# NUKEFACT #6

#### THE REACTOR RATE DIAGRAM

###### last update March 2, 1997

The general equation for source multiplication, as presented in NUKEFACT #3, defined steady state reactor fission power from shutdown to full power. Now we look at transient power and a very unique, and useful, way for its representation. Operationally, the transient state occurs when moving from one steady state condition to another, and involves an ongoing change in neutron population, and power, with time.

Since reactor operation is primarily at and around the condition of criticality, the area of reactor behavior deserving focus is the off-critical condition. For the single delay group model, without an extraneous neutron source, transient power can be represented as:

where:
P(t) = power at time t
P(o) = initial power
rho-dot = reactivity rate, Delta-rho/sec
rho = reactivity
Delta-t = time transpired, seconds
beta = precursor yield fraction
Lambda = effective precursor decay constant, sec-1

On the right hand side of this equation the portion of the exponent in square brackets is the reactor rate, which is a measure of the rapidity and direction of power change with time. In the U.S., PWRs express reactor rate as startup rate, with base-10, while BWRs use reactor period, with base-e.

...................................... STARTUP RATE ................................ REACTOR PERIOD

where:
SUR = startup rate, decades per minute
T = reactor period, seconds

Two properties of reactor rate are important:

1. The algebraic sign of the rate signifies the direction of power change with time. A positive rate is associated with reactor power increase. A negative rate is associated with power decrease.

2. The magnitude of the rate indicates the rapidity of power change. For Startup Rate, the larger the value, the faster the power change with time, whether increasing or decreasing. For Reactor Period, which in actuality is an inverse rate, the smaller the value, the faster the power change with time.

During power change with time, the reactor rate, whether expressed as startup rate or reactor period, is either a stable rate or a transient rate. If the reactor rate is constant with time, as associated with exponential power change, the rate is referred to as a stable reactor rate. If the reactor rate is not constant but is changing with time, as for non-exponential power change, the rate is referred to as a transient reactor rate. For an off-critical reactor rate to be stable, reactivity must be constant with time (rho-dot must be equal to zero). Likewise, for an off-critical reactor rate to be transient, reactivity must be changing with time (rho-dot is unequal to 0).

The Reactor Rate Diagram, as illustrated below, is a graphically display of the reactor rate equation. The vertical axis is the reactor startup rate in decades-per-minute. The startup rate scale is linear. The horizontal axis represents reactivity scaled in linear graduations, from
-40x10-4 to +40x10-4.

..............................................THE REACTOR RATE DIAGRAM................................

The equivalent reactor rate diagram scaled for reactor period displays the same the curves.

Each Diagram consists of three curves, one for the stable rate and two for transient rate (ramp-out and ramp-in). The solid curve represents the stable rate, rho-dot = 0.0 delta-rho/sec. The two dashed curves shown are the transient reactor rates for rho-dot = ± 2x10-4 Delta-rho/sec. The shape of the roughly parallel rate curves is one of sharp upward curvature as reactivity moves from left (negative) to right (positive). In effect, the ramp curves form a band around the stable rate curve. A positive ramp translates the stable rate curve upward and a negative ramp translates the stable rate curve downward.

At rho = 0, the stable startup rate is zero DPM. For this single point, power is constant with time. This condition is criticality.

For a specified reactivity condition, i.e. with rho and rho-dot defined, the Rate Diagram provides the precalculated rate at which reactor power is changing with time.