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Vent sizing for gas-generating runaway reactions
Vent sizing for gas-generating reactions
runaway
Jasbir Singh Hazard Evaluation EN5 5TS, UK
Laboratory
Limited,
50 Moxon
Street,
Barnet,
Her&
Exothermic runaway reactions that generate non-condensible gas as the temperature increases, as is typical of decompositions for example, can reach extremely high rates of pressure rise necessitating emergency relief of the process vessel containing the reactant. Sizing of a relief device using presently recommended methods (e.g. DIERS) frequently leads to extremely large and expensive vents. This paper presents a methodology that leads to a simple but much improved method for vent sizing, fully allowing for two-phase release of the gas-liquid mixture. A number of examples are presented which lead to interesting conclusions about the influence of plant variables. Keywords: runaway reactions; vent sizing; two-phase release
Exothermic chemical reactions that are capable of exponentially accelerating in temperature and pressure are termed runaway reactions. The maximum pressure that can be generated by such a reaction is frequently much higher than the process equipment is designed to withstand, and so the vessel is fitted with a vent. The sizing of the vent depends crucially on the pressure-temperature relationship during the reaction. T by If the pressure P is related to the temperature an Antoine-type relation:
lnP=A,++ where A, and B1 are constants, then the system is a pure vapour pressure type. In this case, pressure relief will result in flashing off of vapour and the liquid temperature will fall as a consequence of the need to provide the latent heat for this phase change. When the temperature is controlled in this manner, the system is described as ‘tempered’. The DIERS research project developed elegant vent sizing relationships that are safe but not too conservative for vapour pressure systems’. Another category of systems, described as gasgenerating [or gassy), produce a pressure rise as a result of non-condensable gas generated during the reaction. In such cases, pressure and temperature do not follow an Antoine-type relationship. More importantly from a venting point of view, the removal of gas produces no cooling and the reaction temperature continues to rise despite gas removal. In effect, the gas production (and hence pressure rise) may be considered quite separately from the temperature rise. O!SC-4230/94/060461-11 0 1994 Butterworth-Heinemann
The vent sizing equation recommended for these gassy reactions is based on the maximum gas generation rate developed by the reaction. The vent is sized for two-phase flow, large enough to accommodate the maximum gas rate. The resulting relationship is simply:
where A is the vent area (m2), m, is the mass of reactants (kg), m, is the mass of sample (kg) used experimentally to measure the gas generation rate Qs (m3 s-i), G is the two-phase mass flux (kg me2 s-l), and Y is the specific volume of the two-phase mixture vented (m” kg-‘). The gas generation rate can be measured experimentally in a closed vessel:
where V, is the volume of gas space in the experimental equipment (m’), T, is its temperature (K) and T, is the reactant temperature at the maximum gas rate (Pa s-l). The pressure P,,, is the maximum (dP/dr),,,,, permissible reactor pressure (Pa). Combining Equations (1) and (la) and assuming that v = V,lm, A=
mFV,T, Gm,VJJ’,
dP ( dt ) mal.e
(2)
where V,, is the reactor volume (m’). The above approach ensures a safe vent size but frequently an overly conservative one. This lead to either unnecessary reductions in reactor batch size or expensive plant modifications to accommodate the large vent.
Ltd
J. Loss Prev. Process Ind., 7994, Volume
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481
Vent sizing for gas-generating
runaway
reactions:
This paper describes a new approach to relief sizing for gassy reactions which produces safe results but substantially smaller vents. The rigorous solution results in a differential equation that requires numerical but simpler approximate analytical integration, expressions are also presented.
Acquisition
of reaction
data
It is common practice, particularly following the DIERS work, to base the vent size on data obtained from bench-scale adiabatic calorimetry equipment. The relevant equations (for example Equation (1)) can then be developed specifically to take advantage of this data. Commercial devices suitable for obtaining such data have been described elsewherez,3; an example that illustrates the essential features is PHI-TEC II shown in Figure 1. This device consists of a sample cell containing mass m, of sample (about 100 g) surrounded by three ‘guard’ heaters. The sample temperature T and the pressure P are measured continuously. The heaters are computer-controlled such that their temperature is equal to that of the reactants, at all times, thus ensuring adiabatic conditions. The entire assembly is housed in a large pressure vessel (-4 litres capacity). A feature of the instrument is that a facility is also provided for pressure compensation between the inside and outside of the test cell. This is achieved by injection of nitrogen gas into the pressure vessel as the pressure in the test cell increases. This capability allows very thin-walled metal cells to be used, which
J. Singh minimizes the thermal capacity of the cell in relation to that of the test sample. When gassy reactions are examined, it is frequently convenient to use an ‘open’ test cell with a small vent in the top. The gaseous products of reaction fill the pressure vessel as the reaction proceeds. This is simply a convenience which allows the pressure of the gas generated by the reaction to be measured in a larger volume than the test cell. The temperature of the gas (in the vessel) will be essentially ambient since the heavy vessel will not heat up as the reaction proceeds. The reactant temperature will be unaffected by this change since the gas does not provide any latent cooling. An important feature of the design is the fact that the sample cell is extremely light, thus the thermal capacity of the metal is small compared with that of the sample. This allows the data from the tests to be applied directly to large-scale plant provided scaling can be properly achieved. A mathematical description of the data obtained from the set-up in Figure I will now be developed, firstly to describe a closed thermal runaway reaction and then to describe the effect of venting.
Mathematical
l l
VESSEL
@
TEST SAMPLE
Q
DIRECT STIRRING OPTION
of gassy reaction
The most common and mathematically most convenient reactor type is a closed batch system. The gas generated by reaction will be at the same pressure and temperature as the liquid and the amount of gas produced will be proportional to the liquid quantity. It is also convenient to make the following assumptions: l
CONTAINMENT
description
that the gas is insoluble in the liquid that the liquid has a negligible vapour pressure that the liquid volume remains constant
The reaction of interest is of course an exothermic one, and the rate of reaction and hence the amount of gas generated increase with temperature. The reaction is assumed to be irreversible for the present purposes. Consider the reactor in Figure 2 which illustrates the main variables relating to the gas/liquid mixture as the reaction proceeds. If the gas in the vessel obeys the ideal gas law, then the pressure P is related to gas volume Vg and temperature T, by: P=y+
nRT
where n is the number of moles of gas and R is the gas constant. For a fixed volume, differentiation gives: NITROGEN EXHAUST
Figure 1 The PHI-TEC adiabatic of reactive materials
482
calorimeter
J. Loss Prev. Process Ind.,
for the assessment
1994, Volume
The subscript c refers to the fact that pressure rise in a closed vessel is being considered. The objective is to obtain the data required in Equation (4) from an instrument such as PHI-TEC II
7, Number
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Vent sizing for gas-generating -VENT
FLOW (F)
VENT
-(MAXIMUM
OPENING
PRESSURE
(P.)
ALLOWABLE PRESSURE Pn)
and, REACTANT MASS (m, j VOLUME (V,) TEMPERATURE (T,)
at any other
n, =
[
V PFT
e pressure
n=
Main variables
in gassy reaction
P:
1 D
Then, by difference, the generated by the reaction
Figure2
J. Singh
where the subscript e refers to experimentally determined values in the large vessel. Thus, knowing the vessel volume and temperature (which are fixed), the number of moles of gas produced by the reaction can be obtained from the pressure measurements. If the initial pressure in the large vessel (prior to reaction) is Pi, the initial amount of gas Iii is given by:
[4e = [pi&]
-
reactions:
[2]=[%&]e
VOID SPACE (V, )
-
runaway
number of moles of gas n, at any pressure P, is:
[(P-Pi)&]e
The above equations involving n are limited to situations where the amount of reactant is m,, the sample size used in the test. If the reactant quantity used in a large-scale plant is mrr then Equations (5) and (6) become, respectively:
model
dn dt = and
/THERMOCOUPLE _-OPEN
VENT
>RESSURE v’ESSEL -
(7)
CONTAINMENT
VOLUME V, PRESSURE P. TEMPERATURE
and rate by:
therefore substitution into Equation (4) for the of pressure change in a large-scale unit is given
+(P-Pi,$z
T.
(9)
c --.-
/ TEST CELL SAMPLE MASS TEMPERATURE
GUARD
HEATERS
MAGNETIC STIRRER = =
m. T,
Figure 3 Arrangement
of test apparatus
for gassy reaction
test
in Figure 1. A simplified sketch of the experimental and the variables being measured is arrangement shown in Figure 3. As a result of the set-up in Figure 3, the gas temperature (in the large containment vessel) remains virtually constant, i.e. it does not follow the liquid temperature. By applying the gas law to the vessel, we obtain:
Using Equation (9), it is possible to calculate the explosion pressure trace in a large-scale reactor using data obtained from a suitable small-scale test. The variables with the subscript e are test-dependent. Note in particular that T, and dT,ldt are not affected in this manner: this is because the thermal capacity of the small test cell in Figure 1 is negligible. This means that the experimental temperature profile is exactly the same as that expected at the large scale where the temperature runaway will occur under essentially adiabatic conditions.
Venting of a gas-generating
reaction
Rigorous solution If the reactor system illustrated in Figure 2 is fitted with a vent that opens at a specified pressure P,, and the vented fluid is a two-phase mixture of quality x
J. Loss Prev. Process Ind.,
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483
Vent sizing for gas-generating
runaway
reactions:
(quality being defined as the mass fraction of gas), then the vent opening will affect the rate of pressure rise as described by Equation (9) in two ways: l
l
the the the the
mass of reactants will be reduced and hence rate of pressure rise will be slowed down vented two-phase mixture will directly reduce pressure.
The reduction in gas generation can be determined by simply replacing the mass of reactants m, in Equation (9) by: m = m, - F(t-t,)(l-x)
(10)
where m is the reactant mass at time t, t, is the time at which the vent opens and F is the venting rate (kg s-l). The pressure change due to venting (i.e. removal of gas/liquid) can be calculated from: dP dt
c-1 = -
PFY V,
J. Singh A = FIG
(13)
where G is the mass flux through the vent. The method proposed by Tangren et a1.5 may be used to evaluate G (see Appendix 2). This essentially requires only a knowledge of the void fraction a of the twophase mixture entering the vent in order to calculate the mass flux. Analytical equations A safe vent size is assured if the pressure rise can be arrested when the gas generation rate is maximum. Conservatively, it may be assumed that the maximum rate occurs at the maximum allowable pressure in the reactor, i.e. when P = P,, the rate parameter fi is given by:
(11) max,c
(see Appendix 1 for derivation). Thus, for P > P, and t > t,, the rate of pressure change in a vented vessel, (dP/dt),, becomes: (m, - F(l - x)(t - t,)) - FFu
g
c
from Equation
= m,/? - Fp (t-t,)
(1-x)
(9):
- $Fu !z
(12)
max.e is the maximum rate of pressure where (dPldt),,,,, rise measured experimentally, and P,,,, and T, are the pressure and temperature, respectively, at that point in time. If the pressure in the vented reactor is to cease to rise at this point, then the required vent rate is obtained by putting (dP/dt), = 0 in Equation (12). Therefore:
where
e
+ (P - Pi).%
Note that the time t, for a foaming liquid is the point at which the vent opens, but, for other types of reactants, it can be &me later time at which liquid starts to be vented out of the reactor. This is considered later in more detail. The parameter p represents the rate of pressure rise in a closed vessel per unit mass of reactant. The dT,Jdt term is normally much smaller than T,(dPld& and is frequently neglected, so that
Equation (12) describes the pressure-time relationship after the vent opens, for any selected vent rate F. If the fluid vented is a homogenous gas-liquid froth, the specific volume may be estimated from:
m, y= [ vg+v=
1
mrVOPm
F=
(14a) v +
/3,,,AtVo( 1 -x)
The value of At can be calculated of Equation (12):
(
rn,p - GFv Ar - F(1-x)p2 g )
where AP = P,,,-P,, p between P, and P,, and Combining Equations quadratic equation which value of overpressure BP.
-1
where V, is the liquid volume. be calculated from:
J. Loss Prev. Process
The vent area A may
Ind., 1994, Volume
(14)
where At is the time taken to go from P, to the maximum pressure P,,, in the real venting incident. Since V, = crVo, the above equation may also be written as:
AP =
484
m,& + &Ar (l-x)
F = (P,,,IV,)v
A, = $?(l
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+ ,&(l-x)/4P,
by integration
-(At)’
(15)
is the average value of j3 F = P, + APl2. (14a) and (15) leads to a can be solved for At for any The resulting equations are:
Vent sizing for gas-generating
C1=AP + = m&n X=BIP¶l? and Y = PJP,
Solutions to this quadratic show that the result is highly insensitive to AP. It follows that AP may be assumed to be 0; this immediately leads to an analytical solution for F (for x = 0):
For practical purposes: x = Pvf&
l
(17)
where K is the correction
+
J. Singh
factor:
- (E)“/(%),l
Comparing Equation (17) with the DIERS equation (Equation (2)), the vent area is reduced by a factor K, which is equal to 1 + 2(1-x)/(1+x). Thus, when (dP/dt) Q (dP/dt), (i.e. x 2- 0), the relief area is a factor of 3 smaller than that predicted by the DIERS method. The reason for this reduction is that, under these conditions, the vent is open long before the maximum gas generation rate is reached, thus a considerable quantity of reactant is vented out before the worst condition is reached. Conversely, when (dPldt), = (dPldt),, the equation reduces exactly to the DIERS result. Note that both (dP/dt), and (dP/dt), are experimentally measured rates. The experimental pressure at which (dP/dt), is selected is the following:
(18) It is also necessary to correct for any differences in pad pressure of inert gas above the reacting sample. The choice of (dPldt), is crucial. For a truly foamy system, it corresponds to the point of vent opening. For other systems, it is the point at which liquid
of the pressuretime
Using experimental data of the type described earlier, the pressure-time (P - t) profile of a reactor undergoing thermal runaway may be calculated very simply using Equation (9) (before vent opening) and Equation (12) (after the vent opens). The period after vent opening may be treated in a number of ways depending on the type of relief device (rupture disc or valve) and the type of discharge, possibilities being:
l
it follows that:
A-
K=l+
Detailed evaluation profile
l
(dPldt)v ^-p (dP/dt), Since the relief area A = F/G,
reactions:
discharge from the reactor begins, which may be later (i.e. at a higher value of dP/dt). Equation (17) assumes that the rate of pressure rise with time is linear and uses the arithmetic mean as the average. This grossly overestimates the true average, and hence leads to short venting times and correspondingly large vent sizes. Therefore, in general, the equation is conservative in its prediction of the vent size, but the sizes will normally be smaller than those calculated from the DIERS equation.
1
aAP (1 -x)/P,
runaway
all gas venting, due to total disengagement between liquid and gas partial disengagement a foamy system, leading to an homogenous liquid/ gas release immediately after the vent opens.
A software package VENTSIZE, developed by the Hazard Evaluation Laboratory Limited, provides all of the above options (in addition to other capabilities). In the case of a partially disengaging system, VENTSIZE uses the chum-turbulent model6 to predict the onset and cessation of liquid entrainment (see Appendix 3) for two-phase flow. In this option, the venting phase depends crucially on two variables: l l
the void fraction in the reactor (cy) the gas velocity through the vessel
The void fraction is of course determined by the liquid level; a high level will increase the likelihood of twophase flow. The gas velocity is governed by the reaction rate as measured by the gas generation (hence dP/dt). Typically, all gas venting normally occurs when the vent first opens (since the gas generation rate will be relatively slow), but this will change to two-phase relief as the worst condition (high dP/dt) is approached. The equations for mass flux determination under conditions of all gas flow are given in Appendix 4.
Application
to actual decomposition
data
Selected reaction data The increase in pressure and temperature following the decomposition of a liquid nitrate compound is shown in Figure 4. This shows the characteristic way in which the pressure and temperature rise: very slowly at first and then increasingly rapidly. The final 20 bar rise in pressure take less than 1 min compared with nearly 75 min for the first 20 bar. The rate of pressure rise as a function of temperature is shown in Figure 5;
J. Loss Rev.
Process Ind., 1994, Volume
7, Number
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485
Vent sizing for gas-generating
runaway
reactions:
J. Singh Vent sizing using the simple analytical equations The runaway data presented in Figures 46 were obtained using a low thermal mass test cell and a sample size (m,) of 80 g; the pressure was measured in a gas space (Ve) of 100 ml. The data were used to calculate the relief size for all the systems to be considered in this paper for a reactor with the following characteristics: l
TIME[min)
Figure4 Runaway reaction temperature versus time
experimental
120
F
100
‘= B g
80
g B f B rd
60 40 20
l
data:
pressure
and
._-----I_____
O!
I
Figure 5 Runaway reaction experimental rise versus temperature
volume = 5 m3, cross-sectional area = 1 m2 relief set pressures = 3, 10 or 20 bara reactant density = 1000 kg mm3 = 101 bar min-’ at 182.3”C, (dP/dt),,,,, 41.85 bar
... E!i l
175.0
2110.1 0
data: rate of pressure
the shape is rather similar to that for a first-order Arrhenius reaction. If the rate data are plotted as a function of time, Figure 6 is obtained: this re-emphasizes the highly non-linear nature of the change in pressure (and temperature) with time. The maximum rate of pressure rise (-101 bar min’) is reached after about 75.3 min, yet the rate is less than 1 bar min-’ at 70 min into the reaction.
l
Using the above data, the DIERS method (Equation (2)) and the new analytical expression (Equation (17)) were used to calculate the relief size for a range of overpressures. The results are plotted in Figure 7 where the relief area predicted by Equation (17) is compared with that obtained from the DIERS equation. The area reduction factor is between 3 and 1 (i.e. no reduction), depending on the relief opening pressure and the reactor void fraction. The factor that determines the reduction in area is the pressure (and related dP/dt) at which the vent first opens compared with the point at which the maximum dP/dt is realized. When these two points approach (x = l), the ratio of vent areas approaches 1; when they are very different (x = 0), the ratio of vent areas approaches 3.0. For example, the effect of an increase in reactor void fraction is to reduce the pressure rise; this means that a given relief pressure is reached later in the exotherm (i.e. closer to (dP/dt),,,) as the void fraction is increased. This leads to a vent area increasingly closer to the DIERS value as shown in Figure 7. An increase in the vent opening pressure (for a given void fraction) also moves the point at which the vent opens closer to the worst condition. The trend in Figure 7 will be generally true in all cases but the actual figures will depend on the experimental data. Figure 7 should not be regarded as being generally applicable.
3.50 3.00
Figure 6 Runaway rise versus time
486
reaction experimental
data: rate of pressure
J. Loss Prev. Process Ind., 1994, Volume
Figure 7 Vent area ratio versus system (two-phase relief)
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reactor
void fraction:
foamy
Vent sizing for gas-generating
Table 1 Comparison of dPldt Conditions
of time-weighted
at vent opening
and arithmetic
Average
dfldt
reactions:
J. Singh
averages
(bar min-‘1
Time (min)
Pressure (bara)
dPldt (bar min-‘1
Simple
Time-weighted
30.2 50.2 70.0 72.0 74.5 75.0 75.34
2.6 4.2 9.5 11.5 11.4 24.6 41.9
0.06 0.12 0.84 1.33 8.74 21.1 101
60.5 50.6 50.9 51.2 54.9 81.1 101
0.88 1.51 6.09 9.22 28.62 50.0 101
30 FRICTIONLESS VENTAREA[sq m] Figure 8 Comparison disc size for (foamy) bar)
Comparison with the rigorous method to establish that the relief area predicted by the new analytical equation presented in this paper is safe even though it is frequently much smaller than the value obtained from the DIERS equation. Firstly, it is worth recalling that the reactant mass depletion predicted by the analytical equation is based on a simple arithmetic average of the rates of pressure rise at the point of vent opening and (dP/dt),,,. It is clear overfrom Figure 6 that this will always substantially predict the actual average rate. In Table I, the actual time-weighted average (dP/dt) is compared with the simple arithmetic average of the end values of dP/dt for a number of different vent opening times. The difference between the two averages can be quite substantial: for a vent that opens 30.2 min after the exotherm ‘onset’, the arithmetic average is higher than the actual average by a factor of over 50. Further verification of the analytical equation may be obtained by comparing the results from the more detailed technique as represented by Equation (12). This was carried out using the experimental data presented in Figures 4 to 6 with the VENTSIZE software. In order to use this program, an initial relief size is specified and the program then evaluates the corresponding pressure-temperature profile. In this way, the overpressure corresponding to a given relief size may be evaluated. In Figure 8, the results of several such calculations are compared with the predictions from the analytical equation. For a given overpressure, the vent area predicted from the analytical equation is usually much larger than that from the more accurate simulation. This is certainly true at reactor void fractions below 0.5; however, when the void fraction is about 0.9 or higher, the two methods converge. At this point, the simple DIERS method and the new analytical equation also converge (as is apparent from Figure 7). The simulations from which the results in Figure 8 were extracted were carried out assuming that there is a breather vent on the reactor which, by relieving gas at slow rates, delays the opening of the rupture disc until the reaction rate is quite fast. Using this facility, the rise in reactor pressure did not start until
It is important
runaway
of simple and detailed methods: rupture two-phase relief (relief set pressure = 3
72 min into the runaway, at which time the temperature was already 109°C and the experimental dP/dt = 1.3 bar min-‘. Inclusion of a delay in this manner will increase the amount of reactant present as the worst self-heat condition is approached. This is clearly a conservative assumption. Effect of vapour-liquid disengagement The discussion above has been confined to the case of homogeneous two-phase relief where the liquid-gas mixture vents as a foam. This is a correct assumption in some cases, but frequently some separation between the liquid and gas can be expected. This behaviour will delay the removal of liquid from the reactor, and, in the case of ‘gassy’ reactions, will increase the vent size needed. Using the same reaction as in the last example, and a relief set pressure of 3 bara, vent sizes for a number of different cases were evaluated assuming the churn-turbulent model for gas/liquid disengagement, and the results are plotted in Figure 9. The relief area for a foamy system is typically less than half that for one where gas-liquid disengagement may be expected. The primary reason for this difference is that the reactor empties continuously after the vent opens for foamy systems; however, with churn2.00
1.60
7 g
1.20
s
k.2 i
060
s 0.40
0.00 C
Figure 9 Relief area calculated of foamy and churn-turbulent
by detailed models
J. Loss Prev. Process Ind., 1994, Volume
method:
comparison
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487
Vent sizing for gas-generating ~‘20%
runaway
reactions:
J. Singh
Busting 0%~ Set Pressure = 3 bar o
j Vent Area = 0.0635
,verprersure =025:bor (opprox]
sq.
m&es
i
4000 2 E
3ooo
B = 2000 K-! s
1000
0
,
7.
Ji
73
TIUE [An) Figure
10 Mass depletion following and churn-turbulent systems
relief: comparison
Znfluence of initial reactor void fraction The above result, comparing foamy and churn-turbulent systems, is not really unexpected, but what is surprising is that changes in initial void fraction (see Figure 9) have little effect on the relief area (except when the void fraction is around 0.7 or higher). This is true both for the homogeneous (foamy) and churnturbulent systems. Certainly, reactors with a lower initial load (i.e. higher void fraction) are inherently less hazardous and should therefore need a smaller vent. The simple analytical methods (Equations (2) and (17)) comply with this expectation and predict a reduced vent size with increasing void fraction (see Figure 8 for example). The reason that the more rigorous method produces a somewhat different dependence on (z becomes apparent from a more detailed evaluation of the venting incident. The pressure-time profiles produced by VENTSIZE for four different cases are presented in Figure II :
1
71.00
I
72.M
73.00
c TIME
14.00
’
I
i.e. for void fractions (a) of 0.2, 0.5, 0.7 and 0.9. Up to 72 min into the reaction, the breather vent is able to hold a pressure of 1 bar, at which point the pressures start to rise. For a = 0.2, the rupture disc opens about 20 seconds later, while for a = 0.9, the disc opens over 3 min later. When the bursting disc opens, the pressure starts to fall immediately for all except the (Y = 0.9 case. For cy = 0.2, the pressure falls to -1.85 bar before rising again, while the pressures do not fall as far for LY= 0.5 and a = 0.7. All start to rise at about 74.7 min. The explanations for this behaviour may be found in Figure 12 where the change in reactant mass is plotted as function of time for the four cases considered in Figure II. The point of relief opening is marked by an immediate fall in the reactant mass. As an example, let us compare the cases of a = 0.2 and a = 0.5. For cy = 0.2, there is a higher initial reactant mass, but, as a consequence, the vent for this system opens earlier. At the time that the vent for the (Y = 0.5 reactor opens, both the pressure and reactant mass for a = 0.2 are lower (see Figures 11 and 12). Therefore, at this point (i.e. when the vent for LY= 0.5 has just opened), the u = 0.2 reactor is already the less hazardous one. The mass and mass flux are compared in Figure 13 for these two situations. When
’
72
76.00
[min)
Figure 11 Effect of initial void fraction on pressure following venting: foamy gas-liquid mixture (bursting
488
7
75.00
change disc)
J. Loss Prev. Process Ind., 1994, Volume
75
[min]
Figure 12 Effect of initial void fraction on reactant mass change following venting: foamy gas-liquid mixture (bursting disc)
of foamy
turbulent systems, liquid reactant is only removed when the gas generation rate is sufficiently high. This is clearly illustrated in Figure IO, where, despite the fact that the relief size is 2.3 times greater, the churnturbulent case results in a much higher mass in the reactor when the maximum dPfdt condition is reached.
0.00
74 TIME
73
74 TIUE [min]
75
76
Figure 13 Effect of initial void fraction on mass in reactor and mass flux through vent (vent area = 0.0035 mZ)
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Vent sizing for gas-generating
(bar)
Reactor void fraction (n)
Foamy (relief area = 0.0035 m*)
Churn-turbulent (relief area = 0.008 m2)
0.05 0.20 0.50 0.70 0.80 0.90
0.24 0.28 0.71 0.81 0.64
0.27 0.25 0.24 All gas venting All gas venting All gas venting
reactions:
J. Singh
larger than that for a bursting disc because a valve will minimize the reactant loss during periods of slow reaction. A comparison of the results is presented in Figure I4 (for churn-turbulent two-phase flow). The area needed for a relief valve (a = 0.2) can be 5 to 6 times larger than for a bursting disc if the required overpressure is low. As the permitted overpressure increases, the area required for a relief valve approaches the bursting disc result. The main reason for the difference in results between a bursting disc and a relief valve is that the valve, by its very nature, will re-seat and hold the reactor pressure close to the set value (3 bar in the above example), whereas a bursting disc will let it fall freely. This is shown in Figure 15 for reactors with an initial void fraction of 0.2, a relief valve orifice of 0.0155 m* and a bursting disc of only 0.008 mz. When the worst condition (maximum dPldt) is reached, the pressure rises from the set pressure (3 bar) in the case of the relief valve, whereas the bursting disc reactor pressure rises from a lower value (in this case only - 1.5 bar). If the two situations were to reach the same maximum pressure, then clearly the orifice size for the relief valve would need to be much larger.
Table2 Effect of initial reactor void fraction on overpressure (relief set pressure = 3 bar with breather vent) Overpressure
runaway
the vent opens for (Y = 0.2, the mass flux is close to 13500 kg m-* s-l; when the vent for LY= 0.5 opens, the mass flux is about 9000 kg m-* SC’, but, by this same time, that for the former case has dropped to around 6000 kg rnp2 s-l. After a couple of minutes, the two reactors end up virtually identical in terms of the reactant charge: hence, when the maximum dP/dt point is reached (-75.3 min), they undergo the same pressure rise. Returning again to Figure 12, the reactant mass for all four cases is almost the same when the maximum dPldt point is approached, yet, for a! = 0.9, a higher pressure is reached than for (Y = 0.2. The explanation for this lies in the fact that, although the pressure rise will be comparable for these two cases, they are not both at the same pressure just before the maximum dPldt point is approached: the highest pressure is seen for a = 0.9 because the bursting disc has not yet failed. Table 2 shows that the maximum pressure in the reactor occurs when a! is approximately 0.8. These precise results are of course specific to the data being considered and will not apply for other situations. However, the general trends will hold true.
Conclusions A mathematical description of an exothermic runaway reaction involving a gas-generating chemical reaction has been presented: this permits complete description of large-scale incidents involving both closed and vented vessels using only small-scale experimental data. The method takes advantage of certain useful properties of gas-generating reactions which mean that virtually no physical or chemical property data about the reacting system are needed. The methodology has been incorporated into a software program (VENTSIZE) which reads in experimental data and allows various scenarios to be investigated. In this paper, the influence of a range of important parameters on the vent size needed to protect a process vessel has been studied, Since twophase discharge of a gas-liquid mixture is frequently
Comparison between relief valve and bursting disc important The choice of relief device is a particularly variable in the case of gas-generating reactions. The relief area needed for a relief valve may de considerably
1.60 1.40 1.20 P z g
loo 0.80
g
0.60
o
0.40 0.20 0.0 [b
IO
0.010
0.020
0.030
0.040
FRICTIONLESS VENT AREA
0.050
0.060
0.
[sq m]
Figure 14 Comparison of relief valve and rupture turbulent two-phase flow (set pressure = 3 bar)
disc: churn-
3
72.0
72 5
73.0
73.5
74.0 74.5 TIUE [min)
75.0
75.5
76.0
Figure 15 Relief valve and bursting disc comparison: turbulent flow (set pressure = 3 bar)
J. Loss Prev. Process Ind., 1994, Volume
7, Number
churn-
6
489
,,__A
_:_:--
La_
_^^
_^_^_^
a:---
_..^e.._,A.,
r~..,.*;~..r.
I. Singh
considered important for reacting systems, this aspect has been evaluated in detail. It has been found that homogeneous two-phase relief, frequently considered to be a ‘worst case’, can in fact be less onerous than relief under the assumption of churn-turbulent two-phase relief. A relief area which is higher by a factor of about 2 is required for the churn-turbulent model. The two systems converge as the permitted overpressure (after the vent opens) increases. For both homogeneous and churn-turbulent systems, it has been found that the initial reactor void fraction may be much less important than is frequently assumed. Void fractions between 0.6 and 0.05 produce virtually identical results. At higher void fractions, the two-phase homogeneous venting assumption may lead to a small increase in the required relief size with void fraction increase (for a fixed overpressure) before a reduction of size is predicted. A comparison of the vent area needed for a relief valve as opposed to a bursting disc has also been carried out. In general, a bursting disc is found to be preferable over a relief valve in that over-pressure during venting is reduced for a given installation. Some of these results are surprising but they can all be explained quite simply and logically from consideration of the detailed changes in pressure, reactant mass and flow rate (as a function of time) as the venting proceeds. The detailed model description has also been simplified to generate a simple analytical equation which may be used to evaluate conservative relief sizes. This equation is comparable with the vent sizing equation proposed by DIERS but with a simple correction factor that is a function of the void fraction, the relief set pressure and the experimental dPldt. The equation produces vent sizes that are up to 3 times smaller than those predicted by the DIERS equation. Comparison with the more detailed model shows that the analytical equation is quite conservative except at high reactor void fractions (LY= 0.9) where it converges to the correct result as given by the detailed model. These conclusions apply to calculations using the homogeneous two-phase relief assumption. It is also necessary to compare the results of the analytical equation and detailed model for the churnturbulent mode of venting as this can be more onerous than homogeneous venting. Here too, the analytical equation is conservative but the safety margin is much reduced, particularly at high reactor void fraction.
References 1 Leung, J. C. AIChE 1. 1986, 32(10) 2 Singh, J. in ‘International Symposium on Runaway Reactions’, CCPS/AIChE, New York, 1989 3 hung, J. C., Fauske, H. K. and Fisher H. G. Thermochim. Acta 1986, 104(13) 4 Fauske, H. K. and Leung, .I. C. PlantlOpns Pro& 1987. 6(2) 5 Tangren, R. F., Dodge C. H. and Seifert, H. S. J. Appl. Physics 1949, 20(7). 631-645
490
J. Loss
Prev.
Process Ind.,
1994, Volume
6 Leung,
J. C. AIChE
J. 1987, 33(6)
Nomenclature Area (m’) Cross-sectioned area of the reactor (m’) Constants in Antoine equation or in quadratic equation Empirical correlating parameter in calculation of superficial gas velocity Venting rate (kg s-‘) Acceleration due to gravity (m s-*) Mass flux through vent (kg mm2 s-l) = [2(1-X)/(1+X)] + 1 Mass of reactants (kg) Mass of experimental test sample (kg) Initial mass of reactants in large reactor (kg) Mass of gas (kg) Molecular weight Moles of gas Initial amount of gas (moles) Pressure (Pa) Pressure corresponding to onset of choked flow (Pa) Pressure in gas space of large vessel (Pa) Initial gas pressure in test apparatus (Pa) Maximum permissible reactor pressure (Pa) Pressure at which vent opens (Pa) = Pm - P, (Pa) Gas generation rate (mJ s-‘) Gas constant (J mol-’ K-l) Time (s) Time at which vent opens (s) Temperature (K) Experimental gas space temperature (K) Reactant (liquid) temperature (K) Single bubble rise velocity (m s-‘) Volume fm? Gas spa& in’experimental apparatus (m3) Volumetric gas flow through the vent (m’ s ‘) Gas space in reactor (m’) Liquid volume in reactor (m’) Total reactor volume (m’) Gas quality (mass fraction) entering vent Void fraction in reactor Rate of pressure rise per unit mass of reactant (Pa s-l kg-‘) Value of fi at maximum gas generation rate (Pa s-l kg-‘) Value of p at vent opening condition (Pa s-’ kg-‘) Ratio of specific heats for gas Pressure ratio Critical pressure ratio for choking conditions Specific volume of two-phase mixture (m’ kg-‘) Density (kg m-‘) Surface tension (N m-‘) = PJP.. = m.&, (Pa 6-l) Superficial gas velocity
Appendix
1
Pressure loss due to venting Consider a vessel containing a gas of volume V, and a liquid of volume V,. The gas is ideal, with molecular weight M,, and is insoluble in the liquid. The system is at temperature T and pressure P (which is entirely due to a non-condensable gas pressure). Initially, when the gas and liquid are enclosed in a vessel, the ideal gas law gives:
7, Number
PV, = nRT p=-
nRT V,
6
Vent sizing for gas-generating When both liquid and gas are removed pressure changes as follows:
(vented),
the ?&
(Al) The first term on the right-hand side relates to the effect of gas removal and the second relates to the effect of liquid removal (gas expansion). The gas removal term can be simplified as follows if the mass of gas is M,:
RT __=-_dn VP dt
RT 1 V, M,
=
reactions:
J. Singh
[2.016 +(g$7]-o.“4
unless P,,, < P,, where
p, = P,t, 77c in which case + is the atmospheric
where P,,, exit.
dM, dt
runaway
pressure at the nozzle
Appendix 3
RT pp dVs V, M,,, dt
Equations for prediction of onset of two-phase relief using the churn-turbulent model (Ref. 6)
but
The transition from all gas flow to two-phase flow in the churn-turbulent regime occurs if the following condition for the superficial gas velocity !P is met:
PM,
83 = RT hence
RT dn _ _=V, dt
p
dl/,
V,
dt
The pressure corresponding follows:
642)
reduction due to liquid removal (and gas expansion) may be simplified as
where CO is an empirical correlating parameter. For churn-turbulent flow, C, = 1.5. The dimensionless superficial vapour velocity is calculated from: q=&
nRT dVZ V,” dt =-- P dV,
V,
(A3)
dt
Substituting Equations (Al) gives:
(A2)
and (A3)
into Equation
X m
where Vr is the volumetric gas flow through the vent (m” s-l) and A, is the cross-sectional area of the reactor (m2), where the single bubble rise velocity U, is given by: u, = 1.53
afg
[
1
(Pr - 4) o.25 P?
where pf is the liquid density, pe is the gas density, uf is the surface tension, and g is the acceleration due to gravity.
but
..+!!L+!%)
Appendix 4
hence
Calculation of G for all gas or vapour JIow If
Appendix 2 Equations for G for non-flashing two-phare flow (Ref. 5) The mass flux through by:
a frictionless
then the mass flux through
an ideal nozzle is given
nozzle is given
_
(l-h)V]i’”
otherwise
in which 77 = v~, where
where y is the ratio of specific heats for gas.
J. Loss Prev. Process
Ind., 1994, Volume 7, Number
6
491
runaway
Jasbir Singh Hazard Evaluation EN5 5TS, UK
Laboratory
Limited,
50 Moxon
Street,
Barnet,
Her&
Exothermic runaway reactions that generate non-condensible gas as the temperature increases, as is typical of decompositions for example, can reach extremely high rates of pressure rise necessitating emergency relief of the process vessel containing the reactant. Sizing of a relief device using presently recommended methods (e.g. DIERS) frequently leads to extremely large and expensive vents. This paper presents a methodology that leads to a simple but much improved method for vent sizing, fully allowing for two-phase release of the gas-liquid mixture. A number of examples are presented which lead to interesting conclusions about the influence of plant variables. Keywords: runaway reactions; vent sizing; two-phase release
Exothermic chemical reactions that are capable of exponentially accelerating in temperature and pressure are termed runaway reactions. The maximum pressure that can be generated by such a reaction is frequently much higher than the process equipment is designed to withstand, and so the vessel is fitted with a vent. The sizing of the vent depends crucially on the pressure-temperature relationship during the reaction. T by If the pressure P is related to the temperature an Antoine-type relation:
lnP=A,++ where A, and B1 are constants, then the system is a pure vapour pressure type. In this case, pressure relief will result in flashing off of vapour and the liquid temperature will fall as a consequence of the need to provide the latent heat for this phase change. When the temperature is controlled in this manner, the system is described as ‘tempered’. The DIERS research project developed elegant vent sizing relationships that are safe but not too conservative for vapour pressure systems’. Another category of systems, described as gasgenerating [or gassy), produce a pressure rise as a result of non-condensable gas generated during the reaction. In such cases, pressure and temperature do not follow an Antoine-type relationship. More importantly from a venting point of view, the removal of gas produces no cooling and the reaction temperature continues to rise despite gas removal. In effect, the gas production (and hence pressure rise) may be considered quite separately from the temperature rise. O!SC-4230/94/060461-11 0 1994 Butterworth-Heinemann
The vent sizing equation recommended for these gassy reactions is based on the maximum gas generation rate developed by the reaction. The vent is sized for two-phase flow, large enough to accommodate the maximum gas rate. The resulting relationship is simply:
where A is the vent area (m2), m, is the mass of reactants (kg), m, is the mass of sample (kg) used experimentally to measure the gas generation rate Qs (m3 s-i), G is the two-phase mass flux (kg me2 s-l), and Y is the specific volume of the two-phase mixture vented (m” kg-‘). The gas generation rate can be measured experimentally in a closed vessel:
where V, is the volume of gas space in the experimental equipment (m’), T, is its temperature (K) and T, is the reactant temperature at the maximum gas rate (Pa s-l). The pressure P,,, is the maximum (dP/dr),,,,, permissible reactor pressure (Pa). Combining Equations (1) and (la) and assuming that v = V,lm, A=
mFV,T, Gm,VJJ’,
dP ( dt ) mal.e
(2)
where V,, is the reactor volume (m’). The above approach ensures a safe vent size but frequently an overly conservative one. This lead to either unnecessary reductions in reactor batch size or expensive plant modifications to accommodate the large vent.
Ltd
J. Loss Prev. Process Ind., 7994, Volume
7, Number
6
481
Vent sizing for gas-generating
runaway
reactions:
This paper describes a new approach to relief sizing for gassy reactions which produces safe results but substantially smaller vents. The rigorous solution results in a differential equation that requires numerical but simpler approximate analytical integration, expressions are also presented.
Acquisition
of reaction
data
It is common practice, particularly following the DIERS work, to base the vent size on data obtained from bench-scale adiabatic calorimetry equipment. The relevant equations (for example Equation (1)) can then be developed specifically to take advantage of this data. Commercial devices suitable for obtaining such data have been described elsewherez,3; an example that illustrates the essential features is PHI-TEC II shown in Figure 1. This device consists of a sample cell containing mass m, of sample (about 100 g) surrounded by three ‘guard’ heaters. The sample temperature T and the pressure P are measured continuously. The heaters are computer-controlled such that their temperature is equal to that of the reactants, at all times, thus ensuring adiabatic conditions. The entire assembly is housed in a large pressure vessel (-4 litres capacity). A feature of the instrument is that a facility is also provided for pressure compensation between the inside and outside of the test cell. This is achieved by injection of nitrogen gas into the pressure vessel as the pressure in the test cell increases. This capability allows very thin-walled metal cells to be used, which
J. Singh minimizes the thermal capacity of the cell in relation to that of the test sample. When gassy reactions are examined, it is frequently convenient to use an ‘open’ test cell with a small vent in the top. The gaseous products of reaction fill the pressure vessel as the reaction proceeds. This is simply a convenience which allows the pressure of the gas generated by the reaction to be measured in a larger volume than the test cell. The temperature of the gas (in the vessel) will be essentially ambient since the heavy vessel will not heat up as the reaction proceeds. The reactant temperature will be unaffected by this change since the gas does not provide any latent cooling. An important feature of the design is the fact that the sample cell is extremely light, thus the thermal capacity of the metal is small compared with that of the sample. This allows the data from the tests to be applied directly to large-scale plant provided scaling can be properly achieved. A mathematical description of the data obtained from the set-up in Figure I will now be developed, firstly to describe a closed thermal runaway reaction and then to describe the effect of venting.
Mathematical
l l
VESSEL
@
TEST SAMPLE
Q
DIRECT STIRRING OPTION
of gassy reaction
The most common and mathematically most convenient reactor type is a closed batch system. The gas generated by reaction will be at the same pressure and temperature as the liquid and the amount of gas produced will be proportional to the liquid quantity. It is also convenient to make the following assumptions: l
CONTAINMENT
description
that the gas is insoluble in the liquid that the liquid has a negligible vapour pressure that the liquid volume remains constant
The reaction of interest is of course an exothermic one, and the rate of reaction and hence the amount of gas generated increase with temperature. The reaction is assumed to be irreversible for the present purposes. Consider the reactor in Figure 2 which illustrates the main variables relating to the gas/liquid mixture as the reaction proceeds. If the gas in the vessel obeys the ideal gas law, then the pressure P is related to gas volume Vg and temperature T, by: P=y+
nRT
where n is the number of moles of gas and R is the gas constant. For a fixed volume, differentiation gives: NITROGEN EXHAUST
Figure 1 The PHI-TEC adiabatic of reactive materials
482
calorimeter
J. Loss Prev. Process Ind.,
for the assessment
1994, Volume
The subscript c refers to the fact that pressure rise in a closed vessel is being considered. The objective is to obtain the data required in Equation (4) from an instrument such as PHI-TEC II
7, Number
6
Vent sizing for gas-generating -VENT
FLOW (F)
VENT
-(MAXIMUM
OPENING
PRESSURE
(P.)
ALLOWABLE PRESSURE Pn)
and, REACTANT MASS (m, j VOLUME (V,) TEMPERATURE (T,)
at any other
n, =
[
V PFT
e pressure
n=
Main variables
in gassy reaction
P:
1 D
Then, by difference, the generated by the reaction
Figure2
J. Singh
where the subscript e refers to experimentally determined values in the large vessel. Thus, knowing the vessel volume and temperature (which are fixed), the number of moles of gas produced by the reaction can be obtained from the pressure measurements. If the initial pressure in the large vessel (prior to reaction) is Pi, the initial amount of gas Iii is given by:
[4e = [pi&]
-
reactions:
[2]=[%&]e
VOID SPACE (V, )
-
runaway
number of moles of gas n, at any pressure P, is:
[(P-Pi)&]e
The above equations involving n are limited to situations where the amount of reactant is m,, the sample size used in the test. If the reactant quantity used in a large-scale plant is mrr then Equations (5) and (6) become, respectively:
model
dn dt = and
/THERMOCOUPLE _-OPEN
VENT
>RESSURE v’ESSEL -
(7)
CONTAINMENT
VOLUME V, PRESSURE P. TEMPERATURE
and rate by:
therefore substitution into Equation (4) for the of pressure change in a large-scale unit is given
+(P-Pi,$z
T.
(9)
c --.-
/ TEST CELL SAMPLE MASS TEMPERATURE
GUARD
HEATERS
MAGNETIC STIRRER = =
m. T,
Figure 3 Arrangement
of test apparatus
for gassy reaction
test
in Figure 1. A simplified sketch of the experimental and the variables being measured is arrangement shown in Figure 3. As a result of the set-up in Figure 3, the gas temperature (in the large containment vessel) remains virtually constant, i.e. it does not follow the liquid temperature. By applying the gas law to the vessel, we obtain:
Using Equation (9), it is possible to calculate the explosion pressure trace in a large-scale reactor using data obtained from a suitable small-scale test. The variables with the subscript e are test-dependent. Note in particular that T, and dT,ldt are not affected in this manner: this is because the thermal capacity of the small test cell in Figure 1 is negligible. This means that the experimental temperature profile is exactly the same as that expected at the large scale where the temperature runaway will occur under essentially adiabatic conditions.
Venting of a gas-generating
reaction
Rigorous solution If the reactor system illustrated in Figure 2 is fitted with a vent that opens at a specified pressure P,, and the vented fluid is a two-phase mixture of quality x
J. Loss Prev. Process Ind.,
1994, Volume
7, Number
6
483
Vent sizing for gas-generating
runaway
reactions:
(quality being defined as the mass fraction of gas), then the vent opening will affect the rate of pressure rise as described by Equation (9) in two ways: l
l
the the the the
mass of reactants will be reduced and hence rate of pressure rise will be slowed down vented two-phase mixture will directly reduce pressure.
The reduction in gas generation can be determined by simply replacing the mass of reactants m, in Equation (9) by: m = m, - F(t-t,)(l-x)
(10)
where m is the reactant mass at time t, t, is the time at which the vent opens and F is the venting rate (kg s-l). The pressure change due to venting (i.e. removal of gas/liquid) can be calculated from: dP dt
c-1 = -
PFY V,
J. Singh A = FIG
(13)
where G is the mass flux through the vent. The method proposed by Tangren et a1.5 may be used to evaluate G (see Appendix 2). This essentially requires only a knowledge of the void fraction a of the twophase mixture entering the vent in order to calculate the mass flux. Analytical equations A safe vent size is assured if the pressure rise can be arrested when the gas generation rate is maximum. Conservatively, it may be assumed that the maximum rate occurs at the maximum allowable pressure in the reactor, i.e. when P = P,, the rate parameter fi is given by:
(11) max,c
(see Appendix 1 for derivation). Thus, for P > P, and t > t,, the rate of pressure change in a vented vessel, (dP/dt),, becomes: (m, - F(l - x)(t - t,)) - FFu
g
c
from Equation
= m,/? - Fp (t-t,)
(1-x)
(9):
- $Fu !z
(12)
max.e is the maximum rate of pressure where (dPldt),,,,, rise measured experimentally, and P,,,, and T, are the pressure and temperature, respectively, at that point in time. If the pressure in the vented reactor is to cease to rise at this point, then the required vent rate is obtained by putting (dP/dt), = 0 in Equation (12). Therefore:
where
e
+ (P - Pi).%
Note that the time t, for a foaming liquid is the point at which the vent opens, but, for other types of reactants, it can be &me later time at which liquid starts to be vented out of the reactor. This is considered later in more detail. The parameter p represents the rate of pressure rise in a closed vessel per unit mass of reactant. The dT,Jdt term is normally much smaller than T,(dPld& and is frequently neglected, so that
Equation (12) describes the pressure-time relationship after the vent opens, for any selected vent rate F. If the fluid vented is a homogenous gas-liquid froth, the specific volume may be estimated from:
m, y= [ vg+v=
1
mrVOPm
F=
(14a) v +
/3,,,AtVo( 1 -x)
The value of At can be calculated of Equation (12):
(
rn,p - GFv Ar - F(1-x)p2 g )
where AP = P,,,-P,, p between P, and P,, and Combining Equations quadratic equation which value of overpressure BP.
-1
where V, is the liquid volume. be calculated from:
J. Loss Prev. Process
The vent area A may
Ind., 1994, Volume
(14)
where At is the time taken to go from P, to the maximum pressure P,,, in the real venting incident. Since V, = crVo, the above equation may also be written as:
AP =
484
m,& + &Ar (l-x)
F = (P,,,IV,)v
A, = $?(l
7, Number
6
+ ,&(l-x)/4P,
by integration
-(At)’
(15)
is the average value of j3 F = P, + APl2. (14a) and (15) leads to a can be solved for At for any The resulting equations are:
Vent sizing for gas-generating
C1=AP + = m&n X=BIP¶l? and Y = PJP,
Solutions to this quadratic show that the result is highly insensitive to AP. It follows that AP may be assumed to be 0; this immediately leads to an analytical solution for F (for x = 0):
For practical purposes: x = Pvf&
l
(17)
where K is the correction
+
J. Singh
factor:
- (E)“/(%),l
Comparing Equation (17) with the DIERS equation (Equation (2)), the vent area is reduced by a factor K, which is equal to 1 + 2(1-x)/(1+x). Thus, when (dP/dt) Q (dP/dt), (i.e. x 2- 0), the relief area is a factor of 3 smaller than that predicted by the DIERS method. The reason for this reduction is that, under these conditions, the vent is open long before the maximum gas generation rate is reached, thus a considerable quantity of reactant is vented out before the worst condition is reached. Conversely, when (dPldt), = (dPldt),, the equation reduces exactly to the DIERS result. Note that both (dP/dt), and (dP/dt), are experimentally measured rates. The experimental pressure at which (dP/dt), is selected is the following:
(18) It is also necessary to correct for any differences in pad pressure of inert gas above the reacting sample. The choice of (dPldt), is crucial. For a truly foamy system, it corresponds to the point of vent opening. For other systems, it is the point at which liquid
of the pressuretime
Using experimental data of the type described earlier, the pressure-time (P - t) profile of a reactor undergoing thermal runaway may be calculated very simply using Equation (9) (before vent opening) and Equation (12) (after the vent opens). The period after vent opening may be treated in a number of ways depending on the type of relief device (rupture disc or valve) and the type of discharge, possibilities being:
l
it follows that:
A-
K=l+
Detailed evaluation profile
l
(dPldt)v ^-p (dP/dt), Since the relief area A = F/G,
reactions:
discharge from the reactor begins, which may be later (i.e. at a higher value of dP/dt). Equation (17) assumes that the rate of pressure rise with time is linear and uses the arithmetic mean as the average. This grossly overestimates the true average, and hence leads to short venting times and correspondingly large vent sizes. Therefore, in general, the equation is conservative in its prediction of the vent size, but the sizes will normally be smaller than those calculated from the DIERS equation.
1
aAP (1 -x)/P,
runaway
all gas venting, due to total disengagement between liquid and gas partial disengagement a foamy system, leading to an homogenous liquid/ gas release immediately after the vent opens.
A software package VENTSIZE, developed by the Hazard Evaluation Laboratory Limited, provides all of the above options (in addition to other capabilities). In the case of a partially disengaging system, VENTSIZE uses the chum-turbulent model6 to predict the onset and cessation of liquid entrainment (see Appendix 3) for two-phase flow. In this option, the venting phase depends crucially on two variables: l l
the void fraction in the reactor (cy) the gas velocity through the vessel
The void fraction is of course determined by the liquid level; a high level will increase the likelihood of twophase flow. The gas velocity is governed by the reaction rate as measured by the gas generation (hence dP/dt). Typically, all gas venting normally occurs when the vent first opens (since the gas generation rate will be relatively slow), but this will change to two-phase relief as the worst condition (high dP/dt) is approached. The equations for mass flux determination under conditions of all gas flow are given in Appendix 4.
Application
to actual decomposition
data
Selected reaction data The increase in pressure and temperature following the decomposition of a liquid nitrate compound is shown in Figure 4. This shows the characteristic way in which the pressure and temperature rise: very slowly at first and then increasingly rapidly. The final 20 bar rise in pressure take less than 1 min compared with nearly 75 min for the first 20 bar. The rate of pressure rise as a function of temperature is shown in Figure 5;
J. Loss Rev.
Process Ind., 1994, Volume
7, Number
6
485
Vent sizing for gas-generating
runaway
reactions:
J. Singh Vent sizing using the simple analytical equations The runaway data presented in Figures 46 were obtained using a low thermal mass test cell and a sample size (m,) of 80 g; the pressure was measured in a gas space (Ve) of 100 ml. The data were used to calculate the relief size for all the systems to be considered in this paper for a reactor with the following characteristics: l
TIME[min)
Figure4 Runaway reaction temperature versus time
experimental
120
F
100
‘= B g
80
g B f B rd
60 40 20
l
data:
pressure
and
._-----I_____
O!
I
Figure 5 Runaway reaction experimental rise versus temperature
volume = 5 m3, cross-sectional area = 1 m2 relief set pressures = 3, 10 or 20 bara reactant density = 1000 kg mm3 = 101 bar min-’ at 182.3”C, (dP/dt),,,,, 41.85 bar
... E!i l
175.0
2110.1 0
data: rate of pressure
the shape is rather similar to that for a first-order Arrhenius reaction. If the rate data are plotted as a function of time, Figure 6 is obtained: this re-emphasizes the highly non-linear nature of the change in pressure (and temperature) with time. The maximum rate of pressure rise (-101 bar min’) is reached after about 75.3 min, yet the rate is less than 1 bar min-’ at 70 min into the reaction.
l
Using the above data, the DIERS method (Equation (2)) and the new analytical expression (Equation (17)) were used to calculate the relief size for a range of overpressures. The results are plotted in Figure 7 where the relief area predicted by Equation (17) is compared with that obtained from the DIERS equation. The area reduction factor is between 3 and 1 (i.e. no reduction), depending on the relief opening pressure and the reactor void fraction. The factor that determines the reduction in area is the pressure (and related dP/dt) at which the vent first opens compared with the point at which the maximum dP/dt is realized. When these two points approach (x = l), the ratio of vent areas approaches 1; when they are very different (x = 0), the ratio of vent areas approaches 3.0. For example, the effect of an increase in reactor void fraction is to reduce the pressure rise; this means that a given relief pressure is reached later in the exotherm (i.e. closer to (dP/dt),,,) as the void fraction is increased. This leads to a vent area increasingly closer to the DIERS value as shown in Figure 7. An increase in the vent opening pressure (for a given void fraction) also moves the point at which the vent opens closer to the worst condition. The trend in Figure 7 will be generally true in all cases but the actual figures will depend on the experimental data. Figure 7 should not be regarded as being generally applicable.
3.50 3.00
Figure 6 Runaway rise versus time
486
reaction experimental
data: rate of pressure
J. Loss Prev. Process Ind., 1994, Volume
Figure 7 Vent area ratio versus system (two-phase relief)
7, Number
6
reactor
void fraction:
foamy
Vent sizing for gas-generating
Table 1 Comparison of dPldt Conditions
of time-weighted
at vent opening
and arithmetic
Average
dfldt
reactions:
J. Singh
averages
(bar min-‘1
Time (min)
Pressure (bara)
dPldt (bar min-‘1
Simple
Time-weighted
30.2 50.2 70.0 72.0 74.5 75.0 75.34
2.6 4.2 9.5 11.5 11.4 24.6 41.9
0.06 0.12 0.84 1.33 8.74 21.1 101
60.5 50.6 50.9 51.2 54.9 81.1 101
0.88 1.51 6.09 9.22 28.62 50.0 101
30 FRICTIONLESS VENTAREA[sq m] Figure 8 Comparison disc size for (foamy) bar)
Comparison with the rigorous method to establish that the relief area predicted by the new analytical equation presented in this paper is safe even though it is frequently much smaller than the value obtained from the DIERS equation. Firstly, it is worth recalling that the reactant mass depletion predicted by the analytical equation is based on a simple arithmetic average of the rates of pressure rise at the point of vent opening and (dP/dt),,,. It is clear overfrom Figure 6 that this will always substantially predict the actual average rate. In Table I, the actual time-weighted average (dP/dt) is compared with the simple arithmetic average of the end values of dP/dt for a number of different vent opening times. The difference between the two averages can be quite substantial: for a vent that opens 30.2 min after the exotherm ‘onset’, the arithmetic average is higher than the actual average by a factor of over 50. Further verification of the analytical equation may be obtained by comparing the results from the more detailed technique as represented by Equation (12). This was carried out using the experimental data presented in Figures 4 to 6 with the VENTSIZE software. In order to use this program, an initial relief size is specified and the program then evaluates the corresponding pressure-temperature profile. In this way, the overpressure corresponding to a given relief size may be evaluated. In Figure 8, the results of several such calculations are compared with the predictions from the analytical equation. For a given overpressure, the vent area predicted from the analytical equation is usually much larger than that from the more accurate simulation. This is certainly true at reactor void fractions below 0.5; however, when the void fraction is about 0.9 or higher, the two methods converge. At this point, the simple DIERS method and the new analytical equation also converge (as is apparent from Figure 7). The simulations from which the results in Figure 8 were extracted were carried out assuming that there is a breather vent on the reactor which, by relieving gas at slow rates, delays the opening of the rupture disc until the reaction rate is quite fast. Using this facility, the rise in reactor pressure did not start until
It is important
runaway
of simple and detailed methods: rupture two-phase relief (relief set pressure = 3
72 min into the runaway, at which time the temperature was already 109°C and the experimental dP/dt = 1.3 bar min-‘. Inclusion of a delay in this manner will increase the amount of reactant present as the worst self-heat condition is approached. This is clearly a conservative assumption. Effect of vapour-liquid disengagement The discussion above has been confined to the case of homogeneous two-phase relief where the liquid-gas mixture vents as a foam. This is a correct assumption in some cases, but frequently some separation between the liquid and gas can be expected. This behaviour will delay the removal of liquid from the reactor, and, in the case of ‘gassy’ reactions, will increase the vent size needed. Using the same reaction as in the last example, and a relief set pressure of 3 bara, vent sizes for a number of different cases were evaluated assuming the churn-turbulent model for gas/liquid disengagement, and the results are plotted in Figure 9. The relief area for a foamy system is typically less than half that for one where gas-liquid disengagement may be expected. The primary reason for this difference is that the reactor empties continuously after the vent opens for foamy systems; however, with churn2.00
1.60
7 g
1.20
s
k.2 i
060
s 0.40
0.00 C
Figure 9 Relief area calculated of foamy and churn-turbulent
by detailed models
J. Loss Prev. Process Ind., 1994, Volume
method:
comparison
7, Number
6
487
Vent sizing for gas-generating ~‘20%
runaway
reactions:
J. Singh
Busting 0%~ Set Pressure = 3 bar o
j Vent Area = 0.0635
,verprersure =025:bor (opprox]
sq.
m&es
i
4000 2 E
3ooo
B = 2000 K-! s
1000
0
,
7.
Ji
73
TIUE [An) Figure
10 Mass depletion following and churn-turbulent systems
relief: comparison
Znfluence of initial reactor void fraction The above result, comparing foamy and churn-turbulent systems, is not really unexpected, but what is surprising is that changes in initial void fraction (see Figure 9) have little effect on the relief area (except when the void fraction is around 0.7 or higher). This is true both for the homogeneous (foamy) and churnturbulent systems. Certainly, reactors with a lower initial load (i.e. higher void fraction) are inherently less hazardous and should therefore need a smaller vent. The simple analytical methods (Equations (2) and (17)) comply with this expectation and predict a reduced vent size with increasing void fraction (see Figure 8 for example). The reason that the more rigorous method produces a somewhat different dependence on (z becomes apparent from a more detailed evaluation of the venting incident. The pressure-time profiles produced by VENTSIZE for four different cases are presented in Figure II :
1
71.00
I
72.M
73.00
c TIME
14.00
’
I
i.e. for void fractions (a) of 0.2, 0.5, 0.7 and 0.9. Up to 72 min into the reaction, the breather vent is able to hold a pressure of 1 bar, at which point the pressures start to rise. For a = 0.2, the rupture disc opens about 20 seconds later, while for a = 0.9, the disc opens over 3 min later. When the bursting disc opens, the pressure starts to fall immediately for all except the (Y = 0.9 case. For cy = 0.2, the pressure falls to -1.85 bar before rising again, while the pressures do not fall as far for LY= 0.5 and a = 0.7. All start to rise at about 74.7 min. The explanations for this behaviour may be found in Figure 12 where the change in reactant mass is plotted as function of time for the four cases considered in Figure II. The point of relief opening is marked by an immediate fall in the reactant mass. As an example, let us compare the cases of a = 0.2 and a = 0.5. For cy = 0.2, there is a higher initial reactant mass, but, as a consequence, the vent for this system opens earlier. At the time that the vent for the (Y = 0.5 reactor opens, both the pressure and reactant mass for a = 0.2 are lower (see Figures 11 and 12). Therefore, at this point (i.e. when the vent for LY= 0.5 has just opened), the u = 0.2 reactor is already the less hazardous one. The mass and mass flux are compared in Figure 13 for these two situations. When
’
72
76.00
[min)
Figure 11 Effect of initial void fraction on pressure following venting: foamy gas-liquid mixture (bursting
488
7
75.00
change disc)
J. Loss Prev. Process Ind., 1994, Volume
75
[min]
Figure 12 Effect of initial void fraction on reactant mass change following venting: foamy gas-liquid mixture (bursting disc)
of foamy
turbulent systems, liquid reactant is only removed when the gas generation rate is sufficiently high. This is clearly illustrated in Figure IO, where, despite the fact that the relief size is 2.3 times greater, the churnturbulent case results in a much higher mass in the reactor when the maximum dPfdt condition is reached.
0.00
74 TIME
73
74 TIUE [min]
75
76
Figure 13 Effect of initial void fraction on mass in reactor and mass flux through vent (vent area = 0.0035 mZ)
7, Number
6
Vent sizing for gas-generating
(bar)
Reactor void fraction (n)
Foamy (relief area = 0.0035 m*)
Churn-turbulent (relief area = 0.008 m2)
0.05 0.20 0.50 0.70 0.80 0.90
0.24 0.28 0.71 0.81 0.64
0.27 0.25 0.24 All gas venting All gas venting All gas venting
reactions:
J. Singh
larger than that for a bursting disc because a valve will minimize the reactant loss during periods of slow reaction. A comparison of the results is presented in Figure I4 (for churn-turbulent two-phase flow). The area needed for a relief valve (a = 0.2) can be 5 to 6 times larger than for a bursting disc if the required overpressure is low. As the permitted overpressure increases, the area required for a relief valve approaches the bursting disc result. The main reason for the difference in results between a bursting disc and a relief valve is that the valve, by its very nature, will re-seat and hold the reactor pressure close to the set value (3 bar in the above example), whereas a bursting disc will let it fall freely. This is shown in Figure 15 for reactors with an initial void fraction of 0.2, a relief valve orifice of 0.0155 m* and a bursting disc of only 0.008 mz. When the worst condition (maximum dPldt) is reached, the pressure rises from the set pressure (3 bar) in the case of the relief valve, whereas the bursting disc reactor pressure rises from a lower value (in this case only - 1.5 bar). If the two situations were to reach the same maximum pressure, then clearly the orifice size for the relief valve would need to be much larger.
Table2 Effect of initial reactor void fraction on overpressure (relief set pressure = 3 bar with breather vent) Overpressure
runaway
the vent opens for (Y = 0.2, the mass flux is close to 13500 kg m-* s-l; when the vent for LY= 0.5 opens, the mass flux is about 9000 kg m-* SC’, but, by this same time, that for the former case has dropped to around 6000 kg rnp2 s-l. After a couple of minutes, the two reactors end up virtually identical in terms of the reactant charge: hence, when the maximum dP/dt point is reached (-75.3 min), they undergo the same pressure rise. Returning again to Figure 12, the reactant mass for all four cases is almost the same when the maximum dPldt point is approached, yet, for a! = 0.9, a higher pressure is reached than for (Y = 0.2. The explanation for this lies in the fact that, although the pressure rise will be comparable for these two cases, they are not both at the same pressure just before the maximum dPldt point is approached: the highest pressure is seen for a = 0.9 because the bursting disc has not yet failed. Table 2 shows that the maximum pressure in the reactor occurs when a! is approximately 0.8. These precise results are of course specific to the data being considered and will not apply for other situations. However, the general trends will hold true.
Conclusions A mathematical description of an exothermic runaway reaction involving a gas-generating chemical reaction has been presented: this permits complete description of large-scale incidents involving both closed and vented vessels using only small-scale experimental data. The method takes advantage of certain useful properties of gas-generating reactions which mean that virtually no physical or chemical property data about the reacting system are needed. The methodology has been incorporated into a software program (VENTSIZE) which reads in experimental data and allows various scenarios to be investigated. In this paper, the influence of a range of important parameters on the vent size needed to protect a process vessel has been studied, Since twophase discharge of a gas-liquid mixture is frequently
Comparison between relief valve and bursting disc important The choice of relief device is a particularly variable in the case of gas-generating reactions. The relief area needed for a relief valve may de considerably
1.60 1.40 1.20 P z g
loo 0.80
g
0.60
o
0.40 0.20 0.0 [b
IO
0.010
0.020
0.030
0.040
FRICTIONLESS VENT AREA
0.050
0.060
0.
[sq m]
Figure 14 Comparison of relief valve and rupture turbulent two-phase flow (set pressure = 3 bar)
disc: churn-
3
72.0
72 5
73.0
73.5
74.0 74.5 TIUE [min)
75.0
75.5
76.0
Figure 15 Relief valve and bursting disc comparison: turbulent flow (set pressure = 3 bar)
J. Loss Prev. Process Ind., 1994, Volume
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6
489
,,__A
_:_:--
La_
_^^
_^_^_^
a:---
_..^e.._,A.,
r~..,.*;~..r.
I. Singh
considered important for reacting systems, this aspect has been evaluated in detail. It has been found that homogeneous two-phase relief, frequently considered to be a ‘worst case’, can in fact be less onerous than relief under the assumption of churn-turbulent two-phase relief. A relief area which is higher by a factor of about 2 is required for the churn-turbulent model. The two systems converge as the permitted overpressure (after the vent opens) increases. For both homogeneous and churn-turbulent systems, it has been found that the initial reactor void fraction may be much less important than is frequently assumed. Void fractions between 0.6 and 0.05 produce virtually identical results. At higher void fractions, the two-phase homogeneous venting assumption may lead to a small increase in the required relief size with void fraction increase (for a fixed overpressure) before a reduction of size is predicted. A comparison of the vent area needed for a relief valve as opposed to a bursting disc has also been carried out. In general, a bursting disc is found to be preferable over a relief valve in that over-pressure during venting is reduced for a given installation. Some of these results are surprising but they can all be explained quite simply and logically from consideration of the detailed changes in pressure, reactant mass and flow rate (as a function of time) as the venting proceeds. The detailed model description has also been simplified to generate a simple analytical equation which may be used to evaluate conservative relief sizes. This equation is comparable with the vent sizing equation proposed by DIERS but with a simple correction factor that is a function of the void fraction, the relief set pressure and the experimental dPldt. The equation produces vent sizes that are up to 3 times smaller than those predicted by the DIERS equation. Comparison with the more detailed model shows that the analytical equation is quite conservative except at high reactor void fractions (LY= 0.9) where it converges to the correct result as given by the detailed model. These conclusions apply to calculations using the homogeneous two-phase relief assumption. It is also necessary to compare the results of the analytical equation and detailed model for the churnturbulent mode of venting as this can be more onerous than homogeneous venting. Here too, the analytical equation is conservative but the safety margin is much reduced, particularly at high reactor void fraction.
References 1 Leung, J. C. AIChE 1. 1986, 32(10) 2 Singh, J. in ‘International Symposium on Runaway Reactions’, CCPS/AIChE, New York, 1989 3 hung, J. C., Fauske, H. K. and Fisher H. G. Thermochim. Acta 1986, 104(13) 4 Fauske, H. K. and Leung, .I. C. PlantlOpns Pro& 1987. 6(2) 5 Tangren, R. F., Dodge C. H. and Seifert, H. S. J. Appl. Physics 1949, 20(7). 631-645
490
J. Loss
Prev.
Process Ind.,
1994, Volume
6 Leung,
J. C. AIChE
J. 1987, 33(6)
Nomenclature Area (m’) Cross-sectioned area of the reactor (m’) Constants in Antoine equation or in quadratic equation Empirical correlating parameter in calculation of superficial gas velocity Venting rate (kg s-‘) Acceleration due to gravity (m s-*) Mass flux through vent (kg mm2 s-l) = [2(1-X)/(1+X)] + 1 Mass of reactants (kg) Mass of experimental test sample (kg) Initial mass of reactants in large reactor (kg) Mass of gas (kg) Molecular weight Moles of gas Initial amount of gas (moles) Pressure (Pa) Pressure corresponding to onset of choked flow (Pa) Pressure in gas space of large vessel (Pa) Initial gas pressure in test apparatus (Pa) Maximum permissible reactor pressure (Pa) Pressure at which vent opens (Pa) = Pm - P, (Pa) Gas generation rate (mJ s-‘) Gas constant (J mol-’ K-l) Time (s) Time at which vent opens (s) Temperature (K) Experimental gas space temperature (K) Reactant (liquid) temperature (K) Single bubble rise velocity (m s-‘) Volume fm? Gas spa& in’experimental apparatus (m3) Volumetric gas flow through the vent (m’ s ‘) Gas space in reactor (m’) Liquid volume in reactor (m’) Total reactor volume (m’) Gas quality (mass fraction) entering vent Void fraction in reactor Rate of pressure rise per unit mass of reactant (Pa s-l kg-‘) Value of fi at maximum gas generation rate (Pa s-l kg-‘) Value of p at vent opening condition (Pa s-’ kg-‘) Ratio of specific heats for gas Pressure ratio Critical pressure ratio for choking conditions Specific volume of two-phase mixture (m’ kg-‘) Density (kg m-‘) Surface tension (N m-‘) = PJP.. = m.&, (Pa 6-l) Superficial gas velocity
Appendix
1
Pressure loss due to venting Consider a vessel containing a gas of volume V, and a liquid of volume V,. The gas is ideal, with molecular weight M,, and is insoluble in the liquid. The system is at temperature T and pressure P (which is entirely due to a non-condensable gas pressure). Initially, when the gas and liquid are enclosed in a vessel, the ideal gas law gives:
7, Number
PV, = nRT p=-
nRT V,
6
Vent sizing for gas-generating When both liquid and gas are removed pressure changes as follows:
(vented),
the ?&
(Al) The first term on the right-hand side relates to the effect of gas removal and the second relates to the effect of liquid removal (gas expansion). The gas removal term can be simplified as follows if the mass of gas is M,:
RT __=-_dn VP dt
RT 1 V, M,
=
reactions:
J. Singh
[2.016 +(g$7]-o.“4
unless P,,, < P,, where
p, = P,t, 77c in which case + is the atmospheric
where P,,, exit.
dM, dt
runaway
pressure at the nozzle
Appendix 3
RT pp dVs V, M,,, dt
Equations for prediction of onset of two-phase relief using the churn-turbulent model (Ref. 6)
but
The transition from all gas flow to two-phase flow in the churn-turbulent regime occurs if the following condition for the superficial gas velocity !P is met:
PM,
83 = RT hence
RT dn _ _=V, dt
p
dl/,
V,
dt
The pressure corresponding follows:
642)
reduction due to liquid removal (and gas expansion) may be simplified as
where CO is an empirical correlating parameter. For churn-turbulent flow, C, = 1.5. The dimensionless superficial vapour velocity is calculated from: q=&
nRT dVZ V,” dt =-- P dV,
V,
(A3)
dt
Substituting Equations (Al) gives:
(A2)
and (A3)
into Equation
X m
where Vr is the volumetric gas flow through the vent (m” s-l) and A, is the cross-sectional area of the reactor (m2), where the single bubble rise velocity U, is given by: u, = 1.53
afg
[
1
(Pr - 4) o.25 P?
where pf is the liquid density, pe is the gas density, uf is the surface tension, and g is the acceleration due to gravity.
but
..+!!L+!%)
Appendix 4
hence
Calculation of G for all gas or vapour JIow If
Appendix 2 Equations for G for non-flashing two-phare flow (Ref. 5) The mass flux through by:
a frictionless
then the mass flux through
an ideal nozzle is given
nozzle is given
_
(l-h)V]i’”
otherwise
in which 77 = v~, where
where y is the ratio of specific heats for gas.
J. Loss Prev. Process
Ind., 1994, Volume 7, Number
6
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