Prof.
J. Skufca,
Setup. A ½ in steel bar is wrapped will be insulated
with a piece of foam taken from a pool floatation toy.
The intention is to study flow of heat in the
bar by placing one end of the bar in ice water while keeping the other
end at
room temperature.
(See
photos.) The piece of insulation is 87cm in length and
provides approximately one inch of insulation for the bar.
It will be wrapped around the bar, with a
small part of the bar sticking out the end.
(Having a part of the bar stick past the insulation was required
to
stabilize the test apparatus when it is placed in the ice bath.) From the end of the insulation (assumed at x = 0), thermocouple probes are placed
at (approximately) x =10, 20, 30 and
40 cm. The temperature measuring system
employed was two Pasco GLX Passport data loggers, each with two
thermocouples. The only calibration of
the temperature sensors was a single point calibration such that they
all read
the (nearly) the same value at the start of the experiment. The setup was prepared and then allowed to
stabilize in temperature over the next 12 hours.
Conduct
of Experiment.
At
the start of data logging, the insulated bar end was placed in an ice
water
bath, while the other was kept exposed to room temperature. Data was recorded at one-second intervals for
4000s. The bar was removed from the ice
water at time t = 4000s, quickly dried with a towel, and then left
exposed to
room temperature at both ends. Data
logging continued until time t = 12000s.
The data was recovered from the loggers and stored as an EXCEL
file.
Physical
parameters. The thermal diffusivity of steel depends
strongly upon the specific type of steel, which was not known. We believe a reasonable estimate is
.
No published data
could be found for the characteristics of the foam.
Sources
of Experimental Error. There are three primary sources of error for
the recorded data: (1) the calibration of the thermocouples may not be
exact,
(2) the thermocouple probe is about 8mm in length, so its “position”
along the
bar cannot be exact, and (3) the probe is only taped to the outside of
the rod,
and, as such, it is reading a skin temperature (not a bulk temperature)
and it
may suffer inaccuracies due to inadequate thermal contact.
As additional factors that affect the
accuracy of any models, the room temperature is assumed constant,
though it was
actually 21.38 C at beginning of experiment and about 21.5 at the end,
and was
not monitored during the experiment.
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Experimental
apparatus:
Upper left: steel rod with thermocouples attached at 13,23,33,43 cm
from the
right end. The last 3 cm are to be immersed in ice water. Lower left:
the bar
now wrapped in insulating jacket. The insulation is a hollow foam
cylinder sold
as a pool toy. Right: The rod, one end immersed in ice water, the
thermocouples
attached (in pairs) to recording devices.
Basic
Dirichlet representation.
The
typical interpretation of this problem is that it could be modeled by
the
standard one-dimensional heat equation with Dirichlet boundary
conditions:
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where
L = 87 cm Students
should find that they are not able to replicate the data.
An example figure is shown below. (All
numerical solutions computed in MATLAB
using a Crank-Nicholson implicit scheme.)
Figure
1 - Colored
curves are the data, the black (x) are the
model. Altering thermal diffusivity does
not correct the mismatch.
Model
fix 1 – sides not
perfectly insulated. As
a first improvement to the model, we assume that there is heat loss via
the
lateral boundary, modeled as a convective loss term:
(2)
where
and
Tr is the room temperature, assumed to be
constant. However, we have no a
priori knowledge of how to choose γ2.
Additionally, it is not necessarily the case
that we fully understand the boundary conditions. However,
is we move the origin to the
position of the first thermocouple, take the bar to be of length L = 77 cm and use the DATA
from the first thermocouple as the
boundary condition,
we may experiment with various parameter values to find a reasonable
fit (by
eye). We find that
cm–2
gives reasonable results. (See Figure
2.)
Figure
2
Solution assuming convection,
using x = 10cm data as a left boundary condition. The
colored curves are measure data, black x's are the
simulation. Here we take γ2 =
0.0018 .
Although
this new term describing heat loss through the wall
appears to work well when using the data from 10cm as a boundary, when
we
simulate on the full bar, with original Dirichlet boundary conditions,
we are
unable to get a good match to data (see below).
Figure
3 With
Dirichlet boundary, too much heat is lost to the ice water,
causing all the simulation curves to be too cold (during heat removal). After time 4000, we see that the heat returns
to the bar too quickly, resulting in simulation temperatures well above
measured.
Finding
a good boundary condition. To further improve the simulation, we note
that our data using x = 10cm as the
left side (with boundary condition taken from data), we have very good
results. So the heat flow on the bar and
loss out the
side is well modeled. However, when we
run simulations assuming u(0,t)
= 0, we have too much heat loss out the end.
To improve the model, we assume a left side boundary of the form
(3)
where
describes a heat
transfer property between the bar and its environment, and Te
is the environmental temperature at the left end of
the bar. When the bar is in the ice
water (until time 4000) the transfer is to the water, which is assumed
at 0
degrees, and
appears to fit the
data. After time 4000, the transfer is
to the air. Since we were not blowing
air past the end of the bar, the thermal resistance of the still air
can be
significant, and choosing
seems to match the
data and is reasonable in magnitude when compared to the transfer rate
to
water.
Figure
4 Using a
convection term at the left end provides a
reasonable way to achieve the desired temperature profile at x = 10cm. With that profile
correct, the match at the
rest of the data set is quite reasonable.