# Feedback Delay – Time Constants

## Feedback Delay – Time Constants

In physics and engineering, the time constant is the parameter characterizing the response to a step input of a first-order, linear time-invariant system. The time constant is usually denoted by the Greek letter τ (tau). It is obvious, that thermal time constants, which characterize time required to warm an object by another object, are of importance for reactor stability. In general, the heat transfer from the body to the ambient at a given time is proportional to the temperature difference between the body and the ambient. The time constants that determine the time delays for reactivity feedbacks depend on the specific reactor design. For LWRs the following time constants are usual:

• The time constant for heating fuel is almost zero, therefore the fuel temperature coefficient is effective almost instantaneously.
• The time constant for heat transfer out of a fuel pin varies from a few tenths to a few tens of seconds. The time for heat to be transferred to the moderator is usually measured in seconds (~5s). The presence of the surface film increases the time constant for the fuel element.
• The time constant for equalization of temperatures within the primary loop depends strongly on length of primary piping and the flow velocity. It is usually measured in seconds (<10s)

Knowledge of delays of reactivity feedbacks is very important in the study of reactor transients. A large reactivity insertion will cause a reactor to go on such a short period that the power changes significantly over time spans that are short compared to the number of seconds needed to transfer heat from fuel to coolant. Especially in case of all reactivity initiated accidents (RIA), over short time spans the amount of heat transferred to the coolant is not enough to increase its temperature appreciably. Therefore, the fuel temperature coefficient will be the first and the most important feedback, that will compensate the inserted positive reactivity. The time for heat to be transferred to the moderator is usually measured in seconds, while the fuel temperature coefficient is effective almost instantaneously. Therefore this coefficient is also called the prompt temperature coefficient because it causes an immediate response on changes in fuel temperature.

References:
Nuclear and Reactor Physics:
1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
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7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.
4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

Reactor Physics