In the burnup equations, the atomic number density N(r, t) and the neutron flux ф(~, t) are generally a function of space and time. Nuclear fuel does not uniformly burn at the location in actual nuclear reactors and neutron flux varies with time (neutron flux should be increased according to depletion of fissile nuclides to maintain a specified reactor power). For simplicity, the space-independent burnup equations are solved assuming a constant neutron flux with time.

In Eq. (1.1) for 235U, both sides are divided by N25 and this equation can be integrated from time 0 to t (separation of variables) to give Eq. (1.15).

N2Kt)=N2K0)e-^ (1.15)

As an example, the hypothetical case is considered for a pure 235U-fueled reactor operated at a constant neutron flux of 2.0 x 1014n/cm2s for 1 year. The depletion rate of 235U during the operation is calculated, using the cross section of Fig. 1.1, in the following.

Next, the burnup equation of 236U [Eq. (1.2)] is considered which can be solved in several ways. Here the term of N26(t) is transposed into the left-hand side and each term is multiplied by the integrating factor exp [сг%6фі] to obtain

JN26

Є °™фі +N2G{t) {of фе °™фі ) =<Tf0iV25 (t) e °™фі ot.

It should be noted that the left-hand side is a time-differentiated form of N26(t)exp [аТфі]. It follows therefore that

— IN26 (t) e =а2ЪфЫ2Ъ (t)e °™фі

dt.

Equation (1.16) is given by integrating both sides of the above expression from time 0 to t and substituting Eq. (1.15).

<r25

(1.16)

CJа (Уа

The burnup equations of 238U and 239Pu [Eqs. (1.5) and (1.14)] are identical to those of 235U and 236U. As a consequence, the solutions are given by the next expressions.

N2&(t)=N2S(0)e~a™*t (1.17)

sy 28

(1.18)

(7a (Ja

Continuing for the burnup equation of 240Pu, the term of N40(t) is transposed into the left-hand side and each term is multiplied by the integrating factor exp [аа°фь] to obtain

— IN40 (t) e =crc490iV49 (t) e а*°фі dt.

Equation (1.19) is obtained by integrating both sides of the above expression from time 0 to t and substituting Eq. (1.18).

Fig. 1.2 Depletion of 235U and buildup of Pu isotopes for a representative LWR fuel composition [4]

(1.19)

241 242

The same method can be applied to Pu, Pu, and the nuclides downstream in the burnup chain. Equations (1.17)—(1.19) can be perceived to have a regularity, and the general solution to them is known as the Bateman solution [3].

Figure 1.2 shows the depletion behavior of 235U and the production behavior of Pu isotopes for a representative LWR fuel. While 235U exponentially decreases, the neutron capture of U produces Pu, which in turn is converted by neutron capture into 240Pu and higher Pu isotopes. It is seen that the buildup of Pu isotopes is approaching an equilibrium state.