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Calculating the power distribution

The primary goal when we calculate the flux solution is being able to predict the power distribution in the reactor core.

We have to emphasize that solving the eigenvalue problem (1.91) only gives us the shape of the power distribution. Using the numerical methods presented in chapter 2 to solve the eigenvalue problem we obtain a flux solution, $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$, with the property

$$\Vert \hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex} \Vert _{\infty} = 1$$ (1.72)

To determine the true flux, $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}_c$, in the reactor core we have to specify the level of the fluxes. In other words, we have to calculate a multiplicative constant $\phi_{max}$ such that

$$\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}_c = \phi_{max} \hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$$ (1.73)

where $\phi_{max}$ is measured in units of [neutrons/ $({\mbox{s}}\cdot{\mbox{cm}}^2)$]. Theoretically, it would be best to determine the flux level which resulted in a critical reactor $(\lambda_0 = 1)$. This could be accomplished by the coupling of the neutronics with the thermal-hydraulics model of the whole reactor. In practice [14], however, we determine the scaling factor $\phi_{max}$ such that the total generated power corresponds to some user specified value P_thtot^1.26. Before showing how this is done we have to describe the relation between the fluxes and the generated power. The total generated power, P_thtot, can be expressed as follows

$$P_{th_{tot}} = \int\limits_{z_{c_{bot}}}^{z_{c_{top}}} q_f'''(z) dV_f$$ (1.74)

where q_f''' is the volumetric heat generation rate in the fuel in units of $[ \mbox{W}/\mbox{cm}^3 ]$, dV_f is a differential fuel volume and the limits z_cbot and z_ctop refer to the coordinates of the bottom and top of the reactor core respectively. q_c''' may be written in terms of the fluxes as

$$q_f'''(z) = \phi_{max} E_{fiss} \sum\limits_{g'=1}^G \Sigma_f^{g'}(z)\phi^{g'}(z)$$ (1.75)

where E_fiss is the energy, in [J], released per fission. E_fiss for fission of different fissile isotopes is listed, for reference, in Table 1.1. Note that these values are slightly higher compared to values quoted by other authors. This difference results from a definition of E_fiss which includes not only the direct energy release from fission but also the indirect energy releases. These indirect energy releases include contributions from long-lived fission and neutron-capture products. Also note that even though the total generated power is expressed in terms of a differential fuel volume, only a fraction of P_thtot is generated within the fuel. This subject is treated at the end of this section.

We write the differential fuel volume, dV_f, as

dV_f = A_f dz (1.76)

where A_f is the area of the fuel in a cross section of the core perpendicular to the z-axis. Noting A_f is inevitably independent of z and introducing (1.95), (1.94) is given by

$$P_{th_{tot}} = \phi_{max} E_{fiss} A_f \int\limits_{z_{c_{b... ...{top}}} \sum\limits_{g'=1}^G \Sigma_f^{g'}(z) \phi^{g'}(z) dz$$ (1.77)

In a discrete representation of space we have divided the z-axis up in regions as shown in Figure 1.2 (see p. ). Exchanging the order of the integral and the summation we can write the power generation in a region k, P_th,k, as follows

$$P_{th,k} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\script... ... \int\limits_{z_k}^{z_{k+1}} \Sigma_f^{g'}(z) \phi^{g'}(z) dz$$ (1.78)

In analogy with the discretization procedure for the diffusion equation (see section 1.5, p. ) we divide this integral up in two and using the homogenized fission cross sections (see (1.20), p. ) we can write (1.98) as

$$\begin{eqnarray*} P_{th,k} & = & \phi_{max} E_{fiss} A_f \sum\limits_{g'=1}^G ... ...} ( \phi_k^{g'} + \phi_{k+1}^{g'} ) + {\cal O}(h_k^2) \right) \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

We neglect higher order terms and write the 2nd order correct approximation for P_th,k

$$P_{th,k} \approx \phi_{max} E_{fiss} A_f \left( \frac{h_k}{... ...erline{\Sigma}{}_{f,k}^{g'} ( \phi_k^{g'} + \phi_{k+1}^{g'} )$$ (1.79)

The total power generation, P_thtot, may be stated as

$$P_{th_{tot}} = \sum\limits_{k \in I_{core}} P_{th,k}$$ (1.80)

where I_core is the subset of all regions lying in the core. Inserting the approximation (1.100) into (1.101) yields

$$P_{th_{tot}} \approx \phi_{max} \frac{1}{2} E_{fiss} A_f \s... ...erline{\Sigma}{}_{f,k}^{g'} ( \phi_k^{g'} + \phi_{k+1}^{g'} )$$ (1.81)

We are now able to determine multiplicative constant $\phi_{max}$ in terms of P_thtot and other calculated or specified quantities in the following manner

$$\phi_{max} = \frac{2 P_{th_{tot}}}{E_{fiss} A_f \sum\limits... ...rline{\Sigma}{}_{f,k}^{g'} ( \phi_k^{g'} + \phi_{k+1}^{g'} )}$$ (1.82)

Since we now know the true flux in the reactor core we are able to calculate the linear heat generation rate, $q_k' \equiv q'(z_k)$, measured in [W/m] within the core by noting that

$$\begin{eqnarray*} q_k' &=& \frac{2}{h_{k-1} + h_k} \int\limits_{z_{k-\frac{1}{2... ... \sum\limits_{g'=1}^G \overline{\Sigma}{}_f^{g'} \phi^{g'} dz \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

which can be approximated by the second order correct expression

$$q_k' \approx \frac{2}{h_{k-1} + h_k} \phi_{max} E_{fiss} A_... ...{g'} + h_k \overline{\Sigma}{}_{f,k}^{g'} \right) \phi_k^{g'}$$ (1.83)

The linear heat generation rate, q_k', is used extensively in the thermal-hydraulics analysis of the reactor core.

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