GT2017-64039

SCALING HEAT TRANSFER COEFFICIENTS MEASURED UNDER LABORATORY CONDITIONS TO ENGINE CONDITIONS T I P Shih and C S Lee School of Aeronautics and Astronautics Purdue University West Lafayette IN U S A K M Bryden Dept of Mechanical Engineering Iowa State University and DOE Ames Lab Ames IA U S A ABSTRACT Almost all measurements of the heat transfer coefficient HTC or Nusselt number Nu in gas turbine cooling passages with heat transfer enhancement features such as pin fins and ribs have been made under conditions where the wall to bulk temperature Tw Tb is near unity Since Tw Tb in gas turbine cooling passages can be as high as 2 2 and vary appreciably along the passage this study examines if it is necessary to match the rate of change in Tw Tb when measuring Nu whether Nu measured at Tw Tb near unity needs to be scaled before used in design and analysis of turbine cooling and could that scaling for ducts with heat transfer enhancement features be obtained from scaling factors for smooth ducts because those scaling factors exist in the literature In this study a review of the data in the literature shows that it is unnecessary to match the rate of change in Tw Tb for smooth ducts at least for the rates that occur in gas turbines For ducts with heat transfer enhancement features it is still an open question This study also shows Nu measured at Tw Tb near unity needs to be scale to the correct Tw Tb before it can be used for engine conditions By using steady RANS analysis of the flow and heat transfer in a cooling channel with a staggered array of pin fins the usefulness of the scaling factor Tw Tb r from the literature for smooth ducts was examined Nuengine computed under engine conditions was compared with those computed under laboratory conditions Nulab and scaled by Tw Tb r i e Nulab scaled Nulab Tw Tb r Results obtained show the error in Nulab scaled relative to Nuengine can be as high as 36 6 if r 0 7 and Tw Tb 1 573 in the fully developed region Thus Tw Tb r based on smooth duct should not be used as a scaling factor for Nu in cooling passages with heat transfer enhancement features To address this inadequacy a method is proposed for generating scaling factors and a scaling factor was developed to scale the heat transfer from laboratory to engine conditions for a channel with pin fins NOMENCLATURE A cross sectional area of cooling passage Cf skin friction coefficient Cf w 0 5 rV2 Cp constant pressure specific heat of the cooling fluid D diameter of pin fins h heat transfer coefficient HTC H height of pin fin and cooling channel k thermal conductivity of cooling fluid k k Tb m mass flow rate of cooling flow Nu Nusselt number Nu hD k Nulab Nusselt number obtained with Tw Tb near unity Nuengine Nusselt number obtained with engine relevant Tw Tb P static pressure Pb back pressure at the exit of cooling channel Pr Prandtl number Pr Pr Tb q wall heat flux ReD Reynolds number ReD VD ReD in Reynolds number at x 1 25 T temperature Tb bulk temperature of the cooling flow Tc bulk temperature at cooling channel s inlet Tw wall temperature U friction velocity w 0 5 V velocity magnitude x coordinate in the streamwise direction y coordinate in the spanwise direction y normal distance from wall y normalized distance from channel wall y U y z coordinate in the direction normal to the channel wall Greek Symbols r density of cooling fluid dynamic viscosity of cooling fluid Tb w wall shear stress Proceedings of ASME Turbo Expo 2017 Turbomachinery Technical Conference and Exposition GT2017 June 26 30 2017 Charlotte NC USA GT2017 64039 1Copyright 2017 ASME INTRODUCTION Internal cooling passages inside gas turbine blades and vanes are subjected to extremely high heat transfer rates The heat flux q w on the cooling passage s wall could be as high as 450 W cm2 at a Reynolds number of 104 and as high as 3 000 W cm2 at a Reynolds number of 105 if the hydraulic diameters is 1 cm the wall temperature is 1 000 oC and the temperature of the air at the inlet of the cooling passage is 300 oC With such high heat fluxes the bulk temperature in the passage could vary appreciably from 300 oC to 700 oC or higher depending on the length of the passage and the maximum temperature the material of the turbine can sustain If the bulk temperature of the cooling air increases from 300 oC to 700 oC then the Reynolds number of the air flow in the passage could decrease by 30 because viscosity increases with temperature The Prandtl number will also vary as the bulk temperature increases along the passage but it varies considerably less Though the wall to bulk temperatures Tw Tb in gas turbine cooling passages could be as high as Tw Tb 1 000 273 K 300 273 K 2 2 most experimental studies measure the heat transfer coefficient HTC with Tw Tb near unity see for example Refs 1 to 4 In most experimental studies the wall and bulk temperatures differ from about 6 oC to 20 oC where one of the temperatures is near the ambient temperature There are two reasons why most experiments do not reproduce the Tw Tb that occur in engines The first is that those conditions are hard and expensive to reproduce in the laboratory Second many measurement techniques do not work in the harsh environments associated with those conditions But by measuring HTC at Tw Tb near unity means that the entire passage is characterized by essentially the same Reynolds ReD and Prandtl Pr numbers and this may not tell the whole story especially if the data is to be used to design turbine cooling under engine conditions where Tw Tb is not near unity Thus some questions are as follows First since Tw Tb can vary appreciably along the cooling passage does the rate of change in Tw Tb or ReD along the cooling passage affect the measured HTC or Nusselt number Nu Second can Nu measured with Tw Tb near unity be applied to engine conditions where Tw Tb is not near unity Third if Nu must correctly account for Tw Tb could Nu measured with Tw Tb near unity be scaled to engine conditions where Tw Tb is not near unity A literature search was performed to see what has been done in answering the above three questions On the first question how the rate of change in Tw Tb affect the HTC or Nu nothing was found in the literature It seems that no one has asked this question However the literature search found data that could be used to answer the question Experimental studies of heat transfer have been made in smooth ducts where the magnitude of Tw Tb and its rate of change along the cooling passage are in the range that are applicable to gas turbine engines see for example Refs 5 to 7 On the second question though no one addressed it explicitly there is literature on the role of Tw Tb in Nu see e g Refs 8 to 11 indicating Tw Tb is an important parameter However there are two schools of thoughts In one school the effects of Tw Tb is thought to be automatically accounted for in correlations such as Nu c ReDa Prb 1 where c a and b are constants and b takes on different values depending on whether the wall is being heated or cooled if the transport properties in ReD and Nu are evaluated at the right reference temperatures In the other school of thought the effect of Tw Tb needs to be explicitly accounted for by correlations such as see for example Refs 5 11 Nu c ReDa Prb Tw Tb g 2 If Eq 1 is correct then Nu measured with Tw Tb near unity can be applied to engine conditions If Eq 2 is correct then it cannot On the third question could Nu measured with Tw Tb near unity be scaled to engine conditions where Tw Tb is not near unity nothing was found that explicitly addressed it However Nu for cooling passages with heat transfer enhancement features such as pin fins and ribs are often normalized with respect to Nu for a smooth duct with the same ReD and Pr see for example Ref 12 that is Nu Nus 3 where Nu is the Nusselt number for the cooling passage with heat transfer enhancement features and Nus is for the same duct without those features i e for the smooth duct If Eq 2 is used then Eq 3 becomes Nu Nus cReDaPrb Tw Tb g cReDaPrb Tw Tb g s c cs 4 which tacitly assumes Tw Tb g for the duct with heat transfer enhancement features is the same as Tw Tb g s for the smooth duct Thus a fourth question is could the scaling factor for the smooth duct be applied to the duct with heat transfer enhancement features Based on this literature review the objective of this paper is to answer the following questions Is it necessary to match the rate of change in Tw Tb when measuring HTC or Nu in addition to matching ReD and Pr Should Tw Tb be accounted for explicitly as shown in Eq 2 or is Eq 1 adequate Could scaling factors for Tw Tb from smooth ducts be used to scale Nu in ducts with heat transfer enhancement features If the scaling factor from smooth duct cannot be used then is there a method to determine a scaling factor A computational study on a test problem will be used to seek answers to the above questions Relevant existing data in the literature will also be used The remainder of this paper is organized as follows First the test problem used to seek answers is described Next the problem formulation the numerical method of solution and the grid sensitivity study are explained Afterwards results are presented to address the objectives of this study 2Copyright 2017 ASME Fig 1 Schematic of the pin fin problem studied PROBLEM DESCRIPTION The test problem studied is a cooling passage made up of a channel with a staggered array of pin fins see Fig 1 The geometry and flow conditions for this cooling passage are very similar to the experimental study performed by Metzer et al 13 In Fig 1 all dimensions are expressed in terms of D D 0 2 inches 5 08 mm the diameter of the pin fin As shown in Fig 1 the channel has a length of 50D and a height of H H D with 20 rows of pin fins that are arranged in a staggered fashion The first pin fin is located at x 1 5D and the spacing between the centers of pin fins is Sx 2 5D along x and Sy 2 5D along y Upstream of the channel shown in Fig 1 is attached another channel of length 100D and height H that has no pin fins The purpose of the upstream duct is to ensure the flow approaching the pin fins has a fully developed turbulent profile A channel of length 5D and height H was also appended downstream of the channel shown in Fig 1 to ensure that there are no reverse flows at the outflow boundary The part of the channel with pin fins shown in Fig 1 x 0 to x 50D is referred to as the test section For this problem the cooling fluid is air and enters the channel s inlet at x 100D with uniform temperature Tc 300 K and uniform velocity Uc The magnitude of Uc was adjusted to give rise to Reynolds numbers ReD of 4 650 6 000 6 300 6 800 8 900 and 10 500 at x 1 25D The back pressure at the exit of the channel Pb was fixed at 1 atm For this channel walls are adiabatic from x 100D to 0 and from x 50D to 55D From x 0 to 50D where there are pin fins a constant temperature Tw was imposed To simulate laboratory conditions where Tw Tb is near unity Tw was set at 303 K To simulate engine conditions where Tw Tb can be up to 2 2 the following Tw were investigated 390 500 800 1 050 and 1 800 K Thus Tw Tb at x 0 where the thermal boundary layer starts ranges from 390K 300K 1 3 to 1 800K 300K 6 0 However since Tb increases along the test section Tw Tb at locations where the flow and heat transfer become fully developed are considerably less 2 or less which is more representative of what takes place in gas turbine cooling passages Because of the symmetry of this test problem in geometry and operating conditions only a symmetric portion of the channel was simulated Table 1 summarizes all cases studied Table 1 Summary of Cases Studied T c the bulk temperature at the channels inlet PROBLEM FORMULATION AND METHOD OF SOLUTION For the test problem described in the previous section density can vary appreciable in every cross section and along the duct This is because of the large difference between the wall and the bulk temperatures Also with 20 rows of pin fins pressure drop along the duct is appreciable Thus the compressible form of the conservation balance equations are needed With the large temperature variation in and along the passage temperature dependent properties need to be accounted for as well In this study the governing equations employed are the ensemble averaged continuity full compressible Navier Stokes and energy equations for a thermally perfect gas with temperature dependent transport properties 14 The effect of turbulence was modeled by the shear stress transport SST model of Mentor 15 The SST model was selected because it has been shown to provide good predictions of flow and heat transfer in cooling passages with pin fins 16 Solutions to the governing equations described above were obtained by using the ANSYS Fluent code 17 Since only steady state solutions were sought the pressure based SIMPLE pressure velocity coupling scheme was used 18 All fluxes for density momentum and energy at the cell faces were interpolated by using the second order upwind scheme for numerical stability and low numerical diffusion The pressure equation was also approximated by a second order scheme Each steady state solution is obtained by iterating until all residuals for all equations plateau At convergence the scaled residuals were always less than 10 5 for the continuity less than 10 6 for the three components of the velocity less than 10 7 for the energy and less than 10 5 for the turbulent kinetic energy the dissipation rate of turbulent kinetic energy and omega a ratio of the turbulent dissipation to the turbulent kinetic energy 3Copyright 2017 ASME BULK TEMPERATURE AND HTC Since heat transfer rate is characterized by the heat transfer coefficient HTC or the Nusselt number Nu which in turn requires specification of the bulk temperature this section describes how the bulk temperature HTC and Nu are defined and computed The bulk temperature at any cross section is defined by 9 11 T C 1 The results obtained for the Nusselt number under engine conditions are shown in Figs 9 to 13 Figure 9 shows how Tw Tb varies along the passage as a function of Tw Tb at the passage inlet denoted as Tw Tc From Fig 9 it can be seen that even when Tw Tc 6 0 Tw Tb is about 1 6 when x D 35 where the flow and heat transfer is fully developed Figures 10 and 11 show how variations in Tw Tb along the passage Fig 11 Regionally averaged Prandtl number along the passage as a function of Tw Tc Cases 6 1 ReD in 10 500 Fig 12 Regionally averaged Nusselt number along the passage as a function of Tw Tc Cases 6 11 ReD in 10 500 affect the local ReD and Pr From Fig 10 it can be seen that though ReD at the inlet is 10 500 it drops to 4 650 6 000 6 300 6 800 or 8 900 at x D 35 depending on the value of Tw Tc It is for that reason those ReD were studied for the lab conditions since the lab and the engine conditions must have the same ReD From Fig 11 it can be seen that Pr does not vary appreciably in the temperature range studied Pr over the temperatures studied ranges from 0 679 to 0 717 If Pr is raised to the power of 0 174 then Pr0 174 ranges from 0 935 to 0 944 which produces a relative difference of 0 9 in Nu Thus 7Copyright 2017 ASME matching Pr between the lab and engines conditions is less important Figure 12 shows the variation of the regionally averaged Nusselt number along the passage and Fig 13 shows the correlation obtained for that data and it is given by Nu 0 19ReW X Z HPrX H a b X X 6 13 The correlation given by Eq 13 is valid for the fully developed region of the channel with pin fins in the following range ReD from 4 000 to 10 000 Pr from 0 679 to 0 717 and Tw Tb from 1 07 to 1 70 Fig 13 Nu under lab condition Eq 12 and engine conditions Eq 13 along with the scaling factor Eq 14 Fig 14 Comparing Nulab Nuengine and Nuscaled at the same ReD Table 2 Nulab Nuengine and Nuscaled at the Same ReD Scaling The Nu correlation given by Eq 13 for engine conditions differs from the one given by Eq 12 for lab conditions This difference can be as large as 12 9 if Tw Tb 1 573 This difference can be seen visually in Fig 14 Here it is noted that when Tw Tb in Eq 13 is set to unity Eq 13 does not reduce to Eq 12 as shown in Fig 13 Though those two lines are parallel they do not coincide This indicates the changes in density velocity and local accelerations induced by Tw Tb can significantly change the nature of the surface heat transfer that is what occurs under lab conditions can be quite different from what occurs under engine conditions The question now is how to scale HTC or Nu measured under lab conditions to engine conditions which involves not just matching ReD and Pr but also Tw Tb at the duct s inlet One way is to multiple Nu obtained under lab condition Nulab by a scaling factor suggested by experimental studies of smooth circular ducts such as those described in Refs 5 6 7 and 10 namely Tw Tb r where r can be 0 5 0 55 and 0 7 Figure 14 and Table 2 show how Nuscaled Nulab Tw Tb r compare with Nu obtained under engine conditions Nuengine where ReD and Pr are matched Pr is not listed because Pr did not change appreciably as previously noted From that figure and table it is clear that the scaling factor given by Tw Tb r does not work In fact the corrections increase the errors instead of reducing them If r 0 7 and Tw Tb 1 573 then the error in Nulab scaled 8Copyright 2017 ASME relative to Nuengine can be as high as 36 6 in the fully developed region Thus scaling depends on the geometry That is every geometrical configuration may require a different scaling factor and using scaling factor from smooth ducts as suggested by Eq 4 can cause significant errors Thus a method is needed to obtain the scaling factor when there are heat transfer enhancement features One way is by computational fluid dynamics CFD The errors in CFD based on RANS is due to turbulence modelling However the effects of Tw Tb on transport properties and