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Particle Technology Laboratory Prof. Sotiris E. Pratsinis Sonneggstrasse 3, ML F13, ETH Zentrum Tel.: +41-44-632 25 10 http://www.ptl.ethz.ch 151-0902-00 Micro- and Nano-Particle (MNP) Technology FS17 Exercise 6: Forces on Single Particles Problem 1 The dynamic viscosity f and the density ρf of a fluid can be determined by measuring the stationary settling velocities of two spherical particles of different sizes in this fluid. The first particle has a diameter of 10-3 m and a density of 1.18 g/cm3. The second particle has a diameter of 10-2 m and a density of 4.00 g/cm3. The first particle settles in the Stokes’ regime with a velocity of 10-3 m/s. The second particle settles in the Newton regime with a velocity of 0.89 m/s. Calculate the dynamic viscosity f and the density ρf of the fluid. Problem 2 Carbon black agglomerates with mobility, dm , and primary particle diameter, dp , of 750 and 25 nm, respectively, are filled into a silo at a height of 2 m. In the continuum regime, their number of primary particles, np , is related to dm and dp by (Dastanpour & Rogak, 2016): 1.92 dm 1 np 4.4 0.85 0.03g,p dp where g,p = 1.3 is the geometric standard deviation of the primary particle diameter. a) In the absence of coagulation, how long will it take until all carbon black agglomerates have settled? b) Agglomerates with monodisperse primary particles will settle faster or slower than those with g,p = 1.3? c) What is the error in the settling time calculation assuming carbon black spheres of the same dm ? Data: Viscosity of air (25°C): 1.861×10-5 Pa s Carbon black density: 1.8 g/cm3 1 Reference: Dastanpour R, Rogak SN. The effect of primary particle polydispersity on the morphology and mobility diameter of the fractal agglomerates in different flow regimes, J. Aerosol Sci. 2017; 94: 22-32. Problem 3 The steady state settling velocity (terminal velocity) of a particle can be utilized to determine its diameter. A sphere of density 2500 kg/m3 falls freely under gravity in a fluid of density 700 kg/m3 and viscosity 0.5×10-3 Pa s and reaches a settling velocity of 0.15 m/s. a) Calculate the diameter of the sphere. b) What would be the edge length of a cube of the same material falling in the same fluid at the same steady state settling velocity? Problem 4 Spray drying is commonly used in the food or chemical industry for the production of easily soluble, powdered products such as instant coffee, detergents or pharmaceutics. In spray drying a slurry (suspension) is atomized into a flow of hot gas (Figure 1). The liquid evaporates from the droplets leaving solid particles at the outlet of the drying chamber. The drying time is governed by heat- and mass transfer processes, which depend on the relative velocity between the gas and the droplets. Along with the settling velocity of the droplets it determines the height of the drying chamber. Figure 1. Example of a spray drying unit with cyclone separator product powder recovery. www.malvern.co.uk/. a) The atomizer of the spray dryer breaks the feed suspension into fine droplets and thereby enlarges the surface area enabling fast drying. Calculate the surface area of 1L slurry that is sprayed into spherical 50 m droplets. b) The atomizer generates polydisperse droplets. The droplet size distribution changes during the drying process when converting droplets into particles as shown in Figure 2. The droplet size distribution results in a distribution of the steady-state settling velocity, which has to be accounted for in the design of 2 Passage P the unit. Estimate the range of settling velocities, by calculating the settling velocity of entering droplets with a size corresponding to the 10 and 90% passage. Outlet (Particles) Inlet (Droplets) Figure 2: Droplet size distribution in the spray dryer. Data: Gas temperature: 300 °C, viscosity Gas (300 °C) = 29.3×10-6 m2/s, Gas density Gas (300 °C) = 0.924 kg/m3 Droplet density ≈ density of water: w = 1000 kg/m3 3