7.57 A copper sphere is to be quenched in an oil bath whose temperature is 50PC

Question

7.57 A copper sphere is to be quenched in an oil bath whose temperature is 50PC The sphere's radius is 30 mm, and the convection coefficient is h 300 W/(m2·°C). Assume the sphere and the oil properties are constant. These properties are given in the following table. The sphere's initial temperature is 400 C Oil Sphere 8900 385 400 Property Density (kg/m3) Specific heat c(kg C) Thermal conductivity kW/(m- 7900 400 Assume that the volume of the oil bath is large enough so that its temperature does not change when the sphere is immersed in the bath. Obtain the dynamic model of the sphere's temperature T. How long will it take for T to reach 130 C? 7.58 Consider the quenching process discussed in Problem 7.57 the oil bath volume is 0.1 m3 dynamic model of the sphere's temperature and the bath temperature. How long will it take for the sphere temperature to reach 130 C? . Neglect any heat loss to the su surroundings and develop a

Explanation / Answer

7.8)

If we neglect heat loss from oil bath and specimen to atmosphere, It is obvious that thermal energy given up by the sphere by decreasing it's temperature results in the thermal increase of quenching bath resulting from its temperature increase.

Initial temperature of sphere = 400 C

sphere radius = 30 mm.

Sphere volume = V = (4/3)*3.14*0.033 = 1.13*10-4.

Mass of the ball = mb= Density * volume = 8900*1.13*10 -4 = 1 kg.

Heat capacity of sphere = Cs = 385 J/kg-k

Heat capacity of bath = Cb = 400 J/kg-k.

Volume of oil bath = Voil = 0.1 m3.

Mass of oil = moil= Volume *density = 0.1*7900 = 790 kg.

Heat gave up by the ball by reaching 130 C = Q = mC*dT = 1*385*(400-130) = 103950 J.

All the above heat is given to oil bath. As a result, oil bath temperature raises. The final temperature of the bath is calculated from

From heat balance equation for oil bath

Q = mC*dT = 103950 = 790*400*(T - 50).

Final temperature of the bath = T = 50.33 C.

Time = t = 103950/9790*400) = 0.29 secs.

103950= 790*400*(

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