Heat transfer distribution and shadowgraph study for impinging underexpanded jets

Applied Thermal Engineering 115 (2017) 41–52

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Research Paper

Heat transfer distribution and shadowgraph study for impinging underexpanded jets Ravish Vinze a, S. Chandel a, M.D. Limaye b, S.V. Prabhu c,⇑ a

Department of Mechanical Engineering, D.I.A.T., Pune, India R&DE (E), DRDO, Pune, India c Department of Mechanical Engineering, I.I.T., Bombay, Mumbai, India b

a r t i c l e

i n f o

Article history: Received 5 July 2016 Revised 4 December 2016 Accepted 10 December 2016 Available online 21 December 2016 Keywords: Nusselt number Under-expanded jets Recovery factor Shadowgraph photography NPR Contoured nozzle

a b s t r a c t In the present study, the influence of impinging underexpanded jets on local heat transfer is studied for nozzle pressure ratio (NPR) ranging from 2.4 to 5.1. To measure the local temperature distribution, a thin metal foil technique with Infrared camera is used. The adiabatic wall temperature is taken as the reference temperature for calculating local Nusselt number and recovery factor. The flow structure distributions captured with the shadowgraph technique are compared with the local Nusselt number and recovery factor distributions. Shadowgraph images show that the shock structure in the flow region plays an important role in governing the local heat transfer distribution over the plate. To propose a generalized correlation for local heat transfer for underexpanded jets, three contoured nozzles of exit diameter of 3.6 mm, 5.67 mm and 8.37 mm are studied. Proposed correlations for the local heat transfer show good agreement with the experimental results for larger nozzle to plate distances. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Impinging turbulent jets have number of applications like in manufacturing industries - drying of objects, cooling or heating of surfaces, in aviation – vertical landing and takeoff aircraft, furthermore in defense and space technologies - missile or shuttle exhaust. The profile of the device from which these jets originates depends on applications, available space and cost of manufacturing. In most of the applications, the working fluid is at a higher pressure than the ambient. In such case the fluid expansion takes after jet originates from nozzle. These under-expanded jets have different flow structure than the subsonic compressible jets due to presence of shocks. These shocks significantly influence heat transfer in limited impingement area. The strength of the shocks depends upon nozzle pressure ratios (NPR), jets diameter, nozzle to plate distances. There are many prior studies on underexpanded jets available which focus on the influence of flow device design and Mach number. However, most of these studies are concerned with flow visualization and numerical simulation to understand the flow physics. Few studies are concerned with heat transfer aspects of underexpanded jets. In the present study, attempt is ⇑ Corresponding author at: Department of Mechanical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India. E-mail addresses: [email protected] (R. Vinze), [email protected] (S.V. Prabhu). http://dx.doi.org/10.1016/j.applthermaleng.2016.12.046 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.

made to correlate the heat transfer with flow structure visualization of underexpanded jets. It is essential to understand the jet behavior with different initial conditions by studying the heat transfer from jet impingement. Yuceil and Otugen [1] and Yuceil et al. [2] studied under expanded free jet for nozzle pressure ratios ranging from 1 to 20 and Mach number varying from 1 to 3, by measuring centerline velocity decay rates and PIV analysis. A correlation is proposed for calculation of Mach number, jet diameter, density for a jet under fully expanded condition. Heat transfer from impinging underexpanded jets has been studied by many authors thoroughly in various experimental studies. In most of these studies, fluid dynamics was studied by capturing shadowgraphs and Schlieren images, showing shock structures for a wide range of pressure ratios. Crist et al. [3], Henderson [4], Donaldson and Snedeker [5] and Donaldson et al. [6], Lamont and Hunt [7,8] provided comprehensive study to understand influence of underexpanded jets for various boundary conditions. From all these studies, it is conclusive that the flow structure is unaffected from specific heat ratios, nozzle geometry and absolute pressure. Lamont and Hunt [8] suggested a correlation for pressure distribution for under-expanded jets impinging on inclined and perpendicular plates. Meola et al. [9] investigated the jet’s instability in terms of adiabatic temperature and the pressure for the compressible flow. They found that Mach number of around 0.7 is the critical Mach

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R. Vinze et al. / Applied Thermal Engineering 115 (2017) 41–52

Nomenclature A Cp Cd c D d E h I k l MD _ m Nu Nuo NPR Pe Ps P1 p Pr q qconv qnat qjoule

exit area of the nozzle, m2 specific heat of air at constant pressure, kJ/kg K coefficient of discharge velocity of sound, m/s diameter meter of the supply pipe, m diameter of the nozzle, m enhancement factor heat transfer coefficient, W/m2 K current, A thermal conductivity of air, W/m K length of pipe, m design Mach number, M = ve/c mass flow rate, kg/s
 Nusselt number, hd k Nusselt number at the stagnation point Nozzle pressure ratio (Ps/P1) theoretical nozzle exit pressure, Pa supply pressure, Pa ambient pressure, Pa perimeter, m Prandtl number, (lCp/k) heat transfer rate, W/m2 heat carried out by convection from impinging jet, W/ m2 heat carried out by convection from back side of plate, W/m2 total heat supplied, W/m2

number, beyond which instability is invariably present in the jet. Yaga et al. [10] carried out experimental and numerical analysis for the circular (d = 10 mm) and rectangular nozzles (aspect ratio of 3) for under-expanded jets (pressure ratios of 3 and 4.5). It is concluded that the total temperature is a function of total pressure and nozzle to plate distance (z/d). Kim et al. [11,12] and Yu et al. [13] studied the heat transfer due to under-expanded supersonic and sonic jets issued from the convergent- divergent nozzle and convergent nozzle for a wide range of the nozzle pressure ratios from 2.84 to 8.62. The surface pressure and adiabatic temperature measurement was carried out along with the visualization of shock structures. This study was focused on the heat transfer augmentation at the stagnation point and at the jet periphery for smaller separation distances. It was found that turbulence diffusion from the shear layers around the jet edge region induces higher heat transfer rates while existence of low temperature region along the jet edge and at the stagnation point attributes to vortex induced temperature separation. Katanoda et al. [14], Rahimi et al. [15] and Ramanujachari et al. [16] reported experimental result for velocity, pressure drop measurements and shadowgraph for full-expanded and under-expanded impinging jets from axisymmetric supersonic nozzles for Mach number of 1.5 and 5.1. The important conclusion of this work is that the usual method of representing Nusselt number as a function of Reynolds number is inadequate for compressible flows where the dimensional analysis shows that the nozzle Mach number or pressure ratio may also be included. However, they didn’t suggested any correlation. Ewan and Moddie [17] carried out investigation to study flow structure and velocity profile of under-expanded jets. They introduced analytical model to represent the decay of axial velocity from underexpanded sonic jets. Experiments were carried out using shadowgraphs and laser doppler anemometry to determine the near field jet structure and the axial velocity distribution for the complete field over a range of exit nozzle diameters, exit

qloss qradðbÞ qradðf Þ ue R Re r Tj Taw Td Te Tw T0

ve

V z

heat loss by radiation and convection from the plate, W/ m2 heat loss by radiation from the back side, W/m2 heat loss by radiation from the front side, W/m2 velocity at nozzle exit calculated theoretically for choked flow, m/s recovery Factor _ pldÞ Reynolds Number, ðqv e d=l ¼ 4m= radial distance from the stagnation point, m jet temperature, K adiabatic wall temperature, K jet dynamic temperature, K theoretical fluid temperature at nozzle exit, K wall temperature, K jet total temperature, K exit velocity, m/s voltage, V nozzle to plate distance, m

Greek symbols c specific hest ratio q supply density of fluid, kg/m3 qe density of fluid at nozzle exit calculated theoretically (qe = Pe/RTe), kg/m3 l viscosity of fluid, Pas

pressure ratios and jet gases. Inman et al. [18] interpreted the surface pressure profiles using the planar laser-induced fluorescence, which also exhibited the flow structures for the underexpanded supersonic and sonic, free and impinging jets issued from convergent and convergent divergent nozzles. It was concluded that the formation of recirculation region depends upon nozzle to plate distance and nozzle pressure ratio. Limaye et al. [19,20] compared underexpanded jet originated from two contoured nozzle and standard orifice for different NPR and z/d. It is inferred from these study, that heat transfer increases about 25% in case of orifice compared to contoured nozzles. Yu et al. [21] carried out experiments to visualize and analyze under expanded free jets in the case of IC engine injection application. They introduced relations to find mach disk location for under-expanded jets. Pressure distribution and heat transfer distributions were correlated in these studies. Suzuki et al. [22] studied fluid flow structure and oscillations of under-expanded impinging jets. Schlieren photographs were used to visualize fluid flow. Mitchell et al. [23] and Buchmann et al. [24] conducted 3D particle tracking velocimetry for supersonic impinging jets. They reported that the particle velocity and impingement angle are affected by the gas flow, which is in a way depending on the nozzle pressure ratio and stand-off distance. Higher pressure ratios and stand-off distances lead to higher impact velocities and larger impact angles. The literature on underexpanded jets highlights that there is a need for correlations on heat transfer for underexpanded jets to help design engineer to predict the Nusselt number over the surface. Hence, the proposed objectives of this study are as follows
To measure the local Nusselt number and recovery factor distribution for the under-expanded jets.
To capture shock structure over the smooth plate at different nozzle pressure ratios (NPR) by shadowgraph imaging and to correlate this with heat transfer.

R. Vinze et al. / Applied Thermal Engineering 115 (2017) 41–52


To study the influence of diameter of the nozzle on the heat transfer characteristics for under expanded jets.
To provide a correlations for local heat transfer for any diameter at different NPR (Mach number). 2. Experimental setup and data reduction The arrangement of the experimental setup used for present study is shown in Fig. 1a. Compressed air is supplied by compressor (capacity 10 bar at 50 g/s), passes through gate valve, air filter, pressure regulator and calibrated venturi flow meter (b = 0.6). Air filter and pressure regulator are installed to filter the air and to maintain the downstream pressure as desired. The flow is controlled by two needle valves which are installed at upstream and downstream sides of flow meter. The stagnation temperature of air is measured upstream to the nozzle by calibrated K-type thermo couples. The output of the thermocouples is measured by ‘Meco’ millivoltmeter of 0–20 ± 0.5% V and 0 –300 ± 0.5% The Reynolds number (Re) is calculated based on nozzle diameter (d) _ as and mass flow rate (m)

Re ¼

qVd 4m_ ¼ l pld

ð1Þ

Contoured nozzle is installed at the end of supply pipe as shown in Fig. 1a. A pressure gauge just before the nozzle exit is installed to

43

measure supply pressure. The diameter of the nozzles used in present study is 3.6 mm, 5.67 mm and 8.37 mm with contraction ratios (D/d) of 3–7 respectively. The design of these nozzles follow convergent profile as reported by Gibbing [25] and this same profile was also used by Limaye et al. [19] and Rahimi et al. [15] for their investigation of under expended jets. This profile of contoured nozzle (Fig. 2) enables us to achieve smooth pressure drop across nozzle contraction. The local wall temperature (Tw) is measured through IR camera (Ti200-Thermoteknix) in form of thermal images based on its emissivity (0.98), which is positioned at the opposite side of impinging pipe across target plate as shown in Fig. 1a. The target plate of dimensions of 150 mm  130 mm, and 0.082 mm of thickness is used for present study. 8 mm of foil on either side is clamped between bas bars, which ensures firm grip and provides rigid surface for jet impingement. The thickness of foil (0.082 mm) with respect to length and width is infinitesimal which ensure uniform heat flux condition as described by Lytle and Webb [26]. Hence, the local wall temperature recorded on the surface is considered to be same as that on impinging plane. The surface on which thermal images are captured is painted black using a thin coat of Tempil Pyromark ‘Matt finish’ of high emissivity in the order of 0.98. This helps us to achieve higher spatial resolution of temperature compared to thermocouples. The emissivity value is

a. (1) Air filter (2) Air compressor (3) Air receiver (4) Needle valves (5) Air filter (6) Pressure regulator (7) Orifice (8) Differential manometer (9) U-tube manometer (10) Contoured nozzle (11) Impingement assembly (12) Traverse system (13) Infra red camera. (14) Computer

b. (1) Camera (2) Light Rays (3) Traverse system (4) Nozzle (5) Lenses (6) Light Source (7) Impingement plate (8) Table. Fig. 1. Experimental set-up.

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Heat transfer coefficient :

qconv T w  T aw

ð5Þ

where Tw = Wall temperature (°C), Taw = adiabatic wall temperature (°C)

8.37 mm

Nusselt number :

25.4 mm

Nu ¼

hd k

ð6Þ

where k = Thermal conductivity of air (W/mK) The recovery factor (R) is defined as the increase in wall temperature (Tw) due to conversion of kinetic energy into thermal energy. This temperature increase takes place due adiabatic compression and frictional heating as reported by Kim et al. [11].

Fig. 2. Contoured nozzle profile.

Recovery factor :

experimentally calibrated by method explained by Katti and Prabhu [27]. A set of step down transformer (220–16 V) and variace is used to supply AC power to the test plate. The voltage and current for the heater are measured by ‘Meco’ digital meters of 0–20 ±0.5% V and 0–300 ± 0.5% A, respectively. At suitable position voltage taps are fixed in each the bus bars. A traverse system is used to set different nozzle-to-plate distances. Power losses from the exposed surface of the target plate due to natural convection and radiations are estimated experimentally and are included in the calculation of Nusselt number (Eqs. 2–4). The setup for shadowgraph imaging is shown in Fig. 1b. Two Plano-convex lenses of 100 mm diameter and focal length 150 mm are mounted on a bench with traverse mechanism. Light ray originated from light source passes through a slit, then through a pair of Plano-convex lenses. At the other end of lens, a DSLR camera, which is positioned after focal length (Nikon D550, 15MP) captures image. The high resolution of the camera enables us to achieve high accuracy in measurements. In order to calculate the adiabatic temperature (Taw) and the Nusselt number (Nu) simultaneously, each experiment consists of capturing the thermal images for five different heat fluxes (including zero flux condition) provided to the target plate. For each heat flux, five thermal images are recorded and average of temperature recorded by these images on each pixel is taken as the wall temperature to the corresponding heat flexes. After obtaining all the temperatures corresponding to a heat flux, curve fitting is performed for each pixel of the image using MATLAB. Slope of the curve fitting gives reciprocal value of heat transfer coefficient (h) and Y- axis intercept gives value of adiabatic wall temperature (Taw). The radiation and convective heat losses accounted in heat transfer calculations Eqs. (2–4). The convective heat transfer rate between the impinging jet and the target plate, qconv, is estimated as follows:

Total convective heat transfer :

h¼

qconv ¼ qjoule  qloss

R¼

ðT aw  T s Þ ðT aw  T 0 Þ ¼1þ Td Td

ð7Þ

The dynamic temperature is defined as (Ps/P1)

Dynamic temperature : ("

Ps P1

MD ¼

Ps ¼ Pe

 1þ

c1 c

c1

Td ¼

v 2e 2C p

h

¼

i

c1

M 2D i 1 þ c1 M 2D 2 2

h

ð8Þ

)0:5 2 1 c1 #

ð9Þ

c c1

ð10Þ

2

Coefficient of discharge (Cd) for contoured nozzle is calculated with following relation-

Cd ¼

Actual mass flow rate mesured by v enturi meter ½Aqe ue 

ð11Þ

The values of coefficient of discharge at a given NPR for different diameter nozzles are presented in Table 1. It may be concluded that the coefficient of discharge is affected more for smaller diameter nozzle over the range of NPR. However, for larger diameter nozzle, Cd remains almost constant. This may attributed to the thicker boundary layer for smaller diameter nozzle which reduces effective diameter for jet. Uncertainties in measurements for present study are calculated based on method suggested by Moffat [28]. The typical uncertainty in measurement of mass flow rate ranges from 5.95% to 3%. In calculation of Reynolds number the uncertainty is from 6.6% to 3.2%. In measurement and calculation of wall temperature, Nusselt number and Recovery factor the uncertainties are 5%, 7.2% and 2.6% respectively.

ð2Þ

2.1. Comparison with previous studies

Total heat loss due to natural convection and radiation : qloss ¼ qradðf Þ þ qradðbÞ þ qnat Total heat supplied :

ð3Þ

qjoule ¼ VI

To validate the experimental methodology, present results for different contoured nozzles are compared for NPR = 2.4 at nozzle to plate distance (z/d = 6 and 10) with Rahimi et al. [15] and Limaye et al. [19] as shown in Fig. 3. At z/d = 6, comparison shows very good match with the previous results. The difference in the Nusselt

ð4Þ

where V = Supply voltage (V), I = Supply current (Amp) Table 1 Experimental parameters covered in the present study. NPR Ps/P1

2.40 3.05 3.75 4.40 5.10

Under-expansion ratio Pe/P1

1.27 1.61 1.98 2.32 2.69

D = 3.6 mm

D = 5.67 mm

D = 8.37 mm

Cd

Re

Cd

Re

Cd

Re

0.45 0.48 0.51 0.52 0.52

52,100 68,800 90,400 110,500 132,000

0.60 0.61 0.61 0.62 0.64

106,000 136,000 167,000 199,000 240,000

0.65 0.66 0.64 0.64 0.64

164,000 209,000 257,000 302,000 351,000

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R. Vinze et al. / Applied Thermal Engineering 115 (2017) 41–52

physics at lower nozzle to plate distances. The underexpansion ratio (Ps/Pe) and range of Reynolds numbers maintained in the present study for different diameter nozzles are reported in Table 1. The underexpantion ratio (Ps/Pe) is calculated based on isentropic pressure drop calculated between pipe and nozzle exit as given by Eq. (10). 3.1. Under-expanded free jets flow structure study The heat transfer characteristics of under expanded impinging jets are affected by variations in the flow structure in free jet before impingement. This is because of the existence of shock cell structure in the jets. The shock cell are formed due to sudden expansion of fluid. Fig. 4 shows shadowgraphs of free jets for different NPRs (NPR = 2.4–5.1), for 8.37 mm diameter nozzle. Similar results are found for nozzles of other diameters covered in this study. It is observed that expansion waves from nozzle tip are generated in each case. These waves are reflected back from constant pressure jet boundary as compression wave and finally collapse and forms barrel shock. At the tip of barrel shock, Mach disk is formed, behind this region, subsonic flow exists. Similar observations are reported by Kim et al. [12] and Yu et al. [13]. With increase in NPR (Mach number), the length of a shock cell and Mach disk size increases. The first Mach disk position (First cell structure length) for all nozzle configurations at corresponding NPR are reported in Fig. 5 along with other results reported in previous studies [11,12,14,30]. The shock cell lengths measured in the present study compare reasonably well with those reported in the literature. The shock cell structures are repeated until fluid is completely expanded. Across the Mach disk, flow changes from supersonic (MD > 1) flow to subsonic (MD < 1) flow, due to expansion and compression of fluid as reported by Kim et al. [12]. This process of expansion and compression continues in downstream direction until the flow becomes subsonic by turbulent mixing and viscous effects of fluid. From shadowgraph images for different nozzle, it may be concluded that the effect of diameter may be visible in the form of thinner shear layer, curvature of shear layer.

Fig. 3. Comparison of local heat transfer distribution over a smooth surface for NPR = 2.4 at z/d = 6.

number distribution for z/d = 10 at around r/d  1 may be attributed to the jet instability in the stagnation region because of which the surface temperature measured by different investigators may not match with each other. 3. Results and discussion The influence of nozzle pressure ratio (NPR) and nozzle diameters are investigated by conducting experiments for contoured nozzle of different diameters at various NPR (2.4–5.1). The local heat transfer distributions are correlated with shadowgraph images, captured for lower z/d (z/d 6 4) to understand the flow

NPR

2.4

3.05

3.75 Fig. 4. Free jet shadowgraphs for 8.37 mm Nozzle.

4.4

5.1

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R. Vinze et al. / Applied Thermal Engineering 115 (2017) 41–52

Fig. 5. Shock cell length at different NPRs for different nozzle diameter.

3.2. Study on impinging under expanded jets The shock structure formed in the free jet or in flow region before impingement governs fluid physics over the impingement plate for low nozzle to plate distances (z/d < 4). To bring out this fluid physics, experiments are conducted to measure heat transfer distribution and to capture shock cell structure over the test plate by shadowgraph photography technique. Fig. 6 shows shadowgraph images for under-expanded jet impinge upon a smooth flat plate by under expanded jets of 8.37 mm diameter. The shadowgraph results for other nozzle diameters (d = 3.6 mm and 5.67 mm) are found to be similar. Heat transfer measurements are reported in Figs. 7–9 for different NPR and nozzle to plate distances. From Fig. 6. it can be concluded that for lower z/d (z/d < 4) plate shock is present in the stagnation region just above test plate. For z/d of one nozzle diameter at NPR = 2.4 the jet impinge after shock-cell/Mach disk formation in the jet. For other NPR (3.05–5.1) jets impingement takes place before formation of shock-cell. The relative position of plate shock and shock cell over the surface affect local pressure significantly as concluded by Yu et al. [13]. These results show good agreement with pressure distribution reported by Kim et al. [11,12].

z/d

NPR = 2.4

NPR = 3.05

For NPR 4.4 and 5.1, in the stagnation region along with plate shock and reflected shock, a secondary shock layer is also present for z/d = 1 and 2. This secondary plate shock generates periodic fluctuations in surface pressure, reason for these was explained by Kim et al. [11,12] and Yu et al. [13] in details. They reported that in the wall jet region (r/d > 1) the surface pressure fluctuates till it reaches equilibrium. This fluctuation in density or pressure is attributed due to repeated reflection of waves from the jet edges by the upper boundary of the wall jets and plate surface. With the increase in NPR, this structure also becomes denser and plays vital role in governing heat transfer for low z/d. From Fig. 6, it is also conclusive that at z/d = 2, for NPR 2.4–3.75, jets impinge on plate after formation of first shock-cell structure, this results in lower strength of plate shock in these cases. However, for NPR 4.4 and 5.1, the jet impinges on the plate before the formation of shock-cell and is similar to that of z/d = 1 case. A secondary shock layer is present, this layer indicates fluctuation in local pressure at z/d = 2 for higher NPR. For z/d = 4, as the jet moves downstream towards the test plate, the plate shock is absent for lower NPR (NPR 6 3.75) as shown in Fig. 6. This suggests that the fluid pressure has reached nearly ambient pressure. For higher NPR (NPR P 3.75) where the fluid is still expanding downstream, plate shock is captured in shadowgraph images. Kim et al. [12] defined a term expansion length, in a underexpanded jet downstream to this length fluctuation in pressure remains marginal. This may be the reason for the absence of shock cell in case of lower NPR for z/d > 4 cases. 3.3. Local heat transfer distribution Figs. 7–9 show heat transfer and recovery factor distribution over the test plate for jet impinging at different nozzle to plate distances. In the present work, similar to Rahimi et al. [15], the Nusselt number is normalized with respect to Reynolds number (Re0.52) for respective nozzle configurations. It is observed that for n = 0.52, heat transfer distribution for different nozzle diameters collapses on each other in all jet region. The Nusselt number is normalized to bring out effect of nozzle diameter on local heat

NPR = 3.75

NPR = 4.4

1

2

4

Fig. 6. Shadowgraphs for impinging jets at different NPR for nozzle of 8.37 mm diameter.

NPR = 5.10

R. Vinze et al. / Applied Thermal Engineering 115 (2017) 41–52

transfer, importantly in the stagnation region. Figs. 7 and 8 show the pressure distribution as reported by Limaye et al. [19] along with Nusselt number and recovery factor for different nozzle to plate distances at different NPR of 2.4, 3.75 and 5.1. The Nusselt number distribution for various NPR and z/d shows that maximum Nusselt number is measured in the stagnation region (r/d < 1). For lower z/d (z/d 6 4), it is observed that the Nusselt number steeply decreases from stagnation point to the edge of the stagnation region. This drop is induced due to presence of secondary vortex ring at the edge of stagnation region which is induced due to the presence of a vortex at the nozzle exit. Impinging jet structure governs the pressure and flow distribution over the plate which decides the nature of the Nusselt number variation. For lower NPR (NPR 6 3.75) at z/d 6 2, the jet impinges on the plate after the formation of first shock cell (Fig. 6). Due to the formation of the shock cell, the flow velocity over that surface may be higher compared to that for cases where the formation of shock cell is not yet completed (For NPR 4.4 and 5.1 in Fig. 6). Kim et al. [12] and Yu et al. [13] reported similar results highlighting influence of shock cell and pressure distribution on het transfer for different NPRs. For higher NPR (NPR > 3.75) and z/d = 2, jet impinges before the formation of a shock cell (as shown in Fig. 6). In such cases, jet is not fully expanded and results in lower rate of heat transfer. This observation shows good agreement with pressure distribution measurements reported by Limaye et al. [19], where for NPR = 5.1 at z/d = 2, the surface pressure is lower compared to that at NPR = 2.4 and 3.75 which shows that total pressure is grater for lower NPR cases. In the transition region (1 6 r/d P 3), the working fluid accelerates rapidly which results in increase in heat transfer. For lower z/d

47

cases (z/d 6 4) for all nozzle configurations, a secondary peak in heat transfer is measured around r/d  2.5 as shown in Figs. 7–9. In the wall jet region, heat transfer drops uniformly due to the reduction of flow velocity. In the wall jet region (r/d P 3), the normalized Nusselt number is independent of diameter of the nozzle for a given NPR and z/d as shown in Figs. 7–10. As the nozzle plate distance increase (z/d P 6), the jet’s pressure reduces due to expansion and the velocity profile becomes uniform. For z/d P 6, secondary peak in the Nusselt number distribution is not present. So, it may be concluded that the normalization of Nusselt number with respect to Reynolds number (Re0.52) takes care of the influence of diameter. 3.4. Recovery factor distribution Figs. 7–9 also report the distribution of adiabatic wall temperature in terms of recovery factor (Eq. (7)) for various NPR and nozzle to plate distances. For lower z/d (z/d 6 2), the recovery factor remains around unity at the stagnation point and then decreases (around 15%) steeply up to one nozzle diameter i.e., up to the edge of stagnation region. This drop in the recovery factor may be attributed to enhanced cooling effect due to the sudden change in the flow direction and due to the acceleration of expanding fluid, as reported by Kim et al. [11] and Goldstein et al. [29]. With increase in NPR, cooling effect also increases. As the fluid flows away out of stagnation region, the recovery factor increases up to around unity (R  1) at around 2.5 nozzle diameter distance from stagnation point (r/d  2.5). This may be because of turbulent mixing and entrainment of surrounding air in jet flow which increases flow temperature and also recovery factor as reported by Kim et al.

Fig. 7. Nusselt number, recovery factor and pressure distribution over the plate at NPR = 2.4 and 3.05.

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R. Vinze et al. / Applied Thermal Engineering 115 (2017) 41–52

Fig. 8. Nusselt number, recovery factor and pressure distribution over the plate at NPR = 3.75 and 4.4.

Fig. 9. Nusselt number, recovery factor and pressure distribution over the plate at NPR = 5.1.

[11] and Goldstein et al. [29]. The secondary peak in heat transfer coincides with increase in recovery factor as shown in Figs. 7–9. For higher NPR (NPR P 3.75), a secondary decrease in the recovery factor is observed for low z/d. This may be because of the secondary vortex induced in the shock layer in the vicinity of the stagnation region. The surface pressure distribution for lower nozzle to plate

distance (z/d 6 2) as reported by Kim et al. [11] and shadowgraphs imaging results shows good agreement with these observations. Fig. 10 shows the Nusselt number distribution over the test plate for different nozzle to plate distances (z/d) and NPR of jet issuing from an 8.37 mm diameter nozzle. Similar results are observed for other nozzle diameters covered in this study. For

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R. Vinze et al. / Applied Thermal Engineering 115 (2017) 41–52

Fig. 10. Nusselt number distribution for different z/d for 8.37 mm diameter.

nozzle to plate distance (z/d 6 2), a secondary peak in heat transfer is observed. With the increase in the NPR, the Nusselt number increases except for lower nozzle plate distance (z/d 6 2). The reason for this behavior is explained with the help of shadowgraphs and pressure distribution as explained in Section 3.3. For lower z/d up to z/d 6 4, the heat transfer distribution is nonuniform because of instability of jet, as jet impinges on plate within the expansion length as reported by Kim et al. [13]. For z/d P 4, the heat transfer distribution shows consistent drop in the heat transfer in the radial directions. This is due to the absence of shock cells in jet for higher jet to plate distances (z/d). For all these cases, highest rate of heat transfer is measured at the stagnation point. Fig. 11 shows the effect of NPR on recovery factor distribution at respective z/d. The distributions for recovery factor for all three nozzle diameters are identical. For lower nozzle to plate distances z/d 6 4, recovery factor distribution (R) shows a steep drop around the edge of stagnation region. This phenomenon occurs due to enhanced cooling effect due to the sudden change in the flow direction and due to the acceleration of expanding fluid as explained earlier. With the increase in NPR, this effect also increases. With the increase in z/d, the recovery factor increases up to unity (R  1). This increase in recovery factor may be due to viscous heating taking place in the impingement region (r/d  1–1.5) as reported by Kim et al. [11] and Goldstein et al. [29]. Stagnation point Nusselt number (Nuo) variation with respect to z/d is reported in Fig. 12. The distribution of Nusselt number for different nozzle configuration is similar at a given nozzle pressure ratio (NPR). For 8.37 mm diameter nozzle, highest value of stagnation point Nusselt number is measured and for 5.67 mm and 3.6 mm diameter nozzle subsequently lower values are measured. At NPR = 2.4, the variation in the stagnation point Nusselt number (Nuo) over different z/d is lesser compared to that measured at higher NPRs. This may be attributed to the lower strength of the jet at lower NPR (NPR = 2.4).

3.5. Correlation for local Nusselt number From local Nusselt number distribution for z/d (1 6 z/d 6 12), it is observed that for the range of NPR covered in present study, Nusselt number distributions are similar for larger nozzle to plate distances (z/d P 6). Nusselt number decreases uniformly in the radial direction from the stagnation point. These distributions are similar to the distribution reported for low Reynolds number and incompressible jets reported by Katti and Prabhu [27]. Figs. 13 and 14 show comparison among experimental results, Nusselt number predicted from Katti and Prabhu [27] correlations and modified correlations in the present study over different regions. 3.5.1. Local heat transfer in stagnation region (r/d 6 1) Katti and Prabhu [27] proposed correlations for local Nusselt number distribution (for z/d = 0.5–8) for incompressible jets. To predict local Nusselt number in the stagnation region (r/d 6 1) proposed correlations is given in Eq. (12), where a1 and b1 are coefficients, value of these coefficients changes with respect to z/d. 0:5

Nu ¼ a1 Re Pr

0:33

 z 0:11 d

" 1


r 2
z 0:2 #1:2 d

d

b1

ð12Þ

Coefficients reported by Katti and Prabhu [27] predict local Nusselt number in the range of ±45% for z/d 6 4 and ±20% for z/d P 6 of experimental results. In the present study, these coefficients are modified by using regression, and these modified coefficients can predict local Nusselt number within 40% for z/d 6 4 and 15% for z/d P 6 of experimental results (Figs. 13a and 14a). Table 2 shows modified coefficients for Eq. (12) for nozzle to plate distance z/d = 1–12. It may be concluded that for z/d P 6, local heat transfer for compressible jets can be predicted by incompressible jets correlation proposed by Katti and Prabhu [27]. This may be attributed due to similar velocity profiles for both incompressible and compressible jets for z/d P 6, where jets are fully expanded. However,

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Fig. 11. Recovery factor distribution for different z/d for 8.37 mm diameter.

a.

b.

c.

Fig. 12. Stagnation Nusselt number variation for different z/d and different NPR.

Fig. 13. Comparison between experimental results and correlations over different region.

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does not show close agreement with experimental results. However, for z/d P 6, the predicted Nusselt number is within uncertainty limit.

a.

Nu ¼ 0:6635 Re0:52 Nu ¼ 1:14 Re0:52

b.

c.

Fig. 14. Comparison between experimental results and correlations over different region.

Table 2 Modified coefficients for Eq. (12). z/d

1

2

4

6

8

10

12

a1 b1

1.15 4.3

1.26 3.9

1.5 2.7

1.7 2.9

1.67 2.7

1.95 2.75

1.85 2.3

 z 0:0243  r 0:1166 d

d

 z 0:2389  r 0:3093 d

d

for z=d 6 4

for z=d P 6

ð13Þ ð14Þ

3.5.3. Local heat transfer in wall jet region (r/d P 3.5) It is also observed that wall jet region exist for radial distance more than 3.5 nozzle diameter from stagnation point (r/d P 3.5), where the boundary layer becomes turbulent. In this region, the heat transfer behavior is similar to that for flow over flat plate. Katti and Prabhu [27] observed that heat transfer rates estimated from analytical approximation are lower than the experimental results. Hence, they included enhancement factor (E) in analytical expression which was obtained using regression. The analytical correlation proposed by Katti and Prabhu [27] is given in Eq. (15) and the enhancement factor (E) is reported in Table 3. These factors over predict the local Nusselt number by 100% in comparison with the experimental results for z/d 6 4 and for z/d P 6. Hence, for underexpanded compressible jets, enhancement factor (E) is recalculated by regression as reported in Table 3. These modified enhancement factors predict Nusselt number within 15% in comparison with experimental results as shown in Figs. 13c and 14c. From modified enhancement factors (where E 6 1.5) for Eq. (15), it may be concluded that for underexpanded jets in the wall jet region, fluid flow becomes almost equivalent to the flow over a flat plate. This may be due to loss of momentum of fluid in the transition region. Overall, it may be concluded that the proposed correlation predicts experimental results within the uncertainty limit for z/d P 6.

Nu ¼ 0:0436ðEÞRe0:8 Pr0:33

 z 0:0976  z 1:0976 d

d

ð15Þ

4. Conclusions Table 3 Enhancement factors for Eq. (15). z/d

1

2

4

6

8

10

12

‘E’ from Katti and Prabhu [27] Modified ‘E’

2.8 1.45

2.6 1.4

2.4 1.3

2.35 1.35

2.3 1.2

1.2

1.1

for z/d 6 4, due to highly non uniform distribution for compressible jets, present correlations do not show good agreement (±40%) with the experimental results. 3.5.2. Local heat transfer in transition region (1 6 r/d 6 3.5) For compressible jets, transition region exists from 1 to 3–3.5 nozzle diameters from stagnation point compared to that for incompressible jets for up to 2.5 nozzle diameters. In the transition region (1 6 r/d 6 3.5), the correlation reported by Katti and Prabhu [27] over predict local Nusselt number in the range of 0–60% with respect to experimental results for z/d 6 4 and that in the range of 10 to 50% for z/d P 6. In the present study, correlations are modified using regression to predict local Nusselt number is reported by Eqs. (13) and (14). For z/d 6 4 and for z/d P 6, Eqs. (13) and (14) predict local Nusselt number comparable to experimental results within ±25% and ±15% respectively as shown in Figs. 13b and 14b. In the transition region, for z/d 6 4 may due to highly non uniform distribution of Nusselt number, correlation

Experiments are performed to investigate the effect on heat transfer from under-expanded jets issuing from different diameter nozzles, for different NPR at different nozzle to plate distances. In the present study NPR covered raging from 2.4 to 5.1. Following are the conclusions from the experimental study
Nusselt number distribution for different NPR are non-uniform especially in the stagnation region for z/d 6 4. This is attributed due to the presence of shock-cell and plate shock in flow region. As the NPR increases, the length of shock cell increases and accordingly distribution of Nusselt number changes.
Shadowgraph images for different NPR show that for lower z/d (z/d 6 4), plate shock exists in the fluid flow just above the plate. This plate shock significantly influences the local heat transfer over the plate.
Normalization of Nusselt number with respect to Reynolds number (Re0.52) considers the account of the influence of diameter.
Recovery factor is influenced by the change in NPR. With the increase in NPR, recovery factor shows significant change (around 10%) in the stagnation region for lower z/d. However, in case of higher z/d, recovery factor remains almost constant around unity.
For higher z/d (z/d P 6), due to the absence of shock structure in the jet stream and uniform velocity profile, maximum heat

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transfer and recovery factor is measured at the stagnation point. For higher z/d (z/d P 6), the Nusselt number decreases uniformly from the stagnation point in the radial directions.
In the wall jet region, the Nusselt number of compressible jets is almost half of that predicted from incompressible jet’s correlations. However, in the stagnation region and transition regions, the Nusselt number of compressible jets is higher compared to that predicted from the incompressible jet’s correlations.

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