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Heat Transfer by Impingement of Circular Free-Surface Liquid Jets

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Heat Transfer by Impingement of Circular Free-Surface Liquid Jets 18th National & 7th ISHMT-ASME Heat and Mass Transfer Conference January 4-6, 2006 Paper No: IIT Guwahati, India Heat Transfer by Impingement of Circular Free-Surface Liquid Jets John H. Lienhard V Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge MA 02139-4307 USA email: lienhard@mit.edu Abstract Q volume flow rate of jet (m3/s). 3 Qs volume flow rate of splattered liquid (m /s). 2 This paper reviews several aspects of liquid jet im- qw wall heat flux (W/m ). pingement cooling, focusing on research done in our r radius coordinate in spherical coordinates, or lab at MIT. Free surface, circular liquid jet are con- radius coordinate in cylindrical coordinates (m). sidered. Theoretical and experimental results for rh radius at which turbulence is fully developed the laminar stagnation zone are summarized. Tur- (m). bulence effects are discussed, including correlations ro radius at which viscous boundary layer reaches for the stagnation zone Nusselt number. Analyti- free surface (m). cal results for downstream heat transfer in laminar rt radius at which turbulent transition begins (m). jet impingement are discussed. Splattering of turbu- r1 radius at which thermal boundary layer reaches lent jets is also considered, including experimental re- free surface (m). sults for the splattered mass fraction, measurements Red Reynolds number of circular jet, uf d/ν. of the surface roughness of turbulent jets, and uni- T liquid temperature (K). versal equilibrium spectra for the roughness of tur- Tf temperature of incoming liquid jet (K). bulent jets. The use of jets for high heat flux cooling Tw temperature of wall (K). is described briefly. Tsf liquid surface temperature (K). u, v Nomenclature liquid velocity components in radial, axial di- rection of cylindrical coordinates (m/s) u A2n constants in Eq. (5). rms turbulent velocity fluctuation in jet (m/s). d du e uf B dimensionless velocity gradient, 2 u dr bulk velocity of incoming jet (m/s). f u Cc contraction coefficient for liquid jets, jet cross- h velocity of liquid sheet averaged across thick- h sectional area divided by nozzle area. ness (m/s). u Cf skin friction coefficient. m free surface velocity of liquid sheet (m/s). u r d jet diameter, fully contracted (m). e( ) radial velocity just above boundary layer (m/s). V G(Pr) boundary layer function of Prandtl number, max centerline velocity of incoming jet (m/s). ρu2 d/σ Eq. (12). Wed jet Weber number, f . h local heat transfer coefficient, qw/(Tw − Tf ), 2 (W/m K). Greek Letters h(r) thickness of axisymmetric liquid sheet (m). 2 k thermal conductivity of liquid (W/m·K), or rms α thermal diffusivity of liquid (m /s). surface roughness (m). δ, δt momentum, thermal boundary layer thickness k∗ dimensionless surface roughness, k/d. (m). δ l distance between nozzle and target plate (m). rms local rms surface roughness of turbulent jet Nud local Nusselt number based on jet diameter, (m). η qwd/k(Tw − Tf ). dimensionless wavenumber. p local pressure in liquid (Pa). θ polar angle of spherical coordinates. ν 2/ p∞ ambient pressure (Pa). kinematic viscosity of liquid (m s). ξ Qs/Q pstgn stagnation pressure (Pa). fraction of impinging liquid splattered, . ρ / 3 pe(r) pressure distribution along the wall (Pa). density of liquid (kg m ). σ P2n Legendre function of 2n order. liquid-gas surface tension (N/m). 2 Pr Prandtl number of liquid. φ velocity potential (m /s). 1 Figure 1: Laminar impinging jet: Red =51, 000, d =5.0 mm, sharp-edged orifice, adiabatic target. 1. Introduction Liquid jet impingement cooling offers very low Figure 2: Impinging jet configuration. thermal resistances and is relatively simple to imple- ment. Liquid jets are easily created using a straight flow and heat transfer characteristics are described in tube or a contracting nozzle, and this nozzle can general terms by the usual results for the stagnation be aimed directly toward the region of a heat load. zone. The flow field can be divided into an outer When the jet strikes the target surface, it forms a region of essentially inviscid flow and an inner viscous very thin stagnation-zone boundary layer which of- boundary layer region. fers little resistance to heat flow. Convective heat 2 The analytical solution of the stagnation zone transfer coefficients can reach tens of kW/m K. boundary layer is a classical problem [1, 2, 3], whose These high heat transfer coefficients make liquid results depend primarily on the radial velocity gra- jet impingement attractive in situations where a high dient of the inviscid flow near the stagnation point. heat load must be removed while maintaining a min- To adapt stagnation zone boundary layer results to imum temperature or temperature difference within impinging jets, this gradient must be determined. the system. For example, in some semiconductor Thus, analysis of the stagnation zone requires first laser systems, junction temperatures must be held ◦ 2 a solution for the inviscid flow of the jet, and then below 150 C while heat loads may reach 10 MW/m . application of the classical boundary solutions for the Much attention has been given to jet impingement flow and temperature fields. Together, these lead to cooling of electronics. expressions for the wall heat flux and the Nusselt An impinging jet defines its own flow field, often number. without the need for added channeling or target mod- ifications. Jets are particularly useful when cooling 2.1 Inviscid Outer Flow systems must not add interfering hardware or make The inviscid flow field of an impinging jet is de- structural changes to the cooled object. For exam- termined by: a free streamline boundary condition at ple, a fixed nozzle at the end of a processing line the liquid surface; an impermeable wall onto which can cool each successive item passed under it; and in the jet impacts; and assumed forms of the inlet and some automotive engines, oil jets cool the undersides outlet velocity profiles (Fig. 2). If the inlet profile of the piston crowns. is irrotational (e.g., uniform), the velocity field can Liquid jets can also carry extremely high heat be obtained using potential flow theory (∇2φ = 0); fluxes, if the velocities are such as to produce a high otherwise, the Euler equations must be solved. stagnation pressure. Small diameter water jets at In all cases, the stagnation-zone flow has radial speeds near 130 m/s have removed heat loads of up to velocity distribution at the wall given by 400 MW/m2. Liquid jets are well-suited for cooling u r Cr ··· ,r→ very localized, high-flux heat sources. e( )= + 0 (1) The jets of interest in the present article are un- for C =(due/dr|r=0) a constant radial velocity gra- submerged jets, those that travel through a gas be- dient. This result is necessitated by the kinematics tween the nozzle and the target (Fig. 1). These jets of any stagnation zone (for irrotational flow, see [4]); are only similar to submerged jets in the stagnation the constant C, however, depends on the specific in- region, and then only when the submerged jet is less viscid flow considered. For later use in heat transfer than about five diameters in length. analyses, it is convenient to nondimensionalize the wall gradient: 2. Stagnation Zone Theory d due B ≡ 2 (2) Near the point of impact, an impinging jet’s fluid uf dr r=0 2 The inviscid flow near the stagnation point has a wall pressure distribution given by Table 1: Coefficients of velocity potential [5]. ρ 2 2 ∞ pe(r)=p∞ + uf − ue(r) (3) Wed 50 25 16.7 2 A2 −1.831 −1.881 −1.944 −2.015 ρ 2 = p − ue(r) (4) A4 2.365 2.858 3.469 4.213 stgn 2 A6 0.5906 −0.01553 −0.8006 −1.825 p∞ p for the ambient pressure and stgn the stagnation A8 −14.81 −19.03 −24.30 −30.78 uf pressure. If the jet is nonuniform, refers to the A10 13.35 20.42 29.38 40.61 centerline velocity of the jet away from the target; A12 50.74 68.74 91.31 119.3 if surface tension pressure is significant, only Eq. 4 applies [5]. Measurements of the wall pressure dis- tribution have often been used to determine ue(r). The potential flow is independent of Reynolds Table 2: Velocity gradients at the stagnation point number and scales with the inlet speed and jet diam- during laminar circular jet impingement. eter. Low levels of turbulence in the incoming jet are likely to have only slight effects on the mean velocity Investigators B/2 profile l/d Wed distribution outside the wall boundary layer, so the ≈ . ∞ inviscid solutions should apply to either laminar or Schach [8] 0 88 uniform 1.5 ∞ turbulent jets, if those jets have the specified inlet ve- Shen [9] 0.743 uniform 1.5 locity profiles. In addition, the boundary conditions Strand [10] 0.903 uniform 1.0 ∞ on the free streamline (no shear stress, pressure con- Liu, Gabour, 0.916 uniform 1.0 ∞ stant at p∞) will apply for steady jets of any density; Lienhard [5] 0.981 uniform 1.0 50 thus, solutions that have been obtained in the con- 1.06 uniform 1.0 25 text of nonmixing gas jets (no entrainment of sur- 1.16 uniform 1.0 16.7 rounding fluid) apply equally well to unsubmerged Scholtz and 4.69 parabolic 0.05 ∞ liquid jets. The similarity between submerged jets Trass [11] to 0.5 and free liquid jets obviously fails once shear layer in- stabilities at the freestreamline cause the submerged jet to begin mixing with the surrounding fluid.
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