COOLING CURVES

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The Thermal Analysis

This chapter aims to provide basic information related with the simulation of solidification in castings about heat transfer mechanisms, material properties, boundary conditions, the heat conduction equation and the numerical methods.

 Heat transference

When a system is at a different temperature than its surroundings, the Nature tries to reach thermal equilibrium. To do so, as the second law of thermodynamics explains, the thermal energy always moves from the system of higher temperature to the system of lower temperature.
This transfer of thermal energy occurs due to one or a combination of the three basic heat transport mechanisms: Conduction, Convection and Radiation.

 Conduction

Is the transference of heat through direct molecular communication, i.e. by physical contact of the particles within a medium or between mediums. It takes place in gases, liquids and solids. In conduction, there is no flow of any of the material mediums.
The governing equation for conduction is called the Fourier’s law of heat conduction and it express that the heat flow per unit area is proportional to the normal temperature gradient, where the proportionality constant is the thermal conductivity: q = −kA ∂T (2.1)/∂x
Where q is the heat flux perpendicular to a surface of area A, [W]; A is the surface area through which the heat flow occurs, [m2] ; k is the thermal conductivity, [W/(mK)]; T is the temperature, [K] or [°C]; and x is the perpendicular distance to the surface traveled by the heat flux.

 Convection

Is the heat transfer by mass motion of a fluid when the heated fluid moves away from the heat source. It combines conduction with the effect of a current of fluid that moves its heated particles to cooler areas and replace them by cooler ones. The flow can be either due to buoyancy forces (natural convection) or due to artificially induced currents (forced convection).
The equation that represents convection comes from the Newton’s law of cooling and is of the form:
q = −hA(T∞ − Ts ) (2.2)
Where h is the convective heat transfer coefficient [W/(m2K)]; T∞ is the temperature of the cooling fluid; and Ts is the temperature of the surface of the body.

Radiation

In general, radiation is energy in the form of waves or moving subatomic particles. Among the radiation types, we are specifically interested in the Thermal radiation. Thermal radiation is heat transfer by the emission of electromagnetic waves from the surface of an object due to temperature differences which carry energy away from the emitting object.
The basic relationship governing radiation from hot objects is called the Stefan-Boltzmann law: q = εσA(T1 4 −T2 4 ) (2.3)
Where ε is the coefficient of emissivity (=1 for ideal radiator); σ is the Stefan-Boltzmann constant of proportionality (5.669E-8 [W/(m2K4)]); A is the radiating surface area; T1 is the temperature of the radiator; and T2 is the temperature of the surroundings.
The three of the previously mentioned heat transport mechanisms can be expressed by the model law that state that a flux is proportional to a difference in driving potential divided by a resistance,

Material properties

Thermal conductivity (k)

Is the ability of a material to conduct heat. It is defined as the quantity of heat, Q, transmitted during a period of time t through a thickness L, in a direction
normal to a surface of area A, due to a temperature difference T, under steady state conditions and when the heat transfer is dependent only on the temperature gradient.
Density is a temperature and pressure dependent material property. In solids and liquids is just slightly affected by these factors but in gases is strongly dependent in both of them.

 Specific Heat (cv and cp)

In general, is the measure of the heat energy required to increase the temperature of a unit quantity of a substance by a defined temperature step. For example, how much heat must be added to increase the temperature of one gram of water by one Celsius degree.

Initial and Boundary conditions

Initial conditions and boundary conditions are needed, together with the heat conduction equation, to fully define a transient thermal problem. If the given problem is in steady-state, there is no necessity to define initial conditions
The initial conditions represent the initial temperature distribution throughout the body. In casting processes, the initial condition is assumed to be constant throughout the mould, is also assumed to be constant for the melt in the mould filling simulation, where the temperature will be a superheating temperature. For the solidification simulation, the initial condition is given by the temperature field immediately after filling.
For simplicity, when there is no interest in the mould filling simulation, a constant temperature throughout the melt after filling can be assumed and the superheating temperature of the melt can be used as initial condition for the solidification simulation.
Next, five types of boundary conditions relevant for the modeling of casting processes are introduced together with their mathematical representation

Convection boundary condition

The heat flux across the bounding surface is proportional to the difference between the temperatures of the surface T(P,t) and the surrounding T∞ (t) cooling medium. It is defined by the Newton’s convective law of cooling.

1 Introduction
1.1 BACKGROUND
1.2 PURPOSE AND AIMS
1.3 DELIMITS
2 Theoretical background
2.1 THE THERMAL ANALYSIS
2.2 THE STRESS ANALYSIS
3 Implementation
3.1 PROCESS SUMMARY
3.2 PROCEDURE
4 Cylinder Results
4.1 GEOMETRY
4.2 MESH
4.3 BOUNDARY CONDITIONS
4.4 COOLING CURVES
4.5 THERMAL COLOR SPECTRUMS
4.6 STRESS CURVES
4.7 STRESS COLOR SPECTRUMS
4.8 SIMULATION TIME FOR THE CYLINDER
5 Original Hub Results
5.1 GEOMETRY
5.2 THERMAL AND STRESS CURVES POINTS PLACEMENT
5.3 MESH
5.4 BOUNDARY CONDITIONS
5.5 COOLING CURVES FOR THE ORIGINAL HUB
5.6 THERMAL COLOR SPECTRUMS
5.7 STRESS CURVES FOR THE ORIGINAL HUB
5.8 STRESS COLOR SPECTRUMS
5.9 SIMULATION TIME OF THE ORIGINAL HUB
6 Optimized Hub Results
6.1 GEOMETRY
6.2 THERMAL AND STRESS CURVES POINTS PLACEMENT
6.3 MESH
6.4 BOUNDARY CONDITIONS
6.5 COOLING CURVES FOR THE OPTIMIZED HUB
6.6 THERMAL COLOR SPECTRUMS
6.7 STRESS CURVES FOR THE OPTIMIZED HUB
6.8 STRESS COLOR SPECTRUMS
6.9 SIMULATION TIME OF THE OPTIMIZED HUB
7 Original and Optimized Hub Comparison
7.1 MISES
7.2 MAXIMUM PRINCIPAL STRESS
7.3 MINIMUM PRINCIPAL STRESS
8 Conclusions and discussions
9 References
10 Appendix
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