http://numericalmethods.eng.usf.edu/experiments
Due April 16, 2010 at 11:50 AM
Project: Cooling the Aluminum Cylinder
Background:
A solid aluminum cylinder treated as a lumpedmass
1
system is immersed
in a bath of iced water.
Let us develop the mathematical model for the problem.
When
the cylinder is placed in the iced water bath, the cylinder loses heat to its surroundings by
convection.
Rate of heat lost due to convection =
(
29
(
29
a
A
h
θ
θ
θ

.
(1)
where
=
)
(
t
θ
temperature of cylinder as a function of time,
o
C
)
(
θ
h
= the convective cooling coefficient, W/(m
2

o
C)
=
A
surface area, m
2
=
a
θ
ambient temperature of iced water,
o
C
The energy stored in the mass is given by
Energy stored by mass =
mC
θ
(2)
where
m = mass of the cylinder, kg
C = specific heat of the cylinder, J/(kgK)
From an energy balance,
Rate at which heat is gained ─ Rate at which heat is lost
= Rate at which heat is stored
gives
(
29
dt
d
mC
hA
a
θ
θ
θ
=


(3)
The ordinary differential equation is subjected to
0
)
0
(
θ
θ
=
where
=
0
θ
initial temperature of cylinder,
o
C
Assuming the convective cooling coefficient,
h
to be a constant function of temperature,
the exact solution to the differential equation (3) is
1
It implies that the internal conduction in the trunnion is large enough that the temperature throughout the
trunnion is uniform.
This allows us to assume that the temperature is only a function of time and not of the
location in the trunnion.
This means that if a differential equation governs this physical problem, it would
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 Spring '08
 Kaw,A
 Thermodynamics, Heat, Heat Transfer

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