Quenching of a billet 32 pts a solid sphere initially

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12.Quenching of a billet (32 pts.)A solid sphere initially at temperatureT0(radiusR, massm, heat capacityˆCpˆCV, thermal conductivityk) is totally immersed in a stagnant liquid of massMgreatermuchmand temperatureT. Heat transfer from the surfaceof the sphere to the fluid is described in terms of a heat transfer coefficient,Q=hA(T|r=R-T).For sufficiently small Biot numbers, BihR/klessmuch1, it is reasonable to assume that the temperature withinthe sphere is uniform.Reduce the appropriate macroscopic balances on the following page to determine the temperature of thesphere as a function of time,T(t), in terms of parameters defined above.8
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Macroscopic BalancesMass:dmtotdt=-ΔwMomentum:dPtotdt=-ΔparenleftBigg(v2)(v)w+pSparenrightBiggu+mtotg-FfsEnergy Balance:ddt(Utot+Ktot+ Φtot) =-ΔbracketleftBiggparenleftBiggˆH+12(v3)(v)+ghparenrightBiggwbracketrightBigg+Q+WmMechanical Energy Balance (steady state):ΔparenleftBigg12(v3)(v)parenrightBigg+gΔh+integraldisplay21dpρ=ˆWm-ˆEvAdditional information:f(Re) =braceleftBigg16/ReRe21000.0791/Re1/42100<Re<105ΔˆH=braceleftBiggˆCpΔT+ Δp/ρΔˆUincompr. liquids and solids, const.ρ,ˆCpintegraltext21ˆCp(T)dTideal gasˆEv= 2(v)2LDf(Re)for a straight pipe with circular cross-section9
13.Reaction in a fluidized bed reactor (35 pts.)A tubular fluidized bed reactor (radiusR) contains catalyst particles that cause speciesAto react viaAproductsas illustrated in the figure below.zvz= constantcA(z=0) =cA0RA=-(ka)1’’cAAlthough the reaction is actually heterogeneous, the apparent homogeneous reaction rate (occurring over avolume containing many particles) is given byRA=-(kprimeprime1a)cA,whereais the surface area per unit volume. Here,RAis the rate of appearance ofAper unit volume.The fluid flows through the reactor with velocityvzin thezdirection. The presence of the particles causesthe velocity profile to be uniform (i.e.,vz=constant). The average velocity of particles is zero.Assuming that the inlet concentration ofAin inert liquidBiscA0, that diffusion is negligible relative toconvection (in which case the convective molar flux can be writtenNA,z=cAvz), that the process occurs atconstant temperature, and that mass transfer limitations to reaction are negligible, use a shell mass balance todetermine the steady-state concentration of speciesAas a function of position,cA(z).10

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An Introduction to Physical Science
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Chapter 4 / Exercise 01
An Introduction to Physical Science
Shipman/Wilson
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