Small Diameter Film Cooling Hole Heat Transfer: The Influence of the Hole Length

THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS y^ 345 E. 47 St., New York, N.Y. 17 91-GT-344 The Society shall not be responsible for statements or opinions advanced in papers or in discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. M Discussion is printed only if the paper is published in an ASME Journal. Papers are available ]^L from ASME for fifteen months after the meeting. Printed in USA. Copyright 1991 by ASME Small Diameter Film Cooling Hole Heat Transfer: The Influence of the Hole Length G. E. ANDREWS, F. BAZDIDI-TEHRANI, C. I. HUSSAIN and J. P. PEARSON Department of Fuel and Energy University of Leeds Leeds, LS2 9JT, UK ABSTRACT The overall surface averaged heat transfer was determined for air passing through arrays of small diameter holes drilled at 9 through thin metal walls. The influence of the wall metal thickness, L, was investigated for a range of hole diameters, D, and pitch, X. was varied from.43 to 8.3 using 13 different test geometries. It was found that although the influence of was significant, there was only a ±% data scatter on a correlation of the results that ignored the influence of for.8<<. The results showed that the heat transfer was dominated by the hole approach flow and this surface area A was the appropriate heat transfer area for the determination of the heat transfer coefficient. The dominant parameters that affected the heat transfer were G and X/D. An improved correlation for a range of was achieved if the heat transfer surface area was taken as the sum of A x and A h, the hole internal surface area. NOMENCLATURE A htotal internal hole surface area, m 2. A xtotal hole approach surface area, Eq.5, m 2. Cp Specific heat of the wall material. D Hole internal diameter, m. G Coolant mass flow rate per unit plate area A x, kg/sm 2 h Convective heat transfer_coefficient based on the hole surface area, A h, W/m 2 K. h tconvective heat transfer coefficient based on the area A h+a x. h xconvective heat transfer coefficient based on the surface area A x. k Thermal conductivity of the coolant, W/mK. L Hole length, m. m Mass of the test plate heat transfer section, kg. N Number of holes per unit surface area A 5, m -2. Nu Nusselt number based on h and the hole diameter D, ho/k. Nu xnusselt number based on h x and hole diameter D, h xd/k. Nu tnusselt number based on h t. Pr Prandtl number. Hole ynolds number based on D and the coolant velocity in the hole. Tc Coolant air supply temperature, K. Tw Mean wall temperature, K. Twi t Transient wall temperature at time t, K. Tw iinitial mean wall temperature, K. Time constant, Eq. 4. X Hole Pitch, m. 1. INTRODUCTION Small diameter holes are extensively used for film cooling of gas turbine combustor walls and turbine blades. The geometry is usually that of an array of small diameter holes drilled in a relatively thin metal wall and the influence of this wall thickness was studied in the present work. Full coverage discrete hole film cooling, which is usually referred to as effusion cooling, offers a relatively simple technique for the efficient cooling of gas turbine combustor and turbine blade walls. Many gas turbine combustors use effusion cooling for the cooling of local hot spots on the wall, where a locally high coolant flow rate for the hot spot area is used. Design considerations for the combustor wall applications have been presented by the authors (Andrews and Mkpadi, 1984, and Andrews et al 1985a). The hole length to diameter ratio is rarely in excess of 8 and may be less than unity in some turbine blade cooling applications. A range of from.4 to was investigated in the present work. Heat transfer data is required for the situation of an array of holes drilled in a metal plate if a prediction of the blade or combustor wall temperature is to be made, and the objective of the present work was to provide this data. This type of prediction procedure also requires data on the film adiabatic cooling effectiveness and film convective heat transfer coefficient. Transpiration wall cooling offers the optimum wall cooling performance with the minimum air requirements. Andrews and Asere (1984) and Andrews et al (1986a) have shown for a transpiration cooled wall that at low coolant flow rates both the film and Presented at the International Gas Turbine and Aeroengine Congress and Exposition Orlando, FL June 3-6, 1991

internal wall cooling were important, but that at high coolant flow rates the film cooling was dominant. Thus, the optimum design of effusion cooling at low coolant flow rates must also maximize both the film and internal wall cooling components of the overall heat transfer. The bulk of the literature on discrete hole film cooling has been directed at obtaining adiabatic cooling effectiveness data and the many variables have been correlated by L'Ecuyer and Soechting (1985). Much less information is available on the film heat transfer coefficient and it is often assumed that simple flat plate convective heat transfer coefficients would apply. It has been shown by Hay and Lampard (1984) that this is not valid for single rows of holes and by the authors for arrays of holes (Andrews et al 1985b,c and 1986a,b). In spite of this large research effort on film cooling, there has been little comparable effort on the convective heat transfer from the wall to the film cooling air as it passes through the wall. 2. PREVIOUS WORK Convective heat transfer due to a hole in a wall is often treated as a pipe entry region heat transfer situation and conventional short pipe heat transfer correlations are used. These have been reviewed by the authors (1986b) and are normally expressed as a ratio of heat transfer coefficient for a short hole to that for fully developed pipe flow. These results are generally given as a function of the hole length to diameter ratio,. However, these correlations are only for the internal hole heat transfer and exclude any heat transfer due to the acceleration of the coolant air into the hole, as this surface was insulated in all the experiments. The hole approach surface heat transfer was investigated by Sparrow and Carranco Ortiz (1982) and the authors have shown that their correlation predicts that the approach surface heat transfer will be much greater than the internal hole heat transfer for practical film cooling geometries (Andrews et al 1985b,c and 1986a,b). Andrews et al (1985b, 1985c) have demonstrated the importance of film and internal wall cooling for a limited range of full coverage effusion cooling geometries. The heat transfer due to the passage of the coolant through the wall has been evaluated in an extensive series of investigations by Andrews et al (1986b, 1987, 1989). This has shown that the overall heat transfer was much greater than that due to the internal hole heat transfer. The additional heat transfer was on the hole approach surface and general correlations for this effect were determined, ignoring any influence of. The present work and the previous work of Andrews and Bazdidi-Tehrani (1989) was carried out with a plenum chamber air feed to the array of film cooling holes. This is the situation in most industrial gas turbine combustor wall cooling situations. In other situations there is often a crossflow of air passed the cooling holes, to dilution jets in combustors or to downstream film cooling holes in turbine blades. Byerley et al (1988a,b) have investigated this situation for single holes using both a 9 and an inclined hole. They showed that enhanced heat transfer on the hole approach surface was also important in this configuration. Also the influence of the crossflow was only important at fairly narrow crossflow passage heights. Peerhossaini et al (1989) have also investigated the heat transfer in single 9 film cooling holes with 1<<5 and D in the range.5-.75mm, similar to plates J and X in the present work. They used a high pressure high temperature transient measurement technique and concentrated mainly on the internal hole heat transfer. They found a much higher exponent for the heat transfer of 1.24, compared with.8 found by other investigators of tube flow heat transfer. Andrews et al (1989) correlated the heat transfer data from 19 film cooling hole array geometries by Eq. 1. Nux = 2.44 (X/D)-1.43 '.57 Pr o.33 (1) The equivalent correlation for hx was given by Eq. 2. h x = 32.3 X -o.29 D -.14 57 (2) Both equations were valid for <<43,, 1.9<X/D<21.4, 76<N<269 m -2 and.8<<9.9. All the data was obtained at a constant hole length of 6.3 mm and the range of arose from variations in the hole diameter through variations in N or X/D. The data scatter on the correlation was ±%. All the heat transfer was assigned to the approach surface area and no account of the internal hole surface area or was made. The relatively small effect of is shown in Fig.1 which plots the correlation as a function of hole for a hole of. Most of the data fell within the ±% data scatter and there was no consistent effect of. The results of Andrews et al (1989) included a range of data at constant but variable X/D, where N was varied from 76 to 269 m -2. This also showed a data scatter on the correlation of ±%. The main contribution to this was the use of a constant value for the exponent, which actually showed a significant dependence on the hole array geometry. In spite of this evidence of a relatively small influence of on the hole heat transfer, it would be preferable to have direct evidence of the insensitivity of the heat transfer to through variations of L at constant array geometry and this was the objective of the present work. The lack of dependence of the overall heat transfer on is contrary to the expectations of conventional pipe flow heat transfer and most previous correlations of short hole heat 7 L - 6.35mm - Andrews and Bazdidi-Tehrani (1989) Constant D, variable N and X A Constant X, variable D, N-436m -2 4 Constant X and N v2riable D N=269m 3 + /- % ^X Mean A A.1.2.5 HOLE Fig.1 The influence of L!D at a constant L of 6.35mm on the overall heat transfer due to the holes for the data of Andrews and Bazdidi-Tehrani (1989), where D was varied at constant X and at constant X/D by varying N. t 2

transfer have included an effect (Andrews et al, 1986b). The present work investigated smaller wall thickness than 6.3mm and in addition to investigating the effect will enable data to be obtained closer to current aero and industrial gas turbine combustor and blade wall thickness. The 6.35mm thickness of the previous work originated in an application to very large industrial gas turbine wall cooling where a 6.3mm wall thickness was to be used for the combustor construction. 3. EXPERIMENTAL TECHNIQUES 3.1 Effusion Geometries The effusion cooling geometries were chosen so as to investigate the influence of for a range of number of holes and X/D using two wall thicknesses of 6.35 and 3.3mm. Also one hole configuration was investigated in addition at a.69mm thickness. The range of geometries investigated is detailed in Table 1 and the hole configuration and thermocouple positions are shown in Fig.2. If the data for the larger wall thickness has been reported previously (Andrews and Bazdidi-Tehrani, 1989) then the same plate identification number or letter has been used in Table 1. All the test geometries were manufactured using conventional drilled holes, except for plates T and U which were laser drilled. A plate similar to plate 4 was also tested with spark eroded holes (plate C). Table 1 Test Geometries Plate N D X L X/D 152mm m -2mm mm mm Square Array 1 269 1.31 6.1 6.35 4.7 4.85 25 x 25 Z1 269 1.32 6.1 3.37 4.6 2.55 25 x 25 Z2 269 1.62 6.1.69 3.8.43 25 x 25 J 17222.76 7.6 6.35. 8.32 x X 17222.77 7.6 3.31 9.9 4.3 x T 9688.94.2 6.35.8 6.77 15 x 15 U 9688 1..2 3.32.2 3.32 15 x 15 4 436 1.41 15.2 6.35 11. 4.5 x P 436 1.31 15.2 3.18 11. 3.18 x Q 76 2.88 3.5 6.35.6 2. 5 x 5 R 76 2.86 3.5 3.34.7 1.17 5 x 5 E 436 3.27 15.2 6.35 4.7 1.94 x S 436 3.24 15.2 3.35 4.7 1.3 x 3.2 Mean Surface Averaged Heat Transfer Coefficient Determination A 152mm square Nimonic 75 test wall was used with an array of equispaced equal diameter holes. This was bolted to an internally insulated air supply plenum chamber. Each test plate was instrumented with at least five type K mineral insulated grounded junction thermocouples vacuum brazed to the exit side of the test plate on the centre line of the plate midway between the holes. In addition some test plates had thermocouples brazed to the feed side of the plate to investigate temperature gradients, as shown in Fig.2. The authors have previously shown that these thermocouples showed only small temperature differences. The variation in the heat transfer coefficient between the thermocouples at various axial locations and on the hole inlet and outlet surfaces was small, as shown in Fig.3 for plate I. The maximum variation in h between the highest and the lowest value of h was ±5% of the mean h. This was in agreement with the findings of the more complex film cooling situation where even at high temperatures the test plate was at a nearly uniform temperature with maximum temperature +- 3.2 mm --, i o o e oy + X o a o e a o + + + o e o 1-.8 mm + -1.8 mm o 1 o e++e to o o 25.4mm + o o e o o o o o a o o Effusion cooling holes * Thermocouple flush with the exit(flame) side + Thermocouple on the coolant approach side Diameter X Hole pitch Hole,^^ ^D length, L I Effusion hole Fig.2 Effusion wall test configuration 7 h W/m 2 K 3 II x h (AVERAGE) coolant approach aide + h (AVERAGE) coolant exit side V h (CENTRJU therm station 3 e h ledme1 therrel station i TEST PL ATE I^ N = 269 m -2 D = 1.31 mm X/D = 4.7 = 4.85 a t.1.1 1. G kg/sm 2 Fig.3 The influence of the test wall thermocouple location and wall thickness on h. X 3

a^e v Tg 927 K. Tw - 356-62 K (thermal station 41 e Tg 717 K, Tw-335-493K If TEMPERATURES MEASURED 2mm FROM THE LEADING EDGE A 4 1 v x 1 V e..4.6 6 1. 1. L4 6 (Kg/sm 2 ) Fig.4 Temperature difference through the thickness of the wall as a percentage of the mean wall to coolant temperature difference. Plate I in a high temperature effusion film cooling test. differences of less than 3% of the mean plate temperature (1985b). The variation in the temperature gradient through the wall thickness as a percentage of the coolant to wall temperature difference is shown in Fig.4 for wall I in a high temperature film cooling test. The actual temperature difference was approximately 5C over a wide range of G for Tw decreasing from 33C-83C as G was increased. This relative increase in the temperature gradient was due to the increase in Biot number with G discussed below. The small differences between the thermocouples and their unequal number on each test plate made the evaluation of the mean h time consuming and unnecessary. Less data scatter occurred in the results if one thermocouple was used to determine h rather than the mean of all the thermocouples, and the plate centre thermocouple was used for this purpose (Andrews et al 1988b). This gave a value for h very close to the mean h, as shown in Fig.3, and was well removed from the small edge conduction heat losses of the end two thermocouples. A transient cooling technique was used to determine the mean heat transfer coefficient, h. The development of this technique was fully described by Andrews et al (1986b) and Andrews and Bazdidi-Tehrani (1989). The test plate was heated in the absence of any coolant air flow using a conventional insulated uniform heat flux mat heater. Prior to heating, the air flow controls were set to give the desired flow rate, so that once the plenum chamber was removed from the heated plate one valve was opened and the coolant flow established. The test plate was then cooled by the air flow and the temperatures of all the thermocouples were recorded as a function of time using a fifty channel multiplexer into a microcomputer. The heat transfer in this situation of near uniform wall temperature is that of a low Biot number, where temperature gradients within the metal are small, as discussed above. The Biot number, based on the measured values of h, increased with G due to the increase in h with G. However, for the 6.3mm thick Y 4 6 walls the Biot number was below.1 for all geometries for G<.6 kg/sm 2 and below.2 for all G. For the thinner walls the Biot number was reduced in proportion to the wall thickness. It was less than.1 at all G for all the 3.2mm wall thickness geometries. The wall cooling at low Biot numbers is governed by the classical first order differential equation with a solution that gives an exponential fall in temperature with time. The mean heat transfer coefficient, h, may be found directly from the time constant of the first order system. As the start of the transient cooling was not well defined, due to the time necessary to establish a uniform coolant flow, the time constant was not determined from the usual 63.2% of the temperature change. A simpler procedure was used, which avoided any uncertainty in the initial and final temperature measurements and minimised the consequences of data scatter. This method used successive temperature data points to calculate the instantaneous rate of temperature change. The first order differential equation could than be used directly, as in Eq.3, to determine the time constant, r, as in Eq.4. TW = T c, - r AT W / At (3) r =mcp/ha (4) Each pair of successive temperatures was used to calculate AT / At and this was associated with the mean of the two temperatures. This mean temperature, T W, was than plotted against AT / At and a least squares fit was made to the data to yield the gradient, which was r. This procedure was carried out on the microcomputer for each plate thermocouple. If the centre temperature based 'r was significantly different from the mean, the test was repeated, but this was rarely necessary. Typical transient results for plates 1, Z1 and Z2 are shown in Figs. 5 and 6. Equation 4 shows that h may be calculated directly from r as the other parameters are all known constants for a particular plate material and geometry. The choice of the value of A is particularly significant and is related to the physics of the flow Ii Ii- I.- X-1 A L/-4.85 v L/-2.55 + U..43 Li'D -.43 +++++ vvvvvv ++++ v t/d =2.5vv Ade + eaaaa vv o - 4.85 v ++ v e V A V + V A V A A V D N = 269 m 2 g e 25 TIME. t (sec.) Xioi Fig.5 Typical transient temperature records in a dimensionless format for plates 1, 71 and Z2. 4

xi 1 IA '4.85 v UD.2.55 + ^.43 h W/m 2 K 7 8a v + 7 + GRADIENT - TIME CONSTANT 2 4 6 8 12 i4 16 i8 22 24 26.Tw/At ( C/sec.) X'i Fig.6 Evaluation of the time constant by plotting the average temperabure difference as a function of of the temperature gradient. process. In the present work the use of the hole internal surface area, A h, is compared with the hole approach surface area, A X, used by Sparrow (1982) and with the summation of both areas, A. The use of the hole surface area implies that the internal hole convective heat transfer is dominant while the use of the approach surface area implies that the main convective heat transfer is on the approach surface. By combining the two surface areas equal weighting per surface area is given to the two heat transfer locations. 4. EFFECT USING THE INTERNAL HOLE AREA A h TO DETERMINE h 4.1 N=269 m-2 and X/D=4.6, Plates 1, Z1 and Z2. + =.43 + ++ Z1 + vv GO.5 N = 269 m -2 12 v dt + pride 2.55p d ri d Qd = 4.85 / d d.1.1 1 G kg/sm 2 Fig.7 The variation of h with G at constant N for three hole 4.85, 2.55 and.43. Nu Pr 1 N=269 m -2 =.43 + + T T + L /D= Z1 v'v^ ye ^ O ^^ 9 d-b^f -t - - =4.85 1- d'd_ 12 Test plates 1, Z1 and Z2 had essentially the same design with the plate thickness as the only major difference. The results for h obtained from Eq.4 using A h as the heat transfer area are shown in Fig.7 as a function of the mass flow per surface area, G, and in Fig.8 as Nu v.. These results appear to indicate a very large effect of the hole. If the results at the two smaller are normalised, at a constant G or, to that at an of 4.85 then the measured heat transfer ratios are 1.5 for an of 2.55 and 3.4 for and of.43. These measurements are compared with several correlations for short hole heat transfer in Table 2 (Andrews et al, 1986b). Table 2 Comparison of the influence of, normalised to an of 4.85. Present Mills McAdams Al Arabi Le Grives 1962 1954 1982 1979 2.55 1.5 1.21 1.14 1.18 1.9.43 3.4 1.39 2.1 2.23 1.19 Apart from the results of Mills, which include data down to an of.25 the other correlations are not applicable to the low of the present work. For example McAdam's correlation is valid for >5. The 1UU, Fig.8 Correlation 6f 3 ^he heat transfer data for three as Nu/Pr v.. results of Mills (1962) show that the present results indicated a much higher influence of. However, this influence results directly from the assumption in Eq.4 that all the heat transfer occurs inside the holes and no other surface area participates. A much lower influence of results if the hole approach surface area A is used to determine the heat transfer coefficient, h X, from Eq. 4. A second feature of Figs. 7 and 8 is that the and G exponent, y, is. for all three. The correlations of tube heat transfer have a exponent of.8. If the tube heat transfer was dominant then the exponent should be.8. None of the correlations for the influence of on tube heat transfer involve an influence of on the exponent. This is further evidence that the short hole pipe flow heat transfer does not dominate the overall heat transfer. 5

H1 7 4 m a 3 N = 17222 m -2 D =.76mm = 4.3.5 v J /v o = 8.32 X^y v Nu Pr.33 1 N = 436 m -2 1 D'= 1.4 mm P v =3.18.56 4i = 4.5 oi Fig. 9 Influence of the wall thickness on the heat transfer correlation for plates J and X. m cn N = 9688 m -2 D =.95mm.59 ^-4 e^ -T =3. 4 =6.77 U, 5 Fig.11 Influence of on the heat transfer correlation for plates 4 and P, using the internal hole area as the only heat tranfer area. Nu Pr.33 7 1 4 in = 76 m^ 29 D = 2.87mm R 3 =1.17 / Q =2..45 Fig. Influence of on the heat transfer correlation for plates T and U 4.2 N=17222 m-2 and X/D=, Plates J and X This geometry was investigated at two plate thicknesses giving an difference of 8.3 and 4.3. The results are shown in Fig.9 and both plates had a exponent, y, of.5 with no effect of on this. The shorter hole had much higher heat transfer with a ratio to the longer hole of 1.66. This compares with a ratio of 1.18 from the data of Mills (1962) as correlated by Andrews et al (1986a). Thus the apparent influence of on the heat transfer in the present work is a factor of 3.7 higher than that found by Mills (1962). 4.3 N=9688 m-2 and X/D=.5, Plates T and U This geometry with laser drilled holes was investigated at two plate thicknesses giving an range of 6.77 to 3.32. The results are shown in Fig. and there was again no influence of on the exponent, y, which was equal to.59. The heat transfer increased by a factor of 1.37 due to the shorter compared with a factor of 1.21 based on the din,, 5 Fig.12 Influence of on the heat transfer correlation based on the hole surface area for plates R and Q. results of Mills (1962) as correlated by Andrews et al (1986b). The present results were thus a factor of 1.76 higher than the effect of Mills. 4.4 N=436 m-2 and X/D=11.3, Plates 4 and P This geometry was investigated at two plate thicknesses giving an range of 4. to 2.43. The results are shown in Fig.11 and there was again no influence of on the exponent, y, which was equal to.56. The heat transfer increased by a factor of 1.33 due to the shorter compared with a factor of 1.23 based on the results of Mills (1962) as correlated by Andrews et al (1986b). The present results were thus a factor of 1.43 higher than the effect of Mills. 4.5 N=76 m-2 and X/D=.6, Plates Q and R This geometry was investigated at two plate thicknesses giving an range of 2. to 1.17. The results are shown in Fig.12 and there was again no C7

W U- N I- 2.2 2. 1.8 W 2 1.6 J W PRESENT RESULTS MILLS (1962) Nu x.33 _2 Pr N = 269 m D = 1.4 mm.5 =4.8 1 5 2 =.43 f 2 q 1.4 J 1.2 Id y \ 2 4 6 8 Fig.13 Normalised heat transfer coefficients for identical hole geometries of different wall thickness, plotted as a function of the hole. All the heat transfer assumed to act over the area Ah to give a direct comparison with the short hole heat transfer results of Mills (1962). influence of on the exponent, y, which was equal to.45. The heat transfer increased by a factor of 1.55 due to the shorter compared with a factor of 1.13 based on the results of Mills (1962) as correlated by Andrews et al (1986b). The present results were thus a factor of 4.2 higher than the effect of Mills. The results for each pair of test plates of different thickness are summarised in Fig.13. This presents the normalised heat transfer of the thinner plate relative to the thicker plate plotted as a function of. The results of Mills are given as a direct comparison. Fig. demonstrates the much greater influence of that the present results appear to indicate compared, with the results of Mills. Also shown is the quite variable magnitude of the effect in the present work, as there is no consistent gradient to the apparent effect. The strong disagreement with the results of Mills indicates that it may not be appropriate to assume that all the heat transfer occurred within the short holes. 5. EFFECT USING THE WALL SURFACE AREA A,, TO DETERMINE h,, This procedure for determining the heat transfer coefficient from the measured time constant assumes that all the heat transfer can be ascribed to the hole approach surface area per hole, A, given by Eq. 5. The resultant heat transfer coefficient will be denoted as h and the Nusselt number based on this h and the hole diameter will be denoted by Nu x. A x = Xz -.25 a DZ (5) 1 Fig.14 Influence of on the heat transfer correlation based on the hole approach surface area, A, as the heat transfer area in the heat transfer co6fficient h x, for plates I, Z1 and Z2. The definition of h does not influence the exponent y, as this is set bythe influence of G. This will be demonstrated by using the same exponents in the linear correlation of the h X data as for the data expressed above in terms of h. 5.1 N=269 m-2 and X/D=4.6, Plates 1, Z1 and Z2. The results are shown in Fig.14 for the three test plates of this hole configuration with different of.43, 2.25 and 4.85. Comparison with Fig.8 shows a much better data correlation and a relatively small influence of. These results indicate that the heat transfer is dominated by the hole approach surface heat transfer as the air accelerates into the hole inducing a flow around the hole inlet with the surface velocity decreasing in the radial direction from the hole lip. This type of flow will be termed a squish flow following the terminology for a similar phenomena in IC engines. The influence of the hole was significant and was such that the longer the hole the greater the overall heat transfer. The increase in heat transfer between the of.43 and 2.25 is relatively small at 7% but a much larger increase of 32% was found between 2.25 and 4.85. The reason for this larger influence of for the 2-5 range is not known, but would not be expected from the heat transfer data of Mills (1962). The maximum difference between the smallest and largest was 4%, which is the data scatter range in the previous work, correlated by Eqs. 1 and 2 (Andrews and Bazdidi-Tehrani, 1989). Thus the effect might have been responsible for part of the data scatter in this previous work. The range of in the previous work was.8-9.9 and the present.43 falls out of this range and may increase the data variability due to an effect. These results indicate that the approach surface and hole heat transfer may be additive. If the.43 results are assumed to be primarily due to the hole approach flow heat transfer then the data of Mills (1962) can be used to predict the increase in heat transfer due to the effect. However, this is difficult to do using the present measurement technique 7

as the measured mean wall temperature at any instant is due to the combined heat transfer effects of the approach flow and the internal hole flow. An alternative technique is to add the two heat transfer areas A and A h and to use the combined area, A t, in Eq.4. This is investigated below. 5.2 N=17222 and X/D=, Plates J and X Nu x Pr.33 N = 9688 m-2 D =.95 mm.59 These test plates have the same hole configuration with an of 8.32 and 4.3. The heat transfer results are shown in Fig. 15 which demonstrates a much smaller influence of than in Fig.9. The increase in heat transfer for an increase from 4.3 to 8.32 was approximately 14%. This also indicates an additive effect of the hole heat transfer to the dominant effect of the approach surface heat transfer. However, the small influence of is well within the ±% data scatter of the correlation in Eqs. 1 and 2. 5.3 N=9688 m-2 and X/D=.5, Plates T and U The results are shown in Fig.16, where the smaller influence of compared with that in Fig.8 is demonstrated. However, the influence of was an increase in heat transfer of 25% between an of 3.32 and 6.77, compared with 37% in Fig.. Thus for this geometry there was still a considerable influence of compared with the 14% effect for a similar geometry with N=17222 m -2 as discussed above. However, the increase was similar to that in Fig.14 for an change from 2.55 to 4.85, where the increase was 32%, as discussed above. For this geometry the laser drilled holes may have contributed to the effect as the hole shape was found to vary with the hole thickness. Nu x 5 2 T=6.7 =3.32 uu U Fig.16 Influence of on the heat transfer correlation based on h for plates T and U. x Pr.33 DC =4.5 N = 436 m 2 D = 1.35 mm 4 '.56 ' P =3.18 C 5.4 N=436 and X/D=11.3, Plates 4, C and P The results are shown in Fig.17 and a fairly large effect of was found. For the drilled plates there was a 4% increase in heat transfer at the higher. However, this was reduced to 3% if the spark eroded test plate results were used at the higher. Again this data range would lie at the two extremities of the data scatter from various test wall geometries that was correlated by Eqs. 1 and 2. Nu x Plate C was the same as 4 but with spark eroded holes (D=1.38 mm) 32 5 2 5 1 Fig.17 Influence of on the heat transfer correlation based on h for plates 4 and P. x J5 Pr.33 m 2 N = 17222 m 2 Dr.76 mm =8.32.5 Y =4.3 X 11 Is i Fig.15 Influence of on the heat transfer correlation based on h for plates J and X. x 8 This quite variable effect of is considered to be more associated with manufacturing difficulties between the two test plates than with a genuine influence of. The difference between the drilled and spark eroded test plate results for essentially the same hole geometry is a good illustration of how hole manufacturing techniques can influence the heat transfer. Spark eroded holes have a rougher internal surface than drilled holes and inlet effects can be different. With the relatively small drilled holes used in the present work it is very difficult to achieve consistency in the hole shape and size over a test plate and between test plates with different thicknesses. Variations in the hole size and inlet conditions inevitably occur and these will be different for different geometries and for different manufacturing techniques such as laser drilling. The hole size in the present work was based on an average of all the holes measured on both the inlet and outlet faces of the wall.

5.5 N=76 m -2 and X/D=.6, Plates Q and R The results are shown in Fig.18 and comparison with Fig.12 shows a much smaller influence of, 11% compared with 55% for an increase from 1.17 to 2.. All these results indicate that a correlation of the results based on the hole approach surface area is preferable to that involving the hole area and that the neglect of any effect is a relatively small although significant influence. This was also concluded in the previous work which correlated the influence of X/D and N (Andrews et al, 1987, Andrews and Bazdidi-Tehrani, 1989). The changes in heat transfer due to the changes in at constant hole configuration are summarised in Fig.19, which may be compared with Fig.13. The first obvious difference is that there is an opposite influence of. With the use of A as the heat transfer area in Fig.13, h increases as ^ h decreases as Eq.4 is associating the measured heat transfer with a smaller area. Figure 19 demonstrates the consequences of completely ignoring the effect and the hole heat Nu x Pr.33 )D N = 76 m -2 D = 2.87 mm.45 Q I/D=2. =1.17 R transfer surface area. These effects in Fig. 19 are within the data scatter previously found for designs with different N and X/D, but the same hole size and (Andrews and Bazdidi-Tehrani, 1989). However, apart from the data in Fig.14, which shows no consistent effect of, all the other pairs of test plates with different show a consistent increase in heat transfer as the is increased. An attempt to reduce this effect by including the hole surface area, A h, with A x in Eq.4 is discussed below. 5.6 Data Correlation Based on h The present data for test plates at different thicknesses has been combined with the previous data sets which were correlated by Eqs. 1 and 2 (Andrews and Bazdidi-Tehrani, 1989). The previous correlation was based on 19 different geometries, all with a wall thickness of 6.35mm. Seven additional geometries are included in the present work, 6 with a thickness of approximately 3.2mm and one with a thickness of.69mm. A correlation of all the 26 data sets was made based on the approach surface area, A, and h and this is given by Eq.6 and the data agreement with thecorrelation is shown in Fig.. Nu x = 2.28 (X/D)-'.43 R e o.57 p r a.33 (6) for.8<<, <<43, 1.9<X/D<22 and 76<N<269 m -2 The only difference from Eq.1 is the value of the constant, the equivalent constant for the correlation of h is 3.2 in Eq.2. These constants are a 7% reduc3tion on those in Eqs. 1 and 2 and this is to partially take into account the lower heat transfer at the lower of some of the present work. The only in the present work that was outside the range of the previous correlation was plate Z2 with an of.43. Fig. also shows that this lies well outside the correlation. This is surprising in view of the 32 5 2 5 ^ Iig.lB Influence of L/ on the heat transfer correlation based on h for plates Q and R. x 1.4 I Nux 1.3 / ---Nut H W J w 1.2 1.1 I 1. 2 4 6 8 Fig.19 Normalised heat transfer in terms of Nu and Nu t as a function of for id6ntical hole geometries of different wall thickness. -2 (S)a1.3 v () L/.2. + (R) L/ i.17 X (E) 'i.94 o N)L/.3.32 x 4-4. i (P)L/^2.43 X bt-9.32 o M -4.3 tt)-6.77 f] =4.5 v (Zi) LID-2.55 + (221 -.43 +/-25% +.57 -f 3 4.3 v + z ++ + =.43 + + 9 i 5 4 d + in 2 3 4 5 Fig. Correlation of the present data on variable hole using A as the heat trasfer area to determine h and Nu from the measured time constants. x X

good agreement of the effect in Fig.14. However, the reason is because there was a significant difference in the mean hole size for this plate giving a 17% reduction in the X/D resulting in a 24% reduction in the correlation due to the X/D term. The data for this geometry at higher X/D fell towards the bottom end of the data scatter in Fig., leaving these lower results outside the data range of the other test geometries. The data points shown in Fig. are only for the present 13 test plates. However, when all 29 data sets are included the total data scatter was increased from the previous ±% to ±25%. The high value of the exponent on the X/D term gives a misleading impression of the importance of the X/D in the heat transfer. Eq.2 shows that at a constant coolant mass flow per surface area the influence of X and D is relatively small. This correlation, derived from Eq.1 and 6 predicts only a 3% influence of the difference in diameters between plates Z1 and Z2. It is considered that comparison of the heat transfer at a constant coolant mass flow is the most realistic method. Comparison of the h data as a function of G shows that the present data range is only ±25% of the mean before any correction is made for any influence of X or D. Some of this data range is due to the influence as shown in Fig.2'. This plots the h data as a function of for a constant G of.6 kg/sm 2. A similar plot of the data at constant ynolds number is shown in Fig.22.. However, this variation in h with has to be compared with the data range for the influence of N at constant D and, which is indicated in Figs. 21 and 22 (Andrews and Bazdidi- Tehrani, 1989). As discussed earlier and shown in Fig.1, all the data scatter in the correlation cannot be attributed to the effect, although this is significant. Figures 21 and 22 show that for a constant of approximately 2 the four data points at constant show a wide variation covering the whole range of the data scatter. In Fig.21 the differences are smaller at approximately half the data range. The problem in the data correlation using equations such as Eq.6 is that the X/D and exponents are a compromise and each test wall has a variation on these exponents. The variation in the exponent in the present work was from.45 to.59 for example. Thus, inclusion of the plate thickness influence into the correlation would only be justified if an improvement was made to h W/m K G =.26 kg/sm` +/-25% 1- CONSTANT VARIABLE N AND X L=3.3mm, L=6.35mm STRAIGHT LINES JOIN THE DATA FOR h FOR THE SAME N, X R AND D I=4.4.1.2.5 1 2 5 Fig.21 The influence of on h at constant G=.26kg/sm 2 and for constant N, X and X D. O CL = x 3 = LINES JOIN THE DATA FOR THE SAME N, X AND D +/-25% V 1- CONSTANT VARIABLE N AND X L /b =4.4 L=3.3mm Q L=6.35mm 1 r----, i-+-.1.2.5 Z 5 Fig.22 Influence of on the heat transfer correlation for a constant of and for constant N, X & D the X/D and exponents to make them a function of the hole geometry. The data pairs for the different plate thicknesses have been linked by straight lines in Figs.21 and 22. Although there is some variation in the slope of the influence of the three data points for the geometries of plates 1, Z1 and Z2 give an exponent of.25. The other results are similar to this. A possible method of including the influence of is thus to modify the constant in Eqs.1, 2 and 6 and to include an effect. However, this exponent for is really an exponent for the variation of L. The influence of D is taken into account by the existing terms in Eqs.1, 2 and 6 and the introduction of a further D-.25 term into these equations invalidates the correlation. Consequently, it is considered that the influence of the hole length should be accounted for by including the hole internal area in the heat transfer area in Eq.4. 6. EFFECT USING THE HEAT TRANSFER SURFACE AREA OF A,, + A h TO DETERMINE h t By combining the areas A h and A, as the area used in Eq.4 to determine h t, part of the internal hole heat transfer may be accounted for. This procedure assumes that these surfaces are equally effective in the heat transfer process. The approach flow heat transfer may only dominate the overall process because of the larger area involved. The ratio of the approach flow surface area to the hole internal area will thus give an indication of geometries where the hole heat transfer may be significant. This area ratio is plotted as a function of in Fig.23 for the present geometries and for the previous geometries of Andrews et al (1987). The area ratio is shown as a function of X/D in Fig.24 for the geometries of Andrews et al (1987) and Andrews and Bazdidi-Tehrani (1989). These results show a relatively complex relationship between the area ratio and other geometrical parameters. A small hole does not necessarily result in a high area ratio, for at low X/D and low area ratios near unity arise. However, at higher X/D for the same small the area ratio becomes very large. For the present geometries Fig. 23 shows that the area ratio was greater than 5 for all geometries with an X/D of approximately.5, irrespective of the plate

3 A x ^N=76m A 3 A x X/D =.5 25 i L=3.2mm -j- =4.4 N \^ N=436 L=6.33 X/D' ' N=436 mm \ i L=6.35mm 15 am X/D=4.7 M ZN=436m-' N=436 i "N=96 6Q9 Z.Q=17222 N=269 2 4 6 8 Fig.23 The heat transfer area ratio A /A as a function of for a range of xhote array geometries. thickness, and hence the approach flow heat transfer will dominate. The smaller the number of holes used the larger is the area ratio. Thus the geometries with an X/D of 4.7 and N=269m -2 have area ratios near unity for the larger and the hole heat transfer may be significant here. 6.1 N=269 m-2 and X/D=4.6, Plates 1, Z1 and Z2. The results for the three plates of this geometry with different thicknesses are shown in Fig.25. Comparison with Fig.14 shows a much smaller influence of. Thus, for this geometry, which Fig.23 shows both surface areas to be of similar magnitude, the use of the summation of A h and A for the heat transfer area in Eq.4 has lead to an improved correlation. 6.2 Correlation of ht at Constant G The results in Fig.25 were repeated for the other geometries with a reduced influence of. However, because the geometries in Figs.15-18 were for an X/D of, Fig.23 shows that the area ratio was always high and the inclusion of the Ah in the heat transfer area made only small reductions in the influence of. The heat transfer coefficient, h t, is shown as a function of in Fig.26 for a constant G of.26 kg/sm 2. Comparison with Fig.21 shows a reduced influence of for all the geometries. However, in spite of this reduced influence of there was still a significant data scatter between the results at a fixed and the total data scatter range was not reduced by this improved correlation of the influence. The data correlation for ht is given by Eq.7, but there was no change in the overall ±25% data 5 / 5 15 X/D. Fig.24 The heat transfer area ratio A /A h as a function of X/D, at constant and a constant N. Nu t Pr.33 5 2 N = 269 m -2 D = 1.35 mm.5 =.43 =4.85 =2.55 22 5 3 2 5 4 Fig.25 Influence of on the heat transfer correlation using A t to determine h t, for Plates I, Z1 and Z2. range of Eq.6, as shown in Fig.27. The only advantage of Eq.7 was to include the thin plate data for plate Z2 and hence extend the validity of the correlation down to an of.4. Nu t =.78 (X/D) -1. 16 R e o.ao p ro.33 (7) valid for.4<<, <<43, 1.9<X/D<22, 76<N<269m -2 11

h t W/m 2 K G =.26 kg/sm` THE LINES JOIN THE DATA FOR CONSTANT N, X AND D ii1. =4.4 CONSTANT OF 4.4 ATCONSTANT N & X L = 3.3 mm L = 6.35 mm.1.2.5 1 2 3 5 Fig.26 Influence of on h t at a constant G of.26 kg/sm 2 (S) -i. a (a)l/.2.2 + (P) UD-1.2 X (E L/-1.9 M L/.3.3 I QJ L/-4.5 A ai)-2.4 x ti1-8.3 a -4.3 (TILJD-6.8 A (1J L4.8 V (Iil L/.2.8 + (M L/.. 4 with the X/D exponent reduced to.76. As the based on D is by far the most common practice it has been preferred in correlating the present results. If Eq.7 is expressed as a correlation for h t in terms of G, X and D, then the dependence on X is negligible with an exponent of.4. The exponent for the hole diameter is -.44 and this shows that D is the dominant design parameter which controls h at constant G. This is because D controls the maximum velocity on the hole approach surface, as well as the hole velocity and the maximum local velocity controls the convective heat transfer at constant G. Thus, it is appropriate to use D as the characteristic dimensional parameter as it controls both the surface and hole heat transfer at constant G. This is quite a different conclusion to that in Eq.2 for the correlation of h, where both X and D are significant parameters. The x effect of X is thus predominantly accounted for by the inclusion of A,, in the heat transfer surface area. CONCLUSIONS 7.1 For a constant hole array geometry there was a significant influence of the hole length on the overall wall heat transfer for an array of film cooling holes. However, the magnitude of this effect was relatively small and within the ±25% data scatter of a correlation of all the data. 7.2 The use of the hole internal surface area to derive h from the measured time constant gave an apparently very large influence of and a exponent quite different from that of tube flow entry heat transfer correlations. The heat transfer coefficient increased markedly as was reduced if A h was used as the heat transfer surface area. This was not a realistic result and there must have been another surface area involved in the heat transfer. It was concluded that the hole surface area was not the dominant heat transfer area. - z L 3 4 5 Fig.27 Heat transfer correlation for the influence of based on Nu } for the range of geometries in the present work. This correlation has only a minor change, compared with Eq.6, in the exponent but a more significant change in the X/D exponent. The reason for this is the link between X/D and the approach surface to hole area ratio A X/A h shown in Fig.24. Thus at low X/D the heat transfer surface area, A t, was increased, compared with A, by a much greater fraction than at high X/D. This effectively changes the dependence of the heat transfer on X/D. The correlation of Eq.7 is based on the hole diameter as the characteristic dimension in Nu and. In view of the importance of the approach surface area in the overall wall heat transfer it may be thought preferable to use X as the characteristic dimension. However, this only makes an algebraic change in Eq.7 7.3 The use of the hole approach surface area in the determination of h results in a good correlation of the results with only a small influence of. 7.4 The present data for a range of plate thicknesses may be correlated, together with previous data for other geometries, by Eq.6 to within ±25%. This correlation ignores any influence of. Nu. = 2.28 (X/D) -1.43 R e o.57 p r o.33 (6) for.8<<, <<43, 1.9<X/D<22 and 76<N<269m -2. 7.5 The influence of was reduced if the hole surface area, A, was added to A in the derivation of the heat trans'er coefficient, lt, from the measured time constant. However, this correction was only significant for geometries with a ratio of A x/ah close to unity. The resultant correlation was given by Eq.7 Nu t =.78 (X/D) -1.16 R eo.6o p r o.33 (7) valid for.4<<, <<43, 1.9<X/D<22, 76<N<269m -2. 12

ACKNOWLEDGEMENTS We would like to thank the UK Science and Engineering search Council for a research grant in support of this work (G/D/5329). We would like to thank GEC-Ruston Gas Turbines for the manufacture of some of the test geometries and M.F. Cannon for valuable technical advice. Le Grives, E., Nicolas, J.J. and Genot, J., 1979, 'Internal aerodynamics and heat transfer problems associated to film cooling of gas turbines', ASME Paper 79-GT-57. Mills, A.F., 1962, 'Experimental investigation of turbulent heat transfer in thermal entrance region of a circular conduit', J.Mech.Eng.Sci., Vol.4, No.1, pp. 63-77. REFERENCES Peerhossaini, H., Simon, G., Lemasson, P. and Andrews, G.E. and Mkpadi, M.C., 1984, 'Full coverage Bardon,J.P., 1989, 'Convective heat transfer in small discrete hole wall cooling: discharge coefficients', diameter film cooling short holes at high ynolds Trans. ASME, J.Eng.Power, Vol.6, pp.193-192. numbers', ASME Co-Generation Conference, Int. Gas Turbine Institute, Vol.4, pp.227-233. Andrews, G.E., Asere, A.A., Hussain, C.I. and Mkpadi, M.C., 1985a 'Full coverage impingement heat transfer: Sparrow, E.M. and Carranco Ortiz, M., 1982, 'Heat the variation in pitch to diameter ratio at a constant transfer coefficients for the upstream face of a gap' AGARD CP 39, Heat transfer and cooling in gas perforated plate positioned normal to an oncoming turbines, pp. 26.1-26.13. flow', Int.J.Heat Mass Trans., Vol.25, pp.127-135. Andrews, G.E., Gupta, M.L. and Mkpadi, M.C., 1985b, 'Full coverage discrete hole film cooling: cooling effectiveness', ASME Paper 85-GT-47, Int.J.Turbo Jet Engines, Vol.2, pp. 199-212. Andrews, G.E., Asere, A.A., Gupta, M.L. and Mkpadi, M.C., 1985c, 'Full coverage discrete hole film cooling: the influence of hole size', ASME Paper 85-GT-47, Int.J.Turbo Jet Engines, Vol.2, pp. 213-225. Andrews, G.E., Asere, A.A., Mkpadi, M.C. and Tirmahi, A., 1986a, 'Transpiration cooling: contribution of film cooling to the overall cooling effectiveness', ASME Paper 86-GT-136, Int.J.Turbo Jet Engines, Vol.3, pp.245-256. Andrews, G.E., Alikhanizadeh, M., Asere, A.A., Hussain, C.I., Khoshkbar Azari, M.S. and Mkpadi, M.C., 1986b 'Small diameter film cooling holes: wall convective heat transfer', ASME Paper 86-GT-225, Trans. ASME, J.Turbomachinery, Vol.8, pp.283-289. Andrews, G.E., Alikhanizadeh, M., Bazdidi-Tehrani, F, Hussain, C.I. and Koshkbar Azari, M.S., 1987, 'Small diameter film cooling holes: the influence of hole size and pitch', ASME Paper 87-HT-28. Andrews, G.E. and Bazdidi-Tehrani, F., 1989, 'Small diameter film cooling hole heat transfer: the influence of the number of holes', ASME Paper 89-GT-7. Byerley, A.R., Ireland, P.T., Jones T.V. and Ashton, S.A., 1988a, 'Detailed heat transfer measurements near and within the entrance of a film cooling hole', ASME Paper 88-GT-155. Byerley, A.R., Ireland, P.T., Jones, T.V. and Graham, C.G., 1988b, 'Detailed heat transfer measurements near the entrance to an inclined film cooling hole', 2nd UK National Heat Transfer Conference. Hay, N., Lampard, D. and Saluja, C.L., 1985, 'Effects of cooling films on the heat transfer coefficient on a flat plate with zero mainstream pressure gradient', ASME Paper 84-GT-4. L'Ecuyer, M.R. and Soechting, F.O., 1985 'A model for correlating flat plate film cooling effectiveness for rows of round holes', AGARD CP 39, Paper 19. 13