1 Heat flow – glass coupling 1.1 Heat flow
Figure 1 shows the geometry of the furnace with burners. Burners blow fire using Navier-Stokes equations. I use dimensionless Navier-Stokes equations for convection-radiation.
Figure 1: Illustration of cross-sectional view of the considered furnace domain.
∇·u = 0,
Du+∇p− 1∇2u = Θe, (1)
Dt Re
DΘ− 1 ∇2Θ = − 1 ∇·qR, Dt PrRe PrRe
where e = (0, 1)T is the unit vector associated with the gravitational force, u is the velocity, Θ is the temperature, Pr and Re are Prandtl and Reynolds numbers respectively. The material derivative
DW
Dt of a generic function W is defined as
DW = ∂W + u · ∇W. Dt ∂t
The dimensionless radiative heat flux is given by
−∇·qR = 1 B(Θ)−φ, (2)
τPl
S2
φ is the total incident radiation defined as φ(x) =
The dimensionless function B is given by
I(ω, x) dω.
B(Θ) = 4Θ4·
The radiative transfer equation can be rewritten in dimensionless as
τω · ∇I + κI = κB(Θ), (3)
Results from this part of study can be seen in Figure 2 below:
Figure 2: Temperature view for the operation of three burners.
1.2 Modelling thermo-elastodynamics problems
The heat coming from the fluid (fire) part will be coupled and passed to the structure part using equation 7. Then the calculated solid (glass) temperature will be used to solve equations below.
ρ∂2us − μ∇2us − (λ + μ)∇(∇ · us) = −k(3λ + 2μ)∇Θs + ρgVe, (4) ∂t2
μ∇2us + (λ + μ)∇(∇ · us) = k(3λ + 2μ)∇Θs − ρgVe, (5)
σs =μ(∇us +(∇us)T)+λ(∇·us)I−k(3λ+2μ)ΘsI, (6)
where kc is thermal conductivity, Θs is glass change in temperature, us is the displacement vector, (λ, μ) are the lame ́ constants, k is the coefficient of thermal expansion, ρ is the glass density, g is the gravity, V is the glass volume, e = (0, 1)T which is the unit vector associated with the gravitational force, σs is the stress tensor and I is the identity tensor.
The domain (solid) is subjected to an external heat on the boundaries. We will assume that there is a heat applied from the upper surface of the glass sheet and that has a certain temperature (like 850 K). The rest of the structure faces (sides and the lower surface) are at lower temperature but the temperature can be increasing with the increase in the external source from above. Figure 7 shows the simple domain with an external heating source from the upper surface (like an oven or microwave).
Figure 3: Illustration of cross-sectional view of the considered glass domain.
2
Before passing the temperature from the fluid part to the glass structure, a heat transfer boundary condition must apply. The equation is
k ∂Θs +α(Θ −Θ )+εσ(Θ4 −Θ4)=0. (7) c∂n f s f s
Here, Θf is the fluid temperature, α is the convective heat transfer coefficient, n is face normal, σ is Stefan-Boltzmann constant and ε is the emissivity coefficient. The equation can be solved using any non-linear method (like Newton-Raphson method). Once the temperatures of the solid are calculated on the surface of the domain, we then can pass it to the equations above to do the calculations and deformations shown in Figure 4
Figure 4: Illustration of cross-sectional view of the glass deformation.