Local heat transfer for jet impingement onto a concave surface including injection nozzle length to diameter and curvature ratio effects

Accepted Manuscript Local heat transfer for jet impingement on a concave surface including injection nozzle length to diameter and curvature ratio effects Vijay S. Patil, R.P. Vedula PII: DOI: Reference:

S0894-1777(17)30224-8 http://dx.doi.org/10.1016/j.expthermflusci.2017.08.002 ETF 9170

To appear in:

Experimental Thermal and Fluid Science

Received Date: Revised Date: Accepted Date:

20 February 2017 15 July 2017 2 August 2017

Please cite this article as: V.S. Patil, R.P. Vedula, Local heat transfer for jet impingement on a concave surface including injection nozzle length to diameter and curvature ratio effects, Experimental Thermal and Fluid Science (2017), doi: http://dx.doi.org/10.1016/j.expthermflusci.2017.08.002

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Local heat transfer for jet impingement on a concave surface including injection nozzle length to diameter and curvature ratio effects Vijay S. Patil Department of Energy science and engineering, I. I. T., Bombay, Mumbai (India) Email: [email protected] R. P. Vedula Department of Mechanical engineering, I. I. T., Bombay, Mumbai (India) Email: [email protected]

Address for correspondence: Mr. Vijay S. Patil, C/O- Prof. R. P. Vedula, Department of Mechanical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai – 400 076 India. E-mail: [email protected] Telephone: (+91) 22-25764584

Abstract

The effect of jet nozzle length to diameter ratio on local heat transfer coefficient measurements for a row of circular jets impinging on a concave surface for varying target spacing are reported here. The nozzle length to diameter ratio (L/d) and the nozzle to target spacing (H/d) were varied from 0.2 to 6 and 0.67 to 8 respectively. Three curvature ratios, defined as the ratio of jet diameter to target surface diameter (d/D), equal to 0.1, 0.2 and 0.3 were studied and the jet to jet pitch to diameter ratio (P/d) was kept constant at 4.0. The Reynolds number was varied between 10000 and 50000 and wall static pressures for some cases were measured for obtaining a better understanding of the heat transfer coefficient variations.

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Higher stagnation zone Nusselt numbers were observed for the jet nozzles with small L/d at small H/d values whereas at larger H/d values nozzle length was observed to affect the data only marginally. The difference between Nusselt number values, between H/d=2 and H/d=8, was observed to progressively reduce with increasing L/d ratio and this was true for both the local stagnation point as well as the spanwise averaged stagnation line values. The maximum local Nusselt number values were observed to shift from the geometric impingement location to nearby locations for very small nozzle diameter to target distance ratios (H/d≤1) for larger curvature ratios. A constant heat transfer impingement area with constant pumping power criterion is presented to compare the performance of the different configurations studied. The longer nozzle lengths and smaller curvature ratios are observed to perform better based on this criterion. A correlation is presented for the overall averaged Nusselt number, with a validity within the parameter range studied, with a ±10% error band. Keywords: Concave surface, Jet impingement, Local heat transfer. Nomenclature Aj

Area of jet nozzle (m2)

Ap

Target plate area (m2)

Cp

Pressure coefficient (refer eq. (5))

d

Diameter of nozzle (m)

D

Diameter of target plate (m)

hS,Y

Local convective heat transfer coefficient (W/m2 K) (refer eq. 2)

H

Nozzle to target spacing (m)

I

Input current (amp)

k

Thermal conductivity of the jet fluid (W/mK)

ksteel

Thermal conductivity of stainless steel foil (W/mK)

KL

Loss coefficient (refer eq. (5))

l

Height of the target surface and nozzle plate (m)

L

Nozzle thickness or length (m)

ṁ

Mass flow rate through each jet nozzle (kg/s)

n

Number of jet nozzles in injection plate

Nu

Nusselt number 2

Nu (S, Y)

Local Nusselt number (refer eq. 4)

Nu (S, Y )

Spanwise averaged Nusselt number (refer eq. 4)

Nu av( )

Overall averaged Nusselt number (refer eq. 4)

P

Jet nozzle pitch (m)

P1

Pressure at the jet exit region (N/m2) (refer fig. 1(b))

Pc

Plenum chamber pressure (N/m2)

Ps

Static surface pressure (N/m2)

q

Wall heat flux (W/m2) (refer eq. 3)

qloss

Combined convective and radiative heat loss (W/m2)

Q

Volume flow rate (m3/s)

r

Target plate curvature radius (m)

R

Normalized heat transfer coefficient (refer eq. 9)

Re

Reynolds number (= ρ V d /μ)

Reeq

Equivalent Reynolds number (refer eq. 8)

S

Curvilinear coordinate (refer fig. 1(b))

t

Thickness of stainless steel foil (m)

TS,Y

Local target plate temperature (K)

T∞

Ambient temperature (K)

Tj

Jet exit temperature (K)

V

Input voltage (Volt)

V

Average jet velocity (m/s) = ṁ/ρAj

X

Planer length measured by camera (m)

Y

Spanwise coordinate (refer fig. 1(b))

Greek Symbols ρ Fluid density (kg/m3) μ

Fluid dynamic viscosity (Ns/m2)

λ

Jet nozzle pitch to jet diameter ratio 3

α

Angle between two location in curvilinear direction on target surface

θ

Angular distance for averaging heat transfer coefficients

1. Introduction Heat transfer behaviour for jet impingement on a cylindrical concave surface has been a subject of interest due to its relevance in cooling of the leading edge of gas turbine blades. Chupp et al. [1] reported heat transfer data for a single row of round jets impinging on a concave surface with Reynolds number varying between 3000 to 10000 and the spanwise averaged data was reported for different curvilinear locations. A correlation was presented for the overall averaged Nusselt number as a function of Reynolds number, target spacing, jet to jet spacing and curvature ratio. Metzger et al. [2] reported the effect of target spacing and jet nozzle pitch on heat transfer using a single line of circular jets impinging on a concave surface upto a maximum Reynolds number equal to 6300. The maximum heat transfer coefficient was obtained with H/d=1 for all the jet to jet spacings that were investigated. However, the optimum target spacing for a two dimensional jet was reported to be 3.5 slot widths. Katti and Prabhu [3] investigated the effect of target spacing on local heat transfer coefficients with single row of impinging jets over a cylindrical concave surface with Reynolds number from 3000 to 15000 for curvature ratio equal to 0.167. The Nusselt number contours were reported to be stretched along the curvilinear direction. Martin et al. [4] reported the effect of jet and surface temperature difference on averaged stagnation point Nusselt number for round jets impinging over a concave cylindrical surface with Reynolds number varying between 5000 and 20000. The variation of H/d=2 to 8, P/d=2 to 8 and d/D= 0.18 and 0.28 was used in investigation. It was reported that the effect of temperature difference values between jet and surface on the Nusselt number was small. Hrycak [5] reported correlations for local and averaged Nusselt number for row of jets impinging on semi cylindrical concave surface for Reynolds numbers varying from 2500 to 30000. The H/d and S/d were varied from 2 to 8, 1 to 7 respectively and d/D equal to .025 and .073 were used in the study. It was reported that the Nusselt number values for flat plate impingement, after certain modifications, could also be used for concave surface heat transfer coefficient estimation. Tabakoff and Clevenger [6] reported spanwise averaged heat transfer coefficients for slot jet, single jet and array of jets impinging onto a concave cylindrical surface. The best heat transfer performance was reported for S/d between 5 and 8.8 and H/d=1. The influence of shifting the jet from the center of the symmetry was also studied and it was reported that while the heat transfer coefficients at the center decreased slightly, the overall average increased slightly. Iacovides et al. [7] reported heat transfer behaviour for submerged water jets impinging onto a concave cylindrical surface at Re=15000 with H/d=3.125, d/D=0.16 and S/d=4. The Nusselt number contours were reported to be roughly elliptic due to interaction between adjacent jets and the shape of the target surface. Small secondary peaks midway between the jet impingement zones were also reported. Imbriale et al. [8] studied the heat transfer behavior for jets impinging onto a 4

concave surface with an airfoil leading edge shape. The S/d was varied between 5 and 15 and the jet inclination was varied between 00 and 500. The average and downside region Nusselt number values for small jet pitch were reported to be higher than those for perpendicular impingement onto the surface. Bunker and Metzger [9] reported the detailed curvilinear and spanwise Nusselt number distributions for jets impinging on a concave surface with changing leading edge sharpness, Reynolds number, hole spacing and target spacing. Reducing hole spacing and target spacing were reported to increase Nu values. Katti et al. [10] reported the wall static pressure distribution with Reynolds number equal to 20000 for varying target spacing and curvature ratio. Pressure coefficients larger than unity were observed close to the stagnation region, and secondary peaks were reported at the interaction zones between adjacent jets. The above studies did not explicitly mention the development length for the jets emerging from the nozzles. Rama Kumar and Prasad [11] reported a computational study for flow and heat transfer with a row of circular jets impinging on a concave surface with Reynolds number varying between 5000 and 67800. The nozzle to target spacing, hole to hole spacing were varied for two curvature ratios equal to 0.165 and 0.19 and two nozzle development length to diameter ratios (L/d) of 0.32 and 0.377. The k-ω turbulence model was used and the averaged heat transfer coefficients were reported to be highest for H/d=1. Primary and secondary peaks were observed in local pressure and heat transfer coefficient distributions. Taslim et al. [12] studied the effect of surface roughness on heat transfer over a concave surface with four different geometries using nozzle length to diameter ratio equal to 1.54. Average Nusselt number data was reported for Reynolds numbers up to 40000 and an increase in heat transfer coefficient with increase in roughness was reported. Fenot et al. [13] reported the effect of H/d, P/d and d/D on the local heat transfer and effectiveness distribution for a row of hot jets impinging on a concave surface at Reynolds number values equal to 10000 and 23000. A long developing length was provided for the jets and detailed local Nusselt number data for d/D=0.1 was presented. Trinh et al. [14] compared velocity profile, effectiveness and heat transfer for round orifice, cross shaped orifice and elongated tube with jets impinging on flat plate. A vena contracta was observed at the exit of the round orifice whereas for an elongated tube, a fully developed flow profile was observed. Nusselt number secondary peaks were observed for small target spacing (H/d≤3), which showed higher values with round orifice as compared to the elongated tube orifice. Ashforth and Jambunathan [15] studied the effect of L/d for unconfined and semi-confined jets impinging on a flat surface. They reported the potential core length to be longer for jets with fully developed exit both for the no confinement as well as the semi-confinement situations. Bu et al. [16, 17] compared the heat transfer with single, two and three rows of jets impinging onto a concave surface of the shape of an aircraft wing leading edge, for the Reynolds numbers ranging from 50000 to 90000. The circumferential angles between jets were varied from -600 to 600 and the H/d was varied between 1.74 and 20. The L/d and S/d were kept constant at 25 and 0.75 respectively. An optimal H/d of 4 to 5.75 for single row of jets for maximum Nusselt number at stagnation point was reported. They presented a correlation for variation of the ratio of the local Nusselt number to the stagnation point Nusselt number as a function of circumferential distance 5

from the center of symmetry. Ying et al. [18] reported the influence of surface curvature effect on heat transfer with a single jet impinging onto a concave cylindrical surface. The d/D, L/d and H/d were varied from 0.005 to 0.03, .67 to 2 and 3.3 to 30 respectively. The jet Reynolds number was varied between 27000 and 130000. A decrease in average heat transfer with increase in surface curvature was reported. Cornaro et al. [19] reported flow visualization studies for a jet with a long developing length impinging on a concave surface. The effect of relative curvature, nozzle diameter, nozzle to surface distance and Reynolds number on the flow structure was reported. The location of vortex breakdown resulting in the jets transitioning from laminar to turbulent zones, which depends on jet to target spacing, was reported. Lee et al. [20] reported heat transfer results for a jet impinging onto a concave hemispherical surface for Reynolds numbers between 11000 and 50000. The H/d and d/D were varied from 2 to 10 and 0.034 to 0.089 while the L/d was kept constant at 58. Small target spacings showed secondary peaks at a curvilinear distance of approximately 2 times the nozzle diameter. These peaks monotonically decreased and disappeared with increase in target spacing and were attributed to the turbulence resulting due to transition from laminar to turbulent flow in the wall jet region. Several studies exist where a jet directly emerges from a plenum and others where a long developing length for the jet is provided. The stagnation point heat transfer coefficients for these two sets of studies are reported to be different. In this study the nozzle length to diameter ratio has been systematically varied and its influence on heat transfer coefficients is reported. Local heat transfer data for three different curvature ratios is also presented. In addition, the static pressure on the target plate was also measured which fortified the observations from heat transfer measurements. A criterion based on constant pumping power and constant target surface area is developed and a comparison based on this methodology is presented for the different configurations of the present study.

2. Experimental Setup and Methodologies A schematic of the experimental apparatus used in the current study is shown in the figure 1a. A screw compressor, with an associated air filtration system and pressure fluctuation damping reservoir, supplied air at the required flow rate to the test section. The exit of the compressor reservoir was connected to long piping with appropriate flow control valves and subsequently to a venturimeter. The differential pressure between the throat and inlet sections of the venturimeter, measured using a calibrated differential pressure transducer, and the system pressure measured by a pressure gauge connected to the throat section were used for calculating the mass flow rate. The flow then entered a developing section which gradually transformed the circular cross-section into a rectangular one and ended in a honeycomb structure which was connected to the plenum chamber. Four calibrated chromel-alumel (K- type) thermocouples placed normal to the flow direction were used to measure the temperature of the air in the plenum chamber. The plenum chamber was attached to a semi-cylindrical nozzle plate with holes of the correct dimensions machined in it. The nozzle plate was detachable from the plenum to enable study of different hole geometry configurations. The air exiting from the nozzle plate 6

impinged on a concave semi cylindrical stainless steel target plate. The coordinate axes used for data representation are shown in figure 1b, with ‘S’ and ‘Y’ being the coordinates along the curved surface and spanwise direction respectively, of the plate. The origin is located on the target surface at its center point. The spent fluid was restricted to exit in the ‘S’ direction by blocking the path in the extreme spanwise direction by end plates between the nozzle and target plates. A pressure tap, P1, on the top end was used to measure the pressure of the nozzle exit region. The plenum could be moved away or towards the target plate in order to obtain the required spacing between the jets and target plate.

(a) (b) Fig. 1 a) Schematic of Experimental setup b) Arrangement of nozzle and target plate. Three nozzle diameters equal to 10 mm, 20 mm and 30 mm as shown in figure 2a were used in the study and the target surface diameter was kept constant at 100 mm giving three curvature ratios (d/D) equal to 0.1, 0.2 and 0.3 respectively. The nozzle length was varied to get the different L/d ratios and the pitch was kept constant equal to 4 times the jet diameter. The wall thickness acts as the nozzle length for short nozzles whereas an additional tubular element was provided to increase the flow path distance of the long nozzle. The inlet and exit geometries were therefore maintained the same for all nozzles as shown in the sectional view of two nozzle plates with different L/d ratios in Figure 2b. All nozzle plates were manufactured with a single row of jets.

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(a) (b) Fig. 2 a) Nozzles with different hole diameter and b) Nozzles with different hole length. A target plate was fabricated using a stainless steel foil of thickness 0.1 mm. Copper bus bars soldered to the ends of the foil were connected to a high current low voltage power source so that the foil could be heated by passing alternating current (AC) through it. The minimum and maximum heat flux values over the entire range of experiments were 3500 and 8000 W/m2 respectively. The surface temperature varied between 400 C and 900 C in the circumferential direction and with the temperature difference being not more than 4 0 C in spanwise (vertical) direction. The jets impinged on the concave surface of the target plate while the convex side of target plate was painted black to have an emissivity value as close to unity as possible, to enable temperature measurement over this surface using a thermal camera. The thickness of the high emissivity black paint applied using a commercially available spray can was approximately equal to 0.02+0.002mm over the surface. The thermal camera (Micron M7600PRO) captures a planar image of the temperature distribution on the convex surface and that a correction was applied to transform the temperature values into the physically existing cylindrical geometry by using the following expression, X S  r    r  sin 1   (1) r  Where, S is curvilinear length, r is curvature radius, X is planar length and α is angle between two location in curvilinear direction on target surface. The difference in temperatures across the thickness of the measurement surface calculated using the one dimensional heat conduction equation with uniform heat generation was negligibly small and therefore the measured outer surface temperature was considered equal to the inner surface temperature within limits of measurement uncertainty. The local heat transfer coefficient (hS,Y) was calculated after accounting for the thermal conduction effects within the foil by using the finite volume method for the energy balance at any location (S,Y):

(2) 8

h S,Y

 TS 1,Y  2TS ,Y TS ,Y 1  TS ,Y 1  2TS ,Y T qt  k steel  S 1,Y  dS 2 dY 2   TS ,Y  T j 

 t  qloss 

V I (3) Ap where, q is wall heat flux, t is the thickness of the stainless steel foil, TS,Y is the local target plate temperature, TS-1,Y, TS+1,Y , TS,Y-1 and TS,Y+1 are the neighboring temperatures in the curvilinear and spanwise directions. Tj, V, I, Ap ,qloss are the jet exit temperature , input voltage, input current, target plate area and the heat loss from the surface exposed to the ambient, respectively. The heat loss was experimentally obtained by measuring the target plate temperature after giving a small power value to the target plate with the impingement side insulated. The input power value is the heat loss to the ambient at steady conditions, and several experiments with the wall temperature varying between the maximum and the minimum values encountered in the experimental program were conducted. A calibration curve was obtained between the heat input and the difference between the plate and ambient temperatures, which was used to compute the local heat loss from the convex side of the experimental setup during calculation of the heat transfer coefficient. The radiative loss was subtracted from the total loss to obtain the loss due to natural convection and was locally accounted for, at different locations on the test surface. The maximum temperature variation over the surface at any location in the vertical direction was approximately 40C and the average surface temperature was used to compute the convective loss at each location in the vertical direction. The error in the convective heat loss term was therefore computed by using a +40C uncertainty in temperature measurement during calculation of the uncertainty in the measured heat transfer coefficients. The inner wall was shiny with an emissivity equal to about 0.2 and a radiation loss correction was applied by dividing the surface into 18 equal segments assumed isothermal at the mean temperature in the vertical direction. The radiation interaction between any two segments was assumed two dimensional since the circumferential length of the elements was much smaller than the height and the standard radiosity formulation was used to estimate the radiation correction. This correction was less than 8% of the total loss correction and affected the results only marginally. The following expressions were used to obtain local Nusselt number, Nu (S, Y) , spanwise q

averaged Nusselt number varying in the curvilinear direction, Nu (S, Y ) and overall averaged Nusselt number, Nu av( ) : hd Nu (S, Y)     k ,

Y  /2

hd Nu (S, Y )    dy k  Y  , Y

Nu av( ) 

 hd   ddy k   /2

 

Y 

(4)

An acrylic surface having the same curvature as the steel target plate used for the heat transfer measurements was used as the target surface for the pressure measurement since the steel target plate was too thin for proper attachment of pressure taps. The measurements were 9

obtained for d/D= 0.1 and 0.2 only due to constraints with respect to the experimental apparatus. The static pressure difference between various locations on the target plate and the jet exit region, (Ps-P1) and the pressure difference between the plenum chamber and the jet exit region, (Pc –P1) were measured using U-tube manometers with water as a working fluid. The loss coefficient, KL and the Pressure coefficient, CP were defined as:  P P  K L   C 12   0.5V    ,

P P  C P   S 12   0.5V   

(5)

The thermal camera was calibrated using an isothermal copper block heated by an internal heater unit which had a calibrated thermocouple embedded in it. The thermocouples were calibrated using a temperature controlled water bath apparatus. The uncertainties in temperature measurement by the thermal camera and the thermocouples were obtained to be ± 1.2oC and ±0.50C respectively with a 99.7% confidence level. Error analysis as explained by Coleman and Steele [21] showed maximum uncertainty of 10% and 13% in the measurement of Nusselt number and pressure coefficient values respectively. About 10% of the experiments were repeated on a random basis to confirm the repeatability of the experiments. The different parameters that were studied are given in table 1 below. Table 1 Geometrical and flow Parameters used in the investigation. d/D

H/d

L/d

Re

0.1

1, 2, 4, 6, 8

0.6, 1.5, 3, 6

10000, 23000, 35000, 50000

0.2

1, 2, 3, 4

0.3, 1.5, 3

10000, 23000, 35000, 50000

0.3

0.67, 1, 2, 4

0.2, 1.5, 3

10000, 23000, 35000, 50000

3. Results and Discussions 3.1. Curvature ratio (d/D) = 0.1 The local Nusselt number distribution in spanwise direction and the spanwise averaged Nusselt number along curvilinear direction for H/d=5 with Re=10000 and 23000 are shown in figures 3a and 3b respectively. The results reported by Fenot et al. [13] are also shown and the match is noticed to be within the limits of uncertainty providing confidence in the experimental procedures and methodologies being used in the present study. The minor differences can be attributed to small differences in geometry and flow conditions. The local Nu distribution in the curvilinear direction for three different L/d ratios equal to 0.6, 1.5 and 6 for H/d equal to 2 is shown in figure 4a. The corresponding spanwise variation is 10

shown in figure 4b, but only for the smallest and largest L/d ratio for the sake of clarity. The nozzle with the smallest L/d ratio produces the highest Nu values with the difference between the Nusselt number values at the stagnation point for L/d of 0.6 and 6 equal to approximately 20%. The spanwise distribution shows a peak in the region in between the jets for Re=50000 case, which is significant for the larger L/d case, whereas for shorter L/d, the peak though noticeable is very small. The peaks in between the jet impingement locations are not noticeable for the lower Reynolds number for both L/d ratios. Figures 5a and 5b show the pressure coefficients in curvilinear and spanwise direction respectively for H/d equal to 2 for the nozzles with L/d=0.6 and 6. The curvilinear distribution shows that the peak pressure coefficient for L/d=0.6 is greater than unity whereas the corresponding value for L/d=6 is nearly unity. The spanwise distribution shows that there is a second peak pressure approximately midway between the two jets for both the L/d ratios shown in the figure.

(b) Fig. 3 a) Local Nu variation in spanwise direction between two jets b) Spanwise averaged Nusselt number distribution in curvilinear direction with H/d=5, d/D=0.1, P/d=4, L/d=20. (a)

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(a) (b) Fig. 4 Effect of L/d on local Nu distribution in a) curvilinear direction at Y/d=0 and b) spanwise direction at S/d=0 for d/D=0.1 and P/d=4 with H/d=2.

(a) (b) Fig. 5 Pressure coefficients along a) Curvilinear direction at Y/d=0 and b) Spanwise direction at S/d=0 for short and long with H/d=2, Re=50000, d/D=0.1, P/d=4. Figures 6a and 6b show curvilinear and spanwise Nusselt number distributions with H/d=4 and it can be observed that the values are almost equal for all the L/d values. Figures 7a and 7b show the curvilinear and spanwise pressure distribution for the H/d=4 case for the largest and smallest L/d ratios and again the differences are seen to be very small.

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The Nusselt number and pressure coefficient curves complement each other and suggest the likely flow behavior as the fluid jets exit the nozzles, due to which the heat transfer coefficients behave differently for different L/d ratios. It is well known that when a fluid tries to exit through a long nozzle from a large plenum the jet contracts to an area smaller than the physical nozzle area, and expands till it reaches the nozzle walls and becomes fully developed. However, if the nozzle walls are not long then the jet expands freely into the surroundings since no walls exist to arrest this expansion at exit. A similar phenomenon is likely to exist here also although the surface curvature effects could affect the contraction ratios. The jet from smaller L/d ratio nozzle therefore emerges with a contraction zone which is likely to have higher local momentum and therefore the Nusselt numbers are high when the impingement surface is close enough to be within the influence of this region. The pressure coefficient at the impingement point for smaller L/d ratio is therefore much higher than unity whereas for the high L/d ratio, it is nearly unity. However, when the distance between the jet and target plate is increased, the jet spreads and the local momentum reduces due to entrainment from the surroundings, and the pressure coefficient for the short nozzles reduces. The pressure coefficients thus become nearly equal for both the long and short nozzles for higher jet to target spacing. The differences in the Nusselt number values also therefore reduce to very small values. The second peak in Nusselt number at the midway point between the jets in spanwise direction is due to the complex interaction between the jet flows after impingement.

(a)

(b)

Fig. 6 Effect of L/d on local Nu distribution in a) curvilinear direction at Y/d=0 and b) spanwise direction at S/d=0 for d/D=0.1 and P/d=4 with H/d=4.

13

(a) (b) Fig. 7 Pressure coefficients along a) Curvilinear direction at Y/d=0 and b) Spanwise direction at S/d=0 for short and long with H/d=4, Re=50000, d/D=0.1, P/d=4

(a)

(b)

Fig. 8 Effect of L/d on a) local and b) Spanwise averaged stagnation point Nusselt number values for d/D=0.1 and P/d=4. Figures 8a and 8b show the local and spanwise averaged stagnation point Nusselt number variation with varying L/d for H/d equal to 2 and 8 for Reynolds numbers equal to 10000 and 50000. The Nusselt number is noticed to continuously fall with increasing L/d for the smallest 14

target spacing, whereas for the largest spacing (H/d=8), the Nusselt number continuously rises with increasing L/d. The area contraction at the nozzle impingement zone reduces with increase in L/d for the lower H/d configuration, resulting in a progressive reduction in the Nusselt number values. The jets emerging from the smaller L/d nozzle for larger H/d expand more rapidly in radial direction and therefore the momentum at the impingement location is low, whereas the jets with longer L/d have a fuller profile with a well defined potential core and the jet momentum reduction is smaller till it impinges on the plate. The Nusselt number therefore increases with increasing L/d ratio for the larger H/d case. The difference between the Nu values between a pair of H/d ratios at a given L/d ratio is therefore more significant for smaller L/d. The concavity of the target surface influences the entrainment of the surrounding fluid into the impinging jet affecting the magnitude of the difference between heat transfer coefficients. The detailed local Nusselt number distribution is shown in figure 9, 10 and 11 for only two Reynolds numbers, for the sake of brevity, for the short and long nozzles. The elliptical nature of the Nusselt number contours is due to a combination of the interaction of neighboring jets in the spanwise direction and the concavity of the target surface in the curvilinear direction. The general trends discussed in detail earlier can be observed here also and the nature of contours for all the other cases, not presented here, is similar to these.

Fig. 9 Local Nu contours for Re of 10000 for a) L/d=0.6 and b) L/d=6 with H/d=2, d/D=0.1 and P/d=4.

15

Fig. 10 Local Nu contours for Re of 50000 for a) L/d=0.6 and b) L/d=6 with H/d=2, d/D=0.1 and P/d=4.

Fig. 11 Local Nu contours for Re of 50000 for a) L/d=0.6 and b) L/d=6 with H/d=8, d/D=0.1 and P/d=4.

16

3.2. Curvature ratio (d/D) = 0.2 Figure 12a shows the variation of local Nusselt number distribution for different H/d ratios for a larger curvature ratio d/D= 0.2 for L/d equal to 0.3 for Re=50000 and 10000. The stagnation region Nusselt number values are nearly constant and increase as the H/d reduces to 2 similar to the smaller curvature ratio case. However, further reduction in H/d is noticed to cause a decrease in the local stagnation Nusselt number value. However at H/D=1 even though the stagnation value reduces, the Nusselt number rises along the curvilinear direction after impingement due to the acceleration of the fluid as it flows out in the narrow region between the jet exit and the concave target plate. Figure 12b shows the spanwise local Nusselt number distribution at the geometric impingement location, S/d=0 and the values are noticed to drop rapidly on either side of the stagnation region due to the absence of the concave curvature in this direction. The Nusselt number values in the region between the jets are nearly equal for H/d=3 and 4, whereas there is a rapid rise in the Nusselt number values as H/d reduces below 3. The closeness of the nozzle to the concave target plate impedes the ease with which the fluid flows outward in the curvilinear direction after the impingement due to which the jet to jet interaction in the spanwise direction becomes stronger with reducing H/d values. Figure 12c shows the spanwise averaged Nusselt number distribution and in contrast to the local Nusselt number values, it is observed that the spanwise averaged values for H/d=1 are higher than H/d=2 values. At H/d=1 even though the local stagnation value reduces, the values in between the jets rise due to significant interaction between the jets resulting in higher spanwise averaged values.

(a)

(b)

(c)

Fig. 12 Effect of target spacing on a) curvilinear local Nu at Y/d=0, b) spanwise local Nu values at S/d=0 and c) Spanwise averaged Nu distribution with d/D=0.2 for L/d=0.3. Figure 13a shows the local Nusselt number distribution in curvilinear direction for L/d equal to 3. The variation in stagnation point Nu values is small for this higher L/d when 17

compared to the L/d equal to 0.3 case even for H/d lower than 2. Figures 13b and 13c are the corresponding spanwise local values at the stagnation line and the spanwise averaged values along the curvilinear direction respectively. Figures 14a and 14b show the effect of nozzle length on local curvilinear and spanwise pressure coefficient at Re=50000 for H/d=1 and H/d=2. The region close to the impingement location is observed to have almost constant pressure values in the curvilinear direction due to combined effect of curvature and the proximity of the impingement nozzle to the target surface. This is in contrast to the spanwise distribution in which the pressure falls steeply after the geometric impingement location since the obstruction to the flow is smaller in this direction. A similar characteristic was observed in the local Nusselt number values which remained relatively constant in the impingement region in the curvilinear direction but rapidly reduced in the spanwise direction. Figures 15 and 16 show Nusselt number contours with H/d=1 and H/d=2 for L/d=0.3 and L/d=3 respectively. The Nusselt number contours are similar for decreasing H/d for the larger L/d case. However for L/d=0.3, the nature of the contours are significantly different in the stagnation region due to the strong jet to jet interaction as discussed above.

(a)

(b)

(c)

Fig. 13 Effect of target spacing on a) curvilinear local Nu at Y/d=0, b) spanwise local Nu values at S/d=0 and c) Spanwise averaged Nu distribution for d/D=0.2 with L/d=3.

18

(a)

(b)

Fig. 14 Effect of nozzle length on a) Curvilinear pressure coefficient at Y/d=0 and b) Spanwise Local Pressure coefficient for H/d=1 and H/d=2, with d/D=0.2 and Re=50000.

Fig. 15 Local Nu contours for Re of 50000 for a) H/d=1 and b) H/d=2 with L/d=0.3, d/D=0.2 and P/d=4.

19

Fig. 16 Local Nu contours for Re of 50000 for a) H/d=1 and b) H/d=2 with L/d=3, d/D=0.2 and P/d=4. 3.3. Curvature ratio (d/D) = 0.3 Figures 17a and 17b show the effect of H/d on local and spanwise averaged Nusselt number values respectively with L/d equal to 0.2 and 3. The stagnation Nu values again rise uptil H/d=2 for L/d=0.2 and then start reducing with further reduction in the H/d as observed for the d/D=0.2 case. The longer nozzles i.e. L/d=3 also follow the pattern indicated for the smaller d/D ratios with the Nu values remaining nearly same for all the target spacings tested. In addition, very similar to the d/D=0.2 case, the spanwise averaged Nu values continue to remain at the highest levels for the lowest H/d values for which measurements were made. The local Nusselt number contours are shown in figure 18a and 18b for H/d of 0.67 and 2 respectively. It is clearly visible that for H/d= 2 Nu values stretch significantly in curvilinear direction and highest values are seen at the stagnation region. However for H/d of 0.67, the maximum values of the Nusselt numbers are no longer at the geometric impingement location but are spread around the stagnation region in the spanwise and curvilinear directions. Therefore even though the local Nu values along curvilinear direction for H/d of 0.67 are seen to be less, the spanwise averaged Nu values are very nearly equal to and slightly greater than the values for H/d= 2.

20

(a)

(b)

Fig. 17 Effect of H/d on a) Local Nu values at Y/d=0 and b) Spanwise averaged Nu values along curvilinear axis with Re=50000, d/D=0.3, P/d=4 for L/d=0.2 and L/d=3.

Fig. 18 Local Nu contours for a) H/d=0.67 and b) H/d=2 with Re=50000, d/D=0.3, L/d=0.2 and P/d=4. 21

4. Constant pumping power for constant target surface area The curvature ratio, d/D, in the current investigation was changed by changing the jet diameter and keeping the target area constant. A change in the jet diameter or nozzle length results in a change in the pressure drop across the jet nozzle when the Reynolds number is kept constant. A comparison for the different cases is proposed based on the constant pumping power criterion for equal heat transfer area. A particular geometry is arbitrarily taken as the reference case. An equivalent Reynolds number for the reference geometry is computed using a selected geometry keeping the pumping power constant. The ratio of the heat transfer coefficients of the selected geometry and that for the reference case at the equivalent Reynolds number provides a basis for comparison. The equivalent Reynolds number is computed as follows:  V 2 Pumping power  nQP  nQK L   2 

   

(6)

Where, n is the number of jets per nozzle plate. Therefore, assuming that the target plate has a height equal to P/2 on either side of the first and last injection holes: 3        ld  Pumping power  K L    2   Re3 2  8      Pd 

 

(7)

Where, KL is flow loss coefficient.

In the current study, the working fluid, the height of the target surface and the P/d ratio are identical. The expression for evaluating the equivalent reference case Reynolds number using any other configuration with a specified Reynolds number at constant pumping power therefore becomes:

Re    eq

Re

1

3 K L   d ref       K L ref   d 

2

3

(8)

The angular distance over which the Nusselt number averaging is performed is kept same to keep the target area the same. The ratio of the heat transfer coefficients for a given configuration and the reference case therefore becomes: k   Nu av( )  d  R (9) k  Nu   av( ) d  eq

22

The configuration with d/D = 0.3 with L/d=0.2 is arbitrarily chosen as the reference case and a value of R greater than unity indicates better performance compared to this case. The measured loss coefficients for the different configurations used in the current study are shown in Figure 19. 10

10

d/D=0.2, L/d=0.3

d/D=0.2, L/d=1.5

d/D=0.3, L/d=0.2

d/D=0.3, L/d=1.5

1 6000

KL

d/D=0.1, L/d=1.5

KL

d/D=0.1, L/d=0.6

1 6000

60000

Re

Re

60000

Fig. 19 Loss coefficient at different Reynolds number.

(a)

(b)

Fig. 20 Normalized heat transfer coefficient with a) H/d=2 and b) H/d=4 based on constant pumping power for different nozzles over a target area with θ= 90 0. The target surface area with an included angle θ=900 is considered as the constant area for comparison of the different cases. Figure 20 shows the normalized heat transfer coefficient 23

value ‘R’ for different nozzles. The elongated nozzles for d/D=0.1 are noticed to give better constant pumping power heat transfer performance as compared to other nozzles. Experimental data was not available for obtaining the Nusselt numbers at all equivalent Reynolds number values and therefore the correlation proposed in the next section was used for all calculations even though some Reynolds number values were marginally outside the validity of correlation. 5. Correlation for Nusselt number The data obtained from the experiments was averaged over a specific angular distance, θ, as shown in figure 1b to provide correlations that are averaged in the longitudinal and curvilinear directions. Since the Nusselt number data were seen to get affected for small values of H/d with changing L/d, separate correlations to isolate the influence of this parameter were obtained. Since only the stagnation quantities get affected this was done only for the θ =250 case. The correlations obtained using the linear regression methodology are given below: Nu av( )  0.225 Re

0.619

Nu av( )  0.243 Re

0.648

H   d 

0.101

H   d 

0.255

d   D

0.033

d   D

0.097

L   d   L   d 

0.012

for

0.67 ≤ H/d ≤ 2

(10)

for

2 ≤ H/d ≤ 8

(11)

0.02

(a) (b) (c) 0 Fig. 21 Correlation for averaged Nusselt number for strip with included angle = 25 , a) 0.67 ≤ H/d ≤ 2, b) 2 ≤ H/d≤ 8 and c) 0.67 ≤ H/d ≤ 8. Figures 21a and 21b show a comparison of the predicted (using the correlation) and measured Nu values and it is noticed that the match is good with a deviation of ± 5% from the 24

mean value for over 95% of the data. Figure 21c shows the plot for a correlation where data points for all H/d values are included and the deviation is about ±10%. Combining all the parameters together gives only a small loss in accuracy and therefore a single correlation can be considered to be adequate over the entire range of data obtained and the overall averaged Nusselt number correlations valid over the range of the following parameters, 10000 ≤ Re ≤ 50000, 0.1 ≤ d/D ≤ 0.3, 0.67 ≤ H/d ≤ 8, 0.2≤L/D≤6.0 with P/d=4 for 250, 450 and 900 are: Nu av( )  0.24 Re

0.637

H   d 

0.159

d   D

H   d 

0.169

H Nu av( )  0.232 Re 0.666  d 

0.159

Nu av( )  0.233 Re

0.647

0.0095

L   d 

d   D

0.149

d   D

0.312

0.013

L   d 

0.008

L   d 

0.004

for

(θ= 250)

(12)

for

(θ= 450)

(13)

for

(θ= 900)

(14)

The influence of L/d is noticed to reduce with increasing included angle since the data in the stagnation zone is affected the most, due to a change in the value of this parameter.

6.

Conclusions

Local heat transfer measurements with a single row of jets, with P/d=4, impinging on a concave surface have been presented. The length of the jet nozzles was varied and the short nozzles are observed to give higher Nusselt number values when the target spacing is small, but for larger target spacing the heat transfer coefficients are almost same. The Nusselt number values in between the geometric impingement location progressively increase with reduction in the H/d value. The Nusselt numbers at the geometric impingement location increase till the target distance is reduced to H/d=2 beyond which there is a sharp reduction for shorter nozzles but for longer jet nozzles this is not observed. The spanwise averaged values increase monotonically for both long and short nozzles with reduction in the H/d values. A methodology for the heat transfer performance on the basis of constant pumping power and constant target area has been presented and based on this criterion the long nozzles, in general, are observed to perform better. A correlation for the average Nusselt number has been presented which is valid up to Re=50000 with a ±10% error compared to the measured values.

References

25

1. R. E. Chupp, H. E. Helms, P. W. McFadden, T. R. Brown “Evaluation of internal heat transfer coefficients for impingement cooled Turbine airfoils” Journal of Aircraft, 3(1969), 203-208. 2. D. E. Metzger, T. Yamashita, C. W. Jenkins, ‘Impingement cooling of concave surface with lines of circular air jets’, Journal of Engineering for power, 7(1969), 149-158. 3. V. V. Katti, S. V. Prabhu, ‘Local Heat Transfer Distribution from a Row of Impinging Jets to a Cylindrical Concave Surface’, International Conference on Fluid Mechanics and Fluid Power, IIT Madras, Chennai, December 2010, FMFP10-HT-34. 4. E. L. Martin,L. M. Wright, D. C. Crites, ‘Impingement heat transfer enhancement on a cylindrical, leading edge model with varying jet temperatures’, Proceedings of ASME TURBOEXPO 2012, June 2012, Copenhagen, Denmark. 5. P. Hrycak, ‘ Heat transfer from a row of impinging jets to concave cylindrical surfaces’, International jounal of heat mass transfer, 24(1981), 407-419. 6. W. Tabakoff, W. Clevenger, ‘Gas turbine blade heat transfer augmentation by impingement of air jets having various configurations’, Journal of engineering and power, 1972, 51-60. 7. H. Iacovides, D. Kounadis, B. E. Launder, J. Li, Z. Xu, ‘Experimental study of the flow and thermal development of a row of cooling jets impinging on a rotating concave surface’, Journal of turbo machinery, 127(2005), 222-229. 8. M. Imbriale, A. Laniro, C. Meola, G. Cardone, ‘Convective heat transfer by a row of jets impinging on a concave surface’, International journal of thermal sciences, 75(2014), 153163. 9. R. S. Bunker, D. E. Metzger, ‘Local heat transfer in internally cooled turbine airfoil leading edge regions: Part I – Impingement cooling without film coolant extraction’, Journal of Turbomachinery, 112(1990), 451-458. 10. V. V. Katti, S. Sudheer, S. V. Prabhu, ‘Pressure distribution on a semi circular concave surface impinged by a single row of circular jets’, Experimental thermal and fluid science, 46(2013), 162-174. 11. B. V. N. Rama Kumar, B. V. S. S. S. Prasad, ‘Computational flow and heat transfer of a row of circular jets impinging on a concave surface’, Heat Mass transfer, 44(2008), 667678. 12. M. E. Taslim, L. Setayeshgar, S. D. Spring, ‘An experimental evaluation of advanced leading edge impingement cooling concepts’ Proceedings of ASME TURBOEXPO 2000, May 2000, Munich, Germany. 13. M. Fenot, E. Dorignac, J. J. Vullierme, ‘An experimental study on hot round jets impinging a concave surface’, Int. J. Heat and Fluid Flow, 29(2008), 945-956. 14. X. Trinh, M. Fenot, E. Dorignac, ‘The effect of nozzle geometry on local convective heat transfer to unconfined impinging air jets’, Experimental thermal and fluid science, 70(2016), 1-16.

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15. A. Ashforth-Frost, K. Jambunathan, ‘Effect of nozzle geometry and semi confinement on potential core of a turbulent axisymmetric free jet’, Heat and mass transfer, 23(1996), 155162. 16. X. Bu, L. Peng, G. Lin, L. Bai, D. S. Wen, ‘Experimental study of jet impingement heat transfer on a variable-curvature concave surface in a wing leading edge’, Heat and Mass Transfer, 90(2015), 92–101. 17. X. Bu, L. Peng, G. P. Lin, L. Bai, D. Wen, ‘Jet impingement heat transfer on a concave surface in a wing leading edge: Experimental study and correlation development’, Experimental thermal and fluid science, 78(2016), 199-207. 18. Z. Ying, G. Lin, X. Bu, L. Bai, D. Wen, ‘Experimental study of curvature effects on jet impingement heat transfer on concave surfaces’, Chinese journal of aeronautics, 30(2017), 586-594. 19. C. Cornaro, A. S. Fleischer, R. J. Goldstein, ‘Flow visualization of a round jet impinging on cylindrical surface’, Experimental Thermal and Fluid science, 20(1999), 66-78. 20. D. H. Lee, Y. S. Chung, S. Y. Won, ‘The effect of concave surface curvature on heat transfer from a fully developed round impinging jet’, Heat and Mass transfer, 42(1999), 2489–2497. 21. H. W. Coleman, W. G. Steele Jr., ‘Experimentation and uncertainty analysis for engineers’, John Wiley, New York 1989 (chapter 3).

27



Local Nu for single row of jets impinging into concave target surface



Influence of varying nozzle length to diameter and curvature ratios on local Nu



Peak Nu at large curvature ratio and small target spacings not at impingement point



Performance comparison at constant pumping power and constant heat transfer area



Correlations for overall averaged Nu and fixed area over wide Re range

28