1.2 Mathematical Modeling of Induction Heating Processes
1.2.2 Basic Model of the Induction Heating Process
Equation (1.5) through Equation (1.8) as well as Equation (1.15) are valid for general three-dimensional electromagnetic and thermal fields and allow one to find all of the required design parameters of the induction system. Although there is a considerable practical interest in three-dimensional problems, most engineer-ing problems in induction heatengineer-ing tend to be reduced to consideration of two- or one-dimensional fields.
It can be shown that, for the great majority of induction heating applications, it is possible to simplify the mathematical model further by some typical assump-tions. For example, it is possible to take the material properties as piecewise continuous and to neglect hysteresis and magnetic saturation. It should be men-tioned here that for such induction heating applications as heating prior to forging, rolling, and extrusion, a heat effect due to hysteresis losses does not typically exceed 7% compared to the heat effect due to eddy current losses. Therefore, an assumption of neglecting the hysteresis is valid.
Assuming that the currents have a steady-state quality, we can conclude that the electromagnetic field quantities in Maxwell’s equations are harmonically oscillating functions with a single frequency. Thus, a time-harmonic electromag-netic field can be introduced. In other words, an assumption of harmonically oscillating currents with a single frequency means that harmonics are absent in the impressed and the induced currents and fields.
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Theory and Industrial Application of Induction Heating Processes 13 For many induction heating applications, the quantities of the magnetic field (such as magnetic vector potential, electric field intensity, and magnetic field intensity) may be assumed to be entirely directed. This allows one to reduce the dimensionality of governing equations. Moreover, most mathematical models of induction heating tend to be handled with a combination of the following assump-tions:
• Neglecting nonlinearity by averaging process parameters on the cor-responding temperature intervals;
• Mathematical description of heated workpieces as regular bodies (plate, cylinder, rectangle, sphere);
• Simplified description of the geometric input data for induction heating system;
• Taking into account nonuniformity of temperature distribution only along one or two coordinate axes (reducing a three-dimensional tem-perature field to a one- or two-dimensional form).
Under these assumptions, the set of Equation (1.5) through Equation (1.8) and Equation (1.15) can be reduced to the following equations in differential form:
; (1.17)
. (1.18)
Here,
∇2 is the Laplacian;
x is a spatial coordinate;
ω = 2πf, where f is a frequency of coil current;
a is average value of temperature conductivity of heated material.
Equation (1.17) is a one-dimensional linear Helmholtz’s equation with respect to complex magnetic field intensity . Equation (1.18) is a one-dimensional linear heterogeneous equation of heat transfer with respect to temperature t(x,τ) at velocity V = 0. Equation (1.17) and Equation (1.18) can be solved separately.
F(x,τ) is a function described distribution of internal heat source density induced by eddy currents per unit time in a unit volume. F(x,τ) can be obtained by solving Equation (1.17) as:
, (1.19)
∇2H x( , )τ = jωµ σ0 H x( , )τ
∂ τ
∂τ τ
γ τ
t x a t x
c F x ( , )
( , ) ( , )
= ∇2 + 1
H
F x( , )τ = 1σEm 2
2 DK6039_C001.fm Page 13 Thursday, June 8, 2006 8:39 AM
14 Optimal Control of Induction Heating Processes where
. (1.20)
Here, Em denotes an amplitude value of the electric field intensity.
By using well-known Expression (1.16) and solutions of Equation (1.17),25 it is possible to obtain the following expressions with respect to F(x,τ) for the axially symmetric case of unlimited plate:
(1.21) and cylindrical workpiece of infinite length:
(1.22) Here,
P0(τ) is active power absorbed by unit surface of heated body;
X is a cylinder radius or a half of slab thickness;
ber(z), bei(z), ber′(z), bei′(z) are Kelvin’s functions and their first derivatives;
l is a relative value of spatial coordinate in a plate depth or cylinder radius (l = x/X);
ξ is a specific parameter defined as:
, (1.23)
where δ is a current penetration depth, which can be calculated as follows10:
. (1.24)
For the typical induction heating process frequency of coil current is constant (ξ = const). Then, total internal heat power absorbed by unit volume of heated body can be written as:
(1.25)
E H
= − ∂x
∂ 1 σ
F l P P
X W l W l ch l
1 0 0
1 1
( , , ) ( ) 2
( , ); ( , ) ( ) co
ξ = τ ξ ξ = ξ − ss( )
( ) sin( 2 ) ;
2 2ξ 2
ξ ξl ξ
sh −
F l P P
X W l W l ber l
2 0
0
2 2
2
( , , ) ( )
( , ); ( , ) ( )
ξ = τ ξ ξ =ξ ′ ξ ++ ′
′ + ′
bei l ber ber bei bei
2( ) ( ) ( ) ( )ξ ( ).
ξ ξ ξ ξ
ξ= X 2 /δ
δ= 2/ (µ ωσ0 )
P( )τ =P0( ) /τ X.
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Theory and Industrial Application of Induction Heating Processes 15 There is a squared relationship between specific power P(τ) and inductor voltage.10 The value of P(τ) is in the range of 0 to Pmax corresponding to maximum inductor voltage:
. (1.26)
Substituting F(x,τ) in the forms of Equation (1.21) and Equation (1.22) into Equation (1.18), one can obtain the following one-dimensional linear heteroge-neous equation of heat transfer considered further as a basic mathematical model of induction heating process:
. (1.27) Equation (1.27) is written with respect to relative units of temperature θ(l,ϕ) and power of internal heat sources u(ϕ). These relative values can be calculated according to the formulas:
; (1.28)
. (1.29)
In Equation (1.27),
Γ = 0 or Γ = 1 for the plate or cylinder, respectively.
W(ξ, l) is determined according to Equation (1.21) or Equation (1.22).
is the Fourier number.
ϕ0 is the total time required for heating.
tb is a basic temperature.
The formulation of a problem requiring the solution of a partial differential equation also requires the specification of appropriate boundary conditions and initial conditions. Specification of the dependent variable or its time derivative at time zero is referred to as an initial condition. The initial temperature condition refers to the temperature profile within the workpiece at time ϕ = 0:
. (1.30)
0≤P( )τ ≤Pmax,τ≥0
∂θ ϕ
∂ϕ
∂ θ ϕ
∂
∂θ ϕ
∂ ξ ϕ
( , ) ( , ) ( , )
( , ) (
l l
l l
l
l W l u
= 2 2 +Γ +
)), l∈[ , ];0 1 ϕ∈[ ;0 ϕ0]
θ ϕ( , ) ( , )τ λ
max
l t x t
P X
= − b
2
u P
( ) P( )
max
ϕ = τ
ϕ= aτ X2
θ( , )l 0 =θ0( )l
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16 Optimal Control of Induction Heating Processes The initial temperature distribution is usually uniform and corresponds to the ambient temperature. In some cases, the initial temperature distribution is non-uniform due to residual heat after the previous technological process. The con-dition in Equation (1.30) is required only when dealing with a transient heat transfer problem where the temperature is a function not only of the space coordinates but also of time.
For physical problems, the term “boundary” literally means on the physical boundary of the region in space in which the solution is sought. The three most common boundary conditions for induction heating problems are:
1. Value of temperature is specified on the boundary of the heated body.
This boundary condition is also known as a Dirichlet condition or a boundary condition of the first kind.
2. Normal derivatives of temperature are specified on the boundary. This condition is known as a Neumann condition or a boundary condition of the second kind.
3. A linear combination of conditions 1 and 2 is specified on the boundary.
Convective boundary conditions in heat transfer are of this type. This condition is known as a Robbins condition or a boundary condition of the third kind.
If the heated body is geometrically symmetrical along the axis of symmetry, the Neumann boundary conditions can be formulated as:
, (1.31)
. (1.32)
The condition in Equation (1.31) implies that the temperature gradient in a direction normal to the axis of symmetry is zero. In other words, no heat exchange takes place at the axis of symmetry. This boundary condition can also be applied in the case of a perfectly insulated body.
In Expression (1.32), q(ϕ) represents a relative value of heat losses and can be determined as:
. (1.33)
Here, Qs(ϕ) is a flow of heat loss from a surface of the heated body (i.e., during quenching or as a result of workpiece contact with cold rolls or water-cooled guides, etc.).
∂θ ϕ
∂ ( ,0 )
l =0
∂θ ϕ
∂ ϕ
( , ) 1 ( )
l =q <0
q Q
P X
( ) s( )
max
ϕ = ϕ
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Theory and Industrial Application of Induction Heating Processes 17 For most induction heating problems, boundary conditions include the heat losses due to convection. In this case, the boundary condition of the third kind can be expressed as:
, (1.34)
where
∂θ/∂l is the temperature gradient in a direction normal to the surface at the point under consideration.
Bi = αX/λ is the Biot number.
α is the convection surface heat transfer coefficient.
θa(ϕ) denotes a relative value of ambient temperature ta(τ) calculated as:
. (1.35)
The heat losses at the workpiece surface are highly variable because of the nonlinear behavior of convection losses.
Equation (1.27) with boundary conditions in Equation (1.31), Equation (1.32), and Equation (1.34) are the most popular equations for mathematical modeling of the heat transfer processes in induction heating and heat treatment applications.
Let us assume that power of internal heat sources u(ϕ) can be changed almost in an arbitrary way. In this case, the solution of Equation (1.27) under initial and boundary conditions (1.30) through (1.32) and (1.34) can be written in the form of Duhamel integral as25:
. (1.36)
Taking into consideration appropriate boundary conditions of the second (Φ2, Λ2) and the third (Φ3, Λ3) kinds, functions Φ(l, ϕ) and Λ(l, ϕ) can be calculated as25:
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18 Optimal Control of Induction Heating Processes
; (1.38)
; (1.39)
. (1.40)
In Expression (1.37) through Expression (1.40), the fundamental functions K(µnl), K1(µnl) should be calculated using following expressions:
; (1.41)
(1.42) where J0(µnl) and J1(µn) are Bessel functions of the zero and first orders, respec-tively.
Under boundary conditions of the third kind, fundamental numbers µn, n = 1, 2, … represent roots of the equation:
. (1.43)
Under boundary conditions of the second kind, fundamental numbers µn, n = 0, 1, 2, … represent roots of the equation:
. (1.44)
In Expression (1.37) through Expression (1.40):
, n = 0, 1, 2, ….
(1.45) The finite and sufficient number of terms of series should always be used in given expressions.
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Theory and Industrial Application of Induction Heating Processes 19
1.3 TYPICAL INDUSTRIAL APPLICATIONS AND