Simulation and Control Framework
3.3 Heat Transfer System
3.3 Heat Transfer System
The melting process is triggered when the temperature reaches the melting point. Thus, we require a thermal transfer simulation for the heat energy flow in the fluid elements, which are SPH particles in our approach. We have modeled several types of thermal exchange types as listed here:
• Thermal radiation from heat source to surface particles,
• Thermal conduction from particle to its neighbor particles,
• Thermal dissipation from surface particles to cold air.
Figure 3.3: Thermal transfer system flowchart.
3.3 Heat Transfer System 19
Figure 3.4: Thermal simulation in our approach.
The thermal information is stored on each particle and heat energy can be exchanged with the neighborhood in the particle supporting range. Every particle has the heat properties as follows: total heat energy, temperature, absorb energy, latent heat threshold, conductive rate, emissive rate, and adopt heat energy coefficient. The total heat energy shows the current heat energy amount in this particle and can be converted into temperature using Equation 3.19:
Ti = Qi
miCi (3.19)
where Tiis the temperature, Qiis the heat energy stored in particle i, and Ciis the specific heat capacity. The absorb energy and latent heat threshold are used to interpolate the viscosity and
3.3 Heat Transfer System 20
the elastic stress coefficients for the melting process from solid to liquid. When the temperature reaches the melting temperature, which invokes the phase change condition, particles start ab-sorbing energy to destruct the solid structure to become liquid. The viscosity and elastic stress coefficient are inversely proportional to the ratio of absorb energy and latent heat. Conductive rate defines the conduction between particles and emissive rate represents the emissivity from the surface particles to the air. Finally, adopt heat energy coefficient is used to adjust the amount of heat energy obtained from heat sources per iteration.
Besides, to simulate the thermal radiation and the thermal dissipation from the object sur-face, we have to identify the surface particles first. We make use of the gradient of the color field in Equation ??, which also represents the surface normal direction. By detecting whether the magnitude exceeds a given threshold, we may extract the surface particles and the surface normal as shown in Figure 3.5.
Figure 3.5: Surface particle and its normal.
Thermal Radiation transmits energy from heat sources to the surface particles in a radiative way. In this simulation framework, we only consider the direct radiative ray from the sources.
3.3 Heat Transfer System 21
Indirect radiation (multi-bounce of radiative ray) is ignored in our approach since its effect is insignificant. Therefore, we first detect the visibility by ray casting between surface particles and the heat sources. If the particle pass the visibility test, we compute the angle between the surface normal vector and the vector from particle to heat source position (see Figure 3.6). We than calculate the energy value depending the Equation 3.20.
Figure 3.6: Thermal radiation diagram
∆Q =X
s
δm
d2 cos θQs(Ts, ∆t) (3.20)
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where ∆Q is the energy variation in each iteration, δ is the adopt heat energy coefficient, d is the distance between the source and the particle position, cos θ is the angle we mentioned above, Qsis a function representing the heat energy value generated by the heat source. Qs(Ts, ∆t) ∝ Ts∆t in our implementation.
We have modeled two kinds of heat sources in our simulation. The first one is the point heat source as in the Figure 3.4. Another one is the area heat source, we modeled this as a plane which provides the heat energy for one side. The area heat source is used to model the heat source like a hot pan or hot ground.
Thermal Conduction diffuses the energy between particle and particle. According the physics of thermology, the heat energy flows from region of high temperature to region of lower tem-perature until it reaches the equilibrium state. Therefore, we model the heat conduction only if the particle temperature is higher than its neighborhood and the diffusion range is defined as the particle’s supporting range. The conduction is proportional not only to the ∆T , but also to the mass and the inverse of distance between particles, as shown in the following Equation 3.21 and 3.22:
where Qcis the heat energy through conduction, dij is the distance between particle i and particle j, and λ is the conductivity we have mentioned above. We distribute the energy flow fairly to the neighborhood particles. Equation 3.21 computes the heat energy conducts from par-ticle i to its neighborhood parpar-ticle j, and Equation 3.22 evaluates the weighting of conduction by considering the distance, particle mass, and the difference of temperature of two particles.
Thermal Dissipation is another thermal radiation in which the energy is radiated from the surface to the outside medium. The general equation of thermal radiation can be expressed as
3.3 Heat Transfer System 23
Equation 3.23 because most of the materials in nature are considered as grey bodies, we need to consider the emissivity . This factor has to be multiplied with the radiation spectrum formula before integration. For simplicity, we can simplify the factor as a constant similar to [MGG+10]
did.
Q = · σ · A · T4 (3.23)
Equation 3.23 is the equation of the general thermal radiation in physics, where factor σ is the Stefan-Boltzmann constant, is the emissivity, A is the radiating surface area, and T is the object temperature. Based on Equation 3.23, we formulate the thermal dissipation in our simulation as the following form:
Qemissive = σVp(Tp4− Tair4 ) (3.24)
Equation 3.24 computes the heat energy dissipate from the surface to the contact medium.
We use the particle volume Vp to approximate the area term since it is much easier to compute in our approach.
After all the computation of energy transfer, we may update the temperature of particles through Equation 3.19, also update the viscosity according to the ratio of energy absorbtion and latent heat value.
3.3.1 Previous Thermal Diffusion Method
In the previous methods for object melting problem ([CMVHT02], [PPLT06], [CBL+09], [WLK03]), neither of them had implemented the thermal radiation effect, and some of them use the SPH interpolation scheme to simulate the thermal diffusion between particles. But in the real world, thermal radiation is actually a domination part in object melting. And the thermal conduction usually propagate the heat energy much slower than the radiation. In their demos, we found it is hard to capture the detail drops flowing and the melting behavior caused by the exposure to heat sources. Also, the thermal radiation allows us to determine the melting region and the melting