Numerical modelling of heat transfer
This assignment reinforces the material you are learning in the theory of heat transfer. We will use MATLAB to develop a finite difference model of either a steel, nickel, or titanium square cross section subject to quenching from a temperature of within the region of 950-1050°C. Assume that the quench media temperature is 20°C.
We will then perform an inverse analysis to determine the boundary conditions from experimental data. The quench media will be either water or air. The convective heat transfer coefficient varies for the different surfaces of the component.
The thermal history of a component during a quench is of importance when considering phase transformations and residual stresses. Due to the non-uniform boundary conditions and temperature dependent thermo-physical properties of the alloys of interest, it is necessary to assess the heat transfer using a numerical method. In these assignments you will use the finite difference method which can be easily implemented for the geometry of interest.
Supplementary notes are provided explaining how to implement such a heat transfer model, however to accurately capture the behaviour you will have to extend the description to include a temperature dependent thermal conductivity and include radiation. Lecture 13 from the notes is relevant to the assignment. Examples of finite difference models of heat transfer are also provided.
You have been randomly assigned a material, a geometry, an emissivity, a furnace temperature, and a quench media. The material may be either Inconel 718, Ti-6Al-4V, or the super duplex stainless steel Zeron100 to study. The thermo-physical parameters are defined Table 1, where 𝜌 is the density, 𝑘 is the thermal conductivity, and 𝐶𝑝 is the specific heat capacity. The temperature dependencies are given in Kelvin.
Table 1: The thermo-physical properties of Inconel 718, Ti-6Al-4V, and Zeron100
Inconel 718 Ti-6Al-4V Zeron100
𝜌 𝑘 𝐶𝑝
8344.2 − 0.4𝑇 4430 7830
𝑊/𝑚/𝐾 6.952 + 0.0157𝑇 1.116 + 0.0174𝑇 7.33 + 0.019𝑇
(𝐽/𝑘𝑔/𝐾) 351.21 + 0.216𝑇 546.31 + 0.219𝑇 360.21 + 0.131𝑇
Location of thermocouples
You will be assigned the geometry of the square cross section you need to study. Several thermocouples have been imbedded into the component during either an air quench or a water quench. The location and label of the thermocouple data is presented in Figure 1 where the size of the geometry is given by 𝐿.
Figure 1: Thermocouple locations and labels, where the length of the square cross section is given by 𝐿
Deliverable 1: Model implementation & verification
Due: 20/03/2020, 4pm
Thoroughly documented source code and model calculations are needed for the finite difference solution of the 2D heat transfer problem. The model must calculate the temperature history at the locations illustrated in Figure 1 for your assigned material, geometry, and furnace temperature.
The code will write text files labelled ‘ai1.txt’,’ai2.txt’, ‘ai3.txt’ … ‘ai9.txt’ for the locations shown in Figure 1. The files will contain two columns of data describing the time and temperature, respectively. The temperature history calculated at these locations will be tested to assess the accuracy of your model. Ensure that the temperature history calculated by your model is saved at suitable intervals during the quench. The models must simulate the cooling of the component until the maximum temperature has dropped to 100°C.
Provide the model results for two conditions, with and without radiation heat losses.
Assume a heat transfer coefficient of 680𝑊/𝑚2/𝐾 acts upon all surfaces. The environment temperature is 20°C. You are assigned the emissivity of the component. Latent heat from solid-state phase transformations are accounted for within the temperature dependent specific heat capacity.
Note, the source code will be checked for plagiarism.
Criteria to consider include the following and are ranked in order of importance:
How accurate is your calculation? (Comparison against known solutions that use similar assumptions, correct implementation, model assumptions, sufficient spatial and temporal discretization)
Is the code documented and readable?
How user friendly is the code to use?
How versatile is the code to use?
Computational efficiency (How fast the calculation is to perform.)
You will provide both source code and the generated text files containing the calculated thermal history at the locations illustrated in Figure 1.
Deliverable 2: Model application and inverse analysis
Due: 22/05/2020, 4pm
We want to know the boundary conditions descriptive of either a water or air quench applied to your material. We will provide thermocouple data from the locations illustrated in Figure 1. Perform an inverse analysis using the provided experimental data to approximate the heat transfer coefficient on the top, sides and bottom of the square cross section.
Use the material, geometry, furnace temperature, and emissivity assigned to you.
The results are to be delivered in a technical report where you can assume the audience is familiar with the theory and problem of interest. The report should be no longer than 5 pages.
Section 1: Model assumptions and implementation
Describe briefly the model being implemented, the assumptions, and material parameters used. Describe any steps taken to verify the implementation.
Section 2: Inverse analysis
Describe the method used for the inverse analysis, and present the results for the determination of the boundary conditions. A figure is needed that compares the thermocouple data with the calibrated model predictions.
Section 3: Discussion
Discuss any limitations of the model and implications of the assumptions you have made. Section 4: Conclusion
Briefly summarise the main features of the model and key findings of the work.