The onset of two-phase venting viaentrainment in liquid-filled storage vesselsexposed to fire

Michael Epstein, Hans K. Fauske, and George M. HauserFauske & Associates, Inc., 16WO70 West 83rd Street, Burr Ridge, II 60521,USA

A simple criterion is developed for predicting the onset of liquid entrainment due to venting-inducedvapour flow in the free-board volume of a storage vessel exposed to fire. Through the use ofpotential flow theory, a solution for the two-dimensional axisymmetric velocity distribution in thefree-board volume of an upright cylindrical vessel is obtained. The illustrative example consideredverifies the hydrodynamic aspects of the entrainment criterion.

(Keywords: vents; storage areas; fire)

The venting of liquid-filled storage vessels subjected toexternal heating by fire may be complicated by two-phase flow effects due to entrainment of the liquid at itssurface by the venting gas. An increase in the vent areaof more than an order of magnitude is required toaccommodate a two-phase discharge relative to a reliefvent designed for all-vapour flow. It is shown here via atheoretical examination of the vapour flow patternwithin a vertical right circular cylindrical vessel that it ispermissible to size relief vents for fire emergencies onthe basis of all-vapour flow.

The situation of interest is illustrated in Figure I.With a strong fire energy flux through the vertical wallto the liquid inside, vapour bubbles form, break awayfrom the inside surface, and rise along the wall. Vapour-liquid drag causes the liquid to move upward with thevapour as a two-phase free-convective boundary layer.An analysis that describes the void distribution andthickness of the boundary layer, both of which increasewith vertical distance along the wall, has been pre-sented ‘. In this earlier paper, vapour carry-under by thereturning core of downward flowing liquid was assumedto occur if the liquid velocity exceeded the terminalbubble rise velocity. Significant liquid swell was predic-ted as a result of this assumption and, depending on theliquid fill level and vessel geometry, the potential fortwo-phase vent flow was concluded to be large.

However, the assumption of vapour carry-under doesnot agree with tests performed with water in an extern-ally heated vessel and reported in a subsequent paper’.Measured swell heights were found to be in agreementwith predictions based on the vapour volume fraction in

Received IO October 1988

0950-4230p9/010045-05$3.00A 1989 Butterworth & Co. (Publishers) Ltd

the boundary layer alone. The increased subcoolingwith liquid depth due to hydrostatic pressure was usedas an argument against the existence of vapour withinthe descending liquid core. The work reported’ suggeststhat in venting assessments of liquid-filled vessels sub-jected to an external heat flux, it is acceptable to ignorethe effects of liquid swell for non-foamy liquid ma-terials. However, the possibility of two-phase flow tothe vent still exists if the velocity of the venting gaswithin the free-board volume is such that a spray ispulled off the liquid surface and carried by the gas to thevent. This paper considers the potential for liquidentrainment by analysing the induced vapour flow in thefree-board volume of a vessel surrounded by fire.

In most analytical studies of the transition fromstratified to the drop flow regime, a one-dimensionalflow model is utilized. Such a description of the flow in aconfined channel of large aspect ratio may be quiteadequate; however, it cannot be extended to describethe flow in the free-board volume of an externallyheated vessel receiving vapour from the boundary layerbelow the liquid surface at the periphery of the vesseland losing vapour through a vent located, say, above thecentre of the vessel. Clearly, the induced flow in thefree-board volume is three-dimensional. Therefore, toassess the potential for liquid entrainment by the vapourflow, it is necessary to resort to a three-dimensional or,more simply, two-dimensional axisymmetric descriptionof the flow.

We first consider the flow field in an ‘infinitely’ largefree-board volume as induced by the flow through thevent only. This will provide a feeling for the extent ofthe spatial region within the free-board volume overwhich the vent exerts its influence, in a fluid mechanical

J. Loss Prev. Process Ind., 1989, Vol2, January 45

The onset of two-phase venting: M. Epstein et al.

>3

3

03

a

e-c-c-

0

00

0

101c

>>

00

c

Figure 1 Sketch of the boiling boundary layer and free-boardvolume vapour flow within an externally heated right circularcylindrical vessel; indicating coordinate system andnomenclature. Also indicated is the descending liquid core

sense, and suggest a simple geometrical criterion forpredicting the onset of two-phase flow within the vent.Then approximate solutions are presented for the flowfield in a free-board volume of finite dimensions, whichfully account for the presence of the boundary layer inthe liquid below as well as the flow out the vent. Thesolutions serve to support the two-phase flow inceptioncriterion mentioned earlier.

Flow field below the vent in an infinitelylarge free-board volume and the liquidentrainment conditionIn the absence of appreciable liquid swell (due to, say,foaming) within a vessel under fire emergency con-ditions, the liquid phase can only reach the vent if thevapour flow far from the vent and near the liquidsurface is sufficient to suspend the liquid against gravity.The critical gas velocity for droplet ‘fluidization’ by thev a p o u r s t r e a m c a n b e e s t i m a t e d b y u s i n g t h eKutateladze correlation 3

(1)

where pp and PI are the densities of the gas and liquidphase, respectively, (T is the surface tension of the liquid,and ucr is the critical horizontal or vertical component

of the velocity above which liquid entrainment orfluidization occurs. This correlation expresses a balancebetween dynamic pressure (or drag) and buoyancyforces that is consistent with droplet stability and isparticularly appropriate when no characteristic dimen-sion of the system is important. The expression indicatesthat gas velocit ies of approximately IO ms-’ arerequired to keep a liquid suspended in the form of adroplet cloud and/or entrain surface liquid by a parallelgas flow. In liquid storage vessels where large overpres-sures can be tolerated, vapour may pass through thevent with a velocity as high as several hundred metresper second. Thus it is of interest to determine whether ornot the velocity field established below the vent canproduce velocities of sufficient magnitude to pull liquidfrom the surface in accordance with Equation (1).

The possibility of fluidization of the boundary layerliquid by the vapour species as it passes through theliquid surface should also be mentioned. Typical vapourseparation velocities are of the order of only 1 m s-’ andtherefore cannot lift the liquid from its surface. There isa horizontal velocity component (radial velocity) of theflow that is directed toward the centre of the vessel; it iszero at the vessel wall, rises to a maximum at the edge ofthe bubble boundary layer and decays back to zero atthe centre of the vessel. Our potential flow analysis (seebelow), when applied to the peripheral region of thevessel, indicates that the inward radial velocity at theliquid surface can exceed uEr within a narrow region justinside the boundary layer. The extent of the liquidsurface over which it exceeds uET is typically so narrowthat the quantity of liquid entrained must indeed benegligible. Even if substantial liquid entrainment occursabove the boundary layer, as already mentioned, liquidcannot enter the vent so long as the vapour velocity atsome location between the surface of the liquid and thevent falls below ucr. The distance from the vent at whichthe velocity ucr is achieved by the flow is now obtainedfrom a simple mass balance.

The flow pattern in the immediate vicinity of the ventis rather complex, but just a short distance away fromthe vent, of the order of one or two vent diameters, thespeed of the vapour is approximately the same at allpoints equidistant from the centre of the vent. Recallthat we are ignoring the presence of the liquid surfaceand the vessel wall, that is the finiteness of the free-board volume. We also neglect the effects of gas com-pressibility in the neighbourhood of the vent. If ‘a’ isthe radius of the vent and I/O is the vapour velocitywithin the entrance plane of the vent, conservation ofmass dictates that the vapour speed q is

qz; f’v,0where s is the distance measured from the centre of thevent to the hemispherical surface over which the velocityis constant and equal to q (see Figure2).

Clearly, entrainment at the liquid surface by thevapour flow induced by the vent becomes hydro-dynamically possible when q exceeds ucr at a distance s

46 J. Loss Prev. Process lnd., 7989, Vol2, January

The onset of two-phase venting: M. Epstein et at.

vent i s i n f e r r ed f rom mass con t i nu i t y , and i sVc = 197 m s-‘. Finally, Equation (3) reveals that liquidentrainment is possible when the height of the free-board volume is reduced below 0.33 m, a distance ofonly 2.8 vent radii. An equivalent interpretation of thisresult is that only 4% free-board volume is required forall-vapour venting.

Figure 2 Hemispherical surface of constant velocity, demon-strating incipient liquid entrainment when q = uCr and s = h

from the vent equal to the height of the free-boardvolume, h. In other words, the incipient entrainmentcondition postulated here is achieved when the hemi-spherical locus of points at which q = ucr touches theliquid surface. This condition is illustrated in Figure 2,and is expressed mathematically by setting q = uCr ands = h in Equation (2):

l/2

Equation (3) enables the question of interest to beanswered: for given vent size and fire heat flux, by howmuch can we reduce the free-board volume of a vesselbefore liquid entrainment and, therefore, two-phaseventing is possible? Equation (3) is especially simple touse, as illustrated below for a vessel of radius R = 5 m ,filled with water to a height L = 8 m, and subject to aspatially uniform fire heat flux 4” = 10’ W m-*. Themaximum pressure that can be tolerated by the vessel isgiven as 20 psig (or 2.39 x lo5 Pa). Using the two-phaseboundary layer theory presented previously’, the boil-ing boundary layer thickness at the liquid surface isfound to be 6 = 0.74 m for water at this pressure. Thevapour velocity V that passes through the liquid surface(‘boiling velocity’) is related to the wall heat flux by

*[R* - (R - Q2]p,V= 27rRL(@‘/hr,) (4)

where hr, is the latent heat of evaporation of the liquid.From Equation (4), we find V= 0.4 m s-r. This velocityis well below the fluidization velocity inferred fromEquation (1)

&r = 12.8 m s- ’ (5)

Thus, as already anticipated, the boundary layer doesnot represent a potential source of two-phase venting.

Based on all-vapour venting and an orifice coefficientof 0.7, a vent of radius a = 0.12 m is required toaccommodate the fire heat flux at an overpressure of20 psig. This vent size is obtained by balancing thechoked flow discharge rate against the vapour flow ratedue to boiling, as given by the right-hand-side ofEquation (4). The velocity of the vapour as it enters the

Analysis of two-dimensional flow in thefree-board volumeThe purpose of this section is to examine the validity ofthe simple entrainment criterion presented in the fore-going from a theoretical point of view. We wish todetermine whether this radial flow model offers a usefulsubstitute for the more rigorous equations of vapourflow in a free-board volume fed by a two-phase boun-dary layer . To this end, we solve for the two-dimensional axisymmetric flow field induced in thefree-board volume of a vertical, right circular cylin-drical vessel subjected to an external fire heat flux.

The flow is assumed to be ideal and incompressibleand adequately described by the Laplace equation incylindrical coordinates, see Figure (1):

a24 1 a4 a*9-g-p+;%+‘=0a2 (6)

where &(r, z) is the velocity potential related to t h eradial and axial components of the velocity through therelations

The boundary conditions for Equation (6) are:

a4ar r=R =I 0 (8)

for 0 < r < afor a < r < R (9)

forO<r<bfor b < r < R (10)

where, b is the radial location of the outer edge of theboundary layer, as shown in Figure 1. Note that theorigin of the axisymmetric coordinate system is locatedat the centre of the vent entrance plane. Clearly, theboiling velocity Vand the venting velocity VO are relatedby virtue of mass continuity:

V, R2-b2-=Va2 (11)

It seems unlikely that an analytical solution ofEquation (6) can be found since it is difficult to satisfyall the required boundary conditions in both rectangularand circular coordinates. An approximate numericalsolution, however, can be obtained by considering the

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The onset of two-phase venting: M. Epstein et al.

velocity potential function

Q = - vo ; 2z0

+ c F (exnnz + e-A’“Z)Jo(Xmr) (12)Ill=, m

where the X,s are the solutions (or zeros) of theequation

J,(X,R)=O (13)

Equation (13) has the properties that it obeys theLaplace equation, Equation (6), and satisfies boundaryconditions (8) and (9). It is possible to satisfy thevelocity conditions at z = h, namely boundary condition(lo), at as many points as the number of terms taken inthe last summation term of Equation (12). A value ofthe unknown coefficient A, is calculated for each pointr,,, along z = h where Equation (12) is forced to satisfyboundary condition (10). Since the best choices of rmare not known, we are at liberty to select arbitrarily.Four equally spaced rms across the boundary layer and20 r,,,s along the liquid surface from r = 0 to r = b werefound to provide an excellent representation of bound-ary condition (IO). One obtains a set of linear equationsfor the A,,,s, which may be solved by matrix inversion.

Once the values of A, are determined, the velocitycomnonents are obtainable from the potential functionby using Equations (7). Hence

u= 2;? 2 Jl(hna)m--l Xm[Jo(hnR)

- mF, A,,,(eXattz + e-‘““)J

-Fe1-““‘LJ,(Xmr)

t (Xrnr) (13)

Jr (Ama)L,[Jo(hnWIZ e

-x~B~ZJo(X,“r)

(14)

The velocity vector q(r, z) is related to its components(u, o) through the relation

q2 = u2 + v2 (15)

All the numerical computations were carried out forthe sample problem of the previous section. ThusR=5m, b=R-6=4.26m, a=O.l2m, andV = 0.4 m s- ’ . Figure 3 presents the streamlines withinthe free-board volume region for three different heightsh, which in order of decreasing h are 2, 1, and 0.33 m.Since the region has symmetry, in Figure 3 we consideronly the half space in the r - z physical plane from thecentre of the vent to the wall of the vessel (i.e., fromr = 0 to r = R, see Figure I ), All geometric quantities arenormalized with respect to R so that the radius of thevessel is unity and the radius of the vent for the sampleproblem is 0.024. Also, the velocity vectors q are

Figure 3 Streamline patterns for flow within the free-boardvolume of an externally heated right circular cylindrical vessel far:a. h/R = 0.4; b, h/R = 0.2; c. h/R = 0.07

normalized with respect to the superficial boilingvelocity V. Thus, for example, the number 2 on the flowfield plots indicates vector velocities q that are twice theevaporation velocity, or 0.8 m s-r. As seen from Figures3a and 3b, the flow rises from the boundary layer at theperiphery of the free-board volume, flows radiallyinward, and turns upward in the vicinity of the vent.The vapour flow slows down after it leaves the liquidsurface above the boundary layer and the normalizedvector velocity falls below unity, the boiling velocity,just above the surface. The flow then picks up speed asit turns inward, the normalized velocity passes throughunity from below and continues to increase as itapproaches the vent. In Figure 3c, which pertains to thesmallest free-board volume height considered here, asignificant flow decrease between the boundary layerand the vent does not arise.

The normalized velocity 32 in Figure 3 corresponds tothe entrainment velocity based on Equation (1). Thesurfaces of constant velocity in the vicinity of the ventcan be best described as flattened hemispheres. Thuscriterion (3), which is based on the assumption ofhemispherical surfaces of constant velocity, yields pre-dictions of free-board volume heights corresponding toincipient l iquid entrainment that are larger thanindicated by the two-dimensional flow theory. Forexample, criterion (3) predicts the onset of entrainmentfor the configuration shown in Figure 3c, whereas thelocus of points that satisfy entrainment lie on a surfacein the two-dimensional flow field that does not quiteextend down to the liquid level. We conclude from

40 J. Loss Prev. Process lnd., 1989, Vol2, January

Figure 3c that reasonable and conservative estimates ofthe minimum free-board volume permissible beforeentrainment sets can be obtained using Equation (3).

ConclusionsBy combining the assumption of radial vapour flow tothe vent of a vessel subjected to an external fire heat fluxwith a correlation for the minimum vapour velocity tobring about liquid entrainment, a criterion for theoccurrence of two-phase venting has been derived. Asolution for the two-dimensional flow field in the free-board volume of a vented, vertical right circular cylin-drical vessel was obtained and supports the notion ofnear radial flow within the vicinity of the vent. Theresults of this study indicate that rather small free-boardvolumes can be tolerated without two-phase venting.

ReferencesGrolmes, M. A. and Epstein, M., Plant~Operations Progress 1985,4 (4), 200-206Fauske, H. K., Epstein, M., Grolmes, M. A., and Leung, J. C.,Planr/Operations Progress 1986, 5 (4). 205-208Kutateladze, S. S., Fluid Mechanics - Soviet Research, 1972, 1 ,2 9 - 4 6

The onset of two-phase venting: M. Epstein et al.

Nomenclature

:,,,= radius of vent= unknown coeffxients in potential flow solution

b = radial location of outer edge of boiling boundary layer atliquid surface

g = gravitational constanth = height of free-board volumehr, = liquid latent heat of evaporationJO, J, = Bessel functions of zero and first-orderL = depth of liquid in vessel4 = flow velocity vector (or speed). I,9 = external fire heat fluxI = radial coordinate measured from centre of ventK = radius of vessels = radius of hemispherical surface of constant flow speed:

Figure 2L4 = radial velocity component of flow&r = critical entrainment velocity; Equation (I)” = vertical velocity component of flowV = vapour flow velocity entering free-board volume (‘boiling

velocity’)VI, = vapour flow velocity into vent

:= vertical coordinate measured from centre of vent= thickness of boiling boundary layer at liquid surface

A,,, = defined such that the product X,R is a root of theequation J I (X.,R) = 0

Pr = vapour densityPI = liquid density

;= liquid surface tension= velocity potential function

J. Loss Prev. Process Ind., 7989, Vol2, January 49