Adjoint flux calculation of natural mode equation by time dependent neutron transport

Figures & data

Figure 1. Schematic view of neutron collision on track.

Table 1. Comparison of k-eigenvalue and natural mode equation.

Forward
Equation	k-eigenvalue Equation (30)(30) $\mathcal{L}{\phi }_k=\frac{1}{k_{eff}}\left({\mathcal{F}}_p+\sum _j{\mathcal{F}}_{d,j}\right){\phi }_k=\frac{1}{k_{eff}}\mathcal{F}{\phi }_k$(30)	Natural mode Equation (7)(7) $\frac{\omega }{V}{\phi }_{\omega }=-\mathcal{L}{\phi }_{\omega }+{\mathcal{F}}_p{\phi }_{\omega }+{\sum }_j\frac{{\lambda }_j}{\omega +{\lambda }_j}{\mathcal{F}}_{d,j}{\phi }_{\omega }$(7)
Eigenvalue	keff: Effective multiplication factor	ω: Reciprocal of reactor period
	The ratio of number of fission neutrons between successive generations	(dP/dt)/P where P is the core power varies exponentially with time.
Eigenfunction	φk	φω
	The fundamental mode becomes dominant as the number of generations of fission chain increases	The fundamental mode becomes dominant as time progresses
Calculation	Power Iteration	Time dependent transport calc. or α-k method [23], etc [21,22].
Measurability	Fundamental mode keff and φk can be measured only in the critical case	Fundamental mode ω and φω are measureable in any state of neutron multiplication
Adjoint
Equation	Equation (31)(31) ${\mathcal{L}}^{†}{{\phi }_k}^{†}=\frac{1}{k_{eff}}{\mathcal{F}}^{†}{{\phi }_k}^{†}$(31) : Adjoint of k-eigenvalue equation	Equation (9)(9) $\frac{\omega }{V}{{\phi }_{\omega }}^{†}=-{\mathcal{L}}^{†}{{\phi }_{\omega }}^{†}+{{\mathcal{F}}_p}^{†}{{\phi }_{\omega }}^{†}+{\sum }_j\frac{{\lambda }_j}{\omega +{\lambda }_j}{{\mathcal{F}}_{d,j}}^{†}{{\phi }_{\omega }}^{†}$(9) : Adjoint of natural mode equation
Eigenfunction	φk^†	φω^†
Physical implication	IFP: Iterated fission probability = Core power sufficient number of generations after a source is placed [9]. The core power is proportional to the amplitude of neutron flux, reaction rates induced by the neutron, the fission neutron emission [11].	I = Amplitude of neutron flux at a time of sufficient length after a source is placed
Calculation	k-power iteration from a specified source. Take the total product of em which is the ratio of the number of fission neutrons between successive cycles. Here, ”cycle” analogous to ‘generation’.	Fixed source calculation of the time dependent neutron transport until time where the neutron population change exponentially with time with a single time constant. Or G-IFP method [16]

Figure 2. Schematic view of MOX-UO₂ fuel assembly used for verification.

Figure 3. Convergence of iterated fission probability for prompt critical case.

Figure 4. Comparison of I_α and I_FP,p for prompt critical case.

Figure 5. Convergence of iterated fission probability for delayed critical case.

Figure 6. Time dependent variation of total neutron flux calculated for 22 source points shown in Figure 2.

Figure 7. Comparison of spatial distribution of flux: φ_k by k-power iteration and φ_ω by TDPI.

Figure 8. Comparison of I by TDPI calculation to iterated fission probability I_FP.

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