University Physics V · topic UPV-16.04
Reactor physics: point kinetics, feedback, and reactor types
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Point kinetics, inhour roots and feedback: predict a finite change
Point kinetics is the approximation that the flux keeps one fixed spatial shape, so a whole core is a scalar amplitude n(t) plus six delayed-precursor concentrations, obeying a linear system whose eigenvalues — the inhour roots — set the reactor period. Use the supplied topic model Y_out = (b-r)/(0.0767*r)+10/r, where Y_out is ONE-GROUP PERIOD (s). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. r is Step reactivity ρ (pcm), measured in pcm. Keep b (Delayed fraction β_eff (pcm)) = 650 pcm fixed. For this exercise use only 29.125 pcm ≤ r ≤ 170.875 pcm on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at r = 60.31 pcm and 80.155 pcm. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δr across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
ONE-GROUP PERIOD against r
Predictions from the supplied Point kinetics, inhour roots and feedback model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): ONE-GROUP PERIOD / s
Horizontal axis (x): r: Step reactivity ρ (pcm) / pcm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| r: Step reactivity ρ (pcm) / pcm | ONE-GROUP PERIOD / s |
|---|---|
| 29.125 | 278.2781138262334 |
| 64.5625 | 118.37866696284665 |
| 100 | 71.80795306388525 |
| 135.4375 | 49.60788501698766 |
| 170.875 | 36.61589201221949 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of r separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At r = 60.31 pcm, substitute into Y_out = (b-r)/(0.0767*r)+10/r: Y₀ = 127.645 s.
- At r = 80.155 pcm, the same fixed-parameter model gives Y₁ = 92.8143 s.
- Signed change: ΔY_out = Y₁ − Y₀ = -34.8306 s; the input change is Δr = 19.845 pcm.
- Average sensitivity is ΔY_out/Δr = -1.75513 s per pcm. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 127.645 s, Y₁ = 92.8143 s; ΔY_out = -34.8306 s; average sensitivity = -1.75513 s per pcm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Point kinetics, inhour roots and feedback: infer an input with uncertainty
Point kinetics is the approximation that the flux keeps one fixed spatial shape, so a whole core is a scalar amplitude n(t) plus six delayed-precursor concentrations, obeying a linear system whose eigenvalues — the inhour roots — set the reactor period. Use the supplied topic model Y_out = (b-r)/(0.0767*r)+10/r, where Y_out is ONE-GROUP PERIOD (s). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. r is Step reactivity ρ (pcm), measured in pcm. Keep b (Delayed fraction β_eff (pcm)) = 650 pcm fixed. For this exercise use only 29.125 pcm ≤ r ≤ 170.875 pcm on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 60.362941 s, with a bounded reading uncertainty ±3.5621 s. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown r by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.014175 pcm, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
ONE-GROUP PERIOD against r
Predictions from the supplied Point kinetics, inhour roots and feedback model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): ONE-GROUP PERIOD / s
Horizontal axis (x): r: Step reactivity ρ (pcm) / pcm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| r: Step reactivity ρ (pcm) / pcm | ONE-GROUP PERIOD / s |
|---|---|
| 29.125 | 278.2781138262334 |
| 64.5625 | 118.37866696284665 |
| 100 | 71.80795306388525 |
| 135.4375 | 49.60788501698766 |
| 170.875 | 36.61589201221949 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(r) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(29.125) = 278.278 s and Y_out(170.875) = 36.6159 s. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives r_line = 29.125 + (60.3629 − 278.278)(170.875 − 29.125)/(36.6159 − 278.278) = 156.946 pcm. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives r = 115.593 pcm; substituting back gives Y_out = 60.362941 s.
- The interpolation error in the input is |r_line − r| = 41.3534 pcm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 56.800841 s and 63.925041 s. Sort the roots even if the response decreases: 110.242 pcm ≤ r ≤ 121.488 pcm.
- The lower and upper input deviations are 5.35001 pcm and 5.89576 pcm. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer r ≈ 115.593 pcm; compatible interval [110.242, 121.488] pcm. Endpoint interpolation gives 156.946 pcm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.