Metal cooling process
Large metal structures, such as steel posts used in bridge construction, are cast in molds and then cooled so that they can be installed. When metals cool, they change their structure. Their state of aggregation changes from liquid to solid. This is called crystallization. In the process, the so-called heat of crystallization is released, which slows down and even reverses the cooling process. If the cooling process is not known precisely enough, this can lead to defective products during casting, making them less stable.
For the company RWP GmbH (Richter Weiß und Partner, named after the founders of the company Wieland Richter and Konrad Weiß), it is therefore important for the casting of metals to
- to know the temperature and the time of crystallization and
- to determine the amount of energy released.
Integration and differentiation are needed to solve these two subproblems. The material gives students real-world applications of high school mathematics, which contributes to their motivation.
Duration: 2 double lessons
Contents: Differentiation, difference quotient, integration, sum of squares, measurement error.
Prior knowledge: Differentiation, difference quotient, integration, sum of squares.
Applicable: For upper grades
Created by: Kirsten Wohak
Registration: Appointments can be made individually via this form.
Image source: Link
List of publications and presentations related to this module:
- Wohak, K.: Didaktisch-methodische Entwicklung und Evaluation von computergestützten Lehr- und Lernmaterialien zur Mathematik inverser Probleme. Dissertation KIT, 2021.
Wohak, K.: Digital und interaktives Lernmaterial zur mathematischen Modellierung am Beispiel vom Abkühlprozess bei Metallen (Vortrag), digital GDM-Monat.
Timetable
Phase | Content | School reference | further math. Contents | media/ materials | Time (min.) |
Entry + Technical introduction | Motivation, introduction to the cooling process of metals, setting up the function for released heat energy during the cooling process, simplification and translation of the problem in math. Model | Exponential function, determination of the surface area via the rectangular sum | Presentation slides | 15 10 | |
Development AB1 | Modeling of released heat energy, calculation of heat of crystallization | Understanding of data, graph-function relationship, integration, sums of rectangles, linear relationships | Formula of heat energy | AB1-SuS | 40 |
Elaboration AB2 | Time determination of crystallization, determination of cooling rate | Graph-function relationship, calculation of the slope over two data points, difference quotient | Oscillation | AB2-SuS | 40 |
Method sheets (Supplementary to sheet 2) | Choice of seven methods for calculation of cooling rate | Mean, derivative, difference quotient, thresholds, distance, inequality | Tangent hyperbolic, plotting, if statements, for loops. | Method(1-7)-SuS | 25 |
Backup + final discussion | Summary of central steps of the cooling process, classification of the data, securing of results, discussion about occurring problems, comparison of the different methods. Methods | Critical reflection | - | Presentation slides | 20 |
Material
In this zip folder you will find all presentations, worksheets with solutions and the lesson plan for this workshop. The zip folder is password protected. We will gladly send you the password by mail upon request.