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Thermal Analysis 333

sink is initially at a temperature of 28°C. Suppose a heat flux input is prescribed
on the bottom face of the heat sink. The heat flux is of a symmetrical square wave
pattern with a period of 60 s (equal time for each half-cycle) and peak magni-
tudes transitioning from 800 W/m to 0 W/m . A fan forces air over all surfaces
2
2
of the heat sink except for the bottom face, where the heat flux is prescribed. The
surrounding air is 28°C with a film coefficient of 30 W/(m °C). Assume that the
2
heat sink is made of aluminum with thermal conductivity k = 170 W/(m K), den-
sity ρ = 2800 kg/m , specific heat c = 870 J/(kg K), Young’s modulus E = 70 GPa,
3
Poisson’s ratio ν = 0.3, and thermal expansion coefficient α = 22 × 10 /°C. Study
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the transient thermal response of the heat sink during the first three cycles.

9.7 The electric kettle base shown below has a thickness of 0.5 cm and is made of struc-
tural steel with thermal conductivity k = 60.5 W/(m K), density ρ = 7850 kg/m ,
3
and specific heat c = 434 J/(kg K). Suppose a heat flux condition is defined through
a serpentine heating element embedded at the bottom of the kettle base. Except
for the imprinted heating element face, all other faces of the kettle base are sub-
jected to convective heat transfer with a bulk air temperature of 40°C and a film
coefficient of 80 W/(m °C). (1) Study the steady-state thermal response of the
2
kettle base if a constant heat flux of 800 W/(m °C) is applied through the heat-
2
ing element. (2) Suppose the heating element is cycled on and off with a period
of 60 s (30 s on-time followed by 30 s off-time). During on-time, the heat flux is
of magnitude 1000 W/(m °C). Study the transient thermal response of the kettle
2
base for a period of 120 s using the steady-state solution as the initial condition.

Isometric view of the two sides of the kettle base

Heating element is shown as
an imprinted face on the
bottom of the kettle base.