Project 1: Cooling of a Hot Steel Cube

Transient conduction with mixed convection boundary conditions

1 Problem Statement

A steel cube of side length

[ L = 70 ]

is initially at

[ T_0 = 1000^ ]

and is cooled by air and an ice-water mixture. The lower 40 mm are submerged in an ice-water mixture at (0^). The surrounding air is at (20^). A constant heat flux of (200 ^2) is applied at the bottom. The process lasts 6 minutes.

2 Geometry and Boundary Conditions

Quantity	Symbol	Value
Cube side length	(L)	(0.07 )
Water level	(h_w)	(0.04 )
Initial temperature	(T_0)	(1000^)
Air temperature	(T_{,a})	(20^)
Water temperature	(T_{,w})	(0^)
Bottom heat flux	(q’’)	(200 ^2)
Simulation time	(t_{end})	(360 )

3 Assumptions

Constant steel properties.
Heat transfer in the steel is by conduction.
External heat transfer is represented by constant heat-transfer coefficients.
Radiation is neglected in the base model.
The ice-water mixture remains at (0^).
No phase transformation of steel is considered.

4 Governing Equation

[ c_p = k^2T ]

[ = ^2T ]

with

[ =. ]

5 Material Properties

Property	Symbol	Value
Density	()	(7850 ^3)
Heat capacity	(c_p)	(600 )
Thermal conductivity	(k)	(35 )

6 Julia Starter Code

using Plots

L = 0.07
T0 = 1000.0
Tair = 20.0
Twater = 0.0

ρ = 7850.0
cp = 600.0
k = 35.0
α = k/(ρ*cp)

println("Thermal diffusivity = ", α)

7 Student Tasks

Implement a 1D or 2D transient heat-transfer model.
Plot temperature fields after 1, 3, and 6 minutes.
Plot center, top, and bottom temperatures versus time.
Compute the average cube temperature.
Compare with a lumped-capacitance model.
Perform a sensitivity study for (h_{water}).
Discuss whether radiation should be included.
Optional: build an ML surrogate or PINN model.

8 Verification

Recommended checks:

Energy balance.
Grid refinement.
Time-step refinement.
Comparison with lumped model for small Biot number.

9 Deliverables

Julia code or notebook.
Temperature plots.
Parameter study.
Short technical report.

--- title: "Project 1: Cooling of a Hot Steel Cube" subtitle: "Transient conduction with mixed convection boundary conditions" --- ## Problem Statement A steel cube of side length \[ L = 70\ \text{mm} \] is initially at \[ T_0 = 1000^\circ\text{C} \] and is cooled by air and an ice-water mixture. The lower 40 mm are submerged in an ice-water mixture at \(0^\circ\text{C}\). The surrounding air is at \(20^\circ\text{C}\). A constant heat flux of \(200\ \text{W/m}^2\) is applied at the bottom. The process lasts 6 minutes. ## Geometry and Boundary Conditions | Quantity | Symbol | Value | |---|---:|---:| | Cube side length | \(L\) | \(0.07\ \text{m}\) | | Water level | \(h_w\) | \(0.04\ \text{m}\) | | Initial temperature | \(T_0\) | \(1000^\circ\text{C}\) | | Air temperature | \(T_{\infty,a}\) | \(20^\circ\text{C}\) | | Water temperature | \(T_{\infty,w}\) | \(0^\circ\text{C}\) | | Bottom heat flux | \(q''\) | \(200\ \text{W/m}^2\) | | Simulation time | \(t_{end}\) | \(360\ \text{s}\) | ## Assumptions 1. Constant steel properties. 2. Heat transfer in the steel is by conduction. 3. External heat transfer is represented by constant heat-transfer coefficients. 4. Radiation is neglected in the base model. 5. The ice-water mixture remains at \(0^\circ\text{C}\). 6. No phase transformation of steel is considered. ## Governing Equation \[ \rho c_p\frac{\partial T}{\partial t} = k\nabla^2T \] or \[ \frac{\partial T}{\partial t} = \alpha\nabla^2T \] with \[ \alpha=\frac{k}{\rho c_p}. \] ## Material Properties | Property | Symbol | Value | |---|---:|---:| | Density | \(\rho\) | \(7850\ \text{kg/m}^3\) | | Heat capacity | \(c_p\) | \(600\ \text{J/(kg K)}\) | | Thermal conductivity | \(k\) | \(35\ \text{W/(m K)}\) | ## Julia Starter Code ```julia using Plots L = 0.07 T0 = 1000.0 Tair = 20.0 Twater = 0.0 ρ = 7850.0 cp = 600.0 k = 35.0 α = k/(ρ*cp) println("Thermal diffusivity = ", α) ``` ## Student Tasks 1. Implement a 1D or 2D transient heat-transfer model. 2. Plot temperature fields after 1, 3, and 6 minutes. 3. Plot center, top, and bottom temperatures versus time. 4. Compute the average cube temperature. 5. Compare with a lumped-capacitance model. 6. Perform a sensitivity study for \(h_{water}\). 7. Discuss whether radiation should be included. 8. Optional: build an ML surrogate or PINN model. ## Verification Recommended checks: - Energy balance. - Grid refinement. - Time-step refinement. - Comparison with lumped model for small Biot number. ## Deliverables - Julia code or notebook. - Temperature plots. - Parameter study. - Short technical report.