The modelling of pool vaporization

The modelling of pool vaporization

Paolo Leonelli, Carlo Stramigioli and Gigliola Spadoni Department of Process and Chemical Engineering, University of Bologna, Viale de1 Risorgimento 2, I-40136 Bologna, Italy

This paper describes a mathematical model which calculates the time dependencies of the flow rate and composition of the vapour emerging from a pool. A large variety of accidental cases can be covered: continuous or instantaneous spills, on confined or unconfined ground, ideal or non-ideal liquid mixtures in boiling or evaporating conditions. The boiling, when present, is modelled through an equation system comprising the Rachford-Rice relation and the energy balance of the pool, which is assumed to be well-mixed. In the case of a volatile pool, inter-facial mass rates are determined taking into account the Stefan flux, and the thermal resistance inside the liquid phase is also considered. In all situations, the energy balance includes the contribution of ground, sun and air. Known experimental data have been used to validate the model.

Keywords: pool vaporization; modelling; energy balance

Most accidents in chemical plants result in liquid releases of toxic or flammable materials which spread on the ground and evaporate, often causing damage to people. To assess the consequences of such accidents, the atmospheric dispersion of the vapour emerging from the pool must be evaluated. As the evaporative rate strongly influences the subsequent dispersion, great effort has been and is still being expended by the scientific community in order to build mathematical models which are able to determine its behaviour.

The numerical codes commonly used in plant risk analysis cover different situations such as instantaneous or continuous spills on confined or unconfined ground, but frequently adopt restrictive hypotheses, in that they either consider the case of one-component liquids’-’ or treat liquid mixtures as pseudo-component mixtures. In cases where multicomponent ideal mixture8 are taken into account, often only the case of volatile pools is examined.

The aim of this paper is to describe a mathematical model which removes these limitations and is able to calculate the evaporative rate for a pool of a non- ideal mixture which is either boiling or evaporating. One-component or ideal mixture liquid spills are included as simplified cases.

At first, the liquid pool was modelled as a well- mixed one, i.e. the physical/chemical properties were assumed to be uniform all over the pool. Later, the difference between surface and bulk temperatures was considered in order to fulfil the experimental data better.

095&4230/94/060443-06 0 1994 Butterworth-Heinemann Ltd

Classification of possible accidental cases

The large variety of accidental events involving liquids can be classified with regard to certain parameters: the substances involved, the process conditions inside the failed vessel or pipe, the duration of the release, and the type and topography of the ground.

The accidental release of the contents of a plant can be assumed to be instantaneous when a large hole develops in a process unit such that the material is released in a very short time; in contrast, continuous outflow is considered to occur when a hole of limited dimensions occurs in the wall of a container or pipe. Confined pools with a constant evaporation area result from spills on bunds, while, if no obstacles to liquid motion are present, a pool with a time-dependent area (unconfined) is derived from liquid spreading.

In addition, depending on both substance and process conditions, boiling or volatile pools may form and a transition may occur from one situation to the other.

Whatever the accidental case, it will be a combi- nation of the above-mentioned scenarios.

General equation of the model

If the temperature and composition of the pool are assumed to be uniform (i.e. a strictly well-mixed pool), then the molar and energy balance equations become:

dN. ‘ = & - tin dt

i = 1, . . . . Nc

and

(1)

J. Loss Prev. Process Ind., 7994, Volume 7, Number 6 443

The modeling of pool vaporization: P. Leone//i et al.

with the initial conditions t = 0; Ni = N,,; i = 1, . . . . NC, Ei = I&,.

The equation system allows instantaneous and continuous releases to be described: in fact, No and Noi represent the amount. of 1iqui.d which escapes instantaneously, whereas N, and Nei represent the molar rates due to a continuous spill. The solubility of air was not taken into account.

If a non-idea1 mixture is considered, liquid molar enthalpies in the energy balance cannot simply be expressed as the product of specific heat (averaged on pure components) and temperature difference, as they contain thermal effects due to mixing. Assuming pure liquid compounds at temperature TSR and atmospheric pressure as the reference state, the molar enthalpies become:

ii = EPL.~.,,, CT, - Ts,) + A&,~,sR

for the input stream, and

fi = CPI_,~ (TP - Ts.4 + ~&X,SR (4)

for the pool, where A&, is the thermal effect due to mixing.

The molar enthalpies of the compounds in the vapour stream emerging from the pool, with the assumption of thermal equilibrium at the interface, are:

fili = @v,i (TP - TSR) + Afivi,s~ (5)

and incorporate the latent heats of vaporization of the pure compounds.

The term AH,,,, depends on the composition of the liquid mixture and this dependence can be determined either by simply fitting experimental data or by resorting to the equation:

(6)

which introduces the need to calculate the activity coefficients y<.

Simpler expressions for Equations (3) and (4) hold if an ideal liquid mixture is considered, since ARM is equal to zero in this case.

In the energy balance, all the thermal fluxes exchanged with the environment (sun, ground, air) must be taken into account due to the impossibility of establishing their relative weight a priori in determin- ing the evaporation rates. The expressions inserted in the mode1 to evaluate the thermal fluxes are syntheti- cally explained below.

Thermal @xes The term QE is the energy inflow due to solar radiation and it is expressed as?

8; = L(1 - a) (7)

where L is the solar radiative flux at ground level and a is the albedo, i.e. the fraction of incident radiation reflected by the body and depending on the sun’s elevation.

Radiative heat transfer theoryq supports the calculation of radiative heat exchange (Q&) between the air and the pool:

The total emissivities of the atmosphere and the pool are assumed to be 0.75 and 0.95, respectively, following the suggestions given in Reference 3 for pools evaporating in open spaces. However, E* = l p when the evaporation occurs in a closed environment (e.g. experiments within a wind tunnel or hood) and the area of the pool is negligible in comparison with the total area of the walls.

The thermal flux due to convective exchange (b&.,) between the pool and the air is obtained by

Q&r.~=hv(L- 7'~) and the analogy

(9)

Nu = 0.036 ReU-8 Pr’” 2x104<Re~7x1@ (10)

has been properly used to determine the value of the heat transfer coefficient hV (Reference 1). The physical chemical properties included in Re, Pr and Nu have been calculated as described in Reference 6, assuming a mixture of air and vapour at an average temperature and composition between those of the interface and the air. Well-known formuIae’0 were used to predict the physical properties of a gaseous mixture using those of its pure components. The pool diameter and the wind velocity at a height of 10 m were assumed as the characteristic length and velocity, respectively.

Finally, particular attention must be devoted to the heat transfer rate from the ground (&a. This results from the solution of an unsteady heat conduction problem in a semi-infinite slab (the ground) with a time- dependent boundary condition (pool temperature). In fact, the temperature of a multicomponent pool is time-dependent for both a volatile pool and a boiling pool. An analytical solution of the problem is achievable by resorting to the Duhamel theorem.

For a confined pool:

&;( = k[T,o - TPWJ Jzt -p-&j=& (11)

For an unconfined pool, due to its spreading, the ground begins to exchange energy with the liquid at different times and the flux becomes:

&,= (I R Woo - TP(O)I 0 Jrrouo x2mdr

-[[&~d~&]x2~dr\~ (12)

444 J. Loss Prev. Process Ind., 1994, Volume 7, Number 6

where t’ is the time at which the pool reach radius r. The correlation between r and t’ derives from the spreading model adopted, as will be shown in a subsequent section. The thermal conductivity (k) and thermal diffusivity (a) of the ground are generally dependent on granulometry and humidity1s2*6; some values are listed in Table I.

Molar fluxes Boiling or non-boiling liquids require quite different physical treatments of the vaporization phenomenon which is reflected in different expressions for interfacial molar fluxes.

Boiling pool. Let us consider a cryogenic boiling mixture accidentally released onto the ground: the heat fluxes from the environment support the boiling and the pool temperature increases as a result. In fact, the more volatile a compound is, the higher its molar flux is, so that the boiling temperature and the composition of less volatile substances in the liquid phase are increased. The energy influx will compensate for loss of enthalpy due to vaporization and allows heating of the pool to enable it to remain at the boiling temperature.

The physical equilibrium at the interface is:

YIi = Ki Xi i= 1,. . . ,N, (13)

At boiling conditions:

NC

z1 YIi = 1

and it follows that:

(14)

(15)

Numerical solution of the resulting system is more easily achievable by resorting to a convenient discretization over a time interval At of the energy and molar balance equations, through which we can calculate the new bubble point of the pool. If zi is the molar fraction in the liquid phase at time t, a non- adiabatic flash is solved from the equations:

The modelling of pool vaporization: P. Leonelli et al.

and

N[A(TP + AT,+,) - @T~,zi)l + NJI 2 ziKz[I-i,i( TP) - fi(T~v Zi)I

i=l 1+ $(Kj - 1) (17)

- Qm,-r(T,=~i) = 0

where QToT, which is computed at time t, includes all thermal fluxes and the energy influx due to a continuous spill.

The vaporization ratio 4 and the temperature variation AT, are thus obtained. The new vapour and liquid compositions are:

zi Ki y’i = 1 + (MKi - 1)

i=l 7.. . > NC (18)

In this way, the model depicts the time dependency of the evaporation rate and vapour composition during the boiling phenomenon, once the equilibrium constant Ki is known. If a non-ideal mixture is considered, Ki is given by:

Ki = 9 yi i=l I-. . 7 NC (20)

where -yi is the activity coefficient for a compound in the mixture. This equation assumes that the gas mixture adjacent to the liquid surface is an ideal one, since its pressure is low (atmospheric pressure).

The choice of model to evaluate yi is strictly related to the mixture of interest in order to achieve the best determination of the vapour-liquid equilibrium. In this study, a non-random two-liquid (NRTL) model was used which allows determination of the dependence on temperature and composition of a mixture of NC components once the coefficients for the binary solutions are known.

Volatile pool. The necessity of taking into account the Stefan flux in modelling the evaporation phenomenon is well established1y6, particularly for very volatile mixtures. In this case, the i-component molar flux emerging from the pool surface is given by”:

Table 1 Thermal properties of ground types

Material

Thermal Thermal conductivity (&) diffusivity (a) (W m-’ Km’) (m2 s-l)

Asphalt 0.70 3.606 x lo-’ Concrete 0.92 4.160 x 1O-7 Soil (average) 0.96 4.590 x 10-7 Soil (sandy. dry) 0.26 1.980 x lo-’

So;aAy;)ist, 8% water, 0.59 3.360 x 1O-7

i= 1,. . . ,NC (21)

where yIi is the molar fraction of the i-component in the vapour at the interface and is given by:

YI, = &xi i=l 9.. . , NC (22) NC

In this case, c yri # 1. i=l

Some models reported in the literature disregard the Stefan flux and suggest the following expression to evaluate the evaporation rate:


The modelling of pool vaporization: P. Leone/Ii et al.

80

60

40

0 0.2 0.4 0.6 0.8 1 Time (h)

Figure 1 Mass flux of ammonia emerging from an ammonia-water pool: (m) Equation (21); (A) Equation (23)

FJii = K& :T (yxi - JJ,i) i=l,. . . ,N, (23)

This assumption is correct for fairly non-volatile mixtures, but in other cases may greatly underestimate the molar fluxes, as is shown in the following example. Figure 1 compares the interfacial molar fluxes obtained from Equations (21) and (23) in the following case: 5000 kg of aqueous solution of ammonia (30% mol) with an evaporating surface of 10 m2; ambient conditions: temperature = 25°C relative humidity = 68%) wind velocity = 2.8 m s-l; substrate = concrete. The initial evaporation rate is considerably underestimated using Equation (23), and only after a long period do the fluxes become nearly equal, when the concentration of ammonia in the pool has decreased so much due to evaporation that its evaporation rate is low and the Stefan flux may be disregarded.

Equation (21) may be solved once the molar transport coefficients (R&) and the physicahchemical properties of the mixtures are known. The expressions generally used to evaluate K& are summarized in Table 2. Equation (a) in Table 2 resorts to the von Karman analogy (turbulent flow) with a power law profile for the wind velocity12; the resulting expression, which has been modified by PasquilP using experimental results on pure compounds, shows a dependence

Table 2 Different expressions for the calculation of molar transport coefficients

PC, = Cl&(R)” where C= fDfeZb; a = G,

n 2n b=-2+n;g=2+n

K& = 8.11 x 10-6(3600$- (2RP SC;-*” (b)

K& = 0.664Re15Q3 $$j ; Re < 320000 (cl

on both the atmospheric conditions, through velocity and stability class, and the diffusion coefficient of the i-component in air. Equation (b) in Table 2 was proposed by Mackay and Matsugu3: they adopted the same analogy, determined the numerical coefficients by experiments on a pool of cumene, and the diffusivity characteristics of other substances were taken into account via the Schmidt number.

It should be noted that these expressions do not allow correct description of the laminar flow conditions, which probably characterize diffusion over a small pool. In this case, the known analogies for a flat plate (Equation (c) in Table 2) must be utilized4.

In the proposed model, the expression derived by Mackay and Matsugu has been adopted (Equation (b)): the choice was suggested by previous comparisons involving turbulent flow, the results of which showed evaporative rates similar but greater (and consequently conservative) compared to those calculated using the Pasquill relation. In addition, the heat transfer coefficient calculated by the Chilton-Colburn analogy utilizing the Mackay-Matsugu expression is in good agreement with the value given by Equation (10).

Obviously, the physical/chemical properties of the gas mixture are calculated using the same hypotheses adopted to evaluate the heat transfer coefficient (hv).

Spreading As far as an unconfined pool is concerned, the spreading of the liquid must be modelled to estimate the total vaporization rate. For this purpose, attention was focused on two types of releases, instantaneous and continuous. In every case, the ground was assumed to be flat, horizontal, non-penetrable and uniformly rough.

When an instantaneous release is dealt with, the liquid is considered as a cylinder that collapses, and the principle of energy conservation gives’:

dR dr = &.08g(k, - h) (24)

where t = 0, R = Ro, and the pool depths h and ho are:

The liquid volume V at time t can be deduced from the global molar balance in integrate form:

&(r)dr + 5 I’Ijh&) dr (26) r=l 0

where the term he, is obviously equal to zero. The pool spreads until it reaches a minimum layer

thickness depending on the roughness of the substrate (hmi,)l; the pool will then shrink as the volume decreases due to evaporation and the depth will remain constant at h,i,. It is worth noting that shrinking of the area may also occur in a confined pool when the amount released is small and the time considered is large



In our model, for a continuous spill, the liquid spread is calculated by resorting to the Bernoulli theorem using the following equation”.

dR - = d2g(h - hrnin) dt

t=O;R=R, (27)

When h = &,, the spreading stops unless the mass influx is greater than the evaporation rate, in which case spreading continues at h = h,i,. The shrinking phase, when present, is described as reported for the case of instantaneous release.

Transition from the boiling to the non-boiling state This section considers a pool at its boiling point. The net heat intake (QAIMB ) due to exchanges with the environment is determined, as well as the mass rate that air is able to carry away by convective mass transfer, and, as a consequence, the amount of energy (Q+& necessary for this convective flux. If QMASS < QnMB, the pool remains in its boiling state, whereas the pool stops boiling if &,ss > BaMB, and &,,, is provided by QAMB and cooling of the pool.

It is worth noting that this check is performed in Referetce 1 by comparing the evaporative mass rate due to QAMB in boiling conditions and the evaporative mass rate due to convective mass transfer. This procedure is correct only for one-component liquids, as a direct proportionality exists in this situation between the mass and enthalpy of a stream once the temperature and pressure are fixed. For a multicomponent mixture, the stream enthalpy also depends on composition, and, in general, the vapour emerging from a boiling pool has a composition different to that of a volatile one.

During the boiling phase, the pool temperature increases and consequently QAMB decreases: thus a transition may occur and the check outlined in this section must be continuously performed.

Comparisons with experimental data The model was tested against available experimental data on the evaporation of ammonia-water solutions”,14. These experiments considered small pools, which were insulated on the bottom and sides. A simulation was made of the following experimental conditions: 2240 g of aqueous solution of ammonia (28.8% by weight) with an evaporating surface of 541 cm2; ambient conditions: temperature = 28”C, relative humidity = 68%, wind velocity = 2.22 m s-l. The pool is volatile in this situation.

For the sake of comparison, the quantities of ammonia evaporated were determined by means of the three methods for calculating molar transport coefficients listed in Table 2. The results are reported in Figure 2. Note that the Mackay-Matsugu expression gives the best fit of the experimental data with a maximum error less than 7%.

A negative water evaporation rate results from simulation for about 2 h: this means that condensation

600 / ,

5500 .

2400 $ -5 300

E 200

5 100

0 0 1 2 3 4 5

Time (h)

Figure 2 Time dependency of the evaporated mass of ammonia: (A) Equation (a) in Table 2; (A) Equation (b) in Table 2; (0) Equation (c) in Table 2; (M) experimental data

of the air humidity occurs and the vapour emerging from the pool is pure ammonia, This phenomenon was observed in the experiment. Comparisons were performed with other experimental data, and, in every case, the maximum error of the evaporation rate was less than 10%. The results obtained in the test cases show that the proposed model may be used with confidence to estimate the source term (which is an important parameter in plant risk analysis).

The time dependency of the pool temperature in the above experiment is reported in Figure 3: the agreement with experimental data is worse than that for evaporated mass. This discrepancy suggested that the hypothesis of uniform temperature inside the pool

-10 0 1

Gime (i) 4 5

Figure 3 Time dependency of pool temperature: (A) Equation (a) in Teble 2: (A) Equation lb) in Table 2; (0) Equation (c) in Table 2; (m) experimental data



I I

t

Figure4 Surface and bulk balances for evaporation

should be removed and that the heat transfer resistance in the liquid phase should be taken into account.

Modified model The hypothesis of uniform temperature inside the pool is certainly valid for a boiling pool, as the liquid phase is well-mixed due to the turbulence of the boiling state. For volatile pools, the hypothesis of uniform temperature is removed and the pool is assumed to be formed by a well-mixed bulk whose temperature may differ from that of a very thin surface layer (see Figure 4).

Heat balances were performed separately for the surface layer and the bulk. The heat required for vaporization at . the pool surface is provided by solar radiation (Qg), convection from the atmosphere (Q&&, radiative heat exchange with the atmosphere (Q;;), free convection in the liquid phase from the bulk (Q&) and enthalpy influx due to mass transfer from the bulk to the surface. The energy balance for the bulk includes conduction from the soil, enthalpy influx due to continuous spills, and energy exchanges with the surface.

The heat balances mentioned above are:

[Q; + Q;; + Q&, + Q;; + fi;@u - A,“,,)

- ,+;(fir - &,,,)] x A = 0 cm

and

hJd$ = A(&- (jtlB) + &(fi, - fi,) (29)

The fluxes & and e&N are:

Q”o, = h,U’, - TSURF) (30)

and

e;; = o(e*G - +GURF) (31) The equilibrium constants, used to determine the composition at the liquid-air interface, are calculated at temperature TSURF. The term Qg is given by:

Q;; = hvu(Tu - Ts,,,) (32) where hv, is a measure of the thermal resistance of the liquid phase.

Combining Equations (28) and (29) gives the global energy balance:

N2=A[Q;+ &+ Q&,, + Q& (33) - i+:‘(Pr - fiu)] + tiJir, - &)

where the heat fluxes are modified as described in Equations (30) and (31).

In the model implemented, Equation (2) was replaced by Equation (33) and Equation (29) was used to fulfil the new variable TSURF.

Convective bulk heat transfer The pool can be considered as a liquid enclosed between two horizontal surfaces. Two cases can occur:

(i) TSURF 2 TB

(ii> T&RF < TB

In the first case, heat transfer is due to conduction, while free convection may occur in the second case. In order to calculate hYR, the following equation is proposed’:

Nu, = c(RaL)m (34) where c and m are dependent on Ra,. The depth of the pool is used as the characteristic length in NuL and Ra,.

A different expression to evaluate hVB, suggested by Kawamura and MackayIS, is:

2k. hv, = z

0

where 6-l = 1 + exp(O.O6(Tu, - 70)), ho is the initial depth of the pool and TuP is expressed in “C. This equation results from fitting of experimental data on pools with a depth of a few centimetres.

For the sake of comparison, the bulk and surface temperatures of the pool were computed by means of Equations (34) and (35), for the case previously mentioned, and the results are shown in Figure 5. The differences are not too large, and Equation (34) is also suitable for pools with a large thickness.

In Figure 6, the time dependencies of the bulk and surface temperatures are compared with available experimental data. The time dependency of the temperature obtained from the original model is also reported. The fitting of experimental data is enhanced if the average of TsURF and TB is assumed to be representative of the measured temperatures.

As far as the evaporation rate is concerned, the results obtained with the modified model are practically coincident with those shown in Figure 2.

Conclusions In this paper, a model is proposed to simulate the time dependency of the rate and composition of the vapour emerging from a multicomponent non-ideal liquid pool. Different situations were considered: confined or unconfined pools of volatile or boiling


30

20

10

0

-10 0 0.5 1 1.5 2 2.5 3

Time (h) Figure 5 Time dependency of bulk and surface temperatures: (WI ~S”RF~ Equation (34); (A) T., Equation (34); (0) T,,,,, Equation (35); (A) TB, Equation (35)

30

20

10

0

-10 0 0.5 1 1.5 2 2.5 3

Time (h) Figure 6 Comparison between calculated temperatures and experimental data: (m) TSURF, Equation (34); (A) T., Equation (34); IO) r,, original model; (0) experimental data

liquids, and continuous or instantaneous spills. At first, the hypothesis of a well-mixed pool was adopted and the results were tested with available experimental data for a highly non-ideal water-ammonia mixture. Good agreement was obtained between calculated and experimental data for the quantities of ammonia evaporated, while larger discrepancies resulted for temperatures. A modification was introduced to enhance the model performance, which takes into account the heat transfer resistance in the liquid phase. The proposed model may be considered as a simple mathematical tool able to describe a complex phenomenon.


Although tested on a binary system, the model can be used for mixtures with any number of components if correct data on physical properties are available.

Acknowledgements This research was funded by the Ministry for the Universities and Scientific and Technological Research, Italy.

References 1

2

3

4

5 6

7

;

10

11

12

13

14

15

Bureau of Industrial Safety (TNO) ‘Methods for the Calculation of Physical Effects Resulting from Releases of Hazardous Materials (Liquids and Gases)‘, CPR 14E, chapter 5, TNO, The Netherlands, 1988 Cremer and Warner Ltd. ‘Vaporisation from Liquid Spills’, Rijnmond Public Authority Report A3, Reidel Publishing Co., London, 1982 Mackay, D. and Matsugu, R. S. Can. J. Chem. Eng. 1973, 51, 434 Mills, M. T. 1990. 29. 69

and Paine, R. J. Reliability Eng. System Safety

Pasq&‘F. Proc. R. Sm. [A] 1943, 182, 75 Shaw, P. and Briscoe, F. ‘Evaporation from Spills of Hazardous Liquids on Land and Water’, SRD R-100, UKAEA, Culcheth, 1978 Studer, D. W., Cooper, B. A. and Doelp, L. C. Plant Opm Proe. 1988. 7. 127 Drivas, P. J. Env. Sci. Technol. 1982, 16, 726 Ozisik, M. N. in ‘Heat Transfer. A Basic Approach’, McGraw Hill, New York, 1985, pp. 452-459, 64M57 Hirschfelder, J. O., Curtiss, C. F. and Bird, R. D. in ‘Molecular Theory of Gases and Liquids’, Wiley, New York, 1964, pp. 514-610 Leonelli, P.: Stramigioli, C. and Spadoni, G. in ‘Proceedings of the Conference on Chemical and Process Engineering’, AIDIC, Milan, Italy, 1993, pp. 293-297 Sutton, 0. G. in ‘Micrometereology’, McGraw Hill, New York, 1953, p 273 Frie, J. L., Page, G. A., Diaz, V. and Kives, J. J. in ‘Proceedings of the International Symposium on Loss Prevention and Safety Promotion in the Process Industries’, SRP Partners, Rome, Italy, 1992, pp. 16.1-16.16 Mikesell, J., Buckland, A., Diaz, V. and Kives, J. in ‘Proceedings of the International Conference and Workshop on Modelling and Mitigating the Consequences of Accidental Releases of Hazardous Materials’, New Orleans, Louisiana, 1990 Kawamura, P. and Mackay, D. J. Hazard Mater. 1987, 15, 343

Nomenclature

zl Albedo Pool surface (m’) Molar heat capacity (J mol-’ K-‘) Diffusivity of the i-component in air (m’ s-l) Gravitational acceleration (m se2) Liquid height (m) Minimum liquid height (m) Convective heat transfer coefficient (W m-* K-I) Molar enthalpy (J mol-I) Enthalpy of mixing (J mol-L)

A& Enthalpy of vaporization (J mol-I) k Thermal conductivity (W m-l K-l) K Partition coefficient e Molar transport coefficient (m s-l) L Solar radiation flux (W m-“)

; Parameter depending on stability class

IQ Number of moles in the pool Molar rate (mol s-l)

NC Component number in the liquid phase Nu Nusselt number P Pressure (Pa) P* Vapour pressure (Pa) Pr Prandtl number QTOT Total energy (I)



e R

R Ra Re Sc I T u V x Y Z

a Y c G-

; + 6

Heat flow (W) Pool radius (m) Universal gas constant (J mol-’ Km’) Rayleigh number Reynolds number Schmidt number Time (s) Temperature (K) Wind speed (m 5-l) Volume (m’) Liquid molar fraction Gas molar fraction Reference height (m)

Thermal diffusivity (m’ s-l) Activity coefficient Total emissivity Stefan Boltzmann constant (W m-’ Km4) Dummy variable Parameter in Equation (21) Vaporization ratio Parameter in Equation (35)

Subscripts A Relative to air

AMB B BP CON

: G 1 I L m MASS P R S SR SURF V Z

0

Relative to the environment Relative to bulk Relative to the boiling point Relative to convection Relative to the input Relative to the core of air Relative to the ground Relative to the i-component Relative to the interface Relative to liquid Mean Relative to mass Rux Relative to the pool Relative to radiation Relative to the sun Relative to the reference state Relative to the surface Relative to vapour Relative to the reference height

Relative to the initial condition

Superscripts ,, Relative to the unit area m Parameter in Equation (34)


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The modelling of pool vaporization