Resfriamento de Esfera em Banho Térmico - Análise Detalhada e Gráficos

PROBLEM 575 KNOWN Sphere quenching in a constant temperature bath FIND a Plot T0t and Trot as function of time b Time required for surface to reach 415 K t c Heat flux when Tro t 415 K d Energy lost by sphere in cooling to Tro t 415 K e Steady state temperature reached after sphere is insulated at t t f Effect of h on center and surface temperature histories SCHEMATIC ASSUMPTIONS 1 Onedimensional radial conduction 2 Constant properties 3 Uniform initial temperature ANALYSIS a Calculate Biot number to determine if sphere behaves as spatially isothermal object 2 o c h r 3 75W m K 0015 m 3 hL Bi 022 k k 17 W m K Hence temperature gradients exist in the sphere and Trt vs t appears as shown above b The exact solution may be used to find t when Tro t 415 K We assume that the oneterm approximation is valid and check later Hence with 2 o 75 Wm K 0015 m hr Bi 0662 k 17 Wm K from Table 51 ζ1 13188 C1 11877 Then o i Tr t T 415 C 320 C r 1Fo 01979 T T 800 C 320 C θ and Equation 553b can be solved for o θ o 1 rsin 1 r 01979 13188 1sin13188 02695 θ θ ζ ζ Then Equation 553c can be solved for Fo o 1 2 2 1 1 1 Fo ln C ln 0269511877 0853 13188 θ ζ 2 3 p 2 2 o o c r 400kg m 1600J kg K t Fo Fo r 0853 0015m 72 s k 17 W m K ρ α Note that the oneterm approximation is accurate since Fo 02 Continued PROBLEM 575 Cont c The heat flux at the outer surface at time t is given by Newtons law of cooling 2 2 o q h T r t T 75W m K 415 320 K 7125W m The manner in which q is calculated indicates that energy is leaving the sphere d The energy lost by the sphere during the cooling process from t 0 to t can be determined from Equation 555 o 1 1 1 3 3 o 1 3 Q 3 02695 1 sin cos 1 sin13188 13188cos13188 0775 Q 13188 θ ζ ζ ζ ζ The energy loss by the sphere with V πD36 is therefore from Equation 547 3 o p i Q 0775Q 0775 D 6 c T T ρ π 3 3 Q 0775 400kg m 0030m 6 1600J kg K 800 320 K 3364J π e If at time t the surface of the sphere is perfectly insulated eventually the temperature of the sphere will be uniform at T Applying conservation of energy to the sphere over a time interval Ein Eout ΔE Efinal Einitial Hence Q ρcVT T Qo where Qo ρcVTi T Dividing by Qo and regrouping we obtain o i T T 1 Q Q T T 320K 1 0775 800 320 K 428K f Using the IHT Transient Conduction Model for a Sphere the following graphical results were generated 0 50 100 150 Time t s 300 400 500 600 700 800 Temperature TK h 75 Wm2K r ro h 75 Wm2K r 0 h 200 Wm2K r ro h 200 Wm2K r 0 0 50 100 150 Time ts 0 30000 60000 90000 Heat flux qrot Wm2K h 75 Wm2K h 200 Wm2K The quenching process is clearly accelerated by increasing h from 75 to 200 Wm2K and is virtually completed by t 100s for the larger value of h Note that for both values of h the temperature difference T0t Trot decreases with increasing t Although the surface heat flux for h 200 Wm2K is initially larger than that for h 75 Wm2K the more rapid decline in Trot causes it to become smaller at t 30s COMMENTS Using the Transient ConductionSphere model in IHT based upon multipleterm series solution the following results were obtained t 721 s QQo 07745 and T 428 K