Calculation of the Accident Pressure in a Containment

APPENDIX

CALCULATION OF THE ACCIDENT PRESSURE IN A CONTAINMENT

2

A2.1 INTRODUCTION An initial release of a steam
water mixture with a high internal energy into the containment takes place in many water reactor accident scenarios. Typically, it is the water of the primary cooling system which causes an initial overpressure and a subsequent pressure transient in the containment itself. The following paragraphs describe some simple methods, with essential data, for calculating the pressure with time in these two phases. A note concerning measurement units used in these calculations has to be added. Owing to the long history of the first creation of the related computer program and of its subsequent improvements and tests, the measurement units do not all belong to the Standard International (SI) system. They have been left as they were, in order not to lose the benefit, in terms of reliability, of the long testing of the program. The strongest discrepancy from the SI units is that large calories (Cal) are used instead of Joules and bars or kg/cm2 instead of Pascals.

A2.2 INITIAL OVERPRESSURE The initial pressurization of the containment is a constant volume phenomenon (the containment volume) and, therefore, in order to calculate the final state parameters (e.g., the pressure), it is necessary to equate the initial and final internal energies of the involved fluids. Here it is assumed that the initial pressurization of the containment is relatively fast, for example, corresponding to the break of an intermediate or large recirculation pipe (an intermediate or large LOCA). Therefore the heat exchanged with objects internal to the containment and between the inside and outside of it can be considered negligible. The initial and final energies of the fluids concerned can be calculated by the following considerations and formulae. Total internal energy 5 Air energy 1 Waterðliquid and steamÞenergy

(A2.1)

Air internal energy; Ua 5 Ma Cv t Cal=kg

(A2.2)

where Cv is the specific heat at constant volume of air (0.172 Cal/kg in normal conditions), Ma is the weight of air in the containment (kg) and t is the temperature in  C.

343

344

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

Specific internal energy of the water
steam mixture; UH2 O 5 MH2 O ðHH2 O 2 JpvÞ Cal=kg

(A2.3)

where MH2 O is the weight of water
steam (kg), HH2 O is the specific enthalpy of water (a function of the mixture quality and of the pressure) (Cal/kg), J is the inverse of the mechanical equivalent of the calorie [J 5 1/(427 kg m/Cal)], p is pressure (kg/m2), and v is the specific volume (m3/kg). The quality, X, of the mixture, before and after the pressurization of the containment, can be calculated from the specific volumes of the water and steam which are known. The weight of water and steam is equal to the released amount (e.g., that of the primary cooling water), while the initial volume is that of the primary system and the final volume is that of the containment. X5

v 2 v1 vfg

(A2.4)

where v1 and vfg are the specific volume of liquid water and the difference between water vapor specific volume and liquid water volume, respectively, and can be obtained from steam diagrams and tables as well as from the approximate formulae (A2.5) and (A2.6) (CNEN, 1976). 

 9:165659e 2 4 3 p3 2 4:159937e 2 1 3 p2 2 ð35:05628 3 pÞ 2 120:077 v1 5 p3 2 251:462p2 2 31207:36p 2 117706:3

and


vfg 5

  22:309098e
3 3 p4 1 4:162979p3 2 857:4263p2 2 14867:06p 2 3998:127  4  p 2 381:89p3 2 7810:05p2 2 3776:419p 1 529:4787

(A2.5)

(A2.6)

were p is pressure (kg/cm2). The specific enthalpy HH2 O is given by Eq. (A2.7). HH2 O 5 Hf 1 XHfg

(A2.7)

where the enthalpies can also be calculated by the approximate formulae A2.8 and A2.9. 

 964:3845p3 1 188946:5p2 1 2470981p 2 1649689 Hf 5 3 p 1 665:0797p2 1 16075:48p 1 26716:57 
 231973:9p3 2 5:284174e7 3 p2 2ð1:191874e9 3 pÞ 2 1:575882e9  Hfg 5  4 p 1 82:67094p3 2 126285:4p2 12315288p 1 2785184

(A2.8)

(A2.9)

The initial values of the internal energies can be calculated directly, while the final ones must be obtained by a trial and error procedure, usually drawing a graph (e.g., in Microsoft Excel). It is possible to start with a tentative tfinal value from where the partial pressure of air is obtained (by the perfect gas law and the initial values) as well as the partial pressure of steam by diagrams,

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

345

200

1 0.8

150

350

0.6

300

100

0.4 250 (kcal kg)

T (°C)

V/P (m3/kg)

400 (kcal kg)

50

0.2

0

0 0

1

2

3

4 5 Pr (kg/cm2)

6

7

8

FIGURE A2.1 Loss of coolant accident pressure in a containment.

tables, or approximate relationships like that of Eq. (A2.10) which is very good between 99 C and 374 C, and discrete above 65 C. 
 9 8
2:284709 3 1026 t3  = 3 1029 t4 1 < 24:241304

2 2:952689 3 1024 t2 1 2:164816 3 1022 t : ; 2 5:712048 3 1021 
 p5 8
9 3 10211 t4 2 3:21231 3 1028 t3 = < 2:066907
 1
2:049397 3 1025 t2 : ; 2 6:895268 3 1023 t 1 1

(A2.10)

The final accident pressure can also be calculated by specific diagrams, such as the one shown in Fig. A2.1, where Pr is the relative accident pressure in the containment (kg/cm2), T is the corresponding temperature ( C) and V/P is the ratio between containment volume and weight of water released (m3/kg). The four curves of the final pressure refer to various values of the specific internal energy of the released liquid. Example: The containment has a free volume of 60,000/m3, into which 250 t of primary water are released, with an average temperature of 300 C. Initially the pressure in the containment is equal to 1 bar. Therefore V/P 5 0.24 m3/kg. The specific enthalpy of the liquid water at 300 C is equal to about 314 Cal/kg (practically coincident with the specific internal energy). Entering these values into the graph, the relative accident pressure equals about 2.7 kg/cm2 and the final containment temperature is about 125 C.

A2.3 CONTAINMENT PRESSURE VERSUS TIME The following describes a simple spreadsheet which can be useful for rough evaluations. Where the assumptions on which it is based do not match those of interest (e.g., an absence of spray systems in the containment) the program can be easily modified.

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APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

A2.3.1 INTRODUCTORY REMARKS During the design of the pressure containment building of a water reactor, the calculation of the transient pressure within it as a consequence of a LOCA is very important. In the first place, the knowledge of the pressure history in the containment, in times subsequent to the rupture, is necessary for the determination of the maximum internal pressure after the accident, which in some cases can be higher than the first initial pressure peak occurring shortly after the break. This, in general, occurs when, for the constructive characteristics of the containment, the dispersion of heat toward the outside is limited. Representative examples of this situation are those containers where an internal liner in reinforced concrete or an external biological shield of the same material which encloses totally or partially the metal container is present (e.g., the Indian Point, Elk River, Connecticut Yankee, Trino Vercellese, and similar plants). In such cases, and in the absence of specific pressure abatement systems, such as cold water spray systems inside the containment, in addition to the first pressure peak in the instants immediately following the rupture, a second pressure peak can occur, higher than the first one, due to the release within the containment of the decay heat of the reactor core and to other possible phenomena, even in the realm of the design basis accidents. The second peak will occur at different times after the accident, according to the particular thermal characteristics of the system. In the second place, the knowledge of the pressure history in the containment is necessary for the evaluation of the release outside it of radioactive substances from the core through the inevitable leaks of the structure. The amount of this release depends, in fact, on the internal pressure.

A2.3.2 CALCULATION METHOD The step-by-step procedure described here is for use on a Microsoft Excel, or similar, spreadsheet. For the generic time interval the amounts of heat exchanged with the containment internal atmosphere on the basis of the conditions existing at the start of the same interval are calculated, assuming that in the interval the temperature of the air
water
steam mixture remains constant. Then the balance of these quantities is made and, on the basis of the current heat capacity of the mixture, the variation of its temperature in the time interval and the corresponding final pressure are evaluated. The initial conditions for the subsequent time interval are then calculated. The method has been developed for simple pressure containment such as that shown in Fig. A2.2 where the heat sources and sinks are solar heat absorbed by the containment (Qs), the heat exchanged with concrete (Qc), the heat exchanged with cold metals (Qmf), the heat exchanged with hot metals (Qmc), core decay heat (Qd), and the heat exchanged by the mixture toward the outside through the containment (Qco). With small and obvious modifications this method can also be adapted to rather different containments, such as double containment.

A2.3.3 HEAT EXCHANGED WITH THE OUTSIDE THROUGH THE METAL CONTAINER The container considered is painted on its surfaces and the thermal resistance of the metal is negligible compared with the resistance between the metal and the air
steam mixture on one hand and external air or water of the external spray system on the other. With these assumptions and in the

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

347

Qcs

Qmf Qc

Qco Qmc

Qd

FIGURE A2.2 Containment scheme.

case where the external spray is not operating, the formulae giving the amount of heat exchanged in the generic time interval and the metal temperature at the end of the same interval are given in Eqs. (A2.11)
(A2.15).    h1 Qco 5 Δτ C1 ðTm 2 Te Þ 2 Qcs h1 1 h2 0 1 Qcs h T 1 h T 1 1 m 2 e B
 Sco C C e
C3 Δτ 2 1 1 C2 B @Tco ð0Þ 2 A h1 1 h2 0 B Tco 5 e
C3 Δτ B @Tco ð0Þ 2

h1 Tm 1 h2 Te 1 h1 1 h2

(A2.12)

h1 h2 Sco h1 1 h2

(A2.13)

h1 Sco C3

(A2.14)

Sco ðh1 1 h2 Þ Cc

(A2.15)

C1 5

C2 5 C3 5

1 Qcs Qcs h1 Tm 1 h2 Te 1 Sco C Sco C1 A h1 1 h2

(A2.11)

where C1 (Cal/min  C), C2 (Cal/ C, kg), C3 (Cal/min) are three convenient calculation quantities, Cc is the specific heat of the concrete (Cal/kg,  C), h1 is the transmission coefficient between the

348

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

containment metal and the mixture (resistance of the paint and of the paint
mixture interface) (Cal/m2 min  C), h2 is the transmission coefficient between the containment metal and external air (resistance of the paint and of the paint
air interface) (Cal/m2 min  C), Sco is the containment surface area exposed to external air (m2), Tco is the temperature of the containment metal ( C), Tco(0) is the container metal temperature at the start of the interval of time ( C), Te is the temperature of the external air ( C), Tm is the temperature of the air-stream mixture within the containment ( C) and Δτ is the time interval (min). In the case where an external spray system operates it is possible to neglect the heat capacity of the containment and the heat released to the outside is calculated on the assumption that the spray water is poured from the top of the containment. The heating of the water itself while it flows along the surface is, moreover, taken into account. Thus Eq. (A2.16) follows:

 Qco 5 Gse CðTm 2 Tse Þ 1 2 e2hSco =Gse Δτ

(A2.16)

where c is the total container metal thermal capacity (Cal/ C), C is the specific heat of the external spray water (Cal/kg,  C), Gse is the flow rate of the external spray (kg/min), h is the transmission coefficient between the mixture and the external spray water (Cal/m2, min,  C), and Tse is the temperature of the external spray water ( C). This equation does not include the solar heat because, if the external spray is operated, this contribution has no influence on the transient.

A2.3.4 HEAT RELEASED BY HOT METALS The hot metals are the primary and secondary systems and the related hot auxiliary systems inside the containment. These plant parts are all thermally insulated by a liner. The heat exchange is calculated assimilating these components to a flat layer of thickness equal to the average value of the thicknesses of all the components themselves, perfectly isolated on one side and lined on the other (toward the mixture) by the usual insulating liner. It is admissible to consider the metal as a capacity without resistance and the liner as a resistance without capacity and, with this scheme, the heat amount and the final temperature are given by Eqs. (A2.17) and (A2.18): Qme 5 hme Sme ΔτðTmc 2 Tm Þ Tmc 5 Tmc ð0Þ 2

hmc Smc ðTmc 2 Tm ÞΔτ; Cmc

(A2.17) (A2.18)

where hmc is the transmission coefficient between hot metals and the mixture (resistance of the isolating liner and the liner
mixture interface) (Cal/m2 min  C), Smc is the hot metal surface area (m2), Cmc is the thermal capacity of the hot metals (Cal/ C), Tmc is the temperature of the hot metals ( C), and Tmc(0) is the temperature of the hot metals at the start of the time interval ( C).

A2.3.5 HEAT EXCHANGED WITH COLD METALS The cold metals are those metallic components which during operation are at about the ambient temperature of the containment. They are lined, on exposed surfaces, by a layer of paint. The model

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

349

used here is a simple capacity (metal) and a resistance (paint and interface paint mixture). Thus Eqs. (A2.19) and (A2.20) follow:

hmf Smf  Qmf 5 Cmf ðTmf ð0Þ 2 Tm Þ e Cmf 2 1 Δτ

(A2.19)

hSmf Δτ

Tmf 5 Tm 1 ðTmf ð0Þ 2 Tm Þe Cmf

(A2.20)



where Cmf is the thermal capacity of the cold metals (Cal/ C), hmf is the transmission coefficient between the metal and the mixture (Cal/m2 min  C), Smf is the cold metal surface area (m2), and Tmf is the temperature of the cold metals ( C).

A2.3.6 HEAT EXCHANGED WITH CONCRETE LAYERS The concrete layers have been modeled as plane insulated layers on one side and in contact, on the other side, with the air
steam mixture through a paint layer. The calculation method is that described in Jakob (1962) which uses the finite difference method for the solution of the heat transfer equations. The concrete layers have been grouped in a certain number of groups, each with an average thickness and an exposed surface equal to the sum of the surfaces of the concrete layers included in the group. The heat exchanged with one of the groups of layers during the generic time interval Δτ is given by Eq. (A2.21): Qc 5 hc Sc ðTm 2 Tc ÞΔτ;

(A2.21) 

where hc is the mixture
concrete transmission coefficient (Cal/m min C), Sc is the concrete surface area (m2), Tm is the temperatures of the mixture ( C) at the start of the interval, and Tc is the temperature of the concrete wall ( C) at the start of the interval. The temperatures, T0 , of the layers in which the concrete has been subdivided at the end of the time interval are calculated by Eqs. (A2.22)
(A2.24): 2

T 01 5

2N M 2 2N 2 2 2 Tm 1 T1 1 T2 M M M

(A2.22)

1 M22 1 Ti21 1 Ti 1 Ti11 M M M

(A2.23)

for the first layer, T 01 5

for the layers between the first and the last, and T 0n 5

1 M21 Tn21 1 Tn M M

(A2.24)

for the last layer. M is an auxiliary calculation nondimensional quantity and is given by Eq. (A2.25): M5

ρc Cc 2Δx2 Kc Δτ

(A2.25)

350

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

where ρc is the concrete density (kg/m3), Kc is the concrete heat conduction coefficient (Cal/m, min,  C), and Δx is the thickness of the concrete layer (m). N is another auxiliary calculation nondimensional quantity and is given by Eq. (A2.26): N5

hc Δx Kc

(A2.26)

The necessary condition for the convergence of the calculation is the one given by Eq. (A2.27) M . 2N 1 2

(A2.27)

The choice of the intervals Δx and Δτ has been made in a way which abundantly satisfies Eq. (A2.27), that is, M B 2(2 N 1 2).

A2.3.7 DECAY HEAT As far as the transfer of the decay heat of the core to the water
steam mixture is concerned, here too the assumptions are made (usual in this type of calculation) of the total and instantaneous transfer of the available energy from the core to the mixture. These assumptions are not likely to be complied with in an accident, especially when it is assumed that the core always remains dry (i.e., no spray or flooding system operates). In reality the heat released is only partially transmitted to the mixture and, moreover, this phenomenon occurs after a delay. The assumption of the total transfer to the mixture of the energy released over time by the core is certainly cautious, while the assumption of an absence of delays in the phenomenon may or may not be cautious according to the aspects of the accident considered. In fact, what can be expected by the assumption of immediate transfer of the heat from the core is a pressure transient characterized at the start by higher values but having a shorter duration. Therefore this assumption is very likely to be conservative for the evaluation of the probability that a second pressure peak higher than the first one in the containment occurs. It will not necessarily be so for the evaluation of prolonged releases of activity from the containment in the absence of pressure abatement systems, for example, spray systems. The core decay heat is essentially composed of the decay heat of the fission products, the decay heat of the decay chain of uranium-239 and neptunium-239 produced by neutron capture by uranium-238, the decay heat of other actinides, the control rods and the structural materials, and the heat generated by the residual fissions and by neutron capture by the fission products. The heat of the residual fissions is generally very small 100 s after shutdown and can be completely neglected for the study of medium-term and long-term transients. The decay heat of the structural materials can also be neglected. As far as the control rods are concerned, the heat released by them is not completely negligible, but it can probably be ignored if a safety factor for the total decay heat of at least 1.1 is used. The decay heats of the fission products have been amply studied and the values used here are those suggested by Shure and Dudziak (1961). They are very close to the values of the ANS (1994) and ISO (1992) curves. Some values of the decay heat of the fission products for infinite irradiation according to Shure are shown in Table A2.1.

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

351

Table A2.1 Decay Heat (Shure and Dudziak, 1961) Time After Shutdown (s)

Decay Power as a Percentage of the Thermal Operating Power

102 103 104 105 106 107 108

3.3 1.87 0.97 0.48 0.268 0.121 0.0515

Table A2.2 Decay Heat (ANS, 1994; ISO, 1992) Time After Shutdown, t (s) 1 10 102 103 104 105 106 107 108

ANS 5.1/94 22

6.066 3 10 4.731 3 1022 3.193 3 1022 1.980 3 1022 9.718 3 1023 5.548 3 1023 2.315 3 1023 7.015 3 1024 1.001 3 1024

ISO 10645 6.005 3 1022 4.738 3 1022 3.220 3 1022 2.031 3 1022 1.028 3 1022 5.705 3 1023 2.364 3 1023 7.461 3 1024 9.666 3 1025

For the time interval 150 , t , 4 3 106 s, which generally covers the time span of interest for this transient, Shure suggests the following approximate analytical expression for the decay heat for an infinite irradiation time, valid with a maximum error of 5%: MðN; tÞ 5 13:01t20:2834

(A2.28)

where M is the percentage of operating power and t is time (s). Table A2.2 lists for various times the total decay power as a fraction of operating power (practically infinite time) according to ANS (1994) and ISO (1992). The decay heat for a finite irradiation time t0, at time t after shutdown, is given by Eq. (A2.29): Mðt0 ; tÞ 5 MðN; tÞ 2 MðN; t 1 t0 Þ

(A2.29)

The decay heat of uranium-239 is an important fraction of the total decay heat. It is directly proportional to the initial conversion ratio of the core. For a conversion ratio equal to 0.5, to an approximation of about 15%, the approximate law [Eq. (A2.30)] holds for the total power within the interval 102 , t , 3 3 105 s after shutdown (i.e., from 100 s to about 3.5 days). Pd 5 14:9t20:278

where Pd is the percentage of the operating power and t is time (s).

(A2.30)

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APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

As usual Eq. (A2.30) gives the decay heat for an infinite operation time. The power for a finite operation time is given by Eq. (A2.31): Pd ðt0 ; tÞ 5 Pd ðN; tÞ 2 Pd ðN; t 1 t0 Þ

(A2.31)

The correction Pd(N, t 1 t0) is not negligible in this type of problem. The expression of the decay heat to be inserted in the program is determined case by case by Eq. (A2.30) or by its equivalent for conversion ratios different from 0.5, and by Eq. (A2.31), on the basis of the value of the core operation time t0. It will be opportune to add a safety factor of the order of 1.15
1.20 in order to take into account the mistakes due to approximate expressions of the type of Eq. (A2.30), and the fact that the control rod decay heat has not been taken into account, and so on.

A2.3.8 HEAT REMOVED BY THE SPRAY SYSTEM INTERNAL TO THE CONTAINMENT If the mechanical work for the introduction of water into the containment is neglected (a reasonable assumption), the energy absorbed by the sprayed cold water in the interval Δτ will be that necessary to bring the specific internal energy of the water from the u0 value (Cal/kg) pertinent to cold water to the value u pertinent to the steam
water system present in the containment. Thus Eq. (A2.32) follows: Qsi 5 Gsi ðu 2 u0 ÞΔτ

(A2.32)

where Gsi is the weight flow rate of the internal spray system (kg/min) and Qsi is the heat absorbed by the internal spray (Cal). In order to use Eq. (A2.32) in the program it is necessary to use an analytical expression of the internal energy, u, of the steam
water mixture as a function of the total volume, V (m3), its weight and the partial pressure of the steam or temperature as given in Section A2.2.

A2.3.9 SOLAR HEAT The solar heat contribution is not negligible in this problem and must, therefore, in general, be included in the calculation. The solar heat impinging on a surface outside the terrestrial atmosphere and normal to the direction of the solar beams, at the average distance from the earth, is 20 Cal m2 min (mean solar constant). This value undergoes a maximum variation of 6 3.5% during the year because of the variation of the distance between the Earth and the Sun. In order to evaluate which part of the mean solar constant is absorbed by a surface at ground level it is necessary to evaluate the effects of the inclination of the surface, the latitude, and the Sun’s declination, as well as of the transparency of the atmosphere and the surface reflection. In a conservative evaluation and on the basis of data in MARKS (1958), pp. 12
114, the following multiplication factors can be assumed in order to take into account the aforementioned effects at about 43 degrees of latitude North (readers will insert a latitude of their interest here): For the surface inclination, the latitude, the Sun’s inclination, and the distance of the Sun from the earth:

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

f1 5 0:4 3 0:965 5 0:386

353

(A2.33)

where 0.4 is the surface inclination and latitude nondimensional coefficient and 0.965 is the distance of the Sun from the Earth nondimensional coefficient. For the transparency of the atmosphere: f2 5 0:6

(A2.34)

If the area of the containment surface exposed to the Sun is indicated with Scs (m ) and the conservative assumption of a unit absorption coefficient of the surface is made, it is possible to calculate the heat absorbed in one minute by the containment by Eq. (A2.35): 2

Qcs 5 20 3 f1 3 f2 3 Scs 5 4:63 Scs Cal=min

(A2.35)

A2.3.10 THERMAL BALANCE IN THE INTERVAL Δτ The variation of the internal atmosphere temperature of the containment, ΔTm, in the time interval Δτ, can be evaluated on the basis of the heat quantities exchanged by it [see Eqs. (A2.11), (A2.16), (A2.17), (A2.19), (A2.21), (A2.31), and (A2.32)] by the expression: ΔTm 5

ΣQ Qd 1 Qmc 2 Qco 2 Qmf 2 Qc 2 Qsi 5 W W

(A2.36)

where Qd comes from Eq. (A2.31) and W is the thermal capacity of the gas
vapor mixture inside the containment (air, water, steam) and can be expressed with sufficient approximation by Eq. (A2.37):
 W 5 Ca 1 PH2 O 1 V 0:002 Tm2 2 0:185 Tm 1 6:05 Cal= C

(A2.37) 

where Ca represents the constant volume thermal capacity of the containment air (Cal/ C), which is assumed to be constant during the transient, PH2 O is the total steam
water weight (kg), which is constant only if the internal spray is not operating, and V is the free volume of the containment (m3). The initial conditions for the subsequent interval will then be calculated by Eqs. (A2.12), (A2.18), (A2.20), (A2.22)
(A2.24).

A2.3.11 CONSIDERATIONS ON THE PERFORMANCE OF THE CALCULATION AND ON THE CHOICE OF THE INPUT DATA When performing this type of calculation it must be remembered that the transient is very sensitive to relatively small errors in the heat amounts. This is due to the fact that in Eq. (A2.36) the effective heat quantity ΣQ is small in comparison with most of the other terms and, therefore, a relatively small error in one of them introduces a large error in ΣQ and therefore in ΔT. This is particularly true in those cases where spray systems are not operating and during a long transient, that is, in those cases where the variation of temperature and pressure with time is slow. Table A2.3 lists the values of ΣQ and the values of the various heat quantities as a percentage of ΣQ for values of the time after the occurrence of the accident in a cases of this type. This situation demands an extremely attentive determination of the input data in the calculation (heat exchange coefficients, area of the surfaces exposed to the atmosphere, and so on) to ensure

354

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

Table A2.3 Heat Rates From Various Sources Time After the Accident

ΣQ (Cal/h)

Qd (%)

Qmc (%)

Qmf (%)

Qc (%)

Qco (%)

30 min 2h 10 h 1 (day) 3 (days)

22900 3380 2180 1730 264

2900 1680 1500 1500 6700

31 26 36 37 135

34 13 11 9.5 7.3

2600 1300 990 720 1660

480 300 500 700 5000

that the various heat quantities exchanged by the mixture are evaluated in a conservative way. The following looks at some input data for the calculation whose determination is usually uncertain.

A2.3.11.1 Heat Transfer Coefficients As far as the heat transfer coefficient between the air
steam mixture in condensation and the various surfaces exposed to it is concerned, various theoretical (Jakob, 1962; McAdams, 1985) and experimental (Kolflat and Chittenden, 1957; Goodwin, 1958; Jubb, 1959; Leardini et al., 1961; Leardini and Cadeddu, 1961; Uchida et al., 1964) studies exist. A value normally accepted for operational water reactors (initial peak overpressure of some bars) is of 200 Cal m2 h/ C, at least until the pressure stays at high values, that is, until the percentage of steam in the containment is significant. In the first instants after the accident the heat transfer coefficient is likely to be higher than the indicated value, by as much as a factor of 10, because of the motion of the air and steam mixture due to the efflux from the reactor pressure boundary. The influence of the value given to the heat exchange coefficient between the air
vapor mixture and the walls on the transient is limited by the fact that generally the walls are covered by paint layers whose resistance has, on the basis of the current evaluations, a value of the order of that of the resistance mixture paint. Moreover, this fact demonstrates the importance of carefully evaluating the thermal resistance of the paint layers in addition to that of the transmission coefficient between mixture and paints. As far as the heat transmission coefficient from the containment outside surface to the atmosphere in the absence of external spray is concerned, it is worthwhile remembering that the contribution of radiation is important. The coefficient values usually range from 5 to 20 Cal m2 h/ C according to the building layout adopted. If the external spray is supposed to operate, the transmission coefficient between paint and spray water is of the order of 500
5000 Cal m2 h/ C.

A2.3.11.2 Choice of the Length of the Time Step and of the Thickness of the Concrete Layers, ΔX A series of tests performed in a typical case has shown that a maximum acceptable value of the step Δτ is about one minute. If a step 10 times lower is used no important differences are noted, while with a step 10 times longer the transient is completely wrong. The choice of the thickness, ΔX, of the concrete layers does not appear as critical as that of Δτ. Indeed, once the necessary stability condition [Eq. (A2.27)] is satisfied with a certain margin, for example putting M B 2(2N 1 2), the transient is not very sensitive to the value of ΔX, especially after the first hours from the start of the accident.

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

355

Hence, if only the long-term transient is of interest, the layers in which the concrete is subdivided can also be very thick.

A2.3.12 EXAMPLE CALCULATION This section describes the sample VBA (Visual Basic for Applications) macro PRESCONT for use with a Microsoft Excel 97 spreadsheet which is available on the Mendeley website (file CONTPRESSURE). A simple containment example is examined without internal or external spray. The decay heat corresponds to a conversion factor of 0.5 [Eq. (A2.30)], an operation time of 15 months and a safety factor of 1.2. Three groups of concrete slabs are considered which can be subdivided for the calculation into a maximum number of 630, 160, and 100 layers. The absolute pressure in the containment before the accident is 1 kg/cm2. The input data are C6 (Cal/ C) C10 (Cal/ C) CAP (Cal/ C) CM and CN CMC (Cal/ C) D (min) H1 (Cal/m2 min  C) H2 (Cal/m2 min  C) HC (Cal/m2min C) HMC (Cal/m2 min  C) HMF (Cal/m2 min  C) IC ICM ICN P (MWt) PH2O QS (Cal/min) SC (m2) SCC (m2) SCCM (m2) SCCN (m2) SMC (m2) SMF (m2) T (s) TA ( C) TE ( C) TF (min) TM ( C) TMC ( C) V (m3)

Thermal capacity of cold metals Thermal capacity of metal containment wall Total thermal capacity of air in the containment Nondimensional constants of the concrete [see Eqs. (A2.25) and (A2.26)] Thermal capacity of hot metals Calculation time step Transmission coefficient between mixture and containment metal Transmission coefficient between the containment metal and external air Transmission coefficient between mixture and concrete slabs Transmission coefficient between hot metals and the mixture Transmission coefficient between cold metals and mixture Number of layers in the first group of concrete slabs Number of layers in the second group of concrete slabs Number of layers in the third group of concrete slabs Steady thermal power of reactor (kg): weight of water released by the break Solar thermal power absorbed by the metal surface of the containment Containment surface area exposed internally to the mixture and externally to air Surface area of first group of concrete slabs Surface area of second group of concrete slabs Surface area of third group of concrete slabs Hot metal surface area Cold metal surface area Current time Containment atmosphere temperature before accident Temperature of the external air Time after rupture at which transient calculation is terminated Initial temperature of the containment mixture after efflux Hot metals initial temperature Internal free volume of the containment

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APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

The results of the first calculation step for this example are: The containment pressure, PR (kg/cm2) 5 1.996362 The heat exchanged with the concrete of the first group, QC (Cal) 5 146,666.8 The heat exchanged with the concrete of the second group, QCM (Cal) 5 1,925,000 The heat exchanged with the concrete of the third group, QCN (Cal) 5 1,925,000 The heat exchanged by the mixture toward the outside through the containment, QCO (Cal) 5 1,466,663.8 The decay heat, QD (Cal) 5 982,505.35 The heat exchanged by the mixture with hot metals, QMC (Cal) 5 66,500 The heat exchanged with the cold metals, QMF (Cal) 5 502,030.25 The current time, T (s) 5 1 The temperature of the containment metal, TCO ( C) 5 32.059002 The temperature of the first layer of the first concrete group, TC1 ( C) 5 52.380951 The temperature of the first layer of the second concrete group, TCM(1) ( C) 5 52.380951 The temperature of the first layer of the third concrete group, TCN(1) ( C) 5 52.380951 The temperature of the mixture, TM1 ( C) 5 91.952075 The temperature of the hot metals, TMC ( C) 5 298.1 The temperature of the cold metals, TMF ( C) 5 50.101512 The program listing follows. Sub PRESCONT() Dim TC(630) As Single Dim TCC(630) As Single Dim TCM(160) As Single Dim TCCM(160) As Single Dim TCN(100) As Single Dim TCCN(100) As Single J51 T50 TA 5 Range(“$f$2”) For I 5 1 To IC TC(I) 5 TA Next I For I 5 1 To ICM TCM(I) 5 TA Next I For I 5 1 To ICN TCN(I) 5 TA Next I TE 5 Range(“$h$2”) TCO 5 (TA 1 TE)/2 TMF 5 TA H1 5 Range(“$d$5”)

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

H2 5 Range(“$f$5”) SC 5 Range(“$h$5”) D 5 Range(“$d$4”) C1 5 H1 * H2 * SC * D/(H1 1 H2) C10 5 Range(“$b$10”) C2 5 H1 * C10/(H1 1 H2) C3 5 SC * (H1 1 H2)/C10 H3 5 H1 1 H2 CMC 5 Range(“$h$6”) CM 5 CMC/D CAP 5 Range(“$h$3”) PH2 5 Range(“$f$3”) TM 5 Range(“$d$2”) V 5 Range(“$d$3”) ProgramStart: W 5 CAP 1 PH2 1 (0.0022 * TM ^ 2 - 0.185 * TM 1 6.05) * V QS 5 Range(“$b$4”) QCC 5 C1 * (TM - TE) - H1 * D/H3 * QS QCO 5 QCC 1 C2 * (TCO - (H1 * TM 1 H2 * TE 1 QS/SC)/H3) * (Exp(-C3 * D) - 1) C4 5 Range(“$f$6”) TMC 5 Range(“$b$3”) QMC 5 C4 * (TMC - TM) * D C6 5 Range(“$d$7”) C7 5 Range(“$b$8”) QMF 5 C6 * (TM - TMF) * (1 - Exp(-C7 * D)) C8 5 Range(“$h$8”) QC 5 C8 * (TM - TC(1)) * D C9 5 Range(“$d$9”) QCM 5 C9 * (TM - TCM(1)) * D C11 5 Range(“$h$9”) QCN 5 C11 * (TM - TCN(1)) * D T 5 T 1 D/2 P 5 Range(“$b$2”) QD 5 172 * P * D * (14.9 * (60 * T) ^ ( 2 0.278) - 0.076) TM1 5 TM - (QC 1 QCM 1 QCN 1 QCO 1 QMF - QMC
QD)/W TCCO 5 (TCO - (H1 * TM 1 H2 * TE 1 QS/SC)/H3) * Exp(-C3 * D) 1 (H1 * TM 1 H2 * TE 1 QS/SC)/H3 C5 5 Range(“$b$7”) TMC 5 TMC - C5 * (TMC - TM) * D TMF 5 TM - (TM - TMF) * Exp(-C7 * D)

357

358

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

CN 5 Range(“$f$4”) CM 5 Range(“$H$4”) TCC(1) 5 2 * CN/CM * TM 1 (CM - 2 * CN - 2)/CM * TC(1) 1 2/CM * TC(2) Id 5 Range(“$d$11”) For I 5 2 To Id TCC(I) 5 TC(I - 1)/CM 1 (CM - 2)/CM * TC(I) 1 TC(I 1 1)/CM Next I IC 5 Range(“$f$10”) TCC(IC) 5 TC(Id)/CM 1 (CM - 1)/CM * TC(IC) TCCM(1) 5 2 * CN/CM * TM 1 (CM - 2 * CN - 2)/ CM * TCM(1) 1 2/CM * TCM(2) Idm 5 Range(“$f$11”) For I 5 2 To Idm TCCM(I) 5 TCM(I - 1)/CM 1 (CM - 2)/CM * TCM(I) 1 TCM(I 1 1)/CM Next I ICM 5 Range(“$h$10”) TCCM(ICM) 5 TCM(Idm)/CM 1 (CM - 1)/CM * TCM(ICM) TCCN(1) 5 2 * CN/CM * TM 1 (CM - 2 * CN - 2)/ CM * TCN(1) 1 2/CM * TCN(2) Idn 5 Range(“$b$12”) For I 5 2 To Idn TCCN(I) 5 TCN(I - 1)/CM 1 (CM - 2)/CM * TCN(I) 1 TCN(I 1 1)/CM Next I ICN 5 Range(“$b$11”) TCCN(ICN) 5 TCN(Idn)/CM 1 (CM - 1)/CM * TCN(ICN) For I 5 1 To IC TC(I) 5 TCC(I) Next I For I 5 1 To ICM TCM(I) 5 TCCM(I) Next I For I 5 1 To ICN TCN(I) 5 TCCN(I) Next I TCO 5 TCCO PA 5 (TM1 1 273)/(TA 1 273) PR 5 10 ^ (17.457 - 2795/(TM1 1 273)
1.6799 * Log(TM1 1 273)) 1 PA

APPENDIX 2 CALCULATION OF THE ACCIDENT PRESSURE

359

T 5 T 1 D/2 Range(“b” & (J * 5 1 15)) 5 T Range(“d” & (J * 5 1 15)) 5 TM1 Range(“f” & (J * 5 1 15)) 5 PR Range(“h” & (J * 5 1 15)) 5 QD Range(“b” & (J * 5 1 16)) 5 QCO Range(“d” & (J * 5 1 16)) 5 TCO Range(“f” & (J * 5 1 16)) 5 QMC Range(“h” & (J * 5 1 16)) 5 TMC Range(“b” & (J * 5 1 17)) 5 QMF Range(“d” & (J * 5 1 17)) 5 TMF Range(“f” & (J * 5 1 17)) 5 QC Range(“h” & (J * 5 1 17)) 5 TC(1) Range(“b” & (J * 5 1 18)) 5 QCM Range(“d” & (J * 5 1 18)) 5 TCM(1) Range(“f” & (J * 5 1 18)) 5 QCN Range(“h” & (J * 5 1 18)) 5 TCN(1) TM 5 TM1 J5J 1 1 If T , Range(“$d$10”) Then GoTo ProgramStart: End If End Sub

If the program crashes for specific cases, it is useful to repeat the calculation using a shorter value of the time step, D. This program can be easily adapted to other cases, for example, by the inclusion of an external and internal spray, activated for a preselected time and duration or by the presence of a second containment.

REFERENCES ANS, 1994. Decay Heat Power in Light Water Reactors. ANSI/ANS-5.1-1994, American Nuclear Society, IL, United States. CNEN, 1976. Raccolta di formulazioni delle proprieta` termodinamiche e del trasporto dell’acqua, Comitato Nazionale per l’Energia Nucleare, SATN-1-76, DISP/CENTR, August 1976. Goodwin, W.W., 1958, Pressure Build-up in a Container Following a Loss of Coolant Accident. ANS Meeting, June. ISO, 1992. Nuclear energy
light water reactors: calculation of the decay heat power in nuclear fuels. ISO 10645. Jakob, M., 1962. Heat Transfer. Wiley, New York. Jubb, D.H., 1959. Condensation in a reactor containment vessel. Nucl. Eng. December. Kolflat, A., Chittenden, W.A., 1957. A new approach to the design of containment shells for atomic power plants. 19th Annual American Power Conference.

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Leardini, I., Cadeddu, M., 1961. Caverns as nuclear power reactor containers. Energ. Nucl. February. Leardini, I., Cadeddu, M., Schiavoni, M., 1961. Tests on a cavern for the determination of temperature and pressure transients in a case simulating a major loss of coolant-type reactor accident. Energ. Nucl. February. MARKS, L.S., 1958. Mark’s Mechanical Engineers Handbook. McGraw-Hill. McAdams, W., 1985. Heat Transmission. R.E. Krieger Pub. Co, USA. Shure, K., Dudziak, J., 1961. Calculating Energy Released by Fission Products, WAPD-T-1309. Bettis Atomic Power Laboratory, Pittsburgh, PN, United States. Uchida, H., Oyama, A., Togo, Y., 1964. Evaluation of Post-incident Cooling Systems of Light Water Power Reactors. A/Conf. 28/P/436, Geneva 1964 Conference on Peaceful Uses of Atomic Energy, UNO, Geneva, 1964.