Table 2, were used in this analysis. The mainstream density was evaluated from the ideal gas equation. The recovery temperature (Eq. (5)) was evaluated from the specified total gas temperature and was used as the effective gas temperature. This is because, for low Mach number flows, the effective gas temperature can be taken to be static temperature; however, at high Mach numbers, such as flow over a large region of the suction surface, it is more accurate to use recovery temperature as the effective gas temperature. An estimate of a turbulent layer flow recovery factor formula $(Λ=Pr3)$ proposed by Lee [26] was used Display Formula

(5)$Tr,g=Ts,g+ΛTo,g−Ts,g$

###### Film Cooling Effectiveness.

The effectiveness of the coolant film diminishes downstream of the cooling hole exit. There have been a vast number of studies, both experimental and numerical, that have been undertaken to measure film effectiveness under a variety of conditions.

From these data, film effectiveness has been correlated with several variables, which include downstream position, coolant and mainstream Re, hole shape and diameter, mainstream gas-coolant density ratio, and specific heat ratio. In this study, the correlation developed by Goldstein [22], Eq. (6), which has an addition of the lateral coordinate, z, (see Fig. 5) was used.

The aerofoil external surface was divided into a number of elements. The values of the mainstream gas parameters, such as density, velocity, and static pressure vary from one grid point to another. A MATLAB2 code was written to compute the film cooling effectiveness at each of these grid points. The values of $xdecay$, $αt$, and $z1/2$ vary with the blowing ratio and streamwise pitch. It was possible to calculate the required values at any given blowing ratio and streamwise pitch by extrapolating the detailed film effectiveness results compiled by Murray et al. [21] and Baldauf et al. [27] obtained through combination of CFD and experiments on a flat plate. Display Formula

(6)$εf=MugD8αtxD+xdecayexp−0.693zz1/22$

For this case of effusion cooling where there are many rows of cooling holes on the blade surface, film effectiveness at each of the succeeding downstream rows of cooling holes is strongly influenced by the film coming from the upstream rows. In this study, Sellers [17] superposition approach (see Eq. (1)) is employed to evaluate the composite film effectiveness of the downstream rows.

The mainstream gas recovery and coolant inlet temperatures were used as boundary conditions. The temperature of the coolant at the exit of each cooling hole, $Tc,ex$ is not known but depends on the wall temperature, coolant temperature, internal heat transfer coefficient and cooling system wetted surface area.

The temperature increase of the coolant, from supply to film cooling hole exit, divided by the maximum temperature increase possible is defined as the convective efficiency,$ηc$ [28]. This parameter is a function of the coolant mass-flow and cooling geometry, as shown in Eq. (7). Assuming a fully developed turbulent pipe flow and using Dittus–Boelter's expression relating Nu and Re ($Nu=0.023Re0.8Pr0.33$) then Eq. (7) can be rewritten so that the $ηc$ becomes a function of only $L/Dh$ (Eq. (8)). Figure 6 shows $ηc$ plotted as a function of Re comparing the convective efficiency of the wall element in this study with three simple duct cooling systems, characterized by L/$Dh$. From the graph, it is evident that the wall element possesses very high convective efficiency corresponding to that of a circular pipe with length L in the range of 20 < L/$Dh<40$Display Formula

(7)$ηc=1−exp−4StLDh$
Display Formula
(8)$ηc=1−exp−0.12Re−0.2LDh$
Display Formula
(9)$Taw=Tr,g−ϵfTr,g−Tc,ex$

An iterative process (the program logic is illustrated in Fig. 7) was necessary owing to the interdependence nature of the metal temperature distribution, the amount of heat picked up by coolant as it flows through the internal passages, the adiabatic temperature decay downstream of the cooling hole, the hot-side and cold-side heat transfer coefficient. Equation (9) was employed to evaluate the coolant exit temperature,$Tc,ex$ which supplies the film.

An initial guess for the metal temperature distribution, $Tm$ (chosen to be a uniform value which was the average of the gas and the coolant inlet temperature) was set. The $ηc$ corresponding to a given wall element geometry and Re, determined from a flow analysis, was read from the $ηc−Re$ database, from which $Tc,ex$ was determined. The latter $Tc,ex$ was then used in Eq. (9) to evaluate adiabatic wall temperature, $Taw$. The calculated $Taw$ was exported to ANSYS steady-state analysis and conjugate simulations executed, as explained in the Conjugate Model Simulation section of this paper, from which volume-average metal temperature, $Tm$ was evaluated. This new updated temperature was then fed into the iteration loop and the iteration process executed till convergence. Convergence was met when $ΔTm/(Tr,g−Tc,in)$ was below 0.01%. To accelerate convergence in subsequent cases, the initial metal temperature was set to be the converged metal temperature of the preceding case. On average, four iterations were required for convergence in all the cases studied.

###### External Heat Transfer Coefficient.

A simple integral approach was used to determine the external heat transfer coefficient around the aerofoil, in the presence of varying free stream velocity. Specifically, Ambrok's procedure for a turbulent boundary layer, Eq. (10), described in Kays and Crawford [29], was used. This procedure of determining external HTC was applied to all regions of suction and pressure surfaces except at the aerofoil's leading edge. The leading edge was modeled as a two-dimensional cylinder Display Formula

(10)$St=0.0287Pr−0.4Tsurface−Tr,g0.25μ0.2∫0xTsurface−Tr,g1.25(ugρg)dx0.2$

###### Conjugate Model Simulation.

A 3D Fourier's law module included in ANSYS 16.2—steady-state thermal module was used for the conjugate simulation. The blade was meshed using ANSYS meshing software and mesh refinement undertaken near the cooling holes to minimize discretization errors. The $Taw$ and external heat transfer coefficient (external heat load) as well as internal heat transfer coefficient (from Eq. (4)) and coolant inlet temperature (internal heat load) were imported into ANSYS finite element method module and interpolated on the external and internal surfaces of the blade, respectively. The summary of this process is shown in Fig. 8. Steps 2, 3, and 4 in Fig. 8 correspond to the steps in the program logic (Fig. 7). Three different total coolant inlet pressure cases; 40, 42, and 44 bar were considered in this analysis for a freestream total pressure of 40 bar and transonic exit conditions. The results are documented graphically in the following section of results and discussion.

## Results and Discussion

###### Film Effectiveness and Adiabatic Wall Temperature.

The Goldstein [22] correlation was used to predict the film effectiveness for each row of holes, which enabled the adiabatic wall temperature to be predicted in a procedure outlined in Fig. 7. The approach introduced by Sellers [17], which superposes the effect of rows of the film, was used to account for the accumulation of film effectiveness. To the authors' knowledge, this is the first time the two-dimensional film distribution has been superposed using the method of Sellers [17], the approach shown to be successful by Murray et al. [21]. Figure 9 shows film effectiveness and its corresponding adiabatic wall temperature on the external surface of the aerofoil.

A similar film cooling effectiveness trend can be seen on both the pressure and suction surface of the blade. On both the pressure and suction surfaces, the metal effectiveness increases from leading to trailing edge. This is because; (a) the film effectiveness builds up downstream as more coolant is injected into the films and (b) the internal heat transfer coefficient increases as the coolant flow through the wall block increases as the pressure difference between the coolant and static pressure increases.

It is interesting to note that the film cooling effectiveness, $ϵf$ on the pressure surface is significantly lower than the level on the suction surface. This bias is known to occur in real turbine designs which include aerofoil curvature and passage secondary flows. The correlation used in the present work is the Goldstein [22] equation based on flat plate data and makes no allowance for curvature or vortices. The different $ϵf$ levels on pressure and suction surfaces is caused by the difference in the mainstream velocity between the two surfaces.

Figure 10 shows the variation of nondimensional film flow rate per hole (obtained by dividing coolant flow rate through each hole by the mean coolant flow rate) as well as film cooling and metal effectiveness as a function of dimensionless streamwise distance, considering a case where the total coolant pressure is equal to 41 bar. There is undoubtedly high coolant mass flow on the suction surface compared to pressure surface—approximately three times in the cooling holes in the vicinity of the transonic mainstream flow. This is undesirable as it causes additional aerodynamic losses. In the near future, the design of the double-wall effusion blade will demand a well-thought means of reducing the amount of coolant ejected into the high-loss regions of the suction surface.

###### External Heat Transfer.

As aforementioned, the external heat transfer coefficient,$hg$ around the aerofoil was calculated from the Ambrok's procedure described in Kays and Crawford [29]. The obtained values of $hg$ were nondimensionalized using an averaged heat transfer coefficient over the whole surface, $hg,ave$. The resulting graph is included in Fig. 10. The highest heat load occurs in the vicinity of the leading edge where stagnation takes place. The suction surface experiences a rapid fall in the heat transfer coefficient in the laminar region of the boundary layer before a sharp rise in the transition region and finally a gradual fall toward the trailing edge. On the pressure surface, there is a gradual fall in the $hg$ from the leading edge to almost the half of the downstream distance followed by a gradual rise toward the trailing edge. It is expected that $hg$ should be higher on the pressure than on the suction surface. However, it is not the case in this study owing to the inability of the modified flat plate correlation used in this study to capture vortices, particularly the Taylor–G $o¨$ rtler vortices (Mayle et al. [30]) that are known to cause an increase in the pressure surface $hg$. A similar observation was made by Chowdhury et al. [31] after employing a simple analytical predictive model to analyze heat transfer coefficients distribution on both suction and pressure surfaces of a gas turbine blade.

###### Internal Cooling Effectiveness.

The success of the internal cooling system arises from the high heat transfer coefficients caused by the combination of the pedestals and the impingement jets. To evaluate the value of internal convection cooling, the model was adjusted to remove the benefits of film cooling (i.e., $ϵf$ was set to zero). The input temperature, on the external surface, while carrying out the conjugate simulation of the blade was set to be the mainstream effective gas temperature, $Tr,g$ not the adiabatic wall temperature, $Taw$. The resulting volume-averaged metal temperature was used in Eq. (12) to calculate the internal cooling effectiveness. The spatially averaged internal cooling effectiveness is plotted against nondimensional coolant mass flow rate in Fig. 11. In order to achieve very low m* values in the graph in Fig. 11, the coolant pressure was varied at each location downstream of the aerofoil so as to allow very low coolant mass flow through the aerofoil Display Formula

(11)$m*=m˙cCpchgAex$

The results show that the overall effectiveness of the double-walled effusion cooled system is dominated by the internal convective heat transfer. Figure 11 makes clear the significant contribution from internal convection to overall cooling effectiveness, with the internal cooling contributing approximately 80% to the overall cooling effectiveness. This contribution is found to be even higher at low $m*$. The high internal convection is attributed to the use of impingement cooling, the significant internal surface area of the impingement plate and pedestals, which manifest themselves in the high internal heat transfer coefficient, calculated from Eq. (4). The dashed lines (calculated with different values of convective efficiency as a parameter) show the cooling efficiency curves of $ηc=0.2,0.4,0.6,and0.8$ calculated from Eq. (13) considering theoretical values of $m*$ and $ϵm$. The internal cooling from this study corresponds to a convective efficiency of approximately $ηc=0.5$ particularly at low m* values (m* was calculated from Eq. (11)). This reflects the convective efficiency results of its unit building cell shown in Fig. 6Display Formula

(12)$ϵm=Tr,g−TmTr,g−Tc,in$
Display Formula
(13)$ϵm=m*ηc1+m*ηc$

It is natural that a high internal convection is beneficial and it results in less coolant air requirements as reported by Colladay [25]. From their analysis and comparison of wall cooling schemes for advanced gas turbine applications, they found out that an increase in the internal convection efficiency resulted in a reduced amount of cooling air required to maintain a given wall temperature. For instance, a full-coverage (effusion) cooling system with an internal convective efficiency of 0.6 required a $m*$ of 2.1 to maintain a wall temperature of 1255 K. But when the internal convective efficiency dropped to 0.2, $m*$ needed to maintain the same wall temperature increased by over 60% to about 3.4. This same observation has been echoed by Holland and Thake [28] in their analysis of high pressure turbine blade cooling.

This balance between internal convection and film cooling for combustor liner geometries is also reported by Andrews et al. [32]. They carried out an experimental investigation on a flat plate geometry to assess the relative performance of effusion and transpiration cooling. In their case, however, there was a greater contribution to overall cooling from film cooling as there was no large internal surface area, and correspondingly high effective internal heat transfer coefficient, influencing the cooling.

###### Overall Cooling Effectiveness.

In this case, both the internal convection and film cooling were taken into consideration. To evaluate the overall metal effectiveness, the conjugate simulation was performed with the external load $hg$ and $Taw$. It should be noted that the procedure followed to calculate the internal cooling and overall cooling effectiveness was similar except that the input temperature onto the blade's external surface in the finite code in the former was the mainstream effective gas temperature, $Tr,g$ while in the later, it was adiabatic wall temperature, $Taw$. The overall metal effectiveness graph is plotted alongside film and internal cooling effectiveness, Fig. 11. Both the overall effectiveness and effectiveness from the internal cooling are within $ηc$ range of 35 to 60%, reflecting the efficiency results from its building cell (shown in Fig. 6). The effectiveness results from this study corresponding to a m* = 2 from Fig. 11 were compared with the effectiveness results from a hypothetical blade (which combines a good level of both film and convection cooling) introduced by Holland and Thake [28]. The authors used analytical equations to estimate the cooling effectiveness of the hypothetical blade. It is interesting to note that the cooling scheme studied here has an effectiveness value that is approximately 20% lower than the hypothetical blade of Holland and Thake [28], at the same nondimensionless coolant mass flow (m* = 2).

## Conclusion

A computationally efficient numerical approach, which permits an assessment of the performance of a complex double-walled effusion cooled turbine blade, has been developed. The modified Goldstein [22] correlation was used to predict the film effectiveness for each row of holes and the film superposition downstream of the rows obtained using the Sellers [17] approach. The internal heat transfer coefficient was evaluated from a validated unit wall element conjugate analysis and the conduction through the blade was simulated using the finite code available in ANSYS steady-state thermal module. The results (which include film, metal effectiveness, and coolant mass flow consumption) have been found to closely match results available from the open literature. The developed novel numerical analysis approach offers a computationally efficient tool that can be used in the preliminary and optimization stages of a gas turbine blade design.

The internal cooling was found to contribute a larger proportion to the overall cooling effectiveness of the double-walled effusion cooled blade and this was attributed to a very large internal surface area brought about by a combination of impingement cooling and a large internal surface area contributed by the pedestals. In addition, there can be a substantial reduction in the cooling air requirements by employing cooling schemes with high internal convection, such as the double wall cooling scheme in this study, combined with effusion cooling.

## Acknowledgements

The authors would like to express sincere gratitude to Rolls-Royce Plc. and Engineering and Physical Science Research Council (EPSRC) for their support as well as the Rhodes Trust for supporting the lead author.

## Nomenclature

• A =

surface area

• $Cd$ =

discharge coefficient

• $Cp$ =

specific heat capacity

• $Cx$ =

chord length

• D =

diameter

• $Dh$ =

hydraulic diameter

• h =

heat transfer coefficient

• K =

thermal conductivity

• L =

length

• $m˙$ =

mass flow rate

• M =

blowing ratio

• Ma =

Mach number

• m* =

nondimensional mass-flow

• Pr =

Prandtl number

• Re =

Reynolds number

• $Sx,Sz$ =

streamwise and spanwise pitches, respectively

• St =

Stanton number

• T =

temperature

• $Taw$ =

• $Tm$ =

volume-average metal temperature

• $Tr$ =

recovery temperature

• $u$ =

velocity

• x, y, z =

Cartesian coordinates

• $xdecay$ =

streamwise film decay factor

• $z1/2$ =

lateral distance at which the temperature difference drops to half its value along the centerline on the hole

Greek Symbols
• $αt$ =

turbulent thermal diffusivity

• $ϵf$ =

film effectiveness

• $ϵm$ =

metal effectiveness

• $ϵo$ =

overall metal effectiveness

• $ηc$ =

convective efficiency

• $Λ$ =

Recovery factor

• $μ$ =

dynamic viscosity

• $ρ$ =

density

Subscripts
• c =

coolant

• ex =

external

• g =

mainstream gas

• in =

internal

• o =

total

• s =

static

## References

Han, J.-C. , 2013, Gas Turbine Heat Transfer and Cooling Technology, CRC Press/Taylor & Francis, Boca Raton, FL.
Andersson, B. , Andersson, R. , Håkansson, L. , Mortensen, M. , Sudiyo, R. , and van Wachem, B. , 2011, Computational Fluid Dynamics for Engineers, Cambridge University Press, Cambridge, UK.
Laschet, G. , Rex, S. , Bohn, D. , and Moritz, N. , 2002, “ 3-D Conjugate Analysis of Cooled Coated Plates and Homogenization of Their Thermal Properties,” Numer. Heat Transfer: Part A, 42(1–2), pp. 91–106.
Laschet, G. M. , Rex, S. , Bohn, D. , and Moritz, N. , 2003, “ Homogenization of Material Properties of Transpiration Cooled Multilayer Plates,” ASME Paper No. GT2003-38439.
Laschet, G. , Krewinkel, R. , Hul, P. , and Bohn, D. , 2013, “ Conjugate Analysis and Effective Thermal Conductivities of Effusion-Cooled Multi-Layer Blade Sections,” Int. J. Heat Mass Transfer, 57(2), pp. 812–821.
Zecchi, S. , Arcangeli, L. , Facchini, B. , and Coutandin, D. , 2004, “ Features of a Cooling System Simulation Tool Used in Industrial Preliminary Design Stage,” ASME Paper No. GT2004-53547.
Hylton, L. D. , Mihelc, M. S. , Turner, E. R. , Nealy, D. A. , and York, R. E. , 1983, “ Analytical and Experimental Evaluation of the Heat Transfer Distribution Over the Surfaces of Turbine Vanes,” NASA/Detroit Diesel Allison; Indianapolis, IN, Technical Report No. NASA CR 168015.
Heidmann, J. D. , Kassab, A. J. , Divo, E. A. , Rodriguez, F. , and Steinthorsson, E. , 2003, “ Conjugate Heat Transfer Effects on a Realistic Film-Cooled Turbine Vane,” ASME Paper No. GT2003-38553.
Kassab, A. , Divo, E. , Heidmann, J. , Steinthorsson, E. , and Rodriguez, F. , 2003, “ BEM/FVM Conjugate Heat Transfer Analysis of a Three-Dimensional Film Cooled Turbine Blade,” Int. J. Heat Fluid Flow, 13(5), pp. 581–610.
Rigby, D. L. , Heidmann, J. D. , Ameri, A. A. , and Garg, V. K. , 2001, “ Improved Modeling Capabilities in Glenn-HT—The NASA Glenn Research Center General Multi-Block Navier–Stokes Heat Transfer Code,” Cleveland, OH, NASA Report No. 20020073073.
Mendez, S. , Nicoud, F. , and Poinsot, T. , “ Large-Eddy Simulation of a Turbulent Flow around a Multi-Perforated Plate,” Complex Effects in Large Eddy Simulations, Kassinos S. C., Langer C. A., Iaccarino G., Moin P., eds., Vol. 56. Springer, Berlin, Heidelberg.
Amaral, S. , Verstraete, T. , Van den Braembussche, R. , and Arts, T. , 2010, “ Design and Optimization of the Internal Cooling Channels of a High Pressure Turbine Blade—Part I: Methodology,” ASME J. Turbomach., 132(2), p. 021013.
Bonini, A. , Andreini, A. , Carcasci, C. , Facchini, B. , Ciani, A. , and Innocenti, L. , 2012, “ Conjugate Heat Transfer Calculations on GT Rotor Blade for Industrial Applications—Part I: Equivalent Internal Fluid Network Setup and Procedure Description,” ASME Paper No. GT2012-69846.
Andreini, A. , Bonini, A. , Da Soghe, R. , Facchini, B. , Ciani, A. , and Innocenti, L. , “ Conjugate Heat Transfer Calculations on GT Rotor Blade for Industrial Applications—Part II: Improvement of External Flow Modeling,” ASME Paper No. GT2012-69849.
Colban, W. F. , Thole, K. A. , and Bogard, D. , 2010, “ A Film-Cooling Correlation for Shaped Holes on a Flat-Plate Surface,” ASME J. Turbomach., 133(1), p. 011002.
L'Ecuyer, M. R. , and Soechting, F. O. , 1985, “ A Model for Correlating Flat Plate Film Cooling Effectiveness for Rows of Round Holes,” AGARD Heat Transfer and Cooling in Gas Turbine, West Palm Beach, FL, p. 12.
Sellers, J. P. , 1963, “ Gaseous Film Cooling With Multiple Injection Stations,” AIAA J., 1(9), pp. 2154–2156.
Andrei, L. , Andreini, A. , Facchini, B. , and Winchler, L. , 2014, “ A Decoupled CHT Procedure: Application and Validation on a Gas Turbine Vane With Different Cooling Configurations,” Energy Procedia, 45, pp. 1087–1096.
Baldauf, S. , Scheurlen, M. , Schulz, A. , and Wittig, S. , 2002, “ Correlation of Film-Cooling Effectiveness From Thermographic Measurements at Engine-like Conditions,” ASME Paper No. GT2002-30180.
Baldauf, S. , Schulz, A. , Wittig, S. , and Scheurlen, M. , 1997, “ An Overall Correlation of Film Cooling Effectiveness From One Row of Holes,” ASME Paper No. 97-GT-079.
Murray, A. V. , Ireland, P. T. , Wong, T. H. , Tang, S. W. , and Rawlinson, A. J. , 2018, “ High Resolution Experimental and Computational Methods for Modelling Multiple Row Effusion Cooling Performance,” Int. J. Turbomach., Propul. Power, 3(1), p. 4.
Goldstein, R. J. , 1971, “ Film Cooling,” Advances in Heat Transfer, Elsevier, New York, pp. 321–379.
Murray, A. V. , Ireland, P. T. , and Rawlinson, A. J. , 2017, “ An Integrated Conjugate Computational Approach for Evaluating the Aerothermal and Thermomechanical Performance of Double-Wall Effusion Cooled Systems,” ASME Paper No. GT2017-64711.
Gurram, N. , Ireland, P. T. , Wong, T. H. , and Self, K. P. , 2016, “ Study of Film Cooling in the Trailing Edge Region of a Turbine Rotor Blade in High Speed Flow Using Pressure Sensitive Paint,” ASME Paper No. GT2016-57356.
Colladay, R. S. , 1972, “ Analysis and Comparison of Wall Cooling Schemes for Advanced Gas Turbine Applications,” NASA/Lewis Research Center, Cleveland, OH, Report Nos. NASA-TN-D-6633.
Lee, T. W. , 2013, Aerospace Propulsion, Wiley, West Sussex, UK.
Baldauf, S. , Schulz, A. , and Wittig, S. , 1999, “ High-Resolution Measurements of Local Effectiveness From Discrete Hole Film Cooling,” ASME J. Turbomach., 123(4), pp. 758–765.
Holland, M. J. , and Thake, T. F. , 1980, “ Rotor Blade Cooling in High Pressure Turbines,” J. Aircr., 17(6), pp. 412–418.
Kays, W. M. , and Crawford, M. E. , 1993, Convective Heat and Mass Transfer, McGraw-Hill, New York.
Mayle, R. E. , Blair, M. F. , and Kopper, F. C. , 1979, “ Turbulent Boundary Layer Heat Transfer on Curved Surfaces,” ASME J. Heat Transfer, 101(3), pp. 521–525.
Chowdhury, N. H. K. , Zirakzadeh, H. , and Han, J.-C. , 2017, “ A Predictive Model for Preliminary Gas Turbine Blade Cooling Analysis,” ASME J. Turbomach., 139(9), p. 091010.
Andrews, G. E. , Asere, A. A. , Mkpadi, M. C. , and Tirmahi, A. , 1986, “ Transpiration Cooling: Contribution of Film Cooling to the Overall Cooling Effectiveness,” ASME Paper No. 86-GT-136.
View article in PDF format.

## References

Han, J.-C. , 2013, Gas Turbine Heat Transfer and Cooling Technology, CRC Press/Taylor & Francis, Boca Raton, FL.
Andersson, B. , Andersson, R. , Håkansson, L. , Mortensen, M. , Sudiyo, R. , and van Wachem, B. , 2011, Computational Fluid Dynamics for Engineers, Cambridge University Press, Cambridge, UK.
Laschet, G. , Rex, S. , Bohn, D. , and Moritz, N. , 2002, “ 3-D Conjugate Analysis of Cooled Coated Plates and Homogenization of Their Thermal Properties,” Numer. Heat Transfer: Part A, 42(1–2), pp. 91–106.
Laschet, G. M. , Rex, S. , Bohn, D. , and Moritz, N. , 2003, “ Homogenization of Material Properties of Transpiration Cooled Multilayer Plates,” ASME Paper No. GT2003-38439.
Laschet, G. , Krewinkel, R. , Hul, P. , and Bohn, D. , 2013, “ Conjugate Analysis and Effective Thermal Conductivities of Effusion-Cooled Multi-Layer Blade Sections,” Int. J. Heat Mass Transfer, 57(2), pp. 812–821.
Zecchi, S. , Arcangeli, L. , Facchini, B. , and Coutandin, D. , 2004, “ Features of a Cooling System Simulation Tool Used in Industrial Preliminary Design Stage,” ASME Paper No. GT2004-53547.
Hylton, L. D. , Mihelc, M. S. , Turner, E. R. , Nealy, D. A. , and York, R. E. , 1983, “ Analytical and Experimental Evaluation of the Heat Transfer Distribution Over the Surfaces of Turbine Vanes,” NASA/Detroit Diesel Allison; Indianapolis, IN, Technical Report No. NASA CR 168015.
Heidmann, J. D. , Kassab, A. J. , Divo, E. A. , Rodriguez, F. , and Steinthorsson, E. , 2003, “ Conjugate Heat Transfer Effects on a Realistic Film-Cooled Turbine Vane,” ASME Paper No. GT2003-38553.
Kassab, A. , Divo, E. , Heidmann, J. , Steinthorsson, E. , and Rodriguez, F. , 2003, “ BEM/FVM Conjugate Heat Transfer Analysis of a Three-Dimensional Film Cooled Turbine Blade,” Int. J. Heat Fluid Flow, 13(5), pp. 581–610.
Rigby, D. L. , Heidmann, J. D. , Ameri, A. A. , and Garg, V. K. , 2001, “ Improved Modeling Capabilities in Glenn-HT—The NASA Glenn Research Center General Multi-Block Navier–Stokes Heat Transfer Code,” Cleveland, OH, NASA Report No. 20020073073.
Mendez, S. , Nicoud, F. , and Poinsot, T. , “ Large-Eddy Simulation of a Turbulent Flow around a Multi-Perforated Plate,” Complex Effects in Large Eddy Simulations, Kassinos S. C., Langer C. A., Iaccarino G., Moin P., eds., Vol. 56. Springer, Berlin, Heidelberg.
Amaral, S. , Verstraete, T. , Van den Braembussche, R. , and Arts, T. , 2010, “ Design and Optimization of the Internal Cooling Channels of a High Pressure Turbine Blade—Part I: Methodology,” ASME J. Turbomach., 132(2), p. 021013.
Bonini, A. , Andreini, A. , Carcasci, C. , Facchini, B. , Ciani, A. , and Innocenti, L. , 2012, “ Conjugate Heat Transfer Calculations on GT Rotor Blade for Industrial Applications—Part I: Equivalent Internal Fluid Network Setup and Procedure Description,” ASME Paper No. GT2012-69846.
Andreini, A. , Bonini, A. , Da Soghe, R. , Facchini, B. , Ciani, A. , and Innocenti, L. , “ Conjugate Heat Transfer Calculations on GT Rotor Blade for Industrial Applications—Part II: Improvement of External Flow Modeling,” ASME Paper No. GT2012-69849.
Colban, W. F. , Thole, K. A. , and Bogard, D. , 2010, “ A Film-Cooling Correlation for Shaped Holes on a Flat-Plate Surface,” ASME J. Turbomach., 133(1), p. 011002.
L'Ecuyer, M. R. , and Soechting, F. O. , 1985, “ A Model for Correlating Flat Plate Film Cooling Effectiveness for Rows of Round Holes,” AGARD Heat Transfer and Cooling in Gas Turbine, West Palm Beach, FL, p. 12.
Sellers, J. P. , 1963, “ Gaseous Film Cooling With Multiple Injection Stations,” AIAA J., 1(9), pp. 2154–2156.
Andrei, L. , Andreini, A. , Facchini, B. , and Winchler, L. , 2014, “ A Decoupled CHT Procedure: Application and Validation on a Gas Turbine Vane With Different Cooling Configurations,” Energy Procedia, 45, pp. 1087–1096.
Baldauf, S. , Scheurlen, M. , Schulz, A. , and Wittig, S. , 2002, “ Correlation of Film-Cooling Effectiveness From Thermographic Measurements at Engine-like Conditions,” ASME Paper No. GT2002-30180.
Baldauf, S. , Schulz, A. , Wittig, S. , and Scheurlen, M. , 1997, “ An Overall Correlation of Film Cooling Effectiveness From One Row of Holes,” ASME Paper No. 97-GT-079.
Murray, A. V. , Ireland, P. T. , Wong, T. H. , Tang, S. W. , and Rawlinson, A. J. , 2018, “ High Resolution Experimental and Computational Methods for Modelling Multiple Row Effusion Cooling Performance,” Int. J. Turbomach., Propul. Power, 3(1), p. 4.
Goldstein, R. J. , 1971, “ Film Cooling,” Advances in Heat Transfer, Elsevier, New York, pp. 321–379.
Murray, A. V. , Ireland, P. T. , and Rawlinson, A. J. , 2017, “ An Integrated Conjugate Computational Approach for Evaluating the Aerothermal and Thermomechanical Performance of Double-Wall Effusion Cooled Systems,” ASME Paper No. GT2017-64711.
Gurram, N. , Ireland, P. T. , Wong, T. H. , and Self, K. P. , 2016, “ Study of Film Cooling in the Trailing Edge Region of a Turbine Rotor Blade in High Speed Flow Using Pressure Sensitive Paint,” ASME Paper No. GT2016-57356.
Colladay, R. S. , 1972, “ Analysis and Comparison of Wall Cooling Schemes for Advanced Gas Turbine Applications,” NASA/Lewis Research Center, Cleveland, OH, Report Nos. NASA-TN-D-6633.
Lee, T. W. , 2013, Aerospace Propulsion, Wiley, West Sussex, UK.
Baldauf, S. , Schulz, A. , and Wittig, S. , 1999, “ High-Resolution Measurements of Local Effectiveness From Discrete Hole Film Cooling,” ASME J. Turbomach., 123(4), pp. 758–765.
Holland, M. J. , and Thake, T. F. , 1980, “ Rotor Blade Cooling in High Pressure Turbines,” J. Aircr., 17(6), pp. 412–418.
Kays, W. M. , and Crawford, M. E. , 1993, Convective Heat and Mass Transfer, McGraw-Hill, New York.
Mayle, R. E. , Blair, M. F. , and Kopper, F. C. , 1979, “ Turbulent Boundary Layer Heat Transfer on Curved Surfaces,” ASME J. Heat Transfer, 101(3), pp. 521–525.
Chowdhury, N. H. K. , Zirakzadeh, H. , and Han, J.-C. , 2017, “ A Predictive Model for Preliminary Gas Turbine Blade Cooling Analysis,” ASME J. Turbomach., 139(9), p. 091010.
Andrews, G. E. , Asere, A. A. , Mkpadi, M. C. , and Tirmahi, A. , 1986, “ Transpiration Cooling: Contribution of Film Cooling to the Overall Cooling Effectiveness,” ASME Paper No. 86-GT-136.

## Figures

Fig. 1

Features of a double-walled effusion cooled concept turbine blade

Fig. 2

Features of a double-walled effusion cooled concept turbine blade including leading edge showerhead cooling holes, pin-fin bank, TE slots and the flow direction

Fig. 3

CFD results of flow velocity contour distribution in the unit cell from Murray et al. [23]

Fig. 4

Double-wall blade with the unit wall element showing the definition of the geometrical parameters

Fig. 5

Outer skin of the blade where numerical analysis is performed

Fig. 6

ηc−Re characteristics compared to that of three simple duct cooling systems, characterized by L/Dh = 20, 40 and 60

Fig. 11

A graph of effectiveness as a function of nondimensional coolant mass flow, m* from this study

Fig. 10

Nondimensional film flow rate per hole, film cooling effectiveness, metal effectiveness and dimensionless external heat transfer coefficient as a function of the blade's dimensionless streamwise location

Fig. 9

(a) Film cooling effectiveness on the blade and (b) its corresponding adiabatic wall temperature at Po,c=40 bar

Fig. 8

Model setup in ansys steady-state thermal module

Fig. 7

Steps in the iterative code used to determine aerofoil wall temperature

## Tables

Table 1 The dimensions of the unit wall element from Ref. [23]
Table 2 Engine-representative operating conditions used

## Errata

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