A 2.5-cm-thick sheet of plastic initially at 21°C is placed between two heated steel plates that are maintained at 138°C. The plastic is to be heated just long enough for its midplane temperature to reach 132°C. If the thermal conductivity of the plastic is 1.1 × 10–3 W/(m K), the thermal diffusivity is 2.7 × 10–6 m2/s, and the thermal resistance at the interface between the plates and the plastic is negligible, calculate: (a) the required heating time, (b the temperature at a plane 0.6 cm from the steel plate at the moment the heating is discontinued, and (c) the time required for the plastic to reach a temperature of 132°C 0.6 cm from the steel plate.
GIVEN
• A sheet of plastic is placed between two heated steel plates
• Sheet thickness (2L) = 2.5 cm = 0.025 m
• Initial temperature (To) = 21°C
• Temperature of steel plates (Ts) = 138°C
• Heat until mid-plane temperature of sheet (Tc) = 132°C
• The thermal conductivity of the plastic (k) = 1.1 × 10–3 W/(m K)
• The thermal diffusivity (?) = 2.7 × 10–6 m2/s
• The thermal resistance at the interface between the plates and the plastic is negligible
FIND
(a) The required heating time (b) The temperature at a plane 0.6 cm from the steel plate at the moment the heating is discontinued (c) The time required for the plastic to reach a temperature of 132°C 0.6 cm from the steel.
ASSUMPTIONS
• The initial temperature of the sheet is uniform
• The temperature of the steel plates is constant
• The thermal conductivity of the sheet is constant
SKETCH
The approximate solutions apply to convective boundary conditions but it can be applied to this problem by letting hc ? ? . Therefore, Bi = ?.
(a) To find the time required to heat the midplane from 21°C to 132°C, first calculate
At center x=0, from we have
for infinite slab
thus the above equation becomes
solving for the time
At center of rubber x=0, we have
for Infinite Slab
For Bi = ?., ? 1 = 1.5708 and C 1 = 1.2732 Thus the above equation becomes
Solving for the time
(b) At 0.6 cm from the steel plate
(c) When the temperature 0.6 cm from the steel plate is 132°C
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