Appunti di Elettrotermia

RESISTOR FURNACES

E = Q_AS - Q_C - P_N t_P + ∫ p(t) dt_tP

        + P_O (t - t_P)

Q_AS = P_N t_P + ∫ p(t) dt - ∫ p_C (t) dt

Q_C = ∫ p_E (t) dt + P_O (t - t_P)

STORED + THERMAL ENERGY

ENERGY LOSS TO

THE ENVIRONMENT

P_O =  

• P_C =  
• LOW CONVECTION
• HIGH CONVECTION

Design of Walls

Anaaling Temperature: 850°C

Treating Chamber: 49, 0.85, 0.65 mm

_{θ i} = 80°C (Max Kieselgur)

_{θ e} = 50°C _{θ e} (Vogtland 01)

_{p e} = 800 W/m²

All Theoretical

Iteration with Thicknesses

A₁ = A_m 8,51 m²

A₂ = 10,18 m²

A_F = A_e = 15,52 m²

⇒ A_Fm = √(A₁ A₂ 8/14 m²)

A_2m = √(A₁ A₂) = 12,19 m²

R_M = _d/_λA2m

P = Q_e/_{R1 + R2 + R3}

P₀ = P₀/A₁ + 14,12 W/m²

! > 800 W/m²

For λ P₀ = 3000 W

⇒ R₁ + R₁ + R₃

A₂ = 23,98 m²

A_2m = 16,57 m²

Induction Heating System

Point Parameters

EM Distribution in Cylindrical Geometry

rot E = - _dH/_p

rot H = _dE/_p - j_w p _H

d²H/_dr² + 1/_r dH/_dr - β² H = 0

Nella forma di un'equazione di Bessel

H = C₁ J₀ (β_r) + C₂ β_y (m)

H₀ = _NI/_l

²(√₅x)= [bex+ βmex] + [beux+ βmuex]

H/H₀ = ^{[ber (m m_r) + β bei (m_r)]} / _{ber (m) + β bei (m)}

THERMAL TRANSIENTS

FOURIER EQUATION:

HEAT FLUX

P = -λ ^dθ/_dm = -λ grad θ = -λ ∇θ

ON A VOLUME dV

∑dΠ = ∑ρidAi = -divP dV

INCREASE OF INTERNAL ENERGY

cy ^dθ/_dt dV

cy ^dθ/_dt = div(λgradθ) + w

k = λ/_cy

THERMAL DIFFUSIVITY

W(ϱ) PARAbolIC DENSITY DISTRIBUTION

BOUNDRY CONDITIONS

θ(ϱ,t)=0 t=0

∂θ/_∂r

r=R

Ψ(η)=^2π2/_P0 ∫ W(n)

SU PARAMETRI ADIMENSIONALI

∂θ/_∂ζ ∂²θ/_∂ζ² +¹/_ϱ²Ψ(η)

⇒Ψ(η)=^W(η)/_W0

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