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NUREG/CR-3719SAND83-2624R3Printed August 1984
Detonation Calculations Using aModified Version of CSQII:Examples for Hydrogen-AirMixtures
R. K. Byers
Prepared bySandia National LaboratoriesAlbuquerque, New Mexico 87185 and Livermore, California 94550for the United States DepartmetfEnergy
under Contract DE-AC04-7600807,89
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SF 2900-Q(6-82)
NOTICEThis report was prepared as an account of work sponsored by anagency of the United States Government. Neither the UnitedStates Government nor any agency thereof, or any of their em-ployees, makes any warranty, expressed or implied, or assumesany legal liability or responsibility for any third party's use, or theresults of such use, of any information, apparatus product orprocess disclosed in this report, or represents that its use by suchthird party would not infringe privately owned rights.
Available fromGPO Sales ProgramDivision of Technical Information and Document ControlU.S. Nuclear Regulatory CommissionWashington, D.C. 20555andNational Technical Information ServiceSpringfield, Virginia 22161
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NUREG/CR-3719SAND83-2624
R3
Detonation Calculations Using a Modified Version of CSQII:
Examples for Hydrogen-Air Mixtures
R. K. Byers
Date Published: August 1984
Sandia National LaboratoriesAlbuquerque, NM 87185
Operated bySandia Corporation
for theU. S. Department of Energy
Prepared forSevere Accident Assessment BranchDivision of Accident Evaluation
Office of Nuclear Regulatory ResearchU. S. Nuclear Regulatory Commission
Washington, DC 20555Under Memorandum of Understanding DOE 40-550-75
NRC FIN No. A-1246
k
ABSTRACT
CSQ is a well-tested and versatile wave propagation computerprogram, a modified version of which has been used to perform anumber of USNRC-supported analyses of detonations of hydrogen-air mixtures in nuclear reactor containment buildings. Themodifications, from a user's viewpoint, are fairly minor, andthis version of CSQ is being prepared for release to interestedorganizations. This report documents the use of CSQ in this form.as well as certain codes which aid in performing the detonationcalculations.
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CONTENTS
Page
I. Introduction 1
I1. Brief Description of CSQ and Related Codes ............ 3
III. Examples and Applications .............................. 5
A. The "Constant Rate - Constant Threshold"Model . ............................................ 6
B. Ignition-Reaction Examples ........................ 9
C. A Sample Containment Calculation ................ 14
IV. Discussion .............................................. 33
References ............................. ................ ...... 35
Appendix ...................................................... 37
V
LIST OF ILLUSTRATIONS
Figure Page
III.i Ratio of Conversion Rate Constant to Sound Speedat C-J State for Relation Given by III.1 .............. 8
II1.2 Momentum (above) and Pressure (below) Profiles,Normalized to C-J Conditions; Conversion RateConstant Given by 111.1 . ...................... 10
I11.3 Momentum (above) and Pressure (below) Profiles,Normalized to C7J Conditions; Conversion RateConstant Twice That Given by II.1 ................... 11
111.4 Momentum (above) and Pressure (below) Profiles,Normalized to q-J Conditions; Conversion RateConstant ýialf That Given by III.1.................. 12
111.5 Normalized Pressure Profiles for a SampleReaction Model,. ......... ................ ......... 13
111.6 Momentum (above) and Pressure (below) Profiles.Normalized to C-J Conditions; Reaction RateConstants Approximating Results of III.1.......... 15
111.7 Momentum (above) and Pressure (below) Profiles,Normalized to C-J Conditions; High Reaction RateConstant..,................. ...... .................. 16
111.8 Momentum (above) and Pressure (below) Profiles,Normalized to C-J Conditions; Low Reaction RateConstant ............................................ 17
111.9 Computational Geometry for Two-DimensionalExample Problem ................................... 18'
111.10 Density (above) and Pressure (below) Profilesin Obstructed Strip, Normalized to C-J Conditions(Two-Dimensional Reaction Model Example) .......... 19
I114l Density (above) and Pressure (below) Profilesin Open Strip. Normalized to C-J Conditions(Two-Dimensional Reaction Model Example).......... 20
iII.12 Boundary Shape for Containment Example.,............ 21
111.13 Density Field at 2 ms in Containment Example ....... 23
111.14 Density Field at 12 ms in Containment Example..... 24
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.aie
111.15 Density Field at 16 ms in Containment Example ..... 25
II.16 Density Field at 28 ms in Containment Example ..... 26
111.17 Density Field at 32 ms in Containment Example ..... 27
111.18 Pressure History at Dome Center in ContainmentExample ........................................... 28
111.19 Pressure History at Center of Drywell Head inContainment Example ............................... 29
111.20 Pressure History at Dome - Wall Joint inContainment Example ............................... 30
111.21 Pressure History about 1.1 m above Bottom ofWetwell in Containment Example .................... 32
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a
ACKNOWLEDGMENT
The author greatly appreciates Jan Frey's efforts in pro-ducing this document.
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I. Introduction
Sandia National Laboratories is engaged in an extensiveprogram, sponsored by the USNRC, involving many safety-relatedaspects of the behavior of hydrogen mixtures in reactor contain-ment buildings [1]. A part of the program is the estimation ofdetonation-caused loads on containment structures. CSQ [2], awell-established computer program which solves continuummechanics problems for two-dimensional motion, has been used ina number of such analyses [3,4,5]. An altered version of theprogram was used in order to make use of constitutive relationsfor hydrogen-air-steam mixtures which were developed as part ofthe overall program [6], and it has been suggested that thisversion of the program be released for use outside Sandia. Thisreport is a brief description of the changes and the way theresulting code may be used.
CSQ solves finite difference analogs for the partialdifferential equations representing the balance of mass,momentum, and energy, together with constitutive relations forthe materials involved. The code incorporates an accuratetreatment of mechanical and thermal equilibrium of mixtures ofas many as ten different materials. One user option, originallyused for (solid) high explosives, is the detonation of regionsof materials. However, this option requires the use of a singleequation of state for both the undetonated and the detonatedmaterials. Furthermore, a time and location for the start ofdetonation are required input, and these limitations were deemedinappropriate for the intended use. For this reason, an alter-nate method of simulating detonations was incorporated in thecode. The method forces a detonation to occur (by convertingunburned to burned material) whenever some quantity (e. g.,pressure) exceeds a threshold value. This very simple model may,however, be easily altered as adequate information becomesavailable, allowing more accurate treatment of detonation initi-ation, as well as quenching or transition from deflagration todetonation.
Section II of this report presents a brief description ofCSQ and the detonation-model modifications, together withdescriptions of ancillary codes which aid in performing CSQdetonation analyses. Section III describes the effects ofvarious changes in the detonation model, as well as some resultsfrom a typical calculation of detonation in a containment. TheAppendix contains a description of how the required equation-of-state information is generated and used to calculate theoreticaldetonation conditions.
1/2
II. Brief Description of CSQ and Related Codes
CSQ solves discrete analogs of the differential equationsrepresenting conservation of mass, momentum, and energy intwo-dimensional motion, in either cylindrical or rectangularCartesian coordinates. Constitutive relations ("equations ofstate") and initial and boundary conditions are combined withthe three conservation laws to complete the system of equations.A rectangular grid is used to discretize the spatial region ofinterest; various geometric options are used to define whichregions of the grid are occupied by a given material. The avail-able boundary conditions include impermeable boundaries, bound-aries which absorb incident stress waves without producing areflection and for which the user may choose whether or not toallow material to enter or leave the mesh, and applied pressureand velocity boundaries [7]. Sets of connected lines internal tothe mesh may also be defined as impermeable.
The numerical method embodies a "pseudo-viscosity" to smoothdiscontinuities, so that the differential equations have solu-tions which match shock wave solutions near a steady waveseparating regions of constant properties [8]. After advancingthe solution by a timestep using a Lagrangian formulation of theequations, the code performs a rezone to the original spatialmesh, so the resulting treatment is essentially Eulerian. Thecurrent version of the program can treat as many as ten differ-ent materials in a problem, with any mixtures assumed to be inmechanical and thermal equilibrium. CSQ incorporates accuratethermodynamics, has been used on a wide class of problems, andhas produced results which compare well with experimental data.
The standard treatment of detonations in CSQ consists ofreleasing the appropriate amount of internal energy in a thinregion (several computational cells) which moves through themesh with the detonation velocity. The undetonated material andthe detonation products must be described by the same constitu-tive relation, and the location and time of initiation must bespecified. In the Sandia hydrogen program, equations of statewere developed for hydrogen--air-steam mixtures and the combus-tion products of those mixtures, and we wished to make use ofthat information. Furthermore, we anticipated a need, as accu-rate information became available, to analyze phenomena such asignition caused by details of a flow field, deflagration-to-detonation transitions, and quenching of a detonation front. Amodified version of CSQ already existed [9], in a form thatcould be easily adapted to such analyses.
CSQ treats mixed regions by maintaining volume fractions foreach material in the cells of the spatial mesh. The modificationconsists of controlling the volume fractions in a way notentirely determined by the motion and the rezoning process. Theconversion of one material to another (or others) may thus be
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specified in terms of any of the quantities calculated by thecode, so the requirement that a detonating material be describedby a single equation of state is removed. In addition, compli-cated chemical processes can, in principle, be modelled in thisway. Finally, zoning complications arise with the standardmethod when propagating detonations through complex geometries,and these complications disappear when it is not necessary tospecify detonation locations and times.
For use in the analyses of interest here, the "tabular"equation of state (EoS) option is employed. With this option,CSQ requires a file containing values of the appropriatethermodynamic variables at points of a temperature-density"mesh". We adapted a computer code already developed in thehydrogen program [6] to create such a file. In providing thisinformation, the principal computational difficulty is indetermining the composition of a given mixture followingcombustion. For each temperature and density, the currentestimate of the average molecular weight is used to calculate apressure, followed by a Newton-Raphson iteration to find themole fractions for the given conditions, assuming mechanical,thermal and chemical equilibrium of the product species. Thisprocedure is repeaeed for each temperature-density pair untilthe computed pressure on successive interactions has not changedby ten parts in one million. The EoS surface data for bothunburned and burned mixtures are then used to create therequired file, or added tp an existing one.
CKEOS2 [10] is a computer program which performs variouscalculations (e. g., isotherms for a range of densities) usingthe same EoS description as is used in CSQ. In order to permitcomparison of CSQ results with other analyses, CKEOS2 has beenmodified to allow the calculation of Chapman-Jouguet (C-J)detonation states (states immediately behind a steady detona-tion wave), without requiring the original and final states toshare an EoS surface. The steadiness condition requires that thesum of the material velocity and sound speeds be equal to thepropagation velocity of the wave, and this requirement is usedin the calculation. The modification also produces thermo-dynamic conditions for an isentrope from the C-J state to theoriginal density.
4
III. Examples and Applications
General Modelling Approach
In principle, modelling combustion phenomena with CSQ shouldbe a fairly simple matter. Having the EoS information for theappropriate mixtures, one must specify some means of describingthe reactions of interest. Reference 1 tabulates 43 reactionsinvolved in the combustion of hydrogen in oxygen or air, andoffers the opinion that fewer than 20 of them are likely to beimportant in determining the overall reaction rate. An attemptto include this kind of detail in a CSQ calculation would bevery expensive in terms of both computer memory and computingtime, and simpler models are clearly desirable. The approachused in this study involves only two materials: one representsthe unreacted mixture, and the 'other, the products of completecombustion of that mixture. Incompletely reacted material is anappropriate mixture of the two components, and a single equationprescribes the reaction rate. The modelling of the effects ofthe many reactions thus depends on the details of the effectivereaction rate equation. The major modification to CSQ uses suchan equation to prescribe the amount of reacted and/or unreactedmixture contained in each computational cell.
For reasons connected with the original development of CSQ.the first material specified in a problem description is treatedseparately from the remaining materials, which are combined toform another material before performing the final cell-averagingprocess. The way the code does this calculation makes it mostconvenient to require that the undetonated material and itscombustion products be components of the second material; thisis the only input restriction on this version of the code.
An effective reaction rate equation for the mass fraction(x) of the unreacted mixture may take the form
dx/dt = F(x,T,v),
with T and v being the temperature and specific volume, respec-tively. In the context of interest here, F is nonpositive forall values of its arguments, and its magnitude increases withtemperature. In addition, F vanishes unless some ignitioncriterion is met (and, of course, if x vanishes). In a particu-lar problem, the form of F, the initial and boundary conditions,and the course of the calculation will determine which ofseveral combustion phenomena are calculated. For example, if thereaction (once initiated) near flame temperatures releasesenergy slowly enough that thermal conduction into the unburnedgas can balance it, a stable deflagration can be produced.Cooling or rapid expansion behind a combustion front may cause apropagating reaction zone to disappear - i.e., the combustionquenches. These two processes are not addressed in this report.
5
However, a strong ignition source or flow disturbances couldcause the reaction to proceed rapidly enough that compressionand consequent additional heating occurs behind the front. Thislast situation may escalate until a steady shock wave is formedwith a constant-thickness reaction zone immediately behind it,which is the basis for the Chapman-Jouguet analysis of a detona-tion.
One other feature of the calculational method differs fromthe ordinary formulation, although it is not an explicit modifi-cation to CSQ. Because the standard approach does not treatchemical reactions, there is no requirement that materialenergies and entropies be related on an absolute scale. Here,however, such a requirement is imposed in constructing the EoStables for corresponding unreacted and reacted mixtures, inorder to account for the energy available from the reaction. Inthe absence of motion and energy transport, the total energy ina cell must remain constant; at the same temperature and pres-sure, the EoS treatment assigns a lower internal energy to thecombustion products, so the result must be an increase intemperature, and hence pressure where the conversion process istaking place. Thus, no explicit energy source ýis required withthis method.
This section describes two reaction - ignition models whichhave been used in CSQ detonation calculations, and presentsresults from various sample problems. The first model is verysimple, and forces complete conversion to be calculated at aconstant rate. If reasonably good approximations of C-J detona-tions are to be obtained, the interaction of the model withCSQ's numerical methods must be considered, and this is alsodiscussed. A slightly more "realistic" model is; also described,which does not force complete combustion of the unburnedmaterial; a model of this form is, conceptually, capable ofproducing all the phenomena outlined in the previous paragraph.The final example is an illustration of the use of CSQ inmodelling detonations in reactor containment buildings.
A. The "Constant Rate - Constant Threshold" Model
As stated previously, the simple model we have used to treatdetonations consists of forcing the conversion of one materialinto another in every cell for which a specified parameterexceeds a threshold value. Convenient variables for the"triggering" quantity are the pressure or temperature; reason-able results are obtained when the threshold value represents ajump of -1-2% of the corresponding jump to the C-J state. Thedetonation may then be initiated by specifying the ignitionregion to satisfy the criterion at the beginning of the problem.As burning proceeds from this region, the original small dis-continuity grows, eventually reaching a state approximating thetheoretical steady detonation conditions.
6
The constant rate at which the conversion takes place, aswell as the threshold value, is inserted in the subroutineTFORM. The mass fraction of unburned material is decreased bythe product of a "burn parameter" (called VDMP in the code) andthe ratio of the timestep to the cell size. The current codingthen places an upper limit of 0.15 on mass fraction change. Thistreatment of combustion is obviously quite unphysical, and theway in which the model parameters interact with CSQ's numericaltechniques and a problem's mesh size has an effect on theresults obtained.
Three simple example problems illustrate the interactionmentioned in the previous paragraph. The problems all consist ofa one-dimensional detonation wave propagating into a dryhydrogen-air mixture, and initiated at the edge of the mesh froma single row of heated cells. The spatial grid is uniform, 10 mmsquares, 3 cells wide and 200 cells in the propagation direc-tion. Each calculation was carried out for approximately 1 ms.The difference between the calculations lies in the value of theratio of the burn parameter (or conversion rate constant) to thesound speed at the detonated state.
CSQ uses a modified Courant timestep control algorithm,based upon local velocities and velocity gradients, mesh size,sound speed, and the artificial viscosity co-efficients. Thetimestep chosen for advancement is the minimum of all thosecalculated on the mesh. Experience has shown that, in order tocalculate a steady wave which is a reasonable approximation of aC-J detonation and its associated release wave, several criteriashould be satisfied. One wants -4 cells in the wavefront, thetimestep to be controlled within one cell of the peak, and themass fraction increment there to be about its maximum allowedvalue. Of course, the calculated peak velocity should also beabout equal to the C-J velocity.
When these criteria are combined with the expression for thetime increment, the ratio (r) of the burn parameter to the soundspeed at the C-J state depends on the Mach number (m) at the C-Jstate, and is given by
r = Afm {B1 + (l+.25Bq)m + [ 1 + (B1 + .25mBq) 2 ]1/2}, (111.1)
where B1 and Bq are the linear and quadratic viscositycoefficients, respectively, and Afm is the maximum allowedmass fraction increment. The conversion rate ratio is normallyabout 0.35, and is shown in Figure II1.1 as a function ofinitial hydrogen mole fraction for B1 = .1, Bq = 2, Afm= .15 (the default values), and two initial temperatures.
7
CONVERSION RATE CONSTANT/SOUND SPEED3.
3.62',
3.601
0
w
z0
wz0U.
3.
3.
3.
3.
3.
3.
3.
3.1.50 2.25 3.00 3.75 4.50 5.25 6.00
x -i0FINITIAL MOLE FRACTION
Figure IIl.1 Ratio of Conversion Rate Constant to Sound Speedat C-J State for Relation Given by III.1
8
The results of the sample problems described above illus-trate the effects of adhering to, and of ignoring, the relationset forth in the previous paragraph. When VDMP is calculated bythe equation given there, C-J conditions are well approximatedafter a propagation distance of 30 - 40 zones, and a fairly goodrelease wave is obtained, as seen in Figure 111.2. Figure 111.3displays the results of doubling the burn parameter; the propa-gation velocity increases and constant peak values are reachedin fewer meshes, but peak values are low compared to C-J values.Decreasing VDMP by a factor of 2 has the opposite effects, asshown in Figure 111.4. The reader should note that although thepeak values of pressure, etc., are altered by changing the burnparameter, the total momentum (or impulse) contained in the waveis not affected.
B. Ignition-Reaction Examples
It was implied previously that CSQ is capable of a reason-ably good treatment of reacting flow, given an accurate model.In the following examples, this capability is demonstrated bythe use of a model which delays ignition until a "history"parameter exceeds a given value, and specifies the reaction rateas a function of the available unburned material and thetemperature. The expressions used are
dH/dt = Max {0,(p-p*)(T - T*)}
and
dx/dt = - klx exp(-k 2 /(T -T
where p and T are the pressure and temperature, respectively,and x is the mass fraction of the unburned mixture. The ignitionvalue of the history parameter H, and the values of the con-stants p* T*, kI, and k2, are arbitrarily chosen toachieve various results; no claim is made that this model or theconstants used in it are particularly realistic. Note, however,that simple global models for reaction kinetics are frequentlycast in a similar form.
Apart from the use of the model described in the previousparagraph, the first example is identical to those described forthe constant threshold - constant rate model, and merely servesto demonstrate that similar results can be obtained with thismethod. As may be seen by comparing Figures 111.5 and 111.2, thereaction model yields a somewhat smoother release wave behindthe detonation front. The peak pressure is very close to the C-Jpressure because the rate constants were adjusted to matchresults of the "good" calculation in the previous section. (Of
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MOM/MOM(C-J. AT 0.1 MS INTERVALS2.400
2.200
2.000
1.900
1 .600
1.400
1.200
a 1.000X.
Z .8030
E .6000
.4000
.2000
0.000
-. 2000 -
S 0. 00
4q
.400 .000 1.20 1.60
DISTANCE (M)
2.00
P/P(C-J; AT 0.1 MS INTERVALSI .500
1.400-
1.300
1 .200
1. 100
1 .000
.9000
o .9000
a: .7000
S.6000
.5000
.4000
.3000
.2000
2 000
0.0000-
a I 0.00
Figure 111.2
.400 .900 1.20 1.60
DISTANCE (Mý
Momentum (above) and PressureNormalized to C-J Conditions;Constant Given by 11I.1
2.00
(below) Profiles,Conversion Rate
10
MOM/MOM(C-J) AT 0.1 MS INTERVALS2.400
2.200
2.000
1.800
1.600
1 .400
1.200
1.000
.8000
.6000
C
4,
K
zw:
.4000
.2000
0.000 H-. 2000
M) 0.00 .400 .900 1.20 1.60
DISTANCE (M)
2.00
P/P(C-J) AT 0.1 MS INTERVALS1.500 -
.400
1.300
1 .200
I. 100
1.000
.9000
o .8000
.7000
m .6000
.5000.4000 4
.3000
.2000
.1000
0.000
a
0.00.400 .800 1.20 1.60
DISTANCE (IM
2.00
Figure I11.3 Momentum (above) and Pressure (below) Profiles.Normalized to C-J Conditions; Conversion RateConstant Twice That Given by III.1
11
MOM/MOM(C-J) AT 0.1 MS INTERVALS2.400
2.200
2.000
1.800
1.600
1.400
1.200
1.000
.8000
.6000
0
4
Z:
z
w
.4000
.2000
0.000 IL
-. 2000 t
M) 0.00 .400 .800 1.20 1.60 2.00
DISTANCE (M)
P/P(C-J) AT 0.1 MS INTERVALS
0
4
wn
Ij.
1.500
1.400
1.300
1.20Q
1.100
1.000
.9000
.8000
.7000
.6000
.5000
.4000
.3000
.2000
.1000
0.000 L-
a 0.00
.400 .800 1.20 1.60 2.00
DISTANCE (M)
Figure 111.4 Momentum (above) and Pressure (below) Profiles,Normalized to C-J Conditions; Conversion RateConstant Half That Given by III.1
12
P/P(C-J) AT 0.1 MS INTERVALS1.300
1.200
1. 100
1.000
.9000
.8000
o .7000I-
S.6000w
S.5000
w0. .4000 -
.3000
.2000
.1000
0.00 .400 .800 1.20 1.60 2.00
DISTANCE (M)
ONED,.22-DRY AIRREACTION MODEL
C80 Is 1 X" 0.
Figure 111.5 Normalized Pressure Profiles for a SampleReaction Model
13
course, if the form of the reaction-rate equation, and thevalues of the parameters in it, were physically accurate, nosuch adjustment would be justified.) In this case, ignition ofthe mixture begins at about .01 ms, and the problem ends at-9 Ms.
Ignition may, obviously, be delayed by adjusting the"critical" value of the history parameter upward, and this isthe case for the next three sample problems. Instead of a row ofheated cells, a constant velocity is defined at the edge of themesh: the state of the material entering the problem is the sameas that of the first "real" cell. A shock of approximately one-tenth the strength of a C-J detonation propagates for about 0.6ms before the detonation begins (Figure 111.6). The detonationwave overtakes the initial shock at about 0.75 ms, and subse-quently reaches fairly steady peak values. Doubling and halvingthe rate parameter k, produce the same relative effects asthey do for the "VDMP" model, as may be seen in Figures 111.7and 111.8.
CSQ may also be used to model ignition caused by localvariations in the flow field. A problem similar to the medium-rate case discussed above illustrates this. The calculation ismade two-dimensional by specifying a centrally located U-shapedobstacle at .25 m from the velocity boundary, with the open endfacing that boundary (Figure 111.9). When the hitherto one-dimensional shock reaches the obstacle at about 0.5 ms, thehistory parameter begins accumulating more rapidly near theobstacle than elsewhere, and ignition begins at approximately0.64 ms. As is seen by comparing Figures III.10 and III.11, theresulting flow is still strongly two-dimensional at 0.7 ms; therapid buildup of the detonation produces virtually one-dimensional wavefronts by 0.1 ms later. (Note that the detona-tion state values used to normalize the profiles are those for awave proceeding into the mixture at its initial state. Theleftward-moving wave exceeds C-J values because it is interact-ing with the initial shock produced by the velocity boundarycondition.) The same type of behavior could, of course, beproduced with the constant rate - constant threshold model.
C. A Sample Containment Calculation
In order to illustrate the way in which CSQ can be used tomodel detonations in containments, some results are presentedfor the General Electric Standard Safety Analysis Report (GESSAR)containment design. The CSQ model for this BWR containment isaxisymmetric, and consists of a large upper compartment with anellipsoidal dome and an approximation to the wetwell region(Figure 111.12). Computational cell size corresponds to 50 cellson the upper compartment radius, and provides a reasonably
14
"OM/MOMIC-J) AT 0.1 "S INTERVALS2.400
2.200
2.000
1 .000
1.600
1.400
o 1.200
i 1.000
z .8000
0K .6000
.4000
.2000
0.000
-. 20000.00 .200 .400 .600 .800
DISTANCE (M)
1.00
P/PIC-J) AT 0.2 MS INTERVALS1.500
1.400
1.300
1 .200
1. 100
1 .000
.9000
o .9000
7 .7000
= .6000
• .5000
.4000
.3000
.2000
.1000 --
0.000 --
Ef 0.00
Figure 111.6 Mo
.200 .400 .600 .900 1.00
DISTANCE (M)
mentum (above) and Pressure (below) Profiles,rmalized to C-J Conditions; Reaction Ratenstants Approximating Results of II.1
NoCo
15
MOM/MOM(C-J) AT 0.1 MS INTERVALS2.400
2.200
2.000
1.800
1.600
1.400
1 .200
1 .000
.8000
.6000
0
4
zwK0K
.4000
.2000
0.000
-. 2000a 0 0.00 .200 .400 .600 .800
DISTANCE (M)
1.00
P/P(C-J AT 0.1 MS INTERVALS1.500
I.400
1 .300
1.200
1.100
.000
.9000
.8000
o .7000
• .6000
a .5000
.4000
.3000
.2000
. 1000
0.000gm 0.00
Figure 111.7 MO!
.200 .400 .600 .800 1.00
DISTANCE (M)
nentum (above) and Pressure (below) Profiles.rmalized to C-J Conditions; High Reaction Ratenstant
No,Cot
16
MOH/MOM(C-J) AT 0.1 MS INTERVALS2.400
2.200
2.000
1 .800
1.600
1 .400
1 .200
1.000
.8000
.6000
0
2
.4000
.2000 -
0.000
-. 2000
f 0.00 .200 .400 .600 .800
DISTANCE (H)
I. 00
P/P(C-J) AT 0.1 MS INTERVALS1.500
1.400
1.300
1 .200
1.100
1.000
.9000
o .9000
= .7000
.6000
.5000
.4000
.3000
.2000
1I000
0.000-
gm0.00
oWoo .200 .400 .600 .900 1.00
DISTANCE (M)
Figure III.8 Momentum (above) and Pressure (below) Profiles,Normalized to C-J Conditions; Low Reaction RateConstant
17
inc identw ave u-shaped
obstacle
Figure 111.9 Computation Geometry for Two-Dimensional ExampleProblem
18
RHO/RHO(C-Jl AT 0.1 MS INTERVALS1.500
1 .400
I .300
1.200
I. 100
1.000
.9000
.8000
• .7000
- .6000
z .5000
4000
3000
2000
1000
0.000
O 0.00 1.00 2.00 3.00 4.00
DISTANCE (")
5.00-I
Xl10
P/P(C-J) AT 0.1 MS INTERVALSi.300
1 .200
1.100
1.000
.9000
.000
.7000
.6000
.5000
.4000
0
w•
U,
.3
.2
.1
Figure 111.10
000 -
000
000
000-0.00 1.00 2.00 3.00 4.00 5.00
-1DISTANCE (M)
X10
Density (above) and Pressure (below) Profilesin Obstructed Strip, Normalized to C-J Conditions(Two-Dimensional Reaction Model Example)
19
RHO/RHO(C-J) AT 0.1 MS INTERVALS1.500 -
1.400
1.300
1.200
I . 100
1.000
.9000
.8000
.7000
.6000
W .5000
.4000
.3000
.2000
.1000
0.000an 0.00 1.00 2.00 3.00 4.00
DISTANCE (H)
5.00
X10
P/P(C-J) AT 0.1 MS INTERVALS1.300
I. 200
1. 100
I .000
9000
.8000
o .7000
" .6000
A .5000
C. .4000
.3000
.2000
. 1000
0.000 -
• 0.00
Figure III.11
1.00 2.00 3.00 4.00 5.00
-1DISTANCE (M) X10
Density (above) and Pressure (below) Profilesin Open Strip, Normalized to C-J Conditions(Two-Dimensional Reaction Model Example)
20
ITIME = 0. CYCLE = 0 TEMP
15 TIM 0. IYCE! IEMP
z0
I-ý-'4X
2
> -11
I-
4M1 -24--
-37
-50 1i3'
I I I
0 -18 -6 6' 18RELATIVE HORIZONTAL POSITION (m)
Figure 111.12 Boundary Shape for Containment Example
21
accurate description of the curvature of the dome. The wetwellregion boundaries are made irregular in an attempt to approxi-mate flow restrictions and equipment. Detonation (using theconstant rate - constant threshold model) proceeds from a heatedcell at the center of the boundary representing the drywell head.
The boundaries of the model are treated as rigid (imperme-able). CSQ is capable of modelling the response of the materialswhich make up the containment structure, but the timestep limi-tations would be so severe as to make such a calculation prohib-itively expensive. For this reason, pressure histories aremaintained at points along the boundary, and may be used toestimate the effects of the loads on the structure.
The entire model is filled with a mixture of hydrogen anddry, sea-level air, with an initial hydrogen mole fraction of0.22 and an initial temperature of 315 K; the resulting pressureand density for the mixture are 0.142 MPa and 1.250 kg/m 3 ,respectively. An isochoric (constant-volume) burn of this mix-ture reaches a temperature of 2364 K and a pressure of 0.885MPa. C-J detonation values are 1.842 MPa, 2589 K, and 2.204kg/m 3 , while an isentrope from the C-J state to the initialdensity ends at 2295 K and 0.923 MPa.
As may be seen in Figures 111.13 and 111.14, the squarezones in the CSQ mesh keep the detonation wave from beingperfectly spherical, but it becomes more nearly so as thecalculation proceeds. (The plotted density of dots in thefigures corresponds to the mass density in the calculation.) Allof the mixture in the upper compartment is detonated by about 16ms, while the wetwell still contains mostly undetonated material(Figure 111.15). At a later time, the pressure waves fromreflections at the wall interact at the compartment centerline,producing a region of very high pressure, as seen in Figure111.16. Figure 111.17 shows the situation still later, whenexpansion from the axial high-pressure region results in asecond strong reflection at the centers of the dome and drywellhead.
The phenomena described in the previous paragraph lead to acommon feature of central-point detonations in large compart-ments: the maximum pressure on the boundary is not the directresult of the arrival of the detonation wave, but of subsequentinteractions between reflected waves. Figures 111.18 and 111.19display pressure histories at the dome center and drywellcenter, respectively, and clearly show the importance of waveinteractions long after complete detonation has occurred. Theeffects of these interactions are also evident where the domeand vertical wall join (Figure 111.20), but at that location,the peak pressure does occur when the detonation wave arrives.
22
TIME =- 2.065E-03 CYCLE = 52 DENSITY"-5
f
E
0
L)I-
LjULUL
LU
_JLUI
2t
-I I
24 f
-37
-50 I I I
-30 -18 -6 6 18 30RELATIVE HORIZONTAL POSITION (in)
Density Field at 2 ms in Containment ExampleFigure 111.13
23
TIME = I.204E-02 CYCLE = 147 DENSITY [T15 I I I
9
E2
z0
<I:
> -11L_1I-IJ
-LJ
wd-24
-37
-50
-30 -18 -6 6 18
RELATIVE HORIZONTAL POSITION (mi)30
Figure 111.14 Density Field at 12 ms in Containment Example
24
TIME = 1.601E-02 CYCLE = 200 DENS ITY15 TIE = i.0E0 CYL = 20 ESiTY
r=
z0
LUJ-JLU
LUJ
Lii
2
-11
-24
-37
-50 1-30
I I I I
0 -18 -6 6 18 30RELATIVE HORIZONTAL POSITION (m)
Density Field at 16 ms in Containment ExampleFigure 111.15
25
TIME = 2.809E-02 CYCLE = 336 DENS ITY15 T IE I.0E0 CYCEI 33 ENIT
E
z0
-IJ
I-
-J
wj
24
11 1
24 f
-37
-50 i I I I I
30 -18 -6 6 isRELATIVE HORIZONTAL POSITION (m)
Density Field at 28 ms in Containment Example
30
Figure 111.16
26
I
I TIME = 3.206E-02 CYCLE = 380 DENSITY
E
z0
I-
LLJ
-LJX
15
2
-11
0
-24 t
-37
-50
-30 0 -18 -6 6 18RELATIVE HORIZONTAL POSITION (m)
30
a
Figure 111.17 Density Field at 32 ms in Containment Example
27
EULERIAN POINT 9] I 800E-0 I. 7. 740E +00 )
LD
CD
0-
UJ0:Da-
I8.000
7.000
6.000
5.000
4.000
3.000
2.000
1.000
0.000
0.00 .400 .800 1.20 1.60 2.00
kTIME (SEC)
GESSAR II.H2-DRY AIR EVERYWHERE(.22)
-1XIO
Figure 111.18 Pressure History at Dome Center in ContainmentExample
i
28
EULERIAN POINT 52 ( 1.800E-01. -1.206E+01)
C0
0
5.500
5.000
4.500
4. 000
3.500
3. 000
0j
LtnV,)
2. 500
2. 000
1 500
1 000
.5000
0.000
0.00 .400 .800 1.20 1.60 2.00
IT TIME (SEC)
GESSAR II.H2-DRY AIR EVERYWHERE(.22)
-1X]O
Figure 111.19 Pressure History atContainment Example
Center of Drywell Head in
29
4
EULERIAN POINT 73 ( 1.746E+01. 9.OOOE-01)3.750
LD+
o 3.500X
3.250
3.000
2.750
2.500
2. 250
2. 000
- 1 .750- -
0-
1 .500
S1 .250
U,
.7500
.5000
2500
0.000 i0. 0 .400 .800 1. 20 1. 60 2.00
TIME (SEC) XlO
GESSAR II.H2-DRY AIR EVERYWHERE(.22)
Figure 111.20 Pressure History at Dome - Wall Joint inContainment Example
30
(The theoretical value of reflected pressure for the detonationwave is greater than 4 MPa; the lower calculated value is causedby the finite zone size.) As expected, the pressure histories inthe wetwell, exemplified by Figure 111.21, show much more rapidearly variations, because of the more complicated geometry ofthe boundary. Pressures are still fairly high when the calcula-tion is terminated at 200 ms, since no flow is allowed out ofthe problem.
Because of the complicated spatial and temporal variationsin loads produced in calculations like the one described above,the CSQ results frequently provide only an indication of whetheranalyses using other methods should be performed. For example,dynamic structural calculations could be carried out using thepressure histories, or approximations of them, as boundary con-ditions, if the CSQ results seem severe enough to warrant suchan investigation.
31
EULERIAN POINT 6 ( 1.386E+01. -2.934E+01)I . 400 ,
- I .300X
1 .200
EIERA PON 10 ( .8EO. 294+
1.000
. 9000
.BOO
0.3000
.6200
.5000
.4000
.8000
.2000
.1000
0.00 400 4000 1.20 1060 2.0
TIME (SEC) X o
GESSAR II.H2-DRY AIR EVERYWHERE(.22)
Figure 111.21 Pressure History about 1.1 m above Bottom ofWetwell in Containment Example
32
IV. Discussion
The example results presented in this report show that amodified version of CSQ can make use of equation-of stateinformation to calculate detonations without the requirementthat the unburned material and the combustion products bedescribed by the same constitutive relations. The examplespresented all deal with calculations involving tabular EoSdescriptions of dry hydrogen-air mixtures, but this is not anecessary restriction. Three examples also demonstrate that,given adequate models for ignition criteria and reactionkinetics, analyses of reasonable sophistication could be carriedout. Finally, calculations with CSQ can yield reasonablyaccurate estimates of dynamic loads exerted on reactor con-tainment boundaries in the event of internal detonations.
33/34
References
1. M. P. Sherman, et al.. The Behavior of Hydrogen DuringAccidents in Light Water Reactors, SAND8O-1495(NUREG/CR-1561), Sandia National Laboratories, Albuquerque,
NM. August 1980.
2. S. L. Thompson, CSQII - An Eulerian Finite DifferenceProgram for Two-Dimensional Material Response - Part I.Material Sections, SAND77-1339, Sandia NationalLaboratories, Albuquerque, NM, January 1979.
3. W. B. Murfin, et al., Report of the Zion/Indian Point Study:Volume 1, SAND8O-0617/1 (NUREG/CR-1410), Sandia NationalLaboratories, Albuquerque. NM, August 1980.
4. R. K. Byers, CSQ Calculations of H2 Detonations in the Zionand Sequoyah Nuclear Plants. SAND81-2216 (NUREG/CR-2385).Sandia National Laboratories. Albuquerque, NM, July 1982.
5. J. C. Cummings, et al.. Review of the Grand Gulf HydrogenIgniter System, SAND82-0218 (NUREG/CR-2530), Sandia NationalLaboratories, Albuquerque. NM, March 1983.
6. M. Berman, ed., Light Water Reactor Safety Research ProgramQuarterly Report, July-September 1980, SAND8O-1304(NUREG/CR-1509). Sandia National Laboratories, Albuquerque,NM, March 1981.
7. J. M. McGlaun, Improvements in CSQII: A TransmittingBoundary Condition, SAND82-1248, Sandia NationalLaboratories, Albuquerque, NM, June,1982.
8. R. D. Richtmeyer and K. W. Morton, Difference Methods forInitial Value Problems, Interscience Publishers (1967).
9. S. L. Thompson. Sandia National Laboratories, privatecommunication.
10. S. L. Thompson, CKEOS2 -- An Equation of State Test Programfor the CHART/CSQ EOS Package, SAND 76-0175, Sandia NationalLaboratories. Albuquerque. NM. May 1978.
11. D. R. Stull and H. Prophet. JANAF Thermochemical Tables,Second Edition, NSRDS-NBS37, National Standard ReferenceSeries. United States National Bureau of Standards,Washington D. C., June 1971.
35
12. IMSL Library Reference Manual, Edition 9. LIB-0009,International Mathematical and Statistical Libraries. Inc..Houston. TX, June 1982.
13. S. L. Thompson, RSCORS - A Revised Scors Plot Program. ,SAND77-1957. Sandia National Laboratories, Albuquerque, NM.May 1978.
36
APPENDIX
The standard CSQ code package includes the following pro-grams, listed with their purpose:
1. CSQGEN - Initialize new problem or rezone an existingcalculation
2. CSQ - Main analysis program
3. CSQLINE - Produce 1-Dimensional plots
4. CSQPLT - Produce 2-Dimensional (plane) plots
5. CSQSURF - Produce 3-Dimensional plots
6. CSQHIST - Produce plots and edits of point histories
7. CSQTAP - Produce edits of CSQ output tape
8. CHDCSQ - Couple output from the I-D Lagrangian code CHARTDto CSQGEN
9. PRECSQ - Preprocess size data to calculate storage arraysizes consistent with problem
Programs 3 through 6 employ the RSCORS [13] plot routines.which is available from Sandia National Laboratories. The firstfive codes in the list use the preprocessor's output to acquirecommon block statements via CDC UPDATE. Complete information onthe codes and their use will be found in the primary reference[2].
37
Equation-of-State (EoS) Tables and Calculations
The code package described above has been expanded toinclude a tabular EoS file which describes 65 gases, the last ofwhich is dry air (EoS number 1885). The remaining tables corre-spond to unburned and burned mixtures of hydrogen and dry air,with initial hydrogen mole fractions spanning the range .18-.58.Unburned mixtures have an odd EoS number from 1 to 6.3; thecorresponding combustion products are described by the nextlarger EoS number. EoS numbers 1-10 describe mixtures withinitial mole fractions from .18 to .26, .02 apart. With oneexception, the range .27-.34 is covered by EoS numbers 11through 40 in increments of .005; EoS numbers 21 and 22 refer toa stoichiometric mixture with an initial hydrogen mole fractionof 0.29524. The r-emaining 12 pairs revert to a mole fractionincrement of .02. The tables for the mixtures cover a densityrange of .0001 to 100 kg/m3. and a temperature range of 300 to4000 K.
The release package also includes a FORTRAN program whichproduces the EoS tables (on file TAPE12) for hydrogen-steam-airmixtures. This code reads unformatted input, one variable to aline. Input consists of, in order, initial steam and hydrogenmole fractions, reference temperature (K), and integers definingthe EoS numbers for unburned and burned mixtures. Followingthese quantities with -1 causes the code to expect input foranother mixture. EoS numbers should be placed on the file inascending numerical order. When the last input set is followedby 1, the code will insert the new tables at the appropriateplace in an EoS file from TAPE66. The program is based on theJANAF tables [11], 1and uses routines from the IMSL library [12].
CKEOS2 [10] is a program which performs various thermo-dynamic calculations with the EoS data used by CSQ, includingthe tabular data described above. An input parameter selects thetype of calculation (e.g., an isotherm for a range of densities),and its value is less than 8 for curves in a plane of 2 thermo-dynamic variables. As in aid in interpreting CSQ results, a needexists for computing Chapman--Jouguet states connecting twoseparate materials, and this is provided in a modified versionof CKEOS2. In order to exercise this option, the user mustrequest at least one calculation for the unburned mixture,followed by a request for a calculation for the burned mixture,with the curve type parameter equal to the standard value plus10. When the code processes the first request for the burnedmaterial, it first produces the standard calculation: theseresults are followed by a set of C-J detonation data. Each pointon the last curve for the unburned material is used as aninitial state for the detonation calculations. The results alsoinclude temperatures and pressures for isentropes from the C-J
38
states to the initial density. The EoS routines will extrapolateto values outside the range of the tables. However, because ofthe reference point used for the entropy, peculiar results maybe obtained for that quantity if the extrapolation distance isvery large. The CKEOS2 reference should be consulted for acomplete set of input instructions.
39/40
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48
A
NRC FORM 335 US NUCLCAR REGULATORY COMMISSION REPORT NUMBER (Asused by TIOC add Vol No. d ncy)
16.3)
BIBLIOGRAPHIC DATA SHEET SAND83-2624NUREG/CR-3719
2 Leave Diank
3 TITLE AND SUBTITLE 4 RECIPIENTS ACCESSION NUMBER
Detonation Calculations Using a Modified Versionof CSQII: Examples for Hydrogen-Air Mixtures 5 DATE REPORT COMPLETED
MONTH MYEARMarch 1984
6. AUTHORISI 7 DATE REPORT ISSUED
MONTH IYEAR
R. K. Byers August 19849 PROJECT, TASK /WORK UNIT NUMBER
8 PERFORMING ORGANIZATION NAME AND MAILING ADDRESS fInclude ZJp COde)
Sandia National LaboratoriesP. 0. Box 5800 10 FIN NUMBER
Albuquerque, NM 87185 • A-1246Division 6444
11 SPONSORING ORGANIZATION NAME AND MAILING ADDRESS (Include Zio Cod.) 12a TYPE OF REPORT
Severe Accident Assessment BranchDivision of Accident Evaluation TechnicalOffice of Nuclear Regulatory Research 12b PERIOD COVERED linclusIi° des)
U. S. Nuclear Regulatory CommissionWashington, DC 20555
13 SUPPLEMENTARY NOTES
14 ABSTRACT (200 words or less)
CSQ is a well-tested and versatile wave propagation computerprogram, a modified version of which has been used to perform anumber of USNRC-supported analyses of detonations of hydrogen-air mixtures in nuclear reactor containment buildings. Themodifications, from a user's viewpoint, are fairly minor, andthis version of CSQ is being prepared for release to interestedorganizations. This report documents the use of CSQ in this form.as well as certain codes which aid in performing the detonationcalculations.
15a K.EY WORDS AND DOCUMENT ANALYSIS 15 blFRl fl0
16
AVAILABILITY
STATEMENT
I6 AVAILABILITY STATEMENT
-49-