Fluid Flow and Mixing Characteristics in a Gas-stirred Molten Metal Bath* By Masamichi SANO* * and Kazumi MORI* Synopsis Fluid flow and mixing characteristics in a molten metal bath are analyzed for inert gas injection through a nozzle at the center of the vessel bottom. It is postulated that the bath consists of two zones; bubble plume zone where gas-liquid mixtures flow upward and annular zone where liquid flows downward. The analysis is made by setting up a steady-state energy balance for the liquid phase. The liquid velocity in the plume zone, the liquid circulating flow rate and the mixing time are calculated for various injecting conditions and correlated as simple functions of gas flow rate, liquid depth and cross-sectional areas of both the plume zone and the vessel. It is found that the cross-sectional area of the plume zone has a significant influence on the circulating flow. Large cross-sectional area of the plume zone is favorable for mixing in the bath. The calculated results of circulating flow rate and mixing time agree with the experimental results obtained previously. I. Introduction Gas injection into molten steel is widely used in steelmaking processes. Fluid-flow phenomena and mixing in the processes have profound influences on the steelmaking reaction rate. Several experimental and theoretical studies have been made on circulating flow and mixing time in the molten steel bath.l-3~ Nakanishi et al.'> obtained a quantitative correlation between mixing time and stirring power of gas. Szekely et a1.2~ interpreted their experimental measurements for tracer dispersion and dissolution patterns of immersed graphite rods in argon-stirred ladles by using a two equation model of turbulent circulating flow. Recently Hsiao et a1.3~ determined the flow velocity of molten steel in argon-stirred ladles by measuring the drag force exerted on a probe immersed in molten steel. In the chemical engineering field, many models of circulating flow in bubble columns are proposed on the basis of momentum balance4-7) and energy balance.8-10> From the models one can calculate the flow velocity and gas holdup. The present study is concerned with analyzing circulating flow and mixing time in a gas-stirred molten metal bath. The analysis is made by setting up a steady-state energy balance for the liquid phase. The liquid velocity in the bubble plume zone and the mixing time are calculated under various injecting conditions. Effects of various factors influencing the fluid-flow phenomena and the mixing time are investigated. II. Theory 1. Work Done by Injected Gas and Effective Stirring Power of Gas Work done by the gas injected into a molten metal is : 0 the expansion work due to pressure and temperature changes near the tuyere, 0 the transfer of kinetic energy from the gas to the liquid near the tuyere, and the work done by the gas bubbles during their rise through the melt. Most of the work 0 and 0 is considered to be done near the tuyere and makes only a little contribution to the liquid circulation. Assuming that the fraction of the work Q and 0 effectively used for the circulation is 7), one obtains the effective stirring power of gas for a molar gas flow rate n.11> ~ = nr TL In p1 P + Tn In Pn + 1- Tn 2 ~L P1 ~L + Tn 1 P n T L PGnl4...(1)*** where, R : gas constant P2: atmospheric pressure PGn, ugn : gas density, velocity at the tuyere exit, respectively. The coefficient i in Eq. (1) is not known for gas injection into molten metals. For the kinetic energy of gas injected into water, 0.06 is given to From this, it may be assumed that is small in the present case. In the following theoretical calculation of the liquid circulation and mixing time, only the first term of the right side of Eq. (1) is taken into account. The static pressure at the nozzle exit P1 and the molar flow rate n are expressed as where, P1= P2+PLgH0...(2) n = VGMPM/R L...(3) PL : liquid density g: acceleration of gravity H0: bath depth VGM : gas flow rate at liquid temperature TL and at logarithmic mean pressure PM. PM = (P1-P2)/ln (P1/P2)...(4) Substituting Eqs. (2), (3) and (4) into Eq. (1) gives Presented to the 101st ISIJ Meeting, April 1981, at The University of Tokyo in Tokyo. Manuscript received April 8, 1982. 1983 ISIJ Department of Iron and Steel Engineering, Faculty of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464. * If the nominal gas velocity at the nozzle exit exceeds the sonic velocity and the gas pressure P n at the exit is higher than the static pressure P1, the gas gives work on the surroundings during pressure decrease from Pn to P1. Here, it is assumed that the pressure decreases from Pn to P1 at a constant temperature Tn and then the temperature rises from Tn to the liquid temperature TL at the constant pressure P1. (169 )
3 (170) Transactions ISII, Vol. 23, 1983 E = VGMPLgHO...(5)* 2. Liquid Circulation Model In industrial injection processes, injected gas distributes over a small portion of total cross-sectional area of a vessel. Such maldistribution of gas gives rise to gross circulation of liquid. The present liquid circulation model is shown in Fig. 1. Gas is injected at the center of the vessel bottom. Bubbles in the central core rise in a stream of concurrent liquid. This core is called " bubble plume zone ".** Outside the core liquid flows down. This downflow zone is called " annular zone ". The diameters of the vessel and the plume zone are represented by D and d, and their cross-sectional areas by A and Ap. The two areas are assumed not to change with the vertical distance upward from the vessel bottom. In the following description, variables of the plume and annular zones are distinguished by subscripts P and A, respectively. 3. Energy Balance Equation The circulating flow produces turbulence in the liquid to dissipate the input energy. At a steady state, the rate of energy dissipation due to the liquid circulation is equal to the rate of input energy, that is the stirring power of gas. Bhavaraju et al.10~ assumed that the rate of energy dissipation is equal to the difference between the rate of kinetic energy associated with the upward flowing liquid in the plume zone and that associated with the downward flowing liquid in the annular zone. The present analysis of the circulating flow in a molten steel bath 1s based on the assumption of Bhavaraju et al.10~ The effects of gas holdup Op and energy dissipation due to bubble slip in the plume zone on the circulating flow are additionally taken into account. Hence, the energy balance equation is given by E = c+ Bs...(6) where Ec is the rate of energy dissipation associated with the liquid circulation, L L _ 'c - PL(1 - Y'p) ~d2 4 ulp' 1 ulp ir(d2-d2) 1 2 - PL 4 ula 2 ula... (7) and EBS is the rate of energy dissipation due to bubble slip.l3) us EBS = 8...(8) us+ulp The slip velocity us in Eq. (8) is related to the gas (bubble) and liquid velocities, ugp and ulp. These velocities are in turn related to the superficial gas and liquid velocities in the plume zone, Up and UL P 14 UGP ULP us=ugp-ulp=, 9 p 1--... ) The slip velocity us has been shown to be independent of the flow rates of the two phases and to depend mainly on the gas holdup Op and the rising velocity of a single bubble ubo. Here, the empirical equation proposed by Davidson and Harrison,i5~ and Griffith and Wallis16~ for large bubbles dispersed in liquids is adopted for describing us. U8-1... --~Ip (10) The following equation is used to calculate UBO,17) ubo = %/0.5dBg...(11)*** where, db : the diameter of dispersed bubbles. By substituting Eqs. (7) and (8) into Eq. (6), the energy balance equation is transformed as follows : (1 -~P) 1Id2 7C(D2-d2) Fig. 1. Liquid circulation model. * If the gas flow rate at 0 C and 1 atm, VG, is used, Eq. (5) becomes
Transactions ISIJ, Vol. 23, 1983 (171) The liquid velocity in the annular zone ula is related to that in the plume zone ulp by the following equation. ULA l K2 1 (p)ul p 1 where, K2= (d/d)2. Substituting Eqs. (5) and (14) into Eq. (13) gives the following equation after some rearrangement: ulp(ulp+us) - A J7( 1(2 2 VGMgH. (15) F(Op, K2) = (1-Op) K2 )2(l _~p)2...(l6 1_ ) Ap = ~rd2/4...(17) In gas injection processes, the cross-sectional area of the bubble plume zone is small as compared with that of the vessel, and the liquid velocity in the plume zone is large. It is assumed that the liquid in the plume zone is well mixed and the mixing time is determined by mixing in the annular downstream zone. The circulation time t~ in the downstream zone is t - ~r(d2-vd c 4 L 2)H =4L(1-PK2) r(d2-d~2)ho...)( 24 where, H: bath depth with gas injection. Substituting Eq. (24) into Eq, (23) gives t 3 ~r(d2-d2)h0 m = 4 VL(1-~PK2)...(25) If the diameter of the plume zone d is known, one can solve Eqs. (9) and (15) as simultaneous equations of ulp and ~isp. In case of ulp»us, that is if the energy dissipation rate due to bubble slip is small, ulp can be expressed as 2 VGMgHO 1/3 ulp = A j...(18) pf(c5p, K2) and the circulation flow rate of liquid is given by V P 2 VL _(1-Op)APULP = _~) 2GMgHA ]1/3...(19) On the other hand, in case of ulp «us, that is if the energy dissipation rate due to bubble slip is large, ulp and V L can be written as ulp = [ 2 VGMgH 1/2 u8apf(~ p, K2) J...(20) 2 VGMgHOAP 11/2 L ( 0P)[ u 8F(~p, K2) j (21) 4. Relation between Mixing Time and Stirring Power of Gas The mixing time may be determined by measuring the difference in tracer concentration between two widely spaced positions in the vessel, one of which is near the point of tracer injection. The time required to reduce a suddenly created concentration difference between the two positions to less than i % of the average concentration is given by'9, t,, i = t~ 1n 100...(22)* 2 where, t~ : circulation time (= V / TL, VL : liquid volume). The mixing time tm is usually defined on the basis of 5 % difference criterion.3'20~ Following this definition, one obtains tm = tm5 =, 3tc...(23). VL In order to clarify the dependence of the mixing time on the injecting conditions, Eq. (25) is transformed by using Eq. (19) obtained for the limiting case of ulp»us.* If the stirring power of injected gas per unit liquid volume ~V is used, is given by V 2Ho~VK4 1/3 L = (1-Op)A PLF(op, K2)...(27) where, A='rD214. Combining Eqs. (25) and (27) leads to 1 -K2 [ K2F(cP, K2) 11/3 t m - 3 K2(l -~p)(l -~pk2) [ 2 j HoPt 1/3 If the stirring power per unit liquid mass EM is used, t 1-K2 [ K2F(,bp, K2) 11/3 m - 3 K2(1-Ybp)(1 -Y5 K2) [ 2 x H2"3...(29) j 3M Previously, on the basis of the assumption that the stirring power effectively used to circulate the liquid is proportional to ~, the following correlations are obtained 11) tm oc LPL )1/3 / sy oc {(PLVL)1 31~v}1l3,...(30) cc (L2\'3 cc ML 2/3 sm 1/3... (31) EM PL where, L : characteristic length of the vessel VL, ML : liquid volume and mass, respectively. From comparison of Eq. (28) with Eq. (30) and Eq. (29) with Eq. (31), it is seen that if Ho is taken as L, Eq. (28) becomes of the same form as Eq. (30) and Eq. (29) as Eq, (31). * Equation (22) is obtained for the decrease of tracer concentration with time in a semi batch reactor in which the contents are well mixed. Applicability of Eq. (22) to the present circulation process will be shown later. ** As shown later, for argon-stirred ladles ulp is larger than US.
(172) Transactions ISI7, Vol. 23, 1983 III. Calculated Results and Discussion 1. Liquid Velocity and Gas Holdup in the Bubble Plume Zone Equations (9) and (15) are solved as simultaneous equations of the liquid velocity ulp and the gas holdup Yb in the plume zone. Figures 2 and 3 show the calculated results for an argon-stirred ladle of 2.2 m diameter and 1.5 m bath depth. The melt weight is 40 ton. The gas-flow rate VG and the diameter of the plume zone d are varied from 0.001 to 1 Nm3/min and from 0.1 to 1 m, respectively. It is seen from Fig. 2 that as the diameter of the plume zone is increased, the liquid velocity ulp increases. In the range of d larger than 0.45 m, the logarithmic plot of up against VG shows an almost linear relationship, the slope of which is about 1/3. This value coincides with that of the exponent in Eq. (18) which is derived by neglecting the energy dissipation due to bubble slip. Calculations show that if d is larger than 0.45 m, the ratio of the energy dissipation rate due to bubble slip to the total energy input rate is nearly constant (=0.2N0.3), being almost independent of the gas-flow rate. This may be reflected in Fig. 2 showing the slope of if 3 for d>0.45 m. In the range of high gas-flow rates and small diameters of the plume zone (d<0.2 m), a linear relationship between ulp and VG on the logarithmic graph is not obtained. Since the superficial gas velocity UGp in this range is very high, the gas holdup Y'P becomes very large. In this case, F(~i5p, K2) given by Eq. (16) decreases considerably with increasing gas-flow rate. This is the reason why ulp in the range tends to be larger than the value expected from the linear relationship for the smaller gas flow rates on the logarithmic graph of Fig. 2. Figure 3 shows the relation between the gas holdup in the plume zone Yb and the superficial gas velocity UGP(=VGM/AP). The solid line in the figure represents the calculated result for molten steel. Experimental data previously obtained for mercury21 are also shown in Fig. 3. It is seen that the gas holdup in mercury is much larger than that in molten steel. The difference in the gas holdup between mercury and molten steel is partly due to the fact that the size of bubbles and the bubble rising velocity in molten steel are larger than those in mercury. However, the difference mainly comes from the different vessel sizes and hence from the different flow patterns in the liquids. In the measurement of gas holdup in mercury21> the vessel diameterr was small, so that bubbles were distributed almost uniformly throughout the vessel. Therefore, it is presumed that circulating flow in the vessel was not formed. On the other hand, in the present calculation for molten steel bubbles are assumed to be locally distributed as shown in Fig. 1. In this case, the gas velocity in the plume zone ugp(=us+ulp) becomes large. This large gas velocity in the plume zone leads to the small gas holdup in molten steel. 2. Circulating Flow Rate of Liquid On the basis of the calculation of liquid velocity in the plume zone in Section III. 1, the circulating flow rate of liquid VL is calculated from the following equation: VL = /(l- 7"p) ~cd2 4 ulp...(32) ` Fig. 2. Relation between and gas-flow rate. liquid velocity in bubble plume The calculated result is shown in Fig. 4. It is noted that VL increases with increasing diameter of plume zone d and with increasing gas flow rate VG in the case of large d (>0.3 m). When d is smaller than 0.3 m, the increase of VL with VG declines in the range of large VG, and at still larger VG, VL begins to decrease in the cases of d=0.1 and 0.15 m. This tendency is also found in the relation between pumping rate and gas flow rate in an air lift pump.22~ The superficial gas velocity UGP at which the increase of VL with VG begins to decrease is about 1 1.5 m/s in Fig. 4 and 1.4 m/s for the air lift pump. Fig. 3. Relation between gas holdup in bubble plume and superficial gas velocity. Fig. 4. Relation between gas-flow rate. liq uid circulating flow rate and
Transactions ISIJ, Vol. 23, 1983 (173) The reason why VL decreases with increasing VG is that the rate of energy dissipation due to bubble. slip increases with increasing VG because of large gas holdup ~5p and hence large slip velocity us (see Eq. (10)). The phenomenon does not occur in an actual inert gas injection into a ladle, where the diameter of plume zone is much larger than 0.15 m. 3. Mixing Time The mixing time tm is calculated by substituting the circulating flow rate of liquid VL into Eq. (25). The result is shown in Fig. 5. In order to compare the present result with the experimental results obtained previously, the axis of abscissa is taken as ~M(ML/pL)-213 (Eq. (31)). The range within which the experimental results fall is bounded by the two dotted lines." As can be seen from Fig. 5, the mixing time tm is strongly dependent on the diameter of plume zone d, and decreases with increasing d. If ~M(ML/pL)-2/3 is smaller than 10, tm is in inverse proportion to ~M(ML/PL)-2' 3 roughly to the one-third power. In the range of large &M(ML/pL)-2"3, tm for d=0.1 and 0.15 m tends to increase with increasing ~M(ML/pL)-213 This is caused by the decrease in the circulating flow rate of liquid. From comparison between calculation and experiment in Fig. 5, it is seen that the calculated result for d=0.25-'o.5 m roughly agrees with the experimental results. The effects of furnace diameter, bath depth, and others on the mixing time will be mentioned in Section III. 5. 4. Approximated Correlating Equations for Liquid Velocity in Plume Zone, Circulating Flow Rate of Liquid and Mixing Time The calculated results obtained in Sections III. 1 to III. 3 are used to obtain approximated correlating equations for the liquid velocity in the plume zone. ulp, the circulating flow rate of liquid VL and the mixing time tm. Equation (15) shows that if cp and K2 are small and hence F(~bp, K2) is near 1, logarithmic plot of ulp against VGMgHO/Ap gives a straight line, the gradient of which is one third. The relation between ulp and VGMgHO/AP obtained by solving the simultaneous equations (Eqs. (9) and (15)) o f ulp and ~i5p is plotted in Fig. 6. The calculation is made for inert gas injection into molten steel at 1 600 C (ladle diameter D=2.2 m, bath depth Ho=1.5 m; D=1 m, Ho=1 m) and water (D=3 m, Ho=1 m). The diameter of plume zone is varied between 0.1 m and 1 m (0.8 m for D=1 m), and the gas flow rate VG between 0.001 Nm3/min and 1 Nm3/ min for molten steel and between 0.05 Nm3/min and 0.3 Nm3/min for water. The logarithmic plot of ulp against VG.MgHO/Ap in Fig. 6 shows that most of the calculated points lie on a straight line. The calculations which are considered to deviate appreciably from the linear relationship are shown by and A. In these cases, (bp or K2 is larger than about 0.3. Thus, it is deduced that if qip< N0.3 and K2< N 0.3, the relation between ulp and VGMgHO/A p may be expressed by the straight line in Fig. 6, independently of liquid physical properties. The straight line leads to the following equation ulp = 1.17( VGMgHo/Ap)0.346...(33) where each variable is expressed by using units of meter and second, and VGM is the gas flow rate at liquid temperature and at logarithmic mean pressure. In the range of gas injecting conditions under which the present calculations are made, ~b p is larger than 0.3 only at VG =1 Nm3/min and d<0.3 m. K2 is usually smaller than 0.3 in gas injection into a ladle. Thus, it is clear that applicability of Eq. (33) to the circulating flow in the ladle is wide. For the circulating flow rate of liquid VL and the mixing time tm, similar calculations are made. For ~bp < N 0.15 and K2< '-0.30, V L is expressed as VL = 1.17 (VGMgHoAP)0.339... (34) For cp < 0.15 and K2 < 0. 10, tm is given by tin = 26.2[(Ho/K2)2/~M]0.337...(35) The unit of the stirring power of gas sm is w/ton, and the units of the other variables are the same as those in Eq. (33). Fig. 5. Relation between mixing time and EM(ML/pL)-2/3. Fig. 6. Relation between liquid velocity in bubble plume and VGM$HO/Ap.
0 (174 ) Transactions ISIJ, Vol. 23, 1983 5. Comparison between Calculation and Experiment 1. Circulating Flow Rate of Liquid Hsiao et a1.3) measured the liquid velocity at the center of the plume zone in water, um. The vessel size used was 1 m in diameter and 1.1 m in height. The value of um is substituted into Eq. (36) to obtain ulp. ulp =Urn exp (--r2/0.01(h +0.8)2)...(36)3) where, r: radial distance from the center of the plume zone. The circulating flow rate of liquid is given by Eq. (37). The result is plotted in Fig. 7. VL = S2ruLPdr ~r...(37)3) The circulating flow rate calculated from Eq. (34) is plotted in Fig. 7. The cross-sectional area of the plume zone AP in Eq. (34) is given by AP = '~ (0.37H )2...(38)* 4 Figure 7 shows that the present calculated result roughly agrees with that obtained by Hsiao et a1.3) 2. Mixing Time Many investigators expressed their experimental data for mixing time as a function of stirring power of gas per unit mass of liquid EM(w/ton).1,3,12,20) The present analysis shows that the mixing time is also dependent on the dimensions of the vessel and the bubble plume zone. If the diameter of bubble plume zone given by Eq, (40) is substituted into Eq. (35), the mixing time is expressed as tm -100 {(D2/H )2/EM}0.337...(41) In Fig. 8, the mixing time calculated from Eq. (41) is compared with the experimental results.3,12,20) The diameter and height of the bath in the experiments of Hsiao et al, are taken as D=2.44 m and H0= 1.35 m for the 60 ton ladle,3) and D=1 m and H =1. l m for the 6 ton ladle.24) Except for the results of Lehrerl2) and Haida et al.20) for the experiment with model slag on the water surface, good agreement between the experiments and the calculation is seen in the figure. In the present paper, only the case of gas injection at the bottom center has been treated. Concerning the effects of tuyere position and presence of slag on the melt surface, future work is needed. Iv. Conclusion A mathematical model has been presented to describe circulating flow in a molten metal bath with inert gas injection. The model is based principally on the energy balance for the liquid phase. In the model the effects of gas holdup in the bubble plume zone and energy dissipation due to bubble slip have been taken into account. The liquid velocity ulp and the gas holdup Y'P in the plume zone, the liquid circulating flow rate VL and the mixing time tm have been calculated for various injecting conditions and correlated as simple functions of gas flow rate, liquid depth and cross-sectional areas of both the plume zone and the vessel. It has been shown that ulp and VL are proportional to and tm is inversely proportional to one-third power of gas flow rate or stirring power of gas. The cross-sectional area of the plume zone has a significant influence on the circulating flow. Large cross-sectional area of the plume zone is favorable for mixing in the bath. The calculated results of circulating flow rate and mixing time agree with the experimental results obtained previously. Fig. 7. Comparison of liquid circulating flow rate between experiment and present theory. Fig. 8. Relation between mixing time and sm (D2/H )-2. * Themelis et aḷ 23~ showed that the diameter of a jet injected into a liquid where, h: distance from the nozzle d : nozzle diameter lip: jet cone angle. For h» d, e~ = 20.5 deg and h = H, d=2h+ tan B~ 2 2 d = 0.371 was given by a e~ The diameter of the plume zone is assumed to bee q ual to that of the jet 'et cone at the bath surface, surface. (39 (40;
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