Memo #2 The Critical dryout condition1 3806 by xzibit

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									1 Memo #2 The Critical dryout condition 1 3/8/06 The purpose of this memo is to summarize the best available correlation for predicting the thermal power in a BWR that causes dryout somewhere in the core. The correlation proposed by Hench and Gillis2 is an improvement on earlier correlations (e.g., Hench-Levy) because it accounts for conditions along the entire length of a rod instead of only at a single elevation. For a bundle containing N rod rods of diameter D and heated length L, the operating condition can be  expressed in any of three ways: the bundle power Q bundle ; the bundle-average linear heat rate
 (LHR) q 'bundle  Q bundle / N rod L ; or the bundle-average heat flux q"bundle  q 'bundle / D . The bundle power is varied and for each value, the dryout condition for each rod is predicted from the correlation. The bundle power at which the dryout condition is just met at a single rod is the “critical bundle power” or the “critical LHR”. The maximum allowable bundle power during operation is a specified fraction of the critical value.

Critical quality As coolant rises in the flow area around a rod (i.e., the subchannel), it eventually reaches an elevation zB where boiling begins. From here upward, all heat absorbed by the coolant is consumed in converting liquid to vapor, thereby increasing the quality of the mixture. As long as the cladding surfaces contact liquid water, heat transfer is good. At a certain quality, called the “critical quality”, the cladding surfaces lose a steady protective water film. When the liquid film periodically disappears and reappears, the temperature swings by  20oC. The resulting thermal stress variations can cause fatigue failure of the cladding. The condition that causesthis phenomenon is related to the quality of the coolant. The critical quality is a function of many variables, most important of which are: i) the coolant mass flux G in the bundle, which is equal to the bundle mass flow rate divided by the flow area; ii) the axial LHR distribution, which is assumed to be the same for all rods in the bundle; iii) the peaking factors for each rod. Boiling Length In order to explain this concept, we consider a single rod designated as rod p. Let: h(z) = specific enthalpy rise of coolant around the rod from inlet to elevation z.

h ( z)  h in 

f p q 'bundle L  m

 0.5



Y

( Y' )dY'

(1)

Y = z/L = dimensionless elevation hin = enthalpy of the subcooled liquid entering the bundle, kJ/kg fp = LHR peaking factor for rod p q ' bundle  bundle-average linear power, kW/m
1 2

See p. 562 of Todreas & Kazimi J. Hench and J. Gillis, “Correlation of critical heat-flux data for application to BWR conditions”, Final report EPRI project NP-1898 (1981)

2
 m  coolant flow rate in the subchannel, kg/s (Y) = axial LHR shape function (assumed to be the same for all rods in the bundle 3)

The product of the peaking factor and the bundle-average LHR is the rod-average LHR. Each rod in the bundle can have its own peaking factor. The thermodynamic quality at elevation z in a rod is defined by:
x h ( z)  h f hg  hf

(2)

where hg and hf are saturation values at system pressure. The quality of the subcooled water entering the bundle is negative. The elevation zB (or Y B) at which boiling starts on the rod is given by x(z B) = 0. Substituting this condition into Eq (2) gives h(z B) = hf and from Eq (1), zB is determined by:
YB

h f  h in 

f p q 'bundle L  m

 0.5



( Y' )dY'

(3)

The boiling length is defined as:

Y = Y – Y B

(4)

The search for dryout extends from Y = 0 to the top of the rod, Y = ½ - YB. The Hench-Gillis correlation2 The critical quality (i.e., the quality at which dryout occurs) is best correlated with a dimensionless form of the boiling length. The result is:
x crit 
 where G  m/A flow boiling length:

0.50 G 0.43 Z 165  115 G 2.3  Z

(2  J)

(5)

is the mass flux in units of Mlbm/ft2 -hr and Z is the dimensionless
Z  D  Y A flow

(6)

where Aflow is the subchannel flow area in m2 . The first term on the right-hand-side of Eq (5) is based on numerous experiments in which all subchannels (rods) behaved identically, Variations in the average LHR of rods in the bundle is accounted for in the J factor in Eq (5).
3

see memo #1 for typical radial and axial LHR variations

3 Application to BWRs In boiling-water reactors, the condition underlying Eqs (5) and (6), namely the, is not valid. To account for lateral nonuniformity of the subchannel heating this requires (a) a detailed analysis of every rod in the bundle and (b) the multiplicative correction factor (2 – J) in Eq (5). The correction factor J for a particular rod is given by: (7) 0.19 J  F (F  1) 2  K G F is a weighted average of the peaking factors of rod p and the rods immediately surrounding it as well as the position of the rod in the bundle: corner, side and central rods behave differently (Fig. 1).

Fig. 1

Rod configurations in a BWR

Rods i and k are immediate neighbors of rod p and the rods labeled j are the next-nearestneighbors. The weighting factors are different for each type of rod. If p is a corner rod, the weighted-average peaking factor is: Fcorner = 0.78fp + 0.095(fi1 + fi2) + 0.03fj1 for rods along the side of the duct, the factor F in Eq (7) is: Fside = 0.69fp + 0.095(fi1 + fi2) + 0.06fk + 0.03(fj1 + fj2) for rods entirely surrounded by other rods (central rods): Fcentral = 0.63fp + 0.063(fi1 +…fi4) + 0.03(fj1 + ….fj4) The last term in Eq (7) accounts for the proclivity of corner rods to reach dryout before the others. This term is set equal to zero for corner rods and to: 0.07 K  0.05 G  0.25 for side rods. For central rods, K is twice the value given by Eq (9). For a given bundle power, the rods are analyzed individually. The enthalpy is calculated as a function of z from Eq (1) and the corresponding quality from Eq (2). The latter is compared to the critical value obtained from Eqs (5) and (7). When these two are equal, dryout occurs on rod p at elevation z. (8c) (8b) (8a)

(9)

4 Calculation methodology Peaking factors The factor J for each rod is calculated from Eqs (7) – (9) using information on its location in the bundle, its peaking factor and those of its neighbors. Fig. 2 shows the procedure.

Select rod (p) corner
Input fp, fi1, fi2,fj1 F: Eq (8a) K: 0

side
Input fp, fk. fi1,fi2,fj1,fj2

central
Input fp, fi1...fi4 fj1....fj4 F: Eq (8c) K: 2xEq (10)

F: Eq (8b) K: Eq (10) Jp: Eq (7) more rods? no Exit

yes

\Fig. 2 Hench_Gillis method of calculating peaking factors for rods in BWR bundles Critical LHR An algorithm for determining the critical LHR is shown in Fig. 3. The starting coreaverage LHR is calculated from the reactor thermal power, the number of rods in a bundle, the number of bundles in the core, and the rod length:  Qcore (10) q 'core  N rod  N bundle  L The “hot” bundle is selected by multiplying q 'core by the core radial peaking factor (Memo #1). The average LHR for this bundle is denoted by q 'bundle . Next, a rod in the bundle is selected (second box in Fig. 3) and the elevation at which boiling first starts is computed. Then a loop increments the elevation. The critical quality is calculated from the correlation and the actual quality from the enthalpy input up to that elevation. The two qualities are then compared. If the critical value has not been exceeded anywhere on the rod, the next rod is chosen. If the critical quality is reached before the end of the rod, the computation restarts at a reduced bundle-average LHR and continues until the critical quality is not attained anywhere in the bundle.

5

The best way to understand this process is to consider an oversimplified case of a bundle consisting of 3 rods subjected to three LHRs. The sequence is illustrated in Fig. 4. The initial bundle-average LHR, q ' A , causes dryout in rod #1 with the lowest peaking factor. For the next lower LHR, q 'B , the critical quality is not exceeded anywhere on rods #1 and #2, but is reached on rod #3. The final LHR, q 'C , does not cause dryout anywhere along any of the three rods. The critical LHR is between q 'C and q 'B .

Fig. 4 Critical quality variation with boiling length for three bundle-average LHRs

6 Sample calculation The following is a calculation of the critical quality and actual quality as functions of elevation for one of the 81 rods in a 9x9 bundle. The rod peaking factors decrease by 20% in a row-by-row fashion from one side to the opposite side, with no transverse variation. This is shown schematically in Fig. 5.

Fig 5 Peaking factors in a 9x9 BWR bundle Not all 81 rods need be analyzed. In row 1, there are two identical corner rods (1X ) and seven identical side rods (1S). The same applies to row 9. In the intervening rows, there are two identical side rods (2S ….8S) and seven central rods. Therefore, each row requires two analyses, so for 9 rows, there are 18 calculations needed. In the following example, only corner rod 1X is analyzed. For the reference BWR in slides 11 – 13 of lecture 2, the following conditions hold: System pressure = 7.2 MPa (saturated at 288 oC). The pertinent enthalpies are (in kJ/kg): Inlet at 279oC, hin = 1231 At saturation: hg = 2771; hf = 1279 The axial LHR shape factor is: where Y = z/L.

(Y)  1.5  2.53 Y

1.2

(11)

In order to use Eqs (1) and (3), integrals of the shape function are needed. Below the midplane (- ½ < Y < 0), integrals from – ½ (the bottom of the fuel stack) to a height Y (negative) are the same as integrals from Y above the midplane to the top of the stack(Y = ½). For Y < 0:

0.5



Y

0.5

(Y' )dY' 


Y

0.5

(Y' )dY' 


Y

(1.5  2.53Y'1.2 )dY'  0.5  1.5 Y  1.15 Y

2.2

(12a)

7

for elevations above the midplane (Y > 0), the appropriate integral is:

0.5



Y

(Y' )dY' 

 0. 5



0

(Y' )dY' 


0

Y

(Y ' )dY'  0.5  1.5Y  1.15 Y 2.2

(12b)

The core thermal power is 3290 MW, from 764 bundles each containing 81 fuel rods with a fuel stack height of 3.71 m. The core-average linear power is:  Q 3290  103 kW q 'core    14.3 N rod N bundle  L 81 764  3.71 m The peaking factor for the central (“hot”) bundle is 1.3 (see Memo #1), so that this is the bundle to be analyzed. The bundle-average linear power is:
q ' bundle  1.3  q 'core  1.3  14 .3  18 .6 kW/m

The coolant mass flux is G = 1370 kg/m2 -s (slide 13, lecture #2), or 1.0 Mlbm/ft2 -hr. The rod OD is 9.5 mm and the pitch is 12.6 mm, giving a subchannel flow area of:

A flow  (12.6 2    9.52 )  106  8.8  105 m 2 4
from these values, the flow rate per subchannel is:  m  G  A flow  1370  8.8 105  0.121 kg / s Dryout analysis of Rod 1X The first step is determination of the boiling length (zB, or in dimensionless terms, YB). Equation (3) becomes 0.9  18.6  3.71 2.2 1279  1231  0.5  1.5 YB  1.15 YB 0.121





Solving this equation for Y B (numerically) gives Y B = -0.34. Next, the J factor in Eq (5) is determined. Following the notation in Fig. 5, the peaking factors of the rod group are: f1X =0.9 From Eq (8a), f1X1 = 0.925 f1X2 = 0.9 f1Xj = 0.925

F = 0.78x0.9 + 0.095(0.9 + 0.925) + 0.03x0.925 = 0.903

The J factor is given by Eq (7):

J  0.903 
With G = 1.0 in Eq (5):

0.19 (0.903  1) 2  0.901 1.0

8

x crit 
and from Eq (6):
Z 

0.55  Z 280  Z
(Y  YB )  1260 (Y  0.34 )

(13)

  9.5  10 3  3.71 8.8  10  5

(14)

Equations (13) and (14) fix the variation of xcrit with Y > -0.34. The final step is determination of the actual equilibrium quality as a function of elevation. The former is obtained from Eq (2):

x

h (Y)  1279 h (Y)   0.86 2771  1279 1492

(15)

For Y < 0, h(Y) is obtained from Eq (1):

h (Y)  1231 
and for Y > 0:

0.9  18.6  3.71 2.2 0.5  1.5 Y  1.15 Y 0.121





(16a)

h (Y)  1231  513  0.5  1.5 Y  1.15 Y



2.2



(16b)

Equations (13)/(14) and (15)/(16) are plotted for rod 1X in Fig. 6 and the quality ratio in Fig. 7. Either plot shows that dryout does not occur on this rod. However, the absence of dryout does not mean that this is an operable rod. Figure. 3 shows that the outlet quality (x = 0.31) is well above the design value, which is usually in the range 0.1 – 0.2. Reduction of core power would rectify this difficulty, but the margin to dryout would increase well above the outlet value of 1.4 from Fig. 7. This means that based on the analysis of this rod, dryout is not a limiting constraint. However, to verify this inference, the rods with the largest peaking factors (rods 9X and 9S in Fig. 5) would require analysis.

Fig. 6 Critical and actual steam qualities in rod 1X

9

Fig. 7 Hench-Gillis plot for rod 1X


								
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