Application of sensitivity analysis in experiments design

APPLICATION OF SENSITIVITY ANALYSIS IN EXPERIMENTS DESIGN

Jerzy Mendakiewicz

Silesian University of Technology, Gliwice, Poland, jerzy.mendakiewicz@polsl.pl

Abstract. The system casting-mould is considered and it is assumed that the aim of experiments is to determine the course of substitute thermal capacity of casting material.

The casting is made from cast iron and the austenite and eutectic latent heats should be identified. To find the optimal location of sensors the methods of sensitivity analysis are applied. In the final part of the paper the results of computations are shown.

1. Governing equations

The energy equation describing the casting solidification has the following form [1, 2]

( )

( ) ( )

: ( ) T x t λ ,

x C T T T x t

t

∈Ω ∂ ∂ = ∇ ∇  (1)

where C(T) is the substitute thermal capacity of cast iron, λ is the thermal conduc- tivity, T, x, t denote the temperature, geometrical co-ordinates and time.

The equation considered is supplemented by the equation concerning a mould sub-domain

( , ) 2

: ^m λ ( , )

m m m m

T x t

x c T x t

t

∈Ω ∂ = ∇

∂ (2)

where cm is the mould volumetric specific heat, λm is the mould thermal conductiv- ity.

In the case of typical sand moulds on the contact surface between casting and mould the continuity condition in the form

λ ( ) ( , ) λ ( , )

: ( , ) ( , )

m m

c

m

T T x t T x t

x T x t T x t

− ⋅∇ = − ⋅∇

∈Γ  =

n n

(3)

can be accepted.

On the external surface of the system the Robin condition

0: λ_{m m}( , ) α _m( , ) _a

x∈Γ − n⋅∇T x t = T x t −T  (4) is given (α is the heat transfer coefficient, Ta is the ambient temperature).

For time t = 0 the initial condition

0 0

0 : ( , 0) ( ) , _m( , 0) _m ( )

t= T x =T x T x =T x (5)

is also known.

In the case of cast iron solidification the following approximation of substitute thermal capacity can be taken into account (Fig. 1)

( )

1

2

,

,

,

, ,

L L

aus

P A L

L A

aus

P E A

A E

eu

P S E

E S

S S

c T T

c Q T T T

T T

c Q T T T

C T T T

c Q T T T

T T

c T T

>



 + < ≤

 −



+ < ≤

=

 −

 + < ≤

 −

 ≤



(6)

where TL, TA, TE, TS correspond to the border temperatures, cL, cS, cP = 0.5 ⋅ (cL + + cS) are the constant volumetric specific heats of molten metal, solid state and mushy zone sub-domain, Qaus =Qaus1 + Qaus2, Qeu are the latent heats connected with the austenite and eutectic phases evolution, at the same time Q = Qaus + Qeu.

Fig. 1. Substitute thermal capacity of cast iron

The thermal conductivity is defined as follows λ ,

λ ( ) λ , λ ,

L L

P S L

S S

T T

T T T T

T T

 >

= ≤ ≤

 <



(7)

where λL, λS, λP = 0.5 ⋅ (λL + λS) are the constant thermal conductivities of liquid state, solid state and mushy zone sub-domain, respectively.

2. Sensitivity analysis

It is assumed that the aim of experiments is to determine the latent heats Qaus1, Qaus2, Qeu of casting material (an inverse problem) and in order to find the optimal location of sensors the sensitivity analysis methods [3-5] are applied.

To determine the sensitivity functions the governing equations (1)-(5) are dif- ferentiated with respect to p1 = Qaus1, p2 = Qaus2 and p3 = Qeu. So, the following ad- ditional problems should be solved

[ ]

[ ] [ ]

2

0

( ) ( , ) ( , )

: ( )

λ ( ) ( , ) ( , )

: λ ( , )

λ ( ) ( , ) λ ( , )

: ( , ) ( , )

: λ

e e

e

m

m m m m

e e

m m

e e

c

m

e e

m m

e

C T T x t T x t

x C T

p t p t

T T x t p

T x t

x c T x t

p t p

T T x t T x t

p p

x T x t T x t

p p

x T

p

 

∂ ∂ ∂ ∂

∈Ω +  =

∂ ∂ ∂  ∂ 

∂ ∇ ∇ 

∂

∂

 

∂ ∂  

∈Ω ∂  ∂ = ∂ ∇ 

∂ ∂

− ⋅∇ = − ⋅ ∇

 ∂ ∂

∈Γ 

∂

 ∂ =

 ∂ ∂



∈Γ − ⋅ ∂ ∇

∂

n n

n

[

]

( , 0) ( , 0)

0 : 0 , ^m 0

e e

x t T x t T x

p p

=

∂

= ∂ = =

∂ ∂

(8)

or

[ ]

2

0

( , ) λ ( )

: ( ) λ ( ) ( , ) ( , )

( ) ( , )

( , )

: λ ( , )

λ ( )

( , ) λ ( ) ( , ) λ ( , )

:

( , ) ( , )

: λ ( , ) 0

e

e

e

e

m e

m m m m e

e m m e

c e

e m e

m m e

Z x t T

x C T T Z x t T x t

t p

C T T x t

p t

Z x t

x c Z x t

t

T T x t T Z x t Z x t

x p

Z x t Z x t

x Z x t

t

 

∂ ∂

∈Ω =∇ ∇ + ∇ ∇ −

∂  ∂ 

∂ ∂

∂ ∂

∈Ω ∂ = ∇

∂

∂

− ⋅∇ − ⋅∇ = − ⋅∇

 ∂

∈Γ 

 =



∈Γ − ⋅∇ =

n n n

n

0 : Z_e( , 0)x 0 , Z_{m e}( , 0)x 0

= = =

(9)

where

( , ) ( , )

( , ) , ( , ) ^m

e m e

e e

T x t T x t

Z x t Z x t

p p

∂ ∂

= =

∂ ∂ (10)

Differentiation of substitute thermal capacity with respect to the parameters p1, p2, p3 leads to the following formulas

( )

1

0,

1/ ,

( ) 0,

0, 0,

L

L A A L

E A

S E

S

T T

T T T T T

C T T T T

p T T T

T T

>

 − < ≤

∂ 

= < ≤

∂  < ≤

 ≤

(11)

( )

2

0, ( ) 0,

1/ ,

0, 0,

L

A L

A E E A

S E

S

T T

T T T

C T T T T T T

p T T T

T T

>

 < ≤

∂ 

= − < ≤

∂  < ≤

 ≤

(12)

and

( )

3

0, ( ) 0,

0,

1/ ,

0,

L

A L

E A

E S S E

S

T T

T T T

C T T T T

p T T T T T

T T

>

 < ≤

∂ 

= < ≤

∂  − < ≤

 ≤

(13)

3. Example of computations

The casting-mould system shown in Figure 2 has been considered. The basic problem and additional problems connected with the sensitivity functions have been solved using the explicit scheme of FDM [1]. The regular mesh created by 25×15 nodes with constant step h = 0.002 m has been introduced, time step

∆t = 0.1 s. The following input data have been assumed: λL =20 W/(mK), λS =

= 40 W/(mK), λm = 1 W/(mK), cL =5.88 MJ/(m³ K), cS =5.4 MJ/(m³ K), Qaus1 =

= 937.2 MJ/m³, Qaus2 =397.6 MJ/m³, Qeu =582.2 MJ/m³, cm = 1.75 MJ/(m³ K), pouring temperature T0 =1300°C, liquidus temperature TL =1250°C, border tem- peratures TA = 1200°C, TE = 1130°C, solidus temperature TS = 1110°C, initial mould temperature Tm0 = 20°C.

10 30

30

50

casting

mould

Fig. 2. Casting-mould system

In Figures 3-5 the distributions of functions Z1 ⋅ 10⁹, Z2 ⋅ 10⁹, Z3 ⋅ 10⁹ in the do- main of casting for different times are shown. It is visible, that maximal values of sensitivity functions for different times appear in different places, but the global maximum corresponds to the point 1 (0, 0) marked in Figure 8. Figures 6 and 7 illustrate the courses of sensitivity functions at the points 1 and 2 shown in Figure 8.

Fig. 3. Distribution of function Z₁ ⋅ 10⁹ for times 30 and 90 second

Fig. 4. Distribution of function Z2 ⋅ 10⁹ for time 60 and 120 second

Fig. 5. Distribution of function Z3 ⋅ 10⁹ for time 120 and 180 second

0 20 40 60 80 100

0 30 60 90 120 150 180

Z_e·10⁹

t [s] 1

2

3

Fig. 6. Courses of sensitivity functions at the point 1 (0 , 0)

On the basis of sensitivity functions analysis the optimal location of sensors can be determined (see. Fig. 8).

0 20 40 60 80 100

0 30 60 90 120 150 180

U_e·10⁹

t [s]

1 2 3

Fig. 7. Courses of sensitivity functions at the point 2 (0.02, 0)

Fig. 8. Optimal position of sensors

Acknowledgement

This work was funded by Grant No N N507 3592 33.

References

[1] Mochnacki B., Suchy J.S., Numerical methods in computations of foundry processes, PFTA, Cracow 1995.

[2] Majchrzak E., Mendakiewicz J., Identification of cast iron substitute thermal capacity, Archives of Foundry 2006, 6, 22, 310-315.

[3] Majchrzak E., Mochnacki B., Identification of thermal properties of the system casting - mould, Mat. Science Forum 2007, 539-543, 2491-2496.

[4] Mochnacki B., Majchrzak E., Szopa R., Suchy J.S., Inverse problems in the thermal theory of foundry, Scientific Research of the Institute of Mathematics and Computer Science, Czesto- chowa 2006, 1(5), 154-179.

[5] Kleiber M., Parameter sensitivity, J. Wiley & Sons Ltd., Chichester 1997.