Introduction

The package SmartCAE HeatTransmission calculates the temporal course of the temperatures within a multilayered wall for various boundary conditions. The height and width of the wall are assumend to be much larger than the thickness of the wall, so that the temperature varies only in the direction perpendicular to the wall surface and can be calculated by solving the one dimensional heat equation along this direction. For simple situations there is also a module which solves the steady state case of the corresponding problem.

The resulting one dimensional heat equation is a partial differential equation with two independent variables and the solution is the temperature as a function of time and position within the wall. This solution is determined by numerical means. The calculation can take into account boundary conditions for convection, radiation and explicit external heat fluxes through the wall surfaces. The temperature dependence of material properties are taken into account exactly. For the steady state case the default is to use only material properties for averaged temperatures, which is very fast and often a good approximation, but optionally the exact dependencies can also be taken into account.

The numerical solution is determined with the powerful differential equation solver of Mathematica which uses the "Method of Lines" with a pseudospectral discretization ("Chebyshev–Gauss–Lobatto points") in the spatial direction and a variable step solver for the time coordinate which can handle stiff systems with good accuracy and low calculation times. With the combination of these methods, for almost all relevant cases a relatively coarse grid provides good accuracy combined with very short calculation times. The underlying solver has built in checks for instabilities and errors which enables to even handle those rare cases where the coarse grid is not sufficient: when the checks suggest that the solution was not accurate, the calculation is restarted with finer and finer spatial grids until the the checks don't trigger accuracy warnings anymore. This is all automatic so that the user does not need any knowledge about the details of these procedures.

Start Palette

If you install the Package, the following start palette should be shown:

If you have installed the Package and use Mathematica, you will find that palette also in the "Palettes" menu:

If you use the Mathematica-Player Pro, there is no Palettes menu and you can find the position of the Palette within the packages installation directory in the subdirectory FrontEnd -> Palettes. Evaluate the following line to find the position on your system:

You can copy this palette to any place in the file system or create links to it and then use the usual way of your operating system (usually a click or double click in a file browser) to open it with either Mathematica or the Mathematica-Player.

The start palette lets you:

show this documentation,

start the heattransmission-program

inspect results of former calculations

define materials,

change settings

or uninstall the package.

Heat Transmission Calculation

1. Start of Program

If you click the "Start" button next to the "Run New Simulation" entry in the start palette, a graphical user interface will be opened that lets you define the parameters for the heat transmission calculation and looks like this:

On the top there is a toolbar with 4 buttons:

The "Load" and "Save" buttons let you save the parameter set to a file or load previously saved parameter sets. You can learn more details about where the files are saved and how you can change that location in the section "Settings" -> "Repositories". With the "Run" and "Abort" buttons you can start a calculation or abort a running one, respectively (these are only existing for the transient case).

In the main part of the window contains 4 sections, the first giving a very short description of how the program works, the other three let you define the parameters of the wall, the boundary conditions on the left and right side of the wall and the time to simulate and the solver details. You can open and close each of these section by clicking on the small triangles or arrowheads left to the section header.

2. Definition of Wall Parameters

The first step is the definition of the parameters of the wall.

You can open the Wall section by clicking on the small triangle right of the Wall section header. You will find some subsections that you can again open with similar little triangles:

For the definition of a wall there are available the following parameters:

StartTemperature:

Initial temperature of the wall. The temperature can be given as a constant value or a function of position within the wall. To define a function choose "Interpolating Function", see section "Defining Functions" for details on how to define a function or import data to be used for the function definition. This parameter does not exist for the steady state case.

Layers:

Tabular data with a row for each layer that defines the Material and thickness of each layer. You can click on the row headings to get a simple menu which lets you add or delete layers. See section "Defining Materials" for further details on how to add and change material definitions.

The wall sections contains some preliminary results that you can use to check the plausability of your input:

Sketch:

Sketch shows a cross section of the wall and a simple 3D-visualization of the layers.

Summary:

Summary gives you an overview of the relevant properties of each layer.

Steady State Case

3. Definition of Boundary Conditions

In the section for the definition of the boundary conditions there are two identical section for the boundary conditions of the left and right wall (the wall must not necessarily be oriented vertically, but depending on the orientation you might need different values for the heat transfer coefficient).

Each section for the boundary conditions of one side of the wall has three more subsections, for the heat transfer by convection and radiation and for an explicitly given heat flux, respectively.

Some of these parameters can be given as constant value or as a function. In the section "Defining Functions" you can find a more detailed description of how to input function-valued parameters.

The default values for the boundary value parameters are such that radiation is effectively switched off due to the default value for the emissivity ϵ=0 and the explicitly given heat flux is also set to zero.

4. Definition of Solving Method

In the section "Solver" you can choose the calculation method. It is possible to choose from "AverageLayer" and "Exact":

If the material properties of all layers do not depend on temperature, both methods will yield identical results. Only if the material properties have a temperature dependence and this dependence and the temperature differences within one layer are large enough, a difference between the two methods can be recognized. When choosing "Exact", the "AverageLayer" solution is also determined since it is so fast and shown along with the exact solution in orange, so one can see the difference of the two. This is an example where the temperature differences in the middle layer can not be neglected:

Note that still the temperatures in the first and third layer are almost identical and so are the heat fluxes through the wall.

5. Results

For the steady state case the outputs given are a graphics of the temperature distribution within the wall, two summary tables and a possibility to get an exact temperature value for a given position.

The graphics of the temperature distributions shows also all non vanishing heat fluxes on the left and right side of the wall. These are given as totals and also broken down into convection, radiation and explicitly given heat flux. The total heat flux must be identical on both sides for the steady state case:

The table "LayersTable" shows a summary of the wall construction including the temperatures in each layer and the average material properties in each layer. When these depend on the temperature they are not known beforehand but part of the solution. Some of these values are also given for the entire wall. Clicking on the "Export" button will let you save that table to an Excel file.

The table "SurfaceTable" shows the conditions on the left and right surface of the wall. Along with the boundary condition parameters the heat fluxes are given, again broken down into convection, radiation and explicitly given heat flux:

There is also the possibility to get an exact temperature value for a given position:

The position can be given in the upper input field (x) in mm, the corresponding temperature is then shown in the lower field in °C.

Transient Case

3. Definition of Boundary Conditions

The section "Ambience" lets you define the boundary conditions at the left and right side of the wall. For each side there is the possibility to define convection, radiation and external heat flux parameters. The parameters which define the boundary conditions can either be given as constants or as functions of time (temperatures, emissivity, heat flux) or temperature (heat transfer coefficient). The default settings are constant values for all parameters: the radiation is switched off by choosing emissivity ϵ = 0 and the external heat flux is also set to 0.

If you want to define time or temperature dependent values for one of the parameter, you must select the corresponding "Interpolating Function" radio button. See section "Defining Functions" for more details on how to define a function and how to import tabular data to be used for the function definition.

4. Definition of SimulationTime

Finaly you will need to define the time you want the problem to be solved for. You will do this in the Solver section, which only has one entry which lets you choose the simulation time. Note that you can choose appropriate units for your problem:

6. Running a Calculation

Once you have defined your input, you can start the calculation with the "Run" button. Depending on your parameters, a calculation might take a few minutes, if you want to abort a calculation you can do so by clicking the "Abort" button. You will find that many problems are solved so fast that you hardly have a chance to press it, though. During the calculation, you will see some status information in the lower left corner of the window. When the calculaction is finished, a new window is opened which lets you inspect the results and is explained in the next section.

7. Results window

The graphical user interface which lets you inspect various results of a heat transmission calculation looks like this:

The window has several sections which visualize the results in various ways and are explained in further detail in the following. On the top there are buttons which let you save the results, load results that you have previously saved, create a printable report of your results or rerun the calculation with the same or similar parameters.

The results window will not only be opened automatically when a calculation has finished but also can be opened from the start palette by clicking on the "Start" button next to "Inspect Simulation Results". You can then load a previously saved result and do some further analysis.

In the upper part of the window there are summaries of your input:

The first subsection in the results section is a summary of the calculation details:

Then you will find a section with some visualizations of the temperature distribution within the wall, amongst them an animation, a contour and a 3D-plot.

The 3D-Plot and the animation will show in separate windows. The 3D-Plot can be interactively rotated, shifted and zoomed into by dragging the mouse, with the Shift and Alt keys pressed, respectively:

To run the animation, you can drag the slider or click onto the small + next to it to get some further control elements and also an input field for the time:

In the following section you can define positions within the wall for which the temperature is shown as function of time. As in other tables, you can add and delete rows by clicking on the row headings which will give you a menu to choose from:

With the "Export"-Button you can export the data for the lines shown to an excel or cvs file for further investigation.

Finally there follow two plots showing the heat content of the wall and each layer as functions of time as well as the heat fluxes due to convection, radiation and explicit external heat fluxes through the surfaces on both sides of the wall:

Last but not least there is the possibility to get exact temperature values for a given time and position:

8. Report

Clicking on the "Report"-Button in the results windows toolbar will create a report for your calculation. The report is a Mathematica notebook which contains all relevant results in a form that is fit for printout. You can either give it away as a mathematica notebook, save it as a PDF-file or send it to a printer. Typically such a report is about 10 printed pages long. Note that only those sections that you have opened in the notebook will be visible in the PDF-file.

Defining Functions

There are many places in the program where you can in principle use not only constant parameters but also function of one variable. To define these, you have to select the "Interpolating Function" radio button:

This will show a table where you can input tabular data and a plot to check that your input is correct:

You can add or delete rows by clicking on the row headers. A new row will be added after the row where you clicked for the menu, if you choose to delete a row it will be the row that you clicked to see the menu:

Instead of defining each data point, you can also import the data from excel files. To do this, click on the button "Import" and a file chooser dialog will be opened. Once you have selected a file, you will see another dialog which lets you define in more detail which data within the file you want to import:

You can choose the data sheet to import from and select the columns and rows you want to import. You will then see the imported data in tabular form and as a plot in the input window:

You should now select the correct units for the data you have imported. In the above example the time values were actually given in seconds, the temperature  was given in Kelvin, so we should select those units from the two popup menus within the column headers:

If the units you need are not available in the menu, you need to transform the data within the excel file to one of the available units.

For functions of time there is an additional "Continuation" menu below the data table where you can choose how the program shall continue the function for data not defined by the tabular data. You can choose from "Constant", "Periodic" and "Linear Extrapolation". You can check the plot to see what this means for your function data, in the following screenshot we have chosen a periodic continuation of the given data instead of the default constant continuation:

Defining Materials

When defining the properties of each layer of the wall, you can choose from a predefined set of materials that are delivered with the package. But of course for many applications you will need additional types of materials. You can add material definitions (or change existing ones) by clicking on the "Start" button next to the "Define Materials" entry within the start palette. This will open a window which looks like this:

It lets you define the properties of a new material and save that material definition by clicking on the "Save" button. You can also change an existing material definition by loading it, making your changes and save it to the same name. If you want to use an existing material as a template for a new one, you can load it, make your changes and save with a new name. See the section Settings -> Repositories for details on where the files are saved.

Note that the material properties MassDensity, ThermalConductivity and SpecificHeatCapacity must all three be given correct values to get reliable results for the transient calculation. This is different from the steady state case, where only the ThermalConductivity must be known. For most materials all three properties do depend on the temperature of the material which must be taken into account to get accurate results, especially when the temperature range is large. This is why you can define all these properties as functions of temperature by supplying tabular data. See the section "Heat Transmission Calculation" -> "Definition Functions" to learn some more details about how to define functions and import data.

Settings

By clicking on the "Start" button next to the "Settings" entry in the start palette, you can change some settings of the program:

Magnification

You can choose from some predifined values for the default magnification of the graphical user interfaces. Choose something appropriate for your screen size and resolution and personal preferences.

Language

You can choose from the available languages in the language menu. Only windows that are opened after you have changed this setting will show their content in the language that you chose.

Repositories

When you will save files, be it Input parameter sets, result files or material definitions, you will only be asked for a name. The program automatically chooses a location within a so called repository where it saves the file. Accordingly it offers you to load appropriate files from within such a repository. By changing the "Repositories" setting you can change the directories which the program uses as repositories. The user interface looks like this:

You must at least have one directory defined where the program will read files from and save to, by default that is the subdirectory SmartCAE in your personal documents directory. You can add as many additional directories where the program will search for files. When saving it will always save the files to the first directory in this list.

If you look at a repository directory where you have saved some files, you will find that it contains some subdirectories into which the actual files are put. It is necessary that this directory structure is preserved to avoid problems with the loading and saving of files. Other than that, these are regular files within your file system.

Support

If you have suggestions or questions you are welcome to contact us:

SmartCAE Stefan Braun
Am Mitterfeld 3
D-81829 München
Tel: +49-(0)89-45108878 10 
Fax: +49-(0)89-43738805

mailto:info@smartcae.de
http://www.smartcae.de