POTENTIAL RUPTURE OF THE MA IN COOLANT LINE IN A NUCLEAR POWER PLANT
BY
I. J. BILLINGTON AND 1,1. GLASS
AUGUST, 1960
TE(HN'S~HE HOGESCHOOL DElFT VL'E.G rUI • vJW 'IJNDE Michi I de ujlerweg 10 - DElFT
1 Z okt.1961
POTENTlAL RUPTURE OF THE MAIN COOLANT LINE IN A NUCLEAR POWER PLANT
BY
1. J. BILLINGTON* AND 1. 1. GLASS**
* Associate, Dilworth, Secor.d, Meagher and Associates Limited, Consulting Engineers, Toronto, Canada. ** Professor of Aeronautical Engineering, Institute of
Aerophysics, University of Toronto.
'.,
'
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This report is based on a study undertaken by Dilworth. Seeord. Meagher and Associates Ltd. in collaboration with Prof. 1. 1. Glass of the Institute of Aerophysics. The study was condueted on behalf of Canadian General Electric Company Ltd .• Civilian Atomie Power Department. and Atomie Energy of Canada Ltd.
The authors are grateful to Canadian General Electric Company Ltd. and Atomie Energy of Canada Ltd. for their permission to publish this work.
The authors wish to express their appreciation to Dr. G. N. Patterson for his interest and encouragement during the course of the work.
An analysis was made of the maximum pressure rise and relevant pressure history th at might occur in the boiler room of a nuc1ear reactor from the complete failure of a main, sixteen-inch, heavy-water, coolant pipe. The boiler room and the 9 ft. x 12 ft. x 50 ft. pressure relief duct were represented by a simplified mathematical model based on one-dimensional, inviscid flow in order to study the transient pressure rise following the failure, and a step-by-step numerical calculation pro-vided the solution. The analysis included the effect of a sealing diaphragm at the upstream end of the pressure relief duct which was assumed to
disintegrate within 10 milliseconds af ter the boiler room pressure reached
1. 5 psig. The pressure-time history determined from this analysis shows
. a rapid rate of pressure rise from the instant of failure levelling off to a peak of 4. 4 psig af ter approximately O. 3 seconds.
Although th is note describes the treatment of a specific problem, it is felt th at the analysis is of general interest in the field of nuc1ear· reactor safety and represents and interesting application of nonstationary-flow theory. The method mayalso prove useful in other industrial applications and may be of some assistance in the development of safety criteria for future nuc1ear power plants.
TABLE OF CONTENTS
Page
NOTATION ii
1. INTRODUCTION 1
2. DEFINITION OF THE EMERGENCY 2
3. PRELIMINARY ASSESSMENT OF THE PROBLEM 3
3. 1 General Considerations 3
3.2 Pressure Rise in the Isolated Boiler Room 3 3.3 Overpressure Required for Steady Flow 5 4. TRANSIENT PRESSURE CALCULATIONS 6 4. 1 Mathematical Model of the Problem 7 4.2 Pressure Rise Calculations 8 4.3 Construction of a Wave Diagram 9
• 4.4 Diaphragm Bursting Calculations 10
5. RESULTS OF NUMERICAL ANALYSIS 12
6. DISCUSSION 14
6. 1 Viscous Effects 14
6.2 Effect of Room and Duct Geometry 15 6.3 Pressure Relief Duct Diaphragm 15
7. CONCL USIONS 16
7.1 Summary of the Problem. Method of Analysis and
Sim plifying As sum ptions 16 7.2 Review of Analysis and Results 17
REFERENCES 19 APPENDIX A 21
..
APPENDIX B 23 APPENDIX C 25 APPENDIX D 28 APPENDIX E 31 APPENDIX F 35A a Cp,Cv D E f h k 1 M
•
m p q R T t u V Vf, Vg W X, x 0<.ra
Area (ft2) NOTATIONVelocity of sound (ft/ sec)
Specific heat at constant pressure and constant volume (BTU /lb oF)
Function defined byequation (54) Internal energy (BTU /lb)
Friction coefficient Enthalpy (BTU /lb)
Heat transfer coefficient (BTU /hr ft2 OF) Length (ft)
Mach number ( U) a
Mass flow (lb/sec or slug/sec) Pressure (psi)
Dynamic pressure (112 f ·u 2 ) (psi) Gas constant (BTU Ilb OF)
Temperature (oR or OF) Time (sec)
Gas velocity (ft
I
sec) Volume (ft 3 ) Specific volume (ft 3 /lb) Weight (lb) Length (ft).<tf
+
1)/«(5 -1) (~-1)/2Y<5
JA
P
Subscripts: a f g w R,LSpecific heat ratio (Cp/Cv)
Viscosity coefficient (slug/ft sec) Density (lb/ft 3 )
Air
Liquid water Steam
Total injected water Right and Left
.In addition, temperatures are used as subscripts to indicate the tempera-ture at which a quantity is to be evaluated. Flow region numbers are also used as subscripts and the symbols i and jare used to denote arbitrary flow regions or times.
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1. INTRODUCTION
The potentially serious consequences of major accidents connected with the operation of nuclear reactors have necessitated detailed consideration of problems related to reactor safety and contain-ment. A number of basic approaches to the safety problems are discussed in some detail in Ref. 1. In general, however, the differences between individual reactor plants are such that standard safety criteria cannot be established and the specific problems of each reactor must be considered during the design phase. Because of the large number of variables
involved in safety studies it is usually necessary to postulate a "worst" accident and to study the safety requirements on this basis.
In the case of the Nuclear Power Demonstration (NPD-2)
reactor, presently being designed by Canadian General Electric Company ~imited for Atomie Energy of Canada Limited, one of the potential
emergencies which must be considered is the failure of a main, high-pressure, heavy-water line in the boiler room of the plant. Such a failure would result in the release into the room of the entire contents of the heavy-water coolant system and would generate large quantities of
st eam which would rapidly raise the pressure in the boiler room tó·
unacceptable levels. Preliminary designs for the NPD-2 reactor included a pressure relief duct (see Fig. 1) running from the boiler room, which was largely underground, into the open air. The dimensions of this duct appeared to be adequate for steady flows, but the nature of the transient pressures following failure of a heavy-water line and the suitability of the duct for these transient conditions was not known.
The problem was further complicated by a requirement that heavy-water which might be spilled from minor leaks be contained and recovered. It was, therefore, desired to close the discharge duct with a diaphragm which could be quickly shattered in the event of a severe pressure increase.
The present report describes some theoretical considerations and a numerical analysis of the emergency pressure conditions in the
NPD-2 boiler room. This analysis was conducted on behalf of the
Canadian General Electric Company Limited, Civilian Atomic: Power Dep}. , and Atomic Energy of Canada Limited.
The probable location of the main coolant line failure and the resulting most severe escape rate of heavy-water were postulated by Canadian General Electric Company Limited and Atomic Energy of
Canada Limited af ter some experimental studies. This information was used as the basis for the analysis. Certain sim plifying assumptions were then applied to the general problem in order to produce a mathematical model which adequantely approximated the physical case and which could be solved by reasonably straightforward numerical techniques to determine
the boiler room pressure transients.
A simpliefied analysis based on the ass urnption of a steady state efflux from the boiler room through a 50 ft. relief duct with a
9 ft. x 12 ft. cross-section indicated that the maximum pressure rise would -be limited to about 2 psig. On the other hand, a similar analysis based .on the assum ption of a closed boiler room with no outflow indicated a very rapid rate of pressure rise in the room leading to a peak pressure in excess of 45 psig. It was evident from these preliminary studies that the problem should be analysed as a transient phenom enon and the ensuing investigation was, therefore, based on this premise.
The numerical analysis of the transient conditions indicated a maximum overpressure of 4.4 psig. The calculations were terminated shortly beyond the point of maximum pressure and do not, therefore, yield a specific pressure time history beyond this point. A qualitative assessment of the various factors influencing the problem indicates, however, that the boiler room pressure would ffilLoff gradually for 2. 7 seconds beyond the peak until the stored mass of coolant was fully expelled and would thereafter decay rapidly to ambient atmospheric
pressure.
Because of the general interest in safety of Nuclear
Reactor Power Plants and other high pressure system s and as an interest-ing application of non-stationary· flow theory, the methods and results of this analysis have been reassembled in the present form . .
2. DEFINITION OF THE EMERGENCY
For the NPD-2 boiler room the "worst" single failure in the system was defined in the following manner:
a) It was assumed that a 16 inch dia. main coolant pipe, initially containing liquid heavy-water (D20 at a temperature of 1030 psia, would be instantaneously severed in such a way that a total cross-section of 322 sq inch would be exposed. From this open area a mixture of water and steam would be discharged at a rate of 25 lb / sec in 2 until a total of 25, 000 Ibs had been discharged, at which time the flow rate was assurned to drop instantaneously to zero.
b) The boiler room in which the failure was assumed to occur was defined as having a free volume of 8.Q,OOO cu. ft. and to be initially filled with dry air at 1000F and 14. 7 psia. A discharge duct 9 ft. wide, 12 ft. high and 50 ft. long was provided in the preliminary design, leading from the boiler room to the outside atmosphere, as shown in Fig. 1. '!he
.
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c) In order to contain coolant lost from minor leaks it was
desired to seal the discharge duct and for this purpose a dia-phragm across the entry to the passage was proposed. For the numerical studies it was specified that the diaphragm
should rupture between 10 and 20 milliseconds af ter the pressure near the primary piping had reached 1. 5 psig.
3. PRELIMINARY ASSESSMENT OF THE PROBLEM 3. 1 General Considerations
Before undertaking any detailed .. nu.merical studies of the transient conditions in the boiler room it was necessary to investigate various aspects of the problem which might influence the approach to the
transient conditions. It was immediately apparent that one important
factor influencing the calculations would be the properties of heavy-water
steam. A search for experimental data concerning D20 vapour (see
Appendix A) was largely unsuccessful and it became necessary to deduce the required properties from the rather meagre information available.
Another consideration which was thought to be of importance was the amount of vaporization which might take place before the fluid was ejected from the fractured pipe. This would have considerable bearing on the ejection velocity of the fluid and consequently would affect the degree of mixing between the water and steam and the air initially in the room. The degree of vaporization of the ejected fluid would also
influence the time required to reach thermal equilibrium and phase
equilibrium between liquid and vapour. A related problem, the time
required for vaporization of a super-heated liquid, was felt worthy of some study and is discussed in more detail in Appendix D. It was finally concluded that instantaneous vaporization rates should be assumed for the transient pressure calculations in the absence of sufficientlyreliable data to justify an;y less conservative assumption.
3. 2 Pressure Rise in the Isolated Boiler Room
It is of pertinent interest to cDnsider the rate of pressure build-up in an isolated closed boiler room due to injection o! water at the specified rate. After addition of a given quantity of water, assuming that
thermal and phase equilibrium is instantaneously achieved, the temperature
and pressure in the room can be calculated by the method of Appendix B. In these calculations it is assumed that the final composition of the
mixing region is saturated steam, liquid water and the original air of the mixing region homogeneously mixed and in thermal and phase equilibrium. The pressure in the mixing region is assumed to be the sum of the partial pressures of the air and the D ~ vapour.
It is apparent that the amount of air that becomes mixed with the injected water will have a significant effect upon the final equili-brium conditions in the room, since the equiliequili-brium temperature is to sorne extent dependent upon the amount of heat that must be transferred from the liquid water to the air and the final pressure is dependent upon the sum of the partial pressures. The two extreme cases that may be considered, and which have been calculated in Appendix Bare zer<>-mixing and complete-mixing. In the former case the steam-water mixture is assumed to remain separated from the air in the room and to expand to an equilibrium volume, at the same time compressing the air isentropically to the same pressure. The result is a room, part of which contains a steam -water mixture and part of which contains air at the same pressure, but the temperatures of the two gases are quite different since heat
transfer from one gas to the other is neglected. In the complete-mixing case the steam-water mixture is assumed to mix homogeneously with the air throughout the room so that the entire room is finally filled with a mixture of uniform composition, temperature and pressure. The actual case in the boiler room under consideration in this study would doubtless lie somewhere between these two extremes, with an intermediate degree of mixing.
. Results of the zero and complete mixing calculations for the closed boiler room are illustrated in Figs. 2, 3 and 4. All these figures are plotted as functions of time and are terminated approximately at 3. 1 seconds, at which time the entire contents of the heavy-water
system considered in the present problem have been emptied into the boiler room. On the scale of the pressure plots of Fig. 2 very little difference between the two cases can be distinguished (see Fig. 8 for an enlarged scale). Actually the zero-mixing case registers pressures lower than
those for complete-mixing during the first 2. 5 seconds and higher pressures thereafter. In both cases the pressure build-up is extremely fast and in the absence of any pressure relief mechanism it is evident that the over-pressure rapidly passes all reasonable structural limits for the reactor building. Figure 3 shows the total weight of steam formed under
equilibrium conditions, and it can be seen that only about one-third of the injected water is actually transformed into the vapour phase. No great difference exists between the complete-mixing and zero-mixing cases.
The temperature variation curves of Fig. 4 exhibit a con-siderable difference between the two cases considered. For complete ... mixing the temperature of the mixture is uniform throughout the room and increases fairly rapidly at the beginning of water injection but less rapidly later. In the zero-mixing case a saturation pressure of at least 14. 7 psi is required from the beginning of the process and this corresponds to a high saturation temperature for the steam. The temperature of the air increases continuously and rapidly as it undergoes an isentropic com-pres sion. At the end of the process it can be seen that there is considerable difference between the temperature of the air and that of the steam-water
, • ,. • , 1
mixture. In an actual case heat transfer from the air to 1:he steam -water mixture would in time remove such a discontinuity in temperature with consequent changes in both pressure and and steam quality. The only point in the process where an apparent equilibrium occurs is the one at about 1. 24 s~conds where the two temperature curves cross over in Fig. 4. Despite the nbnequilibrium nature of this solution, it appears reasonable to neglect heat transfer between air and steam because of the very short tim es involved in the present problem.
3.3 Overpressure Required for Steady Flow
The mixing calculations for an isolated boiler room, dis-cussed in the previous section, show that the incoming water contains
only enough thermal energy to vaporize approximately 35% of Us own mass.
This results in an ave rage steam generation rate of about 2, 800 lb / sec.
On a steady state basis, therefore, it should be necessary to remove only about 2,800 lb/ sec of gas to maintain a constant pressure within the boiler room.
If it is assumed that the boiler room acts as a large settling chamber, th en most of the 60% of the injected water which remains in
liquid f<:>rm will fall to the boiler room floor, and the gas flowing through the relief duct may be considered to be dry. The volume taken up by the water remaining in the boiler room is negligible compared to the volume of the boiler room itself. The possibility and likely effects of water entrainment in the duct flow were studied and the conclusion was reached that this factor could safely be neglected since most of the water wiU have been removed by settling and centrifugal action before the mixture reaches the duct.
The pressure loss coefficients for the exhaust duct in the form shown in Fig. I, have been estimated in Appendix C to be approxi-mately 1. 37 times the dynamic head (q). The overpressures required for various assumed steady state conditions have been calculated in Appendix C and are listed in the table below. For the reasons stated above the case of fuU-water entrainment has not been considered but ther.case of 80% quality steam has been calculated for comparison.
Flow Room Room
Rate Com position Temperature Overpressure
(lb/sec) (OF) Required (psig)
2800 Air-Dry D 2
q
Steam 184 1.7 2800 Air . 244 1.6 2800 Dry D 20 Steam 260 2.3 3500 80% Quality H 20 220 2. 94. TRANSIENT PRESSURE CALCULATIONS
Several immediate conclusions can be drawn from the basic boiler room geometry. In the first place, the length of the relief duct is such that a delay of about O. 1 seconds would occur af ter rupture of the diaphragm. before the first reflected pressure wave from the open end of the duct could reach the boiler room. Furthermore, several such
reflected pulses would be required to start an appreciable mass flow through the duct. Thus the pressure might be expected initially to follow closely the curves of Fig. 2 and then to level off as the--duCtflöw became established.
It can be seen from Fig. 1 that the dimensions of the room itself are sornewhat greater than the length of the duct, with the result that the time required for pressure pulses to cross the room is greater than th at required for them to pass down the length of the duct. Accordingly, there may be considerable delay between the generation of a pressure
pulse at the broken pipe or at the duct entrance.andthe change of pressure caused by such a pulse at the opposite side of the room. It would,
therefore, be possible to register a considerable pressure gradient from one side of the room to the other. Consequently, it was decided that the numerical calculations of the transient pressure conditions should take account of the approximate dimensions of the room as well as those of the duct.
The geometry of the room together with the location of the possible fracture in the heavy-water line suggested that netiher the complete mixing nor the zero mixing cases would adequately describe conditions
during water injection. It was, therefore, decided that a condition of half mixing should be assumed. In this case the injected 'water and steam would mix homogeneously and come to thermal equilibrium with half the air initially in the room. This mixture of air, steam and water would be assumed not to mix with the remaining half of the air in the room.
For the flow in the pressure relief duct it was anticipated that very close agreement between theory and practice would be obtained using one-dimensional flow analysis, providing the conditions in the room were known. For the room, on the other hand, no clear-cut method of analysis was available. Spherical, cylindrical and planar flow were three possible alternatives for representing pressure wave propagation.
Spherical flow theory could be applied on the assumption that the steam-water mixture was added to a spherical area at some point near the centre of the room. In this case a reasonably close approximation to the existing room geometry could be used but the calculations of the reflections and interactions of the spherical waves in the approximately rectangular room would be quite difficult. (Spherical and cylindrical flows are discussed in Refs. 2, 3 and 4.) Cylindrical flow would require the postulation of mixing
,
..
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in a cylindrical region of the room, while planar flow theory would reduce the problem to a one-dimensional analysis. The three methods should all result in pressure disturbances of approximately the same. strength being generated at the boundaries of the mixing region. However, (see Ref. 5) small amplitude spherical waves would decay inversely as the distance travelled and cylindrical waves would decay inversely as the square root of this distance. For the plane wave case no decay would occur and con-sequentlya planar analysis is more conservative. For the geornetry and rate of energy injection considered in this problem the overall difference in the results obtained from the three methods would probably be small.
It was decided to apply the one-dimensional flow theory which has been shown by experiment to provide a good approximation to the true flow in such problems (Refs. 6 and 7) to both boiler room and duct, thus allowing a simplified picture of the entire process to be con-structed num erically and graphically.
4. 1 Mathematical Model of the Problem
A one-dimensional mathematical model of the boiler room and discharge duct was constructed as shown in Fig. 5. This room had an overalllength of 54 ft, approximately the length of the real boiler room in a direetion parallel with the escape duet axis, and a volume of 80, 000
cn. ft. The mathematical rOQm had a uniform cross-section of 1,480 sq. ft. The 50 ft. dis charge duet was c10sed off with a diaphragm as shown.
Sinee one-dimensional flow was assumed, only one
co-ordinate, x, was required to specify position in the room. The position x = 0 eorresponded with the centre of the room as shown in Fig. 5, and the co-ordinates x
=
+27 and x ::: -27 represented the left and right ends of the boiler room respectively. The discharge duet extended for 50 ft from x=
+27 to x = +77.The half-mixing solution discussed in the previous seetion was used and it was assumed that the air initially in the centre half of the room (between x
=
-13. 5 ft and x = +13. 5 ft) defined the initial extent of the mixing region. This assumption will hereafter be referred to asthe "half-mixing assumption" . No mixing was assumed to occur between the steam -water mixture and the air initially in the two ends of the boiler room.
The orie-dimensional nature of t4e problem allows the motion of pressure waves in the room and in the duct to be plotted on a simple distance-time (x, t)-plane wave diagram. The methods employed in the eonstruction of this wave diagram are described in the following sections and the mathematical method is detailed in Appendices E and F.
4. 2 Pressure Rise Ca1culations
In order to perm it the calculation of the transient pressure rise in the boiler room it is necessary to consider the prQcess as occurring in finite steps at time intervals ót. Since the rate of water injection is known, the mass injected during 'time interval ..ót can be calculated and it is assumed that this mass is introduced uniformly throughout the mixing region and instantaneously at the end of the time interval under consideration. It is further assumed th at the air, water and steam within the mixing
region af ter the injection instantly reach thermal and phase equilibrium. The injection of water into the boiler room raises the temperature and pressure of the mixture above that of the unmixed air in the ends of the room. This causes a pressure discontinuity across the two interfaces between the mixing region and the unmixed air in the ends of the room. As this is an unstable condition, compression and expansion waves are generated at the interfaces. The former travel outward to compress the air regions and the latter travel inward to expand the
ml}ung region. The resulting wave pattern in the (x, t)-plane is illustrated in Fig. 6. The equalization of pressure results in a spreading of the
mIxmg region. The initial expansion waves generated at the interfaces cross at the point x
=
0 and continue outward, further reducing thepressure in the mixing region and stopping the growth of this region. The expansipn waves refract at the interface and then travel out into the air regions, following the initial compression waves. The refraction actually produces also a reflected wave which moves back into the mixing region (see Ref. 8) but for the refractions in the present analysis the reflected waves were always extremely weak and were thus neglected. Af ter a further time interval öt, as can be seen in Fig. 6, the procedure is repeated.
In the construction of the wave diagrams in the (x, t)-plane it is assumed that all the waves involved are weak isentropic compressions or expansions, and that these waves are separated by regions of
quasi-steady flow with uniform velocity, pressure, temperature, and composition. It is known from many experimental investigations that these assumptions will give a very close approximation to the actual conditions (Refs. 7, 9 and" 10). The flow regions between waves are numbered 1, 2, 3 etc. iIi Fig. 6. It should be noted that on both sides of the interface between the mixing region and the air region, the pressures and particle velocities
are identical. For example, the pressures and particle velocities in regions 2 and 3 are equal and similarly those in regions 7 and 8. During
con-struction of a wave diagram in the (x, t)-plane the significant flow para-meters in every steady flow region are calculated and can be tabulated.
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In practice the type of (x, t) diagram shown in Fig. 6 is
considerably complicated by the fact that all the outgoing compression and expansion waves reflect from the ends of the boiler room and
ultimately return to the mixing region. On the other hand, the assumption
of a symmetrical boiler room with a centred mixing region as illustrated in Fig. 5 produces a problem which is symmetrical about the centreline
of the room (x = 0). Thus, as long as the diaphragm across the entrance to
the pressure re lief duct is intact, it is only necessary to consider ane end
of the room. This symmetry is apparent in the wave diagram of Fig. 6.
4. 3 Construction of a Wave Diagram
Considering only the half of the room between x = 0 and
x
=
27 during the initial period of pressure build-up, and includingreflected waves from the end of the room, the complete wave diagram
for the early stages of the problem would appear as in Fig. 7. Each
wave reaching the closed end of the room returns a reflected wave of the same type, while each wave reaching the eentreline crosses a symmetrical
wave of the same type coming from the other half of the room and.
therefore, appears to undergo a reflection at the centreline or plane af
symmetry. It can be seen from Fig. 7 that the wave diagram rapidly
becornes very complicated because of the interactions of waves with each
other and with the interface which forms the boundary of the mixing region .
The methods used for calculating the results of all the
interactions in the (x, t)-diagrams are outlined in Appendix F, and are
based upon the methods of Refs. 6, 7 and 9. It can be seen from Fig. 7
that the addition of extra waves as time progresses results in an
increasingly more complex wave diagram. Therefore, some
simplifica-tion of the calculation by elimination and combination of waves was
introduced from time to time. This simplifying process. which is dis
-cussed in detail in Appendix F, inc1uded the omission of reflected waves
from refractions at interfaces and the omission of new interfaces generated
by the head-on collision of waves. Such steps were justified in the present
analysis because of the very weak waves considered.
The construction of an (x, t)-wave diagram for this prablem
does not require that the time increments .6t be all of the same size. This
is illustrated in Fig. 6 where the first time increment is considerably
smaller than the second. Some consideration was given early in the pro-blem to the optimum size of time interval. lf this interval is very small, then the number of steps involved in a calculation become correspondingly large. However, if the interval is large, although the number of steps is reduced, the strength of the individual waves is correspondingly increased and the jumps in pressure and temperature occurring across these waves
also become large. Consequently, the amount of scatter in the numerical
results is directly proportional to the size of the time intervals chosen
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discussed above are only valid for very weak waves. In the limiting case of t
=
0 (i. e., continuous water injection as in the real boiler room) the diagram would show a continuous succession of vanishingly weak pressure pulses, and only the waves resulting fromthe diaphragm rupture would have finite amplitude.Trial calculations were initially made using three different mesh sizes as described in Appendix F. A cornparison of the results of two of these trials is shown in Fig. 8 whichgives the pressure history at the room station x = +20. The increase of scatter with the coarser mesh is quite apparent in this figl,lre. The variation in scatter does not reflect any basic discrepancy between the results calculated with the different mesh sizes since the energy of the system is conserved and the rate of energy addition is identical regardless of the coarseness of the net.
It appeared desirable for the final calculations to introduce water injections at intervals of 0.01 sec in order to determine accurately the time at which the diaphragm rupturing overpressure was reached. Thereafter the mesh size was coarsened considerably in order to reduce the amount of calculation necessary.
4.4 Diaphragm Bursting Calculations
The ,diaphragm was assumed to burst at approximately 10 milliseconds after the overpressure in the vicinity of the break reached
1. 5 psig. This overpressure was reached at O. 09 sec in the step-by-step calculations and consequently the diaphragm was opened instantaneously at t
=
O. 10 sec. The diaphragm bursting calculation and the strength of the resulting shock wave' travelling down the discharge duct were calcu-lated as outlined in Ref. 6. Thereafter the wave diagram had to be con-siderably expanded; the 50 ft length of the discharge duct had to be included at the right end, and the other half of the boiler room had to be added to the left end since the flow in the room itself would no longer be symmetrical about the centreline.The nature of the complete distance-time diagram is
illustrated in Fig. 9. For purposes of illustration this diagram has been greatly simplified (in comparison with that employed in the actual compu-tation) by the inc1usion of only a single mixing point at the beginning of the process. In this diagram compression waves are indicated by heavy lines and expansion waves by : light' lines. The left and right interfaces which form the boundaries of the mixing region are shown as dotted lines 11 and 12 respectively. The diaphragm is assumed to rupture at point Asending an initial shock wave AB down the pressure relief duct and an initial expansion wave AC back into the boiler room. The bursting of the diaphragm also generates an additional interface 13 which separates the cool air which was originally in the duct from the warm air which was originally in the right
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end of the boiler room, and this interface travels down into the pressure relief duct at the local flow velocity. It can be seen that the flow in the interior of the boiler room is completely symmetrical about the centre-line at times before the rupture of the diaphragm but that the symmetry disappears af ter the passage of the initial expansion wave.
Figure 9 illustrates several other charaeteristics of the
wave diagram. Shortly af ter the bursting of the diaphragm an expansiön wave from the far side of the mixing region arrives at the right hand wall of the room (at point D). A weaker expansion DG is reflected into the boiler room and a transmitted expansion DE travels down the discharge duct. Since the initial shock wave has increased the 'speed of sound and
induced a flow velocity in the discharge duct the following expans ion wave
travels at a higher wave speed and ultimately overtakes and coalesces with the initial shock at point E. The resulting weaker shock travels on to the end of the duct (at point F) and passes out into the open air. An
expansion wave FH is immediately generated at the open end of the duct and travels back into the boiler room. Subsequently, other waves
originating in the boiler room arrive at the open end of the duet at times such as J and K in Fig. 9, and these waves in their turn reflect other waves back up the passage. It is of interest to note that the first pressure relieving pulse from the open end of the duct is that which arrives back in the boiler room at point H. Thus the distance AH on the wave diagram of Fig. 9 represents the time delay due to the relief duct length.
In theory the finite expansion waves AC and FH should be shown as fans rather than single waves. Again, however, for the wave strengths considered in this problem it was found that this was arefinement which could be omitted without influencing the results. Representation of these expansions as single lines on the diagram considerably simplified the computational procedures.
As illustrated on Fig. 9 the left interface 11 moves slowly
towards the left wall, compressing the air which is trapped in the back end of the room. The interface 12 at the right end of the mixing region moves more rapidly towards the right end of the room af ter the diaphragm is opened and as air from the room starts to flow into the relief duct. Interface 13 moves rather slowly at first but then more rapidly towards ' the open end of the duct and eventually would pass completely out of the duct. The region between interfaces 12 and 13 contains all the air which was originally in the right hand end of the boiler room. lf the wave diagram were continued far enpugh the interface 12 would also be seen to move
into the relief duct and af ter the pressure peak in the room had been passed interface 11 would be seen to move again to the right.
~ It may be observed from Fig. 9 that the initial expansion wave from the bursting of the diaphragm causes a drop in pressure in the right end of the room immediately after time A, but this effect is not felt at the left en<i until much later (at time C). Similarly, the first reflected pulse from the open end of the relief duct drops the pressure at the right end of the room at time H, but this pulse is not felt in the left end of the room until time L. This illustrates the fact that the
most critical pressure peaks are likely to be experienced in the parts of the boiler room most remote from the duct entrace.
Even the simplified diagram of Fig. 9 is characterized by an ever-growing number of waves since every wave which reaches the area change at point x = +27 ft, either from the right or from the left, produces two waves, one transmitted and one reflected. Theoretically the same is true every time a wave crosses one of the contact surfaces, 11, 12, or 13, but for the weak waves considered here the reflected waves were found to be of negligible strength (see Appendix F). In the complete wave diagram the number of waves present increases very much more rapidly since each succeeding injection of water results in the formation of four new waves which must be added to those already existing. For this reason the combining of waves is a necessary but justifiable
expedient during the computationo
5. RESULTS OF NUMERICAL ANALYSIS
As described above, t
=
0.01 sec was used for the water injection interval in the initial part of the calculations. Having established the point at which the diaphragm should be opened, it was feltworth-while to increase the net size considerably in order to reduce the number of calculations required to cover a long period of time. This was accord-ingly done by first increasing to t
=
0.03 and then to approximately0.05 sec. The 0.05 steps were commenced at time t = O. 17 sec and con-tinued until time t
=
0.36 sec. At this time the mesh was made finer again by reducing t to approximately O. 02 sec in order to reduce the scatter in the results. Having established that further increases in pres-.sure in the boiler room were not occurring, the computations and wave dia-gram were terminated at timé t
=
0.43 sec.The entire (x, t)-diagram included some 400 flow regions. During construction of the diagram the important parameters describing conditions in eacliuniforip flow region (sound speed, air flow velocity, pressure, temperature and gas composition) were recorded.
From the recorded flow parameters it is possible to con-struct the flow history at any location in the assumed one-dimensional boiler room. A typical pressure history (that for station x
=
+20) isshown in Fig. 10. This station was originally in the air region at the right end of the boiler room, but at time
t
=
0.37 sec. the interface 12 moved.
•through this location. Thereafter point x
=
+20 was in the mixing region. This transition is indicated in Fig. 10 by asolid line for air pressures changing to a dotted line for mixing region pressures. It will be noted that the amplitudes of the pressure excursions due to the passage of indivi-dual waves become considerably larger after time t = O. 17 sec when the very coarse net was introduced into the calculations. After reintroduction of a fine net the amplitude decreases again for the last 50 milliseconds . The extremely high peaks, such as those at O. 19, 0.24 and 0.28 sec, occur following the reflection of waves from the right end wall of the boiler room. The high peak occurring at O. 38 sec is due to the sudden addition of further water to the mixing region. The low troughs occurring at such points as0. 26 and 0. 31 sec are due to the reflection of expansion waves from the right end of the boiler room. It should be re-emphasized that the pressures recorded at these high peaks and low troughs do not represent the actual conditions existing at this point of the boiler room. Since water is actually added continuously the true picture would be a continuous variation of
pressure which would be represented by a mean line passing through the irregular curve of Fig. 10.
Since the pressure h~story shown in Fig. 10 represents a location in the room very close to the pressure relief duet entrance, it is of interest to compare the pressure history at a more remote point in the room. Consequently the pressure history at the point x
= -
27 ft (the far wall of the room) has been plotted in Fig. 11. All the flow regions in this case are of longer duration than those shown in Fig. 10 since, ascan be seen in Fig. 9, the flow regions lying between incident and reflected waves are of infinitesimal duration at the wal!. As a consequence the
pressure history at the wall has a coarser appearance than the pressure histories corresponding to points in the interior of the room. Otherwise, the general comments relating to Fig. 10 apply to Fig. 11 also. A
similar plot for x
=
0, t he centre of the mixing region is shown in Fig. 12.It is quite difficult, particularly during the period of coarse net calculations, to determine the exact position of the mean line through a pressure history of the type shown in Figs. 10, 11. and 12. This is
particularly true in the case of x = -27 ft since the highest peaks and lowest troughs occupy a larger proportion of the time. However, mean lines for these cases have been calculated by dividing each history up into a number of small time intervals and averaging the pressure during each interval. Mean pressure lines have been drawn through the average points so calcu-lated .. These mean lines show that the pressure at all stations examined rises initially at approximately the same rate as the equilibrium case, but
at all times lies somewhat below the equilibrium value. The rate of pressure
rise at all points is seen to decrease at times af ter the breaking of the diaphragm. Near the entrace to the pressure relief duet a maximum pressure of 19. 0 psia is reached after about O. 29 sec and the pressure
remains at this level until the end of the period considered in the calculations . At the remote end of the room the pressure rises to approximately 19. 1 psia
after about O. 3 sec and thereafter appears to fall slightly. At the centre of the mixing region a pressure of 19. 1 psia is also attained at about 0.3 sec. Very close comparison of these mean lines is not justified because of the inherent inaccuracies in the averaging procedure. However, it might be expected that the pressure at x = -27 ft would reach a higher
peak than that near the relief duet, and at times greater than t
=
0.37 sec it is reasonable to expect the pressure at x=
+20 ft (which is now in the mixing region) to be slightly higher on the average than pressures in the air region at the end of the room.The results of the step-by-step ca1culations therefore show that the pressure near the walls of the boiler room reaches a maximum of about 19.1 psia after a period of about 0.3 sec. Thereafter no appreciable change is apparent up to time t
=
0.43 when the calculations were discon-tinued. The ave rage flow rate in the pressure relief duct increaseswith time, and the exit velocity at t
=
0.43 is 680 ft/sec, corresponding to a mass flow of over 5,000 lb/sec. Thus the outflow rate substantially exceeds the steam generation rate and an extension of the ca1culations for longer times would be expected to show a gradual decline in pressure.Figure 13 compares the calculated transient overpressure (for x
=
0) with the isolated boiler room eurve of Fig. 2 and with thesteady flow overpressure of 1. 7 psig ca1culated in Appendix C for a dry air-steam mixture. The probable form of the pressure decay at times beyond the end of the present ca1culations is shown in Fig. 13, based on a
simplified inviscid flow ca1culation and an estimated allowance for viscous effects. This gives a transient becoming asymptotic to the steady flow overpressure and a further deeay commencing at t
=
3.1 when water injection ceases. However, for a dynamic analysis of the boiler room structure it was felt that a more severe loading, where the peak pressure was sustained until t=
3. 1 as shown in Fig. 3 should be used.6. DISCUSSION
An assessment of the results will be made in this seetion with a view to estimating-the effects of such items as viscosity, room and duct geometry, and the pressure-relief diaphragm.
6.1 Viscous Effects
The transient analysis described above was based entirely upon inviscid flow theory. Experience has shown that for the type of pro-cess described here inviscid theory provides a very close approximation to the true-flow. Experimental results presented in Ref. 7 suggest that boundary-layer growth following the establishment of flow in the pressure-relief duct should have a negligible effect in view of the large cross-section.
,
.
'
..
Viscous effects would, however, influence the flows at both the entrance and exit of the duct. In particular these effects are likely to delay and modify the formation of the relfe.cted expansion wave (FH, Fig. 10) when the initial shock wave reaches the open end of the duct (Ref. 6, 7~ 9). Consequently, the pressure will not be relieved as suddenly as with a centred expansion fan.
6.2 Effect of Room and Duct Geometry
The actual NPD-2 boiler room is of rather irregular shape and contains m uch equipment in the form of tanks, boilers, access plat-forms, columns, pipes and other miscellaneous shapes. The arrange-m ent of ihe acce.ss platforms is such that the upper portion of the boiler room, in which the initial failure of the hi~h-pressure coolant system must occur, -.is to a large extent shielded from the lower portion of the boiler room. ' This arrangement is likely to restrict mixing of the steam and the air and suggests that the air initially in remote parts of the boiler room is likely to remain there (as in the transient calculations of this report). The presence of large quantities of metal in the room is also likely to cause considerable condensation of steam af ter the failure.
It has previously been observed that the length of the pre-ssure relief duct is an important parameter affecting the flow starting phenomena. The importance of cross-section was also studied by com-parison of the flow-starting performance of a 9 ft x 12 ft duct with that of the 9 ft x 21 ft duct. A simple wave diagram for the starting process is shown in Fig. 14 and described in Appendix F. The boiler room pressures in the two cases are compared in Fig. 15 and it is seen that the increase of cross-section would not greatly improve the flow starting characteristics of the passage.
6. 3 Pressure Relief Duct Diaphragm
The characteristics of the diaphragm for sealing the
pressure relief duct have a very important bearing upon the whole problem. The ideal diaphragm would be one that provided a perfect seal at pressures below the critical value and which disintegrated co.mpletely and instantly when the room pressure exceeded the critical \Talue. In practice any mechanical method of diaphragm opening or breaking results in a time delay th at must be added to the time required to establish flow in the duct and this can cause appreciable pressure build-up in the present problem.
In the numerical calculations an allowance of 10 millisec was made between the time at which the overpressure in the vicinity of the break reached 1. 5 psig and the time at which the diaphragm was
assumed to shatter. A study of the calculations suggests that the limiting case of zero delay would reduce the peak boiler room overpressure by about O. 2 psig.
7. CONCL -q,SIONS
7.1 Summary of the Problem, Method of Analysis and Simplifying Assumptions
The salient features of the problem, the method of analysis and the relevant simplifying assumptions employed in this investigation are summarized as follows:
1. The characteristics of coolant line failure and resultant efflux of heavy-water and steam from the fractured line as postulated by Canadian General Electric Company
Limited result in an average steam generation rate of about 2800 Ibs per sec for a period of 3. 1 sec.
2. The geometrical configuration of the boiler room is as shown in Fig. 1. The relief duct is equipped, at the upstream end, with a diaphragm which is assumed to disintegrate to the point of negligible resistance to wave motion or flow within 10 millisec af ter a pressure rise of
1. 5 psi above ambient atmospheric pressure sensed in the vicinity of the break.
3. The analytical rnethod is based on a simplified mathe-matical model assuming one-dimensional flow generated
~
by plane pressure waves both in the boiler room proper and in the relief duct.
4. The method of computation is based on the assumption of coolant injection into the boiler room in finite steps af ter discrete time intervals.
5. Simplifying assumptions have been made whereby the pro-cesses of vaporization and thermal equilibrium are con-sidered as instantaneous. Further simplifying assumptions have been made whereby the injected water mixes only with that portion of the air which initially occupied the centre half of the boiler room and this mixing process is instantaneous.
6. The geometry of the model neglects possible effects of interference of boiler room equipment.
7. 2 Review of Analysis and Results
It is apparent that the severity of the emergency pressure conditions experienced in the boiler room, and hence the form of safety precautions required, depend very largely upon the nature of the initial accident. It is, therefore, most important in safety studies of this kind that the probabiiity of various types of failure be carefully considered before the worst conceivable accident is postuiated. In the present case the nature of the accident was studied both theoretically and experim entally by Canadian General Electric Company Limited and Atomic Energy of Canada Limited and the coolant inflow rate used in this analysis was felt to be representative of the most severe emergency possible in the NPD-2 boiler room.
The numerical analysis reported here indicates that in an isolated, sealed boiler room the overpressure would reach a value of about 45 psig within approximately 3. 1 sec following the initial accident. However, provision of the 50 ft relief duct and diaphragm would reduce the peak overpressure to 4.4 psig. This peak overpressure will be of
short duration as shown in Fig. 13. The adequacy of the boiler room struc-ture is thus dependent up on its elastic and inertial properties under the influence of a transient pressure loading.
The numerical results indic~te that for the room geometry and rate of energy injection considered here there are no significant pressure gradients across the room. This result suggests that for low injection rates similar problems could be simplified by treating the boiler room as a reservoir at uniform pressure.
Lack of adequate data, particularly with respect to vapori-zation rates and properties of heavy-water (D 2Q in the gaseous phase, introduced some degree of uncertainty into the numerical calculations.
However, the assumptions made at these points in the analysis are felt to be of such a nature as to produce a somewhat pessimistic estimate of boiler room overpressure. In the event that further data were available
it does not appear that the consequent revisions would produce significant changes from the results established here.
The general cornplexity of this problem and the number of variables involved suggest that the only path to more precise determina-tion of emergency condidetermina-tions is provided by experimental study. Such experiments are of considerable value both as a verification of a given analysis and as a means of establishing design criteria for future reactor plants. In the case of NPD-2, considerable testing with small scale
m odels has been co~ducted by Atomic Energy of Canada Lim ited in Toronto and the results lend some support to the present analysis since they have yielded peak overpressures of the same magnitude a's those calculated in th is report.
The provision of a suitable pressure relief diaphragm can be seen to be a very im portant part of the overpressure safety program.
The diaphragm must be a reliahle _ seal at low overpressures and must
be quickly shattered when the critical room pressure is reached. The fragments of the shattered diaphragm must not inhibit the wave formation
or flow development in the duet. Previous experience at the Institute of
Aerophysics suggests that glass would be a very suitable diaphragm material for the low overpressures considered here. A study of the shattering times and the shattering characteristics of glass diaphragtns suitable for the NPD-2 relief duet wiU be the subject of a separate investigation.
The various aspects of the problem studied here have suggested several avenues for further study and research that should be of considerable interest both from academie and design points of view. Among these is the large-diaphragm problem mentioned above and the
determination of the properties of heavy-water vapour. The investigation
• 1. Anon 2. H. L. Brode 3. 1. 1. Glass 4. D. W. Boyer 5. G. A. Bird 6. 1. 1. Glass W. Martin G. N. Patterson 7. I. 1. Glass J. G. Hall 8. 1. J. Billington I. 1. Glass 9. G. Rudinger 10. A. H. Shapiro 11. J. Howison REFERENCES
Reactor Safety and Containment. Power Reactor Technology, Vol. 2, No. 3, p. 22,
(1959).
Numerical Solution of Spherical Blast Waves .
J _. Appl. Phys. Vol. 26, No. 6, p. 766 (1955).
Aerodynamics of Blasts. UTIA Review No. 17, Institute of Aerophysics, University of
Toronto (1960).
Spherical Explosions and Implosions.
UTIA Report No. 58, Institute of Aerophysics, University of Toronto (1959).
The Decay of Plane, Cylindrical and Spherical Shock Waves. Report No. HSA 10, Weapons Research Establishment, Salisbury South Australia (1957).
A Theoretical and Experimental Study of the Shock Tube. UTIA Report No. 2, Institute of Aerophysics, University of Toronto (1953). Handbook of Supersonic Aerodynamics,
Section 18, Shock Tubes.Navord Report 1488 (Vol. 6), U. S. Government Printing Office Washington (1959).
On the One-Dimensional Refraction of a Rarefraction Wave at a Contact Surface. UTIA Report No. 31, Institute of Aero-physics, University of Toronto (1955).
Wave Diagrams for Unsteady Flows in Ducts. Van Nostrand (1955).
The Dynamics and Thermodynamics of Compressible Fluid Flow. Ronaid Press (1953).
Properties of Heavy Water. Appendix E, Report NEI-77, Atomie Energy of Canada Ltd.
12. H. H. Sogin 13. C. K. Rush D. Quan 14. D. Fraser K. C. Pettit E. H. Bowler 15. P. Griffith J. A. Clark W. M. Rohsenow 16. P. Dergarabedian 17. L. J. Briggs 18. W. H. McAdams 19. I. J. Bi11ington 20. G. F.C. Rogers Y. R. Mayhew
A Design Manual for Ther.mal Anti-Icing Syste.ms. WADC Technical Report 54-313
(1954)
Aircraft Gas Turbine Ice Prevention.
Preprint No. 7/10, Canadian Aeronautical
Institute, Annual General Meeting, May 1957. Criteria for the Design, Assessrnent and Control of Icing Protection Systems. Preprint No. 369, Institute of the Aero-nautical Sciences, 20th Annual Meeting
(1952).
Void Volumes in Super-cooled Boiling
Syste.ms. ASME Paper No. 58-HT-19,
ASME-AICHE Joint Heat Conference,
Chicago, 111. August 18-21, 1958.
The Rate of Growth of Vapour Bubbles in
Superheated Water. J. Appl. Mech. 20
p. 537 (1953).
Limiting Negative Pressure of Water. J. Appl. Phys. 21, p. 721 (1950).
Heàf Trans.mission. McGraw-Hi11, NewYork,
(1942).
An Experimental Study of the One-Dimensional Refraction of a Rarefaction Wave at a
Contact Surface. J. Aeronaut. Sci., 23, 11, p. 997 (1955).
Engineering Thermodynamics, Work, and
ApPENDIX A
Properties of. Heavy- Water (D2.Q
Very little information on the properties of heavy-water vapour is availab'le in the literature. The most useful information that could be found was contained in Ref. 11, which provides a table of satura-tion properties of heavy-water in the liquid and vapour phases. The internal energy, a property required for the present analysis, was not specifically listed in this table and it was, therefore, necessary to calcu-late the vapour phase internal energy from the following equation.
(1)
where hg is the vapour phase entha~py, Pg and V gare the corresponding pressure and specific volume of the vapour and
a.
185 is a numerical .constant required for compatibility of the dimensions used in this analysis. Internal energies were calculated as a function of temperature using Equa-tion (1). The liquid phase internal energy was found to vary by a negligible amount from the liquid phase enthalpy, and therefore it could be read
directly from the tables of Ref. 11. Other quantities required for the mixing and thermal equilibrium calculations could all be obtained directly, or by interpolation, from Ref. 11.
In the step-by-step numerical calculations, it was
necessary to have some additional information not contained in Ref. 11. This included the specific heats Cv and Cp, the gas constant Rand the specific heat ratio ~. These four quantities are mutually dependent so that determination of any two of them automatically provides the values of the remaining two. The value of R could be obtained without
difficulty, since this value depends only upon the molecular weight of
the gas and this is weIl known. Accurate determination of any of the other three quantities, however appeared much less hopeful. The specific heat at con!3tant pressure could be obtained from steam tables if data were available for superheated D20 steam, but no such data could be found.
It was finally decided to assume that the specific heat ratio, ~ , for D20 was the same as that for H20 . . {f depends mainly upon the distribution
. J
of energy between the varlOUS degrees of freedorn of the molecule, and thus upon the molecular structure. Since the substitution of heavy
hydrogen for ordinary hydrogen in the molecules should not rnaterially effect the molecular structure, this appears to be a reasonable assumption. The specific heat ratio was, therefore, assumed to be 1. 30 (see Ref. 20, for example). In summary the constants used for heavy-water may be listed as follows:
R = 0.099 BTU /lh oF
Cp = 0.429 BTU /lh oF
(2)
Cv = 0.330 BTU /lh oF
..
'APPENDIX B
Equations of Mixing and Equilibrium Complete Mixing
Consider a volume V (see Fig. 16) initially containing Wa Ibs of air at 14.7 psia and 1000F.
Suppose a volume Vw of water at 1030 psia and 5300F is injected into the room. Assume that no heat or
mass transfer occurs through the boundaries of volume V. Then, since there is no mechanical work done during the subsequent evaporation, and neglecting the kinetic energy of the injected fluid it follows from the First Law of Thermodynamics that the total internal energy of the system must remain unchanged as a result of the phase change and mixing
(Ref. 20). Thus, if the final equilibrium temperature in the volume is T, the energy balance is:
WaC va (100-T)
+
Wf (Ef530 - EfT)+
Wg (Ef530 ~ EgT)=
0 (3) The equilibrium conditions may be determined in the following manner. For a given temperature, T, the values of EfT, EgT and V gT can be determined from Ref. 11. Then, neglecting volum e V w relative to volume V, the weight of saturated steam present at th at temperature is:(4) where V gT is the specific volume of the gas in ft3/ lb . Now all quantities in Eq. (3) are known except Wf, so that Wf can be determined. Then the total water injected is:
(5) Then, on the assumption of instantaneous mixing in the volume V, the time, t, at which these conditions occurred can be calcu-lated from the known rate of water injection, 8050 Ibs / sec:
t
=
sec (6)(80, 000) (8050)
The complete mixing curves of Figs. 2, 3 and 4 were calculated in this manner, taking V
=
80,000 ft3 and the flow rate as 8050 lb/sec.Zero Mixing
The zero mixing case must be calculated somewhat
differently. It is assumed that the injected water comes to phase equili-brium with itself, at the same time compressing the air originally in the room into a smaller volume as shown in Fig. 17. It is further assumed th at no mass or heat transfer occurs across the interface separating the two gases.
Suppose the steam to occupy a length 1w and the air to occupy a length la of the boiler room (see Fig. 17). Then the adiabatic pressure rise of the air to a pressure p is given by:
't
~p1a = 14.7 (1a
+
1w) (7) and for equilibrium, p must also be the saturation pressure of the steam. The energy balance equation th en becomes,where Ta and T are the final temperatures of the air and steam-water regions respectively.
For a given value of T the saturation values of p, EfT and V gT can be found from Ref. 11. Knowing p, the values of la and 1w can be obtained from Eq. (7) and the air temperature, Ta' can be found from the adiabatic temperature ratio. Wf can then be obtained from Eq. (8) and Ww and t can be found as before, from Eqs. (5) and (6). The zero mixing curves of Fig. 2, 3 and 4 were obtained in this way.
Properties of Mixtures
The properties of mixtures of steam and air can be expressed as follows (see Ref. 20):
p OE Pa
+
Pg (9)where Pa and Pg are the partial pressures of air and steam respectively and p is the mixture pressure: .
WaCva
+
WgCVg C v=
Wa+
Wg (10) =WaCPa+
WgCPg Cp :, Wa+
Wg (11) R = Cp-
Cv (12) ~ = Cp/Cv (13)'
.
. ,
APPENDIX C Steady Duct Flow
Sinee continued addition of mass and energy in the form of hot water wil! result in eonstantly varying composition, pressure and temperature of the gas in the room, there wil! never be any true
"equilibrium" flow in the duet. Nevertheless, it is of value to consider the boiler room pressures required to drive gas out through the duct at the same rate at which it is being generated.
Pressure Loss Coefficient for Duet
The tota1 pressure loss coefficient for the duet illustrated in Fig. 18 can be divided into an entry 10ss, a frietiona110ss and an exit 10ss and each of these ean be estimated from experimenta1 and empirica1 data (see Ref. 12 for examp1e).
For the duet entry with the 900 corners shown the entry 10ss is approximately: Similar1y So that ~Pin1et/q
=
0.3 .ó.Pf . rlC lon t' / q= O. 07
DPexit/ q=
1. 0 (~P/q)Tota1 ~ 1. 4 (14) (15) (16) (17) Overpressure Ca1eulationsThe dynamic head, q, can be expressed in psia as follows:
2
q = (1. 08 x 10- 4 ) , m (18)
PA
2where m,
f
and A have dimensions 1b/sec, 1b/ft3 and ft2 respective1y. The equation of state re1ates the various properties of the boiler room gas:p = 5. 4fRT (19)
where R is expressed in units BTU /1b oR. The duet tota1 pressure 10ss can be expressed as follows:
p - 14. 7
=
1.4'1
(20) ./Now, combining Eqs. (18), (19) and (2ó)yields:
p - 14, 7
=
0.0008 m 2RT
pA2 (21)
. Solution of Eq. (21), for a given mass flow and temperature, provides the required boiler room pressure p.
Assurning that 35 percent of the injected water is vaporized, the steam generation rate will be about 2800 lb/ sec, and for complete mixing the average gas composition by weight (considering the total
steam formed and the total air initially present) will be 10 parts D20 to
7 parts air. For this mixture R, calculated fr om Eq. (12), will be 0.086
BTU/lb oR. At a mass flow rate of 2800 lb/sec and an average tempera-ture of 1840F (from complete mixing curve of Fig. 4), and assuming zero water entrainment, the required value of p is 16.4 psia.
For 2800 lb/ sec of air at T
=
2440F (average from zeromixing air curve of Fig. 4), Eq. (21) gives p = 16. 3 psia. For 2800 lb/sec of dry steam at T
=
2600F (average from steam zero mixing curve ofFig. 4) Eq. (21) gives p = 17.0 psia. Heavy-Water Entrainment in Duct Flow
In practice it is possible that some of the liquid heavy-water will be carried out through the discharge duct in the form of drop-lets entrained in the flowing airstream mixture. However, in the present case, there is considerable justification for assuming that a large propor-tion of the liquid in the room will form into droplets and fall to the floor. This assumption appears to be verified by experiments conducted by Atomic Energy of Canada Limited on a small scale model. The presence of large masses of metal in the boiler room is also likely to remove some D 20 from the atmosphere by condensation. Finally, the liquid water
which does remain suspended in the air-steam mixture is unlikely to be homogeneously distributed through the flow and will more probably exist in the form of a c10ud of small droplets . Many of these small droplets would be centrifuged out of the flow as the gas was accelerated into the passage.
A survey of meteorological and aircraft anti-icing literature provides some information on maximum water content and droplet size of liquid water in naturally occurring cloud formations . Ref. 13 gives the normal d;roplèt size in such clouds as between 10 and
60 microns in diameter; while Ref. 14 quotes a theoretical maximum liquid water content for clouds at 10 g/m 3 but indicatesthatactual mea-surements have not produced evidence of cloud formations with liquid water contents greater than 4 g/m 3 . In the present case the liquid water content in the boiler room and atrnosphere will increase approximately
