Dane Sabo
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71 lines
3.0 KiB
TeX
71 lines
3.0 KiB
TeX
\section*{Problem 8}
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\subsection*{Part A}
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The one delayed-group model assumes all six precursor groups can be lumped into
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a single effective group with average parameters. This introduces errors:
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\begin{itemize}
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\item Each group has different decay constants (\(\lambda_i\)) ranging from
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0.012 to 3.01 s\(^{-1}\), corresponding to half-lives from 0.2 to 80 seconds
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\item The one-group model cannot capture the multi-timescale behavior - early
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time dynamics are dominated by fast-decaying groups, late time by slow groups
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\item Effective parameters (\(\beta_{eff}\), \(\lambda_{eff}\)) are only
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approximate averages that work reasonably for long-term behavior but miss
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short-term details
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\end{itemize}
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This weakness matters most for short transients where individual group dynamics
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are important, less so for long-term steady-state calculations.
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\subsection*{Part B}
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For six delayed groups with prompt jump approximation:
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Power is given by:
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\[N(t) = \frac{\Lambda}{\beta - \rho} \sum_{i=1}^{6} \lambda_i C_i(t)\]
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Precursor concentrations evolve as:
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\[\frac{dC_i(t)}{dt} = \frac{\beta_i}{\Lambda}N(t) - \lambda_i C_i(t), \quad i = 1,\ldots,6\]
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In matrix form, define the state vector:
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\[\mathbf{C}(t) = \begin{bmatrix} C_1(t) \\ C_2(t) \\ C_3(t) \\ C_4(t) \\ C_5(t) \\ C_6(t) \end{bmatrix}\]
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The precursor equation becomes:
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\[\frac{d\mathbf{C}}{dt} = \mathbf{A} \mathbf{C}(t)\]
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where the matrix \(\mathbf{A}\) is:
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{\tiny
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\[\mathbf{A} = \frac{1}{\beta - \rho} \begin{bmatrix}
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\beta_1 \lambda_1 - \lambda_1(\beta - \rho) & \beta_1 \lambda_2 & \beta_1 \lambda_3 & \beta_1 \lambda_4 & \beta_1 \lambda_5 & \beta_1 \lambda_6 \\
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\beta_2 \lambda_1 & \beta_2 \lambda_2 - \lambda_2(\beta - \rho) & \beta_2 \lambda_3 & \beta_2 \lambda_4 & \beta_2 \lambda_5 & \beta_2 \lambda_6 \\
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\beta_3 \lambda_1 & \beta_3 \lambda_2 & \beta_3 \lambda_3 - \lambda_3(\beta - \rho) & \beta_3 \lambda_4 & \beta_3 \lambda_5 & \beta_3 \lambda_6 \\
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\beta_4 \lambda_1 & \beta_4 \lambda_2 & \beta_4 \lambda_3 & \beta_4 \lambda_4 - \lambda_4(\beta - \rho) & \beta_4 \lambda_5 & \beta_4 \lambda_6 \\
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\beta_5 \lambda_1 & \beta_5 \lambda_2 & \beta_5 \lambda_3 & \beta_5 \lambda_4 & \beta_5 \lambda_5 - \lambda_5(\beta - \rho) & \beta_5 \lambda_6 \\
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\beta_6 \lambda_1 & \beta_6 \lambda_2 & \beta_6 \lambda_3 & \beta_6 \lambda_4 & \beta_6 \lambda_5 & \beta_6 \lambda_6 - \lambda_6(\beta - \rho)
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\end{bmatrix}\]
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}
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\subsection*{Part C}
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The prompt jump approximation error is likely smaller than the one-group error.
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The prompt jump assumes prompt neutrons equilibrate instantly (valid when
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\(\Lambda \ll\) timescales of interest). For a 50 \(\mu\)s generation time and
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transients on the scale of seconds, this is excellent.
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The one-group approximation loses the multi-timescale structure of the six
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groups, which significantly affects transient shape, especially in the first
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10-20 seconds where fast groups dominate.
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For this problem (low reactivity, second-scale transient), prompt jump
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introduces \(<1\%\) error while one-group can introduce 10-20\% errors in peak
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timing and shape.
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