The practive of baling tires was introduced to reduce the space taken up by tires at disposal sites.  Tires pose a health threat in occupying disposal sites because of insect and rodent infestation, as well as gas accumulation.  To alleviate these problems, tire disposal sites often choose to shred or bale tires, the later of which entails stacking them in rows of similar tires laid flat on their side-wall.  After the rows are assembled, the bales are then compressed and bound with wire, resulting in approximate dimensions of 5 feet by 4 feet by 3 feet, with an estimated weight of 1 ton.

Though tire bales are presently used for retaining walls and sound barriers, their usage in residential and/or commercial construction purposes require that the insulating properties of the composite material be found.  Determining whether or not the baled tires have sufficient insulating properties for use in construction projects includes finding a model pertinent to these specific situations.  This modeling process involves finding equations for heat transfer, then applying these expressions to experimental procedures to be conducted on tire bales.  The model presented in this report is an empirical heat transfer, or thermal diffusivity model, which includes mathematical calculations based on thermal diffusivity, thermal conductivity and specific heat.

Research into the thermal properties of all necessary individual materials comprising is also required.  Accordingly, we have proposed the modeling of an "Effective" thermal diffusivity, described as the diffusivity of the comprehensive tire bale.  In order to model the effective measure of diffusivity for this non-homogeneous composite material, accurate accounts of thermal properties for both air and rubber, as well as the ratios of air and rubber in a generalized tire bale are necessary.  Difficulty lies in the estimation of these thermal values because of the non-uniform processes by which tires are both manufactured and baled.

where alpha is thermal diffusivity, k is the thermal conductivity, rho is the material density, and c is specific heat.

Density is found from the mass and volume of a given bale, whereas thermal conduvtivity and specific heat must be predicted.  Two main parameters are included in ther predictions of thermal conductivity and specific hear.  Essentially, predictions calculations must account for the ratio of air to tire in the bale, as well as the ratios of the various components in tires.  When these factors are combined, and the mass and volume of a tire bale are established, and effective diffusivity constant can be obtained.

Next, we investigated the thermal properties of the materials included in tires and aftermany resources were consulted, the following results were acheived:

Table 1
 

a)  Chemical Ruber Company Handbook of Chemistry and Physics, 1996/97 Edition, Chemical rubber Publishing Co., Cleveland, OH, 1997.
b) Handbookof Physical Properties of Liquids and Gases, 2nd Edition, Vargaftik, N.B., Hemisphere Publishing Corp., Washington, 1975.
c) ASHRAE Handbook of Fundamentals, American Society of Heating, Refrigeration and Air-Conditioning Engineers, Inc., Atlanta, GA, 1981.
d) ASHRAE Handbook of Fundamentals, American Society of Heating, Refrigeration and Air-Conditioning Engineers, Inc., Atlanta, GA, 1989.
e) Physical Properties of Hydrocarbons, Vol. 3, Gallant, Robert W. and Yaws, Carl L., Gulf Publishing Company, Houston, 1993.
f) Handbook of Thermal Conductivity, Vol. 1-4, Yaws, Carl L., Gulf Publishing Company, Houston, 1995.
All unmarked thermal conductivity values came from the Handbook of Thermal Conductivity.

In forming the modeling package, there are significantly different approaches based on the way in which data is received for input.  As such, we restricted our model based on a specific form of experimental procedure.  In doing so, the following assumptions were considered necessary, beginning with that of undeterminable value.  Since the manufacture and composition of tire bale have significant error in ther respective processes, in addition to a lack of known data about various tire compositions, we first made an assumption about the effect of surface processes on a modeling system.

We made the presumption that both convective and radiative heat transfer processes would have insignificant impact on the heat transfer through the solid, which limits the error imposed on boundary conditions in the differential analysis that follows.  This condition also requires that experiments be conducted in a heat transition range between zerp and one-hundred-fifty degrees Fahrenheit.  Essentially, this serves to further limit the error from additional variables introduced by boundary effects such as off gassing or physical deformation.  We therefore decided that the experiment we could best model should ne conducted such that we can assume a constant uniform temperature around the exterior surface of the tire bale while undergoing data gathering.  The suggested procedure should thus be conducted as follows:

• The tire bale should be allowed to come to an equilibrium room temperature, while resting on elevated platforms requiring as little surface contact as possible.
   
• The tire bale will likely be oriented such that the narrowest dimensional depth is vertical, thus remaining as short and stable as possible.
   
• Next, the sealed room containinf the tire bale should then be brought up to a final temperature by way of ambiently heating the room uniformly.
   
• Thermocouples delivering temperature readings at discrete time intervals should be drilled into the vertical center of the bale in the form of a matrix laying parallel to the floor.
   
• The matrix of thermocouples should furthermore be such that spaces between the thermocouple nodes in both dimensional directions are equidistant, which allows for rapid data processing and computing (Figure 1).
   
• Additionally, the thermocouple matrix should be positioned such that it is squarely centered with respect to its exterior walls, which allows for equidistant heat transfer estimations with respect to presumably constant termperature against the exterior surface.
   
• Finally, the time intervals between temperature readings should likewise be equal.  All of these equivalence conditions facilitate an expeditious implementation of a finite difference estimation for effective thermal diffusivity of a non-homogeneous composite material.

Figure 1 — Thermocouple Placement Withing tire Bale
 

The use of input data files requires that we first establish the structure and organization that must be adhered to on the part of those storing the data.  To begin with, we have configured the computer implementation to read data from Microsoft Excel spreadsheets as seen in Figure 2 below.

Figure 2 

Spreadsheet columns A, B and C represent discrete positions of time, relative x-position and y-position, respectively.  These can be interpreted as a time-interval iteration counter in the case of the time column (A), and relative dimensional indices in the cases of the x-position and y-position columns (B  and C).  Column D, on the other hand, represents the decimal value of the temperature reading at a given row's relative positions and time index.

Fundamental unsteady-state linear heat transfer can bemodeled through the second-order partial differential equation that follows, where alpha represent the thermal diffusivity of the material in question, and T represents a differentiable function of the variables x (relative position) and t (elapsed time).  As will be discussed in this section, we have employed the standard process of Fourier sine series expansion to solve this partial differential equation for the actual temperature distribution with respect to relative one-dimensional position and elapsed time.

By way of separation of parameters, we see that:

Since each ordinary differential equation (ODE) - that of X and T, respectively - are equal to some constant value, we can state that they are independent function.  Therefore, the actual dual-parameter temperature distribution equation can be modeled as the product of the solutions to each ODE, where we have replaced the constant C with the arbitrary value negative lamba squared.

By solving each ODE separately, we see that:

We also note the initial and boundary conditions for the one-dimensional model (see Assumptions — Data Gathering for further explanation):

From these equations, we note that the Fourier series equivalent for T(x,t) can be written in the form that follows,(8) where the newly introduced variable L represents the depth through which heat is transferred:

At this point, let us first denote the quantity (T-sub-i minus T-sub-infinity) as delta-sub-i.  The next step involves the following integration, which yields:

Substituting these values back into our Fourier series expansion, we see that the series with the A term drops out of the equation entirely, due to a constant value of zero for all values of n.  We also not that the series term with the (1 - cos(n pi)) in it equals zero when the value of n is even.  Thus, we can reduce our one-dimensional temperature distribution model as follows:

The temperature distribution equation described in the previous section accounts for the modeling of heat transfer in one-dimension, which can be described as the perpendicular heat transfer through an infinite wall. However, due to this project's focus on the properties of tire bales, we are forced to effectively model how the bale will perform as a three-dimensional solid, exposed to heat sources on all sides. Therefore, referring to Holman (1997),(9) we observe that the three-dimensional modeling of heat transfer through a rectangular parallelepiped is directly related to the product of ratios obtained from solving for the linear temperature distribution in each of the three dimensional directions. Where Tx, Ty and Tz are the temperatures obtained from their respective one-dimensional Fourier series expansions, the following relationship holds true:

Therefore, the modeling of the volumetric temperature distribution within a given tire bale can be achieved by employing this composition of linear models. This result subsequently facilitates the linear graphing of temperature distribution at given time readings, but with respect to the accuracy of the three-dimensional model. Figure 3 shows an example of the graphical display of temperature distribution at various times, and furthermore serves to verify the basis of our modeling expression. The graph below was produced with the equations shown above, analyzed and graphed with Mathematica.(10)

Figure 3

The figure shows the family of time-constant curves for the one-dimensional temperature distribution, where the bottom-most curve corresponds to the temperature distribution as t approaches zero and the top curve approaches a flat line as t approaches infinity. For more information, complete documentation of the analysis performed in Mathematica is available in Appendix B.

The following represents the algorithm by which this model can diagrammatically predict the behavior of heated solids over time, depending on the material-specific thermal diffusivity. This algorithm will be vital to a computer implementation that produces a graph like that in Figure 3, as well as predicts the temperature at any given position over a specified amount of time.

Algorithm: Compute Temperature at Point (x, y, z) and Time (t)

1. Calculate temperature from X-direction linear model.
2. Calculate temperature from Y-direction linear model.
3. Calculate temperature from Z-direction linear model.
4. Determine the effective temperature at the point (x,y,z,t) by composing the results from steps I - III by way of the aforementioned composition of linear model temperature difference ratios.

The basis of modeling this component of the project has to do with interpreting discrete data points gathered from individual experimental procedures. As such, there are various numerical methods available to estimate differential values. The most readily adaptable method for this situation is the finite difference method for heat conduction in two directions, as outlined by Holman (1997).(11) This basic equation relates the temperature at a discrete position (i, j) after (t + 1) time intervals based on an average of the temperature of the current node as well as the temperatures of the surrounding nodes at time interval (t), as seen in Figure 4. Note that the variable alpha in the following expression represents the thermal diffusivity of the material in question.

Figure 4

You may notice the use of matrix indices with positive and negative positions relative to the nodal temperature being determined. This suggests that our model would function correctly only for the nodes within the interior of the nodal matrix, leaving a significant number of data points unused. Thus, we have chosen to avert this difficulty by altering each temperature reading matrix such that it incorporates the presumed-constant exterior surface temperature (see Assumptions and Limitations for more information about the experimental procedure configuration). In the augmented matrix, the number of nodes in each of the x- and y-directions is two greater than the number of thermocouples, which allows us to operate on all of the thermocouple nodes while leaving each of the exterior nodes unchanged over time. This concept is pictured as a top-down view for a single time reading in Figure 5, where the shaded nodes represent a constant ambient temperature applied to the tire bale and the unfilled nodes represent the time-dependent temperatures at the thermocouples within the tire bale.

Figure 5

Based on this experimental configuration and the equation in Figure 4, we can solve for multiple estimated values of the material's effective diffusivity. In fact, by incorporating the presumably constant exterior temperature nodes around the bale, we can calculate the following number of diffusivity values:

Our model relies on determining an effective diffusivity by way of averaging this list of estimated diffusivity values. Moreover, the greater the number of time readings and sensors, the greater the accuracy of the averaging process, which fits well with our presumption that it should take a relatively long time to completely heat a given tire bale.

The one consideration we must take while using this method is that the function becomes unstable under the following condition. If the modulus of stability is greater than 0.25,

then the second law of thermodynamics is violated, such that the temperature of a discrete node decreases when heat is applied to it. The fact that this error condition must be checked prior to calculating the heat transfer differential at each individual node illustrates why we have placed restrictions on equidistant spacing within the nodal network, as well as equal time intervals during individual experiments.

Though this model is intended for use with particular experimental procedures, as mentioned previously in this report, there exists the problem of testing the accuracy of this modeling approach prior to its use. However, tables of temperature readings over subsequent times are not readily available in a form suitable for use with this modeling approach. Thus, we generated our own test data by calculating temperature matrices similar to those pictured in Figure 5 from homogeneous materials for which the thermal diffusivity (a) is known. For example, by generating a series of temperature reading matrices for a material such as solid natural rubber, we can then apply some percent of random error (noise) to those temperature matrices and subsequently use them as input to the component of our model that extrapolates an effective thermal conductivity. Though this only tests the algorithms and computer code involved with these processes, this procedure will assure the computer implementation is functionally ready for use when test data become available from experiments conducted on tire bales.

As described previously in this report, our proposed model could be configured in as many ways as there exist to denote procedurally unique research findings. This selection of modeling approaches has been significantly defined by the method in which data shall be gathered and recorded. As such, this model is structured only to work within the guidelines for data gathering explained in the section titled Assumptions and Limitations. Structuring our model in this manner allows the finite difference algorithm to work for any discrete number of time readings and thermocouple nodes in the x- and y-directions. Furthermore, the computer implementation of this algorithm can be feasibly adapted to facilitate data retrieval from a three-dimensional network of thermocouples, rather than a planar matrix of data points.

Finally, this model can be readily adapted to operate when the spacing intervals for data points are not equidistant, or time intervals are not equivalent. These complexities, however, would each add another dimension to the vector space of experimental data points represented in spreadsheet format, and would furthermore exponentially compound the number of computations necessary for data processing. For this reason, the computer implementation of this modeling system has been restricted with the equivalence conditions previously discussed.

The primary goal for the computer implementation of our model is to provide the user with a conveniently interfaced modeling tool to interpret diffusivity values from research data. The program has been designed to find thermal diffusivity values as well as simulate temperature readings for given conditions. This software tool has been specifically formulated to evaluate the thermal properties of tire bales, yet can be extrapolated to other materials.

Essentially, the computer component of this project is to read data from a Microsoft Excel spreadsheet where each data point reflects the temperature at a given three-dimensional position within the tire bale, and at a given distinct elapsed time after the ambient heat source has been applied. The exact configuration for the Excel data files is given in the above section Assumptions and Limitations, Figure 2.

We have chosen to use Visual Basic to implement the software component for two fundamental reasons, first of which is because of Visual Basic's straightforward compatibility with Excel files, both reading from and writing to spreadsheets with automated controls. More importantly, however, Visual Basic is presently used by the probable end-user of this application, which allows us to tailor the software such that it can readily be updated or altered due to future project developments or needs.

In light of the relative lack of original project guidelines, the following specifications were determined after research of the necessary thermal parameters and experimental procedures as outlined in the previous sections of this report.

The two primary goals for the functionality of the program are determining the diffusivity from data files and graphing the temperature distribution. The first of these requires data files with experimentally given temperatures at a discrete position and time. The latter functional specification is dependent upon implementation of the solution to the heat transfer equation, described in the above modeling section. This heat transfer expression, as implemented in the program, is formatted using non-metric English units, due to extensive use of these units within the experimental framework.

The most crucial component of the system specifications is the ability of the program to operate within the Microsoft® Windows environment. This facilitates the software's manipulation of Excel® data files exclusively, which is compatible with the format of the experimental data files.

We were asked to deliver an executable file specific to the experimental application described in detail in this report, as well as a README.txt file, which contains installation instructions. We also were asked to deliver the Visual Basic project (.vbp) and form (.frm) files. We also formatted these files to facilitate future adaptations to the program.

The rudimentary functional requirements of the software are such that only a few screens or windows are necessary. Moreover, each screen will require access to some or all of the same set of data. Thus, the data shall be configured in a two-tiered object-oriented format, with each procedural form (window) representing a child class of the main menu window. This structure allows the main menu to pass only the necessary data to the child forms, where either a temperature distribution will be determined and graphed or an effective diffusivity will be calculated from data stored in an Excel spreadsheet.

If the Temperature Distribution option is selected from the main menu window, the application will subsequently prompt the user for the necessary information to produce either a specific temperature at a given point and time, or a graphical model of a one-dimensional temperature distribution (Figure 3). The information gathered from the user includes any combination of the following:

•    x, y and z dimensional sizes of the tire bale,
•    x, y and z coordinates for which temperature is desired,
•    an elapsed time after application of the heat source,
•    initial temperature within the tire bale,
•    temperature of the applied ambient heat source, and
•    diffusivity value for the tire bale

The Temperature Distribution selection allows calculation of either a temperature at a discrete point, or the temperature distribution with respect to one-dimension. To determine a discrete point temperature distribution, all coordinate values as well as a final time must be entered. The outcome is a specific temperature value for the tire bale at the discrete point and after an elapsed period of time. Alternatively, calculation of one-dimensional temperature distribution is displayed in graph format. The family of time constant one-dimensional temperature distributions are found by selecting a specific final time, as well as which dimension to vary.

Conversely, if the Determine Diffusivity option is selected from the main menu window, the application will prompt the user for a Microsoftâ® Excel input data file containing temperature readings established in previous experiments. As is conveniently facilitated through Visual Basic©, this will be done by way of a standard Windowsâ File Open dialog box. The program is currently organized to read data from a spreadsheet with four columns representing time, relative position, and temperature. For more information, see Assumptions and Limitations, Figure 2. The software will then process the data points by way of the finite difference algorithm described in the Mathematical Modeling section to determine an effective diffusivity from various points and times within the experimental tire bale. After these calculations, the software will then display the ascertained effective diffusivity value.

The research conducted with respect to thermal properties indicates tire bales are in fact an insulating material. This conclusion was gathered by investigating the composition of tires, finding an approximate value for the thermal properties of tires. The thermal properties found for tires were then combined with the thermal properties of air in a ratio expression to evaluate effective thermal properties for the tire bale. To test the hypothetical values found for effective thermal properties, experiments must be conducted. The mathematical procedures outlined above provide methods for analyzing these tests. These mathematical expressions have been proven to accurately define heat transfer and temperature distribution within solids. Modeling techniques were used to effectively implement the relationships of these thermal values into a software environment. Applying the mathematical calculations into the computer program outlined above will provide data-based conclusions about the insulating properties of tire bales. The program described will not only give diffusivity values, but also accurately estimate the temperature distribution in an experimental tire bale.

Based on the methodologies outlined in this paper, and with further analysis of the error encountered at various stages of the data gathering process, our model will effectively interpret diffusivity data for tire bales as a non-homogeneous material. The user will now have the ability to easily interpret the insulating properties of a tested bale, which is the most significant purpose of this project. The processes described above will enable tests to be conducted to accurately analyze the diffusivity of a tire bale. From this study, tests can be further modified and conducted to investigate other thermal and insulating properties of a tire bale so as to help legitimize the use of tire bales in construction projects.

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