Heat Transfer Mathematical Modelling Numerical Methods and Information Technology Part 12 doc

Population Balance Model of Heat Transfer in Gas-Solid Processing Systems 429 () ( ) ( ) (1) (2) 0, 0, ,(1),01 in in in in in nT M TT M TT τφ δ φ δ ϕ = −+− − ≤≤ (85) where M 0,in =10 8 /m 3 is the total number of particles in a unit volume, and φ=0.6 is the ratio of particles having different temperatures. Fig. 4 shows that there may remain rather significant inhomogeneities in the temperature distribution of particles in the bed. These inhomogeneities are decreasing with increasing interparticle collision frequency therefore the particle-particle interaction play important role in homogenization of temperature inside the population of particles. 10 -5 10 0 0 2 4 6 8 data1 data2 data3 data4 data5 data6 data7 σ 2 (τ) τ 1.0 ·10 2 5.0 ·10 2 1.0 ·10 3 2.5 ·10 3 5.0 ·10 3 7.5 ·10 3 1.0 ·10 4 S pp Fig. 4. Time evolution of the variance of temperature distribution of the particle population Fig. 5 presents the time evolution of the temperatures of characteristic parts of the bubbling bed, i.e. mean temperature of particles, the temperatures of the bubble phase and the gas in the emulsion phase, as well as the temperatures of the wall and liquid in the jacket. In this case the input temperature of gas was 180 ºC and the initial temperature of gas in the emulsion and bubble phases were of the same values. It is seen how the temperature profiles vary in interconnections of the different parts of the bed. In steady state, under the given heat transfer resistance conditions of the particle population, the wall and liquid is heated practically only by the gas in the emulsion phase while the bubbles flow through the bed without losing heat assuring only the well stirred state of the emulsion phase. 10 -5 10 0 0 50 100 150 200 τ T(τ) T e T w T l T b T Fig. 5. Time evolution of the temperatures of characteristic parts of the bubbling bed in inter- relations with each other Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 430 In Fig. 5, bubble temperature transient is shown only for the bed output indicating here a rather sharp front. Naturally, the temperature front of bubbles evolves progressively along the bed as it shown in Fig. 6 at different axial coordinates showing that in transient states the bubble phase also plays role in the heating process. However the bubble evolves transient states 10 -6 10 -4 10 -2 10 0 10 2 80 100 120 140 160 180 data1 data2 data3 data4 data5 T b (ξ) τ 0.1 0.3 0.6 0.8 1.0 ξ Fig. 6. Progression of evolution of the temperature front of the bubble phase along the axial coordinate of the bed 10 -5 10 0 20 25 30 35 40 45 data1 data2 data3 data4 τ T 0.5 1.0 5.0 10.0 S pw Fig. 7. Time evolution of the mean temperature of particle population as a function of the particle-wall collision frequency Eq. (76) shows clearly that when the number of particles is constant, i.e. under steady state hydrodynamic conditions interparticle heat transfer does not influence the mean value of temperature of the particle population but, as it is demonstrated by Fig. 4, it affects the variance. However, as Fig. 7 gives an evidence of that the mean value of temperature of the particle population depends on the particle-wall collision frequency. This figure indicates also that because of the heat transfer interrelations of different parts of the bed oscillations may arise in the transient processes which becomes smoothed as the particle-wall collision frequency decreases. At the same time, in this case the particle-wall collision frequency affects also the variance of temperature distribution of the particle population as it is shown in Fig. 8. It is seen that with increasing particle-wall collision frequency the variance de- Population Balance Model of Heat Transfer in Gas-Solid Processing Systems 431 creases. i.e. increasing particle-wall heat transfer intensity gives rise to smaller inhomoge- neites in the temperature distribution of particles. 10 -5 10 0 0 2 4 6 8 10 data1 data2 data3 data4 τ σ 2 (τ) 0.5 1.0 5.0 10.0 S pw Fig. 8. Time evolution of the variance of temperature distribution of the particle population as a function of the particle-wall collision frequency 9. Conclusion The spatially distributed population balance model presented in this chapter provides a tool of modeling heat transfer processes in gas-solid processing systems with interparticle and particle-wall interactions by collisions. Beside the gas-solid, gas-wall and wall-environment heat transfers the thermal effects of collisions have also been included into the model. The basic element of the model is the population density function of particle population the motion of which in the space of position and temperature variables is governed by the population balance equation. The population density function provides an important and useful characterization of the temperature distribution of particles by means of which temperature inhomogeneities and developing of possible hot spots can be predicted in particulate processes. In generalized form the model can serve for cognitive purposes but by specifying appropriate symmetry conditions useful applicative, i.e. purpose-oriented models can be obtained. The second order moment equation model, obtained from the infinite hierarchy of moment equations generated by the population balance equation, as an applicative model can be applied successfully for analyzing the thermal properties of gas-solid processing systems by simulation. The first two moments are required to formulate the heat balances of the particulate system while the higher order moments are of use for characterizing the process in more detail. Applicability of the second order moment equation model was demonstrated by modeling and studying the behavior of bubbling fluidization by numerical experiments. It has proved that collision particle-particle and particle-wall heat transfers contribute to homogenization of the temperature of particle population to a large extent. The particle-particle heat transfer no affects the mean temperature of particle population and, in fact, no influences any of temperatures of the system whilst the particle-wall heat transfer collisions exhibits significant influence not only on the steady state temperatures but on the transient processes of the system as well. It has been demonstrated that the second order moment equation Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 432 model can be effectively used to analyze both the dynamical and steady state processes of bubbling fluidization 10. Acknowledgement This work was supported by the Hungarian Scientific Research Fund under Grant K 77955. 11. Symbols a – surface area, m 2 c – specific heat, J kg -1 K -1 D – dispersion coefficient, m 2 s -1 , bed diameter, m e – coefficient of restitution f – probability density function h – enthalpy, J; heat transfer coefficient W m -2 K -1 K – aggregate rate coefficient of heat transfer k – thermal conductivity, W m -1 K -1 m – mass, kg m k – normalized k th order moment of particle temperature N – number of particles, no m -3 n – population density function, no m -3 K -1 p p – weight parameter p w – weight parameter Pe – Peclet number q – volumetric flow rate, m 3 s -1 S – frequency of collisions, s -1 T – temperature, K t – time, s u – linear velocity, m s -1 v – volume, m 3 V – volume, m 3 x – axial coordinate, m t mean residence time, s . mean value 1 - Heaviside function Greek symbols β – aggregate heat transfer coefficient ρ – density, kg m -3 δ – Dirac delta function κ – parameter ω – random variable characterizing collision heat transfer μ k – k th order moment of particle temperature ε – volumetric fraction ξ – dimensionless axial coordinate τ – dimensionless time Population Balance Model of Heat Transfer in Gas-Solid Processing Systems 433 σ 2 – variance of the temperature of particle population Subscripts and superscripts c – critical value e – emulsion phase; coefficient of restitution g – gas in – input b – bubble max – maximal value min – minimal value p – particle pg – particle-gas pp – particle-particle pw – particle-wall w – wall wb – wall-gas 12. 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Eng. Sci., 35, 1741-1748 17 Synthetic Jet-based Hybrid Heat Sink for Electronic Cooling Tilak T Chandratilleke, D Jagannatha and R Narayanaswamy Curtin University Australia 1. Introduction Modern lifestyle is increasingly dependant on a myriad of microelectronic devices such as televisions, computers, mobile phones and navigation systems. When operating, these devices produce significant levels of internal heat that needs to be readily dissipated to the ambient to prevent excessive temperatures and resulting thermal failure. With inadequate cooling, internal heat builds up within electronic packages to raise microchip temperatures above permitted thermal thresholds causing irreparable damage to semiconductor material. Devices would then develop unreliable operation or undergo complete thermal breakdown from overheating with reduced working life. In electronic industry, 55 percent of failures are attributed to overheating of internal components. Therefore, effective dissipation of internally generated heat has always been a major technical consideration for microelectronic circuitry design in preventing overheating and subsequent device failure. In recent years, the modern microelectronic industry has shown a dramatic growth in microprocessor operating power, circuit component density and functional complexity across the whole spectrum of electronic devices from small handheld units to powerful microprocessors, as evidenced by Figs. 1 and 2. 0 50 100 150 200 250 300 350 400 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 Year Chip Power Dissipation (W) Maximum Minimum Fig. 1. Evolution of chip power dissipation (Chu, 2003) Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 436 Fig. 2. Component density and maximum processing performance trends Source: International Technology Roadmap for Semiconductors (ITRS)-2008 These industry trends into the future are very much poised to increase the microprocessor internal heat loads well beyond the capabilities of established cooling technologies making them rapidly inadequate to meet the predicted intense heat dissipation demand. For this, the future of electronic product design and development is critically hinged upon the availability of more enhanced or effective heat dissipation methods. This chapter presents a novel electronic cooling technique to address this current technological shortfall. 1.1 Thermal management techniques In electronic system design, thermal management involves the use of appropriate heat transfer technology to remove internally generated heat as effectively as possible to retain component temperatures within safe operating limits. Literature identifies two specific stages for thermal management process. The first stage considers heat conduction from integrated circuit and to the encased package surface while the second stage deals with the global rejection of heat from the system to the ambient. Thus, the thermal management technologies can be broadly divided into two groups: (a) Technologies for enhancing heat flow integrated circuits to package surfaces, such as thermoelectric devices and heat pipes; (b) Technologies for enhancing heat exchange between the electronic package and the ambient, such as heat sinks, microchannels and fluid jet cooling. The work entailed in this chapter contributes to the latter group. Thermal management techniques can be passive or active mechanisms. Passive techniques (e.g. convective heat sink, heat pipe) do not require additional energy input for operation; however their poor heat transfer capabilities overshadow this advantage. Intrinsically, active techniques (e.g. micro-refrigerator, microchannel heat sink) have better thermal performance, but are discredited by the extra operating power needs or higher pressure drop penalties. In spite of these limitations, active heat sinks firmly remain the most thermally effective and preferred option for future high-powered microcircuitry cooling applications, whilst passive heat sinks are being confined to low heat loads. As an active thermal management technique, heat sinks utilising micro or mini fluid passages are highly regarded by the electronic industry to be the current frontier technology for meeting high heat dissipation demand. It has been estimated (Palm, 2001) that the Synthetic Jet-based Hybrid Heat Sink for Electronic Cooling 437 industry application of microchannel heat sinks would increase by 10 fold within the next 5 years in view of its high cooling potential achievable. A major drawback of microchannel heat sinks is their inherently high pressure drop characteristics, particularly at increased fluid flow rates necessary to deliver large cooling loads. Motivated by the application needs of the electronic industry, the research on microchannel thermal behaviour has extensively progressed through numerical modelling and experimentation (Lee et al., 2005; Qu & Mudawar, 2002; Lee & Garimella, 2006). The primary focus of such research has been to predict and validate thermal performance. Much less attention has been directed for developing effective thermal enhancement strategies for micro-scale channels. The use of internal fins in microchannels has been identified to be a very promising passive enhancement option for single phase mini and microchannels (Steinke & Kandlikar, 2004) although the increased pressure drop is a design concern. This is well supported by a comprehensive treatment on such internal fins and possibilities for thermal optimisation (Narayanaswamy et al., 2008). Whilst passive enhancement techniques have conceivable application potential, active methods are known to be more thermally effective and relevant for future cooling needs. Active heat sink systems can be made a more attractive proposition if an innovative approach could reduce operating power or pressure penalties without impacting on thermal performance. A hybrid heat sink incorporating a pulsating fluid jet offers a unique thermal enhancement option of this nature, as discussed below. The proposed method utilises a special pulsing fluid jet mechanism called synthetic jet for enhancing convective heat transfer process in fluid flow channels of an active heat sink. This arrangement operates with unprecedented thermal performance but without the need for additional fluid circuits or incurring pressure drop, which are key features that set it apart from other traditional methods. 1.2 Synthetic (pulsed) jet mechanism and its behaviour A synthetic jet is created when a fluid is periodically forced back and forth through a submerged orifice in such a way that the net mass discharged through the orifice is zero while generating large positive momentum in the jet fluid stream. A simple synthetic jet actuator is schematically shown in Fig. 3. Fig. 3. Schematic diagram of a synthetic jet actuator Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 438 The actuator comprises of an oscillating diaphragm that resides within a cavity and induces a periodic fluid flow through a submerged orifice. In its upward motion, the diaphragm forces the fluid to be squirted out through the orifice with very large momentum. During downward motion, the diaphragm draws low-momentum fluid from the surroundings back into the cavity. Thus, over an operating cycle, the jet delivers very high net outflow of fluid momentum with no net change of fluid mass within the cavity. Owing to this unique feature, this jet flow is known as a synthetic jet or Zero-Net-Mass-Flux (ZNMF) jet (Smith & Glezer, 1998). A synthetic jet impinging on a heated surface is capable of generating very high localised cooling because of the large fluid momentum imparted. As evident, this mechanism does not require additional fluid circuits to operate or introduce extra pressure drop to the flow field, which are major technical advantages. Synthetic jets are formed from the same working fluid in which they are deployed. This has significant benefits for microelectronic circuitry where air-cooling is preferred to prevent possible electrical short-circuiting and leakage faults. Smaller synthetic jet size permits high-flux clustered cooling without the need for fluid circulatory circuits unlike steady jets thereby reducing energy consumption and production cost. The diaphragm motion is practically achieved by a piston or acoustic loudspeaker or piezo-electric unit to obtain the desired amplitude or frequency. Synthetic jets have been primarily studied in the context of pulsating jet actuators impinging on submerged surfaces in quiescent fluid media without any cross flow interactions. Such studies indicate outstanding thermal characteristics for localised cooling with synthetic jets. Significant examples of those are by (Campbell et al., 1998) who have demonstrated that synthetic air micro jets were effective cooling arrangements for laptop processors. (Mahalingam & Rumigny, 2004) illustrated the effectiveness of synthetic jets for high power electronic cooling by developing an integrated active heat sink based on this mechanism. (Gillespie et al., 2006) provide the results of an experimental investigation of a rectangular synthetic jet impinging on a unconfined heated plate exposed to the ambient where characteristics of the jet and plots of Nusselt numbers are available. (Pavlova & Amitay, 2006) have conducted experimental studies on impinging synthetic jets for constant heat flux surface cooling and compared its performance with a steady or continuous jet where there are no velocity fluctuations. They concluded that for the same Reynolds number, synthetic jets are three times more effective than the corresponding continuous jets. (Utturkar et al., 2008) experimentally studied synthetic jet acting parallel to the flow within a duct. The synthetic jet was placed at the surface of a heated duct wall and aligned with the bulk flow such that the jet assisted the bulk flow. In a 100 mm square channel with a 30 mm synthetic jet, they obtained a 5.5 times enhancement for a bulk flow velocity 1 m/s. This enhancement reduced to approximately 3 times when the bulk velocity was increased to 2.0 m/s. Their numerical simulation matched reasonably well with only one test condition. (Go & Mongia, 2008) experimentally studied the effect of introducing a synthetic jet into a low speed duct flow to emulate the confined flow within in a typical notebook. The interaction of these two flows was studied using particle image velocimetry (PIV) and measurements on the heated duct wall. They found that the synthetic jet tends to retard or block the duct flow while a 25 percent increase in thermal performance was observed. Numerical studies on thermal performance of synthetic jet with cross flow interaction are also very limited in published literature. Such significant work is presented by (Timchenko et al., 2004) who investigated the use of a synthetic jet induced by a vibrating diaphragm to enhance the heat transfer in a 200 μm microchannel. Their two-dimensional (2-D) transient [...]... micro combustor is larger than that 456 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology of a large-scale counterpart For a can-type combustor, combustor volume and surface area are V = πr2L and A = 2πrL, respectively Here, r and L are radius and length of the combustor Therefore, the heat loss increases more rapidly than the heat generation as the combustor size is... Vol 128 , No 9, 897-907 Qu, W & Mudawar, I (2002) Experimental and Numerical Study of Pressure Drop and Heat Transfer in a Single Phase Microchannel Heat Sink, International Journal of Heat and Mass Transfer, Vol 45, No 2, 2549-2565 Steinke, M E & Kandlikar, S G (2004) Single Phase Heat Transfer Enhancement Technique in Microchannel and Minichannel Flows, International conference of Microchannel and. .. 4350 454 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology Erbas, N & Baysal, O (2009) Micron-level actuators for thermal management of microelectronic devices, Heat Transfer Engineering, Vol 30, No 1-2, 138-147 FLUENT User Guide Manual 6.2.16, 2004 Gillespie, M B.; Black, W Z.; Rinehart, C & Glezer, A (2006) Local Convective Heat Transfer From a Constant Heat Flux... 10 kHz and A = 50 µm 447 448 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology Fig 11 Effect of diaphragm amplitude illustrated by velocity contours at t = ½ T Relative strengths of the synthetic jet and the channel flow drag determine the extent of cross-flow interference and the boundary layer disruption at the heated wall This is illustrated in Figs 11(a) and 11(b)... Figs 8, 10 and 11 450 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology Local Nusselt Number,Nu 25 t = T/6 t = T/3 t = T/2 t = 2T/3 t = 5T/6 t=T 20 15 10 5 0 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 x/do Fig 13 Local Nusselt number at heated wall with stagnant flow in heat sink over one cycle Vi =0 m/s, f = 10 kHz and A = 50 μm 14 Local Nusselt Number,Nu 12 10 8 t =... 60 70 Jet Reynolds Number Fig 16 Degree of thermal enhancement in heat sink due to synthetic jet mechanism 452 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology For increasing channel velocity, Fig 17 shows the thermal performance and pressure drop of a heat sink without synthetic jet mechanism If this heat sink were to deliver, for example, 4.3 times thermal enhancement,... between the jet and the cross-flow heat sink fluid stream During the diaphragm upward motion, a high-velocity fluid jet is discharged through the cavity orifice into the flow passage Determined by diaphragm amplitude, sufficiently strong jet momentum enables the jet to penetrate the micro passage flow to reach the heated Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology. .. Jet-based Hybrid Heat Sink for Electronic Cooling Fig 8 Time-lapsed velocity contours over one diaphragm cycle 445 446 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology Fig 9 Velocity vectors near the orifice at time t=T/2 (maximum expulsion) f = 10 kHz, A = 50 µm Note: Length of arrows and colours indicate velocity magnitude (m/s) Synthetic Jet-based Hybrid Heat Sink for... dimensional, unsteady nature of turbulence is considered as superposing random perturbation on streamwise velocity component as follows: 460 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology u = U inlet (1 + I Φ ) (11) here Φ is a probability function which returns real number randomly in the range of −1 ≤ Φ ≤ 1 and I is the fluctuation intensity U inlet is the fuel or oxidant... However, the k − ε − f μ model and LES predict the 462 Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology flow recirculation region in front of the fuel jet but its size and location are different each other Generally, for turbulence models it is hard to predict the turbulent flow fields with high fidelity having both free shear flow and wall bounded one according . the surfaces of immersed heat transfer exchanger elements. Chem. Eng. Proc. 18, 199-223. Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 434 Mihálykó,. synthetic jet mounted on heat sink in cross-flow configuration Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 440 fluid jet and its vortices interact. heated Heat Transfer - Mathematical Modelling, Numerical Methods and Information Technology 444 -30 -20 -10 0 10 20 30 0 255075100 Flow Time, t (μs) Diaphragm Amplitude, A (μm) -16 -12 -8 -4 0 4 8 12 16 Axial