Developments in Heat Transfer Part 3 docx

Developments in Heat Transfer 70 6.1 Effect of axial magnetic field on convective heat transfer In case of flow of liquid metals in heated channels under the influence of a uniform axial magnetic field shows a decrease of convective heat transfer at low and moderate Hartmann numbers whereas the convective heat transfer and hence Nu increases at higher Hartmann numbers as shown in Miyazaki (1988). It was stated in section 4 that an axial magnetic field does not affect the mean velocity distribution so the modification of convective heat transfer is due the variation in the turbulent fluctuations in time and space. Reynolds number Hartmann Number =0 Nu/Nu B 3 (2.5 5) 10 − ⋅ 360 0.83 4 (1 2) 10−⋅ 700 0.50 4 (3 4) 10−⋅ 1400 0.30 4 110⋅ 3600 2.75 Table 2. Variation of Nusselt Number with Axial Magnetic Field, Miyazaki (1988) The values of Nu for various value of Hartmann numbers is shown in table 2. It can be seen that the values of Nu decreases from its value, =0 Nu/Nu B = 1, at Ha = 0. The decrease in Nu value for small and moderate Ha is more when the Reynolds number is high because of the higher turbulence content in the flow. At lower values of Reynolds numbers, the flow will be inherently laminar and therefore the reduction in Nu due to suppression of turbulent fluctuations will be low. At high values of Hartmann numbers, the Nusselt number was found to increase, violating the earlier theories and studies, see Miyazaki (1988). Miyazaki attributes the increase in the Nusselt number is due to the increase in turbulence levels in the flow as the effect of buoyancy can be ruled out because the flow direction upwards. 6.2 Effect of transverse magnetic field on convective heat transfer The studies in the field of effect of magnetic field on convective heat transfer in ducts subjected to transverse magnetic field can be classified into two cases 1. Absence of high velocity jets near the side walls 2. Presence of high velocity jets near the side walls 6.2.1 Case 1: Absence of high velocity jets near the side walls In case of ducts having Hartmann walls with zero conductivity, high velocity jets will not be formed near the side walls. Gardener and Lykoudis (1971b) performed experiments with flow of Mercury in horizontal electrically insulated pipe subjected to transverse magnetic field. It was found that the velocity profile near the Hartmann wall becomes flat with increase in magnetic field as discussed in section 4.1 and the velocity profile near side walls becomes round as discussed in sections 4.2.1 and 4.2.2. The mean velocity distribution is not much different with the increase in magnetic field, so the modification of turbulence phenomenon by the magnetic field will affect the convective heat transfer predominantly for this case. The Nusselt number distribution near the Hartmann and side walls for a range of Reynolds numbers and Hartmann numbers is shown in figure 12. The decrease of Nusselt number with increase in magnetic field is lesser at lower Reynolds number because the turbulence content in the flow at low Reynolds number will be lesser. Magneto Hydro-Dynamics and Heat Transfer in Liquid Metal Flows 71 0 10 20 30 40 10000 100000 1000000 Nu No magnetic field Hartmann wall - Ha 375 Side wall - Ha 375 10 4 10 5 10 6 1 10 20 30 40 No magnetic field Hartmann – Ha 375 Side – Ha 375 Re Nu 0 10 20 30 40 10000 100000 1000000 Nu No magnetic field Hartmann wall - Ha 375 Side wall - Ha 375 10 4 10 5 10 6 1 10 20 30 40 No magnetic field Hartmann – Ha 375 Side – Ha 375 Re Nu Fig. 12. Nusselt number with magnetic field intensity, Gardener and Lykoudis (1971b) The reduction in Nusselt number with increase in magnetic field is because of the reduction in turbulence quantified using turbulence kinetic energy as shown in figure 13. It was found that the turbulent kinetic energy decreases both near the Hartmann and side walls with increase in magnetic field where r/R = 0 represents the centre of the duct and r/R = 1 represents the walls. The damping force within the Hartmann layer is much higher than at the side region due to the high local electric current density. The turbulence in core is suppressed initially and then the turbulence in the Hartmann layer followed by the turbulence near the side wall. 0 5000 10000 15000 20000 25000 30000 35000 0.00 0.20 0.40 0.60 0.80 1.00 r/R Turbulent Kinetic Energy m 2 /s 2 No magnetic field Hartmann wall - Ha 47 Side wall - Ha 47 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 Turbulent Kinetic Energy x 10 4 m 2 /s 2 0 5000 10000 15000 20000 25000 30000 35000 0.00 0.20 0.40 0.60 0.80 1.00 r/R Turbulent Kinetic Energy m 2 /s 2 No magnetic field Hartmann wall - Ha 47 Side wall - Ha 47 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 Turbulent Kinetic Energy x 10 4 m 2 /s 2 Fig. 13. Turbulent kinetic energy vs. r/R for Re = 50,000, Gardener and Lykoudis (1971a) A correlation for Nusselt number values is created from various experimental results by Ji and Gardener (1997) and is given using the following relation as a function of Peclet number Pe and Hartmann number Ha () () 0.811 1.5 0.00782Pe Nu 7 1 0.0004Ha Pef =+ + (27) () ( ) 592 Pe 0.3 4.75 10 Pe 2.10 10 Pef −− =+× −× Developments in Heat Transfer 72 6.2.2 Case 2: Presence of high velocity jets near the side walls In case of ducts having Hartmann walls with finite conductivity, high velocity jets will be formed near the side walls. The side layers with high velocity jets (M shaped profile, figure 8 case 3) carry high mass flux is the prime reason for increase of heat transfer near the side walls. Miyazaki et al. (1986) performed experiments to determine the heat transfer characteristics for liquid metal Lithium flow in annular duct with electrically conducting walls under the influence of transverse magnetic fields. The Nusselt number plotted with magnetic field is shown in figure 14. It can be seen that the Nusselt number increases near the side walls and decreases near the Hartmann walls. A singular rise of Nusselt number can be seen near both the walls. 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 0.00 0.20 0.40 0.60 0.80 1.00 Magnetic FIeld, Tesla Nu Side Walls Hartmann Walls Fig. 14. Nusselt number plotted with magnetic field intensity, Miyazaki (1986) This effect of heat transfer enhancement near the side walls is caused by the generation and development of large scale velocity fluctuations in the near wall area. The reduction in Nusselt number near the Hartmann walls is created due to the turbulence reduction as shown in figure 15. 0.0 0.2 0.4 0.6 0.8 0.00 0.20 0.40 0.60 0.80 1.00 Magnetic FIeld, Tesla Side Walls Hartmann Walls RMS of Temperature Fluctuation ( 0 C) 0.0 0.2 0.4 0.6 0.8 0.00 0.20 0.40 0.60 0.80 1.00 Magnetic FIeld, Tesla Side Walls Hartmann Walls RMS of Temperature Fluctuation ( 0 C) Fig. 15. RMS of temperature fluctuation with magnetic field intensity, Miyazaki (1986) Magneto Hydro-Dynamics and Heat Transfer in Liquid Metal Flows 73 7. Application of numerical codes A difficulty in experimental study of the flow of liquid metals arises as the visualization is not possible because of the opaque nature of liquid metals. Application of closed form analytical solutions is limited to simple cases where the equations are not very complex. This makes the application of numerical simulations useful for the study of liquid metal magneto-hydro-dynamic flows. An example of application of a numerical code to explain the mechanisms affecting heat transfer for flow subjected to transverse magnetic field is explained using a series of simulations given in Rao and Sankar (2010), see figure 16. x y 300 T 4 T 5 T 7 T 8 B 30 7.6 15.8 1.65 1.1 Heater Pin Flow Direction Fluid Elements Solid Elements (a) (b) Height of the first cell = 1μm x y 300 T 4 T 5 T 7 T 8 B 30 7.6 15.8 1.65 1.1 Heater Pin Flow Direction Fluid Elements Solid Elements (a) (b) Height of the first cell = 1μm Fig. 16. (a) Schematic of model (b) Details of the computational mesh, Rao and Sankar (2010) A numerical study is conducted in an annular duct formed by a SS316 circular tube with electrically conducting walls and a coaxial heater pin, with liquid Lithium as the working fluid for magnetic field ranging from 0 – 1 Tesla The Hartmann and Stuart number of the study ranges from 0 – 700 and 0 – 50 respectively. The Reynolds number of the study is 10 4 . It was shown that the convective heat transfer and hence the Nusselt number decreases near the walls perpendicular to the magnetic field due to reduction in turbulent fluctuations with increase of magnetic field. It was observed that the Nusselt number value increases near the walls parallel to the magnetic field as the mean velocity increases near the walls. A singular rise was observed near both the walls near Stuart number ~ 10 which is due to the increase of turbulence levels in the process of changing from turbulent to electromagnetically laminarized flow, see figure 17. When a very low Reynolds number ~ 300 is used, the reduction in Nusselt number near the Hartmann walls is less as shown in figure 18. This shows that the reduction in Nusselt number near the Hartmann walls for the high Reynolds number study is due to the reduction in turbulent fluctuations. The Nusselt number was found to increase near the side walls as the mean velocity increases near the walls. When an insulating duct is used the Nusselt number near the parallel walls did not increase for the case with insulating walls as Developments in Heat Transfer 74 in the case with conducting walls showing the contribution of the ‘M’ shaped velocity profile in the Nusselt number increase near the parallel walls. The Nusselt number near the perpendicular walls was found to decrease at a higher rate in case of insulating walls than that of the study with conducting walls as shown in figure 19. 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 Hartmann wall Side wall Nu/Nu B=0 0.0 1.7 7.5 16.9 30.1 47.0 Tesla St 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 1.2 1.4 Hartmann wall Side wall 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 Hartmann wall Side wall Nu/Nu B=0 0.0 1.7 7.5 16.9 30.1 47.0 Tesla St 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 1.2 1.4 Hartmann wall Side wall Fig. 17. High Reynolds number with conducting walls, Rao and Sankar (2010) Nu/Nu B=0 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 Hartmann wall Side wall Nu/Nu B=0 0.00 1.78 7.52 16.93 30.10 Tesla St 0.0 0.2 0.4 0.5 0.8 0.8 1.0 1.2 1.4 1.6 Hartmann wall Side wall Nu/Nu B=0 0.8 1.0 1.2 1.4 1.6 0.0 0.2 0.4 0.6 0.8 Hartmann wall Side wall Nu/Nu B=0 0.00 1.78 7.52 16.93 30.10 Tesla St 0.0 0.2 0.4 0.5 0.8 0.8 1.0 1.2 1.4 1.6 Hartmann wall Side wall Fig. 18. Low Reynolds number with conducting walls, Rao and Sankar (2010) 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.00 0.20 0.40 0.60 0.80 1.00 Side wall Hartmann wall Nu/Nu B=0 0.00 1.78 7.52 16.93 30.10 47.04 Tesla St 0.00 0.2 0.4 0.6 0.8 1.0 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 Hartmann wall Side wall 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 0.00 0.20 0.40 0.60 0.80 1.00 Side wall Hartmann wall Nu/Nu B=0 0.00 1.78 7.52 16.93 30.10 47.04 Tesla St 0.00 0.2 0.4 0.6 0.8 1.0 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 Hartmann wall Side wall Fig. 19. High Reynolds number with insulating walls, Rao and Sankar (2010) Magneto Hydro-Dynamics and Heat Transfer in Liquid Metal Flows 75 8. Application of liquid metal MHD studies in nuclear fusion reactors International Thermo-nuclear Experimental Reactor is an international organization formed in 1985 comprising of researchers from US, EU, China, Japan, India, Korea and Russia working towards development of a test reactor (TOKOMAK) which is expected to be developed by 2020. The test reactor will be installed in France where the head office of ITER is situated. Salient details of the reactor to be developed are shown in figure 20. The reactor height will be close to 100 ft and would weigh around 38000 tons. The cryostat is the external chamber around the TOKOMK which maintains high vacuum inside it to reduce the heat load from atmosphere through conduction and convection. The fusion of Deuterium and Tritium happens inside the plasma chamber. The magnets are used to confine the plasma created inside the plasma chamber using a magnetic field of 4-8 Tesla. Plasma Chamber Cryostat Central Solenoid Toroidal Magnets Plasma Chamber Cryostat Central Solenoid Toroidal Magnets Fig. 20. Details of the TOKOMAK Tritium breeding modules are used in fusion reactors to produce Tritium by reacting Lithium with neutrons a byproduct of the nuclear fusion reaction. The two basic breeder concepts developed by ITER are liquid breeder and solid breeders. The advantages of liquid breeder over solid breeder are the high Tritium breeding ratio and the Lead-Lithium eutectic can also act as a coolant inside the breeding module which is subjected to high heat Developments in Heat Transfer 76 from plasma and the heat generated in itself due to bombardment of neutrons. The major disadvantages of liquid breeders over solid breeders is the pressure drop in the form of Lorentz force and the reduction in convective heat transfer characteristics of the liquid metal when it is flowing in the presence of intense magnetic field produced by the cryogenic super-conducting magnets. Wong et al. (2008) has mentioned about the various liquid metal breeders being developed around the world details of which is given in the table 3. All the liquid breeder design uses Lithium as the breeding material though most of them use a eutectic of Lead and Lithium because of the lower electrical conductivity and the neutron multiplication ability of Lead. Country Name of TBM Liquid Metal Used US DCLL – Dual Coolant Lead Lithium PbLi EU HCLL – Helium Cooled Lithium Lead PbLi Korea HCML – Helium Cooled Molten Lithium Li India LLCB - Lead-Lithium Cooled Ceramic Breeder PbLi China DFLL – Dual Functional Lithium Lead PbLi Table 3. Details of the liquid TBM developed in the various countries, Wong et al. (2008) First Wall Pb-Li Inlet Poloidal View Exploded View PbLi First Wall Pb-Li Inlet Poloidal View Exploded View PbLi First Wall Pb-Li Inlet Poloidal View Exploded View PbLi Fig. 21. Details of LLCB- TBM, Wong et al. (2008) Magneto Hydro-Dynamics and Heat Transfer in Liquid Metal Flows 77 Indian Lead lithium Cooled Ceramic Breeder (LLCB) – The design description of LLCB is given in Rao et al. (2008). The details of the exploded and cut section views of the LLCB – TBM is shown in figure 21. The two coolants used in LLCB are Helium and a eutectic of Lead- Lithium, Pb-Li. The two coolants are of different molecular properties as Pb-Li has very low Prandtl Number of the order 10 -2 and Helium gas has Prandtl number of ~0.65. The thermal diffusivity of the two fluids were different as the main temperature difference for Helium in straight ducts were concentrated at the viscous sub layer where as the temperature difference for Pb-Li was also present in the mean core region. The material of construction of the cooling channels is Ferritic-Martensitic Steel (FMS) having electrical conductivity of the order 10 6 1/Ώ-m, so the pressure drop associated with the flow was very high. Hence a coating of Alumina (Al 2 O 3 ), which has very low electrical conductivity (~10 -8 1/ Ώ-m) is used on the wet surfaces of the cooling channels This makes the configuration similar to the rectangular channel of Shercliff’s case with all walls insulating i.e. d A = 0 and d B = 0 and hence as mentioned in 4.2.1, the velocity profiles will not have a high velocity jet near the side walls. So the effect of turbulence modification is more significant on the heat transfer characteristics as mentioned in section 6.2.1. The flow will be electromagnetically laminarized and the heat transfer capacity of the Pb-Li deteriorates at high Hartmann numbers. 9. Nomenclature 2a Distance between Hartmann walls 2b Distance between side walls B 0 Magnetic field c Speed of light C p Specific heat d A Electrical conductivity of wall AA d B Electrical conductivity of wall BB D Displacement current E Electric field H Magnetic field strength Ha Hartmann Number j Electric charge k Thermal conductivity k eff Effective thermal conductivity k Τ Turbulent thermal conductivity L Characteristic length N Interaction parameter Nu Nusselt number p Pressure Pr Prandtl number Pr m Magnetic Prandtl number q ''' Volumetric heat generation S Source term Re Reynolds number Developments in Heat Transfer 78 Re m Magnetic Reynolds number t Time T Temperature U Axial velocity U c Centre line velocity U 0 Mean velocity σ Electrical conductivity of fluid η Non-dimensionalized distance in y direction ξ Non-dimensionalized distance in x direction ν Kinematic viscosity of fluid eff ν Effective viscosity τ ν Turbulent viscosity ρ Density of fluid μ Dynamic viscosity of fluid * μ Magnetic permeability c ρ Electric charge density 10. Acknowledgement We would like to acknowledge Altair Engineering India Pvt. Ltd., for providing an opportunity to do the associated work 11. References Alfven, H. (1942). Existence of electromagnetic-hydrodynamic waves. Nature, Vol.150, (1942), pp.405-406. Davidson, H. W. (1968). Compilation of thermo-physical properties of liquid Lithium NASA Technical Note, Washington. D. C., 1968. Evtushenko, I. A.; Hua, T. Q.; Kirillov. I. R.; Reed, C. B. & Sidorenkov, S. S. (1995). The effect of a magnetic field on heat transfer in a slotted channel. Journal of Fusion Engineering and Design, Vol. 27, (1995), pp. 587-592. Fink, D. & Beaty, H. W. (October 1999). Standard handbook for electrical engineers (14 th Edition). McGraw Hill, ISBN 0070220050. Gardener, R. A. & Lykoudis, P. S. (1971a). Magneto-fluid-mechanic pipe flow in a transverse magnetic field Part 1 Isothermal flow. Journal of Fluid Mechanics, Vol.47, (1971), pp 737-764. Gardener, R. A. & Lykoudis, P. S. (1971b). Magneto-fluid-mechanic pipe flow in a transverse magnetic field Part 1 Heat Transfer. Journal of Fluid Mechanics, Vol.48, (1971), pp. 129-141. Happel, J. & Brenner, H. (1981). Low Reynolds Number Hydrodynamics, Springer. ISBN 9001371159. [...]... FTs for sample ATG1G is also shown Shaded bands behind the plot indicate the apatite and zircon FT age distributions of the granitic rocks that intrude into the Hida Belt (Matsuda et al 1998) Error bars show 2σ uncertainty in age 84 Developments in Heat Transfer 3 Thermal modelling associated with frictional heating and estimation of frictional strength In order to estimate the frictional strength of... if no tray is used These relationships are still valid in case the solution is directly poured in a tray  93 Heat Transfer in Freeze-Drying Apparatus Firstly, the heat transfer coefficient between the heating shelf and the bottom of the vial (Kv′) is calculated as the sum of three terms: ' K v = Kc + Kr + K g (3) corresponding to the various heat transfer mechanisms between the fluid and the vial bottom,... be inserted both in vials placed in the core and at the edge of the lot According to their position over the shelf, the vials of a lot can be classified in various groups depending on the different heat transfer mechanisms involved In particular, in case vials are confined by a metal or plastic band, we can generally identify five zones (see Figure 1), which are characterized by different heat transfer. .. and the heating plate, as the resistance to heat transfer of the stainless steel tray is negligible In addition, as already observed by Bruttini et al (1991) for bulk freeze-drying of a solution of cloxacillin monosodium salt, it is confirmed that, for freeze-drying in trays, the heat is mostly transferred by the conduction through the thin layer of gas that separates shelf and tray surfaces In case... gas) introduced in the chamber The drying chamber can be isolated from the condenser by means of a valve that is usually placed in the duct connecting the chamber to the condenser (Mellor, 1978; Jennings, 1999; Oetjen & Haseley, 2004; Franks, 2007) 92 Developments in Heat Transfer Although it is generally considered a “soft” drying process, because of the low operating temperatures, the heat transfer. .. other samples in the vicinity Assuming that the thermal anomaly is cause by frictional heating during a single earthquake, the frictional coefficient and the ancient depth of gouge samples are evaluated by the thermal 82 Developments in Heat Transfer modelling to satisfy the constraints given by the FT thermochronological data with respect to the geometry and alignment of the gouges in the outcrop... described by a nonlinear equation with the following structure: 94 Developments in Heat Transfer ' K v = C1 + C 2 ⋅ Pc (7) 1 + C 3 ⋅ Pc where: ⎧ ⎪ 3 ⎪C 1 = Kc + 4κ T ( eS + ev ) ⎪ ac 2 73. 2 ⎪ Λ0 ⎨C 2 = 2 − ac T ⎪ ⎪ ⎛ 2 73. 2 ⎞ ⎪C 3 = ⎜ Λ 0 ac ⎟ ⎜ λ 2−a ⎪ T ⎟ c ⎝ 0 ⎠ ⎩ (8) ' At this point, K v can be calculated, whichever are the processing conditions, provided that ac, Kc and ℓ (and thus C1, C2 and C3) are known... activation history, In: Thermochronological Methods: From Paleotemperature Constraints to Landscape Evolution Models, Lisker, F., Ventura, B., Glasmacher, U A (Eds.), 33 1 -33 7, The Geol Soc., London, Special Publications, 32 4, ISBN 978-1-86 239 -285-4, London, UK 6 Heat Transfer in Freeze-Drying Apparatus Roberto Pisano, Davide Fissore and Antonello A Barresi Dipartimento di Scienza dei Materiali e Ingegneria... in this step the pressure in the drying chamber is lowered, thus causing ice sublimation This phase is usually carried out at low temperature (ranging, in most cases, from -40°C to -10°C) and, as sublimation requires energy, heat is transferred to the product through the shelf, by acting on the temperature of the fluid flowing in the coil inserted in the shelf 3 Secondary drying: when the sublimation... container bottom) is less important than that due Heat Transfer in Freeze-Drying Apparatus 1 03 to the variability associated to transport phenomena involved in C1 For all these reasons, we assume that the only responsible for the uncertainty on Kv is the parameter C1 Nevertheless, it is worth noticing that the uncertainty, or variability, on ac and ℓ (and thus on C2 and C3) is implicitly included in . that intrude into the Hida Belt (Matsuda et al. 1998). Error bars show 2σ uncertainty in age Developments in Heat Transfer 84 3. Thermal modelling associated with frictional heating and. not increase for the case with insulating walls as Developments in Heat Transfer 74 in the case with conducting walls showing the contribution of the ‘M’ shaped velocity profile in the. Developments in Heat Transfer 70 6.1 Effect of axial magnetic field on convective heat transfer In case of flow of liquid metals in heated channels under the influence of