Assignment 3 Steady State Heat Transfer MTRL 361 - 2023

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CHAPTER 12

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Material Science

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Dec 6, 2023

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THE UNIVERSITY OF BRITISH COLUMBIA Department of Materials Engineering MTRL 361 Assignment #3 – 1D Steady Sate Conductive Heat Transfer through Multiple Interfaces Date: Monday, October 13 th , 2023 Due: Sunday, October 29 rd , 2023 (11:59 pm, 23:59) Continuous Slab Casting of Steel In a conventional continuous slab casting process, heat is transported out of the system through the mould's external surfaces (mould/env. interface) to water while the slab is located within the mould. See schematic diagram Figure 1. Figure 1 – Schematic Diagram of Slab Casting Mould area At the midpoint of the wide face, it is reasonable to assume that the heat transport is 1-D. Furthermore, adopting an Eulerian frame of reference (i.e., the observer is not moving with the strand), the temperature field is invariant with time for a location at a specific distance from the meniscus. 1/4
In this problem, you will need to solve for the temperature distribution in the system, given the metal temperature at the centerline of the casting and the temperature of the cooling water within the mould. The conductivities of the various materials and the heat transfer coefficients at the interfaces will be given. Your calculations need to be completed in Python, and your executable file for Layers 1 through 3 must be uploaded to Canvas for you to receive a mark . Each question should be completed in a separate computational cell in Python, with the question number – e.g., Layer 1 Q1.1 – clearly labelled at the beginning of the computational cell. Theory and Background An easy way to solve this type of problem is to use the thermal circuit method. For more details on the thermal circuit method, please refer to class notes. Referring to Figure 2, the following resistances and approximate temperatures are identified in the system at some (unknown) location below the meniscus. You may assume that the strand ½ thickness is 50 mm. The material properties of the various components and the interfacial heat transfer coefficients are given in Table 1. Figure 2 – Schematic diagram of the series resistances making up the overall resistance to heat transfer through a section of a continuous casting slab and the resulting steady state temperature distribution[1,2]. 2/4
[1] Y. Meng and B.G. Thomas: Metall. Mater. Trans. B , 2003, vol. 34B, pp. 707-25. [2] J. Birat, M. Larrecq, J. Lamant, and J. Petegnief: in Mold Operation for Quality and Productivity , A.W. Cramb and E. Szekeres, ed., ISS, Warrendale, P A, 1991, pp. 3-14. Table 1 – Model Input Parameters Description Value Units Thermal conductivity of copper Mould 350 W/m/K Thermal conductivity of plain carbon solid Steel. 45 W/m/K Thermal conductivity of semi-solid plain carbon steel 45 x 1.5 W/m/K Thermal conductivity of liquid steel 45 x 5 W/m/K Water/copper interface heat transfer coefficient 20,000 W/m 2 /K Solid steel/mould interface heat transfer coefficient 2,000 W/m 2 /K Input Parameters Layer- 1 1. Complete the following tasks/answer the following questions: 1.1. Evaluate the heat flux across the interface between the cold face of the mould (assumed to be at 100 from the graph) and the cooling water (assumed to be at 20 ) from the graph. Give your answer in MW/m 2 . [5 marks] 1.2. Plot the temperature distribution across the interface. You may assume the boundary layer thickness that forms at the interface between the mould and water is 0 mm, owing to the high water flow rates that are normally used. [5 marks] 1.3. Do a sensitivity analysis on the effect of varying the interface heat transfer coefficient in the range of 15,000 to 25,000 W/m/K on the heat flux and plot it using increments in the heat transfer coefficient of 1,000 W/m/K [5 marks] . Layer- 2 2. Add in the resistance associated with the copper mould. Extract the approximate hot face temperature of the mould from Figure 2. Complete the following tasks/answer the following questions: 2.1. Plot out the temperature profile in the system, including within the mould and across the mould/water boundary layer. [5 marks] 2.2. Compare the heat flux across the mould and across the convective boundary layer at the mould/water interface. [10 marks] 3/4
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2.3. Do your results for 2.1 and 2.2 make sense, with reference to Figure 2? If so, why, if not, why? [5 marks]. Hint: Do a sensitivity analysis on the effect of varying h_interface_water on T_mould_cold. [5 marks] . At what approximate value of h_interface_water do the results come into alignment with Figure 2? [5 marks] Confirm in an updated plot. [5 marks] Why vary h_interface_water and not k_mould? [5 marks] Layer- 3 3. Add in the liquid metal, the mushy zone, the solid shell and the interface between the solid shell and the mould. Complete the following tasks/answer the following questions using the data provided in Figure 2 and Table 1: 3.1. Plot out the temperature profile across the liquid, mushy region, solid shell and mould and include the changes in temperature across the two interfaces. [20 marks] 3.2. There are obviously differences between the resulting distribution of temperature predicted in 3.1 and Figure 2. If you were to pick one parameter to change to achieve better alignment, what would it be (you cannot change the dimensions)? Justify why you have chosen this parameter. [5 marks] 3.3. Use a sensitivity analysis to estimate the value you would use for that parameter. [10 marks] 4/4