Heat Conduction Experiment

Prof. J. Skufca, Clarkson University

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.

 

 

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.

 

 

Modeling the heat flow

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:

,

                                                                                (1)

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.)

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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.)

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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).

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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.

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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.