4844
Ind. Eng. Chem. Res. 1998, 37, 4844-4853
A Newly Developed Algorithm Describing the Fog Formation of Hydrogen Fluoride (HF) in Air Nicholas A. Mandellos,† Zoe S. Nivolianitou,*,‡ and Nicholas C. Markatos† Institute of Nuclear Technology-Radiation Protection, National Center for Scientific Research “Demokritos”, Aghia Paraskevi 15310, Greece, and Department of Chemical Engineering, National Technical University of Athens, Zografou Campus, Athens 157 80, Greece
A new algorithm, based on the thermodynamic model proposed by Schotte (1987), has been developed to simulate the mixing of HF with moist air. This process can deviate considerably from ideal mixing and lead to the formation of a cool gas-fog mixture which disperses as a heavy gas. The new algorithm reduces the calculation time necessary to compute the composition, temperature, and density of the resulting mixture. The results compare very well with experimental data presented by Schotte. Introduction Hydrogen fluoride (HF) is an industrial catalyst of substantial use in the alkylation units of refineries. As a substance HF is very toxic to humans and corrosive according to Mudan,1 so its accidental releases in the atmosphere are of major concern. The behavior of HF as it is released in the atmosphere and is mixed with ambient air is rather complex. The whole mixture is first cooled owing to depolymerization of HF (which can be found in its polymers’ form, namely (HF)2, (HF)6, and (HF)8) and then heated due to the condensation of both water (air humidity) and HF and the heat of solution of HF. These phenomena lead to the mixture temperature lowering and the subsequent rise of the mixture relative density above 1 at the initial stages of the mixing, making the entire fog behave as a heavier-thanair gas, although, per se, hydrogen fluoride is lighter than air. Schotte2 presented a full thermodynamic model describing the phenomenon of fog formation of hydrogen fluoride, kept with inert nitrogen N2 when mixed with moist air on which the present work is mainly based. Several attempts have been made to simulate the dispersion of HF (at these conditions) in the atmosphere. Muralidhar et al.3 describe a two-phase jet model for predicting the HF rainout (capture) in HF/additive releases, building upon the earlier work of Papadourakis et al.,4 who treat the same subject in a broader area of two-phase jets emerging from the release of superheated liquids. Considerable effort has been made to simulate the dispersion of an accidental HF release in a system of computer programs based on Shell’s HEGADAS which has been adapted to incorporate HF thermodynamics, as described by McFarlane et al.5 and Puttock et al.6 The main observation in all these models is that they rely on Schotte’s work. In this paper the physicochemical properties of HFair mixtures have been described on the basis of the work of Schotte and a new algorithm has been developed with the view of being incorporated into computational fluid dynamics programs, for example SOCRATES (see * To whom correspondence should be addressed. † National Technical University of Athens. ‡ National Center for Scientific Research “Demokritos”.
Papazoglou et al.7), so as to study HF dispersion in the atmosphere. However, the mechanisms affecting HF releases described above are significant only at relatively high HF partial pressures and low temperatures such as releases of liquid HF. At high temperatures and low partial pressures, HF vapor approaches ideal behavior and the effects of fog formation are negligible. During the mixing of HF with ambient air a fog is formed having both a liquid and a gaseous phase as a consequence of the depolymerization, the complexing, and the condensation reactions of the gaseous HF polymers and ambient humidity. In dry conditions gaseous HF forms dimers, tetramers, hexamers, and octamers ((HF)2, (HF)4, (HF)6, (HF)8) and is in equilibrium with its polymers according to the following equations
2HF T (HF)2 4HF T (HF)4 6HF T (HF)6 8HF T (HF)8
(1)
In addition gaseous HF and water (H2O) can form a complex (HF‚H2O) according to the reaction
HF + H2O T HF‚H2O
(2)
In the following analysis tetramers are omitted, as their contribution seems quite insignificant in the energy and mass balances. The mass-energy balance describing the fog formation considers adiabatic mixing of M1 mol of monomer equivalent HF per total moles of mixing compounds together with Mn moles of nitrogen (N2) per total moles of mixing compounds. In this way a fog is formed with apparent mole fractions of its compounds M1 for equivalent HF monomer, Mn for nitrogen, Ma for dry air, and M2 for water vapor. For the gaseous phase of the fog the respective apparent mole fractions are M1,gaseous, Mn, Ma, and M2,gaseous, while for the liquid phase the apparent mole fractions for HF and H2O are M1,liquid and M2,liquid, respectively.
10.1021/ie980131x CCC: $15.00 © 1998 American Chemical Society Published on Web 11/19/1998
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4845
Partial Pressures Partial pressure of water vapor for a given relative humidity can be given by a relationship of the Antoine type, as described by Reid et al.,8 which can be transformed into the following expression: 0
P2
[
]
3206.7 7173.79 ) × exp 6.53247 14.7 389.4747 + (32 + 1.8T) relative humidity (3)
where T ) temperature in °C, P20 ) partial pressure in atm, and Pc,H2O ) 3206.7 psia. Assuming that the mixture HF-H2O of the fog liquid phase is in equilibrium with its gaseous phase, an expression giving the partial pressure of HF and H2O in this mixture, as a relationship of the liquid molar fraction x of HF, is given by Vierweg (see Schotte9), as the following expression:
N2 mass per total moles of mixture components is given by the following expression:
Mn ) QM1
(7)
H2O molar mass contained in the initial air quantity can be calculated from the following equations:
P20 M2 ) Ma P - P
0 2
S M2 )
P20
Ma P - P20
(8)
where Ma can be calculated from the mass balance M1 + M2 + Ma + Mn ) 1, substituting M2 from eq 8, as follows:
Ma )
1 - M1 - Mn 1 + P20/(P - P20)
(9)
Considering the gaseous phase of the mixture ideal, the molar masses of compounds can be calculated from their partial pressures:
P1 M1,gaseous ) S Ma + Mn P - P1 - P2
where
M1,gaseous )
P1 (M + Mn) (10) P - P1 - P2 a
P2 M2,gaseous ) S Ma + Mn P - P1 - P2 M2,gaseous ) Plotting of these relationships together with the partial pressure prediction by the Rault Law for ideal solutions,
Pi ) yiP ) xiPis
(6)
Pi ) partial pressure of i component yi, xi ) the compositions of gaseous and liquid phase P ) total equilibrium pressure s
Pi ) partial pressure of pure component i has shown that Rault’s law fails to predict correctly the partial pressure of HF vapor in the range 0-80% w/w with respect to experimental values, while for water vapor partial pressure it could be used up to a composition of 40% (molar). This deviation can be explained by the fact that an (HF, H2O) solution is far from ideal because of the strong polarity of both molecules, leading to bonds creation especially in the intermediate range of HF molar fractions. However, Vierweg’s expressions do present an oscillation around the breakpoints in liquid mole fraction, which in the solution algorithm has been tackled numerically. Mass Terms in the Fog If HF is kept under nitrogen (N2) atmosphere of N2/ HF ratio of Q (mol) (if N2 is absent, then Q ) 0), then
P2 (M + Mn) (11) P - P1 - P2 a
The apparent molar fractions in the gaseous phase of each component are derived by dividing its mass by the total mixture mass. Mass-Energy Balance The analytical expression of the mass-energy balance is given by the following equation:
∆HTa,TMa + ∆HTa,25M2 + ∆HT1,TMn + ∆HT1,25M1 + ∆Hreaction,T1,25M1 + ∆Hreaction,25,TM1,gaseous + ∆H25,TM2,gaseous + ∆H25,TM1,gaseous + ∆H25,TL ) ∆HcondM2,liquid + ∆HcondM1,liquid + ∆HmixM1,liquid + ∆Hreaction,25,TM1,liquid (12) As the route to equilibrium from an initial to a final stage is thermodynamically indifferent, we can consider the following HF-air mixture procedure: Initial Stage. M1 mass of HF monomer equivalent at temperature T1 is kept together with Mn mass of N2 at the same temperature in gaseous phase and are mixed together with Ma mass of dry air and M2 water mass in air also in the gaseous phase. Final Stage. Fog is created of (M1 + M2 + Mn + Ma) apparent mass at temperature T, of which L is the true liquid phase of HF-H2O and 1 - L constitutes the apparent gaseous phase, while the true molar gaseous phase consists of HF polymers, HF‚H2O, complex, H2O vapor, N2, and dry air and is somehow smaller owing to the polymerization and complexing reactions.
4846 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
Enthalpy Terms Schotte2
for the mean According to values cited by specific heat of air, nitrogen, ideal gaseous HF, and gaseous H2O the mean specific heat of HF aqueous solution between 25 and T °C is given by the expression:
where molar fractions of polymers are given by the expressions
y11 )
K2f12 y12 ) Φ1P
Cp ) 18.047 + 0.000397T - (13.974 - 0.0152T)x + (8.528 + 0.00187T)x2 (13) ∆Hreaction for the equilibrium of depolymerization (and complexing, whenever it applies) depends solely on the initial and final polymers concentrations.
∆Hreaction,T1f25 ) H25 - H1 ∆Hreaction,25fT ) H - H25
(14)
(H declares energy content, while ∆H declares enthalpy (heat) of formation from one stage to another). Then the heat of condensation of HF together with its heat of dilution in H2O, taking account the equilibrium at 25 °C, is given by the following expression using also eqs 14:
∆Hcond + ∆Hmix + ∆Hreaction,25°C ) 7231 + 4500(1 - x2) - H25 (15) The relationship giving the enthalpy departure of the HF polymers in equilibrium stage is according to Schotte:2
H)
y12∆H2 + y16∆H6 + y18∆H8 + yc∆Hc y11 + 2y12 + 6y16 + 8y18 + yc
H1 )
y12∆H2 + y16∆H6 + y18∆H8 y11 + 2y12 + 6y16 + 8y18
f1 Φ1P
y16 )
K6f16 Φ1P
y18 )
K8f88 Φ1P
(19)
f1 is the fugacity of HF monomer and Φ1 the fugacity coefficient which, as far as atmospheric conditions are concerned, has typical values in the interval 0.98-0.99. The apparent mole fractions of HF monomer and H2O based on the molecular weight of 20.01 for HF are then known and can be used to calculate the true composition of the mixture:
Y1 )
y11 + 2y12 + 6y16 + 8y18 + yc y11 + 2y12 + 6y16 + 8y18 + 2yc + y2 + yn + ya
Y2 )
yc + y2 y11 + 2y12 + 6y16 + 8y18 + 2yc + y2 + yn + ya (20)
Here y11 + y12 + y16 + y18 + yc + y2 + yN + ya ) 1. Combination of eqs 20 with
P1 ) Y1 P (16)
Substituting the above-mentioned relationships in the mass-energy balance yields an expression which can be solved for liquid mass L:
L ) S/{10519 + (8.05 - 1.09x - Cp)(T - 25) + [4500(1 - x2) - 3288 + H]x} S ) 6.96(T - Ta)Ma + 6.96(T - T1)(Mn + M1) + 8.05(T - Ta)M2 + (H - H1)M1 (17)
P2 ) Y2 P and eqs 18 and 19 yield the following expression:
S1(P - P1)(Kcf1 + Φ1) + (S2 - Φ1P)[Kcf1(P1 - P2) + Φ1P1] ) 0 (21) with
S1 ) f1 + 2K2f12 + 6K6f16 + 8K8f18 S2 ) f1 + K2f12 + K6f16 + K8f18
(22)
HF Polymers and Complex Compositions For the equilibrium stage of reactions 1 and 2 an expression of equilibrium constants is given by Schotte,2 as follows:
yc )
12775.229 - 47.97731 /R K2 ) exp T + 273.16
) ] [( 41927.495 - 138.55185)/R] K ) exp[( T + 273.16 50120.984 K ) exp[( - 165.85264)/R] T + 273.16 6266 - 22.700)/R] K ) exp[( T + 273.16
Kcf1S1P2 Φ1P[Kcf1(P1 - P2) + Φ1P1]
(23)
Temperature and Composition of Fog Liquid Phase
6
8
c
which can be solved for f1 by a trial and error method. So, according to the value of f1 one can estimate yc from the equation
(18)
Taking into account the equations related to massenergy balance (eq 12), one can form a single expression with unknown the values of temperature T, the liquid molar fraction of HF, and the amount of fog formation L. This last quantity is considered unknown as eq 17
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4847
which calculates its value derives from eq 13, so these two are linearly dependent on each other. Therefore two more equations linearly independent are introduced in order to solve for T and x. These two are the ideal gases law and the HF mass balance, as follows:
M1 - xL
P1 w M2 - (1 - x)L P2 P2(M1 - xL) - P1[M2 - (1 - x)L] ) 0 (24) )
with ambient air which leads to the estimation of the fog density, called HFCALCULATIONS. A significant part of the whole algorithm (STEP 2) deals with calculation of the partial derivatives of eqs 21, 24, and 25 versus temperature (T), HF molar fraction in the liquid phase (x), and HF fugacity (f1), which are given by the following equations:
P1 M1 - xL ) w Ma + Mn P - P1 - P2 (M1 - xL)(P - P1 - P2) - P1(Ma + Mn) ) 0 (25) Then eqs 21, 24, and 25 define a non linear 3 × 3 system which can be solved numerically with an extended Newton-Raphson methodology considering the values of T, x, and f1 as the vector elements in a 3 × 3 space. Fog Density The molar volume of the gaseous mixture HF-H2Odry air-N2 can be calculated by the use of the PengRobinson state equation (Reid et al.8)
P)
RT R V - b V(V + b) + b(V - b)
(26)
considering that dry air consists of 79% N2 and 21% O2, and the mixture composition is
Y1 ) P1/P Y2 ) P2/P YN2 ) 1 - YHF - YH2O - YO2 YO2 ) 0.21YAir
(27)
where YAir is calculated from
YN2/YAir ) Mn/Mn P1 + P2 YAir + YN2 ) 1 P
}
P1 + P2 P Mn 1+ Ma
1f YAir )
The liquid mean molecular weight is given from the relationship
M h ) 20.01x + 18.02(1 - x)
(28)
so the total fog mass is
This system is solved by the definition of a 3 × 3 Jacobian matrix with the above presented partial derivatives, the general form of which is as follows for the ith iteration:
W ) 20.01(y11 + 2y12 + 6y16 + 8y18) + 18.02y2 + LM h ya (29) 38.03yc + 28.97ya + 28.02yn + Ma
(i)
B J )
and the fog volume per mole is
Vtot ) V +
LM h ya FLMa
(30)
1000W (g/L) Vtot
(31)
The Algorithm In Figure 1 a schematic description is given of the algorithm developed describing the static mixing of HF
∂F1(i)/∂x
∂F1(i)/∂f1
∂F2(i)/∂T ∂F3(i)/∂T
∂F2(i)/∂x ∂F3(i)/∂x
∂F2(i)/∂f1 ∂F3(i)/∂f1
]
while the correction vector δ of the solution vector Xb with elements T, x, and f1 is of the form
B δ (i) ) [δ1(i), δ2(i), δ3(i)]T
The density of the liquid phase is taken as constant, equal to FL ) 1.2 g/cm3, and the fog density is
F)
[
∂F1(i)/∂T
where
δ1i ) Ti - Ti-1 δ2i ) xi - xi-1 δ3i ) f1i - f1i-1 The resulting linear (3 × 3) system is solved using the CRAMER determinants methodology described in
4848 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
Figure 1. Overview of the HFCALCULATIONS algorithm.
ref 10, which actually transforms a 3 × 3 system into a 2 × 2 one resulting in drastic reduction of CPU time. Comparison of HFCALCULATIONS Results with Experimental Values Very few experiments involving HF releases are reported in the open literature. Among them the most well-known are those of the Goldfish test series in the Nevada test facility mentioned by Hanna et al.11 and Puttock et al.6 as well as the wind tunnel simulations (a following phase of the previous tests) sponsored by the industry cooperative “HF mitigation/assessment” program to assess the effectiveness of vapor barriers in delaying and diluting accidental releases of HF (Petersen et al.12 and Fthenakis et al.13). Fluid bed tests in flow chambers, where different dry powders of low hazard and toxicity are used to mitigate HF aerosol releases instead of water sprays, are also mentioned by Schatz and Fthenakis.14 These tests are more appropriate for checking the dispersion and movement of an HF cloud downwind with the appropriate computational fluid dynamics programs, though HFCALCULATIONS could theoretically be used in simulating Goldfish tests starting from a 100% liquid fraction of HF. The main issue of this work remains the validation of the static HF-air mixing thermodynamic model, and this was done by comparing the HFCALCULATIONS’ prediction with the original experiments conducted by Schotte.2 In these experiments HF (or HF/N) was mixed in a range of concentrations with air containing varying amounts of water vapor. The resulting temperature
Figure 2. Influence of initial mixing temperature of HF with moist air at 80% relative humidity and 768 mmHg on relative fog density.
change was measured as a function of HF concentration and relative humidity. The results of these experiments, together with our model’s predictions, are reported in Appendix I. As the HF fog relative density is the main parameter affecting its dispersion, it is very interesting to study the factors affecting its value, which are ambient temperature, air relative humidity, and presence of inert gases, like N2. Figure 2 presents the influence of the HF and air initial mixing temperature on the value of relative fog density, as estimated by the developed algorithm. It is seen that the relative fog density depends heavily on the mixing temperature especially for mixture temperatures higher than 35 °C, when the fog tends to be neutral. Lower mixing temperatures yield almost al-
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4849
Figure 3. Influence of relative air humidity on relative fog density of HF fog at 768 mmHg and initial mixing temperature of 25 °C.
Figure 4. Influence of HF composition in an HF-N2 mixture on the relative density of the resulting HF fog with moist air (initial mixing conditions 25 °C and 768 mm Hg).
ways heavier-than-air clouds. This conclusion is rather expected, if one recalls that low temperature favors the depolymerization reactions and leads to the system cooling. For further decrease of the mixing temperature, the relative fog density will remain constant to the value of the relative density of fully condensed HF. The mixing temperature is, on the other hand, a very important parameter, as the equilibrium constants of the various HF polymers and HF-H2O complex are very sensitive to temperature values (decrease in temperature leads to increase of depolymerization velocity), the equilibrium being also affected by the quantity of HF in the gaseous phase. As the normal HF boiling point is 20 °C in atmospheric conditions, HF vapor partial pressure is rather high, thus favoring depolymerization reactions. The impact of air relative humidity on the fog relative density as it is estimated by the new algorithm is depicted in Figure 3, where one can observe that lowering of air relative humidity leads to subsequent increase of relative fog density. This is due to the fact that condensation of air humidity on one hand offers heat to the system resulting in the fog temperature rise and on the other it causes the decrease of HF vapor partial pressure, thus limiting the depolymerization reactions and the fog temperature decrease. This is the explanation of all experimental facilities with water curtains, which aim at the mitigation of HF accidental releases, as described in Fthenakis.15 Keeping HF together with inert gases, like N2, leads to the decrease of HF fog relative density, when it mixes with moist air at different relative humidities, as it is depicted in Figure 4. This is due to the fact that inert gases lead to the HF gaseous partial pressure decrease,
Figure 5. Isolines of unity HF fog relative density depicting HF composition versus initial mixing temperature for various values of relative air humidity at 768 mmHg.
a condition limiting depolymerization reactions. On the other hand, most inert gases, like N2, are, per se, lighter than air, so their presence in the fog lead to the decrease of fog relative density. Finally, Figure 5 presents the isodensity lines (relative fog density equals unity) for different values of initial mixing temperature and relative air humidity, as predicted by the new algorithm. This diagram tells whether the fog relative density is greater or lower than the density value for given values of temperature, relative humidity, and total equivalent HF molar fraction in the fog. Points located in the space above the unity isodensity line define that the yielding fog is lighter-than-air. Respectively, points located under this line define the creation of heavier-than-air fogs. If, for example, one decides to mix HF with ambient air in 1:10 ratio (10% mol of HF) at initial mixing temperature of 25 °C, this fog will have relative density equal to unity if and only if the relative air humidity equals 100%. Else, for relative humidity 80%, the fog will be heavier-than-air, as this point lies under the 80% RH isocurve. Accordingly, for initial mixing temperature lower than 25 °C it will also be heavier-than-air, and the same will happen if HF mixes with air in ratio lower than 10% (molar). Efficiency of the Developed HFCALCULATIONS Algorithm Although the theoretical background of the developed algorithm is mainly based on the work of Schotte,2,9 the authors of this paper have chosen a different way of resolving the numerical and physical problem for higher solution efficiency and stability. In his analysis Schotte describes the use of an algorithm for the estimation of x, T and f1 based on two nested Newton-Raphson routines for the estimation of x and f1 in the first place and one external routine for the calculation of the temperature T. This algorithm is based on the verification of the total mass balances of HF components and H2O in fog, as well as on eq 21 and gives very similar results to the ones of HFCALCULATIONS algorithm as it has been shown in Appendix I. The present algorithm is at least 2.5 times faster in terms of CPU time than the algorithm presented by Schotte, as it is explained in Appendix II. A second reason for building a new solution algorithm was the instability in the initial conditions, i.e., the initial values of T, x, f1, when these last ones were estimated in a stepwise procedure, where the error in each step influences the solution of the next making the whole algorithm unstable.
4850 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
In HFCALCULATIONS this danger is significantly limited by the use of the 3 × 3 matrix solution, where operations depend globally on the values of all three variables (T, x, f1) of the solution in each round. HFCALCULATIONS has been developed in Fortran 77 and needs less than 1 s CPU time for a complete solution of the problem on a Pentium 200. Discussion of Results A new algorithm called HFCALCULATIONS describing the temperature and relative density changes during the release of HF into the atmosphere and the subsequent fog formation during mixing with air has been developed, based on the theoretical model proposed by Schotte.2 The significant ability of this algorithm to describe the various mixing phenomena and fog properties has been proved and it is at least 2,5 times faster than Schotte’s algorithm. The theoretical conclusions resulting from Figures 2-5 can be summarized as follows: (i) The temperature of the resulting mixture, together with polymer composition and liquid-phase composition are major factors affecting cloud density. (ii) Fog temperature decrease is mainly due to its depolymerization reactions which are favored in low temperatures. (iii) The fog temperature decrease is mainly balanced by the condensation of water vapor and HF itself. (iv) The HF polymer formation depends mainly on the HF concentration in air. (v) The excess air humidity hinders the creation of HF polymers together with its depolymerization, as it decreases the HF vapor partial pressure and simultaneously heats the system. (vi) The initial mixture temperature of HF with ambient air plays a catalytic role in the relative density of the created cloud as it reduces the HF vapor partial pressure and hinders the depolymerization reactions. Lowering of the initial mixing temperature results in the rise of the relative density of the created fog. (vii) Keeping HF under inert atmosphere causes decrease of HF vapor partial pressure, limits the depolymerization reactions, and leads to the decrease of the fog relative density. The developed HFCALCULATIONS algorithm can be used with any dispersion model to predict the fog movement because of prevailing wind currents. Nomenclature An ) constant in vapor pressure equation Bn ) constant in vapor pressure equation Cp ) specific heat at constant pressure, cal/(K mol) f1 ) fugacity of HF monomer, atm H, H1, H25 ) enthalpy departures of HF from ideal (monomeric) gas at the fog temperature, the initial HF temperature, and 25 °C, respectively, cal/mol ∆H2, ∆H6, ∆H8 ) enthalpies of association of dimer, hexamer, and octamer of HF, cal/mol ∆Hc ) enthalpy of reaction of HF and H2O, cal/mol K2, K6, K8 ) equilibrium constants for dimmer, hexamer, and octamer formation, respectively, atm-(n-1) Kc ) equilibrium constant for formation of H2O complex, atm-1 L ) liquid in fog, mol mixture, molar fraction per mole of mixture M1, M2, MA, MN ) HF, H2O air (dry basis), and N2, respectively, mol M h ) mean molecular weight of aqueous HF P ) total pressure, atm P1, P2 ) partial pressures of HF and H2O, respectively
P20 ) partial pressure of H2O in ambient air, atm Q ) ratio of masses of N2 and HF Qn ) molar fraction of HF in HF-N2 mixture R ) gas constant ) 1.9869 cal/(K mol) for density calculations of 83.14 bar cm3/(kg mol) T, T1, Ta ) fog, initial HF, and air temperature, respectively, °C V ) fog volume, cm3, per mole of gas phase W ) mass of fog, g, per mole of gas phase x ) apparent mole fraction of HF (molecular weight of 20.01) in aqueous HF y11, y12, y16, y18 ) actual mole fractions of monomer, dimmer, hexamer, and octamer of HF, respectively y2, yc, yn, ya ) actual mole fraction of H2O, HF‚H2O complex, N2, and dry air, respectively Y1, Y2 ) apparent mole fractions of HF and H2O, respectively YN2, YO2 ) apparent mole fractions of N2 and O2 in the mixture Greek Symbols F, FL ) densities of fog and aqueous HF, respectively, g/L Φ ) fugacity coefficient of HF in the gas phase
Appendix I: Comparison with Schotte’s Experimental Values and Model The results of Schotte’s experiments, together with our model’s predictions, are reported in Tables 1 and 2. These tables report the final temperature, liquid phase composition, and the amount of HF condensed for HF-air mixtures starting from very low HF concentration till 20% molar. The observed differences between measured and calculated values are very low at only 5.4% maximum which corresponds to a 1.6 °C difference in temperature. Moreover, the temperature change due to the mixing of HF with ambient air is depicted in Figures 6 and 7, where HF mixes with air at a relative humidity of 60% and 80%, respectively. In Figure 6 one can observe the mixture temperature lowering in the 50% relative humidity curve and for very small HF concentrations mixing temperature may lower below zero. This can be explained by the fact that fog formation has not started yet, although the depolymerization reaction goes on and absorbs heat leading to the mixture cooling. Then the temperature rises because of HF condensation and dilution reactions with air humidity and droplet formation until a maximum value, and it starts decreasing again when the heat absorbed by the depolymerization becomes significant. Last, in Figure 8 the density of HF fog relative to ambient air at 80% relative humidity is depicted as a function of HF concentration (comparing experimental data and HFCALCULATIONS predictions). It can be observed that the relative density starts at 1.0 (before mixing) and as HF percentage rises it decreases due to the temperature increase, while in the following steps it rises again exceeding unity as temperature decreases. Appendix II: Efficiency of HFCALCULATIONS Compared to Schotte’s Algorithm In general a Newton-Raphson algorithm is governed by the recursive relationship
xi+1 ) xi +
f(xi) f′(xi)
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4851 Table 1. Experimental Values and HFCALCULATIONS Predictions of Mixing HF and Air at 80% Relative Humidity and 768 mmHg temp, °C fog
xHF
% of HF condensed
tot. % mol of HF
air
HF
Schotte
HFCALC
Schotte
HFCALC
Schotte
HFCALC
0.373 0.696 1.130 1.520 1.900 2.860 5.550 8.850 12.07 15.68 19.63
25.9 26.1 26.0 25.9 25.8 26.0 25.7 25.7 25.5 25.6 25.7
26.6 26.7 26.7 26.8 26.7 25.9 26.2 25.8 25.1 25.0 25.2
29.1 30.8 32.5 33.8 34.2 35.9 35.6 32.0 23.2 12.7 7.0
29.1 31.3 32.9 33.9 34.4 35.5 35.0 31.2 22.7 13.4 7.7
0.261 0.306 0.346 0.370 0.389 0.423 0.490 0.553 0.620 0.687 0.737
0.261 0.306 0.344 0.368 0.387 0.421 0.488 0.551 0.620 0.688 0.738
26.7 29.5 29.8 29.8 29.7 29.3 28.7 28.8 29.5 29.9 29.2
26.5 29.0 29.6 29.5 29.3 29.0 28.5 28.9 29.7 30.4 29.9
Table 2. Experimental Values and HFCALCULATIONS Predictions of Mixing HF-N2 and Air at 80% Humidity and 768 mmHg temp °C fog
xHF
% of HF condensed
tot. % mol of HF
% HF in HF-N2
air
HF-N2
Schotte
HFCALC
Schotte
HFCALC
Schotte
HFCALC
0.197 0.482 0.951 1.410 1.870 2.850 5.260 7.290 10.18 12.76 14.95
48.5 50.7 49.1 50.5 50.3 50.7 50.2 46.9 47.1 47.3 47.3
26.4 26.4 26.3 26.3 26.2 25.8 25.9 25.4 25.4 25.5 25.7
26.7 26.6 26.6 26.4 26.4 25.7 25.6 26.7 26.2 25.8 25.8
28.5 29.7 33.5 35.0 35.9 39.7 41.3 42.7 43.4 42.1 40.8
28.1 31.1 33.9 35.7 36.9 38.4 40.8 42.3 42.8 42.3 41.6
0.215 0.282 0.333 0.362 0.384 0.419 0.469 0.495 0.526 0.552 0.570
0.218 0.280 0.332 0.361 0.383 0.418 0.468 0.495 0.526 0.552 0.571
16.0 21.5 21.8 21.0 20.1 18.6 15.8 13.1 12.2 11.7 11.1
15.0 21.5 21.3 20.6 19.5 18.3 15.6 13.0 12.2 11.8 11.3
Figure 6. Temperature change during mixing of HF and moist air at 50% and 80% relative humidity and 25-26 °C and comparison of experimental data with HFCALCULATIONS predictions.
Figure 7. Temperature change during mixing of (HF-N2, 50% mol) with moist air at 60% and 80% relative humidity and 25-26 °C and comparison of experimental data with HFCALCULATIONS predictions.
where the initial value x0 is assumed and f(x) is the equation (or equation system) which needs to be resolved. When the derivative f′(x) cannot be analytically evaluated, then the easiest way is to evaluate it arithmetically through the expression
We assume that the basic operations number for the calculation of f1(x1,x2,x3),f2(x1,x2,x3), f3(x1,x2,x3) in both algorithms is S and in the most demanding case the F operations regarding the evaluation of each function fi(x1,x2,x3) are executed N times and the Q operations regarding its derivative fi′ evaluation are executed M times. Then one can consider without substantial error that the operations number needed for the evaluation of each function fi(x1,x2,x3) is the same with the operations number Q needed for the derivative evaluation of each function fi′(x1,x2,x3) in an analytical way. Thus one considers Q ) F, and in each NewtonRaphson subroutine described in that way one has in total N(F + Q) ) N(F + F) ) 2NF operations performed. Assuming that one needs the same operations number
f′(xi) )
f(xi + δx) - f(xi) δxi
(32)
Both Schotte’s algorithm and HFCALCULATION solve a nonlinear (3 × 3) system of equations which is defined as
(f1(x1,x2,x3),f2(x1,x2,x3),f3(x1,x2,x3)) ) (0,0,0)
4852 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
demands F operations, the total number of operations needed is 9F and this equals exactly the operations number for the Jacobi determination. Thus the total number of operations performed by HFCALCULATIONS according to eq 33 is
N(S + ∆ + 9F) ) N(S + ∆ + (S/3)) ) N(4S + ∆)
(35)
Comparing the total number of operations in the two algorithms from eqs 34 and 35 one has
Figure 8. Density of HF fog relative to ambient air at 80% relative humidity and 25-26 °C and comparison of Schotte’s algorithm predictions with HFCALCULATIONS ones.
F for the determination of each function f, and having performed in total S operations, one has
F ) S/3
(33)
Schotte’s algorithm defines a 3 × 3 system in which the main variable is x1 while the other two are dependent on the first i.e.,
x2 ) x2(x1)|x3 x3 ) x3(x1)|x2 Considering the value of the other two constant, it creates in this way an external Newton-Raphson loop for the evaluation of the main parameter x1 and two nested loops for the calculations of the other two. Thus for the internal Newton-Raphson routines one will have according to the above analysis 4NF operations for each repetition of the external loop. The same is true for the external Newton-Raphson routine with the clarification that the derivative fi′(x1,x2,x3) cannot be analytically evaluated, as every function fi depends on all values of xi defined by the internal Newton-Raphson routines. Thus the derivative evaluation will be based (in the simplest case) on eq 32 performing twice the F operations and twice the internal routines for the calculation of the other two variables. The total number of operations in Schotte’s algorithm will be according to eq 33
N[2F + 2(4NF)] ) 2NF(1 + 4N) ) 2NS(1 + 4N)/3 (34) In HFCALCULATIONS algorithm, one has only one Newton-Raphson loop which solves the system (f1(x1,x2,x3),f2(x1,x2,x3),f3(x1,x2,x3)) ) (0,0,0) at once, as in the recursive relationship of the Newton-Raphson algorithm
xi+1 ) xi +
f(xi) f ′(xi)
one considers f(xi) as a vector and its derivative the Jacobian of the matrix which f(xi) itself defines, having thus N repetitions of S operations for the estimation of fi, N operations for the solution of the 3 × 3 linear system resulting with this methodology, and N repetitions of Q’ operations for the evaluation of the Jacobian. As all derivatives are analytically evaluated (9 for the 3 × 3 system) and the evaluation of each derivative
(total number of operations in Schotte’s algorithm) - (total number of operations in HFCALCULATIONS) ) N(4S - 2/35 - 8/35N + ∆) (36) The operations ∆ of solving a linear (3 × 3) system are few in comparison with operations S, but nevertheless if one makes the assumptions that S ) ∆, eq 36 results in
(total number of operations in Schotte’s algorithm) - (total number of operations in HFCALCULATIONS) ) NS(13 - 8N)/3 (37) which is positive only when N < 1 and negative for every other N value. This proves that the HFCALCULATIONS algorithm is faster. Comparing the total number of operations in the two algorithms, one can take the ratio of eqs 34 and 35 (total number of operations in Schotte’s algorithm) ) (total number of operations in HFCALCULATIONS) 2(1 + 4N) (38) 15
Considering N)5 as a typical value of repetitions of the Newton-Raphson algorithm, the ratio of eq 38 takes the value 2.8; that is HFCALCULATIONS is at least 2.5× faster than the algorithm proposed by Schotte which is very important when multiple calculations are to be conducted with HFCALCULATIONS integrated in a complicated computational fluid dynamics dispersion prediction model, such us SOCRATES. Literature Cited (1) Mudan, K. S. Use of Toxicity Data in Quantitative Risk Assessment of HF Alkylation Units, AIChE, 1989 Summer National Meeting, Aug 20-34, Philadelphia, PA. (2) Schotte, W. Fog Formation of Hydrogen Fluoride in Air. Ind. Eng. Chem. Res. 1987, 26, 300-306. (3) Muralidhar, R.; Jersey, G. R.; Krambeck, F. J.; Sundaresan, S. A two-phase release model for quantifying risk reduction for modified HF alkylation catalysts. J. Hazard. Mater. 1994, 44, 141183. (4) Papadourakis, A.; Caram, H. S.; Barner, C. L. Upper and lower bounds of droplet evaporation in two-phase jets. J. Loss Prev. Process Ind. 1994, 4, 93-101. (5) McFarlane, K.; Prothero, A.; Puttock, J. S.; Roberts, P. T.; Witlox, H. W. M. Development and validation of atmospheric dispersion models for ideal gases and hydrogen fluoride, Part 1 Technical Reference Manual (TNER 90.015), Part 2 HGSYSTEM Program User’s Manual (TNER 90.016), Shell Research Limited Thornton Research Centre; prepared for The Industry Cooperative HF Mitigation/Assessment Program, Ambient Impact Assessment Subcommitee, Nov 1990.
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4853 (6) Puttock, J. S.; McFarlane, K.; Prothero, A.; Rees, F. J.; Roberts, P. T.; Witlox, H. W. M.; Blewitt, D. N. Dispersion models and hydrogen fluoride predictions. J. Loss Prev. Process Ind. 1991, 4, 16-28. (7) Papazoglou, I. A.; Aneziris, O.; Bonanos, G.; Christou, M. SOCRATES: a computerized toolkit for quantification of the risk from accidental releases of toxic and/or flammable substances. In Integrated Regional Health and Environmental Risk Assessment and Safety Management; Gheorghe, A. V., Ed.; Int. J. Environ. Pollut. 1996, 6 (4-6), 500-533. (8) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The properties of gases and liquids, 3rd ed.; McGraw-Hill: New York, 1974. (9) Schotte, W. Thermodynamic Model for HF Fog Formation; letter to C. A. Soczek C. A. dated Aug 31, 1988; E I du Pont de Nemours & Co., du Pont Experimental Station, Engineering Department, Wilmington, DE 19898, private communication, as reported by MacFarlane (1990). (10) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes: The Art of Scientific Computing; Cambridge University Press: Cambridge, U.K., 1988. (11) Hanna, S. R.; Strimaitis, D. G.; Chang, J. C. Evaluation of fourteen hazardous gas models with ammonia and hydrogen fluoride field data. J. Hazard. Mater. 1991, 26, 127-158.
(12) Petersen, R. L.; Diener, R. Vapour barrier assessment programme for delaying and diluting heavier-than-air HF vapour clouds, wind tunnel modelling evaluation. J. Loss Prev. Process Ind. 1990, 3, 187-196. (13) Fthenakis, V. M.; Blewitt, D. N. Mitigation of hydrofluoric acid releases: simulation of the performance of water spraying systems. J. Loss Prev. Process Ind. 1993, 6 (4), 209-218. (14) Schatz, K. W.; Fthenakis, V. M. Mitigation of hydrogen fluoride aerosols by dry powders. J. Loss Prev. Process Ind. 1994, 7 (6), 451-456. (15) Fthenakis, V. M. Prevention and control of accidental releases of hazardous gases; Van Nostrand Reinhold: New York, 1993.
Received for review February 27, 1998 Revised manuscript received September 8, 1998 Accepted September 9, 1998 IE980131X
