Liquid curtains—I. Fluid mechanics

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 3184, 1988. 0009 Chemical Engineering Science, Vol. 43, No. 12, pp. 3171 Printed in Great Britain. LIQUID 0 CURTAINS-I. FLUID 2509/88 $3.00+0.00 1988 Pergamon Press plc zyxwvut MECHANICS zyxwvutsrqponmlkjihgfedcba J. I. RAMOS Department of Mechanical Engineering, Carnegie-Mellon (Receiued University, Pittsburgh, PA 15213, U.S.A. 30 September 1987; accepted 12 May 1988) Abstract-A mathematical model of liquid curtains which accounts for gravity, surface tension, pressure differences and nozzle exit geometry is presented. Analytical solutions are obtained in the absence of friction and compared with the results of numerical calculations; differences of at most three percent in the convergence length, are obtained between the numerical and analytical results even for nozzle exit angles of thirty degrees. It is shown that the convergence length is a monotonically increasing function of the Froude number, initial thickness to initial radius ratio, pressure difference and nozzle exit angle. The convergence length increases as the Weber number is increased. It is also shown that very small pressure differences between the gas enclosed by and the gas surrounding the liquid curtain are required to dramatically increase the cdnvergence length. Pressure differences higher than a critical value which is related to the Froude and Weber numbers, are shown to result in flow divergence. INTRODUCTION The analysis presented in this paper has practical implications in the design of protection systems for laser fusion reactors (Hovingh, 1977) and in the design of cylindrical curtain chemical reactors (Roidt and Shapiro, 1985). Inertial confinement fusion (ICF) reactors produce pulsed fusion power by focusing lasers or charged particle beams at frequencies of about 1 Hz into a small pellet composed of deuterium and tritium. The rapid heating and compression of the pellet causes it to explode with an attendant release of high energy alpha particles, neutrons, electrons, X-rays and other ionic debris. These particles, radiation and shock waves would subject a reactor chamber, or first wall, to intolerable cyclic loads and stresses. To alleviate this problem, a thick, recirculating annular jet or “water fall” of liquid lithium that flows between the fuel pellets and the reactor chamber wall can be used. The liquid lithium jet serves three purposes. First, it acts as a shield for the first wall. Second, it has very good heat transfer characteristics. Third, it serves as the tritium fuel breeder. The geometry of the waterfall must be accurately known when designing the fusion reactor chamber. The total exposed area of the waterfall is important for determining the effectiveness of the lithium liquid curtain in condensing the vaporized lithium that is found after each fuel pellet explosion. Disturbances caused by external perturbations or liquid turbulent motion may cause the annular jet to break up and disperse before it reaches the bottom of the reactor. Under certain circumstances holes or “blinking eyes” can develop in the liquid curtain (Roidt and Shapiro, 1985). The diameter of these holes oscillates with time. Liquid curtains of relevance to ICF reactors have been studied by Hovingh (1977), Hoffman ef al. (1980) and Esser and Abdel-Khalik (1984). The formulations of Hovingh (1977) and Hoffman et al. (1980) are based on those of water bells (Boussinesq, 1869a, b; Taylor, 1959a, b). Taylor (1959a, b) and Lance and Perry (1953) wrote the momentum equations for a liquid curtain element along and normal to the streamlines and accounted for gravity, internal and external pressure, surface tension and air drag. Baird and Davidson (1962) used a similar formulation, neglected gravity, assumed a negligible film thickness and showed that when the Weber is larger than one, long thin sheets can be obtained. However, if the Weber number is smaller than one, a round jet is obtained. Dumbleton (1969) showed that gravity has an important effect on the stability and shape of water bells. The studies of Boussinesq (1869a, b), Taylor (1959a, b), Lance and Perry (1953), Baird and Davidson (1962) and Dumbleton (1969) are based on the solution of the momentum equations along and normal to the streamlines. In the analyses of Hovingh (1977) and Hoffman et al. (1980) these equations were projected onto a fixed coordinate system. Cylindrical chemical reactors (CCR) have considerable relevance for the direct reduction of zirconium from zirconium tetrachloride and sodium (Roidt and Shapiro, 1985), stack emission scrubbing for pollution control, reaction and control of toxic wastes, gas-liquid and liquid-liquid chemical reactions, scrubbing of radioactive and non-radioactive particuiates and soluble materials, and improved injection and reaction in advanced Kroll combined reduction-distillation furnaces. A cylindrical curtain can be used in the direct reduction of zirconium from zirconium tetrachloride and sodium as follows (Roidt and Shapiro, 1985). A curtain of liquid sodium can be formed through an annulus. Within the volume enclosed by the liquid curtain, an atmosphere of zirconium tetrachloride is maintained; sodium particles and argon are injected into this atmosphere. The sodium, zirconium and zirconium tetrachloride are collected in a pool of molten zirconium. An appreciable mass can be converted to zirconium during the spray-to-pool transit. Sodium particles with sufficient linear momentum can carry the reduced 3171 3172 J.I. RAMOS zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA zirconium on their surfaces into the pool where the sodium chloride and the excess sodium are vaporized. This CCR concept results in minimum sodium evaporation. By way of contrast, in the absence of the sodium curtain, the temperature of the smaller sodium particles may be raised to the vaporization level during the particle transit because the reduction reaction is highly exothermic, and sodium vapor escapes into the zirconium tetrachloride vapor. The resulting reactions throughout the gas would result in a poor collection efficiency due to the inability to recover the expensive material. A similar situation is encountered when homogeneous reduction reactors are used. The liquid sodium curtain prevents the escape of homogeneously formed zirconium atoms and increases the Fig. 1. Schematic of a liquid curtain. zirconium yield. Instabilities in annular liquid jets can result in axisymmetric oscillations and pinchoff effects which where p is the fluid density, V is the liquid curtain can yield liquid shells (Lee and Wang, 1986; Kendall, velocity in the streamline direction, 0 is the surface al., 1982). These instabilities can be 1986; Kendall et zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA used to form spherical shells (less than 1 mm in tension, R is the local radius of the curtain, b is the local thickness of the curtain, r, is a radius of curvadiameter) for ICF reactors. ture, g is the gravity acceleration, p. and pi are the In this paper we present a steady state analysis of pressures of the fluids which surround and are enliquid curtains which are relevant to both CCR and closed by the liquid curtain, respectively, f, and _& are ICFR concepts. The analysis is based on the theory of the friction forces exerted on the liquid curtain by the “water bells” but with the difference that the governing fluids which surround and are enclosed by the liquid equations are not solved along the streamlines but curtain, respectively, 8 is the angle that the tangent to projected onto the axial and radial directions as in the the liquid curtain forms with the z axis, and s is the papers of Hovingh (1977) and Hoffman et al. (1980). distance measured along the liquid curtain. These equations are solved numerically for a range of In deriving eqs (1) and (2) we have used the midline parameters which are relevant to both CCR and ICFR of the curtain and applied the surface tension, pressure concepts. The liquid curtain shape is also determined differences and gravity on that line. Therefore, we have analytically for negative and positive pressure differimplicitly assumed that the liquid curtain thickness is ences in terms of Airy’s and Bessel’s functions of the small compared with the Gaussian radius of curvafirst kind and fractional order. In addition, analytical ture. If the curtain thickness were not smaller than the and numerical solutions based on the full nonlinear radii of curvature, it would be necessary to account for equations but neglecting viscosity, i.e. Reynolds the velocity variations within the curtain thickness, number effects, are obtained as a function of the the boundary conditions on the liquid curtain surFroude number, liquid curtain initial thickness-tofaces, and the relaxation of the velocity profile as the initial radius ratio, convergence parameter, pressure liquid jet emerges from the nozzle, i.e. the relaxation differences and initial angle in order to assess the from the no-slip condition within the nozzle to stress effects of the nozzle geometry, flow, fluid and body boundary conditions on the liquid curtain surfaces. forces on the convergence length and liquid curtain Equations (1) and (2) can be projected onto the r and shape. z axes to obtain the following equations of motion PROBLEM FORMULATION pbg = -(fi+J,)sin@-cos8 20 cos 8 ~ R L We consider the liquid curtain shown schematically in Fig. 1. By establishing a balance of forces in the 20 tangential and normal directions to the mean stream+-+P.-Pi zyxwvutsrqponmlkjihgfedcbaZYXWV (3) TV line for an element As of curtain, the following equations are obtained (Baird and Davidson, 1962; d2z 2a cos 0 p bdt2 = pgb - (A +f,) cos 0 + sin 0 ___ Dumbleton, 1969; Hoffman et al., 1980) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA R [ V2 20cos B 2c 2a +2aRAs (4) +-+pps-Pi R r, X 2nRbAs = rv > 1 1 + (p, - p,)2zRAs Vz p2nRbAs + pg 2nRbAs sin B (1) = pg 2xRbA.s cos 8 - (A +f,)2nRAs (2) where zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM t is time, z is the axial location of an element of the liquid curtain, and r, is given by 1 - = -$$/[I TV +(gYlY . (5) 3173 Liquid curtaiv-I dz* Equations (3) and (4) have also been derived by =(r = 0) = Fr cos 0,, z*(O) = 0. (17) al. (1980) and Hovingh (1977) for pi - pe Hoffman et zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA =O, andA==f,=O. Along the liquid surface, boundary layers are The continuity equation can be written as formed. These boundary layers are not “conventional”, i.e. they are different from those along solid VbR=V,b,R,, “=[(!$)z+($)‘] (6) walls, since kinematic and stress boundary conditions must be satisfied at the gas-liquid interfaces. The where V,, b, and R, are the initial velocity, thickness kinematic condition states that the inner and outer and radius of the liquid curtain (Fig. 1). surfaces of the liquid curtain are material surfaces Introducing the following nondimensional varialong which the shear stresses are continuous and the ables difference in normal stresses is balanced by surface tension. However, friction effects are expected to be negligible. Since the dynamic viscosity of liquids is in general much larger than that of air (Thompson, 1972), the shear of the ambient fluids may influence the linear momentum of the liquid curtain, but it does not introduce strong variations of velocity across the into eqs (3)(6), the following system results cross-section of the liquid curtain. V*b*R* = OF,. ANALYTICAL (9) b*- d2R* ds2 d2z* __ = Fr dr2 = -C,sin0--kcos8 which can be integrated (10) b*- d2z* dr2 = Frb*-CrcosQ+isinB +; 1 SOLUTIONS In the absence of friction (C, = 0) and for long curtains, i.e. IdR*/dz*l < 1, eq. (11) can be written as dz* = Fr(z + cos e,), dt (18) subject to eq. (17) to yield For long curtains Ir:l B R*, (dR*/dz*) = 0, eqs (10) and (7f) can be expressed +C,sin8 (11) rv I !t!$ (19) z* = Frr(G.+cosBO). % 1, and Cr as v*=g.cm) _(&+“.)A$ where the Froude number (Fr), friction coefficient (C,), pressure coefficient (C,), convergence parameter (N) Substitution of eqs (8) (20b) and (19a) into eq. (20a) and Weber (We) number are defined as zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA yields V2 d2R* = (12) Fr = g, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA C, = (A +f.) 0 -= -;+ C,R* PR:g2 P 0 dr2 Fr2f+fe vf V2 which is to be solved with o _ Pe-pi c, = (Pe-8% zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ) provided by eq. (16). Fr2 pR,2g2 PC R, We=-. (13) 2a At the nozzle exit dR dt(t = 0) = V, sin 8,, R(t = 0) = R, (14) + > (7 + cos 0,) the initial conditions The solution of eq. (21) depends on the value of the pressure coefficient, C,, and is described in the next three subsections. However, if C, = - l/N and 0,, = 0, eqs (21) and (16) can be written as d2R* -= dr2 -;(l - R*);(T + l), 0 dR* x(O) R*(O) = $(t = 0) = v,cos e,, z(t=O) = 0 (1% where 0, is the angle that the annular jet makes with the z-axis at the nozzle exit Equations (14) and (15) can be written in nondimensional form as dR* dt (r = 0) = Fr sin B,, R*(O) = 1 (16) (21) 0 Equation (22) clearly indicates = 0, 1. (22) that S(O)=0 (23) for C, = -l/N, 8, = 0, idR*/dz*I + 1 and C, = 0. Furthermore, differentiation of eq. (22) with respect to r yields d”R*/dr” = 0 for all n > 2. (24) 3174 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA J.I. RAMOS This means that if R*(Az) is expanded in Taylor’s series around 7 = 0 in an analogous manner to what is done in partial differential equations for the Cauchy-Kowaleski theorem (Garabedian, 1964; Courant and Hilbert, 1962), it can easily be shown that R*(Ar) = R*(7) = 1 for T>O. > pi) and C, L* = Frz, condition can be ex- (28) % 1. c$+cosO,) (25) Thus, if c, = -l/N, B, = 0, c, = 0 and IdR*/dz*( + 1, eq. (22) implies that the liquid curtain never converges, i.e. an annular vertical jet is obtained. As the annular jet is continuously accelerated by gravity (eq. 18), its thickness decreases and instabilities may develop on the annular jet surface. The theory presented in this paper cannot, in its present form, deaI with spatial and/or temporal instabilities and the onset of turbulence. In order to study spatial and/or temporal instabilities, it is necessary to develop a theory which accounts for the spatial and temporal dependences of the flow variables. The analysis presented in the previous paragraphs clearly illustrates the existence of a critical value of the pressure difference beyond which the liquid curtain does not converge. We now turn our attention to the analytical solution of eq. (21) subject to eq. (16) for the cases C, = 0 (zero pressure difference, pi = p,), C, > 0 (P, ldR*/dz*l+ I or L*% 1. This pressed, using eq. (l9b), as The roots of eq. (27) can be calculated as follows (Aleksandrov et al., 1969; Abramowitz and Stegun, 1964). Define q= 3Nb, r =. R S, = [r+(i' (29) (1 - R, - Fr sin 0, cos 0,) - cos3 0, (30) s2 = +r2)1/2]113, [r-(q3 +r2)'12]'13. (31) -. Then, the roots of eq. (27) are 7 .-,=(sl (32) +S,)-cosOO i3 II2 7 52 = (33) -~(S1+s2)-E0SQ.++S,-S2) =c, = +r < 0 (P: > P,). 1: C, = 0 If C, =0 or pe = pi, eq. (21) can be integrated subject to eq. (16) to yield -2%Frsin00-cos2f10 0 +s2)-cos0, i3r12 -+sr -s2) (34) Case twice +rFrsin0,+1. (26) The liquid curtain converges whenever the liquid curtain inner radius is zero, i.e. at a time Z, for which R*(7,)b*(7,)/2=0. For thin curtains (b* < R*), the convergence time can be estimated by setting the right-hand-side of eq. (26) to zero. However, we will assume here that the liquid curtain radius at convergence is R*(7,) = R,. Thus, the convergence time is given by eq. (26) as 7% + 37: cos 0, - 6r, Nb 2 R, Frsin8,-6 ?(I where i=( - 1)‘/2. If $+s2 - 0, the roots of eq. (27) are real and, at least, two of them are equal. If q3 + r2 -c 0, all the roots are real, whereas if q3 + r2 > 0, one root is real and the other two are complex conjugate. The real roots of eq. (27) subject to the condition 7,aO can be used to determine the range of the parameters for which the analytical solutions are valid (cf. eq. 28). This results in a hypersurface which is a _ function of O,, b,/R,, Fr, N and R,. 2: C, > 0 If pe > pi, eq. (21) can be integrated introducing the mapping Case analytically by -R,)=O. 0 (7rR)+(T,Q), Q =;+C,R*, (27) An expression similar to eq. (27) has previously been obtained by Hovingh (1977) and Hoffman et al. (1980) for 8, = 0. Hovingh (1977) included only the surface tension from the liquid curtain inner surface, whereas upon Hovingh’s Hoffman et al. (1980) improved theory by including the surface tension from both the inner and outer surfaces. Furthermore, Hoffman ec al. (1980) assumed that at the liquid curtain convergence, R*(L*) = R*(7,) = 0 and R*(L*) = R*(7,) = 0.389, where L* is the convergence length which can be determined from eq. (l9b) once 7, is known. Equations (19b) and (27) can be used to determine the ranges of the parameters 0,. bdR,, N and Fr for which the analytical solutions obtained in this section are valid, i.e. the values of the parameters for which 113 T= Introducing (cos e, + 7). eq. (35) into eq. (21) we obtain d2Q dr2+TQ=0. The solution of eq. (36) is (Arpaci, Q = AT” 2J,,3 (35) + BT”’ (36) 1966; Watson, 1944) J_ L,3 (37) 3175 zyxwvu Liquid curtains-I or Adding eq. (41) to eq. (43) we obtain the solution eq. (40), i.e. of R* L&+@G)‘;” p== & +CAi[r+)“ ’ (cos&,+~)] x (cos 8, + #‘* zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Jllj +DBi[(y)1’3(cos00+r)] (44) x (cos 0, + 7) 3,2] + B(!!E5)‘” where C and D can be determined from eq. (16). Equation (44) can be used to determine the convergence length L* =z*(r,), and the ranges of the parameters 8,, C,, b,/R,, N and Fr for which this x (cos 8, + p zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA equation is valid, i.e. L* % 1. The analytical solutions presented in the previous where J1,3 and J_,,3 are Bessel’s functions of the first paragraphs neglect the radius of curvature r: (eqs 10 kind of orders l/3 and - l/3, respectively, and A and and 11). In order to determine the effects of r,*, B are integration constants which can be determined eqs (10) and (11) were solved numerically by means of from eq. (16). an explicit fourth-order accurate Runge-Kutta use d to calculate the conEquation (38) can be zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA method (Carnahan et al., 1969). Comparisons between vergence time, i.e. the timer, at which R*(r,) - b*(r,)/Z the analytical and numerical results are presented in =O. The converrgence time can then be substituted the next section. zyxwvutsrqponmlkjihgfedcbaZYXWVUTS into eq. (19b) to obtain the convergence length L* = z*(r,). PRESENTATJON OF RESULTS N and C, The range of the parameters 8,, C,, b,/R,, In Figs 2-6, we present some sample results which for which eq. (38) is valid can be determined by using illustrate the liquid curtain geometry represented by eq. (28) once T, is calculated from R*(r,) - b*(r,)/2 = 0. R*, as a function of 0,. N, b,/R,, Fr and C,. The upper part of these figures corresponds to the numerical Case 3: C, c 0 solution of eqs (10) and (11) which were solved numeriIf pi > p., eq. (21) can be integrated analytically as cally by means of an explicit fourth-order accurate follows. We first introduce a change of independent Runge-Kutta method, whereas the lower part correvariable sponds to the analytical solutions presented in the T= cose,+t previous section. Since both the analytical and nu(39) merical solutions depend on five parameters, the which yields effects of each nondimensional group were studied R, d2R* independently. Table l shows the values of the par---ICC, TR* = -2 T. (40) ameters used in the calculations. dT* b, 0 In all the figures presented in this paper, the calcuThe particular solution of eq. (40) is lations were stopped when the radius of the liquid curtain inner surface became zero, i.e. when R*(r,) 1 (41) R&r, = - b*(r,)/2 = 0, where rF is the convergence time which WC,1 defines the convergence length L* = z*(7,). and the homogeneous part of eq. (40) can be written as Figure 2 shows the liquid curtain shape represented by R* = R*(z*), i.e. the liquid curtain mean radius, as a d*R;t R,IC,I 1’3 function of the Froude number. This figure indicates --TT,R,*=O, T,=T ___ (42) that the nondimensional convergence length increases dT: ( > b, with the Froude number, i.e. with the liquid curtain initial velocity or initial linear momentum. where the subscript h denotes homogeneous. Figure 3 presents the liquid curtain shape as a The solution of eq. (42) can be expressed in terms function of the initial thickness to initial radius ratio, of Airy’s functions (Bender and Orszag, 1978; and shows that the convergence length is a monoAbramowitz and Stegun, 1964) as as should be tonically increasing function of b,/R, expected since the gravitational force is proportional R; = to the liquid volumetric flow rate and liquid curtain thickness. Figure 4 shows the liquid curtain shape as a func+ DBi[ r+)‘” T] (43) tion of C,, i.e. the pressure difference across the curtain and indicates that the convergence length increases as where Ai and Bi are Airy’s functions, and C and D are Note that negative values of C, a function of pi-pc. integration constants. correspond to pi>pc (cf. eq. 13a). x (cos 8, + z)l” J _ 1,3 I> CA~[(?)“l~] J. I. 3176 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF RAMOS X FR=lO -- C CR=15 -__ 0 FR=20 n FR=25 . . . . . _... x FR=S + FR=lO @ FR=15 --- -1.5 b I 100 I 200 I 300 Z* I 400 500 0 FR=20 -- + FR=25 I 600 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK Fig. 2. Liquid curtain shape as a function of the Froude number. A BORO=D.025 X BORO=005 ---_ C BORO=O.i -___- qORO=OZ tS -_Z BORO=O4 ____.___.__...___ n BORO-0 -- 025 e BORO=O 05 @ BORO=O -.-.-._ -1.5 ! 0 1 70 I 140 I 210 0 BORO=OZ ---- + BORO=04 -- i t 280 z* Fig. 3. Liquid curtain shape as a function of zyxwvutsrqponmlkjihgfedcbaZYXWVUTS b,jR,. The effects of the convergence parameter on the liquid curtain shape are shown in Fig. 5, which indicates that the convergence length is a monotonically increasing function of N or a monotonically increasing function of the Weber number. Thus, as the magnitude of the surface tension is decreased, N and the convergence length increase. Note that for C, = Cr =0 and zero surface tension, the liquid curtain does not converge, i.e. d2R*/dz2 =0 for all z 20 (cf. eq. 10). Figure 6 shows the liquid curtain shape as a function of 8,, i.e. the angle between the liquid velocity at the nozzle exit and the direction of the gravitational acceleration; positive (negative) angles indicate outward (inward) radial motion at the nozzle exit. Figure 6 indicates that as 0, is increased both the convergence length and the maximum radius of the liquid curtain increase. Since the vertical velocity always increases due to gravity, large values of 8, result in very thin liquid curtains particularly near the location of the maximum radius. Figure 6 clearly illustrates that large convergence lengths and large enclosed volumes can be achieved by using large values of BO. The results shown in Figs 2-6 indicate that large enclosed volumes can be achieved by increasing the initial curtain thickness, the initial velocity and the initial angle, and by decreasing the surface tension, i.e. by using liquids of low surface tension. They also indicate that the differences between the analytical and numerical results are less than 3% for the values of the parameters shown in Table 1. Figures 7-16 show the convergence length for a variety of parameters. These figures correspond to Liquid curtains-I 3177 cP=-0 A 0010 x cP--0.0009 ---_ c Es___ I CP=O -_- H CP-0.0018 ---_-_-..__.*._.. * cP~-o.ooli3 -__ 0 CP=-0.0009 --_ 0009 a !z=L____ 0 CF-=0.0009 ---- + cP=oooi.5 -- Fig. 4. Liquid curtain shape as a function of C,. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO 0 000 0 N=125 X N=625 --_ U N=3125 -____ I N=15625 -__ I N=78125 ._..._.* .__ X N=125 l N=625 8 N=3:25 A-- 0 N=15625 --- + N=78125 -_ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1600 2400 3200 zyxwvutsrqponmlkjihgfedcbaZ Z* Fig. 5. Liquid curtain shape as a function of N. parametric studies which were performed to detervalue of b-/R,, the convergence length increases as 9, mine the volume enclosed by the liquid curtain as a is increased. For b,/R, = 0.1, the convergence length function of b,/R,, N, 0,. Fr and C,. Some of the results for B,=30” is almost sixteen times greater than that presented in Figs 7-16 cover large ranges of these corresponding to e0 = O”, whereas for b,/R, = 0.025, i.e. parameters and include liquid curtains used as prothin liquid curtains, the convergence length for 0, = 30” is about thirteen times greater than that corretection systems in laser fusion reactors as well as chemical reactors. Further parametric studies are presponding to t?,=O” . Figure 8 indicates that the convergence length sented in Ramos (1987). The results shown in Figs increases with 0, and N. For a given nozzle, geometry 7-16 correspond to the numerical solution of eqs (10) and Froude number, N increases as 0 is decreased, i.e. and (11). the convergence length increases as the surface tension Figure 7 shows that the convergence length inis decreased or as the Weber number is increased. creases with the angle at the nozzle exit and with the Figure 8 also indicates that for 43,I - 15”, the coninitial liquid curtain thickness. For a fixed positive vergence length is almost independent of the con(negative) value of 0,. the convergence length increases vergence parameter for the values shown in Table 1. b,fR, is increased, whereas for a fixed (decreases) as zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA J.I. RAMOS 3178 1 200 I 100 0 , 300 I 400 I 500 I 600 A THETAO=-lO X THETAO=-S ---_ 0 THETAO=O -___- I THETAO=S -*- K THETAO=IO ..__.*.*_._._._.* X THETAO=-lO * THETAO=- 8 THETAO=O ---- 0 THETAO= ---- + THETAO=IO -- r 700 z* Fig. 6. Liquid curtain shape as a function of 0,. Table 1. Values of the parameters used in the calculationst Figure zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Fr N bc.lRo 0, (“ 1 CP 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 variable 5 5 5 5 5 5 5 variable variable 5 5 variable variable 5 500 500 500 variable 500 500 variable 500 500 500 variable 500 variable 500 variable 0.1 variable 0.1 0.1 0.1 variable 0.1 0.1 0.1 variable variable variable 0.1 0.1 0.1 0 0 0 0 variable variable variable variable variable 0 0 0 0 0 0 0 0 variable 0 0 0 0 variable 0 0 0 variable 0 variable variable tc, =o. -20 Fig. 7. Convergence 2’0 4’0 length as a function of 6, and be/R,. A BORO=O X BORO=005 ---_ 025 0 BORO=O mm__- I BORO-02 -_- i E BORO=04 .________________. 3179 Liquid curtains-I 6 ,...-.--- .__.......--= 5 4 ] 3 c3 3 s 2- A l- o -440 I -20 I 0 I 20 N=125 X N=625 -em C N=3125 ----_ I N=15625 --_ I N=78125 __________.. I 40 THETA0 Fig. 8. Convergence length as a function of 0, and N. 2.5 A CP=-0.0018 x CP=-0.0009 ---_ 0 cP=o -__-- zyxwvutsrqponmlkjihgfedcb m -_cP=0.0009 -30 -20 -10 0 B CP=O.O018 . .._._.-----.__._. H CP=O.O027 10 THETA0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG Fig. 9. Convergence length as a function of 0, and C,. However, for eO> - 15”, the convergence length inliquid curtain may break up and the above analysis becomes invalid. creases quite rapidly as the convergence parameter is Figure 9 shows the convergence length as a function increased. The results shown in Fig. 8 can be justified as of 0, and C, and indicates that the convergence length is a monotonically increasing function of (pi-p,) and follows. For N - ’ = CP = C, = 0 and (dR*/dz*( Q 1, the BO.Note that small pressure differences may result in solutions of eqs (lo), (ll), (16) and (17) are zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC substantial increases in the convergence length. For C,I -0.002 it was not possible to obtain enclosed z*=rFr[; + cod.,]. (45) R* = 1 + rFr sin 9,. volumes in some cases. Note that the critical value of C, is -l/N for f?,=C,=C,=O and IdR*/dz*l<l. Thus, the liquid curtain does not form an zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC enclosed Figure 10 indicates that the convergence length is a volume if 8,>0” because R* increases monotonically monotonically increasing function of 0, and Fr for 8, with 8, for 0” < 6” I 90”. For 13,= 0, eq. (45) shows that greater than approximately - 1.5”. Note that a fivefold increase in the Froude number corresponds to R*(r)= 1, z*(T)=rFr t + 1 about a fifteen-fold increase in the convergence length (46) ( > for 8,=30”. Figure 10 also indicates that for a given value of 8,s - 15”, the convergence length decreases i.e. a cylindrical curtain is obtained. However, as the as the Froude number is increased. liquid falls due to gravity, its velocity increases and its Figure 11 shows that the convergence length is a thickness decreases. instabilities may develop, the 3180 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA J.I. RAMOS A FR=5 X FR=lO -- 0 FR=15 --- m FR=ZO ri Fig. 10. Convergence FR=25 _..I___._ length as a function of Q, and Fr. 3.2 - 2- A FR=5 X FR=lO -- 0 FR=15 -__ CZ FR=20 Z I.” FR=25 ___. . . . . . , zyxwvutsrqponmlkjihgfedcbaZ 0.; 0.0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Oil 0.; 0.; 0.; BORO Fig. 11. Convergence length as a function of b,/R, monotonically increasing function of the liquid curtain initial thickness and of the Froude number. As the velocity at the nozzle exit and the liquid curtain initial thickness are increased, the effects of surface tension (Weber number) decrease and the liquid curtain tends to preserve a cylindrical shape. However, as the flow is accelerated by gravity, its thickness decreases and aerodynamic instabilities may develop and the liquid curtain may break up. Figure 12 indicates that the liquid curtain convergence length increases with be/R, and as the surface tension is decreased. In the limits 0,-O and N-co, the convergence length is infinite and no closed volume is obtained for a fixed value of b,/R, (cf. eq. 46). The importance of the surface tension in determining the liquid curtain shape and convergence length is clearly illustrated in Fig. 12; a ten-fold increase in and Fr. convergence length is obtained by using liquids whose surface tensions differ by a factor of about one hundred. Figure 13 shows that the convergence length of the liquid curtain increases as pi-pPe and b,/R, are increased. It also indicates that small pressure differences can dramatically increase the convergence length. The effects of the Froude number and convergence parameter on the convergence length are illustrated in Fig. 14. This figure shows that as the Weber number increases (or as the surface tension decreases) for a fixed Froude number, the convergence length increases. Figure 14 also shows that the convergence length increases as the Froude number is increased for a fixed value of the convergence parameter. Figure 15 shows that the convergence length in- Liquid curtains-I ________..-.-..- _.._---,- +.____---- ____*____...__ ...--..------.-n ..-- ..-* _/ x*’ ,_A _---= __-- _/ 3181 __.-E EL- _______.-------- /&;___-: ,-- 1 o’.o 1 0.1 ! 0.2 I 0.3 I 0.4 A N=125 X N=625 --_ c N=3125 -me._ E N=l5625 -__ = N-78125 . . . ..----.__ I 0.5 BORO b,/R, and N. Fig. 12. Convergence length as a function of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP 0 011 A CP--0 0078 x CP--0 -_-_ 0009 G cP=o --se- I cP=0.0009 -_- E CP-0.0018 -.____..-.___...-. X CP=O.O027 1 0.; 0.; 0.i~ 0.5 BORO Fig. 13. Convergence length as a function of b,/R, and C,. creases as the Froude number and (pi-p,_) are increased. Note that very small pressure differences are needed to achieve large convergence lengths. Figure 16 shows that the convergence length increases as N is increased and as C, is decreased. Note that the critical value of C, for which the liquid curtain does not converge corresponds to C, = -0.002 for N =500. Figure 16 clearly illustrates that the convergence length approaches infinite as C, approaches -0.002 for N = 500. The analytical solutions presented in the previous section agree with the solution of the nonlinear equations shown in Figs 2-16 within 3% for the values of the parameters shown in Table 1. This agreement improves as pi -p, is increased or as C, is decreased. The agreement between the analytical and numerical results indicate that even for 8, = 30”, the analytical solutions are good approximations even though the condition IdR*/dz* 16 1 is not satisfied near the nozzle exit. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML CONCLUSIONS A mathematical model of steady state liquid curtains falling under gravity has been developed and analyzed numerically by means of an explicit fourthorder accurate Runge-Kutta method. The model accounts for gravity, surface tension, pressure differences, initial thickness and nozzle exit angle but neglects the friction of the air surrounding and enclosed by the liquid curtain. Analytical solutions were obtained and compared with the numerical solution of the nonlinear equations; differences of at most 3% between the analytical and numerical solutions were obtained in the con- 3182 RAM OS J. I. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Fig. 14. A N=175 x N=625 --_ 0 N=3125 -____ I N=15625 -__ E N=78125 ._...* .__._. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Convergence length as a function of Fr and N. A W--O x CP=-0.0009 ---- 0 CP=O -___- OOi8 q CP-00009 -_- 1.5 ! 0 1 5 I 10 Fig. 15. Convergence I I 20 I 25 = CP=O 0018 _...____________-. X CP=O 0027 1 30 length as a function of Fr and Cr. vergence length and liquid curtain shape even for nozzle exit angles of 30”. These differences indicate that accurate liquid curtain shapes can be obtained by using the analytical solutions, even though these solutions depend on Bessel’s functions of the first kind and fractional order or Airy’s functions which are given by infinite series. The steady state analytical and numerical results reported in this paper show that the convergence length and volume enclosed by the liquid curtain are monotonically increasing functions of the Froude number, pressure difference, convergence parameter, nozzle exit angle and initial thickness-to-initial radius ratio. In terms of physical variables, the convergence length and volume enclosed by the liquid curtain increase with the nozzle exit angle, initial velocity, pressure difference between the gas enclosed by and the gas surrounding the liquid curtain, and initial liquid curtain thickness, but decrease as the surface tension is increased. As the liquid curtain falls its velocity increases and instabilities may develop on its surface. These instabilities may result in liquid curtain breakup. In order to analyse the liquid curtain stability, it is necessary to develop a theory which accounts for the spatial and temporal variations of the flow variables. In the model presented in this paper, the friction on both surfaces of the liquid curtain was neglected but can be easily incorporated by analysing the boundary layers on the liquid curtain surfaces. The boundary layers along the inner and outer surfaces of the liquid curtain are unconventional since at the air-liquid 3183 Liquid curtains-I zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM 2.3 - 2.1El c3 s 1.9- a cP=-0.0016 x cP=-0.0009 ---- q cP=o -_*_- 1.7 - q cP=0.0009 -_- 1.5 1 0 I 100 I 200 8 300 N I 400 n cP=o.oola __..___._.-.----.. H CP=O.O027 I 600 500 zyxwvutsrqponmlkjihgfedcbaZYXW zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG C,. Fig. 16. Convergence length as a function of N and zyxwvutsrqponmlkjihgfedcbaZYXW interfaces the shear stresses must be continuous, the velocity components at the interface must satisfy a kinematic condition, i.e. the radial to the axial velocity ratio is equal to the slope of the gas-liquid interface, and the jump in normal stresses must be balanced by surface tension. As a first approximation the air boundary layer on the external surface can be analysed as if it were the boundary layer along a moving “rigid” surface where the radial and axial velocities are specified at the air-liquid interface. An analysis of the air motion within the volume enclosed by the liquid curtain is complicated by the possible presence of toroidal recirculation zones particularly near the convergence point where an adverse axial pressure gradient is present. In the steady state analysis presented in this paper, the pressure in the volume enclosed by the liquid curtain was asssumed constant. In practice, some air is dissolved in the liquid and the pressure in the enclosed volume decreases with time unless some means are provided to keep it constant. This has a very important effect on the liquid curtain shape which may exhibit an unsteady behavior. The results presented in this paper indicate that as (pi - p,) is increased, the convergence length increases. However, there is a critical pressure difference beyond which the liquid curtain does not converge. This may be explained experimentally as follows. As pi is increased, the convergence length is increased until C, exceeds a critical value; when this occurs, the curtain opens so that pi decreases. Then, the curtain closes until pi increases again and the process is repeated. An analysis of this process must, of course, be timedependent, and a formulation such as the one described in previous paragraphs may be useful. Such a formulation can also be used to study annular jet instabilities and the formation of liquid sheHs (Lee and Wang, 1986; Kendall, 1986; Kendall et aE., 1982). Acknowledgements-This work was supported by the Office of Basic Energy Sciences, Department of Energy, under Grant No. DE-FGO2-86ER13597 with Dr Oscar P. Manley as Technical Monitor. This financial support is deeply appreciated. The author also appreciates the support provided by CRAY Research, Inc. and the Pittsburgh Supercomputing Center through a 1986 CRAY Research and Development Grant. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO NOTATION Airy functions liquid curtain thickness gap width (bb = b,/cos 0,) Ai, Bi b bb BORO be/R, friction coefficient pressure coefficient (= CP in the figures) friction force per unit area Froude number f = FR in the figures) Bessel function convergence length convergence parameter pressure liquid curtain radius radius of curvature arc length along the liquid curtain time Cf f" FlJ L N P R rv s t THETA0 V We Greek 0 P d 7Z 00 velocity along the liquid curtain Weber number letters angle between the symmetry axis and the tangential direction to the liquid curtain density surface tension nondimensional time I. RAMOS J. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF 3184 Subscripts Dumbleton, J. H., 1969, Effect of gravity on the shape of water bells. J. appl. Phys. 40, 3950-3954. Esser, P. D. and Abdel-Khalik, S. I., 1984, Dynamics of e gas surrounding the curtain vertical annular liquid jets. ASME J. Fluids Engng 106, h homogeneous 45-51. i gas enclosed by the curtain Di@-ential Equations, pp. Garabedian, P. R., 1964, Parrtial 6-17. Wiley, New York. 0 nozzle exit Hoffman, M.~A., Takahashi, R. K. and Monson, R. D., 1980, part particular Annular liquid jet experiments. ASME 1. Fluids Engng 102, 344-349. Superscript Hovingh, J., 1977, Stability of a flowing circular annular * nondimensional quantities liquid curtain with a vertical axis subjected to surface tension forces. Internal Memorandum No. SS&A-77-108, Lawrence Livermore National Laboratory, Livermore, REFERENCES California. Handbook of Abramowitz, M. and Stegun, I. A. (eds), 1964, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB Kendall, J. M., 1986, Experiments on annular liquid jet instability and on the formation of liquid sheIls. Phys. Mathematical Functions, pp. 446450. National Bureau of Standards, Applied Mathematical Series 55, Washington, Fluids 29, 2086-2094. D.C. Kendall, J. M., Lee, M. C. and Wang, T. G., 1982, Metal shell Aleksandrov, A. D., Kolmogorov, A. N. and Lavrent’ev, M. technology based upon hollow jet instability. J. &‘ucuum Sci. Technol. 20, 1091&1093. Its Content, Methods and Meaning, A.. 1969, Mathematics: Lance, G. N. and Perry, R. L., 1953, Water bells. Proc. Phys. 2nd Edn, Vol. 1, pp. 266268. The M.I.T. Press, Cambridge, Sot. Lond. Ser. B 66. 1067-1072. Massachusetts. Lee, C. P. and Wang, T. G., 1986, A theoretical model for the Arpaci, V. S., 1966, Conduction Heat Transfer, p. 138. annular jet instability. Phys. Fluids 29, 207&2085. Addison-Wesley, Reading, Massachusetts. Baird, M. H. I. and Davidson, J. F., 1962, Annular jets-I. Ramos, J. I., 1987, Mathematical modelling of liquid curFluid dynamics, Chem. Engng Sci. 17, 467472. tains--I. Steady state analysis. Report CO/87/2, Department of Mechamcal Engineering, Carnegie-Mellon UniMathematBender, C. M. and Orszag, S. A., 1978 Advanced ical Methods for Scientists and Engineers, pp. 6-9 and versity, Pittsburgh, Pennsylvania. 100. McGraw-Hill, New York. Roidt, R. M. and Shapiro, Z. M., 1985, Liquid curtain reactor. Westinghouse R&D Center. ReDort 85M981, Boussinesq, J., 1869a, Theorie des experiences de Savart sur . Pittsburgh, Pen&.ylvania. la forme que prend une veine liquihe apres s’etre choquee contre un plan circulaire. Compt. Rend. Acad. Sci., Paris 69, Taylor, G. I., 1959a, The dynamics of thin sheets of fluid-I. 45-48. Water bells. Proc. R. Sot. Land. Ser. A 253. 289-295. Boussinesq, J., 1869b, Theorie des experiences de Savart sur la Taylor, G. I.. 1959b, The dynamics of thin sheets of fluid--II. forme que prend une veine liquide apres s’etre heurtee Waves on fluid sheets. Proc. R. Sot. Land. Ser. A 253, 296-312. contre un plan circulaire. Compf. Rend. Acad. Sci., Paris 69, 128-131. Dynamics, pp. Thompson, P. A., 1972, Compressible-Juid 641 and 644. McGraw-Hill, New York. Carnahan, B., Luther, H. A. and Wilkes, J. O., 1969, Applied Watson, G. N., 1944, A Treatise pp. 367-380. Wiley, New York. on the Theory of Bessel Numerical Methods, ofMathematical Courant, R. and Hilbert, D., 1962, Methods Functions, 2nd Edn, p. 96. Cambridge University Press, Physics, New York. Vol. 2, pp. 39-56. Interscience-Wiley, New York. C convergence