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Online simulation model of the slab-reheating process in a pusher-type furnace
Applied Thermal Engineering 27 (2007) 1105–1114 www.elsevier.com/locate/apthermeng
Online simulation model of the slab-reheating process in a pusher-type furnace Anton Jaklicˇ b
a,*
, Franci Vode a, Tomazˇ Kolenko
b
a Institute of Metals and Technology, Lepi pot 11, 1000 Ljubljana, Slovenia University of Ljubljana, Faculty of Natural Sciences and Technology, Asˇkercˇeva 12, 1000 Ljubljana, Slovenia
Received 13 September 2005; accepted 10 July 2006 Available online 18 October 2006
Abstract This paper presents an online simulation model of the slab-reheating process in a pusher-type furnace in Acroni d.o.o. in Slovenia. The simulation model is connected to the information system of a hot-processing plant that provides online measuring and charging data of the furnace. The simulation model considers the exact geometry of the furnace enclosure, including the geometry of the slabs inside the furnace. A view-factor matrix of the furnace enclosure was determined using the Monte Carlo method. The heat exchange between the furnace gas, the furnace wall and the slab’s surface is calculated using a three-temperature model. The heat conduction in the slabs is calculated using the 3D finite-difference method. The model was validated using measurements from trailing thermocouples positioned in the test slabs during the reheating process in the furnace. A graphical user interface (GUI) was developed to ensure a user-friendly presentation of the simulation-model results. 2006 Elsevier Ltd. All rights reserved. Keywords: On-line simulation; Reheating furnace; Slab heating; Mathematical model
1. Introduction A computer-controlled hot-rolling process for steel slabs requires high-quality reheated slabs in terms of time, temperature, thermal profile and furnace atmosphere. Each steel slab has to be reheated at a suitable temperature for hot-rolling with the prescribed temperature difference inside the slab. Temperature differences during the whole reheating process should not cause the maximum-allowed thermal stress inside the slab to be exceeded. The slabs are reheated in the gas-fired pusher-type furnace (Fig. 1). The furnace has six control zones: the upper and lower preheating zones, the upper and lower heating zones, and the left and right soaking zones. The material flow through the furnace is discontinuous, with the movement happening in push steps. At each push step the pushing machine pushes all the slabs until the slab at the exit *
Corresponding author. Tel.: +386 31 809 357; fax: +386 1 4701 939. E-mail address: [email protected] (A. Jaklicˇ).
1359-4311/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2006.07.033
drops out from the furnace. The length of the pushing step depends on the width of the discharged slab. The number of slabs inside the furnace can vary and depends on the width of the individual slabs. There are many influences that can affect the reheating process. The time intervals between pushing steps can vary, and delay conditions in the material flow are often present. There are two kinds of delay: scheduled delays, e.g., mill roll changes, meal breaks, etc., or unscheduled delays, e.g., mill breakdowns, etc. Another influence on the reheating process comes from slabs of different dimensions and the different steel grades, which have to be reheated to different discharge temperatures. All these influences significantly affect the reheating process and mean that almost every slab has a different reheating history. Knowledge of the temperature field of slabs during the reheating process is very important for successful furnace control. The temperature fields of the slabs are non-measurable values during the production process. Computer simulation is sometimes the most reasonable way to
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Nomenclature a A c d F F I I K L M N q Q Q S T
heat-transfer coefficient (W m2 K1) surface area (m2) specific heat (J/kg K) furnace-wall/floor-layer thickness (m) view-factor view-factor matrix number of surfaces in a furnace enclosure including slab surfaces identity matrix number of the furnace-wall/floor-layers length (m) number of the furnace-floor surfaces number heat flux (W m2) element of heat-flux vector heat-flux vector width of the water pipe temperature (K)
Greek letters a absorptivity factor e emissivity factor / water flow (m3/s) k heat conduction (W m1 K1) p number PI q density (kg/m3) r Stefan–Boltzmann constant, 108 W m2 K4
slabs
upper EXIT
r = 5.671 ·
pusher
burners
ENTRY
lower
soaking zone
heating zone
Subscripts abs absorption air air cond conduction conv convection floor furnace floor g furnace gas H2O water i index of surface in furnace enclosure in input j index of surface in furnace enclosure k index of layers of furnace wall/floor m index of furnace-floor surface n index of furnace-floor surface out output p pipe rad radiation s slab total total w furnace wall !, ) outgoing , ( incoming () balance
preheating zone
Fig. 1. Pusher-type furnace.
determine such values. The state of the art is one- and twodimensional online calculations of the stock temperature [1–5]. However, the computational power of today’s personal computers means that increasingly complex simulations can be performed in real-time. Thus, complex simulation models can be implemented in the monitoring and control systems of industrial processes, and the first attempts have been made to calculate the stock temperature in three dimensions online [6–8]. An online supervision system based on a three-dimensional simulation model of a steel-slab reheating process
in a pusher-type furnace has been developed. The calculations in the model are based on a mathematical model that includes the main physical phenomena appearing during the reheating process in a natural-gas-fired pusher-type furnace: thermal radiation is the main heat-transfer mechanism, and the geometry of the furnace enclosure has an important role in the heat transfer of the thermal radiation. The furnace enclosure consists of the furnace geometry together with the geometry of the charged slabs inside the furnace. One of the chief mathematical complexities in treating the radiative heat transfer between surfaces is accounting for the geometrical relations involved in how the surfaces view each other. For the whole furnace enclosure they are expressed with a view-factor matrix form. In order to determine the matrix, a separate simulation model based on the Monte Carlo method was developed [9]. This model allows a view-factor determination for a general furnace enclosure consisting of rectangular surfaces, including multiple reflections. In the presented approach the view-factor matrix for a particular furnace enclosure, including the slabs and skid pipes, is calculated only once (the typical calculation time on a Pentium4 PC is three days). The viewfactor matrix is read into the online simulation model of
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the slab-reheating process at its start up. In the models described in [1–5,7,8], simplified approaches for treating the geometries of the furnace enclosures are used. The simplified approaches are acceptable for one- and two-dimensional simulation models. The presented online simulation model is capable of treating the exact geometry of the pusher-type furnace enclosure in thermal radiation calculations.
ing for the temperature profile of the zone from the table. The slab-surface temperature, Ts, is calculated step-bystep, using the finite-difference method. The only unknown temperature in the system is the furnace-gas temperature, Tg.
2. Heat exchange in the furnace
The model takes into account variations in gas temperature along the furnace length and does not account the variations across the width of the furnace. In the gas-fired furnace it is almost impossible to achieve homogenous temperature field of gas in the furnace (burners, flames, gas streams, etc.). Gas temperature field in the furnace is strongly dependent on the loading of the furnace and it varies during the production process. Therefore multi point measurement system of gas temperature field in the furnace with hundreds of sensors should be applied in the furnace enclosure in order to determine gas temperature field in real-time to account for the gas temperature variations across the width of the furnace. The literature describes off-line simulation models which accounts for the flames and streaming of furnace gasses which is quite complex. The goal of this work is to develop real-time simulation model, which is always the compromise between the quality of the simulation results and the calculation speed. In the model the furnace gas temperature is calculated for the individual furnace length segment (slab width) on the base of thermal flux equilibrium on the inner side of furnace walls. The furnace gas temperature, Tg, represents an average value for the furnace length segment. It can be determined by solving an equilibrium equation on the inner furnace wall:
To evaluate the heat exchange between the charge, the furnace gas and the furnace wall, basic algorithms from Heiligenstaedt [10] are used. The model is based on three temperatures: the furnace-gas temperature, Tg, the furnace-wall temperature, Tw, and the slab-surface temperature, Ts (Fig. 2(a)). All three temperatures have to be known for an evaluation of the temperature-dependent heat fluxes in the system. The modification of the model assumes that the furnace-wall temperature can be measured more reliably than the furnace-gas temperature. Therefore, the furnace-wall temperature was taken as the model’s primary value [11]; in contrast to the original model, where this was the furnace-gas temperature. The model was also improved to take into account the geometry of the furnace enclosure, including the slabs. The model assumes that the furnace-gas temperature is homogenous, and so to comply with this assumption the length of the furnace is divided into smaller segments, which are equal to the slab widths. The heat exchange in the furnace segment is shown in Fig. 2(a). The total heat transfer to the slab surface can be estimated by evaluating the partial mechanisms. The furnace gas has the highest temperature, Tg, in the system and emits heat to the furnace wall (qrad gw, qconv gw) and to the slab surface (qrad gs, qconv gs) by radiation and convection. A fraction of the heat to the furnace wall is lost to the outside, through the furnace wall (qcond w), and another fraction is emitted to the slab surface (qrad ws), where it is partly absorbed by the furnace gas (qabs g). The inner furnace-wall average temperature, Tw, is known; it is calculated for a particular control zone on the basis of measurements from the control thermocouple by account-
0 ¼ qrad gw ðT g ; T w ÞAw þ qconv gw ðT g ; T w ÞAw qcond w ðT w ÞAw qrad ws ðT w ; T s ÞAs þ qabs g ðT w ; T s ÞAs
ð1Þ
In Eq. (1), Aw is the area of the wall surface of the furnace segment, As is the area of the slab surface that is surrounded by the furnace-wall segment. The above equation can be solved numerically using the bisection method. The
furnace wall
a qcond w
2.1. Determination of the furnace gas temperature for the length segment
b Tw
qrad gw qconv gw qabs g
Tair
qrad ws
furnace wall
Tg
furnace gas
qcond w
qrad gs qconv gs
Tg
qrad gw qconv gw
Tair
Ts slab
Ts Tw
Tw
qH O 2
qabs g
qrad ws
furnace gas
qrad gs qconv gs Ts slab
Tg
Tair =Tkorndorfer=100 ˚C
qcond floor, m
Fig. 2. (a) The heat exchange in the preheating and in the heating zones, (b) the heat exchange in the soaking zone of the furnace.
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qrad gw(Tg, Tw) is calculated with Eq. (2); it describes the difference between the emitted heat flux from the gas mixture and the absorbed heat flux emitted from the furnace wall. The emissivity of furnace walls is high (0.95) therefore they are treated as black according to the model of the net radiant-heat interchange between a gas containing carbondioxide and water-vapor and its black bounding surface [12]. The values for eg and ag are determined using the model for a grey-gas CO2 and H2O mixture of Hottel found in [13] qrad gw ðT g ; T w Þ ¼ eg ðT g ÞrT 4g ag ðT g ; T w ÞrT 4w
ð2Þ
The qconv gw(Tg, Tw) in (1) can be determined using Eq. (3). The convection heat transfer in a high-temperature reheating furnace has a minor influence on the reheating process, especially in the high-temperature part of the furnace. Different authors recommend different values for agw between 10 and 15 W/m2 K. In our model we used agw = 12 W/m2 K qconv gw ðT g ; T w Þ ¼ agw ðT g T w Þ
ð3Þ
For the evaluation of the furnace-gas temperature, Tg, of the furnace segment (1) the average heat flux between the furnace wall and the slab surface qrad ws(Tw, Ts) is determined by (4): qrad ws ðT w ; T s Þ ¼
ew ðT w ÞrT 4w es ðT s ÞrT 4s As 1 1 þ 1 Aw ew es
ð4Þ
The part of the heat exchange between the furnace wall and the slab surface that is absorbed in the furnace gas qabs g is determined by Eq. (5). According to the Heiligenstaedt’s model [10] the portion of radiant flux from the furnace wall to the charge absorbed in the furnace gas is equal to the radiant flux from the furnace gas to the charge calculated as if the furnace gas had the temperature of the furnace wall. The values for eg and ag are determined using the model for a grey-gas CO2 and H2O mixture of Hottel [13] qabs g ðT w ; T s Þ ¼ eg ðT w ÞrT 4w ag ðT w ; T s ÞrT 4s
ð5Þ
The heat conduction through a K-layer furnace wall qcond w is calculated using Eq. (6). The thermal conductivities kk of individual furnace-wall layers are temperature dependent. Therefore, Eq. (6) is solved iteratively qcond w ¼ P
1
K dk k¼1 kk ðT k Þ
þ a1air
ðT w T air Þ
ð6Þ
z
a
i
y j
z
x
i
b
y
j x Fig. 3. (a) The furnace enclosure for one-row charging, (b) the furnace enclosure for two-row charging.
the 3D temperature field of the slabs is modelled. Therefore, the exact geometry of the furnace has to be treated in the model, including the charged slabs and skid pipes. The furnace geometry, including charged slabs, is divided into I smaller surfaces under the assumption of isothermal surfaces (Fig. 3(a) and (b)). For the surface j in Fig. 3(a) and (b) it is possible to write heat balance Qj () as the difference between the incoming Qj( and the outgoing radiation Qj) [14]: qj Aj ¼ Qj () ¼ Qj( Qj) ¼
I X
ei ðT i ÞrT 4i Ai F i!j ej ðT j ÞrT 4j Aj
ð7Þ
i¼1
The incoming thermal radiation Qj( consists of I thermal radiation contributions of the individual surface elements i. Since some of the surfaces in the enclosure cannot view each other and some can be viewed just partly, the furnace enclosure presents a complex geometry for the view-factor calculation. Fi!j are the view-factors that describe the fraction of the total thermal radiation emitted by surface Ai that is absorbed by the surface Aj, including multiple reflections [14]. For the whole furnace geometry consisting of I surfaces the heat exchange is written in a vector-matrix form: Q () ½I1 ¼ Q(½I1 Q)½I1 ¼ F ½II Q)½I1 Q)½I1
2.2. Thermal radiation heat exchange between the surfaces in the furnace enclosure In the algorithm for determining the average furnace-gas temperature Eq. (4) was used to calculate the thermal radiation from furnace walls with an average temperature Tw to the slab surface qrad ws(Tw, Ts). Eq. (4) gives the heat flux on the centre of the slab surface, which is not sufficient when
¼ ðF ½II I ½II Þ Q)½I1
ð8Þ
The heat-balance vector Q () [I·1] for the whole furnace geometry can be calculated when the view-factor matrix F and the outgoing thermal flux vector Q)[I·1] are known. The calculation of the thermal radiation heat transfer between two surfaces can thus be divided into the energy and the geometry parts [14]. The energy part describes
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the vector of the outgoing thermal fluxes Q)[I·1] with elements ej ðT j ÞrT 4j Aj . The main part of the calculation of the thermal radiation heat exchange between the surfaces in the furnace enclosure is the determination of a view-factor matrix F. Since energy travels in discrete photon bundles along a straight path before interacting with a surface [14], problems in thermal radiation are particularly well suited to the Monte Carlo method. In the case of determining view-factors, a large number, Ni, of photons are emitted from the surface, i.e., i. These photons are emitted from the surface according to probability density functions. The path of each photon, including possible reflections, is traced to its absorption at one of the surfaces of the furnace enclosure. In the model the emission and the reflection of photons are treated as grey and diffuse. Thus, determining view-factors in the furnace enclosure with the Monte Carlo method involves emitting and tracing the history of a statistically meaningful random sample of photons from their points of emission to their points of absorption, including multiple reflections. After the emission of a large number of photons, Ni, from a surface i, the view-factor Fi!j can be directly determined by counting the number of photons Ni!j that have been absorbed at the surface j [14]: N i!j N i!j F i!j ¼ lim ð9Þ N i !1 Ni N i N i 1 In order to determine the matrix, a separate simulation model based on the Monte Carlo method was developed [9]. This model allows a view-factor determination for a general furnace enclosure consisting of rectangular surfaces, including multiple reflections. The matrix F is calculated prior to the simulation for the particular furnace geometry, and the results are stored in data files. One of possible ways to account the absorption in gas at surface to surface radiation heat transfer is during the Monte Carlo determination of view-factors. Then view-factors of the enclosure include also the absorption in the gas. The problem of this method is that the absorption in gas is dependant on the gas temperature, which varies during the furnace operation. Therefore view-factors which account for the absorption are valid just for the case of assumed gas temperatures used during Monte Carlo determination. In the real-time model the absorption of the gas is treated for individual length segment of the furnace in a global fashion (5) according to the Heiligenstaed’s model [10] by using absorptivities calculated by Hottel [12]. The advantage of this method is that the model is relatively simple and allows therefore accounting for the temperature timedependant variations of furnace gas in real-time. Drawback of this method is that the absorption is not treated accurately by path length between individual surfaces. 2.3. The total heat flux on the slab surfaces The total heat flux qtotal,j to the surface elements j (Fig. 2(a) and (b)) can be determined in the same way for
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the preheating, the heating and the soaking zones using Eq. (10). A separate determination is needed only for the parts of the bottom slab surfaces that lie on the skid pipes, and for the bottom slab surfaces in the soaking zone, where the slabs lie on the furnace floor qtotal;j ¼ qrad gs ðT g ; T s Þ þ qconv gs ðT g ; T s Þ þ qrad ws;j ðT 1 . . . T I Þ qabs g ðT g Þ
ð10Þ
The calculation of qrad gs(Tg, Ts) in (10) is analogous to that for qrad gw(Tg, Tw) by (2): qrad gs ðT g ; T s Þ ¼ eg ðT g ÞrT 4g ag ðT g ; T s ÞrT 4s
ð11Þ
The calculation of qconv gs(Tg, Ts) in (10) is analogous to that for qconv gw(Tg, Tw) using (3): qs ðT g ; T s Þ ¼ ags ðT g T s Þ
ð12Þ
The thermal radiation heat exchange between the surfaces in the furnace enclosure Q () [I·1] is calculated once in the calculation step for the whole furnace geometry using Eq. (8). Eq. (8) allows an exact geometrical treatment in the online model. The heat exchange from the furnace surfaces to the jth surface of the slab qrad ws,j in Eq. (10) is then the jth element Qj () of vector Q () [I·1] divided by the area of jth surface Aj: qradws;j ðT 1 . . . T I Þ ¼ Qj () ðT 1 . . . T I Þ=Aj
ð13Þ
When the total heat fluxes to all the slab-surface elements are determined, the 3D heat conduction inside the slab is calculated using the finite-difference method. 2.4. The total heat flux to the bottom slab surfaces lying on the skid pipes In the preheating and the heating zones of the furnace the slabs are lying on the water-cooled skid pipes (Fig. 2(a)). They allow the slabs to be positioned at a suitable height for the heating from both sides. The skid pipes are insulated to minimize the heat conduction from the surface of the slab. To evaluate the heat conduction from the slab surfaces that lie on the skid pipes a simple model was developed (14). It is assumed in the model that the heat conduction is uniform throughout the whole length of the skid pipes. The factor 1/4 in (14) is used from the fact that about 1/4 of skid pipe is covered by slabs, other part is opened to the inside of the furnace. As a result it is roughly assumed that 1/4 of heat to the cooling water comes from the parts of slabs surfaces which lie on the pipes. The model takes into account the basic measured data of the water-cooled skid pipes (Fig. 4) qtotal;j ¼ qH2 O;i 1 cH O qH2 O ð/H2 O =4ÞðT H2 O;out;i T H2 O;in Þ ¼ 2 ; 4 S p Lp i ¼ 1; 2; 3; 4
ð14Þ
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TH2O,out,4 TH2 O,out,3
TH2O,out,2
iz4
TH2O, in
iz3
n
iz2
2
TH2O,out,1
Lp
φ H2O
11
cH2O
Sp
ρH
2O
Fig. 4. In the preheating and heating zones of the furnace the slabs lie on the water-cooled skid pipes.
For the parts of the bottom slab surfaces which lie on the skid pipes it is assumed thermal flux evaluated by Eq. (14) which is based on the heat flow removed from the slab surface by the cooling water. For the parts of the bottom slab surface which do not lie on the skid pipes the radiation heat exchange by Eq. (10) which accounts for individual view factor values as for other slab surfaces. 2.5. The total heat flux to the bottom slab surface in the soaking zone The slabs in the soaking zone of the furnace lie on the furnace floor built of Magmalox blocks (Fig. 2(b)). The furnace floor can be divided into M smaller areas with the assumption of a uniform temperature. Since some of these areas are covered by the slabs we assume that they are mostly reheated from the bottom slab surfaces. In order to determine the furnace-floor temperature for individual areas a special procedure was developed. It is assumed in the procedure that due to the roughness of the furnace floor surface and due to the oxide-scale, which lies on the floor, the slab has contact with the furnace floor at just a few points. Therefore, the main heat-transfer mechanism between the slab and the furnace floor is assumed to be thermal radiation. The furnace floor is divided into M floor-surface elements to fulfil the uniform-temperature condition. For each floor-surface element m, the heat-flux balance equation is derived: qm qm! qcond floor;m ¼ 0
ð15Þ
The incoming heat flux qm can be determined as the sum of the contributions of the surfaces in the furnace enclosure, including the slab surfaces (16), by considering the elements Fi!m of the view-factor matrix F qm ¼
I X
ei ðT i ÞrT 4i
Ai N i!m Am N i
ei ðT i ÞrT 4i
Ai F i!m Am
i¼1
¼
I X i¼1
ð16Þ
In the case of i = m and Fm!m 5 0 in Eq. (16) the self-radiation of the element m is considered. The outgoing heat flux qm! in Eq. (15) can, due to thermal radiation, be calculated using Eq. (17). The temperature Tm is unknown. Therefore, the outgoing thermal flux cannot be determined directly qm! ¼ em ðT m ÞrT 4m
ð17Þ
The heat conduction through the furnace-floor element qcond floor,m is evaluated using Eq. (6) by taking into account the material properties of the furnace-floor layers and the measured outer temperature, Tair = 100 C. The calculation of all three thermal fluxes in Eq. (15) depends on the unknown temperature, Tm, of the furnace-floor element. The bisection method is used in the model to solve the balance equation (15) numerically, which gives the temperature of the furnace-floor element Tm. When the furnace-floor temperatures are known for all M elements of the furnace floor then the total heat flux to the bottom slab surface elements in the soaking zone can be calculated. Considering the above assumption, that the main heat-transfer mechanism is thermal radiation, then the total heat flux to the slab surface j is the thermal radiation heat exchange to the surface j. After the calculation of the thermal radiation heat exchange for the whole furnace enclosure by (8), the qtotal,j is determined by (13) as the jth element Qj () of vector Q () [I·1] divided by the area of the jth surface Aj. 3. Validation of the simulation model The validation of the simulation model is a very important phase in the process of developing the simulation model. The model was validated on the basis of measurements on the pusher furnace in the Acroni Steelworks in Slovenia. The length of the furnace is 24.6 m, the width of the furnace is 6.0 m. The enclosure of furnace walls is divided in smaller surfaces as can be seen in Fig. 3(a) and (b) in order to determine view factor matrix F. There are determined two separate matrixes one for one-row and the other for two-row charging. Side walls, ceiling and furnace floor are divided into length segments which are approximately 1 m wide. The exception is the furnace floor in soaking zone (8 m · 6 m) which is divided into 8 · 12 elements respectively because of calculation of their temperatures on the base of heat balance by accounting for the temperatures of the slabs which lie on the furnace floor. The skid pipes are treated as they have insulation of square crosssection. Each of the skid pipes surfaces is longitudinally divided into approximately 1 m long segments. The surfaces of slabs in the furnace are longitudinally divided into 0.1 m long segments. For one-row charging mode each of the segments which present the upper or the bottom slab surface is additional longitudinally divided into four seg-
A. Jaklicˇ et al. / Applied Thermal Engineering 27 (2007) 1105–1114
ments. In view-factor determination the emissivity of furnace walls is assumed to be 0.95 and the emissivity of the slabs is assumed to be 0.8. Test measurements were performed during two rows charging mode (Fig. 3(b)), where the test slabs were charged on the left-hand side of the furnace. For the heat conduction calculation by finite difference method the test slabs are divided into: • slab 1: l = 2.5 m, 25 · 25 · 20 elements, • slab 2: l = 2.5 m, 25 · 25 · 25 elements,
w = 1.56 m, respectively, w = 1.56 m, respectively.
h = 0.20 m
into
h = 0.25 m
into
The measurements were performed using five trailing thermocouples (Type K, B6 mm, L = 30 m). These five thermocouples were mounted on a test slab, as shown in
1111
Fig. 5(a) and (b). Thermocouple TC1 was mounted 10 mm under the upper slab surface, TC2 was mounted in the slab centre, TC3 was mounted 10 mm above the bottom surface, TC4 and TC5 were mounted on the slab surface. The first measurement was performed on a test slab (material: AISI 316L, thickness 200 mm). The temperatures were measured during the reheating of the slab as it passed through the furnace. The simulation model was compared with the measurements at three points, TC1, TC2 and TC3. The tuning of the model was performed by adjusting the temperature profile of the furnace’s ceiling and sidewalls. Temperature profile of furnace walls (ceiling, side walls) was measured once by optical pyrometer (where possible) during the steady-state production process at known furnace set temperatures. The distance between measured points is approximately 1 m. Points of the profile located
Fig. 5. (a) Measuring points in the test slabs, (b) test slab with the mounted thermocouples.
Acroni - pusher furnace - Comparison between calculation and measurements 24.08.2002 1400 1300 1200 1100
Temperature (°C)
1000 900 800 Calculation upper surface Calculation center Calculation bottom surface Measurement upper surface TC1 Measurement center TC2 Measurement bottom surface TC3 Furnace wall temperature Furnace gas temperature Push interval
700 600 500 400 300 200 100 0 0
20
40
60
80
100 Time (min)
120
140
160
180
200
Fig. 6. Validation of the simulation model for the test slab (material: AISI 316L, thickness 200 mm) after the tuning of the model. The small vertical lines at the bottom of the graph show the push intervals of the furnace.
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Acroni - pusher furnace - Comparison between calculation and measurements 23.02.2003 1400 1300 1200 1100
Temperature (°C)
1000 900 800 Calculation upper surface Calculation center Calculation bottom surface Measurement upper surface TC1 Measurement center TC2 Measurement bottom surface TC3 Furnace wall temperature Furnace gas temperature Push interval
700 600 500 400 300 200 100 0 0
20
40
60
80
100 Time (min)
120
140
160
180
200
Fig. 7. Validation of already-tuned simulation model for the test slab (material: AISI 316L, thickness 250 mm). The small vertical lines at the bottom of the graph show the push intervals of the furnace.
in a particular zone are bonded to the control thermocouple measurement for the particular zone. If the temperature of control thermocouple for the particular zone is increased for i.e., 20 C then points of the temperature profile of the particular zone are also increased for this temperature difference. Values between points of temperature profile are interpolated. During the tuning of the model optionally additional fixed temperature difference between control thermocouple and the points of measured temperature profile is applied to get better agreement with measurements. Tuning is performed for each individual furnace zone. After the tuning of the model, all the parameters of the model were within the real physical values. Good agreement was obtained between the measured and the calculated temperatures at all three comparison points in the whole reheating process (Fig. 6). The second measurement was performed on a test slab (material: AISI 316L, thickness 250 mm). The temperatures were again measured during the reheating of the slab as it passed through the furnace. The heating curves of the already-tuned simulation model were compared with the measurements at three points, TC1, TC2 and TC3. Good agreement was obtained between the measured and the calculated temperatures, even if the test slab had a different thickness (Fig. 7). The good agreement of the bottom-surface temperatures in the soaking zone confirms that the algorithm for the furnace-floor temperature calculation is appropriate. The validation phase shows that the developed algorithms of the simulation model for slab reheating are in good agreement with the real physical behaviour of the reheating process.
4. Implementation of the simulation model For online operation the computer with the simulation model is connected to the information system of the hotrolling plant (Fig. 8). The real-time input data for the model are in an ASCII formatted file ‘‘IMT.DAT’’. The input data consist of input measured data (temperatures of zones, fuel/air consumptions) and charging data (data of the slabs in the furnace, i.e., dimensions, material, etc.). The file is refreshed in 60-s intervals (sampling intervals) on the main process computer, DEC alpha. Every 50 s the simulation model transfers the file from the main process computer to the simulation model computer using
INTERNET Industrial LAN Ethernet
Open VMS DEC
GUI
Main process computer
GUI of the simulation model
PLC
Suse Linux
Simulation model of slab reheating
Token Pass Line (TPL)
mA and digital I/O signals
Fig. 8. The implementation of the simulation model to the industrial network system.
A. Jaklicˇ et al. / Applied Thermal Engineering 27 (2007) 1105–1114
FTP. The transferred file is then analysed in terms of changes with respect to the previous one. During the production process slabs of different widths can be charged; therefore, push steps can be of different lengths. Slab movement in the furnace is treated by changing the position of individual slab on the base of the slab position data in the charging data file. For the view-factor matrix determination (surface-to-surface thermal radiation model) slab surfaces are longitudinally divided into smaller length segments (0.1 m) for which we assume homogenous thermal flux. In the heat conduction model exact slab dimensions are treated in the calculation and the division of elements is finer. Since the division of slab elements for both models is different, both models are coupled by the following three steps: • Temperatures of surface elements in the view-factor matrix are determined as an average value of temperatures of slab surface elements (heat conduction model) according to equally positioned elements. • Calculation of thermal radiation heat flux to each of the slab surface element in the view-factor matrix. • Thermal radiation heat fluxes are assigned to equally positioned slab surfaces in the heat conduction model. If the charging data were change then the push of the slabs occurs. Therefore, the simulation is performed for the time interval from the previous measurement to the moment of the push. Then the calculated temperature fields are written in slab history files and the locations of the slabs are changed for the push length. After that, the sim-
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ulation is performed for the time interval from the moment of the push to the current measurement (to the end of sampling interval). If the new input measured data is found without a change of charging data, then the simulation is performed for the time interval from the previous measurement to the current measurement which is for the whole sampling interval. In this case the calculated temperature fields for an individual slab are written in online data files. The parameters of the simulation are as follows: the slabs are divided into a mesh with elements of approximate dimensions: (Dx = 100 mm, Dy = 10 mm, Dz = 10 mm), the computation interval is Dt = 1 s, and the communication interval is 60 s. These parameters ensure stable solution of the finite-difference method. The graphical user interface (GUI) (Fig. 9) was developed for the user-friendly presentation of the real-time simulation model results. The GUI runs parallel to the simulation model. The data between both programs are exchanged using dynamically refreshed ASCII files. The GUI runs in Real-time or in Archive mode. When the GUI runs in Real-time mode the top view to the furnace is shown in the upper part of the window (Fig. 9). Also shown are all the slabs that currently exist in the furnace. The slab is selected by a mouse click. In the lower part of the window are detailed data of the selected slab: Slab ID data, material, dimensions, etc. The calculated temperature field of the selected slab is presented by three sections using a thermal scale, or in a diagram form for the selected slab elements. The Archive mode allows a preview of the reheating process of already-reheated slabs.
Fig. 9. The GUI of the real-time simulation model of the steel-slab reheating process (two-rows charging mode, real-time view).
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5. Conclusion The presented online simulation model of the steel-slab reheating process in a pusher-type furnace allows the monitoring of non-measurable values (temperature fields of slabs in the furnace). It considers the exact geometry in thermal radiation calculations of the furnace enclosure, including the geometry of the slabs inside the furnace. The algorithms are optimized to allow online simulation. The thermal radiation of the surfaces is calculated online for the whole furnace enclosure. The geometric relations between the individual surfaces in the furnace enclosure are written in the form of a view-factor matrix. The matrix for the whole furnace enclosure, including slabs, is determined by the Monte Carlo method. The matrix is determined once before the simulation. At the beginning of the simulation the model reads the view-factor matrix data. The temperatures of the furnace-floor segments are calculated on the basis of heat balance. Good agreement between the measured and the calculated heating curves shows that the model includes the main physical phenomena occurring during the reheating process in the pusher-type furnace. The model is implemented online in Acroni d.o.o. Jesenice, where it has been used to monitor slab reheating during the regular production process since September 2004. References [1] B. Dahm, R. Klima, Feedback control of stock temperature and oxygen content in reheating furnaces, in: IoM Conference Challenges in Reheating Furnaces, Conference Papers, London, October, 2002, pp. 287–296.
[2] A. Jaklicˇ, T. Kolenko, B. Glogovac, Supervision of slab reheating process using mathematical model, in: 3rd IMACS Symposium on Mathematical Modelling MATHMOD, February 2–4, Vienna. Proceedings, (ARGESIM Report No. 15), Vienna, 2 (2000) 755–759. [3] A. Jaklicˇ, T. Kolenko, B. Glogovac, Simulation of billet reheating process in walking beam furnace, Metalurgija 40 (1) (2001) 23–27. [4] B. Leden, STEELTEMP – a program for temperature analysis in steel plants, Scandinavian Journal of Metallurgy 15 (1986) 215–223. [5] D.F.J. Staalman, Process control in reheating furnaces, in: IoM Conference Challenges in Reheating Furnaces, Conference Papers, London, October 2002, pp. 267–285. [6] A. Jaklicˇ, T. Kolenko, B. Glogovac, A real-time simulation model of billet reheating in the Allino walking-beam furnace, in: Proceedings of the International conference on Refractories, Furnaces and Thermal Insulations, High Tatras, Slovakia, 2004, pp. 237–242. [7] B. Leden, D. Lindholm, E. Nitteberg, The use of STEELTEMP software in heating control, La Revue de Me´tallurgie-CIT 96 (3) (1999) 367–380. [8] B. Leden, STEELTEMP for temperature and heat-transfer analysis of heating furnaces with on-line applications, in: IoM Conference Challenges in Reheating Furnaces, Conference Papers, London, October 2002, pp. 297–307. [9] A. Jaklicˇ, T. Kolenko, B. Zupancˇicˇ, The influence of the space between the billets on the productivity of a continuous walking-beam furnace, Applied Thermal Engineering 25 (5–6) (2005) 783–795. [10] W. Heiligenstaedt, Waermetechnische Rechnungen fuer Industrieoefen, Verlag Stahleisen M.B.H, Duesseldorf, 1966. [11] T. Kolenko, B. Glogovac, A. Jaklicˇ, An analysis of a heat transfer model for situations involving gas and surface radiative heat transfer, Communications in Numerical Methods in Engineering 15 (1999) 349–365. [12] H.C. Hottel, in: W.H. McAdams (Ed.), Heat Transmission, third ed., McGraw-Hill, New York, 1954. [13] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, McGrawHill Book Company, New York, 1981. [14] M.F. Modest, Radiative Heat Transfer, McGraw-Hill Book Company, New York, 1993.
Online simulation model of the slab-reheating process in a pusher-type furnace Anton Jaklicˇ b
a,*
, Franci Vode a, Tomazˇ Kolenko
b
a Institute of Metals and Technology, Lepi pot 11, 1000 Ljubljana, Slovenia University of Ljubljana, Faculty of Natural Sciences and Technology, Asˇkercˇeva 12, 1000 Ljubljana, Slovenia
Received 13 September 2005; accepted 10 July 2006 Available online 18 October 2006
Abstract This paper presents an online simulation model of the slab-reheating process in a pusher-type furnace in Acroni d.o.o. in Slovenia. The simulation model is connected to the information system of a hot-processing plant that provides online measuring and charging data of the furnace. The simulation model considers the exact geometry of the furnace enclosure, including the geometry of the slabs inside the furnace. A view-factor matrix of the furnace enclosure was determined using the Monte Carlo method. The heat exchange between the furnace gas, the furnace wall and the slab’s surface is calculated using a three-temperature model. The heat conduction in the slabs is calculated using the 3D finite-difference method. The model was validated using measurements from trailing thermocouples positioned in the test slabs during the reheating process in the furnace. A graphical user interface (GUI) was developed to ensure a user-friendly presentation of the simulation-model results. 2006 Elsevier Ltd. All rights reserved. Keywords: On-line simulation; Reheating furnace; Slab heating; Mathematical model
1. Introduction A computer-controlled hot-rolling process for steel slabs requires high-quality reheated slabs in terms of time, temperature, thermal profile and furnace atmosphere. Each steel slab has to be reheated at a suitable temperature for hot-rolling with the prescribed temperature difference inside the slab. Temperature differences during the whole reheating process should not cause the maximum-allowed thermal stress inside the slab to be exceeded. The slabs are reheated in the gas-fired pusher-type furnace (Fig. 1). The furnace has six control zones: the upper and lower preheating zones, the upper and lower heating zones, and the left and right soaking zones. The material flow through the furnace is discontinuous, with the movement happening in push steps. At each push step the pushing machine pushes all the slabs until the slab at the exit *
Corresponding author. Tel.: +386 31 809 357; fax: +386 1 4701 939. E-mail address: [email protected] (A. Jaklicˇ).
1359-4311/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2006.07.033
drops out from the furnace. The length of the pushing step depends on the width of the discharged slab. The number of slabs inside the furnace can vary and depends on the width of the individual slabs. There are many influences that can affect the reheating process. The time intervals between pushing steps can vary, and delay conditions in the material flow are often present. There are two kinds of delay: scheduled delays, e.g., mill roll changes, meal breaks, etc., or unscheduled delays, e.g., mill breakdowns, etc. Another influence on the reheating process comes from slabs of different dimensions and the different steel grades, which have to be reheated to different discharge temperatures. All these influences significantly affect the reheating process and mean that almost every slab has a different reheating history. Knowledge of the temperature field of slabs during the reheating process is very important for successful furnace control. The temperature fields of the slabs are non-measurable values during the production process. Computer simulation is sometimes the most reasonable way to
A. Jaklicˇ et al. / Applied Thermal Engineering 27 (2007) 1105–1114
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Nomenclature a A c d F F I I K L M N q Q Q S T
heat-transfer coefficient (W m2 K1) surface area (m2) specific heat (J/kg K) furnace-wall/floor-layer thickness (m) view-factor view-factor matrix number of surfaces in a furnace enclosure including slab surfaces identity matrix number of the furnace-wall/floor-layers length (m) number of the furnace-floor surfaces number heat flux (W m2) element of heat-flux vector heat-flux vector width of the water pipe temperature (K)
Greek letters a absorptivity factor e emissivity factor / water flow (m3/s) k heat conduction (W m1 K1) p number PI q density (kg/m3) r Stefan–Boltzmann constant, 108 W m2 K4
slabs
upper EXIT
r = 5.671 ·
pusher
burners
ENTRY
lower
soaking zone
heating zone
Subscripts abs absorption air air cond conduction conv convection floor furnace floor g furnace gas H2O water i index of surface in furnace enclosure in input j index of surface in furnace enclosure k index of layers of furnace wall/floor m index of furnace-floor surface n index of furnace-floor surface out output p pipe rad radiation s slab total total w furnace wall !, ) outgoing , ( incoming () balance
preheating zone
Fig. 1. Pusher-type furnace.
determine such values. The state of the art is one- and twodimensional online calculations of the stock temperature [1–5]. However, the computational power of today’s personal computers means that increasingly complex simulations can be performed in real-time. Thus, complex simulation models can be implemented in the monitoring and control systems of industrial processes, and the first attempts have been made to calculate the stock temperature in three dimensions online [6–8]. An online supervision system based on a three-dimensional simulation model of a steel-slab reheating process
in a pusher-type furnace has been developed. The calculations in the model are based on a mathematical model that includes the main physical phenomena appearing during the reheating process in a natural-gas-fired pusher-type furnace: thermal radiation is the main heat-transfer mechanism, and the geometry of the furnace enclosure has an important role in the heat transfer of the thermal radiation. The furnace enclosure consists of the furnace geometry together with the geometry of the charged slabs inside the furnace. One of the chief mathematical complexities in treating the radiative heat transfer between surfaces is accounting for the geometrical relations involved in how the surfaces view each other. For the whole furnace enclosure they are expressed with a view-factor matrix form. In order to determine the matrix, a separate simulation model based on the Monte Carlo method was developed [9]. This model allows a view-factor determination for a general furnace enclosure consisting of rectangular surfaces, including multiple reflections. In the presented approach the view-factor matrix for a particular furnace enclosure, including the slabs and skid pipes, is calculated only once (the typical calculation time on a Pentium4 PC is three days). The viewfactor matrix is read into the online simulation model of
A. Jaklicˇ et al. / Applied Thermal Engineering 27 (2007) 1105–1114
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the slab-reheating process at its start up. In the models described in [1–5,7,8], simplified approaches for treating the geometries of the furnace enclosures are used. The simplified approaches are acceptable for one- and two-dimensional simulation models. The presented online simulation model is capable of treating the exact geometry of the pusher-type furnace enclosure in thermal radiation calculations.
ing for the temperature profile of the zone from the table. The slab-surface temperature, Ts, is calculated step-bystep, using the finite-difference method. The only unknown temperature in the system is the furnace-gas temperature, Tg.
2. Heat exchange in the furnace
The model takes into account variations in gas temperature along the furnace length and does not account the variations across the width of the furnace. In the gas-fired furnace it is almost impossible to achieve homogenous temperature field of gas in the furnace (burners, flames, gas streams, etc.). Gas temperature field in the furnace is strongly dependent on the loading of the furnace and it varies during the production process. Therefore multi point measurement system of gas temperature field in the furnace with hundreds of sensors should be applied in the furnace enclosure in order to determine gas temperature field in real-time to account for the gas temperature variations across the width of the furnace. The literature describes off-line simulation models which accounts for the flames and streaming of furnace gasses which is quite complex. The goal of this work is to develop real-time simulation model, which is always the compromise between the quality of the simulation results and the calculation speed. In the model the furnace gas temperature is calculated for the individual furnace length segment (slab width) on the base of thermal flux equilibrium on the inner side of furnace walls. The furnace gas temperature, Tg, represents an average value for the furnace length segment. It can be determined by solving an equilibrium equation on the inner furnace wall:
To evaluate the heat exchange between the charge, the furnace gas and the furnace wall, basic algorithms from Heiligenstaedt [10] are used. The model is based on three temperatures: the furnace-gas temperature, Tg, the furnace-wall temperature, Tw, and the slab-surface temperature, Ts (Fig. 2(a)). All three temperatures have to be known for an evaluation of the temperature-dependent heat fluxes in the system. The modification of the model assumes that the furnace-wall temperature can be measured more reliably than the furnace-gas temperature. Therefore, the furnace-wall temperature was taken as the model’s primary value [11]; in contrast to the original model, where this was the furnace-gas temperature. The model was also improved to take into account the geometry of the furnace enclosure, including the slabs. The model assumes that the furnace-gas temperature is homogenous, and so to comply with this assumption the length of the furnace is divided into smaller segments, which are equal to the slab widths. The heat exchange in the furnace segment is shown in Fig. 2(a). The total heat transfer to the slab surface can be estimated by evaluating the partial mechanisms. The furnace gas has the highest temperature, Tg, in the system and emits heat to the furnace wall (qrad gw, qconv gw) and to the slab surface (qrad gs, qconv gs) by radiation and convection. A fraction of the heat to the furnace wall is lost to the outside, through the furnace wall (qcond w), and another fraction is emitted to the slab surface (qrad ws), where it is partly absorbed by the furnace gas (qabs g). The inner furnace-wall average temperature, Tw, is known; it is calculated for a particular control zone on the basis of measurements from the control thermocouple by account-
0 ¼ qrad gw ðT g ; T w ÞAw þ qconv gw ðT g ; T w ÞAw qcond w ðT w ÞAw qrad ws ðT w ; T s ÞAs þ qabs g ðT w ; T s ÞAs
ð1Þ
In Eq. (1), Aw is the area of the wall surface of the furnace segment, As is the area of the slab surface that is surrounded by the furnace-wall segment. The above equation can be solved numerically using the bisection method. The
furnace wall
a qcond w
2.1. Determination of the furnace gas temperature for the length segment
b Tw
qrad gw qconv gw qabs g
Tair
qrad ws
furnace wall
Tg
furnace gas
qcond w
qrad gs qconv gs
Tg
qrad gw qconv gw
Tair
Ts slab
Ts Tw
Tw
qH O 2
qabs g
qrad ws
furnace gas
qrad gs qconv gs Ts slab
Tg
Tair =Tkorndorfer=100 ˚C
qcond floor, m
Fig. 2. (a) The heat exchange in the preheating and in the heating zones, (b) the heat exchange in the soaking zone of the furnace.
A. Jaklicˇ et al. / Applied Thermal Engineering 27 (2007) 1105–1114
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qrad gw(Tg, Tw) is calculated with Eq. (2); it describes the difference between the emitted heat flux from the gas mixture and the absorbed heat flux emitted from the furnace wall. The emissivity of furnace walls is high (0.95) therefore they are treated as black according to the model of the net radiant-heat interchange between a gas containing carbondioxide and water-vapor and its black bounding surface [12]. The values for eg and ag are determined using the model for a grey-gas CO2 and H2O mixture of Hottel found in [13] qrad gw ðT g ; T w Þ ¼ eg ðT g ÞrT 4g ag ðT g ; T w ÞrT 4w
ð2Þ
The qconv gw(Tg, Tw) in (1) can be determined using Eq. (3). The convection heat transfer in a high-temperature reheating furnace has a minor influence on the reheating process, especially in the high-temperature part of the furnace. Different authors recommend different values for agw between 10 and 15 W/m2 K. In our model we used agw = 12 W/m2 K qconv gw ðT g ; T w Þ ¼ agw ðT g T w Þ
ð3Þ
For the evaluation of the furnace-gas temperature, Tg, of the furnace segment (1) the average heat flux between the furnace wall and the slab surface qrad ws(Tw, Ts) is determined by (4): qrad ws ðT w ; T s Þ ¼
ew ðT w ÞrT 4w es ðT s ÞrT 4s As 1 1 þ 1 Aw ew es
ð4Þ
The part of the heat exchange between the furnace wall and the slab surface that is absorbed in the furnace gas qabs g is determined by Eq. (5). According to the Heiligenstaedt’s model [10] the portion of radiant flux from the furnace wall to the charge absorbed in the furnace gas is equal to the radiant flux from the furnace gas to the charge calculated as if the furnace gas had the temperature of the furnace wall. The values for eg and ag are determined using the model for a grey-gas CO2 and H2O mixture of Hottel [13] qabs g ðT w ; T s Þ ¼ eg ðT w ÞrT 4w ag ðT w ; T s ÞrT 4s
ð5Þ
The heat conduction through a K-layer furnace wall qcond w is calculated using Eq. (6). The thermal conductivities kk of individual furnace-wall layers are temperature dependent. Therefore, Eq. (6) is solved iteratively qcond w ¼ P
1
K dk k¼1 kk ðT k Þ
þ a1air
ðT w T air Þ
ð6Þ
z
a
i
y j
z
x
i
b
y
j x Fig. 3. (a) The furnace enclosure for one-row charging, (b) the furnace enclosure for two-row charging.
the 3D temperature field of the slabs is modelled. Therefore, the exact geometry of the furnace has to be treated in the model, including the charged slabs and skid pipes. The furnace geometry, including charged slabs, is divided into I smaller surfaces under the assumption of isothermal surfaces (Fig. 3(a) and (b)). For the surface j in Fig. 3(a) and (b) it is possible to write heat balance Qj () as the difference between the incoming Qj( and the outgoing radiation Qj) [14]: qj Aj ¼ Qj () ¼ Qj( Qj) ¼
I X
ei ðT i ÞrT 4i Ai F i!j ej ðT j ÞrT 4j Aj
ð7Þ
i¼1
The incoming thermal radiation Qj( consists of I thermal radiation contributions of the individual surface elements i. Since some of the surfaces in the enclosure cannot view each other and some can be viewed just partly, the furnace enclosure presents a complex geometry for the view-factor calculation. Fi!j are the view-factors that describe the fraction of the total thermal radiation emitted by surface Ai that is absorbed by the surface Aj, including multiple reflections [14]. For the whole furnace geometry consisting of I surfaces the heat exchange is written in a vector-matrix form: Q () ½I1 ¼ Q(½I1 Q)½I1 ¼ F ½II Q)½I1 Q)½I1
2.2. Thermal radiation heat exchange between the surfaces in the furnace enclosure In the algorithm for determining the average furnace-gas temperature Eq. (4) was used to calculate the thermal radiation from furnace walls with an average temperature Tw to the slab surface qrad ws(Tw, Ts). Eq. (4) gives the heat flux on the centre of the slab surface, which is not sufficient when
¼ ðF ½II I ½II Þ Q)½I1
ð8Þ
The heat-balance vector Q () [I·1] for the whole furnace geometry can be calculated when the view-factor matrix F and the outgoing thermal flux vector Q)[I·1] are known. The calculation of the thermal radiation heat transfer between two surfaces can thus be divided into the energy and the geometry parts [14]. The energy part describes
A. Jaklicˇ et al. / Applied Thermal Engineering 27 (2007) 1105–1114
the vector of the outgoing thermal fluxes Q)[I·1] with elements ej ðT j ÞrT 4j Aj . The main part of the calculation of the thermal radiation heat exchange between the surfaces in the furnace enclosure is the determination of a view-factor matrix F. Since energy travels in discrete photon bundles along a straight path before interacting with a surface [14], problems in thermal radiation are particularly well suited to the Monte Carlo method. In the case of determining view-factors, a large number, Ni, of photons are emitted from the surface, i.e., i. These photons are emitted from the surface according to probability density functions. The path of each photon, including possible reflections, is traced to its absorption at one of the surfaces of the furnace enclosure. In the model the emission and the reflection of photons are treated as grey and diffuse. Thus, determining view-factors in the furnace enclosure with the Monte Carlo method involves emitting and tracing the history of a statistically meaningful random sample of photons from their points of emission to their points of absorption, including multiple reflections. After the emission of a large number of photons, Ni, from a surface i, the view-factor Fi!j can be directly determined by counting the number of photons Ni!j that have been absorbed at the surface j [14]: N i!j N i!j F i!j ¼ lim ð9Þ N i !1 Ni N i N i 1 In order to determine the matrix, a separate simulation model based on the Monte Carlo method was developed [9]. This model allows a view-factor determination for a general furnace enclosure consisting of rectangular surfaces, including multiple reflections. The matrix F is calculated prior to the simulation for the particular furnace geometry, and the results are stored in data files. One of possible ways to account the absorption in gas at surface to surface radiation heat transfer is during the Monte Carlo determination of view-factors. Then view-factors of the enclosure include also the absorption in the gas. The problem of this method is that the absorption in gas is dependant on the gas temperature, which varies during the furnace operation. Therefore view-factors which account for the absorption are valid just for the case of assumed gas temperatures used during Monte Carlo determination. In the real-time model the absorption of the gas is treated for individual length segment of the furnace in a global fashion (5) according to the Heiligenstaed’s model [10] by using absorptivities calculated by Hottel [12]. The advantage of this method is that the model is relatively simple and allows therefore accounting for the temperature timedependant variations of furnace gas in real-time. Drawback of this method is that the absorption is not treated accurately by path length between individual surfaces. 2.3. The total heat flux on the slab surfaces The total heat flux qtotal,j to the surface elements j (Fig. 2(a) and (b)) can be determined in the same way for
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the preheating, the heating and the soaking zones using Eq. (10). A separate determination is needed only for the parts of the bottom slab surfaces that lie on the skid pipes, and for the bottom slab surfaces in the soaking zone, where the slabs lie on the furnace floor qtotal;j ¼ qrad gs ðT g ; T s Þ þ qconv gs ðT g ; T s Þ þ qrad ws;j ðT 1 . . . T I Þ qabs g ðT g Þ
ð10Þ
The calculation of qrad gs(Tg, Ts) in (10) is analogous to that for qrad gw(Tg, Tw) by (2): qrad gs ðT g ; T s Þ ¼ eg ðT g ÞrT 4g ag ðT g ; T s ÞrT 4s
ð11Þ
The calculation of qconv gs(Tg, Ts) in (10) is analogous to that for qconv gw(Tg, Tw) using (3): qs ðT g ; T s Þ ¼ ags ðT g T s Þ
ð12Þ
The thermal radiation heat exchange between the surfaces in the furnace enclosure Q () [I·1] is calculated once in the calculation step for the whole furnace geometry using Eq. (8). Eq. (8) allows an exact geometrical treatment in the online model. The heat exchange from the furnace surfaces to the jth surface of the slab qrad ws,j in Eq. (10) is then the jth element Qj () of vector Q () [I·1] divided by the area of jth surface Aj: qradws;j ðT 1 . . . T I Þ ¼ Qj () ðT 1 . . . T I Þ=Aj
ð13Þ
When the total heat fluxes to all the slab-surface elements are determined, the 3D heat conduction inside the slab is calculated using the finite-difference method. 2.4. The total heat flux to the bottom slab surfaces lying on the skid pipes In the preheating and the heating zones of the furnace the slabs are lying on the water-cooled skid pipes (Fig. 2(a)). They allow the slabs to be positioned at a suitable height for the heating from both sides. The skid pipes are insulated to minimize the heat conduction from the surface of the slab. To evaluate the heat conduction from the slab surfaces that lie on the skid pipes a simple model was developed (14). It is assumed in the model that the heat conduction is uniform throughout the whole length of the skid pipes. The factor 1/4 in (14) is used from the fact that about 1/4 of skid pipe is covered by slabs, other part is opened to the inside of the furnace. As a result it is roughly assumed that 1/4 of heat to the cooling water comes from the parts of slabs surfaces which lie on the pipes. The model takes into account the basic measured data of the water-cooled skid pipes (Fig. 4) qtotal;j ¼ qH2 O;i 1 cH O qH2 O ð/H2 O =4ÞðT H2 O;out;i T H2 O;in Þ ¼ 2 ; 4 S p Lp i ¼ 1; 2; 3; 4
ð14Þ
A. Jaklicˇ et al. / Applied Thermal Engineering 27 (2007) 1105–1114
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TH2O,out,4 TH2 O,out,3
TH2O,out,2
iz4
TH2O, in
iz3
n
iz2
2
TH2O,out,1
Lp
φ H2O
11
cH2O
Sp
ρH
2O
Fig. 4. In the preheating and heating zones of the furnace the slabs lie on the water-cooled skid pipes.
For the parts of the bottom slab surfaces which lie on the skid pipes it is assumed thermal flux evaluated by Eq. (14) which is based on the heat flow removed from the slab surface by the cooling water. For the parts of the bottom slab surface which do not lie on the skid pipes the radiation heat exchange by Eq. (10) which accounts for individual view factor values as for other slab surfaces. 2.5. The total heat flux to the bottom slab surface in the soaking zone The slabs in the soaking zone of the furnace lie on the furnace floor built of Magmalox blocks (Fig. 2(b)). The furnace floor can be divided into M smaller areas with the assumption of a uniform temperature. Since some of these areas are covered by the slabs we assume that they are mostly reheated from the bottom slab surfaces. In order to determine the furnace-floor temperature for individual areas a special procedure was developed. It is assumed in the procedure that due to the roughness of the furnace floor surface and due to the oxide-scale, which lies on the floor, the slab has contact with the furnace floor at just a few points. Therefore, the main heat-transfer mechanism between the slab and the furnace floor is assumed to be thermal radiation. The furnace floor is divided into M floor-surface elements to fulfil the uniform-temperature condition. For each floor-surface element m, the heat-flux balance equation is derived: qm qm! qcond floor;m ¼ 0
ð15Þ
The incoming heat flux qm can be determined as the sum of the contributions of the surfaces in the furnace enclosure, including the slab surfaces (16), by considering the elements Fi!m of the view-factor matrix F qm ¼
I X
ei ðT i ÞrT 4i
Ai N i!m Am N i
ei ðT i ÞrT 4i
Ai F i!m Am
i¼1
¼
I X i¼1
ð16Þ
In the case of i = m and Fm!m 5 0 in Eq. (16) the self-radiation of the element m is considered. The outgoing heat flux qm! in Eq. (15) can, due to thermal radiation, be calculated using Eq. (17). The temperature Tm is unknown. Therefore, the outgoing thermal flux cannot be determined directly qm! ¼ em ðT m ÞrT 4m
ð17Þ
The heat conduction through the furnace-floor element qcond floor,m is evaluated using Eq. (6) by taking into account the material properties of the furnace-floor layers and the measured outer temperature, Tair = 100 C. The calculation of all three thermal fluxes in Eq. (15) depends on the unknown temperature, Tm, of the furnace-floor element. The bisection method is used in the model to solve the balance equation (15) numerically, which gives the temperature of the furnace-floor element Tm. When the furnace-floor temperatures are known for all M elements of the furnace floor then the total heat flux to the bottom slab surface elements in the soaking zone can be calculated. Considering the above assumption, that the main heat-transfer mechanism is thermal radiation, then the total heat flux to the slab surface j is the thermal radiation heat exchange to the surface j. After the calculation of the thermal radiation heat exchange for the whole furnace enclosure by (8), the qtotal,j is determined by (13) as the jth element Qj () of vector Q () [I·1] divided by the area of the jth surface Aj. 3. Validation of the simulation model The validation of the simulation model is a very important phase in the process of developing the simulation model. The model was validated on the basis of measurements on the pusher furnace in the Acroni Steelworks in Slovenia. The length of the furnace is 24.6 m, the width of the furnace is 6.0 m. The enclosure of furnace walls is divided in smaller surfaces as can be seen in Fig. 3(a) and (b) in order to determine view factor matrix F. There are determined two separate matrixes one for one-row and the other for two-row charging. Side walls, ceiling and furnace floor are divided into length segments which are approximately 1 m wide. The exception is the furnace floor in soaking zone (8 m · 6 m) which is divided into 8 · 12 elements respectively because of calculation of their temperatures on the base of heat balance by accounting for the temperatures of the slabs which lie on the furnace floor. The skid pipes are treated as they have insulation of square crosssection. Each of the skid pipes surfaces is longitudinally divided into approximately 1 m long segments. The surfaces of slabs in the furnace are longitudinally divided into 0.1 m long segments. For one-row charging mode each of the segments which present the upper or the bottom slab surface is additional longitudinally divided into four seg-
A. Jaklicˇ et al. / Applied Thermal Engineering 27 (2007) 1105–1114
ments. In view-factor determination the emissivity of furnace walls is assumed to be 0.95 and the emissivity of the slabs is assumed to be 0.8. Test measurements were performed during two rows charging mode (Fig. 3(b)), where the test slabs were charged on the left-hand side of the furnace. For the heat conduction calculation by finite difference method the test slabs are divided into: • slab 1: l = 2.5 m, 25 · 25 · 20 elements, • slab 2: l = 2.5 m, 25 · 25 · 25 elements,
w = 1.56 m, respectively, w = 1.56 m, respectively.
h = 0.20 m
into
h = 0.25 m
into
The measurements were performed using five trailing thermocouples (Type K, B6 mm, L = 30 m). These five thermocouples were mounted on a test slab, as shown in
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Fig. 5(a) and (b). Thermocouple TC1 was mounted 10 mm under the upper slab surface, TC2 was mounted in the slab centre, TC3 was mounted 10 mm above the bottom surface, TC4 and TC5 were mounted on the slab surface. The first measurement was performed on a test slab (material: AISI 316L, thickness 200 mm). The temperatures were measured during the reheating of the slab as it passed through the furnace. The simulation model was compared with the measurements at three points, TC1, TC2 and TC3. The tuning of the model was performed by adjusting the temperature profile of the furnace’s ceiling and sidewalls. Temperature profile of furnace walls (ceiling, side walls) was measured once by optical pyrometer (where possible) during the steady-state production process at known furnace set temperatures. The distance between measured points is approximately 1 m. Points of the profile located
Fig. 5. (a) Measuring points in the test slabs, (b) test slab with the mounted thermocouples.
Acroni - pusher furnace - Comparison between calculation and measurements 24.08.2002 1400 1300 1200 1100
Temperature (°C)
1000 900 800 Calculation upper surface Calculation center Calculation bottom surface Measurement upper surface TC1 Measurement center TC2 Measurement bottom surface TC3 Furnace wall temperature Furnace gas temperature Push interval
700 600 500 400 300 200 100 0 0
20
40
60
80
100 Time (min)
120
140
160
180
200
Fig. 6. Validation of the simulation model for the test slab (material: AISI 316L, thickness 200 mm) after the tuning of the model. The small vertical lines at the bottom of the graph show the push intervals of the furnace.
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Acroni - pusher furnace - Comparison between calculation and measurements 23.02.2003 1400 1300 1200 1100
Temperature (°C)
1000 900 800 Calculation upper surface Calculation center Calculation bottom surface Measurement upper surface TC1 Measurement center TC2 Measurement bottom surface TC3 Furnace wall temperature Furnace gas temperature Push interval
700 600 500 400 300 200 100 0 0
20
40
60
80
100 Time (min)
120
140
160
180
200
Fig. 7. Validation of already-tuned simulation model for the test slab (material: AISI 316L, thickness 250 mm). The small vertical lines at the bottom of the graph show the push intervals of the furnace.
in a particular zone are bonded to the control thermocouple measurement for the particular zone. If the temperature of control thermocouple for the particular zone is increased for i.e., 20 C then points of the temperature profile of the particular zone are also increased for this temperature difference. Values between points of temperature profile are interpolated. During the tuning of the model optionally additional fixed temperature difference between control thermocouple and the points of measured temperature profile is applied to get better agreement with measurements. Tuning is performed for each individual furnace zone. After the tuning of the model, all the parameters of the model were within the real physical values. Good agreement was obtained between the measured and the calculated temperatures at all three comparison points in the whole reheating process (Fig. 6). The second measurement was performed on a test slab (material: AISI 316L, thickness 250 mm). The temperatures were again measured during the reheating of the slab as it passed through the furnace. The heating curves of the already-tuned simulation model were compared with the measurements at three points, TC1, TC2 and TC3. Good agreement was obtained between the measured and the calculated temperatures, even if the test slab had a different thickness (Fig. 7). The good agreement of the bottom-surface temperatures in the soaking zone confirms that the algorithm for the furnace-floor temperature calculation is appropriate. The validation phase shows that the developed algorithms of the simulation model for slab reheating are in good agreement with the real physical behaviour of the reheating process.
4. Implementation of the simulation model For online operation the computer with the simulation model is connected to the information system of the hotrolling plant (Fig. 8). The real-time input data for the model are in an ASCII formatted file ‘‘IMT.DAT’’. The input data consist of input measured data (temperatures of zones, fuel/air consumptions) and charging data (data of the slabs in the furnace, i.e., dimensions, material, etc.). The file is refreshed in 60-s intervals (sampling intervals) on the main process computer, DEC alpha. Every 50 s the simulation model transfers the file from the main process computer to the simulation model computer using
INTERNET Industrial LAN Ethernet
Open VMS DEC
GUI
Main process computer
GUI of the simulation model
PLC
Suse Linux
Simulation model of slab reheating
Token Pass Line (TPL)
mA and digital I/O signals
Fig. 8. The implementation of the simulation model to the industrial network system.
A. Jaklicˇ et al. / Applied Thermal Engineering 27 (2007) 1105–1114
FTP. The transferred file is then analysed in terms of changes with respect to the previous one. During the production process slabs of different widths can be charged; therefore, push steps can be of different lengths. Slab movement in the furnace is treated by changing the position of individual slab on the base of the slab position data in the charging data file. For the view-factor matrix determination (surface-to-surface thermal radiation model) slab surfaces are longitudinally divided into smaller length segments (0.1 m) for which we assume homogenous thermal flux. In the heat conduction model exact slab dimensions are treated in the calculation and the division of elements is finer. Since the division of slab elements for both models is different, both models are coupled by the following three steps: • Temperatures of surface elements in the view-factor matrix are determined as an average value of temperatures of slab surface elements (heat conduction model) according to equally positioned elements. • Calculation of thermal radiation heat flux to each of the slab surface element in the view-factor matrix. • Thermal radiation heat fluxes are assigned to equally positioned slab surfaces in the heat conduction model. If the charging data were change then the push of the slabs occurs. Therefore, the simulation is performed for the time interval from the previous measurement to the moment of the push. Then the calculated temperature fields are written in slab history files and the locations of the slabs are changed for the push length. After that, the sim-
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ulation is performed for the time interval from the moment of the push to the current measurement (to the end of sampling interval). If the new input measured data is found without a change of charging data, then the simulation is performed for the time interval from the previous measurement to the current measurement which is for the whole sampling interval. In this case the calculated temperature fields for an individual slab are written in online data files. The parameters of the simulation are as follows: the slabs are divided into a mesh with elements of approximate dimensions: (Dx = 100 mm, Dy = 10 mm, Dz = 10 mm), the computation interval is Dt = 1 s, and the communication interval is 60 s. These parameters ensure stable solution of the finite-difference method. The graphical user interface (GUI) (Fig. 9) was developed for the user-friendly presentation of the real-time simulation model results. The GUI runs parallel to the simulation model. The data between both programs are exchanged using dynamically refreshed ASCII files. The GUI runs in Real-time or in Archive mode. When the GUI runs in Real-time mode the top view to the furnace is shown in the upper part of the window (Fig. 9). Also shown are all the slabs that currently exist in the furnace. The slab is selected by a mouse click. In the lower part of the window are detailed data of the selected slab: Slab ID data, material, dimensions, etc. The calculated temperature field of the selected slab is presented by three sections using a thermal scale, or in a diagram form for the selected slab elements. The Archive mode allows a preview of the reheating process of already-reheated slabs.
Fig. 9. The GUI of the real-time simulation model of the steel-slab reheating process (two-rows charging mode, real-time view).
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5. Conclusion The presented online simulation model of the steel-slab reheating process in a pusher-type furnace allows the monitoring of non-measurable values (temperature fields of slabs in the furnace). It considers the exact geometry in thermal radiation calculations of the furnace enclosure, including the geometry of the slabs inside the furnace. The algorithms are optimized to allow online simulation. The thermal radiation of the surfaces is calculated online for the whole furnace enclosure. The geometric relations between the individual surfaces in the furnace enclosure are written in the form of a view-factor matrix. The matrix for the whole furnace enclosure, including slabs, is determined by the Monte Carlo method. The matrix is determined once before the simulation. At the beginning of the simulation the model reads the view-factor matrix data. The temperatures of the furnace-floor segments are calculated on the basis of heat balance. Good agreement between the measured and the calculated heating curves shows that the model includes the main physical phenomena occurring during the reheating process in the pusher-type furnace. The model is implemented online in Acroni d.o.o. Jesenice, where it has been used to monitor slab reheating during the regular production process since September 2004. References [1] B. Dahm, R. Klima, Feedback control of stock temperature and oxygen content in reheating furnaces, in: IoM Conference Challenges in Reheating Furnaces, Conference Papers, London, October, 2002, pp. 287–296.
[2] A. Jaklicˇ, T. Kolenko, B. Glogovac, Supervision of slab reheating process using mathematical model, in: 3rd IMACS Symposium on Mathematical Modelling MATHMOD, February 2–4, Vienna. Proceedings, (ARGESIM Report No. 15), Vienna, 2 (2000) 755–759. [3] A. Jaklicˇ, T. Kolenko, B. Glogovac, Simulation of billet reheating process in walking beam furnace, Metalurgija 40 (1) (2001) 23–27. [4] B. Leden, STEELTEMP – a program for temperature analysis in steel plants, Scandinavian Journal of Metallurgy 15 (1986) 215–223. [5] D.F.J. Staalman, Process control in reheating furnaces, in: IoM Conference Challenges in Reheating Furnaces, Conference Papers, London, October 2002, pp. 267–285. [6] A. Jaklicˇ, T. Kolenko, B. Glogovac, A real-time simulation model of billet reheating in the Allino walking-beam furnace, in: Proceedings of the International conference on Refractories, Furnaces and Thermal Insulations, High Tatras, Slovakia, 2004, pp. 237–242. [7] B. Leden, D. Lindholm, E. Nitteberg, The use of STEELTEMP software in heating control, La Revue de Me´tallurgie-CIT 96 (3) (1999) 367–380. [8] B. Leden, STEELTEMP for temperature and heat-transfer analysis of heating furnaces with on-line applications, in: IoM Conference Challenges in Reheating Furnaces, Conference Papers, London, October 2002, pp. 297–307. [9] A. Jaklicˇ, T. Kolenko, B. Zupancˇicˇ, The influence of the space between the billets on the productivity of a continuous walking-beam furnace, Applied Thermal Engineering 25 (5–6) (2005) 783–795. [10] W. Heiligenstaedt, Waermetechnische Rechnungen fuer Industrieoefen, Verlag Stahleisen M.B.H, Duesseldorf, 1966. [11] T. Kolenko, B. Glogovac, A. Jaklicˇ, An analysis of a heat transfer model for situations involving gas and surface radiative heat transfer, Communications in Numerical Methods in Engineering 15 (1999) 349–365. [12] H.C. Hottel, in: W.H. McAdams (Ed.), Heat Transmission, third ed., McGraw-Hill, New York, 1954. [13] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, McGrawHill Book Company, New York, 1981. [14] M.F. Modest, Radiative Heat Transfer, McGraw-Hill Book Company, New York, 1993.