A New Statistical-Based Correlation for the Rib Fin Effects on the Overall Heat Transfer Coefficient in a Rib-Roughened Cooling Channel

Heat transfer coefficients in the cooling cavities of turbine airfoils are greatly enhanced by the presence of discrete ribs on the cavity walls. These ribs introduce two heat transfer enhancing features: a significant increase in heat transfer coefficient by promoting turbulence and mixing, and an increase in heat transfer area. Considerable amount of data are reported in open literature for the heat transfer coefficients both on the rib surface and on the floor area between the ribs. Many airfoil cooling design software tools, however, require an overall average heat transfer coefficient on a rib-roughened wall. Dealing with a complex flow circuit in conjunction with 180◦ bends, numerous film holes, trailing-edge slots, tip bleeds, crossover impingement, and a conjugate heat transfer problem; these tools are not often able to handle the geometric details of the rib-roughened surfaces or local variations in heat transfer coefficient on a rib-roughened wall. On the other hand, assigning an overall area-weighted average heat transfer coefficient based on the rib and floor area and their corresponding heat transfer coefficients will have the inherent error of assuming a 100% fin efficiency for the ribs, that is, assuming that rib surface temperature is the same as the rib base temperature. Depending on the rib geometry, this error could produce an overestimation of up to 10% in the evaluated rib-roughened wall heat transfer coefficient. In this paper, a correction factor is developed that can be applied to the overall area-weighted average heat transfer coefficient that, when applied to the projected rib-roughened cooling cavity walls, the net heat removal from the airfoil is the same as that of the rib-roughened wall. To develop this correction factor, the experimental results of heat transfer coefficients on the rib and on the surface area between the ribs are combined with about 400 numerical conduction models to determine an overall equivalent heat transfer coefficient that can be used in airfoil cooling design software. A well-known group method of data handling (GMDH) scheme was then utilized to develop a correlation that encompasses most pertinent parameters including the rib geometry, rib fin efficiency, and the rib and floor heat transfer coefficients.


INTRODUCTION
Serpentine cooling channels within turbine airfoils are usually roughened with ribs.These ribs increase the level of mixing of the cooler core air with the warmer air close to the channel wall and restart the boundary layer after flow reattachment between ribs resulting in enhanced convective heat transfer coefficients.Experimental results, reported by many investigators, show as high as a five-fold enhancement in heat transfer coefficients of rib-roughened surfaces when compared with those of smooth (nonribbed) channels.Geometric parameters such as channel aspect ratio (AR), rib height-to-passage hydraulic diameter (e/D h ) or blockage ra-tio, rib angle of attack (α), the manner in which the ribs are positioned relative to one another (inline, staggered, crisscross, etc.), rib pitch-to-height ratio (P/e), and rib shape (round versus sharp corners, fillets, rib aspect ratio (AR rib ), and skewness towards the flow direction) have pronounced effects on both local and overall heat transfer coefficients.Considerable data are available on both the heat transfer coefficient on the passage surface between the ribs and on the rib surfaces.Some of these effects were studied by different investigators such as Abuaf et al. [1], Burggraf [2], Chandra et al. [3], Chandra and Han [4], Han et al. [5][6][7][8], Liou et al. [9,10], Metzger et al. [11][12][13], Taslim et al. [14][15][16][17][18], Taslim and Wadsworth [19], Korotky and Taslim [20], Taslim International Journal of Rotating Machinery and Korotky [21], and Taslim and Lengkong [22,23] and Webb et al. [24].These studies show a considerable variation in heat transfer coefficient from the surface area between the ribs to the rib forward, top, and aft surfaces.Most of the software tools for the design of airfoil cooling circuits, however, have no provisions to handle either the geometric details of the ribs or the variations in heat transfer coefficients along the rib-roughened surface.Therefore, the airfoil cooling circuit designer is limited to assign an equivalent heat transfer coefficient on channel surface that accounts for the area enhancement, heat transfer coefficient variations, and the rib fin effects.Heat transfer area enhancement is a purely geometric analysis that is presented in this paper for a variety of rib geometries.Variations in heat transfer coefficient on the area between the ribs (h floor ) and on the rib surface itself (h rib ) are available for a wide range of rib geometries in open literature.Thus, an area-weighted average heat transfer coefficient is easily evaluated for a wide range of rib geometries.This area-weighted average heat transfer coefficient, however, is based on the approximation that the entire rib surface is at the same temperature as the rib base, that is, a 100% fin efficiency is assumed for the ribs.These ribs at the same time do not fall into the classic fin category for which the overall fin efficiencies are readily available.Therefore, the main objective of this investigation was to generate a correlation for the rib fin effect corrections that encompasses all common rib geometric parameters as well as the common hot and cold side flow conditions.

DERIVATIONS
Details of the geometric derivations are reported in Taslim [25].For a repeated computational domain with a length of P (rib pitch) that includes a rib and half of the floor surface on each side of the rib, as shown in Figure 1, the heat transfer areas are A floor = P − w bot + 2r fillet tan(β/2) sin(α) a surface area between a pair of ribs, (2) The equivalent heat transfer coefficient for a rib is defined as a heat transfer coefficient that, when applied on the base surface of the rib (projection surface), will have the same thermal effects as that of the actual heat transfer coefficient applied on all rib exposed surfaces, that is, Substituting for these areas from (1) through ( 3) and normalizing all rib dimensions with the rib height, e, we have (5) The overall heat transfer coefficient is the area-weighted average of the surface between a pair of ribs (floor) and the rib equivalent heat transfer coefficient, that is, where Pa is the projection area associated with one rib that is the area representing the airfoil cooling cavity surface in most airfoil cooling design software tools.Upon substitution for areas and heat transfer coefficients from (1) through (5) with simplifications,

Correction for the rib fin effects
In applying the h equiv to the rib-roughened projection area on the cooling channel surface, it is assumed that the entire rib surface is at its base temperature.In other words, the rib fin effects are neglected.This assumption may not introduce any significant error for small blockage ratio ribs arranged at high pitch-to-height ratios.However, for high aspect and blockage ratio ribs, the error in the overall heat transfer coefficient can be as high as 20%.To investigate these effects, twodimensional models of several rib geometries were meshed and numerically analyzed for a range of pertinent parameters and realistic boundary conditions.Three-dimensional models are also under investigations that will be reported in the future.The objective is to determine a correction factor that when is multiplied by h equiv , it gives the final heat transfer coefficient, h corrected , that goes into airfoil cooling design software tools.

Numerical models
Figure 2 shows the actual airfoil cooling cavity wall (on the left) and the simplified equivalent wall used in design software tools (on the right).The total heat transfer rate per unit depth from the hot to the cold side for the simple slab case is readily evaluated as Software tools domain The total heat transfer rate from the gas to the coolant side for the actual cooling cavity wall, however, was determined numerically.A finite element package was used to solve the two-dimensional heat conduction equation, ∇ 2 T = 0, for the assigned convective boundary conditions on the hot and cold sides and symmetric boundary conditions on the right and left boundaries where a rib-roughened wall repeats itself.Temperature field for a typical geometry and for one set of assigned boundary conditions is shown in Figure 3. Once the temperature field was calculated, the total heat transfer rate from the gas to the coolant side was determined by where n is the number of finite elements on the hot side of the numerical model and T s,i is the surface temperature of the ith element on the gas side.Note that The corrected heat transfer coefficient was then calculated by equating ( 8) and ( 9) to obtain ( Qdesign = Qnum ): The reported correction factor for the rib fin effect is defined as  Figure 4 and Table 1 show all the geometries that were analyzed.Four rib aspect ratios (0.5, 1, 1.33, and 2) were investigated at four pitch-to-height ratios of 5, 7.5, 10, and 12.5.Each case was run for all sharp as well as round top corners and fillets.The range of the chosen parameters are believed to cover many of today's designs.

RESULTS AND DISCUSSION
The first step in this investigation was to determine which parameters were dominant in rib fin effects.Earlier runs revealed that T cold , T hot , and h hot had little effects on the correction factor.The effects of t wall and k metal within the range of today's designs were also negligible.Therefore, these parameters were eliminated from the list of pertinent parameters.Details of all numerical analyses are given in Taslim et al. [18].The following figures, representing a total of 386 runs, show the variation of the correction factor with respect to different parameters for some selected geometries.The practical ranges of these parameters were determined in consultation with our industrial partners.They were (T hot − T cold ): 700 to 1000 K, T hot : 1450 to 1650 K, h hot : 35 to 280 W/m 2 K, h rib : 88 to 212 W/m 2 K, h floor : 22 to 88 W/m 2 K, k metal : 5 to 9 W/mK.Figure 5 shows the variation of the correction factor with the rib pitch-to-height ratio, P/e, for three rib aspect ratios.Each rib geometry is run for three cases of all sharp corners, round top corners with no fillets and round top corners with fillets.The general trend is that the correction factor increases with P/e and with the rounding of top corners and presence of fillets for all cases.This behavior is expected since higher pitch-to-height ratios correspond to less number of ribs per given area and, as a result, less rib fin effects on the overall heat transfer coefficient.Rib aspect ratio of 1 at P/e = 12.5 had the highest correction factor while the rib aspect ratio of 2 at P/e = 5 had the lowest correction factor.
Figure 6 shows the variation of correction factor with h floor and h rib for a typical rib geometry representing a rectangular rib of aspect ratios 1 with sharp as well as round corners.The general behavior is the same for all cases as they are presented in Taslim [25].The correction factor increases with h floor and decreases with h rib .A physical explanation for this behavior is that higher rib heat transfer coefficients correspond to higher convective heat transfer from the rib surface to the coolant and, as a result, higher temperature  difference between the rib surface and rib base.Furthermore, the ribs with round corners and fillets produce higher correction factors than those of sharp corners and, as it was shown in Figure 5, the correction factor increases with the rib pitchto-height ratio.It should be noted that a higher correction factor means that the rib surface temperature is less affected by the rib fin effect.
Figure 7 represents the results of all 386 cases that were analyzed.High aspect ratio rib cases (AR rib = 2) with sharp corners have the lowest correction factor since, geometrically, they are closer to conventional fins and, as a result, fin effects are more pronounced.The low aspect ratio rib cases specially with round corners and fillets have the highest correction factors, that is, less fin effects since, geometrically, they are far from conventional fins.
The next major task was to develop a correlation that would encompass all pertinent parameters affecting the correction factor.For the development of an effective statistical model to account for the influence of the pertinent nondimesional parameters on output function, the Group method of data handling (GMDH) was used.This method (Ivakhnenko [26], Ivakhnenko and Yurachkovsky [27]) is intended for data files with a limited number of experimental points.It permits combining the influence of all important International Journal of Rotating Machinery  parameters in the entire structural form of the correlation.Since this method is not very known and widely used, it is necessary to describe it in brief.
The GMDH is an original method for obtaining mathematical models for problems of structural-parametric identification, or performing simulations on the basis of ex- perimental data with uncertainty.Such problems consist of obtaining mathematical models which approach unknown functional relationships between different parameters (objects) in a process.The available information about these relationships is presented in implicit forms within a sample (file) of existing data.The GMDH differs from other International Journal of Rotating Machinery  methods by the application of the principles of automatic generation of variants-inconclusive solutions, and by successive selection with the help of external criteria for the definition of models of optimal complexity.The GMDH relates to the group of methods of "cross-model validation" (as distinct from methods of applied regression analysis and methods "with explicit limitation of model complexity") the solution of simulation problems.The GMDH procedure consists of the optimization of the model structure and parameters with the help of expedient enumeration of possibilities for automatically generated model variants within the set class of basic models and with the help of external criteria  based on a sample fragmentation.Using the GMDH algorithms, it is possible to simulate static objects, time series, and dynamic objects (processes) within the following main classes of models: linear, polynomial, additive-multiplicative, harmonic, autoregressive, and so forth, to name a few.During enumeration model possibilities, it may be convenient to use generators of structures of enumerating and multiseries types.In this procedure, the last ones are completely original, and the first ones in some particular cases (under abandoned principle of inconclusive solutions, or freedom of choice) emulate the procedures of test of all regressions, branches and bounds, or step-by-step regression.Therefore, the GMDH is designed for models definition of technical, technological, ecological, and other complex objects and processes using experimental data under the condition of uncertainty, featured by limited numbers of noise observations, incomplete information about structure of significant variables, and internal correlations of object (process).Several successful examples of the GMDH, applied to turbomachines, are explained in Nezym et al. [28,29].

RIB FIN STATISTICAL MODEL BASED ON THE GMDH ALGORITHM
Initial data for the model development (60 out of 386 experimental points), shown in

Development of the model
A multiseries algorithm with the orthogonalization of the principal members was selected.The orthogonality of the model members permits the user to determine the highest degree of the model with N experimental data points.A high-degree polynomial was found as a support function.
For identification of optimal structure of the model formula, a consistency criterion in the form of minimum shift, Ivakhnenko and Yurachkovsky [27], was used at the base of analysis of solution: where with (i) criterion: stability, (ii) number of series: 8, (iii) standard deviation = 0.00432, (iv) correlation coefficient r = 0.999990, (v) maximum relative error = −1.31%for data point 37, (vi) standard relative error = 0.47%.
The calculated correction factors using the proposed correlation ( 14) are compared with those of Table 2 in Figure 14.A maximum relative error of −1.31% for point number 37 was calculated.The solid symbols in that figure demonstrate the model test points (not used for the simulation).These five test points, shown in Table 3, were chosen with the help of the table of random numbers.

CONCLUSIONS
Group method of data handling, a self-organizing method, was used to correlate the parameters affecting the fin efficiency of the ribs in cooling cavities of turbine airfoils or on heat exchangers surfaces.Major conclusions of this study were the following.
(1) Rib fin effects should be taken into consideration in the design of airfoil cooling circuits, heat exchangers, and thermal systems when, in using the software design tools, details of rib geometry and variations of heat transfer coefficients are replaced by an equivalent heat transfer coefficient.(2) Evidenced by the accuracy of the model (standard deviation: 0.00432, standard relative error: 0.47%, correlation coefficient: 0.99396), GMDH is a powerful tool for the correlation of data with complex interrelated parameters.

NOMENCLATURE a:
Channel width (Figure 1) A base : Rib base area (Figure 1) A floor : Heat transfer area between two ribs A rib : Rib total exposed heat transfer area AR: Channel aspect ratio AR rib : Ribaspectratio(2e/(w top + w bot )) D h : Channel hydraulic diameter e: Rib height h corrected : Fin-effect-corrected heat transfer coefficient on a rib-roughened surface (13) h equiv : Rib equivalent heat transfer coefficient (4) h floor : Average heat transfer coefficient on the surface between a pair of ribs h hot : Hotsideaverageheattransfercoefficient h overall : Overall area-weighted average heat transfer coefficient on a rib and on the surface between a pair of ribs h rib : Average heat transfer coefficient on the rib surface h smooth : Fully developed heat transfer coefficient in a smooth channel k metal : Rib-roughened wall thermal conductivity l i : Finite element cell width on the gas side boundary n: Number of finite element cells on the gas side boundary Nu smooth : Nusselt number for a fully developed flow in a smooth channel P: Rib pitch (center to center) Qdesign : Heat transfer from the gas to the coolant calculated by blade design software tools (Figure 2) Qnum : Heat transfer from the gas to the coolant calculated from numerical models (Figure 2) r: Correlation coefficient r fillet : Fillet radius r top : Rib top corner radius R corrected : Equivalent coolant side thermal resistance R hot : Hot side thermal resistance R wall : Rib-roughened wall thermal resistance P rib pitch (center to center) t wall : Rib-roughened wall thickness T f : Film temperature T cold : Cold side average temperature T hot : Hot side average temperature T s : Surface temperature w top : Rib top width w bot : Rib bottom width y: Correction factor defined by ( 14) α: Rib angle of attack β: Rib top angle (Figure 1) fillet /e tan(β/2) + w top /e − 2 r top /e tan(β/2) + 2β r top /e + r fillet /e w bot /e + 2 r fillet /e tan(β/2) .

Figure 3 :
Figure 3: Temperature contours for a typical case.

Figure 5 :
Figure 5: Correction factor variation with P/e.

Figure 6 :
Figure 6: Correction factor variation with h rib and h floor for a rib aspect ratio of 1.

Figure 8 :
Figure 8: Correction factor variation with P/e for the set of chosen data.

Figure 9 :
Figure 9: factor variation with R top /e for the set of chosen data.

Figure 10 :
Figure 10: Correction factor variation with R fillet /e for the set of chosen data.

Figure 11 :
Figure 11: Correction factor variation with AR rib for the set of chosen data.

Figure 12 :Figure 13 :
Figure 12: Correction factor variation with h rib for the set of chosen data.

Figure 14 :
Figure 14: Comparison of actual versus correlated data.
an output value of the model, based on the estimated coefficients of subsample A; (iii) Y B is an output value of the model, based on the estimated coefficients of subsample B, and under the condition ofN A + N B = N; (iv) Y *j is the given data in Table2.An external criterion for stability was used and with the application of the GMDH algorithm, the model formula was obtained in the following form: y = 1.01138−1.83878×10−4 h rib + 1.44299 × 10 −5 (P/e)h floor + 7.34671 × 10 −4 R fillet /e)h rib (P/e) − 8.93306 AR rib (P/e)h floor − 1.59563 × 10 −3 (P/e)h floor h rib − 9.79612 × 10 −3 AR rib h rib (P/e)h floor −1.38097 × 10 −3 AR 2 rib −6.27083 × 10 −6 AR 3 rib h rib (P/e)

Table 1 :
Geometric specifications.T cold = 1000 K, T hot = 1645 K, h hot = 141 W/m 2 K, k metal = 7.25 W/mK, t wall = rib biggest dimension (height or width) Figure 7: Correction factor variation with h rib and h floor for all rib geometries.

Table 2 :
Randomly chosen data points.

Table 2 ,
were chosen with the help of the table of random numbers.Individual influence of the six proposed arguments (P/e, R top /e, R fillet /e, AR rib , h rib , and h floor ) on function Y (correction factor) are presented in Figure8through Figure13.It should be noted that there are 60 data points in each figure.However, some of them are coinciding or are very close to each other for the scale of the plots.All these functions are nonlinear, as corre-lation coefficient r is significantly less than 1 (in the range of 0.15 to 0.38).