... RIS\O¹

RISØ National Laboratory, Denmark.

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... equilibrium²

Vapor is assumed at a saturated state.

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... derivation^1.1

The formal way of deriving the diffusion equation is achieved with aid of the Boltzmann transport equation which describes neutron transport exactly. Making several approximations of the Boltzmann transport equation it is possible to end up with the diffusion equation (which actually is an approximation).

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... equation^1.2

We neglect the presence of delayed neutrons. In fact one could say that we assume the energy spectrum of the delayed neutrons to be that of the promptly emitted neutrons. Since the amount of delayed neutrons is limited to 0.65% we make no serious error by neglecting them.

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...density^1.3

The flux density $\Phi(\hspace{0.2ex}\underline{r}{}\hspace{0.15ex},E)$ is the total number of neutrons per unit volume per unit energy in a neighborhood around r with energy E multiplied with the speed of the neutrons.

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... balance^1.4

The neutron balance may be stated as $\sum\mbox{Losses} = \sum \mbox{Sources}$.

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... (discontinuities)^1.5

In physical terms this statement ensures that all neutrons passing the left side of the interface pass the right hand side of the interface and vice versa.

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... distance^1.6

The extrapolation distance has a mathematical rather than physical description. We observe according to (1.12) that we actually assume a linearly varying flux (by neglecting the curvature of $\phi^g$) in the space governed by $\hspace{0.2ex}\underline{r}{}\hspace{0.15ex} \in [\hspace{0.2ex}\underline{R}{}... ...ex}\underline{R}{}\hspace{0.15ex}+\hspace{0.2ex}\underline{d}{}\hspace{0.15ex}]$ and calculate the necessary flux at $\hspace{0.2ex}\underline{r}{}\hspace{0.15ex}=\hspace{0.2ex}\underline{R}{}\hspace{0.15ex}$ to insure a zero scalar group flux at $\hspace{0.2ex}\underline{R}{}\hspace{0.15ex}+\hspace{0.2ex}\underline{d}{}\hspace{0.15ex}$.

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... core^1.7

The reason for this "coupling" is due to the fact that the energy spectrum of neutrons entering the reflector depend on the core material.

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... current^1.8

The physical importance of the net current is realized by noting that the number of neutrons in energy group g passing through a surface element dS with unit normal vector $\hspace{0.2ex}\underline{n}{}\hspace{0.15ex}$ is

$$\mbox{Number of neutrons in $g$} = \hspace{0.2ex}\underline{j... ....15ex}^g \cdot \hspace{0.2ex}\underline{n}{}\hspace{0.15ex} dS$$

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... paths^1.9

The scattering mean free path is defined by

$$\lambda_s(E) \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scr... ...\hspace{0.15ex},E \rightarrow E') dE'} = \frac{1}{\Sigma_s(E)}$$

and specifies how long a neutron with energy E travels in average before being scattered.

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... leakage^1.10

Cf. [1, page 408]

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... RIS\O^1.11

RISØ National Laboratory, Denmark.

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... densities^1.12

The number density is defined as the number of nuclei per $\mbox{cm}^3$.

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... region^1.13

It does not have to be a genuine mean value, ie a value which is calculated by performing an integration. In practical applications it is customary to define, eg

$$\overline{\Sigma}{}_{s,k}^{g \rightarrow g'} \;\hbox{$=$\kern... ...\triangle$}}\;\Sigma_s^{g \rightarrow g'}(z_k + \frac{h_k}{2})$$

ie use the value in the middle of the region--a reasonable choice for small h_k.

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... region^1.14

If material interfaces were to lie inside a region we had not been able to consider Taylor series to the 5th order since the derivatives from $\frac{d^2\phi^g}{dz^2}$ and up would have been unbounded at the interface point.

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... $\zeta \in [z_k-\frac{h_{k-1}}{2};z_k^-]$^1.15

The superscript - indicates, as before, a left-hand limit.

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... problem^1.16

In fact it is a so-called generalized eigenvalue problem--this is seen more clearly in the matrix formulation of the problem, see section 1.6.1.

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... $\widetilde{\mbox{PTE}}_k$^1.17

The principal truncation error may be evaluated by subtracting the approximation (1.43) from the exact equation (1.40) and neglecting higher order terms.

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... error^1.18

The discretization error is the difference between the exact solution to the continuous space equation and the exact solution to the discrete equation. In other words, the discretization error consists of a in general complex mixture of all the local truncation errors.

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... equations^1.19

This section is based heavily on the excellent reference written by Wachspress [4].

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... mathematically^1.20

In any practical situation it would be very inefficient to calculate the inverse matrix and this is in fact never done.

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... minor^1.21

The principal matrix minor, $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_k$, of matrix $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}$ is defined by

$$\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_k \;... ...ts \\ A_{k1} & A_{k2} & \cdots & A_{kk} \end{array} \right]$$

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... $\lambda_e > \lambda_0$^1.22

Note the theorem also holds for $\lambda_e = \infty$ such that $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex} - (\nu/\lambda_e)\hspace{0.2ex}\underline{\underline{F}}{}\hspace{0.15ex}$ becomes $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ itself.

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... matrix^1.23

A permutation matrix is a square matrix with the property that it has one non-zero element of value 1 per row and column. A permutation matrix is most often constructed by interchanging rows and columns of an identity matrix. A detail of importance is that if $\hspace{0.2ex}\underline{\underline{P}}{}\hspace{0.15ex}$ interchanges the ith and jth row $\hspace{0.2ex}\underline{\underline{P}}{}\hspace{0.15ex}^T$ interchanges column i and j.

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... $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$^1.24

Note that the flux vector corresponding to the node ordering is called $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$ as opposed to the flux vector defined in section 1.6.1 which is called $\hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}$.

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... k^1.25

In fact we never have a full scattering matrix in any physical realistic situation because up-scattering only occurs in the thermal energy region. Consequently, for a fast group $\tilde{g}$ we have that $\Sigma_s^{\tilde{g} \rightarrow g'} = 0 ,\;\;\; \forall(g' > \tilde{g})$. It is possible to utilize this in the solution of the equations if we split up the calculation in a thermal and non-thermal part because we can save some of the calculations in the non-thermal part. This optimization of the computational work is, however, not worth while in the 1-D case - the order of the system we are solving is not big enough.

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...P_thtot^1.26

This procedure presents a problem since the $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$ obtained from a system with $\lambda_0 \ne 1$ has no physical significance and we can be sure that altering the material composition of the reactor in order to achieve $\lambda_0 = 1$ will also alter $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$. It is, however, standard practice to assume that the two $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$'s are identical as a first approximation.

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... gradually^1.27

A gradual change in h_k reduces the truncation error since a uniform grid in general gives a higher accuracy. Consequently, the more uniform the grid the lower the truncation error.

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... mind^2.1

In regard to the mathematical properties see section 1.6.

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... accuracy^2.2

This criterion is actually a relative measure of the change of the eigenvector since the scheme is guaranteeing $\Vert \hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^{(k)} \Vert _\infty = 1$.

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... $\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}_n$^2.3

According to section 1.7.1 we are able to restrict our investigation to one where $\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}_i = \hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_{i+1}$ and where matrices $\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}_i$ and $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_i$ are diagonal.

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... by^2.4

It is in general better to use a relative residual defined by

$$\hspace{0.2ex}\underline{r}{}\hspace{0.15ex} \;\hbox{$=$\kern... ...ace{0.15ex} \hat{\hspace{0.2ex}\underline{x}{}\hspace{0.15ex}}$$

but due the time did not permit to implement this improvement in the code!

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... Matlab^3.1

Matlab is an interpreter which has an extensive instruction set for numerical calculations. It allows you to program complex numerical algorithms in a very short time, which makes it ideal for test purposes. Matlab has the drawback of relatively slow computation which makes a "full" implementation in Matlab inefficient, especially when it is impractical or impossible to vectorize the numerical algorithms.

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... allocation^3.2

Again, this can be done more systematic using $\mbox{C}++$'s constructor and destructor operators.

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... pattern^4.1

The second order accuracy arises from the fact that the points located between the two $\hspace{0.2ex}\underline{h}{}\hspace{0.15ex}_0$ points satisfy the conditions of uniform steplength and constant nuclear cross sections. With these conditions satisfied the local truncation error is ${\cal O}(h^2)$ according to (1.47).

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... [Pa]^5.1

Since we do not account for the influence of the pressure distribution in the downcomer on the liquid density we actually assume that the liquid is incompressible--in practice a very good approximation. This is illustrated by noting that the derivative with respect to p of $\rho$ is given by

$$\frac{1}{\rho}\left(\frac{\partial \rho}{\partial p}\right)_T = \frac{1}{K}$$

where K is the bulk modulus [Pa] and has a value of 300-700 MPa for water at saturation pressure in the range $p_{\mbox{\protect\scriptsize sat}} \in [4;8.5]{\mbox{MPa}}$.

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... requirements^5.2

The subject of flow instabilities and coupled neutronics-flow instabilities are beyond the scope of this text.

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... analysis^6.1

In general, a strong coupling exists between the void fraction and the flux and power distribution in a BWR due to large void reactivity coefficients.

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... ratio^6.2

Note that it is also possible to define the (local) slip ratio as

$$S_{\mbox{\protect\scriptsize local}} \;\hbox{$=$\kern-0.68em\... ...1ex \hbox{$\scriptscriptstyle\triangle$}}\;\frac{u_g}{u_\ell}$$

but this form is impractical since the cross-sectional average $\mbox{$<\!{\frac{u_g}{u_\ell}}\!>$}$ is unknown.

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... Mashima^6.3

This assumption is essentially the same as assuming uniform profiles of $\alpha$ and $u_\ell$ as listed in the beginning of this section!

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... enthalpy^6.4

In this analysis we neglect the contributions from kinetic and potential energies. This simplification is sound in most cases of practical interest.

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...$\mbox{\bf Pe}$^6.5

The Peclet number is defined by

$$\mbox{\bf Pe} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\mbox{\bf Re}\mbox{\bf Pr}.$$

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... number^6.6

The Brinkmann number is defined by

$$\mbox{\bf Br} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\; \mbox{\bf Ec}\mbox{\bf Pr}$$

where $\mbox{\bf Ec}$ and $\mbox{\bf Pr}$ are the Eckert and Prandtl number respectively.

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... relations^6.7

Note the variables of the functions in fact only fit our choice of external constitutive relations.

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... parameter^6.8

The term concentration parameter is also found in the literature.

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... factor^6.9

Note that two friction factors are commonly used in the literature. It is the Darcy-Weisbach (or Moody) friction factor, f, and the Fanning friction factor, C_f. The relation between them is

f = 4 C_f.

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...White^6.10

According to Weismann [11] the Colebrook friction factor originally constructed for circular pipes is probably adequate for a pitch to diameter larger than 1.2 (a standard GE $8\times8$ BWR fuel element has $P/D \simeq 1.3$).

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... roughness^6.11

Drawn tubing normally encountered in reactor design has an equivalent roughness of $1.524\cdot10^{-6}$m.

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... predicted^6.12

Even up-to-date comparisons (1992) [19, p. 227] reveal that the Saha-Zuber is the most accurate correlation.

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... correlation^6.13

In the literature we find some confusion in regard to the correlation. It is the question whether to use the hydraulic or heated equivalent perimeters which causes the problem. The reason for the problem is that the correlation relies both on thermal and hydrodynamic dimensionless parameters. Firstly, the Peclet number, $\mbox{\bf Pe}$, is a hydrodynamic parameter which implies that we have to use the hydraulic equivalent diameter, D_e. Secondly, the expression for $\mbox{\bf Pe}<70000$ can actually be written as $\mbox{\bf Nu}=455$ which implies that we have to use the heated diameter, D_H. Finally a comment in regard to the expression for $\mbox{\bf Pe}>70000$ is appropriate since is can be written as $\mbox{\bf St}=0.0065$. It turns out that Saha and Zuber write the Stanton number, St, defined by [35, p. 342]

$$\mbox{\bf St} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\frac{\mbox{\bf Nu}}{\mbox{\bf Pe}}$$

as

$$\mbox{\bf St} = \frac{ q_w''}{\mbox{$<\!{G}\!>$}C_{p,\ell} \Delta T_D}$$

which implies that they assume

D_H = D_e

. Without this assumption the author believes that the expression for $\mbox{\bf Pe}>70000$ needs a factor (D_H/D_e).

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... number^6.14

During normal BWR operation with water as working fluid $\mbox{\bf Pe}>70000$.

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... ODEs^6.15

Ordinary Differential Equations.

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... $(d \rho_\ell / dz)$^7.1

If we for example assume $\rho_\ell = \rho_\ell(p,h)$ this derivative can be expanded into

$$\frac{d\rho_\ell}{dz} = \frac{\partial \rho_\ell}{\partial p}... ...dp}{dz} + \frac{\partial \rho_\ell}{\partial h} \frac{dh}{dz}.$$

The derivative ( dh /dz) may be expanded further by (6.98) to yield

$$\frac{dh_\ell}{dz} = \frac{\partial h_\ell}{\partial \mbox{$... ...$}}{dz} + \frac{\partial h_\ell}{\partial h_g} \frac{dh_g}{dz}$$

where again the derivative of specific enthalpy of vapor at saturation with respect to z is given by $\frac{dh_g}{dz} = \frac{dh_g}{dp} \frac{dp}{dz}$.

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... relations^7.2

Furthermore, implementation and verification of the code would require a considerable effort. On the other hand, we would not have to implement a numerical method in order to solve the equations (7.1)-(7.4).

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... equation^7.3

Note that we have replaced the factor (P_f/A_c) with (4/D_e) by using the definition of the hydraulic equivalent diameter (6.82). Note that it is in general a good idea to scale the variables in the momentum equation such that they all have magnitudes in the order of 1. However, since we have not encountered numerical problems due to the bad scaling especially of p which lies around 10⁶ Pa we have for simplicity not scaled the momentum equation.

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... $\hspace{0.2ex}\underline{\underline{J}}{}\hspace{0.15ex}_g$^7.4

The Jacobian of a vector function $\hspace{0.2ex}\underline{g}{}\hspace{0.15ex}(\hspace{0.2ex}\underline{x}{}\hspace{0.15ex})$, $\hspace{0.2ex}\underline{\underline{J}}{}\hspace{0.15ex}_g$, is defined by

$$\hspace{0.2ex}\underline{\underline{J}}{}\hspace{0.15ex}_g \;... ...cdots & \frac{\partial g_n}{\partial x_n} \end{array} \right].$$

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... radius^7.5

The spectral radius is the eigenvalue with largest modulus.

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... required^7.6

This makes these methods particularly suitable when function evaluations are very costly to acquire.

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... below^8.1

We only list modules which have not been described previously. The core flow implementation among others also uses the general applicaple modules io.c and memalloc.c which were described in section 3.3 (see p. ).

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... hydraulics^8.2

For a description of the hydraulics in the reactor as a whole see chapter 10.

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... function-functions^8.3

The term function-functions was originally introduced in MATLAB user guides [59] for functions which as input takes another function.

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...$h_{\ell,d}$^8.4

According to (8.12) even though we have stated that we do not have void departure at point z_i it is possible that the void departure criterion is fulfilled when we use the correct properties (ie those which belong to z_i) since h_f is decreasing with decreasing pressure and the pressure is decreasing with z.

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... semi-analytical^9.1

The term semi-analytical denotes that some of the calculations are established numerically.

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... element^9.2

The (average) linear heat generation rate for one fuel rod is taken as

$$q_{\mbox{\protect\scriptsize rod}}'(z) \;\hbox{$=$\kern-0.68e... ...ise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\frac{q'}{63}$$

where we have assumed a standard GE 8 x 8 fuel element with 63 fuel rods.

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... structure^9.3

The void fraction is given by a fraction which contains an exponential function in both the denominator and the numerator.

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... implementations^9.4

Note that we cannot go much further below the lowest mass fluxes in the two figures since this would imply a flow quality at the exit of the core of 1--a case which the code cannot handle correctly.

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... results^9.5

We identify that this difference is very close to the error of the result calculated by the C implementation since the error associated with the Matlab calculated results is negligible.

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... points^9.6

In order to perform the subtraction we select every second element of the vector, $\hspace{0.2ex}\underline{\xi}{}\hspace{0.15ex}^{(r+1)}$, which holds the results from the finer grid.

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... %^9.7

The relative derivation of the local pressure is even smaller since p is in the order of $7\cdot 10^6 {\mbox{ Pa}}$.

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... file^10.1

The syntax of the input file is described in section 10.3.1.

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... coefficient^10.2

The loss coefficient is defined in terms of the core mass flux, $\mbox{$<\!{G}\!>$}_{\framebox[1.5ex]{\raisebox{-2.2pt}[0pt][0ex]{\scriptsize 1}}}$.

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... reductions^10.3

Number of reductions of the steplengths by a factor of 2.

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... ABH-method^11.1

See Duderstadt [45] eq. (10-47) and use that

$${\cal F} = \frac{1}{P_F}.$$

Note that the absorption cross sections should be Doppler-corrected.

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... requirement^11.2

The Davis-Anderson requirement most often seen in the literature is

$$T_w - T_{\mbox{\protect\scriptsize sat}} > \left[ \frac{8 \... ...ect\scriptsize sat}} q_w'' }{h_{fg} \rho_g k_f} \right]^{0.5}$$

The requirement is derived by utilizing the Clausius-Clapeyron equation and if the term v_fg in this equation is approximated by v_g for $\rho_f \gg \rho_g$ we obtain the above mentioned equation which actually is an approximation.

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... boiling^11.3

ie boiling in a stagnant (non-flow) liquid.

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... correlation^12.1

We may note that the S and F function values in the Chen correlation are obtained by linear interpolation in the Tables 11.2 and 11.1. Linear interpolation is adequate in this regard when we consider the large scatter produced by the experimental data used to obtain the functions!

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... method^13.1

This method has many names, for instance, the Thomas algorithm.

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...j^*13.2

This search is done in a sequential manner in the implementation since it is not affordable to transform the search into a binary search for up to several hundreds entries.

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... problem^13.3

At least in the cases of the interpolating functions for $\mbox{$<\!{\alpha}\!>$}(z)$ and q'(z).

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... behavior^13.4

We assume that the moderator state (p^*,T^*) in the cross section tables is chosen close to the actual state in the reactor core such that, for instance, the actual moderator temperature is reasonable close to T^*!

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... sections^13.5

Note that the most accurate formulation of the average cross sections is

$$\overline{\Sigma}{}_{a,k}^g = \frac{1}{z_{k+1}-z_k} \int\limits_{z_k}^{z_{k+1}} \Sigma_a^g(\alpha^*,\overline{T}{}_f) dz$$

but considering the approximate nature of the multi-group equations this formulation is hard to justify.

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... investigation^13.6

We will, however, not go through the contents of this screen output since this would require a very detailed description of the source code.

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...RPV^14.1

RPV is short for Reactor Pressure Vessel.

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...)^14.2

This temperature is taken directly from the assumption (14.2).

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... rods^14.3

The temperature of the fuel is taken as 800 ${}^\circ\mbox{C}$ in all the reflector cross section calculations both in regard to the top reflector and bottom reflector.

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... subcooling^15.1

The inlet subcooling is important in an analysis of the flow stability of the boiling channel.

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...N/A^15.2

N/A denotes Not Available.

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...$-31.145^15.3

Note that the major part of the riser pressure drop is due to the elevational pressure change since the riser is 10 meters high.

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... higher^16.1

The peak grows higher as the flux depression increases since the integral or total power remains the same!

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... analysis^17.1

Sensitivity analysis is very important for developing new and more accurate models or improve existing models, for instance, by issuing experimental investigations.

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... appreciable^17.2

The hydraulics model is based on knowledge acquired during design of forced-circulation BWRs. The drawback is that for instance the separator model not being as critical in forced-circulation is very rough.

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... sense^B1

The procedure of obtaining the coefficients which fit the data in a least square sense will not be addressed here--see for instance [27].

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...White^B2

Note that choosing $\frac{T}{T_0}$ instead of White's $\frac{T_0}{T}$ in our case results in a better fit.

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...VDI_steam^B3

Note that we have a misprint in the VDI-table at $t_{\mbox{\protect\scriptsize sat}} = 130\mbox{${}^\circ\mbox{C}$}$. According to [29, Table A27] the correct value is around $0.688 {\mbox{W}}/({\mbox{m}}\cdot{\mbox{K}})$--this value, however, seems a little too high compared to the VDI-data in the neighborhood.

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... $$t_{\mbox{\protect\scriptsizesat}}(p)$^B4

Calculated from curve fit [24, Table A-2].

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... $$\mu_{\ell,{\mbox{\protect\scriptsizetbl}}}$^B5

Table value from [26, Table 4e].

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... $$k_{\ell,{\mbox{\protect\scriptsizetbl}}}$^B6

Table value from [26, Table 4d].

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...$$\mu_f(t)$^B7

Calculated from curve fit--see section B.2, (B.4)-(B.6).

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...$k_f(t)^B8

Calculated from curve fit--see section B.4, (B.11) and (B.12).

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...$$E_{\mu_f}$^B9

See (B.15) with $=_f$.

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...$E_kf^B10

See (B.15) with $=k_f$.

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... error^B11

Note that both the table value and the value calculated from a curve fit has a measurement error (in addition the curve fit has an interpolation error which, however, is of secondary importance)!

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