Prompt Jump Approximation

• The initial part of the transient can be more elegantly derived by the use of the so called prompt jump approximation.
• In this model it is assumed that the precursors do not change their concentrations and stay at their steady state values.
• Thus

• The neutron balance equation can thus be written as

• Using the integral factor, we get

• On integration, we get

• Using the Initial condition φ = φ₀ at t =0

• Thus, it may be seen that for k_∞ (1-β) < 1,  the two exponential terms decay out very quickly with a period

• Substituting k_∞ = (1/(1-ρ)), and rearranging, we get

• This is exactly the inverse of  ω₂.
• The prompt Jump can also be obtained by noting that  the solution obtained quickly reaches the value

• Substituting k_∞ = (1/(1-ρ)), and rearranging, we get