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Numerical modelling and flow visualization of mixing of stratified layers and rollover in LNG
Numerical modelling and flow visualization of mixing of stratified layers and reliever in LNG J.Q. Shi, C. Beduz* and R.G. Scurlock* Department of Mineral Resources Engineering, Imperial College, London SW7 2BP, UK *Institute of Cryogenics, University of Southampton, Southampton SO9 5NH, UK Received 8 J u n e 1993
A numerical model has been developed to study the mixing of two initially stratified layers which are subjected to a uniform lateral heat flux. An important distinction is made between the free surface and the liquid/liquid interface with regard to the different flow characteristics of the two layers. In the upper layer where warm liquid is cooled at the evaporating surface, the convective circulation is featured by a strong downward core flow; in contrast, the fluid flow in the lower layer is mainly confined to the wall boundary and is much weaker. Flow visualization experiments show that mixing of two stratified layers generally involves two stages in sequence: migration of the interface and rapid mixing between the remaining liquids. The interface movement is due to entrainment mixing at the interface. When the two layers approach density equalization, the interface becomes increasingly unstable and the core flow in the upper layer is able to break into the lower layer. The base to side heat flux ratio appears to be a major factor in determining the mode and intensity of the subsequent mixing at a rollover incident.
Keywords: numerical modelling; f l o w v i s u a l i z a t i o n ; LNG
Nomenclature d g k p Pr O Ra* Ri o Rs AS ue ~s v r
Liquid depth (m) Gravitational acceleration (m s-2) Thermal conductivity (W m-1 K - 1 ) Fluid pressure (N m-2) Prandtl number ( = v/s) Heat flux (W m-2) Modified Rayleigh number [ = (gflxd4dl)/(kctv)] Overall Richardson number Salinity Rayleigh number [ = (gflsASd3)/(ctv)] Initial salinity concentration difference across interface (kg m-3) Entrainment velocity (m s-1) Thermal diffusivity (m 2 s-1) Salinity diffusivity (m 2 s 1) Kinetic viscosity (m E S-1) Diffusivity ratio (~J~)
It is widely known that stratification of distinct layers in storage vessels containing cryogenic liquids may lead to rollover. Liquefied natural gas (LNG), being a multicomponent liquid, is the most common cryogenic liquid in which stratification occurs. LNG consists of mainly methane, higher hydrocarbons and sometimes nitrogen. The stratification could be due to preferential evapora0011-2275/93/121116-09 © 1993 Butterworth-Heinemann Ltd
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Cryogenics 1993 Vol 33, No 12
p Ap
P0
Liquid density (kg m -3) Density difference across density interface (kg m - 3) Liquid density at reference point (kg m-3) Thermal expansion coefficient [ = - (i/p)(ap/aT)]
~s
Salinity expansion coefficient [ = (1/p)(Op/OS)]
N o r m a l i z e d variables (against value enclosed in [ ] )
S T t u, w x, z
Salinity concentration [AS] Temperature [qd/k] Time [0.01 de~s] Velocity components in x and z directions, respectively Is~d] Co-ordinates I-d] Vorticity [~/d 2] Stream function [~]
tion of nitrogen, or filling of a cargo with different density from that of the heel. In 1971, a dramatic incident of rollover was experienced at a LNG terminal tank in La Spezia, Italy 1. Under normal storage conditions heat leaking into the liquid from the surroundings is removed by evaporation at the free surface. A convection loop is established in
Mixing of layers and rollover in LNG : J. Q. Shi et al.
which liquid in a boundary layer flows upwards over the vessel walls, absorbing heat from the surroundings, then across the free surface where the heat is removed by evaporation. In the case of two stratified layers, the boundary layer in the lower layer is unable to reach the surface. Heat exchange between the two layers is by conduction through the interface. If this is insufficient to match the heat leaking in from the vessel walls and the base, superheat will build up in the lower layer. As time progresses, the density of the upper layer increases steadily through weathering and gradually approaches that of the lower layer, which is also changing with time. Depending on the net gain or loss of heat and the changes in its composition, the lower layer may become less dense or denser. In time the densities of the two layers become equal, at which point they mix rapidly to form one uniform layer. This process is known as rollover. When the lower layer liquid is superheated with respect to the tank pressure, as is often the case, rollover is accompanied by a sudden increase in vapour evolution and tank pressure. Based upon this understanding of rollover, several simulation models have been proposed. 2 4 In analyzing the well-known La Spezia rollover incident, which has been well documented by Sarsten x, considerable success has been achieved in predicting the time to rollover. These models assume that heat and mass transfers between adjacent layers are by molecular diffusion only and no movement of the interface is allowed in the approach to density equalization. In other words, there is no entrainment mixing across the interface. Laboratory experiments on mixing of two stratified layers under different heating conditions have been carried out by Nakano et al. ~ and more recently by Morrison and Richardson 6 using liquid freon and butane/' pentane mixtures, respectively. In both studies two modes of rollover were observed, identified by the migration and disappearance of the interface. With only lateral heating into the upper and lower layers, the liquid/liquid interface is observed to migrate to the bottom of the tank. In contrast, when only the base is heated the interface position remains almost unchanged prior to the rapid mixing between the layers. When both side and base heating are applied, the outcomes are not so clear cut. Clearly this heating condition is more realistic. Rollover tests carried out in an inground L N G tank using L N G from two different sources showed that the interface level falls gradually, and then rapidly just before intense mixing between layers takes place 7. There are two mechanisms reported in the open literature concerning the interface migration, namely boundary layer penetration at the wall and entrainment mixing at the interface. The former was proposed by Morioka and Enya s, who used water in their rollover experiments. When the densities of the two layers approach equalization, the boundary layer flow of the lower layer at the wall penetrates into the upper layer so that mixing of the two liquids takes place. This boundary layer penetration was also observed by Nakano et al. in their experiments using liquid freon. But from a later study v in which rollover tests were conducted in an actual storage tank, Sugawara et al. concluded that entrainment mixing across the interface is mainly responsible for the vertical
movement of the interface. The observation that the interface movement is closely related to the stability of the interface and the heating position led Sugawara et al. to suggest that the entraining velocity can be expressed as a function of the Richardson number near the interface. The entrainment mixing mechanism is also reported by Agbabi 9, who carried out an experimental study using cryogens to simulate the mixing of a two-layer L N G system. In this paper, a combined numerical and experimental study of the rollover phenomenon is reported. The aim of the numerical simulation was to achieve a better understanding of the mixing process of two stratified layers in cryogenic liquids, and in particular the interface movement and its driving forces. A numerical model for laminar heterogeneous fluid flows was developed. In the light of the numerical findings, flow visualization experiments were carried out to study rollover events in a specially constructed tank.
N u m e r i c a l simulation Numerical model
Free convective fluid flows in rectangular tanks or vessels are modelled as two-dimensional problems. For a heterogeneous flow system involving two incompressible miscible liquids, the governing differential equations are the Navier-Stokes equations, the continuity equation and the transport equations for thermal energy and the solute concentration. It was decided in the model to use the vorticity stream function method rather than a primitive variables approach. The advantage of this is two-fold: 1 by eliminating the pressure terms in the Navier Stokes equations and introducing the vorticity and stream function, the number of independent variables is reduced by one (from p, u, w, T a n d S to cj, ~, Tand S); and 2, the use of a staggering grid, which is normally required when the pressure terms are retained, is avoided. Hence the set of differential equations (in dimensionless form) to be solved is
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Cryogenics 1 9 9 3 V o l 3 3 , No 12
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Mixing of layers and rollover in LNG: J. Q. Shi et al. 1.0
Heat flux
Heat flux
Z
X
0.0 0.0
0.5
1.0
F i g u r e 1 Solution domain: x = 0.0 and 1.0, non-slip walls; z = 0.0 and 1.0, non-slip adiabatic base and isothermal free surface, respectively
A detailed description of the discretization method and the solution algorithm used is given by Shi 1°. The solution domain (with normalized dimensions), together with the boundary conditions, are shown in Figure 1. In the model, an isothermal free surface is assumed to reflect an evaporating surface. Temperature measurements carried out by Beduz et al. 11 on a pool of evaporating liquid nitrogen showed that there exists at the surface a very thin conducting layer with a steep temperature gradient. Accordingly, very fine control volumes are required at the surface region. The mass loss due to surface evaporation is neglected so that a fixed solution domain is maintained throughout the computation runs.
Numerical results and discussion A two-layer stratification was considered with liquid nitrogen (LIN) in the upper layer and mixtures of LIN and a small amount of liquid oxygen (LOX, which is heavier than LIN) in the lower layer. The fluids were motionless and at the saturation temperature of LIN (77.4 K). A stepwise profile of LOX concentration was assumed across the interface, which was initially at the mid-height of the cavity. A uniform heat flux was then applied to the side walls. Numerical predictions of one simulation run with R a * = l0 T and Rs = 3 x 106 are presented and discussed. The two layers are completely mixed at t = 17.5 (normalized against 0.01d2/c0. To facilitate the understanding of the mixing process, four snapshots are displayed in Figure 2 which illustrate the evolution of the flow at different stages. Each snapshot consists of four contour plots, from left to right, temperature (T), LOX concentration (S), density (normalized as S - TRa*/Rs) and stream function (~). Only the left half of the solution domain is presented because of the symmetry of the flows about the central line. From the contour plots in Figure 2, it can be seen that the mixing is a gradual process, marked by the steady descent of the interface level. The migration of the interface appears to result from the entrainment of heavier liquid into the upper layer where mixing takes place. Upon impinging on the interface, the central (or core) downward flow in the upper layer entrains liquid from the interface as it flows across it. This is clearly shown in
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Cryogenics 1993 Vol 33, No 12
the LOX concentration contours, where the distributions of LOX concentration in the upper layer closely follow the patterns of the stream lines. Entrainment also takes place at the lower side of the interface, whereby liquid is entrained from the interface and mixed into the lower layer. The observed interface movement is therefore the net result of these two opposing mixing processes. This continuing process of entraining and mixing results in a steady contraction of the lower layer. Meanwhile, the core flow is seen to penetrate progressively deeper through the interface as the two layers approach each other in density. Eventually the liquids completely mix to form one convecting layer. The rate of heat loss at the surface as a fraction of the heat input, together with the interface movement, are plotted in Figure 3. It can be seen that the heat loss at the surface rises steadily until it reaches the first peak at t = 14.0. This time period sees a decline in the interface level, the interface descending at an increasing speed. After undergoing a period of oscillations, the surface heat flow rate reaches the highest peak and then declines to the equilibrium value. . At t = 0.5 (Figure 2a), separate convective circulations are established in the two layers. With continuing heating, convective flow in the upper layer becomes progressively stronger, whereas the flow in the lower layer weakens steadily. At t = 0.5, the velocities of the core flow in the upper layer and the boundary layer at the wall in the lower layer reach a maximum at z = 0.83 and 0.30, respectively. The profiles of vertical velocity at these two levels are shown in Figure 4a. The ratio of the velocity between the core flow (profile 1) and the boundary layer (profile 2) is around 1. At t = 12.0, the maximum core flow velocity is increased by a factor of over 3, whereas the maximum boundary layer velocity in the lower layer is halved, as shown in Figure 4b. To explain this contrast in the velocity fields, one needs to examine the temperature distributions in the two layers, since the convections are primarily driven by the temperature differentials. In the upper layer where the surface temperature remains at its initial saturation value, warmed fluid rising from the wall boundary layer is cooled as it flows parallel to the surface. The cooled liquid becomes denser than the liquid surrounding it and sinks into the bulk along the central line as a core flow. With the continuing warming up of the bulk liquid, this temperature differential increases with time. In addition, the core flow is further accelerated by the gradient of LOX concentration near the central line (Figure 2). The situation in the lower layer is quite different. The size of the temperature differential across the interface is restrained by the fact that liquids on both sides are warming up, even though at a different rate. Therefore, as warmed liquid from the wall boundary layer flows along the interface, it is not sufficiently cooled to be driven into the bulk, but rather accumulates at the interface. This results in the formation of a stable thermal stratification with the warmest liquid at the top, as is clearly shown in the temperature contours in Figure 2. This stable stratification clearly has an adverse effect on the rising boundary flow. As has been mentioned above, the interface descends at an increasing speed. This can be related to two factors: steady reduction in the density difference across the interface and a progressively stronger core flow.
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: i g u r e 2 Predicted evolution of f l o w leading to complete mixing of t w o initially stratified layers w h i c h are subjected to uniform lateral heating. (The contour heights have been scaled for )resentation).x ~ 0.0, heated w a l l ; x ~ 0.5, central line of d o m a i n ; z = 0 . 0 , adiabatic base; z = 1 0, free surface
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Mixing of layers and rollover in LNG." J. Q. Shi et al. Surface heat loss ratio
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A rectangular tank was constructed to conduct flow visualization experiments, which is schematically shown in Figure 5. The tank is 20 cm long with a cross-section of 12 cm x 12 cm. Independently controlled resistance heaters are mounted on the side walls and the base. The front window is made of 9 mm thick Perspex and the tank is insulated with thick polystyrene to reduce heat exchange with the surroundings. Visualization of flows is made possible by seeding the flow media with foreign particles with good light reflectability. In this technique, density compatibility between the seed particles and the flow medium is essential. In the light of the findings from the numerical simulations, seeding only the upper layer is desirable so that the predicted movement of the interface can be observed. Liquid freon 11/113 mixtures, which are much easier to handle than volatile cryogens such as LIN and LOX, were used to set up a two-layer stratification with pure freon 11 in the upper layer. However, using freon 11 as test liquid has a drawback, as was discovered during the experiments. The saturation temperature of freon 11 is 23.8°C under atmospheric pressure, which was a couple of degrees higher than the ambient temperature in the laboratory. This implies that when freon 11 stored in a container is heated, its surface temperature may rise according to changes in its vapour (partial) pressure. This surface condition therefore does not conform to the normal storage conditions of cryogenic liquids. To achieve isothermal conditions, the generated freon vapour needs to be able to dissipate rapidly, so that a near constant vapour (partial) pressure can be maintained above the surface. This was achieved by leaving the test tank unsealed at the top, or loosely covered by a Perspex plate. This means, unfortunately, that the boiloff rates could not be measured directly during rollover tests.
(b)
Figure 4 Profiles of vertical velocity in upper (curve 1 ) and lower (curve 2) layers. (a) t = 0.5; (b) t = 12.0 w
Limitation of model Preliminary simulation runs with a layer of pure L1N show that when Ra* exceeds 5 x 10 s, instability starts to occur in the core flow 1°. The core flow has a velocity profile similar to that of a free jet (Figure 4). In general, shear flows away from any solid boundaries are much less stable than those near a wall. The order of critical Reynolds number for free jets is typically 10, compared with typically 1000 for wall boundary layer flows 11. In the two-layer system, the lowest Ra* at which instability starts to occur falls below 10 s, due to the extra driving force for the core flow provided by the solute distribution there. Using the properties of LIN at the normal boiling point (77.4 K) and assuming a liquid depth of 0.1 m, Ra* = l0 T represents a side wall heat flux of the order of
A---q~ v
D c
E
10-3 Wm -2
Although this heat flux value is unrealistically small, the above numerical predictions are confirmed by flow visualization experiments.
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Cryogenics 1993 Vol 33, No 12
Figure 5 Flow visualization system. A, Beam of parallel light; B, reflecting mirror; C, illuminated section; D, test tank; E, viewing window
Mixing of layers and rollover in LNG : J. Q. Shi et al. Experimental procedure
Experimental results
The tank is first filled with freon 11 to ~ 5 cm high and left for some time to reach equilibrium. The mixture of freon 11 and a small amount of freon 113 is then slowly bottom-filled through a thin capillary to reach a total liquid depth of l0 cm. After seeding of the particles, the tank is left for about half an hour before the start of heating. A video recorder is placed in front of the window to record the rollover events.
Two rollover tests are reported here, one with side heating only (test 1), the other with both side and base heating (test 2). In both tests, the pre-filling density difference between the layers was l~o and the input heat flux was .~ 40 Win-2. Flow visualization photographs displaying the evolution of flow in the illuminated plane in these two runs are given in Figures 6 and 7, respectively. These photographs were taken from video recordings
Figure 6 Flow visualization photographs showing mixing of two stratified layers (lateral heating only)
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Mixing of layers and rollover in LNG." J. O. Shi et al.
(2c) t = 140 minutes
Figure 7 Flow visualization photographs showing mixing of two stratified layers (lateral and base heating)
and an exposure time of 4 s was used. The trajectories of the seed particle were captured as bright line segments on the photographs. It should be noted that the length of one trajectory does not necessarily represent the whole distance travelled by this particle in the period of 4 s since the flow was not strictly two-dimensional. A particle could well enter and/or leave the illuminated section during the exposure time. Therefore the mean velocities of the seeding particles directly derived from the lengths of their trajectories could well be smaller than their true values. Figure 8 shows the movement of the interface in the two tests.
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Cryogenics 1993 Vol 33, No 12
Side heating only (test 1). Three distinct stages are observed leading to the complete mixing of the liquids after ~ 200 min. (i) 0-45 rain: A two-layer stratification is clearly observed in photograph la. A close look at the seeded upper layer reveals a step change in the brightness of the picture, with the lower region being less bright. This variation in illumination reflects density stratification in the upper layer, which is apparently caused by mixing during the filling of the lower layer. As a result of this stratification, flow is confined to the brighter region only and there is hardly any visible motion in the liquid below.
Mixing of layers and rollover in LNG." J. Q. Shi et al.
Interface level 70 ~
60
Discussion
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D
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0
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,
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,
100
[
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150
R 200
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Observed interface movements (see text for details)
The above flow visualization results demonstrate that rollover is characterized by the initial gradual descent of the interface level over a period of time, then rapid mixing between the remaining layers. This is in agreement with the results of Sugawara et al. v. The migration of the interface level is a result of entrainment mixing, whereby the heavier liquid is entrained and mixed into the upper layer. As the liquid densities approach equalization, the interface becomes increasingly unstable. Eventually the core flow in the upper layer breaks into the lower layer and the liquids are quickly mixed. Turner 13 used a vertically oscillating grid to study mixing across a density interface. When stirring was applied at the same rate in the two layers, the interface remained sharp and central; otherwise the interface moved away from the region of more vigorous stirring until the entrainment rates on the two sides were balanced. The following relationship was proposed for the entrainment velocity u~ Ue .OCRio 2,,2
U
The core flow in the upper layer can be clearly identified. In the next 30 rain, more and more liquid is brought into the motion through the penetration of the core flow and entrainment mixing (photograph lb). Eventually, the entire upper layer becomes one convecting loop (photograph lc). Throughout this stage the interface between the two main layers remains unmoved. This stage corresponds to AB in Figure 8. (ii) 45-185 rain: In the first 30 min of this stage, the interface hardly moves, indicating that the density difference across the interface is still substantial. The interface then starts to move downwards, slowly at first but gradually gaining in speed. Meanwhile, the interface becomes less and less stable (photographs ld and le). This period corresponds to BD in Figure 8. (iii) 185 200 min: This stage (DE in Figure8) is marked by intensive mixing. With the liquids approaching density equalization, the core flow in the upper layer is able to depress the interface and the latter becomes unstable (photographs if and lg). Eventually the core flow breaks into the lower layer (photograph l h) and within minutes the liquids are completely mixed to form one uniform layer.
Base and side heating (test 2). The two layers are completely mixed after 185 min of heating. An intermediate layer is not observed in this test. This situation can arise if the density of the intermediate layer is higher than that of the seed particles. Two distinct stages (represented by PQ and QR respectively in Figure 8) are observed, identified by migration of the interface (PQ) and rapid mixing between the remaining liquids (QR). These two stages correspond to the last two stages in test 1. In the first stage (0-170 min, photographs 2a-d), the interface descends by 20 mm. Within the next 15 rain, the remaining lower layer (32 mm thick) is completely mixed into the upper layer. The final mixing in this test is more intensive than that in test 1, as reflected in photographs 2d-h.
where Rio is an overall Richardson number, defined as Rio = gp'l-/U 2. U and L are the velocity and length scale characteristic of the flow motion near the interface and p' is the normalized density difference across the interface, p' = Ap/po. Substitution of the expression for Rio into the above equation yields U4 Ue ~ ( g p , C ) 3 : 2
The migration of the interface is therefore controlled by the differential in entrainment rates across the interface. In both the rollover tests the interface moved downwards. This confirms the numerical prediction in the last section that the convective flow is more vigorous in the upper layer than in the lower layer. Here the importance of maintaining an isothermal surface during experiments is emphasized. Compared with test 1, the addition of base heating in test 2 has two effects on the interface movement. First, convective flow in the lower layer becomes more vigorous; this tends to slow the downward advance of the interface. And second, the extra heat put into the lower layer speeds up density equalization between the two layers and thus promotes entrainment mixing across the interface. The observed change in pace of the interface movement in the second test is primarily governed by these factors. The fact that p' is in the denominator in the above expression for u c implies that the second factor will become increasingly significant as the two liquids approach density equalization (p' --, 0). It might therefore be expected that migration of the interface in test 2 would initially be slower than in test 1, but would accelerate to a greater pace when density equalization was approached. This trend is clearly demonstrated in Figu r e 8.
Concluding remarks Flow visualization experiments show that mixing of two stratified layers generally involves two stages in sequence: migration of the interface and rapid mixing between the remaining layers. Comparison of the two
Cryogenics 1993Vol33, No 12 1123
Mixing of layers and rollover in LNG." J. Q. Shi et al. rollover tests discussed above suggests that the base to side heat flux ratio is a major factor in determining the mode and intensity of the subsequent mixing at a rollover incident for a given side wall heat flux and density difference between the layers. A larger base heat flux is likely to lead to more severe rollover. Consider an extreme case in which the heat flux into the base is sufficiently large, so that the entrainment rates on the two sides of the interface are balanced. There will be no interface movement prior to density equalization of the liquids, but the mixing, or rollover, will be most severe. This rollover mode is assumed in the simulation models proposed by Germeles and others in analysing the La Spezia rollover incident. However, rollover is not necessarily a dramatic event. A recent review of the 41 L N G rollover incidents 14 shows that in only a few cases does the peak vapour-evolution rate exceed 20 times the normal evaporation rate. Therefore these models are valid only for the most dramatic rollover incidents, in which the migration of the interface is negligible prior to rapid mixing. This could well be the case in the La Spezia incident since the base heat flux (20.82 W m - 2) was three times the side heat flux (6.94 W m - z ) .
References l
Sarsten, J.A. LNG stratification and rollover Pipeline Gas (1972) 199(11)37 39
1124.
Cryogenics 1993 Vol 33, No 12
2 3 4 5
6
7 8 9 10 !1 12 13
14
Germeles, A.E. A model for rollover Adt: Cryog En,q (1976) 21 326 -336 Chatterjee, N. and Geist, J. Spontaneous stratification in LNG tanks containing nitrogen A S M E Paper 76-WA/PID-6 (1976) Heestand, J. et al. A predictive model for rollover in stratified LNG tanks A1CHE d (1983) 29 199 207 Nakano, R. et aL An experimental study on the mixing of stratified layers using liquid freon, paper presented at GASTECH, session VI, paper 7 (1982) Morrison, D.S. and Richardson, A. An experimental study on the stability of stratified layers and rollover in LNG Proc Low Temperature En,qineerin,q and Cryo,qenics Con[ session 01, paper 1 (1990) Sugawara, Y. et aL Rollover test in LNG tank and simulation model Adv Cryo,q En,q (1983) 29 805 811 Morioka, M. and Enya, S. Natural convection of stratification fluid in vessel (report 1, phenomenon observation) 1HI Report D211 Ishikawajima Harima Heavy Industries Co., Japan (1983) Agbabi, T. PhD Thesis Southampton University, UK (1988) Shi, J.Q. Numerical modelling and experimental study of rollover in cryogenic liquids and liquid freon PhD Thesis Southampton University, UK (1991) Beduz, C. et aL Thermal overfill and the surface vaporization of cryogenic liquids under storage conditions Adv Cryaq En~t (1983) 29, 795 803 Tritton, D.J. Physical Fluid Dynamics Van Nostrand Reinhold, UK Turner, J.S. The influence of molecular diffusion on turbulent entrainment across a density interface J Fluid Mech (1968) 33 639 656 Aeton, A. and Van Meerbeke, R.C. Rollover in LNG storage an industry view Proc 8th Int Co,!f on lique[ied Natural Ga~s session III, paper 12 (1986)
A numerical model has been developed to study the mixing of two initially stratified layers which are subjected to a uniform lateral heat flux. An important distinction is made between the free surface and the liquid/liquid interface with regard to the different flow characteristics of the two layers. In the upper layer where warm liquid is cooled at the evaporating surface, the convective circulation is featured by a strong downward core flow; in contrast, the fluid flow in the lower layer is mainly confined to the wall boundary and is much weaker. Flow visualization experiments show that mixing of two stratified layers generally involves two stages in sequence: migration of the interface and rapid mixing between the remaining liquids. The interface movement is due to entrainment mixing at the interface. When the two layers approach density equalization, the interface becomes increasingly unstable and the core flow in the upper layer is able to break into the lower layer. The base to side heat flux ratio appears to be a major factor in determining the mode and intensity of the subsequent mixing at a rollover incident.
Keywords: numerical modelling; f l o w v i s u a l i z a t i o n ; LNG
Nomenclature d g k p Pr O Ra* Ri o Rs AS ue ~s v r
Liquid depth (m) Gravitational acceleration (m s-2) Thermal conductivity (W m-1 K - 1 ) Fluid pressure (N m-2) Prandtl number ( = v/s) Heat flux (W m-2) Modified Rayleigh number [ = (gflxd4dl)/(kctv)] Overall Richardson number Salinity Rayleigh number [ = (gflsASd3)/(ctv)] Initial salinity concentration difference across interface (kg m-3) Entrainment velocity (m s-1) Thermal diffusivity (m 2 s-1) Salinity diffusivity (m 2 s 1) Kinetic viscosity (m E S-1) Diffusivity ratio (~J~)
It is widely known that stratification of distinct layers in storage vessels containing cryogenic liquids may lead to rollover. Liquefied natural gas (LNG), being a multicomponent liquid, is the most common cryogenic liquid in which stratification occurs. LNG consists of mainly methane, higher hydrocarbons and sometimes nitrogen. The stratification could be due to preferential evapora0011-2275/93/121116-09 © 1993 Butterworth-Heinemann Ltd
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Cryogenics 1993 Vol 33, No 12
p Ap
P0
Liquid density (kg m -3) Density difference across density interface (kg m - 3) Liquid density at reference point (kg m-3) Thermal expansion coefficient [ = - (i/p)(ap/aT)]
~s
Salinity expansion coefficient [ = (1/p)(Op/OS)]
N o r m a l i z e d variables (against value enclosed in [ ] )
S T t u, w x, z
Salinity concentration [AS] Temperature [qd/k] Time [0.01 de~s] Velocity components in x and z directions, respectively Is~d] Co-ordinates I-d] Vorticity [~/d 2] Stream function [~]
tion of nitrogen, or filling of a cargo with different density from that of the heel. In 1971, a dramatic incident of rollover was experienced at a LNG terminal tank in La Spezia, Italy 1. Under normal storage conditions heat leaking into the liquid from the surroundings is removed by evaporation at the free surface. A convection loop is established in
Mixing of layers and rollover in LNG : J. Q. Shi et al.
which liquid in a boundary layer flows upwards over the vessel walls, absorbing heat from the surroundings, then across the free surface where the heat is removed by evaporation. In the case of two stratified layers, the boundary layer in the lower layer is unable to reach the surface. Heat exchange between the two layers is by conduction through the interface. If this is insufficient to match the heat leaking in from the vessel walls and the base, superheat will build up in the lower layer. As time progresses, the density of the upper layer increases steadily through weathering and gradually approaches that of the lower layer, which is also changing with time. Depending on the net gain or loss of heat and the changes in its composition, the lower layer may become less dense or denser. In time the densities of the two layers become equal, at which point they mix rapidly to form one uniform layer. This process is known as rollover. When the lower layer liquid is superheated with respect to the tank pressure, as is often the case, rollover is accompanied by a sudden increase in vapour evolution and tank pressure. Based upon this understanding of rollover, several simulation models have been proposed. 2 4 In analyzing the well-known La Spezia rollover incident, which has been well documented by Sarsten x, considerable success has been achieved in predicting the time to rollover. These models assume that heat and mass transfers between adjacent layers are by molecular diffusion only and no movement of the interface is allowed in the approach to density equalization. In other words, there is no entrainment mixing across the interface. Laboratory experiments on mixing of two stratified layers under different heating conditions have been carried out by Nakano et al. ~ and more recently by Morrison and Richardson 6 using liquid freon and butane/' pentane mixtures, respectively. In both studies two modes of rollover were observed, identified by the migration and disappearance of the interface. With only lateral heating into the upper and lower layers, the liquid/liquid interface is observed to migrate to the bottom of the tank. In contrast, when only the base is heated the interface position remains almost unchanged prior to the rapid mixing between the layers. When both side and base heating are applied, the outcomes are not so clear cut. Clearly this heating condition is more realistic. Rollover tests carried out in an inground L N G tank using L N G from two different sources showed that the interface level falls gradually, and then rapidly just before intense mixing between layers takes place 7. There are two mechanisms reported in the open literature concerning the interface migration, namely boundary layer penetration at the wall and entrainment mixing at the interface. The former was proposed by Morioka and Enya s, who used water in their rollover experiments. When the densities of the two layers approach equalization, the boundary layer flow of the lower layer at the wall penetrates into the upper layer so that mixing of the two liquids takes place. This boundary layer penetration was also observed by Nakano et al. in their experiments using liquid freon. But from a later study v in which rollover tests were conducted in an actual storage tank, Sugawara et al. concluded that entrainment mixing across the interface is mainly responsible for the vertical
movement of the interface. The observation that the interface movement is closely related to the stability of the interface and the heating position led Sugawara et al. to suggest that the entraining velocity can be expressed as a function of the Richardson number near the interface. The entrainment mixing mechanism is also reported by Agbabi 9, who carried out an experimental study using cryogens to simulate the mixing of a two-layer L N G system. In this paper, a combined numerical and experimental study of the rollover phenomenon is reported. The aim of the numerical simulation was to achieve a better understanding of the mixing process of two stratified layers in cryogenic liquids, and in particular the interface movement and its driving forces. A numerical model for laminar heterogeneous fluid flows was developed. In the light of the numerical findings, flow visualization experiments were carried out to study rollover events in a specially constructed tank.
N u m e r i c a l simulation Numerical model
Free convective fluid flows in rectangular tanks or vessels are modelled as two-dimensional problems. For a heterogeneous flow system involving two incompressible miscible liquids, the governing differential equations are the Navier-Stokes equations, the continuity equation and the transport equations for thermal energy and the solute concentration. It was decided in the model to use the vorticity stream function method rather than a primitive variables approach. The advantage of this is two-fold: 1 by eliminating the pressure terms in the Navier Stokes equations and introducing the vorticity and stream function, the number of independent variables is reduced by one (from p, u, w, T a n d S to cj, ~, Tand S); and 2, the use of a staggering grid, which is normally required when the pressure terms are retained, is avoided. Hence the set of differential equations (in dimensionless form) to be solved is
(%
8(M.)
?t +
8(wO
(?x +
( ~i~~
~?z = P r \~ -x}
+
,~25,~ " -PrRa* ~2z/
,?T
?x
+ Pr R s ~ CX
~T ~(uT)(, ¢?(wT) ~2T ¢")2T Ot + ~x + ~z = ? 2 x + g'2z
8S
6q(IAS) O(wX)
~?~O (p2X
/(q,2S
~2S~
024' -{-
g?2 z
=
The vorticity ( 0 and stream function (qJ) are related to the velocity components u and w by ~u 5, = r?z bt ~
~ , W ¢,Z
c~w
~x -=-- - OX
Cryogenics 1 9 9 3 V o l 3 3 , No 12
1117
Mixing of layers and rollover in LNG: J. Q. Shi et al. 1.0
Heat flux
Heat flux
Z
X
0.0 0.0
0.5
1.0
F i g u r e 1 Solution domain: x = 0.0 and 1.0, non-slip walls; z = 0.0 and 1.0, non-slip adiabatic base and isothermal free surface, respectively
A detailed description of the discretization method and the solution algorithm used is given by Shi 1°. The solution domain (with normalized dimensions), together with the boundary conditions, are shown in Figure 1. In the model, an isothermal free surface is assumed to reflect an evaporating surface. Temperature measurements carried out by Beduz et al. 11 on a pool of evaporating liquid nitrogen showed that there exists at the surface a very thin conducting layer with a steep temperature gradient. Accordingly, very fine control volumes are required at the surface region. The mass loss due to surface evaporation is neglected so that a fixed solution domain is maintained throughout the computation runs.
Numerical results and discussion A two-layer stratification was considered with liquid nitrogen (LIN) in the upper layer and mixtures of LIN and a small amount of liquid oxygen (LOX, which is heavier than LIN) in the lower layer. The fluids were motionless and at the saturation temperature of LIN (77.4 K). A stepwise profile of LOX concentration was assumed across the interface, which was initially at the mid-height of the cavity. A uniform heat flux was then applied to the side walls. Numerical predictions of one simulation run with R a * = l0 T and Rs = 3 x 106 are presented and discussed. The two layers are completely mixed at t = 17.5 (normalized against 0.01d2/c0. To facilitate the understanding of the mixing process, four snapshots are displayed in Figure 2 which illustrate the evolution of the flow at different stages. Each snapshot consists of four contour plots, from left to right, temperature (T), LOX concentration (S), density (normalized as S - TRa*/Rs) and stream function (~). Only the left half of the solution domain is presented because of the symmetry of the flows about the central line. From the contour plots in Figure 2, it can be seen that the mixing is a gradual process, marked by the steady descent of the interface level. The migration of the interface appears to result from the entrainment of heavier liquid into the upper layer where mixing takes place. Upon impinging on the interface, the central (or core) downward flow in the upper layer entrains liquid from the interface as it flows across it. This is clearly shown in
1118
Cryogenics 1993 Vol 33, No 12
the LOX concentration contours, where the distributions of LOX concentration in the upper layer closely follow the patterns of the stream lines. Entrainment also takes place at the lower side of the interface, whereby liquid is entrained from the interface and mixed into the lower layer. The observed interface movement is therefore the net result of these two opposing mixing processes. This continuing process of entraining and mixing results in a steady contraction of the lower layer. Meanwhile, the core flow is seen to penetrate progressively deeper through the interface as the two layers approach each other in density. Eventually the liquids completely mix to form one convecting layer. The rate of heat loss at the surface as a fraction of the heat input, together with the interface movement, are plotted in Figure 3. It can be seen that the heat loss at the surface rises steadily until it reaches the first peak at t = 14.0. This time period sees a decline in the interface level, the interface descending at an increasing speed. After undergoing a period of oscillations, the surface heat flow rate reaches the highest peak and then declines to the equilibrium value. . At t = 0.5 (Figure 2a), separate convective circulations are established in the two layers. With continuing heating, convective flow in the upper layer becomes progressively stronger, whereas the flow in the lower layer weakens steadily. At t = 0.5, the velocities of the core flow in the upper layer and the boundary layer at the wall in the lower layer reach a maximum at z = 0.83 and 0.30, respectively. The profiles of vertical velocity at these two levels are shown in Figure 4a. The ratio of the velocity between the core flow (profile 1) and the boundary layer (profile 2) is around 1. At t = 12.0, the maximum core flow velocity is increased by a factor of over 3, whereas the maximum boundary layer velocity in the lower layer is halved, as shown in Figure 4b. To explain this contrast in the velocity fields, one needs to examine the temperature distributions in the two layers, since the convections are primarily driven by the temperature differentials. In the upper layer where the surface temperature remains at its initial saturation value, warmed fluid rising from the wall boundary layer is cooled as it flows parallel to the surface. The cooled liquid becomes denser than the liquid surrounding it and sinks into the bulk along the central line as a core flow. With the continuing warming up of the bulk liquid, this temperature differential increases with time. In addition, the core flow is further accelerated by the gradient of LOX concentration near the central line (Figure 2). The situation in the lower layer is quite different. The size of the temperature differential across the interface is restrained by the fact that liquids on both sides are warming up, even though at a different rate. Therefore, as warmed liquid from the wall boundary layer flows along the interface, it is not sufficiently cooled to be driven into the bulk, but rather accumulates at the interface. This results in the formation of a stable thermal stratification with the warmest liquid at the top, as is clearly shown in the temperature contours in Figure 2. This stable stratification clearly has an adverse effect on the rising boundary flow. As has been mentioned above, the interface descends at an increasing speed. This can be related to two factors: steady reduction in the density difference across the interface and a progressively stronger core flow.
Z O
CO CO
LO ~D ~0 < O
-.L
33
D
C) < o Q
,
,
,
,
I
I
I
/ ,
I
,
I
'
Density
(b) t = 6.0
0.5(0.0)
I
,
a) t = 0 . 5
,
Density
'L
0.5(0.0)
900 ~
ConcentrQt ion
0.5(0.0)
emperature
0.5(0.0)
I
Concentration
~
,
,
,
n
./
0~ I
i
i
I
i
.5
0.5
i
Stream Function
.0)
0.5(0.0)
'
0.5(
I~
Stream Function
Z
).0' U 0.0
].5-
1 .0-
~"
0.0
).0-
)).5. .5-
\ N,
I
,
,
I
\
i
i
i
(d) t = 17.5
0.5(0.0)
I
I
I
Densitg
(c) t = 1 2 . 0
,o~,
Densitg
, ~ l ~ - ~,j 0.5(0.0)
Concentration
0.5(0.0)
Temperature
X
,
:oncentrat Ion
0.5(0.0)
Temperoture
,
i
0.5(0.0)
I
0.5
,
I
,0
i
,
I
0.5
;tream Function
).5( 0.0 ) 0.5(i
Stream Function
: i g u r e 2 Predicted evolution of f l o w leading to complete mixing of t w o initially stratified layers w h i c h are subjected to uniform lateral heating. (The contour heights have been scaled for )resentation).x ~ 0.0, heated w a l l ; x ~ 0.5, central line of d o m a i n ; z = 0 . 0 , adiabatic base; z = 1 0, free surface
).00.0
).5
.0"
I°0, 0.0
~).5'
.0'
Temperature
C~
~b
r---
.'11
5'
Mixing of layers and rollover in LNG." J. Q. Shi et al. Surface heat loss ratio
Interface level
1.4
0.7
1.2
°°
Flow visualization of rollover
Experimental apparatus and testing fluid
"V
1
0.5 0.4
0.8 0.6
0.3
ce level
0.4
0.2
0.2
0.1
0
4
8
12
16
20
,
0 24
13me (Normalized) Figure 3
Predicted surface heat loss and interface movement
Velocity 4-
0.0 2 l:z=
1
.
2:
2: z = 0.15
\
\ I
0.0 (a)
I
I
P
0.5 0.0)
1.0
A rectangular tank was constructed to conduct flow visualization experiments, which is schematically shown in Figure 5. The tank is 20 cm long with a cross-section of 12 cm x 12 cm. Independently controlled resistance heaters are mounted on the side walls and the base. The front window is made of 9 mm thick Perspex and the tank is insulated with thick polystyrene to reduce heat exchange with the surroundings. Visualization of flows is made possible by seeding the flow media with foreign particles with good light reflectability. In this technique, density compatibility between the seed particles and the flow medium is essential. In the light of the findings from the numerical simulations, seeding only the upper layer is desirable so that the predicted movement of the interface can be observed. Liquid freon 11/113 mixtures, which are much easier to handle than volatile cryogens such as LIN and LOX, were used to set up a two-layer stratification with pure freon 11 in the upper layer. However, using freon 11 as test liquid has a drawback, as was discovered during the experiments. The saturation temperature of freon 11 is 23.8°C under atmospheric pressure, which was a couple of degrees higher than the ambient temperature in the laboratory. This implies that when freon 11 stored in a container is heated, its surface temperature may rise according to changes in its vapour (partial) pressure. This surface condition therefore does not conform to the normal storage conditions of cryogenic liquids. To achieve isothermal conditions, the generated freon vapour needs to be able to dissipate rapidly, so that a near constant vapour (partial) pressure can be maintained above the surface. This was achieved by leaving the test tank unsealed at the top, or loosely covered by a Perspex plate. This means, unfortunately, that the boiloff rates could not be measured directly during rollover tests.
(b)
Figure 4 Profiles of vertical velocity in upper (curve 1 ) and lower (curve 2) layers. (a) t = 0.5; (b) t = 12.0 w
Limitation of model Preliminary simulation runs with a layer of pure L1N show that when Ra* exceeds 5 x 10 s, instability starts to occur in the core flow 1°. The core flow has a velocity profile similar to that of a free jet (Figure 4). In general, shear flows away from any solid boundaries are much less stable than those near a wall. The order of critical Reynolds number for free jets is typically 10, compared with typically 1000 for wall boundary layer flows 11. In the two-layer system, the lowest Ra* at which instability starts to occur falls below 10 s, due to the extra driving force for the core flow provided by the solute distribution there. Using the properties of LIN at the normal boiling point (77.4 K) and assuming a liquid depth of 0.1 m, Ra* = l0 T represents a side wall heat flux of the order of
A---q~ v
D c
E
10-3 Wm -2
Although this heat flux value is unrealistically small, the above numerical predictions are confirmed by flow visualization experiments.
1120
Cryogenics 1993 Vol 33, No 12
Figure 5 Flow visualization system. A, Beam of parallel light; B, reflecting mirror; C, illuminated section; D, test tank; E, viewing window
Mixing of layers and rollover in LNG : J. Q. Shi et al. Experimental procedure
Experimental results
The tank is first filled with freon 11 to ~ 5 cm high and left for some time to reach equilibrium. The mixture of freon 11 and a small amount of freon 113 is then slowly bottom-filled through a thin capillary to reach a total liquid depth of l0 cm. After seeding of the particles, the tank is left for about half an hour before the start of heating. A video recorder is placed in front of the window to record the rollover events.
Two rollover tests are reported here, one with side heating only (test 1), the other with both side and base heating (test 2). In both tests, the pre-filling density difference between the layers was l~o and the input heat flux was .~ 40 Win-2. Flow visualization photographs displaying the evolution of flow in the illuminated plane in these two runs are given in Figures 6 and 7, respectively. These photographs were taken from video recordings
Figure 6 Flow visualization photographs showing mixing of two stratified layers (lateral heating only)
Cryogenics 1993 Vol 33, No 12
1121
Mixing of layers and rollover in LNG." J. O. Shi et al.
(2c) t = 140 minutes
Figure 7 Flow visualization photographs showing mixing of two stratified layers (lateral and base heating)
and an exposure time of 4 s was used. The trajectories of the seed particle were captured as bright line segments on the photographs. It should be noted that the length of one trajectory does not necessarily represent the whole distance travelled by this particle in the period of 4 s since the flow was not strictly two-dimensional. A particle could well enter and/or leave the illuminated section during the exposure time. Therefore the mean velocities of the seeding particles directly derived from the lengths of their trajectories could well be smaller than their true values. Figure 8 shows the movement of the interface in the two tests.
1122
Cryogenics 1993 Vol 33, No 12
Side heating only (test 1). Three distinct stages are observed leading to the complete mixing of the liquids after ~ 200 min. (i) 0-45 rain: A two-layer stratification is clearly observed in photograph la. A close look at the seeded upper layer reveals a step change in the brightness of the picture, with the lower region being less bright. This variation in illumination reflects density stratification in the upper layer, which is apparently caused by mixing during the filling of the lower layer. As a result of this stratification, flow is confined to the brighter region only and there is hardly any visible motion in the liquid below.
Mixing of layers and rollover in LNG." J. Q. Shi et al.
Interface level 70 ~
60
Discussion
intermediate layer
ii!ii de & base heating
50 40 30 20 .
N
~side ! heating / ~ only ~ /
D
10 0
\ ,
0
I
50
,
I
,
100
[
~E
150
R 200
Time (minutes) Figure 8
Observed interface movements (see text for details)
The above flow visualization results demonstrate that rollover is characterized by the initial gradual descent of the interface level over a period of time, then rapid mixing between the remaining layers. This is in agreement with the results of Sugawara et al. v. The migration of the interface level is a result of entrainment mixing, whereby the heavier liquid is entrained and mixed into the upper layer. As the liquid densities approach equalization, the interface becomes increasingly unstable. Eventually the core flow in the upper layer breaks into the lower layer and the liquids are quickly mixed. Turner 13 used a vertically oscillating grid to study mixing across a density interface. When stirring was applied at the same rate in the two layers, the interface remained sharp and central; otherwise the interface moved away from the region of more vigorous stirring until the entrainment rates on the two sides were balanced. The following relationship was proposed for the entrainment velocity u~ Ue .OCRio 2,,2
U
The core flow in the upper layer can be clearly identified. In the next 30 rain, more and more liquid is brought into the motion through the penetration of the core flow and entrainment mixing (photograph lb). Eventually, the entire upper layer becomes one convecting loop (photograph lc). Throughout this stage the interface between the two main layers remains unmoved. This stage corresponds to AB in Figure 8. (ii) 45-185 rain: In the first 30 min of this stage, the interface hardly moves, indicating that the density difference across the interface is still substantial. The interface then starts to move downwards, slowly at first but gradually gaining in speed. Meanwhile, the interface becomes less and less stable (photographs ld and le). This period corresponds to BD in Figure 8. (iii) 185 200 min: This stage (DE in Figure8) is marked by intensive mixing. With the liquids approaching density equalization, the core flow in the upper layer is able to depress the interface and the latter becomes unstable (photographs if and lg). Eventually the core flow breaks into the lower layer (photograph l h) and within minutes the liquids are completely mixed to form one uniform layer.
Base and side heating (test 2). The two layers are completely mixed after 185 min of heating. An intermediate layer is not observed in this test. This situation can arise if the density of the intermediate layer is higher than that of the seed particles. Two distinct stages (represented by PQ and QR respectively in Figure 8) are observed, identified by migration of the interface (PQ) and rapid mixing between the remaining liquids (QR). These two stages correspond to the last two stages in test 1. In the first stage (0-170 min, photographs 2a-d), the interface descends by 20 mm. Within the next 15 rain, the remaining lower layer (32 mm thick) is completely mixed into the upper layer. The final mixing in this test is more intensive than that in test 1, as reflected in photographs 2d-h.
where Rio is an overall Richardson number, defined as Rio = gp'l-/U 2. U and L are the velocity and length scale characteristic of the flow motion near the interface and p' is the normalized density difference across the interface, p' = Ap/po. Substitution of the expression for Rio into the above equation yields U4 Ue ~ ( g p , C ) 3 : 2
The migration of the interface is therefore controlled by the differential in entrainment rates across the interface. In both the rollover tests the interface moved downwards. This confirms the numerical prediction in the last section that the convective flow is more vigorous in the upper layer than in the lower layer. Here the importance of maintaining an isothermal surface during experiments is emphasized. Compared with test 1, the addition of base heating in test 2 has two effects on the interface movement. First, convective flow in the lower layer becomes more vigorous; this tends to slow the downward advance of the interface. And second, the extra heat put into the lower layer speeds up density equalization between the two layers and thus promotes entrainment mixing across the interface. The observed change in pace of the interface movement in the second test is primarily governed by these factors. The fact that p' is in the denominator in the above expression for u c implies that the second factor will become increasingly significant as the two liquids approach density equalization (p' --, 0). It might therefore be expected that migration of the interface in test 2 would initially be slower than in test 1, but would accelerate to a greater pace when density equalization was approached. This trend is clearly demonstrated in Figu r e 8.
Concluding remarks Flow visualization experiments show that mixing of two stratified layers generally involves two stages in sequence: migration of the interface and rapid mixing between the remaining layers. Comparison of the two
Cryogenics 1993Vol33, No 12 1123
Mixing of layers and rollover in LNG." J. Q. Shi et al. rollover tests discussed above suggests that the base to side heat flux ratio is a major factor in determining the mode and intensity of the subsequent mixing at a rollover incident for a given side wall heat flux and density difference between the layers. A larger base heat flux is likely to lead to more severe rollover. Consider an extreme case in which the heat flux into the base is sufficiently large, so that the entrainment rates on the two sides of the interface are balanced. There will be no interface movement prior to density equalization of the liquids, but the mixing, or rollover, will be most severe. This rollover mode is assumed in the simulation models proposed by Germeles and others in analysing the La Spezia rollover incident. However, rollover is not necessarily a dramatic event. A recent review of the 41 L N G rollover incidents 14 shows that in only a few cases does the peak vapour-evolution rate exceed 20 times the normal evaporation rate. Therefore these models are valid only for the most dramatic rollover incidents, in which the migration of the interface is negligible prior to rapid mixing. This could well be the case in the La Spezia incident since the base heat flux (20.82 W m - 2) was three times the side heat flux (6.94 W m - z ) .
References l
Sarsten, J.A. LNG stratification and rollover Pipeline Gas (1972) 199(11)37 39
1124.
Cryogenics 1993 Vol 33, No 12
2 3 4 5
6
7 8 9 10 !1 12 13
14
Germeles, A.E. A model for rollover Adt: Cryog En,q (1976) 21 326 -336 Chatterjee, N. and Geist, J. Spontaneous stratification in LNG tanks containing nitrogen A S M E Paper 76-WA/PID-6 (1976) Heestand, J. et al. A predictive model for rollover in stratified LNG tanks A1CHE d (1983) 29 199 207 Nakano, R. et aL An experimental study on the mixing of stratified layers using liquid freon, paper presented at GASTECH, session VI, paper 7 (1982) Morrison, D.S. and Richardson, A. An experimental study on the stability of stratified layers and rollover in LNG Proc Low Temperature En,qineerin,q and Cryo,qenics Con[ session 01, paper 1 (1990) Sugawara, Y. et aL Rollover test in LNG tank and simulation model Adv Cryo,q En,q (1983) 29 805 811 Morioka, M. and Enya, S. Natural convection of stratification fluid in vessel (report 1, phenomenon observation) 1HI Report D211 Ishikawajima Harima Heavy Industries Co., Japan (1983) Agbabi, T. PhD Thesis Southampton University, UK (1988) Shi, J.Q. Numerical modelling and experimental study of rollover in cryogenic liquids and liquid freon PhD Thesis Southampton University, UK (1991) Beduz, C. et aL Thermal overfill and the surface vaporization of cryogenic liquids under storage conditions Adv Cryaq En~t (1983) 29, 795 803 Tritton, D.J. Physical Fluid Dynamics Van Nostrand Reinhold, UK Turner, J.S. The influence of molecular diffusion on turbulent entrainment across a density interface J Fluid Mech (1968) 33 639 656 Aeton, A. and Van Meerbeke, R.C. Rollover in LNG storage an industry view Proc 8th Int Co,!f on lique[ied Natural Ga~s session III, paper 12 (1986)