Heat Transfer Notes PDF

B-KUL-ZA0182 Warmte en Stroming | Thermal-Fluid sciences Prof. Maarten Blommaert  PART III: Heat Transfer 2 This Photo by Unknown Author is licensed under CC BY-NC-ND Summary Chapter 16: Heat transfer mechanisms Three (8constitutive9) laws of transport Co...

B-KUL-ZA0182 Warmte en Stroming | Thermal-Fluid sciences Prof. Maarten Blommaert  PART III: Heat Transfer 2 This Photo by Unknown Author is licensed under CC BY-NC-ND Summary Chapter 16: Heat transfer mechanisms Three (8constitutive9) laws of transport Conduction: Fourier (1D version) 㕑ÿ ýሶ ýĀÿþ = 2 㕘 㔴 㕑ý Convection: Newton ýሶ ýĀÿĄ = ℎ 㔴 ÿā 2 ÿ∞ Radiation: Stefan-Boltzmann ýሶ ÿþ 㕖Ă = 㔀 㔎 㔴ā ÿā4 3  Chapter 17 Heat transfer Stationary heat conduction 4 © McGraw Hill Heat transfer: Stationary heat conduction General Fourier9s law and Steady heat conduction in plane walls Thermal contact resistance Generalized thermal resistance networks Heat conduction in cylinders and spheres Critical radius of insulation 5 Fourier’s law ýሶ þ 㕇 Fourier (1D): ÿሶ = = 2 㕘 㔴 þĆ More generally: Heat flux = vector quantity! Remember: Fourier = actually a vector law: ÿԦሶ = 2 㕘∇ÿ 㔕 㕇 ∇ÿ is a vector that points in direction of steepest increase in T 㔕Ć 㔕 㕇 ∇ÿ = 㔕ć 㔕ÿ 㔕ÿ 㔕ÿ 㔕 㕇 Or (for Cartesian coordinates x-y-z): ÿԦሶ = 2 㕘 ÿԦ 2 㕘 ĀԦ 2 㕘 㕘 㔕ý 㔕þ 㔕ÿ 㔕Ĉ 6  17-1 Steady heat conduction in plain walls Heat transfer through the wall of a house can be modeled as steady and one-dimensional. ➔ Temperature of the wall depends on one direction only, e.g. T(x) Figure 17–1: Heat transfer through a wall is one-dimensional when the temperature of the wall varies in one direction only. 8 © McGraw Hill  17-1 Steady heat conduction in plain walls Heat transfer through the wall of a house can be modeled as steady and one-dimensional. Energy conservation for any slab of wall (steady operation) 㕑 㔸ą 㕎ýý ሶ ሶ ý 㕖ÿ 2 ýĀăĂ = =0 㕑 㕡 ➔ ýሶ wall = constant With ýሶ 㔰 㔚 㔥 㔥 given by Fourier’s law: 㕑ÿ 㕑ÿ ÿԦሶ = 2 㕘∇ÿ ⇒ ÿሶ Ć = 2 㕘 ֞ ýሶ Ć = 2 㕘 㔴 㕑ý 㕑ý Figure 17–2: Under steady conditions, þ 㕇 ➔ = constant (temperature drop is linear) the temperature distribution in a plane þĆ wall is a straight line. 9 © McGraw Hill 17-1 Steady heat conduction in plain walls Total temperature drop can be found from integration dT Qcond, wall = −kA dx L T2 x =0 Qcond, wall dx = −  T =T1 kA dT T1 − T2 Qcond, wall = kA (W) L Replacing T2 by T(x) in the integration gives ÿ1 2 ÿ ý ýሶ cond, wall ሶ ýcond, wall = 㕘 㔴 ⇒ ÿ ý = ÿ1 2 ý x 㕘 㔴 Figure 17–2: Under steady conditions, the temperature distribution in a plane ➔ Once the rate of heat conduction is available, the wall is a straight line. temperature T(x) at any location x can be determined 10 © McGraw Hill 17-1 Steady heat conduction in plain walls Thermal Resistance Concept T1 − T2 Qcond, wall = kA L T −T Qcond, wall = 1 2 (W) Rwall Ā þwall = (ÿ/W) 㕘 㔴 Conduction resistance of the wall: Thermal resistance of the wall against heat conduction. Figure 17–3: Analogy between thermal and electrical resistance concepts. Thermal resistance of a medium depends on geometry and thermal properties of the medium. rate of heat transfer → electric current thermal resistance → electrical resistance Similar to V − V2 Re = L /  e A temperature difference → voltage difference Electrical resistance: I= 1 Re 11 © McGraw Hill 11 17-1 Steady heat conduction in plain walls Newton’s law of cooling Qconv = hAs (Ts − T ) Ts − T Qconv = (W) Rconv 1 þconv = (ÿ/W) ℎ 㔴ā Convection resistance of the surface: =Thermal resistance of the surface against heat convection. Figure 17–4: Schematic for convection resistance at a surface. Limit case of very large convection heat transfer coefficient When the convection heat transfer coefficient is very large (h → ∞), þýĀÿĄ → 0 and ÿā → ÿ∞ That is, the surface offers no resistance to convection, and thus it does not slow down the heat transfer process. This situation is approached in practice at surfaces where boiling and condensation occur. 12 © McGraw Hill 12  17-1 Steady heat conduction in plain walls Stefan-Boltzman law Ts − Tsurr Radiation resistance of the Qrad =  As (T − T s 4 4 surr ) rad s s surr = h A ( T − T ) = surface: Thermal resistance Rrad of the surface against radiation. 1 þrad = (K/W) ℎrad 㔴ā With h radiation heat transfer coefficient Qrad hrad = = (Ts2 + Tsurr 2 )(Ts + Tsurr ) (W/m 2  K) As (Ts − Tsurr ) Or the combined heat transfer coefficient hcombined = hconv + hrad Figure 17–5: Schematic for When ÿsurr j ÿ∞ convection and radiation resistances at a surface. 13 © McGraw Hill  17-1 Steady heat conduction in plain walls Thermal Resistance Network We can define a total resistance for the series: T1 − T 2 Q= Rtotal Figure 17–6: The thermal resistance network for heat with transfer through a plane wall subjected to convection 1 Ā 1 on both sides, and the electrical analogy. þtotal = þconv,1 + þwall + þconv,2 = + + (ÿ/W) ℎ1 㔴 㕘 㔴 ℎ2 㔴 14 © McGraw Hill  17-1 Steady heat conduction in plain walls Once ýሶ is evaluated, the temperature drops can be determined from T = QR (C) E.g. to determine the surface temperature T1: T1 − T1 T1 − T1 Q= = Rconv, 1 1/ h1 A Also the overall heat transfer coefficient U can be used 1 Ā 㔴 = ( 㕊/K) þtotal Figure 17–8: The temperature drop across Such that a layer is proportional to its thermal resistance. Q = UA T (W) 15 © McGraw Hill Summary Steady Heat Conduction in Plane Walls. Thermal Resistance Concept Δÿ Ā 1 þ= (ÿ/ 㕊) þwall = (ÿ/W) þconv = (ÿ/W) ýሶ 㕘 㔴 ℎ 㔴ā 1 Qrad 1 Rrad = (K/W) , with hrad = = (Ts2 + Tsurr 2 )(Ts + Tsurr ) (W/m  K) 2 Ā 㔴 = ( 㕊/K) hrad As As (Ts − Tsurr ) þtotal Note: the bigger the heat exchanging surface, the smaller the resistance to heat flow! Watch out: sometimes R is also defined independent of the size of the surface! I.e.: Δÿ Ā 1 þ= ( 㕚2 ÿ/ 㕊) þwall = ( 㕚2 ÿ/ 㕊) þconv = ( 㕚2 ÿ/W) ÿሶ 㕘 ℎ 1 1 þrad = ( 㕚2 ÿ/ 㕊) Ā= ( 㕊/m2 K) ℎrad þtotal 16 © McGraw Hill  17-1 Steady heat conduction in plain walls Multilayer Plane Walls T1 − T 2 Q= Rtotal Rtotal = Rconv, 1 + Rwall, 1 + Rwall, 2 + Rconv, 2 1 L L 1 = + 1 + 2 + h1 A k1 A k2 A h2 A Evaluation of temperatures for T∞1 and T ∞2 given: T1 − T1 To find Tl : Q = Rconv, 1 T1 − T2 To find T2 : Q = Rconv, 1 + RWall, 1 T3 − T 2 Figure 17–9: The thermal resistance network for heat transfer To find T3: Q = through a two-layer plane wall subjected to convection on both sides. Rconv, 2 17 © McGraw Hill  Voorbeeld: bepaal de U-waarde van een muur bestaande uit: 1cm gips ( 㕘 = 0.52 W/mK) 14cm binnenmuur uit metselwerk ( 㕘 = 0.38 W/mK) 10cm PUR-isolatie ( 㕘 = 0.028 W/mK) Een niet-geventileerde spouw van 1cm (ℎā = 6.6 W/m2 K) 9cm holle gevelstenen ( 㕘 = 0.94 W/mK) Met overgangscoefficienten ℎ 㕖ÿ = 8 W/m2 K; ℎĀăĂ = 23 W/m2 K Oplossing: 1 Ā= , met þ 㕡 㕜 㕡 㔴 1 㕑1 㕑2 㕑3 1 㕑4 1 þĂĀĂ 㔴 = + + + + + + ℎ 㕖ÿ 㕘1 㕘2 㕘3 ℎā 㕘4 ℎĀăĂ þĂĀĂ 㔴 = 0.13 + 0.019 + 0.37 + 3.57 + 0.15 + 0.10 + 0.04 = 4.37 㕚2 ÿ/ 㕊 1 Ā= = 0.229 W/m2 K þĂĀĂ 㔴 18 © McGraw Hill  17-1 Steady heat conduction in plain walls U-values required for passive houses (separate directive for windows) How to reach? ➔ increase PUR from 10 to 17cm Comparison of typical U-values and wall thickness of typical German building stock and Passive House construction. 19 © McGraw Hill Heat transfer: Stationary heat conduction General Fourier’s law and Steady heat conduction in plane walls Thermal contact resistance Generalized thermal resistance networks Heat conduction in cylinders and spheres Critical radius of insulation 20  17-2 Thermal contact resistance Thermal contact resistance þ 㖄 When two surfaces are pressed against each other voids arise that are filled with air. These air gaps of varying sizes act as insulation because of the low thermal conductivity of air. This resistance to heat transfer (per unit interface area) is called the thermal contact resistance Rc. The thermal contact conductance is defined as 㖉 㖄 = 㗏/þ 㖄 1 Tinterface Figure 17–14: Temperature distribution and heat flow Rc = = (m 2 C/W) lines along two solid plates pressed against each hc Q/ A other for the case of perfect and imperfect contact. 21 © McGraw Hill 17-2 Thermal contact resistance The value of thermal contact resistance depends on: Surface roughness, Material properties, Temperature and pressure at the interface. Type of fluid trapped at the interface. ➔ The thermal contact conductance is highest (and thus the contact resistance is lowest) for soft metals with smooth surfaces at high pressure. When does the contact resistance matter? Typical values for þý : 0.000005-0.0005 m2 ⋅ K/W ➔ Compare þý to resistance of 1cm slabs Conclusion Thermal contact resistance is significant for Ā 0.01 m þinsulation = = = 0.25 m2 ⋅ ÿ/W good heat conductors such as metals, but can 㕘 0.04 W/m ⋅ °C be disregarded for poor heat conductors such Ā 0.01 m as insulations. þcopper = = = 0.000026 m2 ⋅ ÿ/W 㕘 386 W/m ⋅ °C 22 © McGraw Hill 17-2 Thermal contact resistance The thermal contact resistance can be minimized by applying: A thermal grease such as silicon oil. A better conducting gas such as helium or hydrogen. A soft metallic foil such as tin, silver, copper, nickel, or aluminum. Fluid at the Interface Contact Conductance, hc, W/m2·K Air 3640 Helium 9520 Hydrogen 13,900 Silicone oil 19,000 Glycerin 37,700 Table 17-1: Thermal contact conductance for aluminum plates with different fluids at the interface for a surface roughness of 10 µm and interface pressure of 1 atm. Figure 17–16: Effect of metallic coatings on thermal contact conductance. 23 © McGraw Hill Heat transfer: Stationary heat conduction General Fourier’s law and Steady heat conduction in plane walls Thermal contact resistance Generalized thermal resistance networks Heat conduction in cylinders and spheres Critical radius of insulation 24  17-3 GENERALIZED THERMAL RESISTANCE NETWORKS Effect of thermal resistances in parallel The heat flux can be calculated as T1 − T2 T1 − T2 ö 1 1 ö Q = Q1 + Q2 = + = (T1 − T2 ) ÷ + ÷ R1 R2 ø R1 R2 ø If one wishes to use an equivalent total resistance T1 − T2 Q= Rtotal It needs to correspond to 1 1 1 R1 R2 = + → Rtotal = Rtotal R1 R2 R1 + R2 25 © McGraw Hill  17-3 GENERALIZED THERMAL RESISTANCE NETWORKS Combined situations are technically not 1D anymore! Two assumptions possible to solve as 1D resistance network: 1) Any plane wall normal to the x-axis is isothermal (i.e., to assume the temperature to vary in the x-direction only). 2) Any plane parallel to the x-axis is adiabatic (i.e., to assume heat transfer to occur in the x-direction only). Following assumption 1 gives: T1 − T R1 R2 Q= Rtotal = R12 + R3 + Rconv = + R3 + Rconv Rtotal R1 + R2 L1 L2 R1 = R2 = k1 A1 k2 A2 L3 1 R3 = Rconv = k3 A3 hA3 26 © McGraw Hill  17-3 GENERALIZED THERMAL RESISTANCE NETWORKS Combined situations are technically not 1D anymore! Two assumptions possible to solve as 1D resistance network: 1) Any plane wall normal to the x-axis is isothermal (i.e., to assume the temperature to vary in the x-direction only). 2) Any plane parallel to the x-axis is adiabatic (i.e., to assume heat transfer to occur in the x-direction only). Following assumption 2 gives: T1 − T þÿÿă 㕖Ą,1 þÿÿă 㕖Ą,2 Q= þtotal = þÿă 㕖Ą,1 + þÿÿă 㕖Ą,2 + R conv Rtotal Ā1 Ā3 1 þÿÿă 㕖Ą,1 = þ1 + þ3 = + þ ýĀÿĄ = 㕘1 㔴1 㕘3 㔴1 ℎ 㔴3 Ā1 Ā3 þÿÿă 㕖Ą,2 = þ2 + þ4 = + 㕘1 㔴2 㕘3 㔴2 27 © McGraw Hill Example 17-6: Heat loss through a composite wall 28 © McGraw Hill Example 17-6: Heat loss through a composite wall Two options: Isothermal assumption adiabatic assumption 29 © McGraw Hill Example 17-6: Heat loss through a composite wall Calculate resistances following the isothermal assumption: 30 © McGraw Hill Example 17-6: Heat loss through a composite wall The equivalent resistance for the parallel part is then: And the total resistance The heat transfer through the wall is then given by: For a wall of 3m x 5m the total heat transfer is then 4.37W ýሶ ĂĀĂ 㕎ý = ⋅ 15m 2 = 263W 0.25m2 31 © McGraw Hill Example 17-6: Heat loss through a composite wall Alternative resistance network based on adiabatic assumption ➔ þĂĀĂ 㕎ý = 6.97ÿ/ 㕊 instead of þĂĀĂ 㕎ý = 6.87ÿ/ 㕊 32 © McGraw Hill Heat transfer: Stationary heat conduction General Fourier’s law and Steady heat conduction in plane walls Thermal contact resistance Generalized thermal resistance networks Heat conduction in cylinders and spheres Critical radius of insulation 33 17-4 Heat Conduction in Cylinders and Spheres Heat transfer through a pipe can be modeled as steady and one-dimensional. If the pipe is sufficiently long, the temperature of the pipe depends only on r ➔T = T(r). Figure 17–23: Heat is lost from a hot-water pipe to the air outside in the radial direction, and thus heat transfer from a long pipe is one-dimensional. 34 © McGraw Hill 17-4 Heat Conduction in Cylinders and Spheres 㕑ÿ 㕑ÿ ÿԦሶ = 2 㕘∇ÿ ⇒ ÿሶ Ā = 2 㕘 ሶ ⇒ ý = 2 㕘 㔴 (W) 㕑Ā 㕑Ā Integrate expression over tube (note: ÿሶ = 㕓(Ā)) r2 Qcond, cyl T2  r = r1 A dr = −  T =T1 k dT with 㔴 = 2 㔋ĀĀ T1 − T2 Figure 17–24: A long cylindrical pipe Qcond, cyl = 2 Lk (W) ln(r2 / r1 ) (or spherical shell) with specified inner and outer surface temperatures T1 and T2. T1 − T2 Qcond, cyl = (W) Rcyl ln(r2 / r1 ) ln (Outer radius/Inner radius) Conduction resistance of the cylinder layer ➔ Rcyl = = 2 Lk 2  Length  Thermal conductivity 35 © McGraw Hill 17-4 Heat Conduction in Cylinders and Spheres Similarly for sphere: T1 − T2 Qcond, sph = Rsph Conduction resistance of sphere shell: r2 − r1 Outer radius − Inner radius Rsph = = 4 r1r2 k 4 (Outer radius)(Inner radius)(Thermal conductivity) 36 © McGraw Hill 17-4 Heat Conduction in Cylinders and Spheres Combined conductive/convective heat transfer? T1 − T 2 Q= Rtotal For a cylindrical layer Rtotal = Rconv, 1 + Rcyl + Rconv, 2 1 In(r2 / r1 ) 1 = + + (2 r1 L)h1 2 Lk (2 r2 L)h2 For a spherical layer Figure 17–25: The thermal resistance network for a cylindrical (or spherical) Rtotal = Rconv, 1 + Rsph + Rconv, 2 shell subjected to convection from both the inner and the outer sides. 1 r2 − r1 1 = + + (4 r12 )h1 4 r1r2 k (4 r22 )h2 37 © McGraw Hill 38 © McGraw Hill Heat transfer: Stationary heat conduction General Fourier’s law and Steady heat conduction in plane walls Thermal contact resistance Generalized thermal resistance networks Heat conduction in cylinders and spheres Critical radius of insulation 40 17-5 Critical Radius of Insulation Adding more insulation to a wall or ceiling always decreases heat transfer since the heat transfer area is constant Why? adding insulation increases the thermal resistance of the wall without increasing the convection resistance. Is this also true for insulating pipes? ÿ1 2 ÿ∞ ÿ1 2 ÿ∞ ýሶ = = þins + þconv In(Ā2 /Ā1 ) 1 + 2 㔋Ā 㕘 ℎ(2 㔋Ā2 Ā) Increasing Ā2 for constant Ā1 : Increases þ 㕖ÿā Figure 17–30: An insulated cylindrical Decreases þýĀÿĄ pipe exposed to convection from the ➔ The heat transfer from the pipe may increase or decrease, outer surface and the thermal resistance network associated with it. depending on which effect dominates. 41 © McGraw Hill 17-5 Critical Radius of Insulation For a pipe, it can be shown that a maximum in heat transfer is achieved at q critical radius of insulation of k rcr, cylinder = (m) h Should we be careful in choosing to insulate pipes? The largest value of the critical radius we are likely to encounter is kmax, insulation 0.05 W/m C rcr, max =  hmin 5 W/m 2 C = 0.01 m = 1 cm Figure 17–31: The variation of heat transfer rate with the outer radius of the insulation r2 when r1 < rcr. Conclusion: we can insulate hot-water or steam pipes freely without worrying about the possibility of increasing the heat transfer by insulating the pipes. But: when insulating electric wires mind that heat losses only decrease when Ā > ĀýĀ ! 42 © McGraw Hill Summary Thermal resistances can be combined in a network to solve for temperatures similar as in electrical networks! Thermal Resistance formulas 1 1 T = QR (C) Rrad = (K/W) Rconv = (C/W) hrad As hAs Conductive resistance for plane walls L Rwall = (C/W) kA Conductive resistance for cylinders and spheres Note: careful with thermal insulation when Ā < ĀýĀ ln(Ā2 /Ā1 ) Ā2 2 Ā1 㕘 þcyl = þsph = Ācr, cylinder = (m) 2 㔋Ā 㕘 4 㔋 Ā1 Ā2 㕘 ℎ Thermal contact resistance 1 T 1 þý Δÿinterface Rc = = interface (m 2 C/W) þ 㕖ÿĂÿĀĀ 㕎ýÿ = = = (°C/W) hc Q/ A ℎý 㔴 㔴 ýሶ 43 © McGraw Hill 45 © McGraw Hill B-KUL-ZA0182 Warmte en Stroming | Thermal-Fluid sciences Prof. Maarten Blommaert  PART III: Heat Transfer 2 This Photo by Unknown Author is licensed under CC BY-NC-ND  Chapter 16 Heat transfer Introduction and basic concepts 8 Heat transfer: introduction and basic concepts Introduction Conduction Convection Radiation Simultaneous heat transfer mechanisms 9 1-3 THERMODYNAMICS AND HEAT TRANSFER Heat: form of energy that can be transferred from one system to another as a result of temperature difference. Thermodynamics is concerned with the amount of heat transfer as a system undergoes a process from one equilibrium state to another. Heat transfer determines how fast these energy transfers go 10  1-3 THERMODYNAMICS AND HEAT TRANSFER 11  Heat transfer Basic objective of heat transfer analysis is often to find: Rate of heat flow (ý)ሶ (nl: warmtestroom): Heat transferred per unit time Rate of heat flux ( 㕞)ሶ (nl: warmteflux): Heat transferred per unit time per unit surface normal to the direction of the heat transfer ýሶ þ 㕞ሶ = 㔴 ÿ2 rate of heat flux (magnitude) These relate to the transferred energy as Or when Qሶ is constant: 12 1-3 Applications of heat transfer 13 13 16-1 HEAT TRANSFER MECHANISMS Heat is always transferred from a hot to a cold medium Heat can be transferred in three basic modes : conduction convection radiation All modes of heat transfer require the existence of a temperature difference 14 Heat transfer: introduction and basic concepts Introduction Conduction Convection Radiation Simultaneous heat transfer mechanisms 15 16-2 CONDUCTION Conduction: The transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones Physical mechanisms behind conduction: In gases and liquids, conduction is due to the collisions and diffusion of the molecules during their random motion. In solids, it is due to the combination of vibrations of the molecules in a lattice and the energy transport by free electrons. Experimental observation: Rate of heat conduction õ ( Area )( Temperature difference ) Heat conduction through a Thickness large plane wall of thickness x and area A. Formally: T1 − T2 T Qcond = kA = −kA (W) x x 16 16-2 Conduction Fourier9s law of heat conduction in 1D: ýሶ þ 㕇 When Δ 㕥 → 0 㕞ሶ = = 2 㕘 㔴 þ 㕥 1D heat flux Thermal conductivity ù J W ù ùW W ù  úû sm 2 m 2 úû  úû mK mC úû 104 ÿĂ = ýāĀýþ 㕎Āþ for typical geometries ➔ But drag scales with ý 2 ! 33 RECAP CH15: external flow For external flows boundary layers become turbulent for Re > Reą j 5 ⋅ 105 㔕Ă Adverse pressure drops ( > 0) trigger 㔕ą flow detachment ➔ Drag increase for general shapes ➔ Turbulent boundary layers delay detachment Rotating cylinders/spheres generate lift! 34 Summary Chapter 16: Heat transfer mechanisms Three modes of heat transfer Conduction: Fourier (1D version) ýሶ þ 㕇 㕞ሶ = = 2 㕘 㔴 þ 㕥 With 㕘 the thermal Conductivity (=material property), with units þ/(ÿ ⋅ 㔾) Convection: Newton ýሶ 㕞ሶ = = ℎ 㕇Ā 2 㕇∞ 㔴 With ℎ the convection heat transfer coefficient, with units þ/(ÿ2 ⋅ 㔾) Remark: ℎ is not a property but depends on many factors Radiation: up next 35 Vraag bij example: why do clouds float and raindrops fall: Waarom moeten we de Fb tekenen bij een druppel in lucht (ik denk dat hier de Buoyancy force mee bedoeld wordt), en wat is dan net het verschil tussen Fb en Fd (hebben beide niet met drukverschil boven en onder de druppel te maken?) Fb: drijfkracht = drukverschil onder/boven t.g.v. zwaartekracht = hier verwaarloosbaar o.w.v. groot verschil in ĀĄ en Āÿ ➔Zie voorbeeld drijfkracht op mens Fd: weerstand/drag = drukverschil onder/boven t.g.v. stroming 36  Intermezzo: Cooling computer chips Problem: Power density of chips increases quickly ▪ 㕞ሶ keeps on increasing! ▪ Same T ! ➔ Need more efficient heat transfer 37 16-3 Convection Convection modes for cooling microchips Forced liquid cooling Free convection Forced liquid convection (& radiation) Forced air convection Liquid evaporation Two-phase cooling with heat pipes 2 2 16-3 Convection ýሶ āāĀă = ℎ Δ 㕇 㔴 ℎ can be increased by: Changing fluid e.g. air → water Making convection more effective E.g. natural → forced Has also drawbacks: noise, energy,… Cooling method Air Liquid (Water) Free convection 5-100 100-1200 Forced convection 10-350 500-3000 W/(m²K) Boiling / 3000-100000 40 16-3 Convection Application: Displays Use only natural convection in the outer layer to minimize noise 41 Heat transfer: introduction and basic concepts Introduction Conduction Convection Radiation Simultaneous heat transfer mechanisms 42  16-4 Radiation Electromagnatic radiation: The energy emitted by matter in the form of electromagnetic waves (or photons) as a result of the changes in the electronic configurations of the atoms or molecules. Very fast: travels at speed of light! Thermal radiation: radiation emitted by bodies because of their temperature, with wavelengths ÿ in the range: ÿ. Ā 㕁 㖎 2 Āÿÿ 㕁 㖎 Infrared radiation from a person 43 16-4 Thermal Radiation Everything above 0K radiates! FIGURE 21–4: Everything around us constantly emits thermal radiation. 44 16-4 Radiation Travels through vacuum (!) Can occur between bodies without losing power separated by a colder medium 45  16-4 Radiation Blackbody radiation Blackbody: The idealized surface that emits radiation at the maximum rate. Qemit,max =  AsTs4 (W) Stefan–Boltzmann law  = 5.670 ô 10−8 W/m 2  K 4 Stefan–Boltzmann constant Radiation emitted by real surfaces Qemit =  AsTs4 (W) Figure 16–14: Blackbody radiation represents Emissivity ε : A measure of how closely a surface the maximum amount of radiation that can approximates a blackbody for which ε = 1 of the be emitted from a surface at a specified surface. 0 ≤ ε ≤ 1. temperature. 47  16-4 Radiation Table 16-6: Emissivities of some materials at 300 K Material Emissivity Aluminum foil 0.07 Anodized aluminum 0.82 Polished copper 0.03 Polished gold 0.03 Polished silver 0.02 Polished stainless steel 0.17 Black paint 0.98 White paint 0.90 White paper 0.92−0.97 Radiation emitted by real surfaces Asphalt pavement 0.85−0.93 (W) 0.93−0.96 Qemit =  A T 4 Red brick s s Human skin 0.95 Wood 0.82−0.92 Emissivity ε : A measure of how closely a surface Soil 0.93−0.96 approximates a blackbody for which ε = 1 of the Water 0.96 surface. 0 ≤ ε ≤ 1. Vegetation 0.92−0.96 48 16-4 Radiation Absorptivity : The fraction of the radiation energy incident on a surface that is absorbed by the surface. 0   1 A blackbody absorbs the entire radiation incident on it ( = 1). Kirchhoff’s law: The emissivity and the absorptivity of a surface at a given temperature and wavelength are equal ⇒ 㗼 = 㔀 Qabsorbed =  Qincident (W) Figure 16–15: The absorption of radiation incident on an opaque surface of absorptivity α. 49  16-4 Radiation Net radiant heat transfer for bodies that are completely enclosed by a much larger body consisting of surfaces at 㕻Ā 㖖ÿÿ Are seperated by a gas that does not intervene with the radiation Black bodies, 104 ) Recall: ÿ 㔷 is typically constant for Re > 104 28 15–4 Drag coefficients of common geometries Drag coefficient for typical geometries for high Re-numbers (Re > 104 ) 29 15–4 Drag coefficients of common geometries Streamlining to reduce drag Droplet shape approaches minimal pressure drag for given frontal area Only useful for high Re-numbers! Example: time trial helmets Upper right: Helmet Lower right: Topview of CFD-calculation Syansuri et al (2018) 30 Example: Power required to maintain car speed Question: On a highway, how much power is required to 1.4 ÿ maintain a speed of regular car/minivan to 120km/h? Assume: rolling resistance negligible þā þ 㕥 1.6 ÿ āሶ = =Ă = Ă 㔷 ă þā þā Regular car (1.6m x 1.4m) ÿ 3 Āă 2 ýā 33.3 ሶ ā = ÿ 㔷 㔴 ă = 0.3 ⋅ 1.6ÿ ⋅ 1.4ÿ ⋅ 1.2 3 ⋅ Ā = 14.9ýþ 2 ÿ 2 Minivan (1.6m x 1.85m) ÿ 3 Āă 3 ýā 33.3 āሶ = ÿ 㔷 㔴 = 0.4 ⋅ 1.6ÿ ⋅ 1.85ÿ ⋅ 1.2 3 ⋅ Ā = 26.2ýþ 2 ÿ 2 31 Example: fuel consumption of cars Question: On a highway, how much % of fuel can you save by 1.4 ÿ choosing to drive slower than 120km/h? Assume: rolling resistance negligible, constant velocity ( 㕎 = 0) 1.6 ÿ 32 Example: fuel consumption of cars For 50 km at a speed of 100km/h ✓ 30% Fuel saved ✓ Time increase from 25 min to 30 min For 50 km at a speed of 140km/h ✓ 40% increase in fuel consumption ✓ Time decrease from 25 min to 21.5 min 33 15–5 Parallel Flow Over Flat Plates Not to scale Āý 㕥 㕅ÿą = 㕥: distance from leading edge ÿ To scale Flow becomes turbulent for Re > Reą j 5 ⋅ 105 34  15–5 Parallel Flow Over Flat Plates Skin friction strongly increases when boundary layer becomes turbulent Local skin-friction 㔏Ą ÿĄ,ą = 1 2 Āă 2 ∞ VS. total friction drag 㔹 㔷, 㕓 㕟 㕖 㕐 㕡 㕖Āÿ ÿ 㔷,ĄÿÿāāÿĀÿ = 1 2 2 㔌 㕉 㔴 36 15-5 Influence of pressure gradient on detachment Detachment most often occurs in diverging flow (reasoning below)… - Favorable pressure gradient 㔕ā 0 㔕ą When the avg. Velocity decreases with x: diverging flow 37 Drag on a car Streamlining: drag↓ 38 15–6 Flow Over Cylinders and Spheres Flow over spheres and cylinders omnipresent! Heat sink Wind turbine, lantern, chimney 39 15–6 Flow Over Cylinders and Spheres Viscous flow around cylinder: (A) Mainly friction drag (Stokes flow) (Re