Analysis of large LNG spills on water part 1: Liquid spread and evaporation

The first part o f this two-part review considers the theoretical and experimental results obtained on liquid spread and evaporation of large LNG spills on water. Both instantaneous spills, in which the spill time is much smaller than the time for complete vaporization, and continuous spills are considered. Also applications o f the correlations are discussed.

Analysis of large LNG spills on water Part 1: Liquid spread and evaporation B. O t t e r m a n

Marine transport of liquefied natural gas (LNG) has been carried out safely since 1959.1 Sound engineering practice and environmental impact regulations, however, require assessment of potential hazards due to accidental spills of LNG on water. Factual assessment, in turn, must be based on sound theoretical predictions and experimental data. To obtain such data, many organizations and corporations worldwide have undertaken research programmes. In the United States these include, but are not limited to: US Coast Guard, American Gas, and American Petroleum Institute. At least two distinct hypothetical accidental situations can be postulated. In the first, LNG is ignited at or close to the point of release, and fire of high intensity and probably of short duration would result. 2 In the second, LNG spreads on water, vaporizes, and forms a flammable cloud which drifts with the wind until it is ignited or dispersed to concentrations below the flammability limit. In light of double wall LNG tanker construction universally employed, 3 postulation of a tanker accident which results in large spillage necessarily implies circumstances in which ignition is more likely than significant flammable vapour cloud travel. Nevertheless, it is the latter case that has received wider attention. The main features which constitute the development and dispersion of a LNG vapour cloud are as follows: LNG being lighter than water and insoluble in water will rapidly spread over the surface of the water (Fig. 1). As it does, it will absorb heat from the water and vaporize. As the LNG spreads, the rate of vapour generation increases because there will be more surface over which heat can be transferred. A vapour cloud is formed that grows in size until all of the liquid has evaporated. The diameter of the cloud will be much greater than its height. Although methane vapour is colourless, the cloud will appear white due to condensation and/or freezing of entrained water vapour.

Heat from water

Fig.1

Spreading and evaporating LNG

pheric pressure methane vapour is more dense than air. Consequently, there is no tendency for the vapour to rise. On the contrary, experimental data 4 indicates that the vapour cloud continues to spread radially; the spreading cold vapour entrains the surrounding air and increases in height. Entrainment increases the cloud bulk temperature, but calculations show 5 that, generally, the density of mixture always remains above that of the diluting air. In instances where the heat input from the surface of the water and/or heat input from the condensing and freezing water vapour is significant, 6 the vapour cloud becomes neutrally buoyant. At this point, it stops spreading radially. Although in principle, it is possible for an LNG cloud to become positively buoyant, this has not been observed experimentally except possibly in one test (No 16) discussed in reference 4.

The initial temperature of the LNG vapour formed is approximately -259°F. At that temperature and atmos-

As the cloud moves downwind, air turbulence further dilutes the cloud. In the absence of ignition, as this process continues, the contents of the cloud will be dispersed in concentrations below the lower flammability limit. Characteristics of this phase are strongly dependent on weather conditions. In gusty weather, the cloud will disperse rapidly; in stable weather, it will persist for longer periods of time.

The author is with the Department of Engineering and Computer Sciences, Hofstra University, Hempstead, New York 11550, USA. Received 7 April 1975.

In summary, the overall physical process of vapour cloud formation and dispersion consists of the following three interrelated phases:

C R Y O G E N I C S . AUGUST 1975

455

1. Liquid spread and evaporation. 2. Vapour spread and heating to the state of neutral buoyancy. 3. Vapour drift and dispersion.

Vopour

In part 1 of this report the theoretical and experimental findings of liquid spread and evaporation are reviewed. The paper concludes with application of the correlations discussed.

~Spreoding LNG

li~iiiii!i~!iiiiiiiiiiiii!!!ii!ili

!iiiiiiiiiiiiiiiiiiii!!i!iiiiiiiiiiiiiiiiiilh(t) I

Instantaneous spills Spills in which the time required for the liquid to leave the containment vessel is much smaller than the time for complete vaporization by contact with water (assuming the spill had been instantaneous) are referred to as instantaneous spills. The equations for the rate of spread, dr/dt, of an instantaneous spill are derived either by considering the LNG pool as a hydrostatically driven intrustion 6,7,8 of a fluid of density p, into a fluid (water) of density Pi, or by equating the spreading and resisting forces of the LNG mass. 2 Both approaches yield approximately the same results provided that the rate of spread is retarded only by inertial resistance. In fact, the latter method is no more exact than the former since the governing continuum and momentum equations are incomplete, requiring an additional boundary condition to specify the solution uniquely. 9 This was first recognized by Fannelop and Waldman 10 in connexion with analysis of oil spills. They proposed that leading edge of the spill moved through the water with a velocity of [gh1 AI] ~, where h I is the thickness of the spill at the leading edge and A l = (,oj - P)/Pl. Their analysis showed that during the spill at early times competition between inertia forces control spreading rate. At later times, the rate of spread is determined by the balance between the gravity force and the retarding viscous forces that act at the oil water interface. At very late times, further oil spread is forced by surface tension and is opposed by viscous forces at the interface. In the case of LNG, the nature of the latter two regimes is not well understood due to existence of vapour and hydrates at the interface. Fortunately, in most practical cases of LNG spilled on water, evaporation of most of the LNG is completed within the gravity-inertia regime and the other spread regimes need not be considered. 2 The equation governing a steady state intrustion of two fluids of infinite extent was obtained by yon Karman 11 in 1940. Use of the steady state Bernoulli equation yields the now famous expression for the velocity of the intruding front, U, U

= [2gA/h21 Y2

(I)

o q

Fig.2

Water

The mean thickness approximation

Thus ½ dt

\ 7rr2 ]

(4)

Hoult 9 points out the physics of the mean height approximation is not entirely clear, since a steady.state intrusion must always break, so that a speed between [ghlh ] Y=and [2gAlh ] ½ is possible. Indeed, a recent correlation by May and Perumal ~2 of experimental data given in references 4 and 5 yields a coefficient of 1.35. Our present knowledge of the leading edge physical conditions for both the height and fluid density ls inadequate. According to May and Perumal the material that is spreading can best be described as a 'foam' whose density is appreciable less than that of LNG. Clearly, instantaneous LNG spills are fundamentally unsteady and, as such, give rise to virtual mass effects. The latter are reflected in inertia change of the water as it resists acceleration by the LNG. Analyses in which the two media are coupled are not yet available. Whether such analyses would change the form of the rate of spread equation, or merely modify its coefficient, is not known. Clearly the best current approach is to modify the coefficient on the basis of experimental data. Integration of (4) requires a knowledge of the rate at which the spreading LNG diminishes with time due to vaporization Both Hoult 6 and Fay 7 neglected the change of volume with time, V= constant, and upon integration obtained

where h2, is the height of the intrusion at some distance from the leading edge. Fay 7 and Hoult 6 model the leading edge of an LNG spill as an intrusion. Consequently, for an aM-symmetric pool (Fig.2) dr

dt

- [2,zx lh] 'a

(2)

where h -

456

r =

t½

(5)

One would expect (5) to have greater validity for oil spills than LNG. The assumption of constant volume for LNG yields (neglecting other problems) an overestimate of the instantaneous spill radius. For the case where the density of LNG is 28.3 lb ft -3, (5) becomes r = 2.59 Vo~ t Y2

V 7Tr2

(3)

where the units of r, Vo, and t are ft, ft 3 (liquid), and s, respectively.

CRYOGENICS. AUGUST 1975

Fay and Hoult apply (5) in the following manner. From heat transfer considerations, they estimate the time, ~-, it takes to vaporize all of the LNG, and by substituting that time into (5) obtain an estimate for the maximum pool radius, R. Thus, R = 2.59 Vo ¼ r ½

and consequently, longer time required for complete vaporization and greater extent of pool spread than Fay's analysis. Specifically, Hoult's analysis 6 yields r Hoult = 9.7 (6)

Assuming, for the moment, that valid heat transfer analyses are employed, this procedure will yield an upper bound for the maximum spill radius. Fay 7 and Hoult in his early study, 6 both assumed that vaporization results in the formation of an ice layer beneath the spill. If this were the case, the rate of vaporization per unit area would decrease with time as the ice thickness beneath the spill increases. By assuming that the temperature varies linearly across the ice from LNG boiling temperature, Tb, at the top to the ice freezing temperature, Tf, at the bottom Fay obtains the following expression for the ice thickness ~5 ½

5 =

(pi~ki + 1 [2

(7)

ciPiAT)

where AT = Tf Tb, and ki, is the thermal conductivity, )q the heat of fusion, ci the heat capacity, and Pi the density of ice respectively. Note that the first term in the denominator represents the heat release due to phase change while the second represents the loss of sensible heat by the ise as it cools. By equating the heat required to vaporize the LNG to that removed from the ice, Fay obtains (after neglecting the variation of ice thickness with radial distance), the following expression for, r 1/3

TFay =

16rrgAipikiAT()q+1/2 ciAT)

RHoul t

Vo 5/12

(11)

8.1 Vo s/12

Thus, according to both of the above analyses, the time to complete vaporization varies with the cube root of the spill volume and rate of vaporization as the 2/3 power of the spill volume. However, Hoult's analysis, because it neglects the sensible heat loss by the ice, predicts a vaporization time which is approximately three times, and maximum pool radii which are twice as great, as those predicted by Fay. Unfortunately, conclusive experimental evidence does not currently exist which would clearly indicate whether ice formation occurs. In general, tests in which around 1 m 3 of LNG was spilled into shallow tanks showed ice formation; Is tests in which around 1 to 10 m 3 of LNG was pumped onto the surface of unrestricted open water did not indicate ice formation. 4,13 It should be noted that in all of the open water tests specific instrumentation capable of monitoring ice formation was not installed. Conclusions that ice did not form were based on visual observations and/or correlation of evaporation rate data. In case of the larger spills, 4 direct observation was impossible. In one Shell Research test, 13 on a day with no wind and completely still water surface, 'very much more than usual' ice is reported. In light of the expected small ice thicknesses (Table 1) uncertainty concerning ice formation is understandable. In recent pool boiling experiments of Jeje and Reid, 14 a thermocouple was placed 5 mm below the water surface and subsequently evaporation tests using nitrogen, LNG, ehtane were conducted. It is reported that: 'For short duration spills (less than one minute), little change is noted in the water temperature 5 mm or more below the surface. The surface temperature for liquid nitrogen remains above 0°C while in LNG and liquid and liquid ethane experiments the surface water temperature dropped below 0°C only when the initial water temperature was low.

(8)

Ice formed very rapidly on the water surface in liquid ethane spills. Ice may have been formed in LNG spillz but, if so, it was transient in nature and remelted quickly. No ice was found at the water surface in liquid nitrogen tests.'

where Vo is in ft 3 of liquid spilled and r in s. Moreover, substitution of (8) into (6) yields the following expression for the maximum pool radius

RFay = 4.70

(10)

In reference 6 Hoult gives r = 14.5 VoJ/3 a n d R = 10.4 Vo s/12 ; however, the author's calculations indicate that the correct values of the coefficients are 9.7 and 8.1, respectively.

Evaluation of physical constants corresponding to an average temperature o f - 1 0 0 ° F yields r = 3.00 Vol/3. However, the coefficient of 3.3 in (8) yields results which are constant with those given by Fay. 7 Substituting this value, the equation reduces to

TFay = 3.3 Vo 1/3

=

Vo1/3

(9)

where R is in fl and Vo is in f13 of liquid spilled.

The observations of Jeje and Reid, as well as other findings which will be discussed subsequently, indicate that for LNG

Table ] gives the ice thickness for different size spills as predicted by Fay's analysis. Note that the ice thicknesses are small varying from 0.29 in for a I0 m 3 spill to 1 in for a 20 000 m 3 spill.

Table 1. Ice thickness as predicted by Fay's 7 analysis

Hoult's analysis, 6 like that of Fay, assumes a linear temperature drop across the ice equal to Tf-Tb. Hoult, however, neglects the sensible heat loss by the ice as it cools. Since the latter is approximately a third of the latent heat removed, this assumption is not fully justified. It is because of this assumption, that Hoult's analysis predicts larger ice growth,

C R Y O G E N I C S . AUGUST 1975

Spill volume, m 3

Ice thickness, in

10

0.29

100 1 000 20 000

0.42 0.62 1.00

457

spills extensive ice formation is not likely; moreover should it occur, water surface temperatures which are near the bulk temperature of LNG are unlikely. This conclusion is supported by calculations in which the water is taken as a semiinfinite slab subjected to a sudden enviromental temperature equal to -259°F. For reasonable values of surface conductance, standard calculations involving the Fourier modulus indicate that it would take about fifteen minutes for the surface temperature to decrease to 90% of the bulk LNG temperature. This estimate is conservative since the calculation neglects latent heat removal. A key assumption in Fay's 7 and Hoult's 6 heat transfer analyses is that the ice surface temperature is equal to the LNG bulk temperature. Since experimental 14 and theoretical estimates indicate that this is not likely to be the case, the analyses of Fay and Hoult discussed above are not valid. Recently, Hoult, 8 as well as others, 2,12 have proposed heat transfer models in which ice formation does not occur. These analyses take account of turbulence at the LNG-water interface caused by relative motion. The fact that the importance of turbulence at the interface had been generally overlooked is surprising, in view of pertinent experimental evidence presented by the Bureau of Mines s in 1970. They reported experiments in which heat flux from water to LNG was measured in the following two different situations. In the first, LNG was poured directly onto the water surface. In the second, a thin aluminium plate was placed on the surface of the water and the LNG was poured onto it. Experimental data shows that the heat flux (and correspondingly the vaporization flux) in the latter experiments were approximately one-half of those observed without the plate. Since the thermal resistance and capacitance of the plate is negligible, the decrease in heat flux must be entirely due to the suppression of interfacial turbulence. The magnitude of the latter will depend primarily on velocity of the spreading LNG which in turn is a function of the volume spilled. Available experimental evidence 4,1s indicates that for large spill bodies on water, interfacial turbulence prevents ice formation, and that a typical vaporization is 0.04 lb sa ft "2. This corresponds to a regression rate of 1 in per minute, and a heat flux of approximately 30 000 Btu h a f t "2. Using the latter value of heat flux, Hoult s obtains an estimate for interfacial convective heat transfer coefficient, which gives

r = 7.9VoU4

(12)

R = 7.3Vo3/8

(13)

According to the above equation, time to complete vaporization varies with the fourth root of the volume spilled. Otterman's analysis also assumes that ice does not form. By equating the time integral of the vaporization flux to the weight spilled, he obtains (see Appendix) 1/2 (2~2) 1 Vo 1/4 r = ~ (gA01/4 /il/2 (14)

where/~ is in in per min, and Vo in ft 3 of liquid spilled. Moreover, the maximum pool radius, in ft, is given by Vo 3/8 R = 7.6

(16)

~1/----8-

Raj and Kalekar 2 have obtained an expression for rate of spread by equating the gravitational spreading force Fg, to the inertial resistance force F I where,

Fg = Irrh2 p gAt

(17) d2r

FI = - C Orr2hp) ~

(18)

According to the authors, the factor C in (18) is introduced to account for the fact that the inertia of the entire LNG layer is a fraction, C, of the inertia of the total mass, assuming that entire mass were accelerated at the leading edge acceleration d2r/dt 2. They assume that C is a constant in time. Equating (17) and (18), they obtain the spread law - Crp h =

gAt

d2r -dt 2

(19)

Simultaneously, they account for vaporization of the LNG with the following.mass conservation equation t

V(t) p = p Vo -7r f

q- -r 2 dr L

o

where L is the latent heat of vaporization. The term containing the integral on the ths represents the total mass lost by evaporation in a time duration t. Simultaneous solution of (19) and (20), together with the expression V = n r2h, yields third order non-linear differential equation. In view of the unknown value of C, four boundary conditions are required to specify the solution. The only boundary condition specified explicitly by Raj and Kalelkar is that V = Vo at t = 0. The remaining three constants are obtained by making their solution (for the case of nonevaporating fluid) identical to that of Fannelop and Waldman. The latter assume a leading edge velocity of [gh 1At ] 1/2. Consequently, this assumption is also present on the Raj and Kalelkar analysis. For the case where q is a constant, Raj and Kalelkar obtain the following expressions for the time to complete vaporization and the maximum radius of spill

r = 0.67

( p2L2V° )

(21)

q2 g A ~ -

where/~ is the pool regression rate. For LNG of density

1/8

28.3 lb ft "3, (14) becomes R = 1.0 12.4 vo 1/4 T --

458

h112

(20)

(15)

(p2L2gA'V°) q2

(22)

In above equations, the group q/Lp, is equal to the regression

CRYOGENICS. AUGUST 1975

rate,/i, of the LNG pool. In terms of/~, (21) and (22) become

r -

8.8 G ~

(23)

by2

7.4 Vo 318 R -

(24)

h~

where,/~ is in in per min, Vo in f13. We now compare the predicted results of the above spread analyses for the case of large instantaneous spills. Specifically, Table 2 lists the time to complete evaporation and maximum radius of LNG pool resulting from 4 000, 10 000~ 24 000, and 100 000 m 3 spills. To convert spill sizes from m 3 to tons, divide the former by two. This calculation is exact for LNG whose density is 28.3 lb ft -3, and approximate for heavier or lighter LNG. As indicated in Table 2, analyses of Hoult (non-ice), Otterman, and Raj and Kalelkar yield almost identical predictions for the maximum pool radius for the case of 1 in rain -1 regression rate. Surprisingly, the constant volume assumption in the spreading model as made by Hoult and average volume assumption made by Otterman, is not in the case of high regression rates, as approximate as first expected. It is also interesting to note that Fay's ice analysis yields about the same maximum pool radius as those in which ice formation is not assumed. However, this agreement is fortuitous. The above discussed spreading models predict that the pool would continue to grow as long as there is liquid. Experimental results, T M 5 however, indicate that LNG pools break up into discontinuous areas prior to complete evaporation. According to May and Perumal 12 this break-up appears to start at the centre of the pool (indicating that the thickness there is smaller than at the leading edge) and the last material to evaporate is a ring at the leading edge. According to Boyle and Kneebone, t 5 pool break-up begins when the quantity of LNG per unit area is approximately 0.16 lb ft 2. The latter is equivalent to an LNG depth of 0.006 ft. May and Perumal 12 have analysed the pool break-up data of Boyle and Kneebone as well as that of the Bureau of Mines s and Feldbauer, 4 and have proposed the following empiral correlation for the pool radius at break-up Rb

=

3.56 V~o'3S

(25)

where R b is in ft and Vo in ft 3 of liquid spilled. Note that the exponent on Vo is approximately equal to that pre-

dicted (0.375) by analyses Hoult, 8 Otterman (Appendix), and Raj and Kalelkar. 2 The coefficient, however, is approximately one-half of the value predicted by these analyses. The net effect is empirally predicted maximum pool radii (Table 3) which are approximately 20% smaller than the corresponding theoretical predictions of Hoult, Otterman, and Raj and KaMkar given in Table 2. This is not surprising since these analyses contain, as discussed previously, assumptions which would probably result in an overestimate of the maximum pool radius. In terms of hazards analysis, however, these theoretical models yield conservative estimates when compared to currently available estimates from empirical predictions. Note that models which assume ice formation yield estimates which are not supportable by currently available experimental data. Recent pool boiling experiments of Jeje and Reid 14 showed that the boiling characteristics of LNG on water are very much different from those of pure methane. Addition of 0.10 to 0.20% ethane, propane, or butane to pure methane can lead to increases in boiling rates of 50% or more, and heat fluxes as high as 55 000 Btu h 1 ft -2 were recorded. According to the P2j and Kalkelkar model, the maximum pool radius varies inversely as/~v, (or q'/'). Consequently, a two-fold increase in the regression rate reduces the maximum pool radius by 19%. Accordingly, the difference between the theoretically and empirically predicted values of maximum pool radius could be mainly due to the uncertainty in the heat flux. During the spreading and evaporation process, LNG 'ages' (fractionates), and the heat flux would vary accordingly. Continuous spills

In cases where discharge of LNG extends in a more or less continuous manner over a period of time that is considerably greater than r, the spill is referred to as continuous. In the absence of ice formation, the pool radius will grow

Table 3. Radius at pool break-up (25) for LNG of density 28.3 Ib ft ~3 Volume spilled, m3

R b, ft

4 000

508

10 000 24 000 100 000

698 957 1 567

Table 2. Comparison of maximum pool radius, and time to complete vaporization as predicted by different analyses Spill volume - liquid 10 000 m 3 24 000 m 3 r, min R, ft ~-, min R, ft

100 000 m 3 R, ft

Analysis

4 000 m 3 r, min R, ft

Fay 7

2.9

656

3.9

960

5.2

1385

8.4

2510

Ice

Hoult 6

8.5

1130

11.4

1650

15.4

2400

25

4300

Ice

Hoult 8

2.5

628

3.2

890

4.0

1250

5.7

2100

I in min "1

Otterman (Appendix)

4.0

650

5.0

915

6.3

1270

8.9

2170

Regression

Raj/Kalelkar 2

2.8

630

3.6

890

4.4

1236

6.3

2110

CRYOGENICS. AUGUST 1975

T, min

Heat transfer assumption

459

Table 4. Maximum pool radius for continuous spills each of 30 min duration

to a maximum steady state value such that the rate of evaporation from the pool equals the rate o f release, Q. That is

Total volume Rate of release spilled, m 3 m 3 min -1 Bbl min -1

Pool radius, ft

(26)

For the case where q = 30 000 Btu h l ft -2, L = 220 Btu lb l , and p --- 28.3 lb ft -3, (26) becomes R = 4.5 0,/2

4 000

133.3

833

130

10 000

333.3

2 086

205

24 000

800

5 000

318

(27) 15

where R is in ft and Q in Bbl rain q . Table 4 gives the maximum pool radii for three hypothetical spills of 4 000, 10 000, and 24 000 m 3, each having a duration o f thirty minutes. Note that the pool radii are approximately one-sixth of those predicted 2 for the case o f instantaneous spills o f the same total quantity. Consequently, the assumption o f instantaneous release produces a very conservative upper bound on the maximum pool size. Note also that in the case of continuous spills, a vapour plume is formed which for the duration o f the spill remains attached to the spill site. Since the rate o f vapour generation is much less than in the case where the same quantity o f LNG is released instantaneously, the distance that LNG vapour travel before they are diluted below the lower flammable limit will also be less. These, as well as other aspects of LNG vapour travel, are discussed in a subsequent paper.

Boyle, G. J., Kneebone, A. 'Laboratory investigations into the characteristics of LNG spills on water', API Project on LNG Spills on Water. Ref 6Z32, Washington, DC (1973)

Appendix If, £, is the vaporization flux then, IV, lb o f LNG will be vaporized in time, r, that is,

i LA (t) dt

W =

(A1)

o

where, A (t), is the instantaneous pool area. Experimental data on oil spills indicate that r 2 = (gA 1 111)y2 t

(A2)

Substitution of (A2) in (A1) yields The initial work on this paper was completed while the author was a member of the technical staff at Cabot Corporation. The author is indebted to Dr F. Feakes and Dr C. W. Shipman o f Cabot for their help.

T

W=

t

L 7r (gAl VOw t dt

(A3)

o

References 1 2

3 4

5 6 7 8 9 10 11 12 13 14

460

Booz, Allen Applied Research, 'Analysis of LNG marine transportation' Report MA-RD-900-74040 (1973) V I I - 2 Raj, P., Kalelkar, A. 'Fire hazard presented by a spreading burning pool of liquefield natural gas on water,' presented at Combustion Institute (USA) Western Section meeting (1973)

Wilson, J. J. Cryogenics 14 (1974) 115 Feldbauer, G. F., May, W. G. et al 'Spills of LNG on water vaporization and downwind drift of combustible mixtures,' Esso Research and Engineering Co, Report No EE61E-72 (1972) Burgess, D., Biotdi, J., Murphy, J. 'Hazards of spillage of LNG into water,' PMSCR Report No 4177, Bureau of Mines, US Dept of Interior (1972) Houit, D. 'The fire hazard of LNG spilled on water', Proc Conference on LNG Importation and Safety, Boston, Mass (1972) 87 Fay, J. A. Combustion Science and Tech (1973) 47 Houit, D. Private communications, Dept of Mechanical Engineering, MIT, Cambridge, Massachusetts (1974) Hoult, D. 'Oil spreading on the sea', Annual Review of Flifid Mechanics, Vol 4 (1972) Fannelop, T. K., Waldman, G. D. AIAA 10 No 4 (1972) 506 Von Karman, Th. Bull Amer Math Soc 46 (1940) 615 May, W. G., Perumal, P. 'The spreading and evaporation of LNG on water,' Contributed Paper, Process Ind Div Annual ASME Meeting (Nov 1974) Kneehone, A., Prew, L. R. 'Shipboard Jettison Tests of LNG onto the Sea', Proc LNG 4, Algiers (1974) Jeje, A. A., Reid, R. C. 'Boiling of liquefied hydrocarbons on water', Proceedings AGA/IGT Research and Dev Conference, Dallas, Texas (1974)

Assuming that { = constant, and that average volume of Vl during the process can be approximated by Vo/2 we obtain

w=L~r

gAj ~ -

/

t dt

(A4)

o

which after integration and substitution o f W = plVo, h = LIp yields ½ 7" =

2x/2

1

Vo '~

(AS)

For LNG whose density is equal to 28.3 lb ft -3 , this equation becomes

r = 12.4

Vo ¼

h~

(A6)

where, h is in in. per unit and Vo in fla. Substitution o:f (A6) into (A2) with V1 = Vo/2 yields

i1o R = 7.6 - -

3/8

hlla

(A7)

C R Y O G E N I C S . AUGUST 1975