**5. Sensitivity analysis**

In this sub-chapter the selected problems from the scope of sensitivity analysis application in the thermal theory of solidification processes are discussed. It should be pointed out that the sensitivity models are the essential part of inverse problems solutions, they allow to rebuilt the solution obtained for assumed input data on the other one (see: considerations and examples presented below), sensitivity methods can be also used on a stage of measurements in order to determine the optimum positions of sensors.

The examples of applications here discussed concern the sensitivity of temperature field in a system solidifying metal-mould with respect to perturbations of external, boundary and initial parameters appearing in the mathematical model of process discussed, at the same time the approach called 'a direct sensitivity analysis' is presented. Both the macro models and macro/micro ones are considered

Let us assume that the searched solution of boundary-initial problem (e.g. transient temperature field in the system metal - mould), is the function of parameters *z*1 , *z*2 ,..., *zn* . They can correspond, among others, to thermophysical parameters of sub-domains, heat transfer coefficient on the outer surface of the system, pouring temperature, initial temperature of mould, geometrical parameters etc. The sensitivity *Uk* of boundary-initial

In this sub-chapter the selected problems from the scope of sensitivity analysis application in the thermal theory of solidification processes are discussed. It should be pointed out that the sensitivity models are the essential part of inverse problems solutions, they allow to rebuilt the solution obtained for assumed input data on the other one (see: considerations and examples presented below), sensitivity methods can be also used on a stage of

The examples of applications here discussed concern the sensitivity of temperature field in a system solidifying metal-mould with respect to perturbations of external, boundary and initial parameters appearing in the mathematical model of process discussed, at the same time the approach called 'a direct sensitivity analysis' is presented. Both the macro models

Let us assume that the searched solution of boundary-initial problem (e.g. transient temperature field in the system metal - mould), is the function of parameters *z*1 , *z*2 ,..., *zn* . They can correspond, among others, to thermophysical parameters of sub-domains, heat transfer coefficient on the outer surface of the system, pouring temperature, initial temperature of mould, geometrical parameters etc. The sensitivity *Uk* of boundary-initial

measurements in order to determine the optimum positions of sensors.

Fig. 13. Temperature field in domain considered.

**5. Sensitivity analysis** 

and macro/micro ones are considered

problem solution with respect to parameter *zk* is defined as a partial derivative <sup>1</sup> *U xt Txtz z z <sup>k</sup>* ( , ) ( , , ,... ) / *n k* (Kleiber, 1997; Dems & Rousselet, 1999). As it was mentioned below the methods of sensitivity analysis allows one 'to rebuilt' the solution of problem considered to the solution concerning the other value of parameter *zk*. So, the set of parameters introduced to the computer program as the input data we denote by *z*<sup>1</sup> 0, *z*<sup>2</sup> 0, ..., *zn*0. Additionally we are interested in the others solutions corresponding to changed input data (e.g. new pouring temperature). The methods of sensitivity analysis give the possibilities to transform the basic solution on the solution for disturbed initial parameters. The transformation results from the Taylor formula, namely

$$T(\mathbf{x}, \ t, z\_1^0, z\_2^0, \dots, z\_k^0 \pm \Delta z\_k, \dots, z\_n^0) = $$

$$T(\mathbf{x}, \ t, z\_1^0, z\_2^0, \dots, z\_k^0, \dots, z\_n^0) \pm \mathcal{U}\_k^0 \, \Delta z\_k + \mathcal{V}\_k^0 \, \frac{\Delta z\_k^0}{2} + \dots \, \tag{64}$$

where *T* is a searched function (e.g. temperature), *x* is a spatial coordinate, *t* is a time, *Uk* 0 denotes the derivative *T*/*zk* at the start point 0, whereas *Vk* 0 is the second derivative with respect to *zk* at the same point. The equations determining the values of *Ui* 0 (the first order sensitivity) and *Vi* 0 (the second order sensitivity) result from the differentiation of equations and conditions describing the physical process considered with respect to parameter *zk* (direct approach) We can notice that formula (64) concerns to the disturbance of distinguished parameter, but one can analyze also the disturbance of the certain set of parameters *zk*, simultaneously.

#### **5.1 Sensitivity analysis of macro models**

Sensitivity analysis with respect to boundary and initial conditions (e.g. heat transfer coefficient α, ambient temperature *Ta* , initial temperatures *T*0, *Tm*0) or mould parameters (*cm*, λ*m*) will be discussed, at the same time the direct approach will be applied. The problem is more complicated in comparison with the others ones, because the mathematical model of solidification is strongly non-linear (Szopa, 2005; Szopa & Wojciechowska, 2003; Szopa et al., 2004; Mochnacki & Szopa, 2009).

At first, the energy equation (8) should be differentiated with respect to parameter *zk*, this means

$$\frac{\partial}{\partial z\_k} \left[ \mathbf{C}(T) \frac{\partial T(\mathbf{x}, t)}{\partial t} \right] = \nabla \left[ \frac{\partial}{\partial z\_k} \left[ \mathcal{L}(T) \nabla T(\mathbf{x}, t) \right] \right] \tag{65}$$

Because

$$
\frac{\partial \mathcal{C}(T)}{\partial z\_k} = \frac{\text{d}\mathcal{C}(T)}{\text{d}T} \frac{\partial T}{\partial z\_k}, \qquad \frac{\partial \mathcal{A}(T)}{\partial z\_k} = \frac{\text{d}\mathcal{A}(T)}{\text{d}T} \frac{\partial T}{\partial z\_k} \tag{66}
$$

therefore (using additionally Schwarz theorem)

$$\begin{aligned} \mathbf{C}(T) \frac{\hat{\mathcal{O}} \mathbf{U}(\mathbf{x}, t)}{\hat{\mathcal{O}}t} &= \nabla \left[ \mathcal{A}(T) \nabla \mathcal{U}(\mathbf{x}, t) \right] + \\ \nabla \left[ \mathcal{A}^\*(T) \mathcal{U}(\mathbf{x}, t) \nabla T(\mathbf{x}, t) \right] - \mathbf{C}^\*(T) \mathcal{U}(\mathbf{x}, t) \frac{\hat{\mathcal{O}} \mathbf{T}(\mathbf{x}, t)}{\hat{\mathcal{O}}t} \end{aligned} \tag{67}$$

Numerical Modeling of Solidification Process 535

0 : 0 0 *<sup>0</sup> m 0*

 *T T t = U = = , = = U z z* 

Below, as an example, the sensitivity method is applied to the analysis of dependence between the pouring temperature and the course of solidification and cooling proceeding in the casting domain. The equation describing the heat diffusion in the casting is nonlinear one, whereas the thermophysical parameters of mould are assumed to be the

The equation describing the thermal processes proceeding in the metal domain oriented in

(, ,) ( )

 

*Txyt C T t T x <sup>y</sup> <sup>t</sup> T x <sup>y</sup> <sup>t</sup> T T x xy y*

 *m mm m m*

*c = +* 

model is differentiated with respect to pouring temperature *T*0. So, for (*x*, *y*) Ω

*C U + C = +* 

λ λ

*T T*

2 2 () : <sup>λ</sup> *m mm <sup>m</sup> m m*

If on the outer surface Γ0 of the system we assume the no-flux condition then the adequate boundary condition takes a form, *Um*/*n* = 0. On the contact surface (for λ = const) one has

*x , y = + c*

 *T T U + U x xy y*

(, ,) (, ,) ( ) ( )

 

2 2

(, ,) (, ,) (, ,) <sup>λ</sup>

*T TT x y t x y t x y t*

 *t x y* 

The boundary conditions on the outer surface of the mould determine the continuity of heat flux between the mould and the environment (the Robin condition), but in practice one can assume in this place the no-flux condition (especially in the case of sand mould). On the

 *t*=0: *T* = *T*0, *Tm* = *Tm*0 (76) In order to analyze the influence of the pouring temperature on the course of casting solidification the direct variant of sensitivity analysis has been used and the solidification

 *T U U U*

 

*t tx x y y*

λ λ

 *U UU*

 *t x y*

2 2

2 2

rectangular co-ordinate system {*x*, *y*} (2D problem) is of the form (c.f. equation (1))

The energy equation for the mould domain is the following

contact surface between casting and the condition (5) is given. The mathematical model is supplemented by the initial conditions:

*T*

where *U* = *T*/*T*0, *CT* = d*C*/d*T*, λ*<sup>T</sup>* = dλ/d*T*. Next

where *Um* = *Tm*/*T*0.

constant.

*m k k*

(73)

(74)

(75)

(77)

(78)

where *U* = *T*/*zk* is the sensitivity of temperature field in a solidifying metal domain with respect to perturbations of parameter *zk*.

Now, the equation concerning a mould sub-domain is differentiated with respect to *zk* under the assumption that *cm* and λ*m* are the constant values (this assumption is not necessary)

$$\begin{aligned} \frac{\partial \mathcal{L}\_m}{\partial \boldsymbol{z}\_k} \frac{\partial \mathcal{T}\_m(\mathbf{x}, t)}{\partial t} + c\_m \frac{\partial \mathcal{U}\_m(\mathbf{x}, t)}{\partial t} &= \\ \frac{\partial \mathcal{L}\_m}{\partial \boldsymbol{z}\_k} \nabla^2 \mathcal{T}\_m(\mathbf{x}, t) + \mathcal{A}\_m \nabla^2 \mathcal{U}\_m(\mathbf{x}, t) \end{aligned} \tag{68}$$

where *Um* = *Tm*/*zk*. Taking into account the equation (2) one can write

$$\begin{split} c\_{m} \frac{\partial \operatorname{\mathcal{L}}\_{m}(\mathbf{x}, t)}{\partial t} &= \lambda\_{m} \nabla^{2} \operatorname{\mathcal{U}}\_{m}(\mathbf{x}, t) + \\ \left( \frac{\partial \operatorname{\mathcal{X}}\_{m}}{\partial \operatorname{\mathcal{Z}}\_{\mathcal{Z}k}} \frac{c\_{m}}{\lambda\_{m}} - \frac{\partial c\_{m}}{\partial \operatorname{\mathcal{Z}}\_{\mathcal{Z}k}} \right) \frac{\partial \operatorname{\mathcal{T}}\_{m}(\mathbf{x}, t)}{\partial t} \end{split} \tag{69}$$

The sensitivity equations become simpler when in place of *zk* the actual parameters (e.g. initial casting temperature, initial mould temperature, heat transfer coefficient or ambient temperature) are introduced.

The following boundary conditions should be considered


$$-\frac{\partial \lambda\_m}{\partial \cdot\_{\mathcal{Z}k}} \frac{\partial \operatorname{\boldsymbol{T}\_m(\mathbf{x},t)}}{\partial \operatorname{\boldsymbol{n}}} - \lambda\_m \frac{\partial \operatorname{\boldsymbol{U}\_m(\mathbf{x},t)}}{\partial \operatorname{\boldsymbol{n}}} = 0$$

$$\frac{\partial \operatorname{\boldsymbol{a}}}{\partial \operatorname{\boldsymbol{Z}}\_{\mathcal{Z}k}} \left[ \operatorname{\boldsymbol{T}\_m(\mathbf{x},t)} \operatorname{\boldsymbol{\cdot}} \operatorname{\boldsymbol{T}\_a} \right] + \operatorname{\boldsymbol{a}} \left[ \operatorname{\boldsymbol{U}\_m(\mathbf{x},t)} \operatorname{\boldsymbol{\cdot}} \frac{\partial \operatorname{\boldsymbol{T}\_a}}{\partial \operatorname{\boldsymbol{Z}}\_k} \right] \tag{70}$$


$$\begin{cases} -\lambda'(T)\mathcal{U}(\mathbf{x},t)\frac{\partial \mathcal{T}(\mathbf{x},t)}{\partial n} - \lambda(T)\frac{\partial \mathcal{U}(\mathbf{x},t)}{\partial n} = \\ \quad - \frac{\partial}{\partial \ z\_k} \frac{\partial \mathcal{T}\_m(\mathbf{x},t)}{\partial n} - \lambda\_m \frac{\partial \mathcal{U}\_m(\mathbf{x},t)}{\partial n} \\ \qquad \qquad \mathcal{U}(\mathbf{x},t) = \mathcal{U}\_m(\mathbf{x},t) \end{cases} \tag{71}$$

One can see that for the constant values of metal and mould thermal conductivities the conditions (70) and (71) are similar to conditions (3) and (5). The initial conditions takes a form

$$\mathbf{t} = \mathbf{0}: \qquad \mathbf{U} = \frac{\partial \operatorname{\mathbf{T}}\_{\mathcal{D}}}{\operatorname{\mathbf{\mathcal{D}}} z\_k} = \mathbf{U}\_0 \, \mathbf{J} \qquad \mathbf{U}\_m = \frac{\operatorname{\mathbf{\mathcal{D}}} \operatorname{\mathbf{T}}\_m \, \mathbf{0}}{\operatorname{\mathbf{\mathcal{D}}} z\_k} = \mathbf{U}\_{m \, \mathbf{0}} \tag{72}$$

If the sensitivity analysis does not concern the initial temperatures then the initial conditions are uniform

where *U* = *T*/*zk* is the sensitivity of temperature field in a solidifying metal domain with

Now, the equation concerning a mould sub-domain is differentiated with respect to *zk* under the assumption that *cm* and λ*m* are the constant values (this assumption is not necessary)

(,) (,)

(,) (,)

(68)

(69)

(70)

(71)

*T xt U xt*

<sup>2</sup> (,) <sup>λ</sup> (,)

λ (,)

*mm m m*

 *c c <sup>T</sup> x t - t z z*

The sensitivity equations become simpler when in place of *zk* the actual parameters (e.g. initial casting temperature, initial mould temperature, heat transfer coefficient or ambient

<sup>λ</sup> (,) (,) <sup>λ</sup>

 *xt xt <sup>T</sup> U = n n <sup>z</sup>*

*m m m <sup>m</sup> <sup>k</sup>*

(,) (,)

(,) (,) <sup>λ</sup> () (,) <sup>λ</sup>( )

*<sup>m</sup> <sup>m</sup> <sup>m</sup> <sup>m</sup>*

 *n n z U x t = x t U*

One can see that for the constant values of metal and mould thermal conductivities the

0 : *<sup>0</sup> m 0*

 *T T t = U , UU U z z*

If the sensitivity analysis does not concern the initial temperatures then the initial conditions

<sup>λ</sup> (,) (,) <sup>λ</sup>

*Txt Uxt TUx t T =*

*n n xt xt T U*

*0 m m 0*

(72)

(,) (,)

*m* 

*k k*

 *T x t - + x t - T T U z z*

*<sup>a</sup> m a <sup>m</sup> k k*

 *xt <sup>U</sup> c U = x t +* 

2 2

*c T xt U xt c zt t*

*<sup>m</sup> m mm*

*m m <sup>m</sup> <sup>m</sup>*

*<sup>m</sup> <sup>m</sup> m m*

*k*

*k*

where *Um* = *Tm*/*zk*. Taking into account the equation (2) one can write

λ

*k*

 *t*

*km k*

 

*z* 

The following boundary conditions should be considered - Robin condition (e.g. external surface of mould):

 

conditions (70) and (71) are similar to conditions (3) and (5).

respect to perturbations of parameter *zk*.

temperature) are introduced.


The initial conditions takes a form

are uniform

$$\text{Let } \mathbf{t} = \mathbf{0}: \qquad \mathbf{U} = \frac{\mathfrak{D}\_{\mathbf{T}\_0}}{\mathfrak{D}\_{Z\_k}} = \mathbf{0}, \qquad \mathbf{U}\_{\text{uv}} = \frac{\mathfrak{D}\_{\mathbf{T}\_{\text{uv}}}}{\mathfrak{D}\_{Z\_k}} = \mathbf{0} \tag{73}$$

Below, as an example, the sensitivity method is applied to the analysis of dependence between the pouring temperature and the course of solidification and cooling proceeding in the casting domain. The equation describing the heat diffusion in the casting is nonlinear one, whereas the thermophysical parameters of mould are assumed to be the constant.

The equation describing the thermal processes proceeding in the metal domain oriented in rectangular co-ordinate system {*x*, *y*} (2D problem) is of the form (c.f. equation (1))

$$\mathbf{C}(T)\frac{\partial T(\mathbf{x},\mathbf{y},t)}{\partial t} = $$

$$\frac{\partial}{\partial \mathbf{x}} \left[ \mathcal{A}(T) \frac{\partial T(\mathbf{x},\mathbf{y},t)}{\partial \mathbf{x}} \right] + \frac{\partial}{\partial y} \left[ \mathcal{A}(T) \frac{\partial T(\mathbf{x},\mathbf{y},t)}{\partial y} \right] \tag{74}$$

The energy equation for the mould domain is the following

$$\chi\_{\mathcal{C}\_{m}} \frac{\partial \operatorname{\boldsymbol{T}}\_{m}(\mathbf{x}\_{\prime} \mathbf{y}\_{\prime} \mathbf{t})}{\partial \operatorname{\boldsymbol{t}}} = \lambda\_{m} \left( \frac{\partial^{2} \operatorname{\boldsymbol{T}}\_{m}(\mathbf{x}\_{\prime} \mathbf{y}\_{\prime} \mathbf{t})}{\partial \operatorname{\boldsymbol{x}}^{2}} + \frac{\partial^{2} \operatorname{\boldsymbol{T}}\_{m}(\mathbf{x}\_{\prime} \mathbf{y}\_{\prime} \mathbf{t})}{\partial \operatorname{\boldsymbol{y}}^{2}} \right) \tag{75}$$

The boundary conditions on the outer surface of the mould determine the continuity of heat flux between the mould and the environment (the Robin condition), but in practice one can assume in this place the no-flux condition (especially in the case of sand mould). On the contact surface between casting and the condition (5) is given.

The mathematical model is supplemented by the initial conditions:

$$t = 0 \colon \quad T = T\_{0\*} \quad T\_m = T\_{m0} \tag{76}$$

In order to analyze the influence of the pouring temperature on the course of casting solidification the direct variant of sensitivity analysis has been used and the solidification model is differentiated with respect to pouring temperature *T*0. So, for (*x*, *y*) Ω

$$\begin{aligned} \mathbf{C}\_{T} \cdot \mathbf{U} \frac{\partial \operatorname{\boldsymbol{\mathcal{T}}}{\partial t}}{\partial t} + \mathbf{C} \frac{\partial \operatorname{\boldsymbol{\mathcal{U}}}}{\partial t} &= \frac{\partial}{\partial \operatorname{\boldsymbol{\mathcal{X}}}} \left( \lambda \left. \frac{\partial \operatorname{\boldsymbol{\mathcal{U}}}}{\partial \operatorname{\boldsymbol{\mathcal{X}}}} \right| + \frac{\partial}{\partial y} \left( \lambda \left. \frac{\partial \operatorname{\boldsymbol{\mathcal{U}}}}{\partial y} \right) \right) + \\ &\frac{\partial}{\partial \operatorname{\boldsymbol{\mathcal{X}}}} \left( \lambda\_{T} \left. \operatorname{\boldsymbol{\mathcal{U}}} \frac{\partial \operatorname{\boldsymbol{\mathcal{T}}}}{\partial \operatorname{\boldsymbol{\mathcal{X}}}} \right) + \frac{\partial}{\partial y} \left( \lambda\_{T} \left. \operatorname{\boldsymbol{\mathcal{U}}} \frac{\partial \operatorname{\boldsymbol{\mathcal{T}}}}{\partial y} \right) \right) \end{aligned} \tag{77}$$

where *U* = *T*/*T*0, *CT* = d*C*/d*T*, λ*<sup>T</sup>* = dλ/d*T*. Next

$$\mathbf{u}(\mathbf{x}, \mathbf{y}) \in \Omega\_{\mathfrak{m}} : \qquad \mathbf{c}\_{m} \frac{\partial \mathbf{U}\_{m}}{\partial \mathbf{t}} = \lambda\_{m} \left( \begin{array}{c} \hat{\boldsymbol{\sigma}}^{2} \mathbf{U}\_{m} \\ \hat{\boldsymbol{\sigma}} \mathbf{x}^{2} \end{array} + \frac{\hat{\boldsymbol{\sigma}}^{2} \mathbf{U}\_{m}}{\partial \mathbf{y}^{2}} \right) \tag{78}$$

where *Um* = *Tm*/*T*0.

If on the outer surface Γ0 of the system we assume the no-flux condition then the adequate boundary condition takes a form, *Um*/*n* = 0. On the contact surface (for λ = const) one has

Numerical Modeling of Solidification Process 537

Now, the considerations concerning the continuous casting process will be presented. The goal of work is to apply the sensitivity analysis tool in order to observe the perturbations of casting solidification due to the perturbations of substitute heat transfer coefficient in primary cooling zone of CCT installation. The basic energy equation describing the thermal

processes in the domain of vertical, rectangular cast slab can be written in the form

( ) λ ( )

λ ( ) λ ( )

where *T* = *T*(*x*, *y*, *z*, *t*), *w* is the pulling rate the cast slab shifts in *z* direction (see: Figure 16).

 *T T T + T y y z z*

 *T T T C T + w = T + t z x x*

(80)

Figure 15 illustrates the isolines of function *U* for the same time.

Fig. 15. Distribution of function *U* for time 6 minutes.

Fig. 16. Vertical continuous casting.

$$(\mathbf{x}', \mathbf{y}) \in \Gamma\_m \colon \quad \left\{ \begin{array}{l} -\lambda \frac{\partial \, \mathrm{U}}{\partial \, m} = -\lambda\_m \frac{\partial \, \mathrm{U} \, \mathrm{m}}{\partial \, m} \\ \mathrm{U} = \mathrm{U}\_m \end{array} \right. \tag{79}$$

The problem is supplemented by the initial condition for *t* = 0: *U* = 1, *Um* = 0.

As an example the bar of rectangular section (1014 cm) made from Cu-Sn alloy (10% Sn) is considered. The casting is produced in the sand mix which parameters are equal λ*m* = 2.28 [W/mK], *cm* = 2.320106 [J/m3K]. The thermal conductivity of the casting material equals λ = 50 [W/mK]. According to literature (e.g. [3]) the substitute thermal capacity can be approximated by the piece-wise constant function *T* > 990°C: *C* = 3.678106 [J/m3K], *T* < 825°C: *C* = 3.678106 [J/m3K], *T*  [825, 990]: *C* = 14.558106 [J/m3K]. In order to assure the differentiation of function *C*, the smoothing procedure has been applied (Mochnacki & Suchy, 1995). The initial temperatures are equal to *T*(0) = 1000°C, *Tm*(0) = 30°C. The quarter of domain is taken into account and its shape is marked in Figure14. In this Figure the temperature field for time *t* = 6 minutes is shown.

The sensitivity analysis shows that the influence of pouring temperature on the temperature field is the most essential in the metal sub-domain (in particular at the initial stages of cooling process) and sand mix layer close to contact surface. In other words, the change of *T*0 = 1000°C to the value from interval [*T*<sup>0</sup> Δ*T*0, *T*0 + Δ*T*0] determines the essential fluctuations of temporary temperature field in these sub-domains.

Fig. 14. Temperature field.

*x y : n n*

As an example the bar of rectangular section (1014 cm) made from Cu-Sn alloy (10% Sn) is considered. The casting is produced in the sand mix which parameters are equal λ*m* = 2.28 [W/mK], *cm* = 2.320106 [J/m3K]. The thermal conductivity of the casting material equals λ = 50 [W/mK]. According to literature (e.g. [3]) the substitute thermal capacity can be approximated by the piece-wise constant function *T* > 990°C: *C* = 3.678106 [J/m3K], *T* < 825°C: *C* = 3.678106 [J/m3K], *T*  [825, 990]: *C* = 14.558106 [J/m3K]. In order to assure the differentiation of function *C*, the smoothing procedure has been applied (Mochnacki & Suchy, 1995). The initial temperatures are equal to *T*(0) = 1000°C, *Tm*(0) = 30°C. The quarter of domain is taken into account and its shape is marked in Figure14. In this Figure the

The sensitivity analysis shows that the influence of pouring temperature on the temperature field is the most essential in the metal sub-domain (in particular at the initial stages of cooling process) and sand mix layer close to contact surface. In other words, the change of *T*0 = 1000°C to the value from interval [*T*<sup>0</sup> Δ*T*0, *T*0 + Δ*T*0] determines the essential

 

*<sup>m</sup> <sup>m</sup>*

(79)

*m*

*U U*

*U = U*

 *=* 

<sup>λ</sup> <sup>λ</sup> (,)

*m*

The problem is supplemented by the initial condition for *t* = 0: *U* = 1, *Um* = 0.

temperature field for time *t* = 6 minutes is shown.

Fig. 14. Temperature field.

fluctuations of temporary temperature field in these sub-domains.

Figure 15 illustrates the isolines of function *U* for the same time.

Fig. 15. Distribution of function *U* for time 6 minutes.

Now, the considerations concerning the continuous casting process will be presented. The goal of work is to apply the sensitivity analysis tool in order to observe the perturbations of casting solidification due to the perturbations of substitute heat transfer coefficient in primary cooling zone of CCT installation. The basic energy equation describing the thermal processes in the domain of vertical, rectangular cast slab can be written in the form

$$\begin{aligned} \text{C (T)} \left[ \frac{\partial T}{\partial t} + w \frac{\partial T}{\partial z} \right] &= \frac{\partial}{\partial x} \left[ \lambda \left( T \right) \frac{\partial T}{\partial x} \right] + \\ \frac{\partial}{\partial y} \left[ \lambda \left( T \right) \frac{\partial T}{\partial y} \right] &+ \frac{\partial}{\partial z} \left[ \lambda \left( T \right) \frac{\partial T}{\partial z} \right] \end{aligned} \tag{80}$$

where *T* = *T*(*x*, *y*, *z*, *t*), *w* is the pulling rate the cast slab shifts in *z* direction (see: Figure 16).

Fig. 16. Vertical continuous casting.

Numerical Modeling of Solidification Process 539

As an example the following task is presented. The rectangular steel slab (0.44%C) with dimensions 0.60.2 m is considered. The pouring temperature equals 1550°C, pulling rate: *w* = 0.017 m/s] The basic heat transfer coefficient in the primary cooling zone equals α = 1500 [W/m2K]. The temperature field at the distance 0.6 m from upper surface of the slab has been observed. The precise results will be shown below. For the nodes located near the corner of cast slab the nodal temperatures for α = 1500 are collected on the left side of the page. The temperatures found directly for α = 1700 are written in the middle of this page, while on the right side one can find the temperatures for α = 1700 obtained on the basis of

The sensitivity analysis constitutes also the essential tool of inverse problems solution, in particular when the minimum of functional (corresponding to the least squares criterion) is searched using the gradient methods. The problems connected with the inverse methods applications in the thermal theory of foundry processes can be found, among others, in (Majchrzak et al., 2007; Majchrzak et al., 2008a; Majchrzak & Mendakiewicz 2009; Mochnacki & Majchrzak, 2006; Mendakiewicz, 2008; Majchrzak & Mendakiewicz, 2007; Majchrzak et al.,

In this chapter only the part of problems connected with the solidification process modeling has been presented because the subject-matter discussed is very extensive. It also the reason that the references contain mainly the papers prepared by the author of this chapter and his

Stefanescu, D.M. (1999). Critical review of the second generation of solidification models for

Mochnacki, B. & Majchrzak, E. (2007a). Identification of macro and micro parameters in

Mochnacki, B. & Suchy, J. S. (1995). *Numerical methods in computations of foundry processes*,

Szopa, R. (1999). Modelling of solidification using the combined variant of the BEM,

Mochnacki, B., Majchrzak, E., Szopa, R. & Suchy, J.S. (2006). Inverse problems in the thermal

Majchrzak, E. & Mochnacki, B. (2007). Identification of thermal properties of the system casting - mould, *Materials Science Forum*, Vols. 539-543, pp. 2491-2496. Majchrzak, E. & Mochnacki, B. (1995). Application of the BEM in the thermal theory of foundry, *Engineering Analysis with Boundary Elements*, Vol. 16, pp. 99-121.

*Metallurgy, Publ. of the Silesian Univ. of Technology*, Gliwice, Poland.

Minerals, Metals and Materials Society 1999, pp. 3-20.

55, No. 1, 2007, pp. 107-113.

*Science*, Czestochowa, 1(5), pp. 154-179.

PFTA, Cracow, Poland.

casting, *Modelling of Casting, Welding and Advanced Solidification Processes VI*. The

solidification model, *Bulletin of the Polish Academy of Sciences, Technical Sciences*, Vol.

theory of foundry, *Scientific Research of the Institute of Mathematics and Computer* 

925 883 812 812 858 815 741 741 855 814 740 740 1259 1201 1046 812 1216 1155 986 741 1214 1155 987 740 1496 1478 1201 883 1492 1468 1155 816 1493 1471 1155 814 1512 1496 1259 925 1510 1492 1216 858 1510 1493 1215 855

The initial condition takes a form: *U*(*x*, *y*, 0) = 0.

the sensitivity analysis and Taylor series:

2008b; Majchrzak et al., 2008c).

co-workers.

**6. References** 

The boundary conditions on the lateral surface of cast slab (the continuous casting mould region) are assumed in the form

$$-\lambda \left( T \right) \frac{\partial \ T}{\partial \ n} = \alpha \left( T - T\_w \right) \tag{81}$$

where α is the substitute heat transfer coefficient, *Tw* is the cooling water temperature. On the upper surface of the casting (free surface of molten metal) the boundary condition of the 1st type (pouring temperature) can be taken into account. On the conventionally assumed bottom surface limiting the domain considered we can put *T*/*n* = 0, this means the adiabatic condition.

The initial condition resolves itself into the assumption, that a certain layer of molten metal directly over the starter bar has a pouring temperature *Tp*. The starter bar allows to shut the continuous casting mould during the plant starting.

The numerous experiments show that conductional component of heat transfer corresponding to the direction of cast strand displacement is very small (in the case of steel castings this component constitutes about 5% of the heat conducted from the axis to the lateral surfaces). So, the equation (80) can be simplified to the 2D one. Let we rewrite this equation using the coordinate system 'tied' to a certain section of shifting casting, namely *x* = *x*, *y* = *y*, *z* = *z wt*. It is easy to check up that we 'lose' in energy equation the component *T*/*z* and we obtain

$$\text{C(T)}\ \frac{\partial \, T}{\partial t} = \frac{\partial}{\partial \, \mathbf{x}'} \Big[\lambda \, \text{(T)}\frac{\partial \, T}{\partial \, \mathbf{x}'}\Big] + \frac{\partial}{\partial \, \mathbf{y}'} \left|\lambda \, \text{(T)}\, \frac{\partial \, T}{\partial \, \mathbf{y}'}\right|\tag{82}$$

This approach is called 'a wandering cross section method' (Mochnacki & Suchy, 1995). We consider the lateral section of the cast strand which position at the moment *t* = 0 corresponds to *z* = 0 (the initial condition has a form *T*(0) = *Tp* ), while the boundary conditions on the periphery of this section are the functions of time. If Δ*t*1 corresponds to the 'hold time' of the section in the continuous casting mould region, then for 0 *t*  Δ*t*1 on the casting periphery the boundary condition for the primary cooling zone (e.g., heat flux) is assumed. For the next interval Δ*t*2 we consider the boundary condition characterizing the heat transfer in the 1st sector of the secondary cooling zone etc.

In order to construct the sensitivity model respect to α we differentiate the governing equations over this parameter (a direct approach is applied)

$$\begin{aligned} \frac{\mathbf{d}\, \mathbf{C}(T)}{\mathbf{d}\, T} \mathbf{U} \frac{\partial T}{\partial t} + \mathbf{C}(T) \frac{\partial \mathbf{U}}{\partial t} &= \\ \frac{\partial}{\partial x} \left[ \frac{\mathbf{d}\, \lambda(T)}{\mathbf{d}\, T} \mathbf{U} \frac{\partial T}{\partial x} + \lambda(T) \frac{\partial \mathbf{U}}{\partial x} \right] + \frac{\partial}{\partial y} \left[ \frac{\mathbf{d}\, \lambda(T)}{\mathbf{d}\, T} \mathbf{U} \frac{\partial T}{\partial x} + \lambda(T) \frac{\partial \mathbf{U}}{\partial x} \right] + \\ \end{aligned} \tag{83}$$

In equation (83) one substitutes again *x'* = *x*, *y'* = *y*, symbol *T*/α = *U* denotes the sensitivity function.

Differentiating the condition (81) one has

$$-\lambda \frac{\partial \mathcal{U}}{\partial n} = T - T\_w + \alpha \mathcal{U} \tag{84}$$

The boundary conditions on the lateral surface of cast slab (the continuous casting mould

λ () ( ) *<sup>w</sup> <sup>T</sup> T T T n* 

where α is the substitute heat transfer coefficient, *Tw* is the cooling water temperature. On the upper surface of the casting (free surface of molten metal) the boundary condition of the 1st type (pouring temperature) can be taken into account. On the conventionally assumed bottom surface limiting the domain considered we can put *T*/*n* = 0, this means the

The initial condition resolves itself into the assumption, that a certain layer of molten metal directly over the starter bar has a pouring temperature *Tp*. The starter bar allows to shut the

The numerous experiments show that conductional component of heat transfer corresponding to the direction of cast strand displacement is very small (in the case of steel castings this component constitutes about 5% of the heat conducted from the axis to the lateral surfaces). So, the equation (80) can be simplified to the 2D one. Let we rewrite this equation using the coordinate system 'tied' to a certain section of shifting casting, namely

> ( ) <sup>λ</sup> ( ) <sup>λ</sup> ( ) *T T T C T = T + T t x x y y*

This approach is called 'a wandering cross section method' (Mochnacki & Suchy, 1995). We consider the lateral section of the cast strand which position at the moment *t* = 0 corresponds to *z* = 0 (the initial condition has a form *T*(0) = *Tp* ), while the boundary conditions on the periphery of this section are the functions of time. If Δ*t*1 corresponds to the 'hold time' of the section in the continuous casting mould region, then for 0 *t*  Δ*t*1 on the casting periphery the boundary condition for the primary cooling zone (e.g., heat flux) is assumed. For the next interval Δ*t*2 we consider the boundary condition characterizing the

In order to construct the sensitivity model respect to α we differentiate the governing

<sup>d</sup>λ () d λ () λ ( ) λ ( ) d d

In equation (83) one substitutes again *x'* = *x*, *y'* = *y*, symbol *T*/α = *U* denotes the sensitivity

λ *<sup>w</sup> U = T + U <sup>T</sup>*

 *n*

*TT U TT U UT UT x T x xyT x x*

(84)

 *wt*. It is easy to check up that we 'lose' in energy equation the

(82)

(83)

(81)

region) are assumed in the form

component *T*/*z* and we obtain

continuous casting mould during the plant starting.

heat transfer in the 1st sector of the secondary cooling zone etc.

*CT T U U CT Tt t*

 

equations over this parameter (a direct approach is applied)

d ( ) ( ) d

Differentiating the condition (81) one has

adiabatic condition.

*x* = *x*, *y* = *y*, *z* = *z* 

function.

The initial condition takes a form: *U*(*x*, *y*, 0) = 0.

As an example the following task is presented. The rectangular steel slab (0.44%C) with dimensions 0.60.2 m is considered. The pouring temperature equals 1550°C, pulling rate: *w* = 0.017 m/s] The basic heat transfer coefficient in the primary cooling zone equals α = 1500 [W/m2K]. The temperature field at the distance 0.6 m from upper surface of the slab has been observed. The precise results will be shown below. For the nodes located near the corner of cast slab the nodal temperatures for α = 1500 are collected on the left side of the page. The temperatures found directly for α = 1700 are written in the middle of this page, while on the right side one can find the temperatures for α = 1700 obtained on the basis of the sensitivity analysis and Taylor series:


The sensitivity analysis constitutes also the essential tool of inverse problems solution, in particular when the minimum of functional (corresponding to the least squares criterion) is searched using the gradient methods. The problems connected with the inverse methods applications in the thermal theory of foundry processes can be found, among others, in (Majchrzak et al., 2007; Majchrzak et al., 2008a; Majchrzak & Mendakiewicz 2009; Mochnacki & Majchrzak, 2006; Mendakiewicz, 2008; Majchrzak & Mendakiewicz, 2007; Majchrzak et al., 2008b; Majchrzak et al., 2008c).

In this chapter only the part of problems connected with the solidification process modeling has been presented because the subject-matter discussed is very extensive. It also the reason that the references contain mainly the papers prepared by the author of this chapter and his co-workers.
