**3. Cooling intensity and numerical models**

In order to design a cooling section, knowledge of the cooling intensity is required for a group of nozzles and nozzle headers. Exact knowledge of the heat transfer coefficient as a function of spray parameters and surface temperature is the key problem for any design work. The cooling intensity is a function of several parameters, mainly, nozzle types, chosen pressures and flow rates, surface temperature of a material, and velocity of a material movement whilst under spray. There is no function available which describes cooling intensity using all the mentioned parameters. This is the reason why real measurement is absolutely necessary.

## **3.1. Experimental procedure**

4 Heat Treatment – Conventional and Novel Applications

**Figure 2.** Flat-jet, full-cone, and solid-jet nozzles with computed water distribution on flat surface

Air 3bar Water 0.06l/s Air 3bar Water 0.20l/s Air 2bar Water 0.06l/s Air 2bar Water 0.20l/s

**Figure 3.** Controllability of water-air mist nozzle for surface temperatures 1000 °C

Three measurements are compared when the only different parameter is the casting speed. The first experiment was stationary with no movement, the second experiment used a velocity of 2 m/s and the last experiment was done for a velocity of 5 m/s. These three experiments used the identical water-air mist nozzle, and the same pressure settings were


**Position in the moving direction [mm]**

**2.3. Influence of product velocity on heat transfer coefficient** 

0

500

1000

1500

2000

**HTC [W/m².K]**

2500

3000

3500

During the in-line heat treatment the product is moving so our testing sample should be also moving through the cooling section. Fig. 6 shows schematically a suitable experiment used for obtaining boundary conditions for a numerical simulation. The hot sample is moving at

Design of Cooling Units for Heat Treatment 7

0

5000

10000

15000

20000

25000

**HTC [W/m².K]; Position [mm]**

30000

35000

40000

45000

50000

prescribed velocity which is similar to real conditions. The sample passes under spray which cools down the hot sample. For a simple shape like plate it is recommended to insulate all surfaces excluding the one on which the cooling intensity is investigated. One or more thermocouples are embedded in the sample and measure temperature during the experiment. The installed thermocouples should not disturb the cooled surface. This is the reason why they should be installed inside the sample, not on the investigated surface. In principle when all surfaces are insulated except the one which is investigated one thermocouple is enough. However, this thermocouple should be as close to the investigated surface as possible. Otherwise the resolution of description of boundary conditions will be degraded. After the measurement an inverse algorithm is used to compute boundary conditions on the investigated surface from the measured temperature history inside the

**Figure 7.** Example of recorded temperature history by one thermocouple inside the sample, computed surface temperature above the installed thermocouple, recorded position of the thermocouple in the cooling section, and computed heat transfer coefficient. Cooling section equipped with five rows of

25 50 75 100 125 150 175 200 225 250

**Time [s]**

Heat transfer test bench presented in [4] is designed so that it enables progression of samples up to the weight of 50 kg with infinitely adjustable speed from 0.1 to 6 m/s (see Fig. 7). On the supporting frame there is a carriage moving, on which the sample under examination with embedded temperature sensors and measuring system is fixed (see Fig. 9). The carriage's progression is provided by a hauling rope through a drive pulley and a motor with a gearbox. The motor is power supplied by a frequency converter with the possibility

flat-jet nozzles.

0

100

200

300

400

500

T measured T surface Position HTC

**T [°C]**

600

700

800

900

1000

sample. An example of recorded temperature history is shown in Fig. 7.

**Figure 5.** Influence of velocity at heat transfer coefficient.

**Figure 6.** Moving sample with embedded thermocouples cooled down by water spray

prescribed velocity which is similar to real conditions. The sample passes under spray which cools down the hot sample. For a simple shape like plate it is recommended to insulate all surfaces excluding the one on which the cooling intensity is investigated. One or more thermocouples are embedded in the sample and measure temperature during the experiment. The installed thermocouples should not disturb the cooled surface. This is the reason why they should be installed inside the sample, not on the investigated surface. In principle when all surfaces are insulated except the one which is investigated one thermocouple is enough. However, this thermocouple should be as close to the investigated surface as possible. Otherwise the resolution of description of boundary conditions will be degraded. After the measurement an inverse algorithm is used to compute boundary conditions on the investigated surface from the measured temperature history inside the sample. An example of recorded temperature history is shown in Fig. 7.

6 Heat Treatment – Conventional and Novel Applications

**Figure 5.** Influence of velocity at heat transfer coefficient.

**Figure 6.** Moving sample with embedded thermocouples cooled down by water spray

Thermocouples Direction of sample movement

Nozzle

**-500 -250 0 250 500** 

Spray

Hot material Insulation

Stationary 2 m/s 5 m/s

**Figure 7.** Example of recorded temperature history by one thermocouple inside the sample, computed surface temperature above the installed thermocouple, recorded position of the thermocouple in the cooling section, and computed heat transfer coefficient. Cooling section equipped with five rows of flat-jet nozzles.

Heat transfer test bench presented in [4] is designed so that it enables progression of samples up to the weight of 50 kg with infinitely adjustable speed from 0.1 to 6 m/s (see Fig. 7). On the supporting frame there is a carriage moving, on which the sample under examination with embedded temperature sensors and measuring system is fixed (see Fig. 9). The carriage's progression is provided by a hauling rope through a drive pulley and a motor with a gearbox. The motor is power supplied by a frequency converter with the possibility 8 Heat Treatment – Conventional and Novel Applications

Design of Cooling Units for Heat Treatment 9

of a smooth change of speed. The direction of the carriage can be reversed and passages repeated in a requiring number. The whole cycle is programmed and controlled through the superior PC. There is a spraying section in the central sector where arbitrary jets configuration can be arranged when distribution of heat transfer coefficients or heat fluxes must be measured. The sample is equipped with thermocouples connected to the data logger. The thermocouples are calibrated before use and the results of calibration are used to eliminate dynamic error in measurement of highly transient thermal processes. Before the actual experiment the carriage with the sample is positioned to the electric heater and it is heated to the required temperature inside the furnace. After the temperature in the sample is stabilized, the heating device is removed, the stand is turned to spraying position, the pump for the water gets going and the carriage's runs through the cooling section. The position of the cooled surface can be horizontal with spraying upper or bottom surfaces or vertical. Signals from the sensors are read by the data logger which moves together along with the sample. At the same time, the signal indicating the actual carriage's position is recorded as well. After performing the required number of passes through cooling zone, data are exported from data logger's internal memory into the computer for further

**3.2. Inverse computation of boundary conditions and numerical models** 

Information from temperature histories in a particular depth under the investigated surface are used as entry parameters for the thermal conduction's inverse task. Inverse task outputs are surface temperature histories, heat flows, and heat transfer coefficients (HTC) as function of time and position. Most often, in mathematical models the boundary condition of the 3'd type is used where heat flow is specified by the HTC value and the cooling water

If the boundary conditions of a solid must be determined from transient temperature measurements at one or more interior locations, it is an inverse heat conduction problem (IHCP) during which the dispersed impulse on boundary must be found. The IHCP is much more difficult to solve than the direct problem. Such problems are extremely sensitive to measurement errors. There are number of procedures that have been advanced for the solution of ill-posed problems in general. Tikhonov has introduced the regularization method [5] to reduce the sensitivity of ill-posed problems to measurement errors. The mathematical techniques for solving sets of ill-conditioned algebraic equations, called single-value decomposition techniques, can also be used for the IHCP [6]. There were extremely varied approaches to the IHCP. These included the use of Duhamel's theorem (or convolution integral) which is restricted to linear problems [7]. Numerical procedures such as finite differences [8][9][10] and finite elements [11] were also employed, due to their inherent ability to treat non-linear problems. Exact solution techniques were proposed by Burggaf [12], Imber and Khan [13], Langford [14], and others. Some techniques used Laplace transforms but these are limited to linear cases [15]. Combined approach is also described in [16]. The improvement in artificial intelligence has brought new approaches, such as genetic

processing.

temperature.

algorithm [17] and neural networks [18][19][20].

**Figure 8.** Heat transfer test bench

**Figure 9.** Examples of boundary conditions measurements on steel plate, rail, and pipe.

of a smooth change of speed. The direction of the carriage can be reversed and passages repeated in a requiring number. The whole cycle is programmed and controlled through the superior PC. There is a spraying section in the central sector where arbitrary jets configuration can be arranged when distribution of heat transfer coefficients or heat fluxes must be measured. The sample is equipped with thermocouples connected to the data logger. The thermocouples are calibrated before use and the results of calibration are used to eliminate dynamic error in measurement of highly transient thermal processes. Before the actual experiment the carriage with the sample is positioned to the electric heater and it is heated to the required temperature inside the furnace. After the temperature in the sample is stabilized, the heating device is removed, the stand is turned to spraying position, the pump for the water gets going and the carriage's runs through the cooling section. The position of the cooled surface can be horizontal with spraying upper or bottom surfaces or vertical. Signals from the sensors are read by the data logger which moves together along with the sample. At the same time, the signal indicating the actual carriage's position is recorded as well. After performing the required number of passes through cooling zone, data are exported from data logger's internal memory into the computer for further processing.

8 Heat Treatment – Conventional and Novel Applications

Moving carriage

Motor with gearbox

Tested sample

Spraying nozzles

Electric heater

Supporting frame

**Figure 8.** Heat transfer test bench

**Figure 9.** Examples of boundary conditions measurements on steel plate, rail, and pipe.

## **3.2. Inverse computation of boundary conditions and numerical models**

Information from temperature histories in a particular depth under the investigated surface are used as entry parameters for the thermal conduction's inverse task. Inverse task outputs are surface temperature histories, heat flows, and heat transfer coefficients (HTC) as function of time and position. Most often, in mathematical models the boundary condition of the 3'd type is used where heat flow is specified by the HTC value and the cooling water temperature.

If the boundary conditions of a solid must be determined from transient temperature measurements at one or more interior locations, it is an inverse heat conduction problem (IHCP) during which the dispersed impulse on boundary must be found. The IHCP is much more difficult to solve than the direct problem. Such problems are extremely sensitive to measurement errors. There are number of procedures that have been advanced for the solution of ill-posed problems in general. Tikhonov has introduced the regularization method [5] to reduce the sensitivity of ill-posed problems to measurement errors. The mathematical techniques for solving sets of ill-conditioned algebraic equations, called single-value decomposition techniques, can also be used for the IHCP [6]. There were extremely varied approaches to the IHCP. These included the use of Duhamel's theorem (or convolution integral) which is restricted to linear problems [7]. Numerical procedures such as finite differences [8][9][10] and finite elements [11] were also employed, due to their inherent ability to treat non-linear problems. Exact solution techniques were proposed by Burggaf [12], Imber and Khan [13], Langford [14], and others. Some techniques used Laplace transforms but these are limited to linear cases [15]. Combined approach is also described in [16]. The improvement in artificial intelligence has brought new approaches, such as genetic algorithm [17] and neural networks [18][19][20].

#### 10 Heat Treatment – Conventional and Novel Applications

All the mentioned algorithms need a precise mathematical model of the tested sample for computing the direct heat conduction problem. Analytical methods may be used, in certain cases, for exact mathematical solutions of conduction problems. These solutions have been obtained for many simplified geometries and boundary conditions and are well documented in the literature [21][22][23]. However, more often than not, geometries and boundary conditions preclude such a solution. In these cases, the best alternative is the one using a numerical technique. For situations where no analytical solution is available, the numerical method can be used. Nowadays there are several methods that enable us to solve numerically the governing equations of heat transfer problems. These include: the finite difference method (FDM), finite volume method (FVM), finite element method (FEM), boundary element method (BEM), and others. For one-dimensional model with constant material properties there exists nice similarity. All of the FDM, FVM, and FEM with tent weighting function equations can be put in a similar form:

$$\frac{d}{dt}(\beta \, T\_1 + \gamma \, T\_2) = \frac{-\alpha}{\Delta \mathbf{x}^2} \left(T\_1 - T\_2\right) + \frac{q\_1 \{t\}}{\rho c \, \Delta \mathbf{x}} \, \tag{1}$$

Design of Cooling Units for Heat Treatment 11

a. *Distinct*: The phase change region consists of solid and liquid phases separated by a smooth continuous front – freezing of water or rapid solidification of pure metal. b. *Alloy*: The phase change region has a crystalline structure consisting of grains and

c. *Continuous*: The liquid and solid phases are fully dispersed throughout the phase change region and there is no distinct interface between the solid and liquid phase –

In a distinct phase change, the state is characterized by the position of the interface. In such cases the class of the so called *front tracking* methods is usually used. However, in cases b)

*d dd l ll*

(4)

*dd ll k g k gk* (5)

(7)

(8)

(9)

(6)

*<sup>g</sup> H s gHs k T <sup>t</sup>* 

where *g* is the phase volume fraction, *s* is the phase velocity, and subscript *d* and *l* refer to solid and liquid phases (or structure A and structure B), respectively [24]. The *k* is (in this

*T T*

*Tref Tref H g c dT g c dT c L*

> *vol H T <sup>g</sup> c H ttt*

 

 *vol <sup>T</sup> c kT q <sup>t</sup>* 

> *<sup>l</sup> <sup>g</sup> q H <sup>t</sup>*

Eq. (4) is non-linear and it contains two related but unknown variables *H* and *T* . It is convenient to reformulate this equation in terms of a single unknown variable with

Neglecting convection effects in Eq. (4) and substituting Eq. (7) results in

*s dd l ll ll*

where *Tref* is an arbitrary reference temperature. To overcome the problem of the non-linear (discontinuity) coefficient of a specific heat a non-linear source term is used. The

 

*l*

solid/liquid interface has a complex shape – most metal alloys.

The phase change process can be described by a single enthalpy equation

*H*

polymers or glasses.

and c) the models use the phase fraction.

case) a mixture conductivity defined as

and *H* is the mixture enthalpy

term *H t* can be expanded as

where

$$\frac{d}{dt}\left(\boldsymbol{\chi}\,\boldsymbol{T}\_{j-1} + \boldsymbol{2}\,\boldsymbol{\beta}\,\boldsymbol{T}\_{j} + \boldsymbol{\gamma}\,\boldsymbol{T}\_{j+1}\right) = \frac{-\alpha}{\Delta\mathbf{x}^{2}}\left(\boldsymbol{T}\_{j} - \boldsymbol{T}\_{j+1}\right) + \frac{\alpha}{\Delta\mathbf{x}^{2}}\left(\boldsymbol{T}\_{j-1} + \boldsymbol{T}\_{j}\right)\,\boldsymbol{\epsilon}\tag{2}$$

$$\frac{d}{dt}(\boldsymbol{\chi}\,T\_{N-1} + \boldsymbol{\beta}\,T\_N) = \frac{-\alpha}{\Delta\mathbf{x}^2} \left(T\_{N-1} - T\_N\right) + \frac{q\_N\left(t\right)}{\rho\boldsymbol{c}\,\Delta\mathbf{x}}\,\tag{3}$$

where and have the values listed in Tab. 1. Equations (1–3) are restricted to temperature-independent thermal properties but the concepts can be extended to T-variable cases. In general for multidimensional models and temperature dependent material properties the simplest equations are obtained for FDM while the complexity of equation for FVM and FEM is several times higher.


**Table 1.** Values of the and of Eq. (1–3)

#### **3.3. Phase change implementation**

Physical processes, such as solid/liquid and solid state transformations, involve phase changes. The numerical treatment of this non‑linear phenomenon involves many problems. Methods for solving the phase change usually use a total enthalpy *H* , an apparent specific heat coefficient *Ac* , or a heat source *q* .

The nature of a solidification phase change can take many forms. The classification is based on the matter in the phase change region. The most common cases follow:


In a distinct phase change, the state is characterized by the position of the interface. In such cases the class of the so called *front tracking* methods is usually used. However, in cases b) and c) the models use the phase fraction.

The phase change process can be described by a single enthalpy equation

$$\frac{\partial H}{\partial t} + \nabla \cdot \left( \mathbf{g}\_d H\_d \mathbf{s}\_d + \mathbf{g}\_l H\_l \mathbf{s}\_l \right) = \nabla \cdot \left( k \nabla T \right) \tag{4}$$

where *g* is the phase volume fraction, *s* is the phase velocity, and subscript *d* and *l* refer to solid and liquid phases (or structure A and structure B), respectively [24]. The *k* is (in this case) a mixture conductivity defined as

$$k = \mathbf{g}\_d k\_d + \mathbf{g}\_l k\_l \tag{5}$$

and *H* is the mixture enthalpy

$$\mathbf{H} = \mathbf{g}\_s \int\_{Tref}^{T} \rho\_d \mathbf{c}\_d d\mathbf{T} + \mathbf{g}\_l \int\_{Tref}^{T} \rho\_l \mathbf{c}\_l d\mathbf{T} + \rho\_l \mathbf{c}\_l \mathbf{L} \tag{6}$$

where *Tref* is an arbitrary reference temperature. To overcome the problem of the non-linear (discontinuity) coefficient of a specific heat a non-linear source term is used. The term *H t* can be expanded as

$$\frac{\partial H}{\partial t} = c\_{vol} \frac{\partial T}{\partial t} + \delta H \frac{\partial \mathbf{g}\_l}{\partial t} \tag{7}$$

Neglecting convection effects in Eq. (4) and substituting Eq. (7) results in

$$\mathcal{L}\_{\text{vol}} \frac{\partial T}{\partial t} = \nabla \cdot \left( k \nabla T \right) + \dot{q} \tag{8}$$

where

10 Heat Treatment – Conventional and Novel Applications

weighting function equations can be put in a similar form:

FVM and FEM is several times higher.

 and 

**3.3. Phase change implementation** 

heat coefficient *Ac* , or a heat source *q* .

where

 and 

**Table 1.** Values of the

 

 

 

of Eq. (1–3)

on the matter in the phase change region. The most common cases follow:

**FDM** 1/2 0 1/2 **FVM** 3/8 1/8 1/2 **FEM** 2/6 1/6 1/2

Physical processes, such as solid/liquid and solid state transformations, involve phase changes. The numerical treatment of this non‑linear phenomenon involves many problems. Methods for solving the phase change usually use a total enthalpy *H* , an apparent specific

The nature of a solidification phase change can take many forms. The classification is based

*dt x x*

All the mentioned algorithms need a precise mathematical model of the tested sample for computing the direct heat conduction problem. Analytical methods may be used, in certain cases, for exact mathematical solutions of conduction problems. These solutions have been obtained for many simplified geometries and boundary conditions and are well documented in the literature [21][22][23]. However, more often than not, geometries and boundary conditions preclude such a solution. In these cases, the best alternative is the one using a numerical technique. For situations where no analytical solution is available, the numerical method can be used. Nowadays there are several methods that enable us to solve numerically the governing equations of heat transfer problems. These include: the finite difference method (FDM), finite volume method (FVM), finite element method (FEM), boundary element method (BEM), and others. For one-dimensional model with constant material properties there exists nice similarity. All of the FDM, FVM, and FEM with tent

> <sup>1</sup> 1 2 12 2 *<sup>d</sup> q t T T TT dt x c x*

 1 1 11 2 2 2 *j jj jj j j <sup>d</sup> T T T TT T T*

> 1 1 2

 

temperature-independent thermal properties but the concepts can be extended to T-variable cases. In general for multidimensional models and temperature dependent material properties the simplest equations are obtained for FDM while the complexity of equation for

*N N NN <sup>d</sup> q t T T TT dt x c x* 

 

*N*

 + 

, (1)

, (3)

, (2)

have the values listed in Tab. 1. Equations (1–3) are restricted to

$$
\dot{q} = -\delta H \frac{\partial g\_l}{\partial t} \tag{9}
$$

Eq. (4) is non-linear and it contains two related but unknown variables *H* and *T* . It is convenient to reformulate this equation in terms of a single unknown variable with

#### 12 Heat Treatment – Conventional and Novel Applications

non-linear latent heat. Song [25] and [26] Comini uses so called Apparent heat capacity. The apparent specific heat can be defined as

$$\mathcal{L}\_A = \frac{dH}{dT} = \mathcal{c}\_{vol} + \delta H \frac{d\mathcal{g}\_l}{dT} \tag{10}$$

Design of Cooling Units for Heat Treatment 13

*mT* are compared with the computed temperature *mT*

. (15)

(16)

The sample is heated before starting the measurement. Cooling is applied on one surface and temperature response inside the sample is recorded. Time-dependent boundary conditions are computed using inverse technique from the measured temperature history (see Fig. 7).

This new proposed approach computes step by step (time step) heat transfer coefficients (HTC) on the investigated surface using measured temperature history inside the cooled or heated solid body. However, this method can be very easily changed to compute any kind of boundary conditions, e. g. heat flux. The method uses sequential estimation of the time varying boundary conditions and uses future time steps data to stabilize the ill-posed inverse problem [28]. To determine the unknown surface HTC at the current time *mt* , the

<sup>2</sup> \*

*i i*

1

Any forward solver can be used e. g. finite volume method described by Patankar [29]. The computational model should include drilled hole, whole internal structure of the embedded

At time zero homogeneous temperature is in the sample and thereby zero heat flux and thereby zero HTC on all surfaces is assumed. Otherwise there cannot be homogeneous temperature. This can be done e. g. by heating in furnace after enough long time. If the initial temperature is not homogeneous some modification of the algorithm is necessary for

The algorithm starts at time index zero when the HTC is equal to zero (see Fig. 10). The algorithm uses forward solver and it computes temperature response at thermocouple position for linearly changing (increasing or decreasing) HTC (see HTC1 and T1 computed in Fig. 10) over few time steps. These time steps are called future time steps *n* and five of them are used in Fig. 10–Fig. 11. Determination of minimum number of necessary future time steps to stabilize sequential algorithm is described in [28]. The computed and measured temperatures histories are compared using Eq. (15) the same one as for sequential Beck

*h*

*t*

should be changed until the minimum of SSE function in Eq. (15) is found. Such a minimum says that the computed temperature history matches the measured temperature history the

When the best slope of HTC is found the forward solver is used to compute temperature field in the next time step using the computed boundary conditions. The algorithm is repeated for next time steps until the end of recorded temperature history is reached (see Fig. 11). For *k*

*v*

*m n*

*i m SSE T T* 

Cooling of more surfaces can also be investigated when more thermocouples is used.

measured temperature responses \*

the first time step.

from the forward solver using *n* future times steps

thermocouple, and temperature dependent material properties.

approach. The slope of linearly changing HTC defined as

best for used linearly changing HTC during *n* future time steps.

where

$$\begin{aligned} \mathcal{L}\_{\text{vol}} &= \mathcal{g}\_d \rho\_d \mathcal{c}\_d + \mathcal{g}\_l \rho\_l \mathcal{c}\_l \\ \mathcal{L} \delta H &= \int\_{Tref}^T \left( \rho\_l \mathcal{c}\_l - \rho\_d \mathcal{c}\_d \right) dT + \rho\_l L. \end{aligned} \tag{11}$$

Neglecting convection effects and substituting into Eq. (4) yields apparent heat capacity equation

$$\mathbf{c}\_A \frac{\partial T}{\partial t} = \nabla \cdot \left(k \nabla T\right). \tag{12}$$

Another approach is total enthalpy. From Eq. (6) it can be written

$$
\nabla T = \nabla H \not\!\mathcal{C}\_{vol} - \delta H \nabla \mathcal{g}\_l \not\!\mathcal{C}\_{vol} \,. \tag{13}
$$

Substitution in Eq. (4) will result in a total enthalpy equation

$$\frac{\partial H}{\partial t} = \nabla \cdot \left(\frac{k}{c\_{vol}} \nabla H\right) + \nabla \cdot \left(\frac{k}{c\_{vol}} \delta H \nabla g\_l\right). \tag{14}$$

#### **3.4. Sequential identification inverse method**

For measurements where installed thermocouple inside the investigated body disturbs also surface temperature as it is very close to investigated surface HTC must be computed directly by an inverse method. Classical and very efficient sequential estimation proposed by Beck [27], which computes heat flux instead of HTC, has several limitations. Thus new sequential identification method was developed by Pohanka to solve such inverse problems. The basic principle of time-dependent boundary conditions determination (heat flux, HTC, and surface temperature) from measured transient temperature history is based on cooling (or heating) of heated (or cold) sample with thermocouple installed inside (see Figure 6). Let us assume one-dimensional inverse problem with 3D model involving installed thermocouple for simplicity:


The sample is heated before starting the measurement. Cooling is applied on one surface and temperature response inside the sample is recorded. Time-dependent boundary conditions are computed using inverse technique from the measured temperature history (see Fig. 7). Cooling of more surfaces can also be investigated when more thermocouples is used.

12 Heat Treatment – Conventional and Novel Applications

apparent specific heat can be defined as

where

equation

non-linear latent heat. Song [25] and [26] Comini uses so called Apparent heat capacity. The

*A vol dH dg c cH dT dT*

*vol d d d l l l T*

*c g c gc*

*Tref*

Another approach is total enthalpy. From Eq. (6) it can be written

Substitution in Eq. (4) will result in a total enthalpy equation

**3.4. Sequential identification inverse method** 

thermocouple for simplicity:

Known dimensions of the sample.

HTC is not dependent on position.

All surfaces are insulated except the cooled one.

*l*

 

(11)

. (12)

*l*

. (13)

. (14)

(10)

.

*H c c dT L*

Neglecting convection effects and substituting into Eq. (4) yields apparent heat capacity

*<sup>A</sup> <sup>T</sup> c kT t* 

*vol l vol T Hc H g c* 

*vol vol Hk k H Hg tc c*

For measurements where installed thermocouple inside the investigated body disturbs also surface temperature as it is very close to investigated surface HTC must be computed directly by an inverse method. Classical and very efficient sequential estimation proposed by Beck [27], which computes heat flux instead of HTC, has several limitations. Thus new sequential identification method was developed by Pohanka to solve such inverse problems. The basic principle of time-dependent boundary conditions determination (heat flux, HTC, and surface temperature) from measured transient temperature history is based on cooling (or heating) of heated (or cold) sample with thermocouple installed inside (see Figure 6). Let us assume one-dimensional inverse problem with 3D model involving installed

 Known thermal temperature-dependent material properties of the sample. Known temperature profile at the beginning of the cooling (usually constant).

 

 

*ll dd l*

This new proposed approach computes step by step (time step) heat transfer coefficients (HTC) on the investigated surface using measured temperature history inside the cooled or heated solid body. However, this method can be very easily changed to compute any kind of boundary conditions, e. g. heat flux. The method uses sequential estimation of the time varying boundary conditions and uses future time steps data to stabilize the ill-posed inverse problem [28]. To determine the unknown surface HTC at the current time *mt* , the measured temperature responses \* *mT* are compared with the computed temperature *mT* from the forward solver using *n* future times steps

$$SSE = \sum\_{i=m+1}^{m+n} \left(T\_i^\* - T\_i\right)^2. \tag{15}$$

Any forward solver can be used e. g. finite volume method described by Patankar [29]. The computational model should include drilled hole, whole internal structure of the embedded thermocouple, and temperature dependent material properties.

At time zero homogeneous temperature is in the sample and thereby zero heat flux and thereby zero HTC on all surfaces is assumed. Otherwise there cannot be homogeneous temperature. This can be done e. g. by heating in furnace after enough long time. If the initial temperature is not homogeneous some modification of the algorithm is necessary for the first time step.

The algorithm starts at time index zero when the HTC is equal to zero (see Fig. 10). The algorithm uses forward solver and it computes temperature response at thermocouple position for linearly changing (increasing or decreasing) HTC (see HTC1 and T1 computed in Fig. 10) over few time steps. These time steps are called future time steps *n* and five of them are used in Fig. 10–Fig. 11. Determination of minimum number of necessary future time steps to stabilize sequential algorithm is described in [28]. The computed and measured temperatures histories are compared using Eq. (15) the same one as for sequential Beck approach. The slope of linearly changing HTC defined as

$$
\upsilon = \frac{\partial h}{\partial t} \tag{16}
$$

should be changed until the minimum of SSE function in Eq. (15) is found. Such a minimum says that the computed temperature history matches the measured temperature history the best for used linearly changing HTC during *n* future time steps.

When the best slope of HTC is found the forward solver is used to compute temperature field in the next time step using the computed boundary conditions. The algorithm is repeated for next time steps until the end of recorded temperature history is reached (see Fig. 11). For *k*

Design of Cooling Units for Heat Treatment 15

**slope of HTC**

The SSE function described by Eq. (15) has only one minimum and is dependent only on one variable – slope of HTC (see Eq. 16). Even more the function is very close to parabolic function near the searched minimum because it is sum of square of temperature differences. Brent's optimization method [30], which uses inverse parabolic interpolation, is perfect

Brent's optimization method is based on parabolic interpolation and golden section. The searched minimum must be between two given points 1 and 2 (see Fig. 12). Convergence to a minimum is gained by inverse parabolic interpolation. Function values of the SSE function are computed only in few points. A parabola (dashed line) is drawn through the three original points 1, 3, 2 on the SSE function (solid line). The function is evaluated at the parabola's minimum, 4, which replaces point 1. A new parabola (dotted line) is drawn through points 3, 4, 2. The algorithm is repeated until the minimum with desired accuracy is found. If the three points are collinear the golden section [30] is used instead of parabolic

SSE

parabola through 1, 3, 2 parabola through 3, 4, 2

**2**

candidate for finding the minimum of the SSE function in Eq. (15).

**Figure 12.** Convergence to a minimum by inverse parabolic interpolation.

**3**

To demonstrate the procedure a real measurement is used. The cooling section consists of five rows of flat-jet nozzles. The heated sample passes repeatedly under the spraying nozzles. Several thermocouples in one row, which is perpendicular to the sample movement direction, were installed in the sample to be able to investigate also the cooling homogeneity across the sample. For simplicity we focus now on only one thermocouple, however, it is easy

**4**

**3.5. Evaluation of boundary conditions** 

interpolation.

**1**

**SSE**

**Figure 10.** Measured temperature history and two computed temperature histories using two different slopes of HTC for *n* future time steps.

**Figure 11.** Real HTC and six optimum linearly changing HTC.

measured time steps only *k n* time steps can be computed owing to the use of future data. This method works perfectly when real HTC is almost linear in time. When the slope is abruptly changing the computed HTC curve is slightly smoother than the real one; the more future time steps are used the smoother is the computed curve of HTC (bigger difference between computed and real HTC) but the sequential identification inverse algorithm is more stable.

The SSE function described by Eq. (15) has only one minimum and is dependent only on one variable – slope of HTC (see Eq. 16). Even more the function is very close to parabolic function near the searched minimum because it is sum of square of temperature differences. Brent's optimization method [30], which uses inverse parabolic interpolation, is perfect candidate for finding the minimum of the SSE function in Eq. (15).

Brent's optimization method is based on parabolic interpolation and golden section. The searched minimum must be between two given points 1 and 2 (see Fig. 12). Convergence to a minimum is gained by inverse parabolic interpolation. Function values of the SSE function are computed only in few points. A parabola (dashed line) is drawn through the three original points 1, 3, 2 on the SSE function (solid line). The function is evaluated at the parabola's minimum, 4, which replaces point 1. A new parabola (dotted line) is drawn through points 3, 4, 2. The algorithm is repeated until the minimum with desired accuracy is found. If the three points are collinear the golden section [30] is used instead of parabolic interpolation.

**Figure 12.** Convergence to a minimum by inverse parabolic interpolation.

## **3.5. Evaluation of boundary conditions**

14 Heat Treatment – Conventional and Novel Applications

Optimum is somewhere here

slopes of HTC for *n* future time steps.

0

**HTC [W/m².K]**

20

40

60

**T [°C]**

80

100

120

**Figure 11.** Real HTC and six optimum linearly changing HTC.

**Figure 10.** Measured temperature history and two computed temperature histories using two different

i=0

i=1 i=2

i=3

0 1 2 3 4 5 6 7 8 9 10 **Time index**

0

i=4

i=5

1000

2000

3000

**HTC [W/m².K]**

4000

5000

T measured T1 computed T2 computed

HTC1 HTC2 6000

measured time steps only *k n* time steps can be computed owing to the use of future data. This method works perfectly when real HTC is almost linear in time. When the slope is abruptly changing the computed HTC curve is slightly smoother than the real one; the more future time steps are used the smoother is the computed curve of HTC (bigger difference between computed and real HTC) but the sequential identification inverse algorithm is more stable.

0 1 2 3 4 5 6 7 8 9 10 **Time index**

> To demonstrate the procedure a real measurement is used. The cooling section consists of five rows of flat-jet nozzles. The heated sample passes repeatedly under the spraying nozzles. Several thermocouples in one row, which is perpendicular to the sample movement direction, were installed in the sample to be able to investigate also the cooling homogeneity across the sample. For simplicity we focus now on only one thermocouple, however, it is easy

to do the procedure for all thermocouples. Temperature record from such measurement is shown in Figure 7. Recorded position (zigzag line) of the thermocouple is shown as well and demonstrates the repeated passes through cooling section. Using the inverse method boundary conditions were computed: surface temperature and HTC. All the shown lines are function of time; however, for numerical simulation we need HTC as function of position and surface temperature. We start with surface temperature from the measurement. See Figure 13 with shown surface temperature drawn using green line as function of position. The green lines represents surface temperatures through which the plate pass during experiment. In the place where the green line is shown we also know HTC from the measurement. HTC values are shown using the color scale. HTC values between green lines are interpolated. This chart shows HTC distribution as a function of surface temperature and position and is the key point for accurate numerical simulation. HTC values above the most top green line are extrapolated and are not accurate as there are no data available from measurement. We should avoid usage of these values during numerical simulation.

Design of Cooling Units for Heat Treatment 17

the cooling rate for the surface temperature is higher than in the center. The results are drawn in CCT diagram, however, you should not that the cooling rate is far away from constant. This is very important because the CCT diagram is only informative and the final structure has to be verified by experimental measurement. There are three major reasons why cooling rate is not constant. One is caused by passing product under separate row of nozzles. The passes are obvious from T surface curve in Fig. 14. You can see drops of temperature when the product is passing under spray followed by reheating due to the internal capacity of the heat in the product. The second reason is mentioned Leidenfrost point. You can see low cooling rate in the center up to 20 s as the surface temperature is above Leidenfrost point and after that cooling rate is increasing and reaching maximum which is almost triple in comparison to value above Leidenfrost point. Decreasing of cooling rate is followed as the surface temperature is getting closer to the temperature of water. The third reason is low diffusivity for big products. The product cannot be cooled down at the same cooling rate on the surface as in the center. The lower is the diffusivity and the bigger is the product the bigger difference is between the cooling rate on the surface and in the

center.

**Figure 14.** Simulation of cooling in the CCT diagram.

**4. Verification conducted at pilot test bench** 

It is important to understand that cooling rate in cooling section in industrial application is far away from constant value using which CCT diagrams are obtained. Verification

**Figure 13.** Prepared boundary conditions for numerical simulation from measurement shown in Fig. 7. Chart shows HTC as function dependent on position in cooling section in the direction of sample movement and on surface temperature of the cooled sample.

#### **3.6. Numerical simulation**

Having prepared boundary conditions we can do numerical simulation of cooling of products of various material properties and of various thicknesses. By repeating the boundary conditions we can simulate long cooling section with more rows of cooling nozzles. An example of such simulation is shown in Fig. 14 and is drawn in CCT. You can see computed temperature at the surface and at the center of the material. It is obvious that the cooling rate for the surface temperature is higher than in the center. The results are drawn in CCT diagram, however, you should not that the cooling rate is far away from constant. This is very important because the CCT diagram is only informative and the final structure has to be verified by experimental measurement. There are three major reasons why cooling rate is not constant. One is caused by passing product under separate row of nozzles. The passes are obvious from T surface curve in Fig. 14. You can see drops of temperature when the product is passing under spray followed by reheating due to the internal capacity of the heat in the product. The second reason is mentioned Leidenfrost point. You can see low cooling rate in the center up to 20 s as the surface temperature is above Leidenfrost point and after that cooling rate is increasing and reaching maximum which is almost triple in comparison to value above Leidenfrost point. Decreasing of cooling rate is followed as the surface temperature is getting closer to the temperature of water. The third reason is low diffusivity for big products. The product cannot be cooled down at the same cooling rate on the surface as in the center. The lower is the diffusivity and the bigger is the product the bigger difference is between the cooling rate on the surface and in the center.

**Figure 14.** Simulation of cooling in the CCT diagram.

16 Heat Treatment – Conventional and Novel Applications

should avoid usage of these values during numerical simulation.

movement and on surface temperature of the cooled sample.

**3.6. Numerical simulation** 

to do the procedure for all thermocouples. Temperature record from such measurement is shown in Figure 7. Recorded position (zigzag line) of the thermocouple is shown as well and demonstrates the repeated passes through cooling section. Using the inverse method boundary conditions were computed: surface temperature and HTC. All the shown lines are function of time; however, for numerical simulation we need HTC as function of position and surface temperature. We start with surface temperature from the measurement. See Figure 13 with shown surface temperature drawn using green line as function of position. The green lines represents surface temperatures through which the plate pass during experiment. In the place where the green line is shown we also know HTC from the measurement. HTC values are shown using the color scale. HTC values between green lines are interpolated. This chart shows HTC distribution as a function of surface temperature and position and is the key point for accurate numerical simulation. HTC values above the most top green line are extrapolated and are not accurate as there are no data available from measurement. We

**Figure 13.** Prepared boundary conditions for numerical simulation from measurement shown in Fig. 7. Chart shows HTC as function dependent on position in cooling section in the direction of sample

HTC [W/m².K] 14

5000

10000

15000

20000

25000

30000

35000

40000 42695

Having prepared boundary conditions we can do numerical simulation of cooling of products of various material properties and of various thicknesses. By repeating the boundary conditions we can simulate long cooling section with more rows of cooling nozzles. An example of such simulation is shown in Fig. 14 and is drawn in CCT. You can see computed temperature at the surface and at the center of the material. It is obvious that

#### **4. Verification conducted at pilot test bench**

It is important to understand that cooling rate in cooling section in industrial application is far away from constant value using which CCT diagrams are obtained. Verification

#### 18 Heat Treatment – Conventional and Novel Applications

functionality of a newly designed cooling system prior to its plant implementation is essential. The design obtained by using the numerical model must be verified and finetuned by further full-scale cooling tests. Pieces of tube, rail, wire or plate of real dimensions with implemented thermocouples are tested in the designed cooling section. The length of a laboratory test bench shown in Fig. 8 and Fig. 9 is limited, hence the sample must be accelerated prior entering the cooling section, to a velocity normally used in a plant, and after pass through cooling section, the direction of movement is reversed. In this way, the sample moves several times under the cooling sections. This cooling process is controlled by computer to simulate running under the long cooling section used normally in the plant. Nozzles, pressures, and header configurations are tested. The design of the cooling and the pressures used are modified until the demanded temperature regime and final structure is obtained. The full-scale material samples are then cut for the tests of material properties and structure.

Design of Cooling Units for Heat Treatment 19

The design of cooling sections used for in-line heat treatment for hot rolling plants is very extensive work. It utilizes laboratory measurement, numerical modeling, inverse computations, and also pilot mill tests. The first step is the search of the best cooling regime for steels for which this is not yet known. The second step is to obtain a selection of technical means in order to guarantee obtaining the prescribed cooling rates. Nozzle configurations and cooling parameters are selected and controllability of the cooling section is checked. The final step of the design is a laboratory test using a full size sample simulating plant cooling. Design based on laboratory measurement therefore minimizes the amount of expensive experimentation performed directly on the plant. Elimination of potential errors and enabling adjustment of control models in the plant is possible after the cooling process is

*Brno University of Technology, Heat transfer and Fluid Flow Laboratory, Brno, Czech Republic* 

[1] Raudenský, M.; Horský, J.; Hajduk, D.; Čecho, L. Interstand Cooling - Design, Control and Experience. *Journal of Metallurgical and Mining Industry*, 2010, No. 2, Vol. 3, pp. 193-202. [2] Hnízdil, M.; Raudenský, M.; Horký, J.; Kotrbáček P.; Pohanka M. In-Line Heat Treatment and Hot Rolling. *In proc. AMPT 2010*, Paris, France, October 2010, pp.

[3] Horký, J.; Raudenský, M.; Kotrbáček, P. Experimental study of long product cooling in hot rolling. *Journal of Materials Processing Technology*, August 1998, Vol. 80-81, pp. 337-340. [4] Pohanka, M.; Bellerová, H.; Raudenský, M.; Experimental Technique for Heat Transfer Measurements on Fast Moving Sprayed Surfaces. *Journal of ASTM International*, Sept.

[5] Tikhonov, A. N.; Arsenin, V. Y. *Solution of Ill-Posed Problems*. Washington, D.C.:

[6] Mandrel, J. Use of the singular value decomposition in regression analysis. *Am. Stat*.,

[7] Stloz, G. Jr. Numerical solutions to an inverse problem of heat conduction for simple

[8] Smith, G. D. *Numerical solution of partial differential equations*. UK: Oxford University

[9] Beck, J. V. Nonlinear estimation applied to the nonlinear heat conduction problem. *Int.* 

[10] Beck. J. V.; Litkouhi, B.; St. Clair, C. R. Jr. Efficient sequential solution of the nonlinear inverse heat conduction problem. *J. Numerical Heat Transfer*, 1982, Vol. 5, pp. 275-286. [11] Bass, B. R. Applications of the finite element to the inverse heat conduction problem

using Beck's second method. *J. Eng. Ind*., 1980, Vol. 102, pp. 168-176.

**5. Concluding remarks** 

tested in laboratory conditions.

Michal Pohanka and Petr Kotrbáček

2010, Vol. 1523, pp. 3-15.

1982, Vol. 36, pp. 15-24.

Winston, 1977. ISBN 0470991240.

Press, 1978. ISBN 0-198-59650-2.

shapes. *Int. J. Heat Transfer*, 1960, Vol. 82, pp. 20-26.

*J. Heat and Mass Transfer*, 1970, Vol. 13, pp. 703-716.

**Author details** 

**6. References** 

563-568.

When heat treatment is performed on larger product such as rail, mainly its head, it is not possible to achieve same cooling rate at surface and in the center of rail head. The cooling rate near surface are much faster and even more reheating can appear and can cause very different material properties (see Fig. 15). As the rail head passed under the spray the surface temperature dropped fast and was followed by reheating due to the heat stored inside the head. The reheating caused lower hardness near the surface as shown in Fig. 14. The center of the head is harder because no reheating occurred in the bigger depth. To avoid this problem the cooling section should be modified. One solution is to use more row of nozzles with smaller row pitch and also nozzles with lower HTC. This can be achieved by smaller pressure or smaller nozzles. Also replacement of flat-jet nozzle by full-cone nozzles can be considered. The Leidenfrost temperature should be also considered. We should be above Leidenfrost temperature or below but definitely not near to avoid big different cooling rates for small changes in surface temperature.

**Figure 15.** Measured temperature histories in a rail head in two depths and measured micro-hardness in rail head after heat treatment.
