THE STABLE REACTOR RATE EQUATIONS
Even though we do not advocate emphasis on equation derivation in Reactor Operator training, we employ some mathematics here in order to demonstrate how the correct equation for reactor rate is obtained. Otherwise, it would just be our word that one particular equation is incorrect vis-a-vis another.
In commercial training of Reactor Operators, the equation for stable reactor rate is sometimes expressed (incorrectly) as:
T = the reactor period, seconds-1
rho = reactivity
beta = the precursor yield fraction
lambda = the single precursor group effective decay constant, seconds-1
l* = l/k
l = the prompt neutron lifetime, seconds
k = the effective multiplication factor
Interestingly, this equation form has a long history of survival. It has been included on the Equation sheet of INPO Test Item Catalogs; it has been erroneously derived in reactor vendor manuals (vintage 1984); and it is currently contained in Instructor Lesson Plans and Student Handouts in many Nuclear Training Centers. Unfortunately, it is just plain wrong.
Equation 29.1 is promoted as representing the stable reactor period over a wide range of reactivity values, from far subcritical to well beyond prompt criticality. For example, this equation is said to define the stable rate for rho = -0.1000 as well as for rho = +0.0200. Not so ... at least for reactivities approaching prompt criticality and beyond ! And, of course, even if it were true, then the rule-of-thumb lambda must be dispensed with in favor of a lambda value appropriate to the extreme reactivity condition. (Note: The Single Group Lambda Effective Diagram is illustrated under Feedback: The Reactor Trainer: Example Screen Displays.) The rule-of-thumb lambdas are valid only for operational reactivities that are slightly off-critical.
The correct relationship between reactor period and reactivity has long been recognized as the classic In-Hour Equation, which is:
betai= the yield fraction for the ith precursor group
lambdai = the decay constant for the ith precursor group, seconds-1
In this form, Equation 29.2 cannot be compared directly to Equation 29.1. However, with some rearranging, such a comparison is possible. First, Equation 29.2 can be simplifed to a single effective precursor group, as follows:
Then by multiplying both sides of the Equation 29.3 by (1 + lambda x T) and solving for the T that appears on the left-hand-side of the resultant equation, the reactor period is:
On comparison of Equation 29.4 with Equation 29.1, it is found that Equation 29.4 contains (l*/T) in the numerator of the first term on the right-hand-side. It is this missing term that makes Equation 29.1 incorrect for reactivities approaching prompt criticality and beyond. Even though l* is extremely small, being of the order of 1x10-4 seconds, which makes l*/T insignificant for normal operational situations, this term becomes very significant for large positive reactivities. Equation 29.4 is the correct relationship for reactor period and reactivity.
APPLICATION of the STABLE RATE EQUATIONS
Equation 29.1: The incorrect equation ... As positive reactivity is increased toward prompt criticality, the first term on the right-hand-side of Equation 29.1, (beta - rho)/(lambda x rho), gradually decreases in magnitude. Once at prompt criticality, with rho = beta, the numerator of the first term on the right-hand-side of this equation, beta - rho, is zero. And, beyond prompt criticality this first term becomes negative so that the equation no longer works. Typical resolution is simply to drop the first term of Equation 29.1, to ignore it, and use the second term, (l*/rho) as representative of reactor period at prompt criticality and beyond. Now I happen to believe that Class Room training places too much emphasis on the condition of prompt criticality to begin with. But to treat prompt criticality in this manner is pure voodoo mathematics, as well as bad example and poor training. Besides it results in a reactor rate that is in error by a huge margin.
Equation 29.4: The correct equation ... As positive reactivity is increased toward prompt criticality, the first term on the right-hand-side of Equation 29.4 gradually decreases in magnitude, but not so much as the incorrect equation because as prompt criticality is approached the (l*/T) term in the numerator becomes significant. Once at prompt criticality, with rho = beta, the numerator of the first term on the right-hand-side of this equation remains positive because of the (l*/T) term. And beyond prompt criticality, the (l*/T) dominates the second term on the right-hand-side of Equation 29.4, (l*/rho). Ah, but there is a catch, you say. We are solving for the reactor period, T, based on a specified value of reactivity, rho, but the unknown, period, appears on both sides of the equation. We must know period in order to determine period. Fortunately, this is not the case. Equation 29.4 is easily solved for T by a legitimate mathematical process called iteration.
THE 1300% ERROR
Figure 29.1 provides a graphical comparison of positive reactor period as determined by Equations 29.1 and 29.4:
The two vertical scales are reactor rate, the left scale being reactor startup rate, extending from -200 DPM to +2000 DPM and the right scale being reactor period, extending from -0.13 seconds to +0.013 seconds. For a given reactivity, the reactor period is first calculated by Equation 29.1 and by Equation 29.4. Reactor period is then converted to startup rate by using the following relation:
As can be seen, both equations give the same rate up to a reactivity of about rho = +0.0045. For larger reactivities, the two curves diverge, with the incorrect Equation 29.1 giving the shorter period. The reactor rate at prompt criticality by Equation 29.1 is +1700 DPM, whereas the actual (correct rate) by Equation 29.4 is +130 DPM. The ERROR is 1300%.
To experience a comparable 1300% error in driving your car at exactly 55mph, a glance at the speedometer would find it to be reading 700 mph. You would immediately be aware that something was radically wrong. Likewise, be aware that something is radically wrong here.
Equation 29.1 is wrong by a wide margin and must be applied piecemeal above prompt criticality to avoid negative reactor rates for large positive reactivities. Equation 29.1 should be banned from all training programs. Having said that, neither is it obvious that the correct expression, Equation 29.4, is needed. In fact, we do not avocate the use of Equation 29.4. Rather, the student and the Reactor Operator need only understand the stable rate equation that applies for all operational transients in the Delayed-Critical region, namely:
In training, too much emphasis is placed on the condition of prompt criticality, and that condition certainly needs no special equation to define the stable rate. Rather, if the operational rate equation, Equation 29.6, is used to generate the Reactor Rate Diagrams as shown in NUKEFACT #6, this is fully adequate for operator understanding of the stable reactor rate. From the Reactor Rate Diagram it is obvious that the ascent of the stable rate curve becomes steeper as positive reactivity increases. The Instructor need only indicate that Equation 29.6 is an approximation, but a very good one for operaional transients, and that if the stable rate curve were extended to prompt criticality , i.e. rho = beta, the true reactor startup rate would be in the range of 130 to 250 DPM, or the true reactor period would be in the range of 0.2 to 0.1 seconds. Inference that the reactor rate at prompt critical is of the order of 1700 DPM, or the period is of the order of 0.01 seconds, is total misinformation.
In 1989, when this 1300% error in Equation 29.1 was pointed out to the Nuclear Regulatory Commission's Avisory Committee on Reactor Safety (ACRS), the response was, " ... periods and startup rates are not really very important. ... You can tell an operator, hey, this isn't exactly right, but for your purposes this is close enough to describe the behavior in this regime or in this region."
In 1996, when the error in Equation 29.1 was pointed out the the Nuclear Regulatory Commission's Operator Licensing Branch, their response was that Equation 29.3 was an "inverted" form or Equation 29.1. As shown above, this is not true. But, in any case, we are left with many training programs unknowingly using a reactor rate equation that is grieviously flawed.