Estimation of surface heat flux in continuous casting mould with limited measurement of temperature

International Journal of Thermal Sciences 118 (2017) 435e447

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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Estimation of surface heat flux in continuous casting mould with limited measurement of temperature Udayraj a, Saurav Chakraborty a, Suvankar Ganguly b, E.Z. Chacko b, S.K. Ajmani b, Prabal Talukdar a, * a b

Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India Tata Steel R & D, Jamshedpur, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 February 2017 Received in revised form 11 May 2017 Accepted 12 May 2017

Continuous casting mould is exposed to high heat flux due to its contact with the hot molten metal. The state of heat transfer in a mould and steel solidification depends on the magnitude of the mould boundary heat flux. In the present study, a mathematical model is developed to determine the heat flux across the hot surface of continuous casting mould from limited temperature measurement. The model is based on two-dimensional inverse heat transfer technique that was solved using the conjugate gradient method. Direct problem which involves two-dimensional conduction in the mould is first solved and validated. The inverse problem was tested by using the simulated temperature data obtained from the solution of a direct problem, where good agreement between the actual and estimated boundary heat flux is found. The Gaussian noises are added to the simulated temperatures to mimic the temperature measurement errors for testing anti-noise ability of the inverse problem model. Further, boundary heat flux is also estimated for a continuous casting mould using actually measured plant data. The model is applied to three test cases with temperature data obtained under different operating conditions and the results are analyzed. It has been observed that the proposed methodology results in accurate boundary heat flux estimation. Higher boundary heat flux are obtained for cases with higher casting speed as compared to cases with lower casting speed. © 2017 Elsevier Masson SAS. All rights reserved.

Keywords: Continuous casting mould Heat flux Inverse heat transfer Conjugate gradient method

1. Introduction In the continuous casting of steel, the heat transfer affects many of the quality and operational problems. Due to the non-uniform heating of the mould wall during casting, the resulting temperature distribution gives rise to differential thermal expansion leading to the mould distortion, cracks and breakout. Temperature distribution in mould depends on heat flux at the hot surface of the mould, cooling water velocity, mould wall material and its thickness, casting speed, and carbon content in the steel [1]. Out of the abovementioned parameters, heat flux distribution in the mould is a critical parameter which could provide important information regarding mould performance and product quality. In order to determine the state of heat transfer in continuous casting mould, so as to improve the operational performance and produce defect-free products, accurate calculation of heat flux forms the basis of an advanced control.

* Corresponding author. E-mail address: [email protected] (P. Talukdar). http://dx.doi.org/10.1016/j.ijthermalsci.2017.05.012 1290-0729/© 2017 Elsevier Masson SAS. All rights reserved.

Several studies on heat transfer behavior in a continuous casting mould have been conducted in the past. Various mould heat flux formulas were developed utilizing the industrial mould conditions and casting practices. Such empirical formulas for the mould heat flux are given as a function of casting speed only [2e4]. Later on, experimental and theoretical studies indicated that the boundary heat flux is a function of many other casting parameters as well [5,6]. Some of the factors which may influence the heat flux variation include steel composition, superheat, mould powder, cooling water volume, nozzle type etc. [7]. These are usually not known and difficult to measure during continuous casting, thereby giving rise to uncertainties and inaccuracy in the heat flux calculation. Absence of any reliable and general correlation for boundary heat flux as a function of casting variables led to the development of mathematical models to estimate boundary heat flux based on inverse heat transfer techniques [8]. In the past, various approaches [9e13] were adopted to develop inverse heat transfer algorithms and computational method. A good amount of literature is devoted to the determination of boundary heat flux [14,15]. A one-dimensional (1D) inverse heat

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Udayraj et al. / International Journal of Thermal Sciences 118 (2017) 435e447

transfer model was developed to determine boundary heat flux [16]. Inverse methodology was used to estimate mould boundary heat flux using experimental temperature data obtained from a mould simulator [17]. Later, Ko et al. [18], Park and Sohn [19] and Zhang et al. [20] developed a heat transfer model for mould considering 1D transient heat conduction and estimated boundary heat flux from experimentally measured temperature in a mould simulator. They also used the commercial 1D inverse heat conduction software [16] for this purpose. A two-dimensional (2D) transient heat transfer model was also developed for mould and boundary heat flux was calculated [20]. Boundary heat flux estimated by the 1D [20] and 2D [21] approaches were compared later [22]. Heat fluxes given by 2D model were found to be 1.2 to 2 times higher than heat fluxes estimated by 1D model. Wang et al. [23] also developed a 2D transient heat transfer model and estimated boundary heat flux along mould width using experimental data. Ranut et al. [24] modeled heat transfer through mould as a 2D steady state problem. Experimental temperature measurements were used along with a mathematical model based on conjugate gradient method (CGM) to estimate boundary heat flux. Good agreement with the experimental observations was found. A comprehensive model was developed by Hebi and Man [25] involving 2D transient heat conduction in the mould in case of a round billet. Thermal resistance between billet and mould along with boundary heat flux were estimated based on inverse heat transfer using experimental temperature data. Yin et al. [26] developed a coupled thermo-mechanical model where estimated heat flux is used as boundary condition for the problem. Heat flux was determined based on inverse heat transfer calculations using various sensors located in the transverse and longitudinal sections of the mould. Gonzalez et al. [27], Dvorkin et al. [28] and Nowak et al. [29] coupled solution of a direct problem involving calculation of temperature distribution with an inverse heat transfer code for determining mould heat flux to model solidification process during continuous casting. It may be mentioned here that in order to accurately compute the heat flux variation in a mould, it is important to receive large number of temperature data from the on-line measurement in the mould. However, such provisions of placing large number of temperature sensors in a real production setup are difficult and tedious task [30], and often limited numbers of measured data are available in practical operating conditions which may not reflect the real heat transfer state of the mould. In some cases, temperature data used for inverse heat flux estimation is required to be measured either at mid-section [26] or near the hot face of mould [17,18,25,30,31] which is again very inconvenient and difficult to realize in practice. The present study proposes a methodology wherein boundary heat flux can be estimated using limited temperature sensors which may be placed conveniently away from hot face of the continuous casting mould, without sacrificing the accuracy of the calculation. Accordingly, in the present work, a mathematical modeling exercise has been undertaken to develop a new methodology wherein a two-dimensional inverse heat transfer algorithm based on conjugate gradient method is used to determine the heat flux variation in a continuous casting billet mould. Input condition of temperature profile is provided using theoretical calculation and also the experimental measurement from the continuous casting shop of Tata Steel, India in the operating condition. The study provides valuable insight towards heat transfer analysis in real continuous casting process. 2. Problem description Present study deals with estimation of boundary heat flux

across the hot surface of a billet mould in contact with the molten steel during continuous casting process. Schematic of the problem is shown in Fig. 1. A 2D longitudinal section of the mould is considered as boundary heat flux varies significantly along this direction. It is to be noted here that the mould is assumed to be straight unlike the tapered mould in actual conditions. Hot (right) side of the mould is in contact with the molten steel and exposed to 00 a heat flux (qs ) below the meniscus. This boundary heat flux is a function of y and varies along the mould height. Convection heat losses takes place above the meniscus as shown in Fig. 1. Height of the meniscus is H from the bottom. Total height of the mould is H þ H1 and width is L. Outer surface (left) of the mould is cooled with the help of water flowing in channel of width, L1. Water enters at a velocity (Uw) and temperature (Tw,in) which results in convection heat loss from the cold side. As cold water flows through the channel, it absorbs heat from the hot mould and its temperature increases. Temperature of the water flowing through the channel is hence a function of distance, y. Objective of the present problem is thus to obtain mould boundary heat flux which varies along the mould height using an inverse heat transfer technique. 3. Mathematical modeling Estimation of boundary heat flux for the mould involves solution of direct problem which includes 2D conduction through mould and solution of inverse problem. Both direct and inverse problems are explained below: 3.1. Direct problem Direct problem involves analyzing temperature distribution throughout the mould with a known boundary heat flux. Heat transfer through mould is considered to be two dimensional as shown in Fig. 1 and steady state condition is assumed. Governing equation for heat transfer in the mould is as follows [1]:

v2 T v2 T þ ¼0 vx2 vy2

(1)

where, T is temperature, x and y are coordinate directions. It can be observed from Fig. 1 that the right boundary of the mould (below meniscus) is exposed to a spatially varying heat flux. It is basically the heat removed from the hot liquid metal during it downward movement where solidification of liquid metal slowly begins. Corresponding boundary condition below meniscus at the right wall of the mould (x ¼ L) is as follows:

Liquid pool

L

Mould

L

ha, Ta

hw, Tw(y)

H+H1 y

Meniscus

x z

H+H1

H Mould y x

Water: Uw, Tw,in

Fig. 1. Schematic of the problem.

qs" (L, y)

Udayraj et al. / International Journal of Thermal Sciences 118 (2017) 435e447

k

vTðL; yÞ 00 ¼ qs ðL; yÞ vx

ð0  y  HÞ

(2)

where, k is thermal conductivity of mould material and qs00 (L, y) is the heat flux at the hot face (below meniscus) of the mould due to its contact with molten steel. Above the meniscus (y > H) at the right boundary (x ¼ L), heat loss due to convection is expressed as:

vTðL; yÞ ¼ ha ½TðL; yÞ  Ta  k vx

ðH < y  H þ H1 Þ

vTð0; yÞ ¼ hw ½Tð0; yÞ  Tw ðyÞ vx

ð0  y  H þ H1 Þ

(4)

where, hw is heat transfer coefficient for the water which is a function of water velocity (or Reynolds number). It is obtained using following well-known DittuseBoelter equation [32]:



hw DH r Uw DH 0:8 cp;w mw 0:4 ¼ 0:023 w kw mw kw

(5)

where, Uw is water inlet velocity (m/s), rw is density of water (kg/ m3), mw is dynamic viscosity of water (Ns/m2), cp,w is specific heat of water (J/kgK) and kw is thermal conductivity of water. DH is hydraulic diameter of the water channel. This correlation is valid for Reynolds number (ReD) greater than 10,000 (turbulent flow) and L/ D > 10. This correlation is used here as for all the cases studied in the present study, ReD > 10,000 and L/D > 10. It can be observed from above correlation that as water inlet velocity (Uw) increases, heat transfer coefficient (hw) also increases. In Eq. (4), Tw is water temperature which is a function of y. It is well-known that the cooling water flowing through the channel cools one side of the mould. During the course, temperature of the water increases along the channel length. In order to consider this change in temperature of water along the channel length, another governing equation for water is required [1]. Applying an energy balance at the interface between mould and water channel, the governing equation for water temperature is:

rw Uw L1 cp;w

dTw  hw ½Tð0; yÞ  Tw ðyÞ ¼ 0 dy

(6)

where, rw, Tw and cp,w are density, temperature and specific heat of the water flowing in the channel. L1 is the width of the cooling water channel. Top and bottom boundaries of the mould are assumed to be adiabatic [1]:

k

vTðx; 0Þ ¼0 vy

k

vTðx; H þ H1 Þ ¼0 vy

ð0  x  LÞ

ð0  x  LÞ

boundary condition depends on the direction of the water flow, whether water is flowing upward or downward. In case of upward flow of water,

Tw ð0Þ ¼ Tw;in

(9)

In case of downward flow of water,

Tw ðH þ H1 Þ ¼ Tw;in

(10)

(3)

where, H1 is the height of the mould above meniscus as shown in Fig. 1. In the above equation, ha is heat transfer coefficient for hot face of the mould above meniscus and Ta is ambient temperature. Left boundary of the wall includes a water channel where cold water flows and takes away the heat of the mould. Heat exchange between mould and water at the left boundary (x ¼ 0) can be represented as:

k

437

3.2. Inverse problem Inverse problem involves estimation of unknown spatially varying mould surface heat flux. In the present study, conjugate gradient method (CGM) with adjoint problem is used, as it is able to predict boundary heat flux accurately for such cases and also computationally efficient. Objective function for inverse problem to 00 estimate unknown heat flux, qs (L, y) is as follows: P h   n i X oi2 h 00 00 Y xp ; yp  T xp ; yp ; qs ðL; yÞ S qs ðL; yÞ ¼

where, P is total number of temperature sensors, p is index representing temperature sensor (p ¼ 1, 2, 3, …, P), Y is actual or experimentally measured temperature by sensors and T is temperature calculated with assumed heat flux, qs00 (L, y), at the same sensor location (xp, yp). It is to be noted that for the solution of inverse problem, water temperature throughout the height of the mould is considered to be constant and same as that of inlet water temperature as the effect of considering variable water temperature is found to be very less on mould temperature distribution. The basic steps involved in the estimation of boundary heat flux using the CGM with adjoint problem are described below. 3.2.1. Sensitivity problem To obtain the sensitivity problem, temperature T(x, y) and un00 known boundary heat flux qs (L, y) are perturbed by DT(x, y) and 00 Dqs (L, y), respectively. Substituting these in Eq. (1) and then subtracting Eq. (1) from the obtained equation results in following sensitivity problem:

v2 DTðx; yÞ v2 DTðx; yÞ þ ¼0 vx2 vy2

(12)

Solution of the sensitivity problem, DT(x, y) is required to obtain search step size (bk). For solving the above equation, necessary boundary conditions are obtained as follows:

vDTðx; 0Þ ¼0 vy

ð0  x  LÞ

vDTðx; H þ H1 Þ ¼0 vy

ð0  x  LÞ

(7)

k

vDTð0; yÞ ¼ hw DTð0; yÞ vx

(8)

k

00 vDTðL; yÞ ¼ Dqs ðL; yÞ vx

Eq. (2)e(4), Eq. (7) and Eq. (8) provides necessary boundary conditions for governing energy equation, Eq. (1). Further, inlet temperature of water at the channel entrance (Tw,in) provides necessary boundary condition for Eq. (6). Application of this

(11)

p¼1

k

ð0  y  H þ H1 Þ

ð0  y  HÞ

vDTðL; yÞ ¼ ha DTðL; yÞ vx

ðH < y  H þ H1 Þ

(13)

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3.2.2. Adjoint problem 00 In the present case, temperature T{xp, yp; qs (L, y)} appearing in Eq. (11) should satisfy a constraint of the solution of a differential equation (direct problem), Eq. (1). Hence, the least square estimator, Eq. (11), is modified by following the Lagrange multiplier approach. A term is added in Eq. (11) which is obtained by multiplying Lagrange multiplier, l(x, y), with the governing equation, Eq. (1), and integrating it over the spatial directions. The resultant modified functional form is given by:

Allowing the terms containing DT(x, y) on the right hand side of Eq. (16) to vanish, following adjoint problem is obtained for the Lagrange multiplier, l(x, y):

h 00 i S qs ðL; yÞ

Associated boundary conditions for the problem are obtained as follows:

ZL

HþH Z 1

¼

h

n  oi2    00 Yðx; yÞ  T x; y; qs ðL; yÞ d x  xp d y  yp dxdy

x¼0 y¼0 HþH Z 1

ZL þ

# v2 T v2 T lðx; yÞ 2 þ 2 dxdy vx vy

x¼0 y¼0

(14) Perturbing T(x,y) by DT(x,y) and qs by Dqs tracting Eq. (14) from the resulting equation results in 00 (L,y)

h

ZL

i

00

HþH Z 1

h

00 (L,y),

and sub-

n oi 00 Yðx; yÞ  T x; y; qs ðL; yÞ

     DTðx; yÞd x  xp d y  yp dxdy HþH Z 1

þ

vlðx; 0Þ ¼0 vy

vlð0; yÞ hw ¼ lð0; yÞ vx k vlðL; yÞ ¼0 vx

"

# v2 DTðx; yÞ v2 DTðx; yÞ lðx; yÞ þ dxdy vx2 vy2

x¼0 y¼0

(15) Performing integration by part for the last two terms of Eq. (15) and substituting them into Eq. (15) while utilizing boundary conditions of the sensitivity problem results in

(18)

ð0  y  HÞ

vlðL; yÞ ha ¼  lðL; yÞ vx k

x¼0 y¼0

ZL

(17)

vlðx; H þ H1 Þ ¼0 vy

"

DS qs ðL; yÞ ¼  2

P h   X v2 lðx; yÞ v2 lðx; yÞ Y x þ þ 2 ; y p p vx2 vy2 p¼1  oi  n   00  T xp ; yp ; qs ðL; yÞ d x  xp d y  yp ¼ 0

ðH < y  H þ H1 Þ

3.2.3. Gradient equation In the process of obtaining the adjoint problem mentioned above, Eq. (19) is left,

h

i

DS q’’s ðL; yÞ ¼

ZH

lðL; yÞ

Dq’’s ðL; yÞ k

dy

(19)

0

Function increment can be written as



i

00

DS qs ðL; yÞ ¼ ZL

HþH Z 1

2

h

n  oi    00 Yðx; yÞ  T x; y; qs ðL; yÞ DTðx; yÞd x  xp d y  yp dxdy

x¼0 y¼0 HþH Z 1



HþH Z 1

h lð0; yÞ w DTð0; yÞdy  k

0

0

HþH Z 1

þ

vlð0; yÞ DTð0; yÞdy þ vx

0 HþH Z 1

lðL; yÞ

00

lðL; yÞ

ha DTðL; yÞdy þ k

H



ZH

Dqs ðL; yÞ k

HþH Z 1

0

ZL

v2 lðx; yÞ vx2

0

vlðx; H þ H1 Þ DTðx; H þ H1 Þdx þ vy

0

ZL 0

HþH Z 1

ZL

þ 0

dy

0

 ZL

vlðL; yÞ DTðL; yÞdy vx

0

v2 lðx; yÞ vy2

DTðx; yÞdxdy

DTðx; yÞdxdy

vlðx; 0Þ DTðx; 0Þdx vy

(16)

Udayraj et al. / International Journal of Thermal Sciences 118 (2017) 435e447

h

i

00

h

00

00

i

h

00

i

Objective function, S[qs" (L, y)] calculation using Eq. (11)

DS qs ðL; yÞ ¼ S qs ðL; yÞ þ Dqs ðL; yÞ  S qs ðL; yÞ ZH ¼

h 00 i 00 VS qs ðL; yÞ Dqs ðL; yÞdy

(20)

Comparing Eq. (19) and Eq. (20), following equation for the gradient of functional S is obtained:

(21)

S [qs"(L, y)] <ε

Initial guessed value of qs" (L, y)

Obtain temperature T(x, y) by solving Eq. (1) - Eq. (10), using qs"(L, y)

00

00

qs ðL; yÞkþ1 ¼ qs ðL; yÞk  bk dk

(22)

where, k is index for number of iteration, b is search step size and d is direction of descent. Search step size is obtained using,

o  i   PP h n 00 k  Y xp ; yp DT xp ; yp ; dk p¼1 T xp ; yp ; qs ðL; yÞ k b ¼ o2 PP n  k p¼1 DT xp ; yp ; d (23) Direction of descent is calculated as below:

i h 00 d ¼ VS qs ðL; yÞk þ gk dk1 k

(24)

where, g is conjugate coefficient. Its value is zero for the first iteration and for other iterations it can be calculated using FletcherReeves expression as follows:

Z

gk ¼ Z

H

Compute gradient using Eq. (21)

y¼0

h 00 io2 VS qs ðL; yÞk1 dy

Compute conjugate coefficient γ using Eq. (25) Solve sensitivity problem to obtain ΔT (x, y) using Eq. (12) and Eq. (13)

(25)

4. Solution procedure It can be observed from Eq. (1) and Eq. (6) that both the governing equations are coupled. Hence, an iterative procedure is adopted to compute the temperature distribution. These equations are solved together with the necessary boundary conditions, Eq. (2) e Eq. (5) and Eq. (7) e Eq. (10). Convergence criterion of 1006 is used for both the temperatures while solving the direct problem. In the present work, finite difference method (FDM) is used to discretize both the governing equations. Second order centraldifference scheme is used to discretize Eq. (1) and backward difference is used to discretize Eq. (6). 45 number of nodes in x-direction and 90 number of nodes in y-direction are used in the present study. For inverse estimation of mould boundary heat flux, the CGM algorithm is used. The CGM algorithm involves solution of adjoint problem, gradient equation and sensitivity problem in an iterative manner. Major steps associated with the CGM algorithm are shown in Fig. 2. First of all, initial values of boundary heat flux 00 qs (L, y) is assumed. Direct problem involving Eq. (1) e Eq. (10) is then solved to obtain the temperature distribution throughout the mould. Depending upon the number and location of temperature

Compute direction of descent dk using Eq. (24)

Fig. 2. Flowchart showing steps associated with the CGM Algorithm.

sensors considered for calculation, the objective function, Eq. (11), is computed using the obtained temperature distribution and convergence of objective function is checked. Stopping criteria condition for the CGM is specified as

h 00 i S qs ðL; yÞ < ε

(26)

where, ε is a small predefined constant value. In case of noise-free temperature data, a relatively small value is assigned for ε. However, temperature data measured during experiments contain some measurement errors. In such cases, discrepancy principal is used to select value of stopping criteria, ε. Value of ε for cases involving measurement error is as follow:

ε ¼ P s2

n h 00 io2 VS qs ðL; yÞk dy

y¼0 H n

Calculate search step size β using Eq. (23)

qs" (L, y) is estimated

No

Calculate new/updated heat flux qs" (L, y) using Eq. (22)

3.2.4. Iterative procedure An initial value of boundary heat flux qs00 (L,y) is guessed. Then, in the next iteration this value is updated as follows [33]:

Yes

Calculate Lagrange multiplier λ (x, y) by solving the adjoint problem using Eq. (17) and Eq. (18)

0

h 00 i lðL; yÞ VS qs ðL; yÞ ¼ k

439

(27)

where, P is number of measurements and s is standard deviation of the temperature measurements. If stopping criterion is not satisfied, the adjoint problem is solved which includes Eq. (17) and Eq. (18) to obtain Lagrange multiplier, l(x, y). Subsequently, gradient, conjugate coefficient (g) and direction of descent (d) are computed using Eq. (21), Eq. (25) and Eq. (24), respectively. Next, sensitivity problem involving Eq. (12) and Eq. (13) is solved to determine DT(x, y). Then the search step size (b) is calculated using Eq. (23) and the previous boundary heat flux value is updated using Eq. (22). Again the temperature distribution is determined by solving Eq. (1) e Eq. (10) while using updated value of the boundary heat flux. Above procedure is repeated till the stopping criterion is satisfied. 5. Results and discussion The mathematical model developed in the present work for direct and inverse problems are validated first. These are then used to estimate mould boundary heat flux based on experimental temperature data obtained from the billet mould in continuous casting shop of Tata Steel, India. 5.1. Validation of direct problem code The developed computer code in MATLAB for direct problem is validated against the results available in the literature [1]. A billet mould of total height (H þ H1) 700 mm and width (L) 96 mm is considered, except when mentioned explicitly. Mould is made up of

440

Udayraj et al. / International Journal of Thermal Sciences 118 (2017) 435e447

copper and its thermal conductivity is 390 W/mK. Meniscus is considered to be 617.6 mm above the bottom (y ¼ H). Water enters in the water channel from the bottom (y ¼ 0) at a speed (Uw) of 7 m/ s and temperature (Tw,in) 30  C. Density (rw), specific heat (cp,w), dynamic viscosity (mw) and thermal conductivity (kw) of water are taken as 999.97 kg/m3, 4180 J/kgK, 79.77  1005 Ns/m2 and 0.6 W/ mK, respectively. Heat transfer coefficient, hw is considered to be 22500 W/m2K. For the validation of the direct problem, mould 00 boundary heat flux at the right boundary, qs (L, y) is taken from the work of Savage and Pritchard [2]:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #ffi u" u ðH  yÞ qs ðL; yÞ ¼ 2680  335t Ucasting 00



.  kW m2

(28)

where, Ucasting is casting speed which is taken as 0.0375 m/s (2.25 m/min), unless otherwise stated. Results obtained from the present direct problem code are shown in Fig. 3. Fig. 3 (a) shows comparison of temperature results obtained using the present code and results available in the literature [1] at the hot and cold face of the mould for different water inlet temperatures (Tw,in). Fig. 3 (b) shows the comparison of hot and cold face temperature variation obtained using the present study and previous study [1] in case of moulds of different thickness (L). Similarly, Fig. 3 (c) shows the corresponding results for different casting speeds (Ucasting). It can be observed from Fig. 3 that the present results agree quiet well with the results available in the literature [1] for wide range of different parameters. 5.2. Boundary heat flux estimation This section explains results related to boundary heat flux estimation using temperature data obtained from the solution of direct problem. A comprehensive study was carried out to analyze effect of number of sensors as well as location and arrangement of sensors throughout the mould. It is observed that as number of sensors increases (in the range 22e54), accuracy of boundary heat flux estimation increases significantly and estimated heat flux is in agreement with the actual boundary heat flux. 36 number of sensors are found to be sufficient in order to obtain boundary heat flux with sufficient accuracy and hence 36 number of sensors are used for further analysis. It is also observed that the location of sensors plays significant role in the present problem. So, the effect of locations or arrangements of sensors is also analyzed further. Three different sensor arrangements as shown in Fig. 4 with 36 number of sensors are studied. Estimated boundary heat flux results obtained using CGM for these three cases are shown in Fig. 5 along with the actual boundary heat flux. It is to be noted from Fig. 5 that the accuracy of boundary heat flux estimation deteriorates as sensors are moved away from the heated wall of the mould. 5.2.1. Effect of measurement error As experimental measurements always include some amount of uncertainty or measurement errors, it is important to analyze effect of measurement error on boundary heat flux estimation using inverse solution. For this, case with sensor arrangement SA1 as shown above in Fig. 4 (a) is considered as accuracy of boundary heat flux estimation is found better for this case as evident from Fig. 5. Temperature data obtained at various sensor locations by solving the direct problem are modified suitably as follows [13]:









Y xp ; yp ¼ Tdirect xp ; yp þ us

(29)

where, Tdirect(xp, yp) is the temperature at location (xp, yp)

Fig. 3. Comparison of the present results and available results [1] on hot and cold face temperature distribution for the cases of different (a) inlet water temperatures, (b) mould thickness, and (c) casting speeds.

determined from solution of the direct problem with the exact boundary heat flux and s is the standard deviation of the measurement error which is assumed same for all the measurements. Standard deviation (s) is related to the measured temperature and its value depends on accuracy of the measuring instrument

Udayraj et al. / International Journal of Thermal Sciences 118 (2017) 435e447

441

Fig. 4. Cases with 36 sensors with sensor arrangements (a) SA1, (b) SA2 and (c) SA3 studied to analyze effect of location of sensors on boundary heat flux estimation.

(thermocouple) and other factors. u is a random variable generated using normal distribution within [-2.576, 2.576] range for the 99% confidence level. In the present study, effect of different values of s (0.1, 0.5 and 1.0) on boundary heat flux estimation is analyzed. Here, s ¼ 0.1 corresponds to a maximum error of 0.183 K in the temperature data. Similarly, s ¼ 0.5 and s ¼ 1.0 corresponds to maximum errors of 1.168 K and 2.012 K, respectively. Results corresponding to different measurement error (different s values) are shown in Fig. 6. It is evident that as measurement error increases, accuracy of boundary heat flux estimation decreases.

Fig. 5. Estimated boundary heat flux with 36 number of sensors in case of different sensor arrangements (a) SA1, (b) SA2 and (c) SA3.

5.2.2. Heat flux estimation with few temperature sensors It is clear from above analysis that boundary heat flux can be estimated accurately if large number of sensors is placed inside the mould. However, in practical situations in a steel plant, it is often not possible to place sufficient number of thermocouples or any other temperature measuring devices in the mould. So, it is

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Udayraj et al. / International Journal of Thermal Sciences 118 (2017) 435e447

Fig. 6. Estimated boundary heat flux for three different cases all involving 36 number of sensors at different locations.

imperative to devise any new method which can be used to estimate boundary heat flux with fewer numbers of sensors or less information. Hence, a new methodology is proposed here which can be used to estimate boundary heat flux accurately with the help of only 5 sensors as shown in Fig. 7. Two configurations with limited number of temperature sensors and different sensor locations are considered here: 1) Configuration 1 with 5 sensors placed inside the mould as shown in Fig. 7 (a) and 2) Configuration 2 with 5 sensors placed inside the mould at different sensor locations as shown in Fig. 7 (b). It can be observed from Fig. 7 that the sensor locations in these two configurations are chosen in such a way that in configuration 2, sensors are placed approximately in between the sensor locations as compared to configuration 1. The procedure for estimating mould boundary heat flux using fewer numbers of sensors is explained next. Initially, zero heat flux is assumed at all the boundary nodes and CGM is applied by following the procedure explained in section 4 to estimate boundary heat flux for configuration 1 with boundary heat flux profile 1 using only 5 sensors shown in Fig. 7 (a). This is

Fig. 8. Estimated boundary heat flux at various steps for configuration 1 with 5 number of sensors in case of heat flux profile 1.

the first step (step 1) and the estimated boundary heat flux after this first step is as shown in Fig. 8. Obtained boundary heat flux over all boundary nodes is then averaged and calculated average value after step 1 (as shown in Fig. 9) is used further as initial condition for step 2. Similar procedure is then followed and the boundary heat flux profile obtained after step 2 is also shown in Fig. 8. This procedure is repeated many times till the changes in the consecutive average values of the estimated boundary heat flux become less than 5%. It is found that the changes in the average values of the estimated heat flux became small after step 6 as shown in Fig. 9. So, the results corresponding to step 7 only are considered for further analysis. It can be observed from Fig. 8 that accuracy of the boundary heat flux estimation improved significantly after each successive step. Obtained boundary heat flux profiles and averages corresponding to all the steps are shown in Figs. 8 and 9, respectively for configuration 1 with boundary heat flux profile 1. It is to be noted from Fig. 8 that the accuracy of the results showing estimated boundary heat flux corresponding to the final

L = 9.6 mm

L = 9.6 mm

82.4 mm Meniscus

82.4 mm Meniscus y = 605.1 mm y = 557.6 mm y = 498.3 mm

H = 700 mm y = 355.9 mm

y = 415.3 mm

y = 272.9 mm

y = 201.7 mm

y = 130.5 mm

y = 47.5 mm

y x

H = 700 mm

y = 11.9 mm x = 5 mm

(a)

y x

x = 5 mm

(b)

Fig. 7. Configurations considered for boundary heat flux estimation using less number of sensors (a) configuration 1 with 5 sensors and (b) configuration 2 also with 5 sensors but at different locations.

Average of estimated heat flux (MW/m2)

Udayraj et al. / International Journal of Thermal Sciences 118 (2017) 435e447

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1

2

3

4 Steps

5

6

7

Fig. 9. Average values of the estimated boundary heat flux at various steps for configuration 1 with 5 number of sensors in case of heat flux profile P1.

step (step 7) is still not good. In order to get a better boundary heat flux estimation, estimated boundary heat flux corresponding to 5 sensor locations after final step (step 7) are selected. These individual values of estimated heat flux at various locations are then arithmetically averaged with the obtained overall average value of estimated boundary heat flux after step 7 and these arithmetically averaged values are considered as final results representing the estimated boundary heat flux. This can be expressed as: Predicted heat flux at the ith discrete location ¼ (heat flux obtained after 7th iteration at the corresponding ith discrete location þ average heat flux for the entire length after 7th iteration)/2 The result showing finally obtained boundary heat flux is shown in Fig. 10. Figs. 8 and 9 explain the procedure to estimate boundary heat flux estimation with one particular boundary heat flux profile (profile 1) for configuration 1 of sensor arrangement. In order to check the robustness and reliability of the proposed methodology to determine boundary heat flux using only few sensors another configuration with less number of sensors (configuration 2 with sensor locations shown in Fig. 7 (b)) is also considered. Moreover, these two configurations are tested for two different boundary heat flux profiles. Similar procedure is followed for configuration 1 with boundary heat flux profile 2 and for the configuration 2 with both boundary heat flux profiles. The final results showing estimated heat flux for both the configurations with different sensor locations

443

(configuration 1 and configuration 2) and for two different heat flux profiles (profile 1 and profile 2) are shown in Fig. 10. Figure shows that the obtained result agrees well with the actual boundary heat flux and is indicative of the existing state of heat transfer in the mould. This proves that the methodology proposed in the present study can be used to estimate boundary heat flux for the problem in hand. Relative error [13] is also calculated in order to demonstrate accuracy of the estimated boundary heat flux as compared to the actual heat flux quantitatively. For configuration 1, relative error (RE) are found to be 1.94% and 2.06% for boundary heat flux profile 1 and profile 2, respectively. Whereas, for configuration 2, RE are found to be 3.39% and 3.02% for profile 1 and profile 2, respectively. It can be observed from above that the RE are higher in case of sensor configuration 2 as compared to configuration 1 for both the boundary heat flux profiles. It is due to the fact that in case of configuration 1, sensor are located closer to the meniscus (as compared to the case of sensor configuration 2) where heat flux gradient is higher. This shows that the accuracy of boundary heat flux estimation can be increased if sensors are placed closer to the meniscus. To further check the ability of the proposed methodology to estimate the boundary heat flux accurately in case of moulds of larger thickness and for the cases when temperature sensors are mounted far away from the hot side of the mould, a separate mould of higher thickness is selected as shown in Fig. 11. Instead of 9.6 mm mould width discussed above and shown in Fig. 7, a mould of 15 mm width is considered as shown in Fig. 11 and boundary heat flux are determined based on the temperature data obtained at only 5 sensor locations. It can be observed from Fig. 11 that the sensors arrangement is same as that of configuration 1 shown earlier in Fig. 7 (a) along mould height. However, these sensors are placed further away from the hot side of the mould (10 mm away from the hot side as compared to 4.6 mm in the previous cases). Moreover, two different profiles of boundary heat flux are considered as shown in Fig. 12 where profile 1 is same as that of previous cases and profile 3 is steeper as compared to previous boundary heat flux profiles considered. These conditions make it a more difficult problem for boundary heat flux estimation and allow checking the robustness of the proposed methodology. It can be observed from Fig. 12 that the estimated boundary heat flux agrees with the actual heat flux reasonably well. RE are found to be 1.98% and 3.89% for boundary heat flux profile 1 and profile 3, respectively. The fact that the proposed methodology is able to estimate the boundary heat flux accurately for the two heat flux profiles in

2800000

L = 15 mm

2600000 82.4 mm Meniscus

Heat Flux (W/m2)

2400000

y = 605.1 mm

2200000 2000000

y = 498.3 mm H = 700 mm

1800000

Actual (Profile 1) Estimated - Configuration 1 (Profile 1) Estimated - Configuration 2 (Profile 1) Actual (Profile 2) Estimated - Configuration 1 (Profile 2) Estimated - Configuration 2 (Profile 2)

1600000 1400000 1200000 1000000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y (m) Fig. 10. Estimated boundary heat flux for configuration 1 and configuration 2 with different only five sensors arranged at different locations for both the boundary heat flux profiles (profile 1 and profile 2).

y = 355.9 mm y = 201.7 mm y = 47.5 mm y x

x = 5.0 mm

Fig. 11. Mould with 15 mm width and 5 sensors placed away from the hot surface considered for boundary heat flux estimation.

444

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4.0

Estimated - Profile 1

3.5 Heat Flux (MW/m2)

that the temperature obtained using the estimated boundary heat flux is in good agreement with the actual temperature data at various locations in the mould. It shows that the estimated boundary heat flux using the proposed methodology can be used to determine temperature distribution in the mould.

Actual - Profile 1 Actual - Profile 3

3.0

Estimated - Profile 3 5.3. Boundary heat flux estimation using experimental temperature data

2.5 2.0

The methodology proposed and discussed in the preceding section to estimate mould boundary heat flux using only few sensors is used to estimate boundary heat flux at the mould surface exposed to molten steel using actual experimental data.

1.5 1.0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

y (m) Fig. 12. Comparison of the actual and estimated boundary heat flux for the mould of 15 mm width with only 5 sensors for two different boundary heat flux profiles.

this case also allows us to conclude that it can be used confidently to estimate the boundary heat flux using only few experimental temperature data for different operating conditions (different type of moulds and various heat flux profiles). One of the reasons for estimating boundary heat flux for the mould used in continuous casting is that it enables us to determine temperature distribution throughout the mould. The temperature distribution can then be used to determine thermal stresses and mould distortions, knowledge of which is very essential in order to maintain proper quality of steel products. So, it is imperative to check whether the estimated heat flux is able to provide proper temperature distribution throughout the mould or not. For this, both the actual and estimated boundary heat flux for configuration 1 with boundary heat flux profile 1 obtained in Fig. 12 are selected and temperature distribution is obtained throughout the mould of 15 mm width (shown in Fig. 11). Temperature variation along the mould height obtained using both the actual and estimated boundary heat flux at x ¼ 5 mm and x ¼ 10 mm are shown in Fig. 13. It is to be noted here that the discrete heat flux data obtained using inverse technique at five sensor locations for the estimated boundary heat flux shown in Fig. 12 is used and a quadratic equation for the estimated boundary heat flux is determined. This quadratic equation representing the estimated boundary heat flux is then used to obtain temperature distribution throughout the mould by solving the direct problem. It can be observed from Fig. 13

200 180 160 140 120 100 80 60 40 20

L = 15 mm 140 mm Meniscus

Sensor 1, y = 810 mm Sensor 2, y = 790 mm Sensor 3, y = 750 mm

x = 10 mm

Temperature (oC)

5.3.1. Experimental data from plant: temperature measurement Experimental data used for estimating boundary heat flux is measured in an actual continuous casting mould at Tata Steel Limited, Jamshedpur, India. Temperature is measured by employing temperature sensors installed inside a copper mould of thermal conductivity 377 W/mK. Height of the mould is 1 m and other details related to the mould are shown in Fig. 14. Five temperature sensors are placed along the height of the mould at a distance of 3 mm from the cold side in the mould of thickness 15 mm as shown in Fig. 14. It is found earlier and discussed in section 5.2.2 that temperature sensors should be placed closer to the meniscus in order to estimate boundary heat flux accurately. So, more number of sensors are placed near the meniscus during experimental measurements. Temperature variations across the mould surface along with all relevant operating conditions are recorded for few days. The experimentally measured temperature data over few days is divided into three cases (case A, case B and case C) of small intervals (approximately 40 min to 1 h) as shown in Table 1. Measured temperature variation with time for all the five sensors for case C is shown in Fig. 15. It can be observed from Fig. 15 that the temperature at all the sensor locations fluctuates about a mean value and the mean temperature at each individual sensor is more or less constant with time. The magnitude of fluctuations in the measured temperature with time are higher for the sensors placed near the meniscus as compared to the sensors placed relatively far from the meniscus. These large fluctuations near the meniscus are due to molten metal level variation and changes in the casting speed [30].

Sensor 4, y = 500 mm x = 5 mm H = 1000 mm

Using Actual Heat Flux Using Estimated Heat Flux 0

0.1

0.2

0.3 0.4 y (m)

Sensor 5, y = 150 mm 0.5

0.6

0.7

y

x

x = 3.0 mm Fig. 13. Temperature profile obtained using estimated boundary heat flux along the height of the mould at x ¼ 5 mm and x ¼ 10 mm as compared to actual temperature at locations corresponding to five sensors for configuration 1 with heat flux profile 1.

Fig. 14. Industrial mould with 15 mm mould width considered for boundary heat flux estimation and location of 5 temperature sensors installed in the mould.

Udayraj et al. / International Journal of Thermal Sciences 118 (2017) 435e447

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Table 1 Three cases and experimentally measured temperature data along with other relevant parameters for the cases. Cases

Case A Case B Case C

Casting Speed, m/min

2.40 2.89 2.45

mw, lpm

2436 2430 2434

Tw,in ( C)

38.1 38.2 38.6

Measured Temperature ( C) Sensor 1

Sensor 2

Sensor 3

Sensor 4

Sensor 5

116 118 117

112 114 113

103 105 104

89 98 95

78 86 80

192 mm

182.8 mm Water channel

184 mm

190.8 mm Fig. 16. Cross-section of the water channel used in case of industrial mould.

It is to be noted from Fig. 14 that the first two sensors (Sensor 1 and sensor 2) are placed close to each other and hence temperature of these two sensors are close to each other which is evident from Fig. 15 and also from Table 1. Temperature sensor 1 which is placed towards the top of the mould near meniscus shows highest temperature value, and temperature sensor 5 which is placed towards the bottom of the mould shows lowest temperature value among all the five sensors. This is because the heat flux or heat transfer to the mould is highest at the top and it decreases towards the bottom of the mould. In order to predict the mould boundary heat flux, temperature values required at the sensors are obtained by individually averaging the temperatures over the time intervals at all the five temperature sensors for the three cases. The average temperatures at all the five sensors for the three cases are shown in Table 1. These temperatures are used to estimate the boundary heat flux along the mould height at the hot side of the mould. 5.3.2. Estimated boundary heat flux for industrial mould using measured plant data The procedure mentioned in section 4 and the proposed methodology explained in section 5.2.2 are used along with the plant measurements discussed in section 5.3.1 to estimate the boundary heat flux for continuous casting mould. There exists a convective boundary condition at the cold side of the mould as water is circulated in the upward direction through a channel. The cross-section of the water channel used in the plant is shown in Fig. 16. To deal with the cooling of the mould by the water, convection boundary condition is assigned at the left surface as explained in Eq. (4) in section 3.1. For this, inlet water temperature (Tw,in) and convective heat transfer coefficient (hw) are required. The water temperatures (Tw,in) are also measured during the experimentation as shown in Table 1 for the three cases. However, convective heat transfer coefficient (hw) is calculated using Eq. (5)

DH ¼

4Ac P

(30)

where, Ac is cross-sectional area (m2) and P is wetted perimeter (m) of the water channel shown in Fig. 16. Estimated boundary heat flux for all the three cases are shown in Fig. 17. It can be seen from Fig. 17 that the obtained boundary heat flux profiles for case A and case C are close to each other whereas it slightly different for case B as compared to the other two cases. One of the reason for this higher heat flux for case B as compared to other two cases is that the casting speed is higher for the case B (2.89 m/min) as compared to case A (2.40 m/min) and case C (2.45 m/min). Moreover, it is found that the estimated heat flux near the meniscus are almost same for all the three cases which is due to the fact that the temperature of the sensors placed near the

3.5 3.0 Heat Flux (MW/m2)

Fig. 15. Measured temperature data showing variation of temperature with time for case C.

based on the operating conditions like water flow velocity (Uw), hydraulic diameter of the channel (DH) and water properties. Water flow velocity (Uw) is calculated using the mass flow of water (mw) which is measured during the experiments and shown in Table 1. Hydraulic diameter of the water channel (DH) is calculated using Eq. (30):

Estimated - CaseA

Correlation - Case A

Estimated - Case B

Correlation - Case B

Estimated - Case C

Correlation - Case C

2.5

2.0 1.5 1.0 0.5 0

0.1

0.2

0.3

0.4 0.5 y (m)

0.6

0.7

0.8

0.9

Fig. 17. Estimated boundary heat flux for all the three cases of industrial mould.

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meniscus are very close to each other for all the three cases as shown in Table 1. Further, the estimated boundary heat flux is also compared with the heat flux profiles obtained using existing correlation for boundary heat flux. For this, the well-known correlation proposed by Savage and Pritchard [2], Eq. (28), is used. Satisfactory agreement between the estimated boundary heat flux and the heat flux obtained using the correlation is found for the three cases as shown in Fig. 17. Slight mismatch in the boundary heat flux obtained using the proposed methodology and the correlation could be due to inherent limitations of the correlation. It is evident from the correlation that the boundary heat flux in the correlation is function of only casting speed whereas in actual industrial practice, boundary heat flux depends on various other operating conditions (Steel composition, mould oscillation, mould thickness, mould material, mould taper, mould powder properties, steel superheat, mass flow rate of cooling water, electromagnetic stirring etc.). Analysis of effect of various operating parameters on boundary heat flux and establishing correlation of boundary heat flux with operating conditions will be carried out in future.

6. Conclusions In the present study, a methodology is proposed to estimate surface heat flux of a continuous casting mould using limited temperature measurement data. The mathematical model constitutes of a direct problem and solution of inverse heat transfer problem. Direct problem involves two-dimensional heat conduction through mould which is solved using the finite difference method. Conjugate gradient method is used for the inverse problem of heat flux estimation. Series of numerical test cases are undertaken to ascertain the accuracy of the present model and the results are critically analyzed. Effect of measurement errors on boundary heat flux estimation is also examined. As it is difficult to install large number of temperature measurement sensors in the mould in real casting process, a new methodology is proposed and discussed in the present work to estimate boundary heat flux using limited number of sensors. A case where only five temperature sensors are placed is selected first to explain the methodology and accuracy of the predicted results is carefully assessed. Good agreement between the estimated and actual boundary heat flux is obtained. In order to check robustness and reliability of the proposed methodology, boundary heat flux is also estimated for the cases with different sensor locations and with different boundary heat flux profiles. Further, a different mould with higher mould thickness is considered where fewer numbers of sensors are placed far away from the hot surface of the mould for which heat flux is to be estimated. It is found that the boundary heat flux can be estimated accurately with the limited number of sensors for all the cases. Relative error in boundary heat flux estimation is found to be less than 4%. Boundary heat fluxes are then estimated for an actual industrial mould using in-process temperature measurement data. Experiments are conducted in the mould over a period of time during casting operation and temperature data obtained at five sensors are divided into three cases. Boundary heat flux is estimated for all the cases using online measurement data. It is observed that the proposed model can accurately predict the heat flux variation across the mould surface for all the cases, thereby establishing the applicability of the model to the industrial process.

Acknowledgement The authors would like to acknowledge the financial support of R&D Division, Tata Steel Ltd., Jamshedpur, India.

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