7.5 Time Discretization and Linearization of the Governing Equation

Back

Equation (7.43) is an ordinary differential equation in t . For the time discretization, a

backward Euler scheme is taken. Accordingly,

D

Dt

_T _n+1

≈ 1

_t

__T _n+1

− _T _n

_ . (7.44)

Evaluating Equation (7.43) at t = tn+1 leads to

1

_t

_C_n+1

__T _n+1

− _T _n

_ + _K_n+1 _T _n+1

= _Q_n+1 . (7.45)

HEAT TRANSFER 311

This nonlinear equation will be solved in an iterative way. Assume _T _n is known and we

want to determine _T _n+1. In the iteration k + 1, we have an approximation _T _(k)

n+1 for

_T _n+1, and we seek a better approximation _T _(k+1)

n+1 that satisfies

_T _(k+1)

n+1

= _T _(k)

n+1

+ __T _(k)

n+1 . (7.46)

Substitution of the approximation _T _(k)

n+1 into _C_ will be denoted _C_(k)

n+1. Linearization

of the improved value _C_(k+1)

n+1 leads to

_C_(k+1)

n+1

= _C_(k)

n+1

+

_ ∂ _C_

∂ _T _

_(k)

n+1

__T _(k)

n+1 . (7.47)

Similar expressions apply to _K_(k+1)

n+1 and _F_(k+1)

n+1 . Evaluation of Equation (5.42) in the

iteration k + 1 yields

1

_t

_C_(k)

n+1

+

_ ∂ _C_

∂ _T _

_(k)

n+1

__T _(k)

n+1

__T _(k)

n+1

+ __T _(k)

n+1

− _T _n

_

+

_K_(k)

n+1

+

_∂ _K_

∂ _T _

_(k)

n+1

__T _(k)

n+1

__T _(k)

n+1

+ __T _(k)

n+1

_

= _Q_(k)

n+1

+

_∂ _Q_

∂ _T _

_(k)

n+1

__T _(k)

n+1 . (7.48)

Collecting terms and neglecting quadratic contributions yields



1

_t

_C_(k)

n+1

+

_ ∂ _C_

∂ _T _

_(k)

n+1

__T _(k)

n+1

− _T _n

_

+

_K_(k)

n+1

+

_∂ _K_

∂ _T _

_(k)

n+1

_T _(k)

n+1

−

_∂ _Q_

∂ _T _

_(k)

n+1



__T _(k)

n+1

= − 1

_t

_C_(k)

n+1

__T _(k)

n+1

− _T _n

_ − _K_(k)

n+1 _T _(k)

n+1

+ _Q_(k)

n+1 . (7.49)

The right-hand side is the residual _R_(k)

n+1 of Equation (7.45) in iteration (k). The dependence

of the capacity and conduction terms on the temperature is usually benign, and the

corresponding temperature-derivative terms in Equation (7.49) are often neglected. In this

way, Equation (7.49) reduces to

1

_t

_C_(k)

n+1

+ _K_(k)

n+1

−

_∂ _Q_

∂ _T _

_(k)

n+1

__T _(k)

n+1

= _R_(k)

n+1 . (7.50)

312 HEAT TRANSFER

The only term that needs further analysis is the derivative of the driving flux with respect

to the temperature. Equations (7.38) and (7.42) yield

_∂ _Q_

∂ _T _

_(k)

n+1

=_

e

_L_T

e

_∂ _Q_e

∂ _T _

_(k)

n+1

=_

e

_L_T

e

_∂ _Q_e

∂ _T _e

_(k)

n+1

_L_e . (7.51)

The derivative of _Q_e with respect to the temperature reduces to the derivative of any

of its entries in Equation (7.37). Concentrating on the first term on the right-hand side of

Equation (7.37),

_∂ _Q_1

ei

∂ _T _ej

_

(k)

n+1

= −

_ ∂

∂Tj

_

Ae

h

_ N

_

k=1

ϕkTk

__ N

_

l=1

ϕlTl − Te

_

ϕi dae

_

(k)

n+1

= −_

Ae

_ ∂h

∂T

_(k)

n+1

(T (k)

n+1

− Te)ϕiϕj dae − _

Ae

h(T (k)

n+1)ϕiϕj dae. (7.52)

In a similar way, one finds for the second term on the right-hand side of Equation (7.37),

_∂ _Q_2

ei

∂ _Tej _

_

(k)

n+1

= −

∂

∂Tj

_

Ae

A

_ N

_

k=1

ϕkTk

_

_

θref +

N

_

l=1

ϕlTl

_

4

− θ4

e

ϕi dae

(k)

n+1

= −_

Ae

_∂A

∂T

_(k)

n+1

_(θref + T (k)

n+1)4 − θ4

e _ ϕiϕj dae

− _

Ae

A(T (k)

n+1)4(θref + T (k)

n+1)3ϕiϕj dae. (7.53)

The dependence of h andAon T is usually benign, such that the first terms in Equations (7.52)

and (7.53) are frequently dropped. The dependence on T of the third and fourth terms in

Equation (7.37) is usually also small. If not, their derivative must also be included. Summarizing,

one obtains

_∂ _Q_ei

∂ _T _ej

_(k)

n+1

= −_

Ae

h(T (k)

n+1)ϕiϕj dae − _

Ae

A(T (k)

n+1)4(θref + T (k)

n+1)3ϕiϕj dae. (7.54)

This yields a contribution of the convection and radiation fluxes to the “stiffness” matrix

in Equation (7.50), comparable to the stiffness contribution of the centrifugal forces and

traction forces in Section 3.3. Notice that the resulting equation, Equation (7.50), does not

contain any explicit reference to θref. Consequently, we can freely choose θref, for example,

as absolute zero.

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