Subcritical Boron Dilution and 1/M Extrapolation

The count rate in the source range is proportional to the subcritical multiplication M = 1/(1 - k), so 1/M is proportional to 1/N. Normalize to the first point: (1/M)_i / (1/M)_0 = N_0 / N_i. At 640 ppm, 1/M = 1500/1500 = 1.00. At 610 ppm, 1/M = 1500/2000 = 0.75. At 585 ppm, 1/M = 1500/2850 = 0.526. The slope between the first two points is (0.75 - 1.00)/(610 - 640) = 0.00833 per ppm. The slope between the last two points is (0.526 - 0.75)/(585 - 610) = 0.00896 per ppm. Extrapolating from 585 ppm using the last slope gives 0.526/0.00896 = 58.7 ppm, so critical boron is about 585 - 58.7 = 526 ppm. Extrapolating from 610 ppm gives 0.75/0.00833 = 90 ppm, or about 520 ppm. The two estimates bracket roughly 520-526 ppm, so a working estimate is 525 ppm. The uncertainty is at least several ppm because the 1/M plot is slightly curved, the count rates have statistical and spatial-effects uncertainty, and the boron worth is approximate. The extrapolation should not be treated as a precise critical boron value; it is a trend indicator that requires additional small dilution steps and stable count-rate confirmation.

At 540 ppm, the estimated margin to critical is 540 - 525 = 15 ppm. With a boron worth of about -8 pcm/ppm, that is only about 120 pcm, or 0.0012 delta-k/k. On paper this is just above the 100 pcm procedural minimum, but it is not an acceptable operating margin because the critical boron estimate has several ppm of uncertainty. A small error in the extrapolation, boron concentration measurement, or mixing could place the reactor much closer to critical than intended. The 1.5% delta-k/k shutdown margin requirement is a separate, larger requirement and cannot be demonstrated by this dilution alone; it depends on rod worth and the assumed stuck-rod condition. The correct judgement is to stop the planned dilution before 540 ppm and revise the target upward, for example to 560 ppm. That would provide about 560 - 525 = 35 ppm, or roughly 280 pcm, of margin, which is more consistent with the intent of a 100 pcm minimum and preserves allowance for extrapolation error. The crew should also verify boron concentration by independent sample, confirm stable source-range counts after each step, and not rely on a single 1/M extrapolation.

For dilution with pure water, the boron mass balance is C_final = C_initial * V_RCS / (V_RCS + V_add). Rearranging gives V_add = V_RCS * (C_initial/C_final - 1). With V_RCS = 320 m3, C_initial = 585 ppm, and C_final = 560 ppm, V_add = 320 * (585/560 - 1) = 320 * (1.04464 - 1) = 14.3 m3. For comparison, reaching 540 ppm would require 320 * (585/540 - 1) = 26.7 m3. The 560 ppm target is chosen because it leaves roughly 280 pcm of estimated subcritical margin instead of about 120 pcm. That additional margin is prudent given the curvature in the 1/M plot, the approximate boron worth, and the need to stop well before critical rather than approaching it closely. The crew should add the dilution in a controlled batch, mix and sample the RCS, and re-evaluate 1/M before any further dilution. If the measured boron concentration or count rate differs from the prediction, the target should be re-derived rather than forcing the planned endpoint.