A R C H I V E S O F M E T A L L U R G Y A N D M A T E R I A L S Volume Issue 3

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1 A R C H I V E S O F M E T A L L U R G Y A N D M A T E R I A L S Volume Issue 3 M. KIELOCH, Ł. PIECHOWICZ, J. BORYCA, A. KLOS NUMERICAL ANALYSIS OF CORRELATION BETWEEN HEAT CONSUMPTION AND THE STEEL LOSS FOR SCALE IN THE CHARGE HEATING PROCESS ANALIZA NUMERYCZNA KORELACJI POMIĘDZY ZUŻYCIEM CIEPŁA I STRATĄ STALI NA ZGORZELINĘ W PROCESIE NAGRZEWANIA WSADU The steel loss and heat consumption are basic parameters that are decisive to the costs of the heating process. Establishing the correlation beteen heat consumption and steel loss ill make it possible to model the indices of heating furnace operation. The article presents a numerical model for the computation of charge heating, the steel loss for scale, and heat consumption. Based on computation results, mathematical functions have been developed, hich describe the correlation beteen heat consumption and steel loss. Strata stali i zużycie ciepła to podstaoe parametry decydujące o kosztach procesu nagrzeania. Ustalenie korelacji pomiędzy zużyciem ciepła a stratą stali pozoli na modeloanie skaźnikó pracy cieplnej piecó grzeczych. W pracy zaprezentoano model numeryczny obliczeń nagrzeania sadu, straty stali na zgorzelinę i zużycie ciepła. Na podstaie ynikó obliczeń opracoano funkcje matematyczne opisujące korelację pomiędzy zużyciem ciepła a stratą stali. List of designations A, B, C, D, N constants, a temperature equalization coefficient, m 2 /s or m 2 /h, a, b, c, d, f, g constants, a 1, a 2, b 1, b 2, c 1,. c 2, d 1, d 2 constants, C 0 Stefan-Boltzmann constant, W/(m 2 K4, c specific heat capacity, J/(kg K or J/(um 3 K, F area, m 2, G exponent defining the degree of deviation from the linear temperature increase in the heating period, L length of the furnace, m, l length of charge, m, M surface temperature increase rate, K/s or K/h, m mass, kg n number of charge ros in the furnace, R overall thermal transmission resistance, (m K/W, s computational thickness of charge being heated, m, s differential element thickness, m, T temperature, K, T 0 metal surface node temperature, K, TECHNICAL UNIVERSITY OF CZESTOCHOWA, FACULTY OF MATERIALS PROCESSING TECHNOLOGY AND APPLIED PHYSICS, THE DEPARTMENT OF INDUSTRIAL FURNACES AND ENVIRONMENTAL PROTECTION, CZĘSTOCHOWA, 19 ARMII KRAJOWEJ STR., POLAND

2 648 List of some more important subscripts T p,t p initial and end charge surface temperature, K, T z substitute temperature, K, t temperature, C, t g furnace gas temperature, C, t p,t p initial and end charge surface temperature, C, t absolute temperature difference, K, W d gas calorific value, J/um 3 or kj/um 3, furnace capacity, kg/s or t/h, X Fe average elementary iron mass fraction of the scale, z surface steel loss for scale, kg/m 2, z increment in surface steel loss for scale, kg/m 2, V furnace gas unit volume, um 3 /um 3, V volume flux, um 3 /s or um 3 /h, Q energy consumption, J or kj, Q heat flux, W, q heat flux density, W/m 2, q unit heat consumption, kj/kg, absolute difference, δ scale layer thickness, m, φ configuration ratio, ψ function dependent on initial temperature, ρ specific mass, kg/m 3, η t technological process efficiency, % λ thermal conductivity, W/(m K, τ time, s or h, τ time interval, s or h i, j, k, n ordinal numbers, m refers to the metal surface, p refers to the heating period, pal. refers to the fuel supplied to the combustion process, pr. refers to heat exchange by radiation, prz. refers to the allotropic change, refers to the soaking period, zg. refers to scale. 1. Introduction Heating of charge before hot plastic orking is a very important process hich influences both the quality of rolled product and the overall costs of orking. In a properly run process, it is crucial to follo the established technology that assures the proper quality of heating and the minimization of incurred costs [1, 2]. Factors determining the costs of heating include: energy intensity (heat consumption and the loss of steel for scale [3, 4]. The proper selection of the heating curve and heating time is the key technology element enabling the minimization of heat consumption and steel loss [1, 2]. The literature on the subject, barring fe exceptions, does not provide any investigation and computation results, or analytical derivations concerning

3 649 the correlation beteen the consumption of heat and steel loss. From the selected orks [5, 6, 7] on the operation of heating furnaces, it can, hoever, be concluded that there is a close correlation beteen these indices. 2. Numerical computation of charge heating and heat exchange in the pusher furnace chamber In order to perform computer simulation of the effect of heating rate, capacity, and technology on heat consumption and steel loss, a mathematical model for charge heating and heat exchange in the chamber of a pusher furnace as developed. For the numerical computation of charge heating, the elementary balances method as used, and for the computation of the temperature field in the furnace chamber the brightness and configuration ratios method. The loss of steel as determined by the substitute temperature method, hile the heat consumption as computed using the zone balances method. The numerical computation procedure as carried out in the environment of a CAS (Computer Algebra System type application [8]. 3. The object of modelling The object of modelling is a pusher heating furnace. It as assumed that the heating chamber of the furnace as represented by a rectangular prism ith a length of L = 28 m, a height of 2H = 2.6 m, and a idth of B = 5.6 m. Also, the furnace as conventionally divided into 5 identical technological zones. For computation purposes, the pusher furnace as reduced to a simple model, in hich the charge moves along the furnace chamber over the length L ith uniform motion countercurrently to the direction of furnace gas movement. It as assumed that the transfer of heat to the charge takes place bilaterally over the hole length L. In vie of the assumed symmetry of phenomena, only heat exchange in the zones above the charge axis is considered. The furnace as assumed to be furnished ith a recuperator, hich used the enthalpy of furnace off-gas to heat the combustion air. For each of the assumed technologies, a furnace chamber temperature field as determined, hich ould assure the assumed heating parameters, i.e.: the end surface temperature t p = 1250 C, and the end temperature difference in the charge cross-section t k = 50 K. 4. Modelling of charge heating and steel loss The numerical procedure of charge heating as done using the elementary balances method [9]. The computation programme as carried out for the folloing inputs: heat flo direction normal to the slab surface (a one-dimensional problem, as for slab ith infinite dimensions, the thermophysical parameters of the slab vary ith temperature, a phase change specified for a given temperature range occurs ithin the charge, oxidation process occurs on the charge surface, the scale layer forming affects the conditions of heat transfer to the charge surface, the initial temperature distribution ithin the charge is knon, the computation ending condition reaching the assumed charge surface temperature and attaining the required final temperature difference on the charge cross-section. Numerical computation involve the need for the discretization of the region. The charge cross-sectional dimension 2s as divided into 11 identical differential elements (Fig. 1. The location of nodes representing individual differential elements as assumed in the geometric centres of gravity of those elements. The heat capacities and capacities of internal sources are concentrated in the differential element nodes, hereas the nodes representing charge surfaces are capacity-less. The resistances of flo are concentrated in the segments beteen nodes. In vie of the symmetry of the phenomenon, only half of the heated charge is considered. In the heating period, the temperature of the node situated on the charge surface as determined from the boundary condition in the folloing form: T 0 = T p + M τ G. (1 The value of the exponent G defines the degree of deviation from the linear temperature increase in the heating period. The variable value of the exponent G, under actual conditions, results in a change of heating technology. The value of the heating rate (M defines the furnace capacity (. The boundary condition for the soaking period assumes the form of: T 0 = T p. (2 The temperatures of the remaining nodes ere determined from the so called explicit differential scheme. The temperature of the sub-surface node after the successive time interval as determined from the relationship:

4 650 Fig. 1. Division of charge into differential elements

5 651 T 1k+1 = T 1k + τ c 1k ρ s 2 [( λ 0k + λ 1k ( T0k T 1k + + λ 2 k +λ 1k 2 (T ] 2k T 1k. (3 The temperature of an arbitrary internal node i as determined from the formula: T ik+1 = T 1k + [ τ λi 1k +λ ik c ik ρ s 2 2 (T i 1k T 1k + + λ i+1 k +λ ik 2 (T ] i+1k T ik. (4 The temperature of the node lying in the charge axis as determined from the equation: τ T nk+1 = T nk + (λ ( c nk ρ s 2 n 1k + λ nk Tn 1k T nk. (5 During heating of steel in the temperature range of C [10], there occurs the allotropic change Fe α Fe γ. The heat of transition is alloed for in the expression defining the specific heat capacity of the element: here: c prz. = c i = c pi + c prz., (6 q prz. T prz. 1 T prz. 2. (7 The value of the heat of transition equals q prz = J/kg [2]. In the numerical computation procedure, a constant time interval of τ=1 s as assumed. It loss of steel in every compartment of time as marked as according to dependence [11, 12]: z N k+1 = zn k + zn k+1, (8 z k+1 = ( z N k + zn k+1 B. (9 The increment in steel loss in time τ as determined from the formulas: z N = A τ α C exp ( DTz, (10 z = [ ( A τ α C exp D ] B. (11 T z The substitute temperature for the given time interval τ as determined from the relationship: ( T z = T p + M τ 1 G 2 τ. (12 For the heating stage: T z = T p. (13 The scale layer surface temperature as determined from equation [12]: T zg. = T 0 + q m δzg. λ zg.. (14 The scale layer thickness as determined from relationship [13]: δ zg. = z ρ zg. X Fe. (15 The effect of temperature on the scale thermal conductivity value as considered in the computation [13]: λ zg. = 1, 5 + 1, (t zg (16 The main objective of the charge heating computation procedure is to determine the total heating time, as ell as to determine the folloing for particular computational zones: charge surface temperature, temperature distribution in the charge cross-section, density of the heat flux floing to the charge, steel loss (amount of scale, and scale layer surface temperature. V pal. 5. Modelling of heat consumption The basis for the determination of the fuel flux fed to the computational zone is the thermal energy balance. The thermal energy balance equation for the computational furnace zone has the folloing form: Qd Qd Qod. Qod. j + stre f. j = j + stre f. j. (17 The total heat flux Q d supplied directly to the zone j consists of: the heat flux released in the gas combustion process, and the heat flux released in the charge oxidation process. The total heat flux Q od. supplied directly to the zone j is composed of: the heat flux penetrating though the alls, the heat flux carried aay ith cooling ater, the heat flux carried out by radiation through openings, and other losses.

6 652 The total heat flux Q d stre f. supplied directly to the zone j from the adjacent zone j +1 is composed of: the heat flux carried in ith furnace gas, the heat flux carried in ith the charge, and the heat flux carried in ith the scale. The total heat flux Q od. stre f. supplied directly to the zone j from the adjacent zone j 1 is composed of: the heat flux carried out ith furnace gas, the heat flux carried out ith the charge, and the heat flux carried out ith the scale. The fuel (natural gas volume flux for an arbitrary computational zone j as determined from the relationship: V pal. j+1 V (c p j t g j c p j+1 t g j+1 + Quż. j + Q str. j Q ch zg. j V pal. j =. W d V c p j t g j (18 The heat flux fed to the computational furnace zone j as determined from the relationship: Q j = V pal. j W d. (19 The unit heat consumption index, ithout alloing for heat recovery, as calculated from the equation: n=25 Q j j=1 q 0 =. (20 It as assumed that in the event of a heat sur- < 0 occurring, it ould be used in other plus ( Q j computational zones ( ideal furnace. By appropriately coupling the computational zone energy balance ith the chamber temperature computation results, a mathematical model of furnace operation could be developed. The useful heat flux for the j-th computational zone as determined from the formula: Q f zg. j = 1 2 (a zg. j c zg. j t zg. j a zg. j 1 c zg. j 1 t zg. j 1. (23 The heat flux carried aay ith sliding rail cooling ater as computed from relationships given in ork [14], hose general form is described by the equation: Q chł. j = F szyn j a ( T piec j 100 n, (24 here the furnace zone temperature as determined approximately from the relationship: T piec = T g + Tść. (25 2 The heat flux lost by the furnace lining as computed from the equation: Q ze. j = F ść. j (tść. j t ot.. (26 R The heat flux lost by radiation through the open furnace doors and openings as determined from the relationship: Q pr. j = C o ϕ j ψ j F ot. j ( T 4 pr. j. ( The opening radiation temperature as determined from relationship (25. In computations, also a constant value of other losses as assumed, and the heat input from the exothermic metal oxidation reaction as alloed for: Q ch zg. j = 1 2 q zg. (a zg. j a zg. j 1. (28 With respect to the hole furnace chamber, the balance items ere determined according to the folloing formulas for unit heat: q u. = ( 1 a zg.n+1 cmn+1 t mn+1, (29 Q uż. j = 1 2 [( 1 a zg. j cm j t m j ( 1 a zg. j 1 cm j 1 t m j 1 ]. (21 The heat loss flux as determined from the relationship: Q str. j = Q f zg. j + Q chł. j + Q ze. j + Q pr. j + Q in. j. (22 The heat flux carried out ith scale as computed from the equation: q f zg. = a zg.n+1 c zg.n+1 t zg.n+1, (30 q chł. = 2 q ze. = 2 q pr. = 2 n=25 Q chł. j j=1 n=25 Q ze. j j=1 n=25 Q pr. j j=1, (31, (32, (33

7 653 q ch zg. = Q zg. a zg.n+1, (34 q in. = 2 n=25 Q in. j j=1. (35 Heat losses ith furnace off-gas ere determined from the formula: q sp. = q 0 q uż. q chł. q ze. q pr. q f zg. q in. +q ch zg.. (36 It as assumed (ith some simplification that part of the heat contained in the furnace gas is used in the recuperator for heating up the combustion air: q po. = η rek. q sp.. (37 The value of the furnace gas aste heat recovery factor (η rek. =0.35 as determined from the author s industrial tests. The heat consumption (after making alloance for the heat recovery process as computed from the relationship: Fig. 2. Effect of technology and furnace capacity on heat consumption By analyzing the obtained computation results it can be concluded that the furnace thermal operation indices and steel loss are correlated. For the technology ith a linear charge surface temperature increase (T1, the loest heat consumption and steel loss values and the highest technological process efficiency values ere obtained. Any deviation in surface temperature from the linear increase (T2, T3 results in an increase in heat consumption and steel loss and a reduction in technological process efficiency. q r = q 0 q po.. (38 The efficiency of the technological process for the furnace equipped ith a recuperator as computed from the relationship: η t r = Q uż. q r. 100%. (39 6. Numerical computation results and their analysis The effect of technology and heating rate on heat consumption and steel loss is illustrated in Fig. 2 and Fig. 3. The presented modelling results apply to the ideal furnace ith an infinite heating rate, and equipped ith a recuperator. Fig. 3. Effect of technology and furnace capacity on steel loss From the computation results it can be found that the heat consumption and steel loss is determined by heating technology, and, for a given technology, by furnace capacity. In industrial practice, the heating process intensity is defined by furnace capacity, hich is determined from the relationship:

8 654 For pusher furnaces: = m τ n. (40 m = 2s l L ρ n. (41 In order to determine the correlation beteen heat consumption and steel loss, computations ere made, depending on furnace capacity. For each set technology and capacity, the temperature increase rate as determined using the charge heating computation program in the heating period, for hich heat consumption and steel loss as determined. It as established that the developed furnace model had a certain approximate capacity value (about 80 t/h, above hich charge heating could be non-rational, and sometimes practically impossible, due to excessive thermal loads occurring in particular computational zones. The results of computation of the effect of capacity on heat consumption and steel loss ere subjected to mathematical analysis. For the analysis of the effect of capacity on heat consumption, the folloing functions ere employed: q = a 1 exp (b 1 + c 1 exp (d 1, (42 q = a 1 + b 1 ln (c 1, (43 q = a 1 + b 1 exp (c 1, (44 q = ( a 1 + b c1 1, (45 q = a 1 + b 1. (46 The analysis of the effect of capacity on steel loss as performed using functions identical to those for heat consumption: z = a 2 exp (b 2 + c 2 exp (d 2, (47 z = a 2 + b 2 ln (c 2, (48 z = a 2 + b 2 exp (c 2, (49 z = ( a 2 + b c2 2, (50 z = a 2 + b 2. (51 The effect of capacity on steel loss as additionally described ith the relationship: ( a2 b2 z =. (52 Using the above formula, simple equations describing the correlation beteen heat consumption and steel loss could be obtained. Based on selected equations, functions ere developed, hich ere used for the analysis of correlation beteen heat consumption and steel loss. From Equations (44 and (49, a relationship in the folloing form can be derived: q = a + b exp [ c ln (d z f ]. (53 From functions (43 and (52, the folloing equation can be obtained: ( d q = a z b + c ln. (54 z f Relationships (44 and (52 yield a function in the form of: ( c q = a + b exp. (55 z d From Equations (46 and (52, the folloing relationship can be constructed: q = a + b z c. (56 Relationships (46 and (51 yield a linear function, as belo: q = a + b z. (57 The values of the coefficients in Equations (53 (57 can be determined either analytically (indirectly or by the direct approximation method. For the determination of the approximation coefficients, internal procedures of the OriginPro R 7.5 program ere used. The accuracy of mathematical description, for each of the equations, as expressed by the correlation coefficient, R, and by the mean relative approximation error. A detailed description of the correlation beteen heat consumption and steel loss is provided in ork [1]. From the approximations q = f (, G and z = f (, G, a relationship of to variables as obtained, hose general form is expressed by formula (53. The coefficients of this relationship are described by formulas [15] belo: a = 652, , 72 G, (58 b = 2689, , 37 G, (59 c = 0, 0291 G2 0, 0928 G 0, , 0993 G, (60

9 655 d = 1 19, , 90 G, (61 3, 31 0, 191 G f = 19, 96 9, 90 G. (62 The mean relative difference beteen the results obtained from the derived relationship and the numerical computation results is 0.20%. The correlation beteen heat consumption and steel loss is represented in Fig. 4. Analytical computations for graphic analysis ere carried out using the derived relationship of to variables. The computer simulation has also demonstrated that it is possible to model the heat consumption via the steel loss. The established relationship q = f (z, G enables the heat consumption to be determined through the steel loss for arbitrary heating technology. The obtained relationships have also been reflected in the results of laboratory tests and industrial trials. The industrial trial results have shon that for any furnace there is a close correlation beteen heat consumption and steel loss. Using this correlation, it is possible to predict heat consumption and other furnace thermal operation indices, hile having the value of steel loss for scale available. Thus, model studies on the effect of heating technology and furnace capacity on heat consumption become possible. REFERENCES Fig. 4. Correlation beteen heat consumption and steel loss for the model furnace 7. Summary From the numerical computation results and their mathematical analysis it can be found that heat consumption and steel loss are interdependent. Technology and heating intensity determine the heat consumption and steel loss, oing to hich a close correlation occurs beteen these indices. For any furnace, a rational temperature distribution in the chamber can be determined, hich, for the assumed capacity, ill assure the loest values of heat consumption and steel loss. Heat consumption and steel loss decrease ith increasing heating intensity (capacity. Lo values of heat consumption and steel loss can be assured by the appropriate selection of technology and running the process at the rational capacity. The most favourable values of these indices can be obtained by using to-stage heating ith a linear temperature increase in the heating period. The implementation of such technology ill require the assurance of the appropriate poer distribution in technological zones, as ell as modifications to the furnace design itself. [1] Ł. P i e c h o i c z, Zużycie ciepła a straty stali procesie nagrzeania sadu, Praca doktorska, Politechnika Częstochoska, Częstochoa [2] M. K i e l o c h, Energooszczędne i małozorzelinoe nagrzeanie sadu staloego, Wyd. WIP- MiFS Pol. Częstochoskiej, Częstochoa [3] Ł. P i e c h o i c z, M. K i e l o c h, J. B o r y c a, Optymalizacja kosztó nagrzeania stali poprzez zmianę szybkości podgrzeania, Noe Technologie i Osiągnięcia Metalurgii i Inżynierii Materiałoej, Częstochoa [4] Ł. P i e c h o i c z, M. K i e l o c h, J. B o - r y c a, Wykorzystanie modeloania do optymalizacji kosztó nagrzeania, XIV Międzynarodoa Konferencja Naukoo Techniczna, Produkcja i Zarządzanie Hutnictie, Częstochoa [5] M. K i e l o c h, T. W y l e c i a ł, Wpły eksploatacji piecó grzeczych na stratę stali i zużycie ciepła, Ogólnopolska Konferencja Naukoo-Techniczna, 63 Częstochoa Poraj [6] M. K i e l o c h, P. K a m i ń s k i, R. S t ę p i ń s - k i, Energooszczędne i małozorzelinoe technologie nagrzeania, Ogólnopolska Konferencja Naukoo-Techniczna, Częstochoa Poraj [7] M. K i e l o c h, Energooszczędne nagrzenie sadu piecach pracujących ze zmniejszoną ydajnością, Ogólnopolska Konferencja Naukoo-Techniczna, Częstochoa Poraj [8] W. P a l e c z a k, Metody analizy danych na przykładach, Wyd. Pol. Częstochoskiej, Częstochoa [9] R. B i a ł e c k i, A. F i c i inni., Modeloanie numeryczne pól temperatury, WNT, Warszaa [10] M. K i e l o c h, S. W y c z ó ł k o s k i, Modeloanie numeryczne pola temperatury piecu przelotoym z uzględnieniem założonej tech-

10 656 nologii nagrzeania, VII Ogólnopolska Konferencja Naukoo-Techniczna, Częstochoa Poraj [11] M. K i e l o c h, Ł. P i e c h o i c z, J. B o r y c a, Correlation beteen heat consumption and steel loss for scale in the to-stage heating process, Archives of Metallurgy and Materials 52, (2007. [12] Ł. P i e c h o i c z, M. K i e l o c h, J. B o r y c a, Model numeryczny nagrzeania sadu z uzględnieniem postaania zgorzeliny, Noe technologie i osiągnięcia metalurgii i inżynierii materiałoej, Częstochoa [13] M. K i e l o c h, Technologia i zasady obliczeń nagrzeania sadu, Wyd. Pol. Częstochoskiej, Częstochoa [14] T. S e n k a r a, Obliczenia cieplne piecó grzeczych hutnictie, Wyd. Śląsk, Katoice [15] Ł. P i e c h o i c z, Współzależność pomiędzy zużyciem ciepła a stratą stali procesie nagrzeania, Noe technologie i osiągnięcia metalurgii i inżynierii materiałoej, Częstochoa Received: 20 January 2010.