A numerical investigation and design optimization of impingement cooling system with an array of air jets

International Journal of Heat and Mass Transfer 108 (2017) 880–900

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A numerical investigation and design optimization of impingement cooling system with an array of air jets Prasanth Anand Kumar Lam, K. Arul Prakash ⇑ Fluid Mechanics Laboratory, Department of Applied Mechanics, IIT Madras, Chennai 600036, India

a r t i c l e

i n f o

Article history: Received 20 June 2016 Received in revised form 24 October 2016 Accepted 7 December 2016

Keywords: Impinging jet array SUPG-finite element method Entropy generation Proper orthogonal decomposition Multi-objective genetic algorithm

a b s t r a c t In the present study, fluid flow, heat transfer and entropy generation in impingement cooling system with an array of air jets for different values of Reynolds number (Re), Velocity Ratio (VR) and Channel Height (H/L) are investigated. The magnitude of overall Nusselt number (Nuov ) and global total entropy generation (Stot;X ) is found to increase with increasing Re; VR and decreasing H/L. Further, spectral and proper orthogonal decomposition analyses are performed to analyze spatio-temporal dynamics of vortex structures for unsteady configurations of impingement cooling system. It is observed that, along the interface of jet (both primary and secondary) and ambient fluid, the destabilizing effect of shear forces overcome the stabilizing effect of momentum diffusion. This results in evolution of counter-rotating vortex rings along the interfaces of jet and ambient fluid due to shear layer instability. Finally, MultiObjective Genetic Algorithm (MOGA) has been implemented to obtain optimum configurations of impingement cooling system where a trade-off between two performance parameters, Nuov and Stot;X is obtained. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Heat transfer enhancement in thermal systems is indeed an important field of engineering research. The reason may be due to its contribution towards evolution of highly efficient cooling technologies to meet-up with rapid growth in energy demand as well as a consequential mitigation of environmental degradation. Hence, for sustainable future, considerable emphasis is in-need to develop modern and enhanced cooling techniques in various thermal equipment. In this regard, jet impingement cooling is considered to be one of the promising heat transfer augmentation techniques among the various single-phase heat transfer processes. This is due to its inherent ability to achieve high heat transfer rates with relatively low pressure drop. As an obvious implication, impinging jets are used in several industrial and engineering applications in automotive, aerospace, electronic and process industries includeing drying of papers and food products, tempering and shaping of glass, annealing of plastic and metal sheets, anti-icing of aircraft wings, cooling of heated components in miniaturized electronic devices and gas turbines etc. A representative review on mechanisms of convective transport and potential applications for impinging jets is reported in last several decades [1,2]. ⇑ Corresponding author. E-mail address: [email protected] (K.A. Prakash). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.12.017 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

In the literature, numerous experimental, analytical and computational studies have been performed and reported on fluid flow, heat and mass transfer distributions under single impinging jets [3–5]. Recently, Zukowski [6] experimentally investigated the variaton of local and average Nusselt number along flat plate with confined impinging slot jet. A correlation for non-dimensional parameter Lx (distance from the jet axis to the location where the second rapid decrease in Nusselt number occurs) with regard to the development of optimal model of air solar collectors is proposed. Kalifa et al. [7] conducted an experimental study to analyze the flow structures and thermal development behavior of a circular air jet impinging on flat plate subjected to a variable temperature using Particle Image Velocimetry (PIV) and Laser Dopler Velocimetry (LDV). The effect of Reynolds number, jet-impingement plate distance and impingement plate temperature are studied and observed a tendency of the fluid to rise upward at high impingement plate temperature under the influence of buoyancy forces. Although single impinging jets yield very high localized heat and mass transfer coefficients in the stagnation zone, the overall cooling performance drops quite rapidly as fluid flows downstream the stagnation region, along the impingement surface. For this reason, impinging jets are often used as arrays in industrial applications including gas turbine cooling [8], food processing [9], mixing in combustion chambers [10], materials processing [11] and electronic cooling [12] etc. Many studies have been reported on fluid flow dynamics and heat transfer under multiple impinging jets

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Nomenclature a Be Br H k L Nu Nu Nuov n p p Pr Re Sh Sh;X Sw Sw;X Stot Stot;X Tc Th T0 T0 t u, v

height of the heat source, m local Bejan number Brinkman number (lV21 =kDT ref ) height of the channel, m thermal conductivity, W m1 K1 length of the heat source, m local Nusselt number surface averaged Nusselt number of individual heat source overall surface averaged Nusselt number of all heat sources directional normal dimensional pressure, N m2 dimensionless pressure Prandtl number Reynolds number local entropy generation due to heat transfer global entropy generation due to heat transfer local entropy generation due to fluid friction global entropy generation due to fluid friction local total entropy generation global total entropy generation dimensional temperature of slot jet inlet, K dimensional temperature of the heat source, K dimensional bulk temperature, K non-dimensional bulk temperature dimensional time, s dimensional velocity components in x and y directions, m s1

[13–16]. For instance, Yang et al. [17] studied unsteady fluid flow structure and heat transfer of an array of ten impinging laminar jets and observed self-oscilation of jets caused by Kelvin-Helmholtz instabilities. The unsteadiness associated with vortex oscillations was quantified from spectrum analysis of different flow quantities (velocity, pressure and temperature) and observed an enhancement in heat transfer caused by unsteadiness in the flow field. Shariatmadar et al. [18,19] performed experimental investigation and measured local Nusselt number distribution along an isothermal flat plate with an array of impinging air-slot jets using MachZehnder interferometer. The influence of jet slot width, jet Reynolds number and jet-to-jet spacing on local and average Nusselt number for different (Single, Double, Triple and Quadrapule) jet arrays was studied and correlations for surface averaged Nusselt number for different jet arrays were derived. Huzayyin et al. [20] carried out an experimental investigation for impingement cooling of an array of flush mounted heat sources by a row of slotted air jets. The effect of thermal wake on downstream heat sources was quantified and correlations were proposed based on jet Reynolds number, position of the block with respect to the jet impingement point and the separation distance between the orifice plate and the impingement surface. Luo et al. [21] performed experimental investigation and evaluated the cooling performance of closed loop microjet cooling system for thermal management of LED chips. The cooling effect under different conditions such as different micropump flow rates and different input powers was investigated and an enhanced cooling performance with increasing pump flow rate was observed. Kim et al. [22] analyzed thermal performance of staggered jet-convex dimple array cooling system and performed optimization for thermal resistance with dimple diameter and height as design variables. It was found that, for the optimal design, a lower thermal resistance was achieved with a slight increase in pressure drop.

Uh u; v V1 V2 VM VR Wh W x, y XP x; y

discontinuous upwind perturbation dimensionless velocity components in x and y directions dimensional primary jet inlet velocity, m s1 dimensional secondary jet inlet velocity, m s1 non-dimensional velocity magnitude ratio of secondary and primary jet inlet velocities (V2 =V1 ) continuous weighting function jet inlet width, m dimensional Cartesian coordinates, m dimensional distance along the periphery of heat source, m dimensionless Cartesian coordinates

Greek letters q density of fluid, kg m3 a thermal diffusivity, m2 s1 h dimensionless temperature m kinematic viscosity, m2 s1 s dimensionless time X non-dimensional area of the computational domain Subscripts 0 reference case max maximum tot total

Thermodynamic analysis of convective heat transfer systems (viz. heat exchangers, thermal energy storage, solar collectors and thermal insulation systems etc.) was first reported by Bejan [23]. In the literature, several studies have been reported on entropy generation under single impinging jet [24–26]. Recently, Chen and Zheng [27] performed numerical analysis on impinging flow confined by planar opposing jets to study the effect of Reynolds number and distance between opposing jets on flow field and entropy generation. It was found that, the total entropy generation increases exponentially with Reynolds number whereas it decreases in power function with increasing jet-jet distance. To the best of authors’ knowledge, investigations on steady state and/or unsteady thermo-fluid dynamics combined with thermodynamic analysis and design optimization for impingement cooling system with an array of air impingng jets has not been reported in spite of its importance in localized and micro-sized cooling devices for the thermal control in electronic applications [21,28]. The objective of present study is to analyze fluid flow, heat transfer and entropy generation of impingement cooling system with an array of air jets, which is a representative of typical configuration encountered in electronic cooling applications [21,28]. A Streamline Upwind Petro-Galerkin (SUPG) based finite element algorithm has been employed to solve Navier-Stokes equations for different values of Reynolds number (Re = 100–1000), channel height (H=L ¼ 0:5–1.0) and velocity ratio (VR = 0.0–1.0). The effects of various design parameters (viz. Re; VR and H/L) on fluid flow, heat transfer and entropy generation are studied. Furthermore, spectral and proper orthogonal decomposition analyses are performed to analyze spatio-temporal dynamics for unsteady fluid flow configurations of jet impingement cooling systems. Subsequently, overall Nusselt number (Nuov ) and global total entropy generation (Stot;X ) are selected as objective functions and optimization of objective functions is performed using

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Multi-objective Genetic algorithm (MOGA) to obtain optimum configuration for impingement cooling system with maximum overall Nusselt number and minimum entropy generation.

The local Nusselt number (Nu) along surface of the heat sources is calculated as

Nu ¼ 2. Mathematical modeling 2.1. Physical domain The physical domain under investigation is a two-dimensional channel containing an array of discrete protruding heat sources along the impingement plate as shown in Fig. 1. The geometrical features of the domain are, L = 0.04 m, a/L = 0.125, S=L = 0.2, W=L = 0.5, Le = 0.48 m, similar to Lam and Prakash [25,26]. Air (Pr = 0.71) impinges normally against the protruding heat sources with uniform velocity and temperature. The heat sources placed on the impingement plate are maintained at high temperature whereas other surfaces are kept at adiabatic condition. 2.2. Governing equations The fluid flow is assumed to be viscous, two-dimensional, time-dependent, incompressible and laminar with constant thermo-physical properties. The effect of gravity, radiation and viscous dissipation are neglected. By employing the aforementioned approximations, the governing equations for conservation of mass, momentum and energy for forced convection in jet impingement cooling system can be expressed as: Continuity Equation

@u @ v þ ¼0 @x @y

ð1Þ

Momentum Equations

! @u @u @u @p 1 @ 2 u @ 2 u þv ¼ þ þu þ @s @x @y @x Re @x2 @y2 @v @v @v @p 1 @ 2 v @ 2 v þu þ þv ¼ þ @y Re @x2 @y2 @s @x @y Energy Equation

@h @h @h 1 @2h @2h þu þv þ ¼ @s @x @y RePr @x2 @y2

ð2Þ

! ð3Þ

x ; W

y¼

y ; W

u¼

V2 T  Tc ;h ¼ ; VR ¼ V1 DTref Re ¼

V1 D

m

;

Pr ¼

m a

u ; V1

v¼

v ; V1

DTref ¼ Th  Tc ;

Nu ¼

ð4Þ

p¼

1 A

Nu dA

p

;

s¼

Stot k=W

2

t ; W=V1

ð7Þ

A

Stot

(   2 ) 2 @T @T ¼ 2 þ @x @y T0 "    2 !  2 # 2 l @u @v @v @u þ þ 2 þ þ @x @y @x @y T0 k

ð8Þ

The first term in Eqn. (8) is entropy generation due to heat transfer irreversibility, whereas second term is entropy generation due to fluid friction irreversibility resulting from viscous dissipation. The dimensionless form of local entropy generation due to heat transfer (Sh ) and fluid friction (Sw ) is written as,

2  2 # @h @h þ @x @y

( "

Sw ¼ v 2

qV21

Stot ¼

Z

where A is the total surface area of the heat source for the calculation of Nu. According to thermodynamics, convective heat transfer in a thermodynamical systems results in irreversibility and generates the entropy. Entropy generation, a modern approach for optimization of a thermal systems, is based on the second law of thermodynamics and is used as a parameter for evaluating the efficiency of thermal system. Entropy generation is due to irreversibilities in thermal system and results in destruction of exergy (or loss of useful work). The associated irreversibilities in these systems are due to non-isothermal heat transfer and fluid friction. The local entropy generation during jet impingement cooling of discrete protruding heat sources is written as [25–27,29–31],

Sh ¼

!

ð6Þ

where n is the unit outward normal to the block surface at the fluidsolid interface, i.e. n = y for the top surface of the heat source, n = x at right and n = +x at left surfaces of the heat source. The local Nusselt number is calculated from left face to right face along the periphery of the heat source. The surface averaged Nusselt number (Nu) is calculated as

"

The dimensionless variables may be defined as

x¼

L @h W @n

2  2 #  2 ) @u @v @ v @u þ þ þ @x @y @x @y

ð9Þ

ð10Þ

The corresponding irreversibility distribution ratio (v) is defined as

v ¼ BrT 0 ¼

; ð5Þ

lV21

T0 kDTref DTref

ð11Þ

In the present study, the irreversibility distribution ratio (v = Br T 0 ) is assumed to be unity, a similar value which was used in earlier studies [25–27].

Fig. 1. Computational domain with boundary conditions.

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The local total entropy generation (Stot ) in enclosure is given by the summation of local entropy generation due to heat transfer and fluid friction.

Stot ¼ Sh þ Sw

ð12Þ

The relative dominance of entropy generation due to heat transfer and fluid friction is given by local Bejan number (Be), a dimensionless parameter defined as

Be ¼

Sh Stot

ð13Þ

The global entropy generation due to heat transfer and fluid friction can be obtained by integrating the local entropy generation rates (Sh and Sw ) over the domain X.

Z

Sh;X ¼

X

Z Sw;X ¼

X

Sh dX

ð14Þ

Sw dX

ð15Þ

The global total entropy generation (Stot;X ) in the field is defined as the summation of global entropy generation due to heat transfer and fluid friction over the whole domain X. The expression for global total entropy generation is given by,

Stot;X ¼ Sh;X þ Sw;X

ð16Þ

The global Bejan number (Beav g ) is given as

Beav g ¼

Sh;X Stot;X

ð17Þ

Therefore, at Be !0.0 the contribution in entropy generation is solely dominated by fluid friction irreversibility. In contrast, Be !1.0 is the opposite limit at which, entropy generation is dominated by heat transfer irreversibility. If Be = 0.5, the entropy generation due to heat transfer irreversibility and fluid flow irreversibility are of equal importance. 2.3. Boundary conditions The following boundary conditions are imposed for the present domain of interest (Fig. 1): a. Along primary jet inlet:

u ¼ 0; v ¼ 1:0;

@p ¼ 0; @n

h ¼ 0;

b. Along secondary jet inlet:

u ¼ 0;

@p ¼ 0; @n

v ¼ VR;

h ¼ 0;

c. Confinement plate except jet inlet width:

u ¼ v ¼ 0;

@p ¼ 0; @n

@h ¼ 0; @n

d. Impingement plate:

u ¼ v ¼ 0;

@p ¼ 0; @n

@h ¼ 0; @n

e. Symmetry plane:

u ¼ 0;

@v ¼ 0; @n

@p ¼ 0; @n

@h ¼ 0; @n

f. Along the heat source:

u ¼ v ¼ 0;

@p ¼ 0; @n

h ¼ 1:0;

883

f. Channel outlet:

@u ¼ 0; @x

@v ¼ 0; @x

p ¼ 0;

@h ¼ 0; @x

2.4. Solution methodology, code validation and grid independence study The equations governing fluid flow and thermal energy are solved iteratively by using Eulerian velocity correction approach. An explicit time-stepping with the Streamline Upwind PetrovGalerkin (SUPG) formulation of Brooks and Hughes [32] is employed to solve laminar forced convection during jet impingement cooling of discrete protruding heat sources. The detailed weak formulation and finite element formulation has been presented in Prakash et al. [33]. The present algorithm has been validated for several experimental and numerical benchmark cases of fluid flow and heat transfer problems such as lid driven cavity, backward facing step [33], impinging jet [25,26], buoyancy driven convection and entropy generation without and with porous medium [31]. In addition, the present algorithm is also validated for twin laminar air jets impinging on to an isothermal flat plate, similar to Sharif [34]. The following boundary conditions are imposed for the validation case: Along both the jet inlets, flow is assumed to be uniform i.e. u = 0.0, v = 1.0, h = 0.0. Along the isothermal impingement plate @h = 0.0), no-slip condi(h = 1.0) and adiabatic confinement plate (@n tions are imposed i.e. u = 0.0, v = 0.0 and at channel outlet p = 0.0 is assumed. The flow parameters are Reynolds number (Re) = 100, normalized channel height is, Hj = 4.0, normalized distance between twin jets is, Lj = 4.0. Figs. 2(a) and (b) illustrate the comparison of local Nusselt number and skin friction coefficient along impingement plate with the results of Sharif [34]. The variation of Nusselt number and skin friction coefficient is found to be in excellent agreement with Sharif [34] as shown in Fig. 2(a) and (b). From these results, the present numerical algorithm is found to be reliable for the prediction of fluid flow and heat transfer in impingement flow configurations with an array of air jets. In order to ensure a grid independent solution for jet impingement cooling system (Re = 1000, VR = 1.0 and H/L = 0.75), a grid sensitivity analysis has been performed, for three different meshes of size 331  31, 366  51 and 401  71 along x and y directions respectively. As one moves from coarse grid (366  51) to fine grid (401  71), the maximum deviation in the overall Nusselt number (NuOv ) and global total entropy generation (Stot;X ) is found to be less than 2:21%. Hence, in the present study the grid consisting of (366  51) nodes is chosen for all computations. 3. Results and discussion In the present study, numerical simulations have been carried out to predict fluid flow, heat transfer and entropy generation in jet impingement cooling system with an array of air (Pr = 0.71) jets, for different values of Reynolds number (Re = 100–1000), nondimensional channel height (H=L ¼ 0:5–1.0) and velocity ratio (VR = 0.0–1.0). The geometrical parameters of the present configuration are representative of typical parameters encountered in electronic cooling applications [12,25,26]. Furthermore, spectral and proper orthogonal decomposition analyses are performed to analyze spatio-temporal dynamics of vortex structures for unsteady fluid flow configurations of jet impingement cooling system. Finally, overall surface averaged Nusselt number and global total entropy generation are selected as objective functions and

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Fig. 2. Comparison of (a) skin-friction coefficient and (b) Nusselt number distribution along impingement plate with Sharif [34] at Hj = 4.0 and Lj = 4.0.

Multi-Objective Genetic Algorithm (MOGA) has implemented to achieve optimum configurations with higher overall Nusselt number and low entropy generation simultaneously.

values of Reynolds number (Re = 100, 300, 500, 800, 1000), nondimensional channel height (H/L = 0.50, 0.75, 1.0) and velocity ratio (VR = 0.0, 0.25, 0.50, 0.75, 1.0).

3.1. Steady state and time averaged fluid flow and thermal fields

3.1.1. Effect of velocity ratio Fig. 3 depicts the streamline patterns for five different values of velocity ratio (VR = 0.0 (Single jet case), 0.25, 0.50, 0.75 and 1.0) at Re = 1000 and H/L = 0.75. From Fig. 3(a), it is observed that, fluid enters from primary jet, located along the confinement plate and

In this section, steady state and time averaged fluid flow pattern, heat transfer and entropy generation for jet impingement cooling system with an array of air jets is presented for different

Fig. 3. Effect of VR on Streamlines for jet impingement cooling system at (a) VR = 0.0 [25], (b) VR = 0.25, (c) VR = 0.50, (d) VR = 0.75 and (e) VR = 1.0 for Re = 1000 and H/ L = 0.75.

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positioned over first heat source. As primary jet impinges onto the first heat source, a counter-rotating vortex (primary recirculation bubble) is formed in the vicinity of both confinement surface and primary jet as shown in Fig. 3(a). This may be due to shear force generated along the interface of high velocity jet and ambient fluid in the channel. It is also observed that, downstream the primary recirculation, flow is biased towards confinement surface caused by constriction in flow passage. This lead to an adverse pressure gradient over the upper surfaces of fourth and fifth heat sources, which may eventually results in formation of single elongated vortex ring as shown in Fig. 3(a). The formation of elongated vortex ring lead to accumulation of heat over fourth and fifth heat source resulting in hot spots. A tertiary recirculation bubble is also observed along confinement surface downstream the elongated vortex ring due to constriction in flow passage. Also, small recirculation bubbles are observed in the cavities between the heat sources due to the shear driven interaction by the main flow on fluid entrapped in the cavities. It is observed that, unlike the case of single jet (Fig. 3(a)), the presence of secondary jet, located on the confinement surface and positioned over third heat source, squeeze the fluid from primary jet. This results in reduction of fluid to bias towards the confinement surface leading to a favorable pressure gradient along the impingement surface. This eventually cause the elongated vortex ring formed over fourth and fifth heat source for single jet case (Fig. 3(a)) to convect downstream the fifth heat source, which may otherwise lead to formation of hot spots over the heat sources as illustrated in Fig. 3(b)–(e). Subsequently,

885

a secondary recirculation bubble is observed along the confinement surface, positioned downstream the secondary jet. It is also observed that, as VR increases from 0.25 to 1.0, the length of secondary recirculation bubble increases. This may be due to enhanced shear force produced by secondary jet on ambient fluid. It is found that, with the presence of secondary jet, primary recirculation bubble extends up to the secondary jet, resulting in, an evolution of hovering vortex. This may be due to generated shear interaction along the interface of primary recirculation bubble and secondary jet as shown in Fig. 3(b)–(e). At VR P0.5, a tertiary recirculation bubble is also observed along confinement surface downstream the secondary recirculation. Also, as VR increases from 0.0 to 1.0, the velocity magnitude is observed to increase due to enhanced convection. Fig. 4(a)–(e) demonstrate the pressure contours for VR = 0.0 (Single jet case), 0.25, 0.5, 0.75 and 1.0 at Re = 1000 and H/L = 0.75. A flow stagnation is observed in the vicinity of first heat source, where pressure builds up resulting in fluid to flow towards adjacent heat sources. It is also observed that, as VR increases from 0.0 to 1.0, the maximum pressure at the stagnation region increases. A low pressure zone is observed in the regions of recirculation as illustrated in Fig. 4(a)–(e). Fig. 5(a)–(e) illustrate the effect of velocity ratio (VR = 0.0 (Single jet case), 0.25, 0.50, 0.75 and 1.0) on local Nusselt number distribution along all the five heat sources at Re = 1000 and H/L = 0.75. It is observed that, as secondary jet has no interaction with first and second heat sources, the variation of Nu is negligible

Fig. 4. Effect of VR on pressure distribution for jet impingement cooling system at (a) VR = 0.0 [25], (b) VR = 0.25, (c) VR = 0.50, (d) VR = 0.75 and (e) VR = 1.0 for Re = 1000 and H/L = 0.75.

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Fig. 5. Effect of VR on local Nu distribution along (a) First, (b) Second, (c) Third, (d) Fourth and (e) Fifth heat source.

with increase in VR as shown in Fig. 5(a) and (b). It is also observed that, on top surface of first heat source (flow stagnation region), for all the values of VR; Nu is maximum due to higher velocity and thermal gradients as shown in Fig. 5(a). Similarly, along top surface of second heat source, a maximum value of Nu is observed at the upper corner of leading edge. Further, a decay in Nu is observed in streamwise direction (0.005 6 XP 6 0.045) due to the development of thermal boundary layer as shown in Fig. 5(b). Along top surface of third heat source, at VR = 0.0 (Single jet case), Nu is found to be maximum at upper corner along leading edge and further reduces in streamwise direction (0.005 6 XP 6 0.045). At VR > 0.0 (i:e: with the presence of secondary jet), a marginal increase in Nu along first half (0.005 6 XP 6 0.025) of third heat source is observed. In streamwise direction, as warmer fluid convects over second half (0.025 6 XP 6 0.045) of third heat source, a marginal reduction in Nu is observed as illustrated in Fig. 5(c). Along fourth and fifth heat sources, Nu is found to be maximum at the upper corner of leading edge and decays in streamwise direction (0.005 6 XP 6 0.045) as depicted in Fig. 5(d) and (e) Along all the side surfaces i.e leading edge (0.0 6 XP 6 0.005) and trailing edge (0.045 6 XP 6 0.05) of all the heat sources, a local maximum value of Nu is observed at both the sharp upper corners (i.e. XP = 0.005 for leading edge and XP = 0.045 for trailing edge). At the other locations of all side surfaces of heat sources except trailing edge of fifth heat source, the variation of Nu is found to be marginal due to low heat transfer from the fluid entrapped in the cavities between the heat sources to the main flow. As VR increases from 0.25 to 1.0, the magnitude of Nu is found to increase, along third, fourth and fifth heat sources due to reduced boundary layer thickness caused by an increased secondary jet velocity. Finally, an enhancement in Nu is observed along third, fourth and fifth heat sources with the presence of secondary jet (VR > 0.0) compared to single jet case (VR = 0.0).

Fig. 6 illustrates the corresponding local total entropy generation contours for VR = 0.0 (Single jet case), 0.25, 0.5, 0.75 and 1.0 at Re = 1000 and H=L ¼ 0:75. It is observed that, in the regions of recirculation and the line joined by locus of maximum streamwise velocities, local total entropy generation is low due to marginal velocity gradients. In contrast, along the solid walls (confinement surface, impingement surface and heat sources) except the regions of recirculation, magnitude of total entropy generation is significantly higher. It is also observed that, as the velocity of secondary jet increases with increase in VR, the shear driven interaction between secondary jet and main flow increases causing an increase in viscous dissipation resulting in an increase of total entropy generation.

3.1.2. Effect of Reynolds number Fig. 7 shows the streamline pattern for five different values of Reynolds number (Re = 100, 300, 500, 800 and 1000) at H=L ¼ 0:75 and V:R ¼ 0:75. It is observed that, as Re increases from 100 to 1000, the primary recirculation bubble formed in the vicinity of both confinement surface and primary jet gain strength resulting in splitting up of tiny recirculation bubbles. As Re increases, the maximum value of velocity magnitude (VMmax ) is observed to increase. The length of secondary recirculation bubble formed along the confinement surface, positioned downstream the secondary jet is found to increase with increase in Re. This may be due to enhanced shear force produced by secondary jet on ambient fluid. At Re P 300, primary recirculation bubble is found to extend up to the secondary jet, resulting in, an evolution of hovering vortex. At Re P 800, a tertiary recirculation bubble is formed along the confinement surface located downstream of secondary recirculation and the size of tertiary recirculation is observed to increase with increasing Re. Along fifth heat source, at the upper corner of

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Fig. 6. Effect of VR on local total entropy generation (Stot;X ) for jet impingement cooling system at (a) VR = 0.0 [25], (b) VR = 0.25, (c) VR = 0.50, (d) VR = 0.75 and (e) VR = 1.0 for Re = 1000 and H/L = 0.75.

trailing edge, flow separates leading to the formation of recirculation bubble and observed an increase in length with increasing Re. Fig. 8(a)–(e) illustrate the local Nusselt number distribution along all five heat sources for various values of Reynolds number (Re = 100, 300, 500, 800 and 1000) at H=L ¼ 0:75 and V:R ¼ 0:75. It is observed that, as Re increases from 100 to 1000, an enhancement in Nu along the top surfaces (0.005 6 XP 6 0.045) of all the heat sources due to reduced thermal boundary layer thickness. It is also observed that, as Re increases, the variation of Nu along the side surfaces (both leading and trailing edges) of all the heat sources, except trailing edge of fifth heat source, is found to be marginal. Along trailing edge of fifth heat source (0.045 6 XP 6 0.05), unlike other heat sources, as Re increases, the variation in Nu is observed to increase. This is due to enhanced shear interaction between the main flow and recirculation bubble downstream the fifth heat source. Fig. 9 depicts the effect of Reynolds number (Re = 100, 300, 500, 800 and 1000) on local total entropy generation during jet impingement cooling of discrete protruding heat sources with an array of air jets at H/L = 0.75 and V:R = 0.75. It is found that, the magnitude of maximum total entropy generation (Stot;max ) increases with increase in Re. Also, active regions of Stot increases due to increase in contributions from both heat transfer and fluid friction irreversibilities.

3.1.3. Effect of channel height Fig. 10 depicts the streamline patterns for three different nondimensional channel heights (H/L = 0.5, 0.75 and 1.0) at Re = 500

(Fig. 10 (i)) and Re = 1000 (Fig. 10 (ii)) when VR = 0.75. It is observed that, as fluid enters from both primary and secondary jets, counter-rotating vortices (primary and secondary recirculations) are formed in the neighborhood of confinement surface and inlet jets. This is due to the generated shear force on ambient fluid in the channel as shown in Fig. 10. It is also observed that, for both Re = 500 and 1000, as H/L increases from 0.5 to 0.75, the primary recirculation to extend up to the secondary jet. This may be due to shear interaction between primary jet and ambient fluid in channel. At Re = 1000, as H/L increases from 0.50 to 0.75, a tertiary recirculation is formed adjacent to the confinement surface is observed as shown in Fig. 10(ii,b). As H/L further increases from 0.75 to 1.0, the primary vortex is observed to gain strength resulting in splitting up of tiny recirculation bubbles. Also, for both Re = 500 and 1000, as H/L increases from 0.75 to 1.0, the size of hovering vortex is observed to increase. At H/L = 1.0, secondary recirculation is observed to merge with the tertiary recirculations forming a single elongated vortex ring adjacent to confinement surface as shown in Fig. 10(ii,c). Fig. 11(a)–(e) demonstrate the effect of channel height (H/L = 0.5, 0.75 and 1.0) on local Nusselt number distribution along all the five heat sources at Re = 500 and 1000 when VR = 0.75. It is observed that, along the upper surface of the first heat source (stagnation region), Nu is found to be maximum due to higher velocity and temperature gradients. It is also observed that, for both Re = 500 and 1000, Nu along the stagnation region decreases with increasing H/L as shown in Fig. 11(a). Along the upper surface of second heat source, for both Re = 500 and 1000, Nu is found to be maximum at the upper corner of leading edge and decays in

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Fig. 7. Effect of Re on Streamlines for jet impingement cooling system at (a) Re = 100, (b) Re = 300, (c) Re = 500, (d) Re = 800 and (e) Re = 1000.

Fig. 8. Effect of Re on local Nu distribution along (a) First, (b) Second, (c) Third, (d) Fourth and (e) Fifth heat source.

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Fig. 9. Effect of Re on local total entropy generation (Stot;X ) for jet impingement cooling system at (a) Re = 100, (b) Re = 300, (c) Re = 500, (d) Re = 800 and (e) Re = 1000.

Fig. 10. Effect of H/L on Streamlines for jet impingement cooling system at (i) Re = 500 and (ii) Re = 1000 when (a) H/L = 0.5, (b) H/L = 0.75, (c) H/L = 1.0.

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Fig. 11. Effect of H/L on local Nu distribution along (a) First, (b) Second, (c) Third, (d) Fourth and (e) Fifth heat source.

Fig. 12. Effect of H/L on local total entropy generation (Stot;X ) for jet impingement cooling system at at (i) Re = 500 and (ii) Re = 1000 when (a) H/L = 0.5, (b) H/L = 0.75, (c) H/ L = 1.0.

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Fig. 13. Map of fluid flow regimes at different Re and VR for (a) H/L = 0.5, (b) H/L = 0.75, and (c) H/L = 1.0. Marker attributes: Circle – Steady state regime; Diamond – Unsteady periodic regime.

the streamwise direction (0.005 6 XP 6 0.045). The variation of Nu along second heat source with increasing H/L is found to marginal as illustrated in Fig. 11(b). Along the third heat source, at both Re = 500 and 1000, due to the presence of secondary jet, thermal boundary layer thickness reduces at 0.005 6 XP 6 0.025 which eventually lead to increase in Nu. In contrast, at 0.025 6 XP 6 0.045, a marginal reduction in Nu is observed as shown in Fig. 11(c). As H/L increases, along both fourth and fifth heat sources, for Re = 500 and 1000, Nu found to reduces as depicted in Fig. 11(d) and (e) respectively. Fig. 12 illustrates the corresponding local total entropy generation contours for H/L = 0.5, 0.75 and 1.0 at Re = 500 (Fig. 12(i)) and Re = 1000 (Fig. 12(ii)) when VR = 0.75. It is observed that, Stot is higher along the flow stagnation and the interface between secondary jet and main flow due to higher velocity and thermal gradients. In contrast, along the recirculation regions and the line joined by locus of maximum streamwise velocities, Stot is minimum due to negligible velocity gradients. It is also observed that, along all the solid surfaces (confinement surface, impingement surface and heat sources) and in the regions of contact between recirculation and main flow, Stot is dominated by fluid friction irreversibility. The magnitude of Stot;max is found to be decreasing with increasing channel height. This may be due to reduction in viscous dissipation causing less contribution from fluid friction irreversibility during entropy production. 3.2. Unsteady fluid flow dynamics and heat transfer 3.2.1. Flow regimes in parametric space Fig. 13(a)–(c) show the consolidated schematic map of occurrence of different flow regimes (i:e. Steady state and Unsteady periodic vortex shedding) for H/L = 0.5, 0.75 and 1.0. From Fig. 13(a), at H/L = 0.5, for all the values of Re and VR fluid flow is observed to reach steady state. At H/L = 0.75, except for C1 configuration (i:e VR = 1.0 and Re = 1000), fluid flow is observed to be steady as demonstrated in Fig. 13(b). At H/L = 1.0, only C2 configuration (VR = 0.75 and Re = 1000), and at VR = 1.0, for C3 (Re = 800) and C4 (Re = 1000) configurations, an unsteady periodic vortex shedding is observed as illustrated in Fig. 13(c). 3.2.2. Spectral analysis Fig. 14(a) and (b) present the temporal fluctuation of streamwise (u) and cross-stream (v) velocities, for all unsteady flow configurations (C1 - C4) of jet impingement cooling system. The streamwise (u) and cross-stream (v) velocities are measured at the velocity probe, located along the horizontal mid-section (streamwise direction) and positioned above the trailing edge of

fifth heat source (see Fig. 1). At Re = 1000, for C1 configuration (i:e H/L = 0.75 and VR = 1.0), the fluctuations in streamwise (u) and cross-stream (v) velocities are observed to be purely sinusoidal, showing a single dominant frequency to represent the oscillatory dynamics of jet impingement cooling system. For other configurations (C2 - C4), the streamwise (u) and cross-stream (v) velocities have several local peaks and troughs leading to a dominant vortex shedding frequency accompanied by several subharmonic frequencies (see Fig. 14(a) and (b)). The corresponding phase portraits (i) and power spectrum (ii) for all unsteady flow configurations (C1 - C4), are presented in Fig. 14(c)–(f) respectively. The phase portraits for all unsteady flow configurations (C1 - C4), represent, a complete repeating orbit, confirming the presence of a periodic vortex shedding. It is also observed that, for all unsteady flow configurations (C1 - C4), when phase portraits are located in first and third quadrants, streamwise (u) and crossstream (v) velocities are in-phase and generally are in the same direction. Also, when the phase portraits are in second and fourth quadrants, both u and v are in out-of-phase, and in opposite direction. Subsequently, a Fast Fourier Transform (FFT) routine is utilized to convert the temporal cross-stream velocity (v) fluctuation to the frequency domain. The power spectrum of cross-stream velocity (v) for C1 configuration (Fig. 14(c,ii)) illustrate an appearance of single dominant vortex shedding frequency, represented as Strouhal number (St) and is found to be St = 0.486. For C2 configuration, a dominant vortex shedding frequency (St = 0.215) accompanied by a sub-harmonic frequency (St = 0.4325) is observed. The frequency of convective motion of vortices is found to be higher at C1 configuration compared to C2 configuration. Similarly, at VR = 1.0 and H/L = 1.0, for both C3 (Re = 800) and C4 (Re = 1000) configurations, a dominant frequency (St = 0.2075 for C3 and St = 0.105 for C4) at the onset of vortex shedding accompanied by several sub-harmonic frequencies is observed. 3.2.3. Temporal evolution of vortical structures and heat transfer Fig. 15 demonstrates the temporal evolution of vortical structures for C4 configuration (i:e H/L = 1.0, VR = 1.0 and Re = 1000) using k2 criterion [35] for one complete vortex shedding cycle at various time instants (A - G) marked in Fig. 14(a). The rotational sign of the vortices are identified by superimposing the sign of vorticity (xz ) at the same location. It is observed that, the vortical structures with different temporal frequencies are evolved from various locations in jet impingement cooling system such as: (1) along the interface of primary jet and ambient fluid (PV), (2) in the vicinity of secondary jet and ambient fluid (SV), (3) hovering vortex observed in the neighborhood of primary recirculation

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Fig. 14. Spectral analysis for different flow configurations (C1–C4) in unsteady regime: temporal fluctuation of (a) streamwise velocity (u), (b) cross-stream velocity (v), phase portrait (i) and power spectra (ii) for all four unsteady periodic configurations, (c) C1, (d) C2, (e) C3 and (f) C4.

and secondary jet (HV), and (4) upper corners of the heat sources (RV). It is observed that, along the interface of high velocity jet (both primary and secondary) and ambient fluid, the destabilizing effect of shear forces may overcome the stabilizing effect of momentum diffusion. This may result in evolution of counterrotating vortex rings, (PV) and (SV) along the interfaces of jet and ambient fluid due to shear layer instability, which is similar to the Kelvin-Helmholtz instability. The vortical structures, PV and SV, may either independently break up into smaller vortices or merge and interact with other downstream flow features. The vortex generated in the vicinity of secondary jet (SV) is found to be convected downstream. Furthermore, vortex near primary jet (PV) is observed to merge and interact with already shed vortex (PV0 ). High shear force generated between the shed vortex (PV0 )

and the secondary jet cause an evolution of clock-wise rotating hovering vortex (HV). Subsequently, due to the flow separation at the upper corners of leading edge along all the heat sources, a clock-wise rotating vortices (RV1 ; RV2 , and RV3 ) produced and convected towards trailing edge of fifth heat source where it grow in size forming a vortex (RV2;3 ) and finally convect downstream as illustrated in Fig. 15. Fig. 16 shows the temporal evolution of isotherms for one complete vortex shedding cycle at different time instants (A - G) marked in Fig. 14(a). It is observed that, when clock-wise vortices (RV1 ; RV2 , and RV3 ) are shed from heat sources, the boundary layer (hydrodynamic and thermal) is detached from solid surfaces leading to a significant increment in thermal gradients causing higher heat transfer. Similarly, along the periphery of heat sources, when

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Fig. 15. Evolution of vortical structures at time instants (A–G) marked in Fig. 14(a) corresponding to one vortex shedding cycle for C4 configuration (Re = 1000, H/L = 1.0 and VR = 1.0).

Fig. 16. Evolution of isotherms at time instants (A–G) marked in Fig. 14(a) corresponding to one vortex shedding cycle for C4 configuration (Re = 1000, H/L = 1.0 and VR = 1.0).

the vortices are attached, a marginal increment in boundary layer (hydrodynamic and thermal) thickness is observed which lead to a significant reduction in thermal gradients. Fig. 17 illustrates the temporal evolution of local Nusselt number measured along all the heat sources for one complete vortex shedding cycle at different time instants (A - G) marked in Fig. 14(a). From Fig. 17, it is observed that, Nusselt number along all the heat sources is maximum when vortices detach the heat source causing a reduction in thermal boundary layer thickness along the periphery of heat source where as, Nu is minimum when

the vortices attach to the heat source, which is in accordance with the observed vorticity (Fig. 15) and temperature (Fig. 16) contours. 3.2.4. Proper orthogonal decomposition In order to further obtain a deeper understanding on spatial and temporal dynamics of vortex structures, Proper Orthogonal Decomposition (POD) in the manner of Sirovich’s snapshot method [36] is employed. The dominant coherent structures identified by POD modes are ordered based on their energy content. In the literature, several studies [37–39] have been reported on detailed

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Fig. 17. Evolution of local Nu distribution along (a) First, (b) Second, (c) Third, (d) Fourth and (e) Fifth heat source at time instants (A–G) marked in Fig. 14(a) corresponding to C4 configuration.

mathematical procedure and potential applications of POD methodology. A brief overview on snapshot POD algorithm is presented as follows: Let U m = U(x; sm ), m = 1, . . ., M be M realizations or snapshots of the fluctuating flow field (U m = ðu0 ; v 0 Þm ) obtained through finite element based CFD simulations. The K-L basis functions (/) are taken to be vector functions with two components (/1 and /2 ) associated with both streamwise (u) and cross-stream (v) velocities obtained at each node of the computational domain. The vector space in which the decomposition is sought has an inner product defined as

Z

ðf ; gÞ ¼

ðf 1 g 1 þ f 2 g 2 Þ D

The K-L basis functions (/) is governed by the integral eigenvalue problem

Z

X

0

Rij ðx; x0 Þ/j ðx0 Þdx ¼ k/i ðxÞ

where

Rij ðx; x0 Þ ¼ hU i ðx; sÞU j ðx; sÞi Now, the fluctuating flow field decomposed into a set of eigenfunctions and temporal coefficients is written as,

Uðx; sÞ ¼

M X ap ðsÞ/p ðxÞ m¼1

Once the eigenfunctions are calculated, the temporal coefficients can be obtained by projecting flow field onto eigenfunctions, so that (/p ; /p ) = 1.0

ap ðsÞ ¼ ðU; /p Þ

The total kinetic energy captured in the average sense is presented as

Energy ¼ hðU; UÞi ¼

XX X ðpÞ hap ar ið/p ; /r Þ ¼ k p

r

Fig. 18(a)–(d) illustrate the Eigenvalue spectra for all four configurations (C1 - C4) with unsteady periodic vortex shedding. From Fig. 18(a), it is observed that, for C1 configuration, a most dominant degenerate pair (i:e modes with almost equal kinetic energy content (aq ðsÞ and aqþ1 ðsÞ)) is sufficient to capture 99% of total kinetic energy whose energy content is much higher than the remaining spatial modes. It is also observed that, the dynamic evolution and interaction of vortical structures in unsteady flow field is relatively simple. The convective motion corresponding to dynamic evolution of vortical structures, SV (counter-clockwise vortex) and RV2 (clockwise vortex) has a single vortex shedding frequency component which is in accordance with temporal fluctuation and power spectrum of cross-stream (v) velocity (See Fig. 14). For C2 configuration, two degenerate pairs (First four modes) are sufficient to capture 99% of total kinetic energy. Finally, for C3 configuration, eight modes and for C4 configuration, ten dominant energy modes respectively are sufficient to capture 99% of total kinetic energy inorder to extract all the vortical structures corresponding to various temporal frequencies. Figs. 19 and 20 illustrate the spatial structures and time histories of temporal coefficients for eight dominant energy modes for Re = 1000, H/L = 1.0, and VR = 1.0. A translational symmetry in spatial structures and temporal coefficients with a phase shift for degenerate pair (i:e modes (aq ðsÞ and aqþ1 ðsÞ) with almost equal kinetic energy content) of modes is observed. This phase shift for degenerate pair of modes represent the convection of periodic vortices in flow configurations, which is actually a consequence

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Fig. 18. Energy content of POD modes for velocity field during impingement cooling of discrete protruding heat sources for all four configurations (a) C1, (b) C2, (c) C3 and (d) C4

of global absolute instability. The modal structures for first two dominant energy modes (Fig. 19(a) and (b)) are prominent downstream the secondary jet corresponding to periodic vortex shedding with single frequency component. It is also observed that, higher modes (Mode 3 to Mode 8) increasingly show smaller modal structures in the flow field and the amplitude of temporal coefficients tend to reduce due to reduction in corresponding Kinetic energy which is illustrated in Figs. 19 and 20. Further, higher modes have several local peaks and troughs in time histories of temporal coefficients leading to a dominant frequency component accompanied by several sub-harmonic frequencies as demonstrated in Fig. 20.

3.3. Global entropy generation and Surface averaged Nusselt number Fig. 21(a)–(e) exhibit the variation of Reynolds number (Re) and velocity ratio (VR) at H/L = 0.75 on surface averaged Nusselt number for jet impingement cooling of discrete protruding heat sources. It is observed that, at H/L = 0.75, as Re increases from Re = 100 to 1000, surface averaged Nusselt number along the all the heat sources increases except fourth heat source at VR = 0.0 (Single jet configuration [25]). At VR = 0.0, along fourth heat source,

Nu4 increases for Re 6 300 but at Re = 500, Nu4 is found to be minimum due to the presence of secondary recirculation. Further, at Re P 800, since secondary recirculation bubble convects downstream in the streamwise direction, an increase in Nu4 is observed. It is also observed that, along third, fourth and fifth heat sources, Nu is found to increase with increasing VR. The magnitude of Nuov for jet impingement cooling system increases with increasing Reynolds number and velocity ratio as illustrated in Fig. 21(f). Fig. 21(g)–(i) show the variation of Reynolds number and velocity ratio on global entropy generation due to heat transfer (Sh;X ), fluid friction (Sw;X ) and global total entropy generation (Stot;X ) for jet impingement cooling of discrete protruding heat sources. It is observed that, for all the vales of Re, as velocity ratio (VR) increases the magnitude of Sh;X ; Sw;X and Stot;X increase due to increase in velocity and thermal gradients. Fig. 22 shows the variation of Reynolds number, velocity ratio and channel height on overall surface averaged Nusselt number (Nuov ) and global total entropy generation (Stot;X ) for jet impingement cooling of discrete protruding heat sources. From Fig. 22, it is observed that, the magnitude of Nuov ; Stot;X for jet impingement cooling system increases with increasing Reynolds number and velocity ratio and decreasing channel height.

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Fig. 19. Vorticity contours of Spatial POD modes for C4 configuration; (a) Mode-1, (b) Mode-2, (c) Mode-3, (d) Mode-4, (e) Mode-5, (f) Mode-6, (g) Mode-7 and (h) Mode-8.

Fig. 20. Time evolution of temporal coefficients for degenerate pair of modes for C4 configuration; (a) a1 ðsÞ–a2 ðsÞ, (b) a3 ðsÞ–a4 ðsÞ, (c) a5 ðsÞ–a6 ðsÞ, and (d) a7 ðsÞ–a8 ðsÞ.

897

15

20

10

10 0 1000

800

5 0 1000

500

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300

100

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0

0.5

0.75

15

Nu3

30

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1

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(g)

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VR

(f) 1000

Stot,Ω

1000

0.25

1

5

(e)

Sψ,Ω

Sθ,Ω

(d) 100

0

0.75

10

0 1000

Re

VR

100

0.5

15

5 0 1000

500

300

(c)

NuOv

Nu5

Nu4

5

Re

500

(b) 10

800

800

Re

10

0 1000

5 0 1000

Re

VR

10

800

500 0 1000

500

Re

300

100

0

0.25

0.5

0.75

1

VR

(h)

800

500

Re

300

100

0

0.25

0.5

0.75

1

VR

(i)

Fig. 21. Effect of Re and VR at H/L = 0.75 on surface averaged Nusselt number and global entropy generation: (a) Nu1 , (b) Nu2 , (c) Nu3 , (d) Nu4 , (e) Nu5 , (f) Nuov , (g) Sh;X , (h) S/;X and (i) Stot;X .

3.4. Surrogate modeling In the present study, Nuov and Stot;X are chosen as objective functions for Multi-objective Genetic algorithm (MOGA) to achieve optimum configurations with maximum overall Nusselt number and minimum entropy generation for jet impingement cooling system with an array of air jets. For the purpose, Non-linear regression is used to derive surrogate models for Nuov and Stot;X with respect to their effective input design parameters (Re; H=L and VR). In order to perform a parametric design, FFD (Full Factorial Design) of DOE (Design of Experiments) has been carried out and the total number of simulations performed with three design variables such as Re (5 Levels), H/L (3 Levels) and VR (5 Levels) are 75 (5  3  5). The equations defining the variation of overall surface averaged Nusselt number (Nuov ) and global total entropy generation (Stot;X ) for jet impingement cooling system with an array of air jets are obtained as: 0:483

Nuov ¼ 0:493Re

ð1:5 þ VRÞ

0:325

Stot;X ¼ 117:10Re

0:746

ð1:0 þ H=LÞ

3:347

ð1:5 þ VRÞ

1:066

ð1:0 þ H=LÞ

5:786

ð18Þ ð19Þ

It is observed that, R2 values for Nuov and Stot;X are 0.9915 and 0.9569, respectively which indicate that the derived surrogate models are adequately significant at 95% confidence limit. Fig. 23 shows the goodness of the surrogate models (Eqs. (18) and (19)) obtained for Nuov and Stot;X , where the confrontation between the values predicted by the correlations from Non-linear regression and finite element simulation is presented. It is observed that, the errors between the numerical values of FE simulation and the ones given by the correlation (Eqs. (18) and (19)) have been calculated and the deviation from the simulated values is within a ±15% band as shown by dashed lines in the Fig. 23. Thus, the results obtained using FE simulation and the correlations from Nonlinear regression are in reasonable agreement and the results from proposed correlation may be used in lieu of actual numerical simulations. 3.5. Optimization Bejan [40] suggested that, if a thermal system is to operate in such a way that their destruction of useful work (or exergy) is minimized, then the conceptual design of such systems must begin

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NuOv

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600

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400 200 0 1000

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1

800

500

Re

300

100

0

0.25

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1

VR

(c)

Fig. 22. Effect of Re and VR on (i) Nuov and (ii) Stot;X for jet impingement cooling system at (a) H/L = 0.5, (b) H/L = 0.75 and (c) H/L = 1.0.

Fig. 23. Confrontation between Finite Element (FE) simulation and Non-linear regression for (a) Nuov and (b) Stot;X during jet impingement cooling of heat sources with an array of air jets.

with the minimization of entropy generation. In other words, thermal system with minimum entropy generation is considered as the optimal design. Subsequently, in a thermal system, enhancement in heat transfer (or Averaged Nusselt number) is often accompanied by an increase in entropy generation. Hence, according to Ruocco [24], the optimization of a heat transfer arrangement usually prescribes the trade-off between the minimization of overall entropy generation, and the maximization of solid-to-fluid heat transfer rate. Recently, Copiello and Fabbri [41] investigated heat transfer performance of wavy fins under laminar forced convection

and to optimize fin geometric profile by means of multi-objective genetic algorithm with objectives to maximize the heat transfer and minimize hydraulic resistance. Further, minimization of entropy generation was also studied and found geometries with lower entropy generation and higher Nusselt number were thermodynamically advantageous as they produce heat transfer enhancement with reduced irreversibility in the apparatus. In the literature, several studies have also been performed on optimization of thermal systems using Multi-Objective Genetic algorithms [42–44]. In the present study, two conflicting objective functions

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(Nuov and Stot;X ) have been optimized simultaneously with respect to the design parameters Re; H=L and VR. The multi-objective optimization problem can be formulated in the following form: Maximize : Nuov = f(Re; H=L; VR) Minimize : Stot;X = f(Re; H=L; VR) Subject to : 100 6 Re 6 1000, 0.5 6 H/L 1.0 and 0.0 6 VR 6 1.0 In all runs a population size has been chosen to be 1000 with crossover fraction = 0.7, Pareto-front population fraction = 0.8, generations = 1000 and function tolerance = 105. Fig. 24 illustrates the Pareto optimal front in functional space comprising non-inferior optimal solutions with maximum overall Nusselt number and minimum entropy generation obtained simultaneously in impingement cooling system with an array of laminar air jets. The reported Pareto frontier comprised 80 optimal configurations where an apparent trade-off between Nuov and Stot;X is found. From Fig. 24 it is observed that, on the Pareto optimal frontier a significant improvement in Nuov is accompanied by a relative increase in Stot;X and vice versa. It is also to be noted that, as each configuration is a global Pareto-optimal solution, none of these configurations are superior to others as both objective functions are to be satisfied simultaneously. Furthermore, out of all the obtained Designs in Pareto-Optimal Frontier (DPOF), six representative DPOFs have been selected (as shown in Fig. 24) to predict the accuracy of Non-linear regression and MOGA. For the purpose, finite element based numerical simulations have been performed for these optimal configurations (Pareto-optimal designs) representing with different values of Re; VR and H/L as shown in Table 1. From Table 1, it is found that, the predicted values of Nuov and Stot;X from finite element based simulations and Pareto frontier of Multiobjective Genetic algorithm (as shown in Fig. 24) are comparable indicating the accuracy of Regression and MOGA is quite good.

899

4. Conclusions A computational study using SUPG-FE method has been carried out to determine the fluid flow, heat transfer and entropy generation of discrete protruding heat sources mounted on impingement surface of the channel with an array of laminar air jets. The results for fluid flow patterns, Nu distribution and entropy generation for different values of Reynolds number (Re), channel height (H/L) and velocity ratio (VR) are presented and discussed. The magnitude of overall Nusselt number (Nuov ) and global total entropy generation (Stot;X ) are observed to increase with increasing Re; VR and decreasing H/L. Further, spectral analysis and proper orthogonal decomposition is performed to analyze spatio-temporal dynamics of unsteady configurations in jet impingement cooling system. It is observed that, along the interface of high velocity jet (both primary and secondary) and ambient fluid, the destabilizing effect of shear forces may overcome the stabilizing effect of momentum diffusion. This results in evolution of counter-rotating vortex rings along the interfaces of jet (both primary and secondary) and ambient fluid due to shear layer instability. The translational symmetry in spatial structure and temporal coefficients with a phase shift is observed for a pair of modes having degenerate (almost equal) eigenvalues. This phase shift for degenerate pair of modes represent the convection of periodic vortices in unsteady flow configuration, which is actually a consequence of global absolute instability. Finally, in order to obtain an optimum configuration with maximum overall Nusselt number and minimum entropy generation, the two performance parameters such as Nuov and Stot;X are selected as objective functions and modeled using Nonlinear regression analysis. A Multi-Objective Genetic Algorithm (MOGA) for jet impingement cooling system has been implemented to obtain optimum configurations encapsulating in the functional space lying on the Pareto-optimal frontier where a trade-off between two performance parameters Nuov and Stot;X is obtained. It is also observed that, for representative designs in Pareto-optimal frontier, the predicted performance parameters (Nuov and Stot;X ) from MOGA are compared with corresponding values from finite element based CFD calculations, indicating a good prediction in accuracy. References

Fig. 24. Pareto-optimal front representation of the non-dominated solutions between both the objective functions (Nuov and Stot;X ).

Table 1 Comparison of predictions (Nuov ; Stot;X ) from Multi-objective Genetic algorithm (MOGA) and Finite Element Method (FEM) for six representative Designs in Paretooptimal Frontier (DPOF). DPOF

1 2 3 4 5 6

Re

1000.00 999.00 999.27 999.00 701.37 100.00

H/L

0.50 0.65 0.75 0.79 0.98 1.00

VR

1.00 0.99 0.95 0.49 0.01 0.00

Stot;X

Nuov MOGA

FEM

MOGA

FEM

17.82 16.02 14.90 12.44 7.64 2.95

17.92 15.72 14.67 14.28 8.57 2.91

2272.99 1279.06 875.18 380.07 73.82 36.83

2619.21 1438.12 993.66 414.45 107.27 46.62

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