Ind. Eng. Chem. Res. 1997, 36, 3075-3080
3075
Checking Safety Relief Valve Design by Dynamic Simulation L. Pellegrini,* G. Biardi, M. L. Caldi, and M. R. Monteleone Department of Industrial Chemistry and Chemical Engineering “G. Natta”, Polytechnic, P.le Leonardo Da Vinci, 32, 20133 Milan, Italy
The dynamic behavior of a system constituted by a continuous stirred tank reactor (CSTR), where an esterification reaction takes place, coupled to a condenser which removes most of the heat of reaction, is studied. Dynamic simulation is used as a tool for checking the reliability of the safety valve on the condenser, which is designed according to the Fauske method. The sensitivity of the system behavior to different valve sizes is analyzed to point out, for a given set pressure of the valve, the proper size. It is proved that the best behavior of the relief system corresponds to the Fauske design. Introduction In recent years, owing to several accidents which have occurred and more and more restrictive laws, the reliability and safety of chemical plants have represented one of the main problems the chemical engineering community has had to face. In this context, in the last years, the temporal evolution of the most meaningful variables of the process has been studied in order to obtain more and more reliable information on the safe running of equipment/plants. The present paper concerns the simulation of the dynamic behavior of a continuous stirred tank reactor in which most of the heat generated by the esterification reaction between acetic anhydride and methanol, often found in literature concerning safety (Friedel and Wehmeier, 1990), is removed by a condenser supplied with a relief valve (Figure 1). Relief systems are installed to prevent excessive process pressure being developed. In reality, despite numerous safety precautions, operational errors or equipment failure, typically the failure of the cooling system, may cause a pressure rise that exceeds beyond the strength level of vessels and pipelines and results in the breaking of equipment and in the consequent release of toxic/flammable fluids. It is true that the best way of avoiding accidents is to prevent them by means of an optimized control strategy. Nevertheless, the installation of relief systems on pressure vessels, and of the relevant downstream equipment to handle the discharge material, is an industrial standard. To support this assertion, it is sufficient to remember the abundance of literature on the matter and, in particular, the computer codes developed for sizing emergency relief systems (the British Plastics Federation Computer Code (Booth et al., 1980) for runaway reactions such as phenol-formaldehyde, Huff’s Computer Simulation (Huff, 1984), the DIERS’ Computer Code SAFIRE (1984)). The final aim of our work is to use dynamic simulation as a tool for checking the reliability of the commonly used safety valve design procedures (Fauske, 1984). For this reason the start of the runaway reaction is simulated by the choice of operating conditions (i.e., too low heat removal) as to make the safety device open. * Author to whom correspondence should be addressed. Tel.: +39-2-23.99.32.37. Fax: +39-2-70.63.81.73. E-mail:
[email protected]. S0888-5885(96)00614-8 CCC: $14.00
Figure 1. System scheme.
The Model The following exothermic (∆H ˜ R ) -66.3 kJ/mol), irreversible, esterification reaction with no added catalyst has been taken into account:
(CH3CO)2O + CH3OH f B A CH3COOCH3 + CH3COOH D C Since the purpose of the work is to study the problems with this particular runaway reaction, the secondary reactions have been ignored. For this reason the reaction rate is a function only of the system temperature and of the concentration of reactants (Friedel et al., 1995)
r ) 3.711 × 107e-9323.2/TcAcB Methanol performs the 2-fold task of reactant and solvent, so its initial concentration has been assumed double that of anhydride. © 1997 American Chemical Society
3076 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
In Figure 1 a sketch of the considered system is reported. The Reactor. The esterification reaction takes place in a continuous stirred tank reactor (CSTR) that can also be refrigerated by means of cooling water flowing into a coil; a level controller prevents the reactor from emptying and exceeding a maximum level, over which the liquid reactant mixture might enter the condenser. The outflow from the reactor QOUT is regulated on the basis of the instantaneous value of the reaction volume according to the following control scheme:
QOUT ) SCCvOUTfC(VR) where
fC(VR) )
{
for VR > VRMAX
1 VR - VRMIN
VRMAX - VRMIN
for VRMIN < VR < VRMAX for VR < VRMIN
0
Since the vapor pressures of acetic anhydride and of acetic acid, in our exercise conditions, are negligible in comparison with those of the other species present in the reactor, their evaporation has been disregarded. The mathematical model adopted is based on the conservation principles of mass and energy, from which we can obtain the following balances:
d(cAVR) ) QINc0A - QOUTcA - k e-E/RT cAcBVR dt d(cBVR) ) QINc0B - QOUTcB dt ˜ BIN + W ˜ BOUT k e-E/RT cAcBVR - W d(cCVR) ) -QOUTcC + k e-E/RT cAcBVR dt ˜ COUT W ˜ CIN + W d(cDVR) ) -QOUTcD + k e-E/RT cAcBVR dt dU dt
∑i W ˜ IN∑yi i
) QIN
∑i H ˜ Vi (T) + W ˜ OUT∑xi i
c0i H ˜ Li (TIN) IN
- QOUT
ciH ˜ Li (T)
-
H ˜ Li (TDEW) -
OUT
UsSups(T - TE) The system temperature is derived by neglecting the difference between the internal energy and the enthalpy, an acceptable hypothesis for the liquid phase, and working out the above equation of energy balance. The hypothesis of vapor-liquid equilibrium has been made which, for a generic species, can be expressed in fugacity terms as fVi ) fLi . For the vapor phase the fugacity factors have been assumed to be 1, as we work at a high enough temperature and at a low enough pressure to consider the mixture and the single components as ideal. In regard to the liquid phase the presence of a minimum azeotrope between the methanol and the methyl acetate (19% in weight of CH3OH and Teb ) 54 °C for P ) 1 bar) has been assumed, while the interactions of the other two components between themselves and with the azeotrope are negligible.
The activity coefficient is computed with Wilson’s equation: NC
ln γi ) -ln
∑ j)1
NC
xjΛij + 1 -
xkΛki
∑ NC k)1 xjΛkj ∑ j)1
The Condenser. The condenser has been introduced with the aim of removing most of the heat generated from the reaction by solvent condensation and of recovering the reaction product which, owing to its volatility, leaves the reactor with methanol. The heat is removed by the water which flows into the tubes while condensation takes place on their exterior. The condenser has been schematized as a single phase, perfectly mixed on the refrigerant fluid side and on the process side, because the difference of temperature on both the extremities is wide enough (see Luyben (1966) for justifying such a hypothesis). To avoid the formation of an explosive mixture between the organic substances that vaporize from the reactor and incidental air pockets that could be present in the condenser after maintenance (Cardillo, 1995), the condenser has been filled up with nitrogen, so that the reaction environment becomes inert; the starting amount of inert gas is such as to exert a pressure of 1 × 105 Pa on the liquid below. Nitrogen is discharged into the atmosphere through the relief valve that opens above a maximum set pressure. For each species present in the gaseous phase the following material balances can be written
dnB nB )W ˜ INyBIN - W ˜ OUTxBOUT - W ˜ SF dt nTOT dnC nC )W ˜ INyCIN - W ˜ OUTxCOUT - W ˜ SF dt nTOT dnI nI ) -W ˜ SF dt nTOT The evaluation of the condensed capacity W ˜ OUT has been based on the film condensation theory (Bird et al., 1960), which has been extended from pure vapor to a mixture, giving the following results:
laminar flow (Res < 350) m′′ )
U0(TDEW - TJ)
∑i yi(Hˆ Vi - Hˆ Li )
turbulent flow (Res > 350)
[
5/12 5/4 -13/12 kf µf Pr5/12 × m′′ = 505/90.0325/3F5/6 f g TDEW - TJ
∑i
yi(H ˆ Vi
-
H ˆ Li )
]
5/4
HJ1/4
The vaporized capacity that leaves the reactor and goes up to the condenser has been calculated on the basis of vapor-liquid equilibrium and on the evaluation of the system pressure as the sum of three contributions
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3077
given by the presence of nitrogen, by the vapors that are given off by the reactor, and by the head losses relevant to the discharge through the valve:
P)
nIRT
1
+
∑i Pv γixi - 2fFGv2G
(1)
i
V
Since, for the ideal gas law
P ) nTOTRT/V
(2)
the temperature expression is obtained by differentiating (1) and (2) and substituting the latter equation for the former, which, coupled to the enthalpy balance on the reactor, gives the vaporized flow rate. The Safety Valve. Since the dynamics of the gas discharge through the relief valve is much quicker than the dynamics of condensation, it is possible to analyze the discharge of compressed gas from a reservoir (Crowl and Louvar, 1990). Writing the mass and energy balances on the reservoir
d (Fj V) ) -W ˆ SF dt G
(
)
P d (Fj VU ˆ ) ) -W ˆ SF U ˆ + dt G FjG
=0 ∫12dP FjG
(3)
the expression of the discharged flow rate is obtained
W ˆ SF ) CCSez
x
2P1FjG1
γ γ-1
[( ) ( ) ] P2 P1
2/γ
-
P2 P1
(1+γ/γ)
which has a maximum at the critical ratio:
() ( ) P2 P1
)
cr
2 1+γ
33 × 105 Pa and is valid, in general, for two-phase discharges; however Fauske asserts that for tempered reactions (at the specified relief set pressure) a homogenous discharge assures a safe design. The valve size, in this case, depends on the adiabatic temperature increase R and on the setting pressure of the valve through the parameter F. The Fauske graph, which can be used, with or without reaction, in every case of uncontrolled heating, gives the section Sez of the relief system according to the following equation:
Sez ) FR
and working out the relation (3) that is obtained by applying Bernoulli’s theorem throughout the discharge duct,
1 2 v + 2 2
Figure 2. System pressure vs time for a valve with Sez ) 70 × 10-4 m2.
γ/γ-1
In these conditions a sonic outflow and a sharp discontinuity from pressure P2 to atmospheric pressure occur. So if P2/P1 > (P2/P1)cr a subsonic outflow can be observed; otherwise, for P2/P1 e (P2/P1)cr a sonic outflow occurs. Relief Valve Design For the determination of safety relief valve sizes, empirical methods based on the capacity of the reactor and/or the self-heating rate can be found in literature; the most used are reported here below. The FIA method (Sestak, 1965) is applied to batch reactors working at pressures less than 5 × 105 Pa and whose capacity is between 0.3785 and 37.85 m3. The equation used by Monsanto (Howard, 1973) is based on the experimental data concerning phenolformaldehyde reaction, and it is applied if the set pressure is a little higher than 1 × 105 Pa and the selfheating rate is about 6 K/min. The Fauske method (Fauske, 1984) can be used for reactors with relief valves set at a pressure less than
(4)
Dynamic Simulation and Results Since the reaction involves acetic anhydride and methanol and it is conducted continuously, the Fauske method is the most suitable for designing the relief valve. For a set pressure of 4.5 × 105 Pa and a self-heating rate of 19 K/min with closed valve, the method gives a section of 95 × 10-4 m2. In order to verify the sensitivity of the system to the relief valve size and, consequently, to check method reliability, the system’s behavior has been simulated using valves with different sections. A Runge-Kutta fourth order integration scheme has been used since the system does not present stiffness problems; different step sizes have been considered for different temporal ranges during the integration interval in order to follow the velocity of change of the significative variables. In every test both the set pressure and that of reclosure that are fixed respectively at 4.5 × 105 and 3.5 × 105 Pa are kept constant, as well as all the other system parameters. Among the valves with a section less than 95 × 10-4 2 m , only the smallest, of the analyzed ones, behaves badly, remaining open even when the system reaches the steady state, as verified by a simulation test conducted until 10 000 s (see Figure 2). As shown in Figure 3, where the system pressure is reported vs time, all the other valves present the same behavior pattern as the one designed with the Fauske method, allowing the system to reach the steady state with a closed valve. The lower the section becomes, the more the closing time of the relief valve and the maximum pressure reached after the opening increase, while the decrease rate of pressure rises with the dimensions. In regard to the discharge flow, one has to distinguish between what is discharged into the atmosphere during the overall opening interval and what is instantaneously
3078 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
Figure 3. System pressure vs time for valves with Sez e 95 × 10-4 m2.
Figure 7. System pressure vs time for different ∆P, with PMAX ) 4.5 × 105 Pa.
Figure 4. Discharge flow rate for valves with Sez e 95 × 10-4 m2 .
Figure 8. System pressure vs time for PMAX g 4.5 × 105 Pa, with ∆P ) 1 × 105 Pa.
Figure 5. System pressure vs time for valves with Sez g 95 × 10-4 m2.
Figure 6. Discharge flow rate vs time for valves with Sez g 95 × 10-4 m2.
discharged: the former increases by reduction of the section, while the latter increases with the section (Figure 4). All the valves with a section greater than or equal to 95 × 10-4 m2 behave properly, reaching the steady state closed; the trend of pressure and of discharge flow is similar to that seen before (Figures 5 and 6). In particular it is important to notice that the instantaneously discharged flow increases considerably
with the valve dimensions and this happens especially in the first phase of the operation, stressing the structure. The results make it possible to say that the relief valve designed with the Fauske method for a given set pressure makes the system work best, since valves with a section less than 95 × 10-4 m2 discharge too much; while from the same point of view, using valves with greater section yields only negligible advantages for what concerns the overall discharged quantity. Moreover we controlled the system’s sensitivity toward the set pressure. For this reason two series of tests, fixing the section of the relief valve at 95 × 10-4 m2, have been carried out. At first the set pressure has been kept constant, changing the closure: the results, reported in Figure 7, let us say that if ∆P between the set pressure and reclosure pressure is equal to 1 × 105/1.2 × 105 Pa the system behaves properly, reaching the steady state with a closed valve (it is clear that closing time increases with the lowering of the least pressure). In the two extreme cases with ∆P less than or equal to 0.5 × 105 Pa or ∆P greater than or equal to 1.5 × 105 Pa the system shows an unacceptable behavior pattern. In the former the valve wavers between the opening and closing position; in the latter the pressure stabilizes itself at a value greater than PMIN, and therefore the system reaches the steady state with an open valve (in this case the relief valve acts like a breaking disk). Tests have been performed keeping ∆P constant but have varied the set pressure (Figure 8); it can be noticed that, with a set pressure greater than 4.5 × 105 Pa, the valve of 95 × 10-4 m2 is undersized and presents wavering behavior. In Figure 9 the case with PMAX ) 2.5 × 105 Pa is reported. Again a bad behavior can be observed: the valve opens sooner than in the preceding situations, because the set pressure is fixed at a lower value; as soon as the valve opens, a sudden pressure downfall
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3079
Notation
Figure 9. System pressure vs time for PMAX ) 2.5 × 105 Pa, with ∆P ) 1 × 105 Pa.
occurs, which makes the safety device close. Everything happens in a short time during which the system is still in a transition state with a big evaporated flow: this causes another increase in pressure that makes the valve open again, and the device also remains in this condition when the steady state is reached. To make really sure that the relief valve designed with the Fauske method makes it possible to work best, analysis has been again carried out for other operative conditions: fixing the set pressure at 4 × 105 Pa, for a self-heating rate of 17 K/min, a section of 90 × 10-4 m2 is obtained from relation (4). When the sensitivity toward the section size is checked again, the simulation gives the same type of results. In the end, it is worth noticing that, if for the system under consideration and for the analyzed working conditions a difference between set and reclosure pressure equal to about 1 × 105 Pa lets the system work best, this is not an absolute statement. Moreover the effective operative ∆P value largely depends on the particular valve type. Conclusions The dynamic behavior of a CSTR where the heat generated by a runaway reaction is removed in a condenser supplied with a relief device has been studied for safety purposes. The Fauske method for sizing safety valves, which is based on the system’s self-heating value, has been applied to the system, and its reliability has been checked by dynamic simulation. It is proven that the Fauske method makes it possible to design the relief valve so that it works in the best possible way, and in particular: (a) the valve does not open again; (b) the final pressure is sufficiently below PMAX; (c) on the whole the valve discharges a limited quantity of material; (d) the flow discharged instantaneously is low enough, so that the structure is not stressed too much.
CC ) contraction coefficient c ) concentration [kmol m-3] E ) activation energy [kJ kmol-1] f ) friction factor, fugacity [Pa] F ) H. K. Fauske parameter fC ) level control parameter g ) gravity acceleration [m s-2] H ) height [m] H ˆ ) mass enthalpy [kJ kg-1] H ˜ ) molar enthalpy [kJ kmol-1] k ) reaction kinetic constant [m3 kmol-1 s-1], thermal conductivity [kW m-1 K-1] m′′ ) specific condensed flow rate [kg m-2 s-1] n ) moles in gaseous phase [kmol] P ) pressure [Pa] Pr ) Prandtl number Pv ) vapor pressure [Pa] Q ) flow rate [m3 s-1] r ) reaction rate [kmol m-3 s-1] R ) gas constant [kJ kmol-1 K-1], self-heating rate [K min-1] Re ) Reynolds number S ) reactor outlet transversal section [m2] Sez ) relief valve area [m2] Sup ) heat transfer area [m2] t ) time [s] T ) temperature [K] U ) system internal energy [kJ], overall heat transfer coefficient [kW m-2 K-1] U ˆ ) mass internal energy [kJ kg-1] v ) velocity [m s-1] VR ) reactor holdup [m3] W ˜ ) molar flow rate [kmol s-1] W ˆ ) mass flow rate [kg s-1] xi ) molar fraction in the reactor liquid phase xiOUT ) molar fraction in the condenser liquid phase yi ) molar fraction in the condenser gaseous phase yiIN ) molar fraction in the reactor gaseous phase ∆H ˜ ev ) molar latent heat of vaporisation [kJ kmol-1] ∆H ˜ R ) heat of reaction [kJ kmol-1] ∆P ) difference between the set pressure and the reclosure pressure [Pa] γ ) heat capacity ratio µ ) dynamic viscosity [kg m-1 s-1] F ) density [kg m-3] Fj ) average density [kg m-3] Superscripts L ) liquid V ) vapor 0 ) initial Subscripts E ) cooling water in the coil f ) film G ) gas i ) i component I ) inert gas IN ) inlet J ) condenser MAX ) maximum MIN ) minimum OUT ) outlet R ) reactor s ) coil SF ) discharged through the relief valve TOT ) total
3080 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 1 ) inlet section of the discharge duct 2 ) outlet section of the discharge duct
Howard, W. B. Reactor Relief Systems for Phenolic Resins. Monsanto Company, 1973.
Literature Cited
Huff, J. E. Computer Programming for Runaway Reaction Venting. In I. Chem. E. Symposium Series No. 85, Institution of Chemical Engineers: Chester, England, 1984.
Bird, R. B.; Stewart, W. E.; Lightfoot, E. L. Transport Phenomena; John Wiley: New York, 1960. Booth, A. D. Design of Emergency Venting System for Phenolic Resin Reactors. Parts I and II. Trans. J. Chem. Eng. 1980, 58, 75. Cardillo, P. Strumenti e metodi per valutare l’infiammabilita` e la reattivita`. Corso di aggiornamento su analisi dei rischi nell’industria di processo, IV Edizione, Politecnico di Milano, 25 gennaio - 24 marzo 1995. Crowl, D. A.; Louvar, J. F. Chemical Process Safety: Fundamentals with Applications; Prentice Hall: Englewood Cliffs, NJ, 1990. Fauske, H. K. Generalized Vent Sizing Nomogram for Runaway Chemical Reactions. Plant/Oper. Prog. 1984, 3, 213. Friedel, L.; Wehmeier, G. Modelling of the vented methanol-acetic anhydride runaway reaction with the SAFIRE program. Proceedings of the Eurotherm Seminar, No. 14, Louvain La Neuve, 1990. Friedel, L.; Kranz, N. J.; Wehmeier, G. Theoretical and Experimental Analysis of Reactor Top Venting of Two-Phase Reaction Systems with Low Liquid Viscosity. Chem. Eng. Process. 1995, 34, 71.
Luyben, W. L. Stability of Autorefrigerated Chemical Reactors. AIChE J. 1966, 12, 662. SAFIRE User’s manual, 1984, Vol. 1-7, Fauske & Associates, Inc. Report FAI/84-5 (DIERS Property Information). Sestak, E. J. Eng. Bulletin N-53, Fact. Ins. Assn., Hartford, CT; prepared by H. Doyle and R. F. Schwab, 1965.
Received for review October 3, 1996 Revised manuscript received February 12, 1997 Accepted February 13, 1997X IE9606146
X Abstract published in Advance ACS Abstracts, June 15, 1997.
