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Single-phase heat transfer

In the single-phase region, heat transfer is often evaluated by the Dittus-Boelter equation (see, for instance, Lahey and Moody [18])

$$\mbox{\bf Nu}_{D_e,m} = 0.023 \mbox{\bf Re}_{D_e,m}^{0.8} \mbox{\bf Pr}_m^{0.4}$$ (11.29)

where the subscript m indicates that the properties are evaluated at the mean or bulk temperature and the dimensionless groups are defined by

l

Nusselt number is a heat transfer group defined by

$$\mbox{\bf Nu}_{D_e} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\frac{h D_e}{k}$$ (11.30)

where h is the convection heat transfer coefficient [W/( ${\mbox{m}}^2\cdot$K)], D_e is a characteristic length (typically an equivalent diameter) [m] and k is the thermal conductivity of the fluid [W/(m$\cdot$K)].

l

Reynolds number is a hydrodynamic group which can be defined as

$$\mbox{\bf Re} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\s... ...tscriptstyle\triangle$}}\;\frac{ \mbox{$<\!{G}\!>$} D_e}{\mu}$$ (11.31)

where $\mbox{$<\!{G}\!>$}$ is the average mass flux [ ${\mbox{kg}}/({\mbox{m}}^2\cdot{\mbox{s}})$], D_e is a characteristic length (typically an equivalent diameter) [m] and $\mu$ is the dynamic viscosity of the fluid [kg/(mcdots)].

l

Prandtl number is defined by

$$\mbox{\bf Pr} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\frac{c_p \mu}{k}$$ (11.32)

where c_p is the specific heat at constant pressure [J/(kg$\cdot$K)], $\mu$ is the dynamic viscosity [kg/(m$\cdot$s)] and k is the thermal conductivity of the fluid [W/(m$\cdot$K)].

Strictly speaking the Dittus-Boelter equation is only valid for circular tubes. However, for other geometries, for instance, rod bundles it is possible to use the Dittus-Boelter equation if we use the hydraulic equivalent diameter, D_e, defined by (6.82) (see p. ) as the characteristic length.

It is, however, recognized by several authors ([11] and [47]) that the complex geometry of rod bundles needs special attention. It is found that the correlation by M.S. El-Genk et al [47]

$$\mbox{\bf Nu}_{D_H,m} = C \mbox{\bf Re}_{D_H,m}^{0.8} \mbox{\bf Pr}_m^{0.33}$$ (11.33)

with

$$C \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscripts... ...box{where}} \qquad 1.1 \le \left[ \frac{P}{D} \right] \le 1.5$$ (11.34)

where P is the pitch of the square rod array [m] and D is the rod diameter [m] predicts experimental data within $\pm 8\%$ for turbulent flow. Note that in the definition of both Re and Nu the characteristic length employed is the heated equivalent diameter, D_H. The heated equivalent diameter is defined by

$$D_H \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\frac{4 P_H}{A_c}$$ (11.35)

where P_H is the heated perimeter [m] and A_c the cross-sectional flow area [${\mbox{m}}^2$].

That is the correlation (11.33) include an effect of the cold wall which surrounds the rod array in the BWR fuel bundle!

Inserting the definitions of the dimensionless groups into (11.33) yields

$$\left[ \frac{h_{1\phi} D_H}{k_\ell} \right]_m = C \left[ \... ...8} \left[ \frac{c_{p,\ell} \mu_\ell}{k_\ell} \right]_m^{0.33}$$ (11.36)

or

$$h_{1\phi} = C \left[ \frac{\mbox{$<\!{G}\!>$} D_H}{\mu_\el... ...{k_\ell} \right]_m^{0.33} \left( \frac{k_\ell}{D_H} \right)_m$$ (11.37)

where the subscript m indicates that the properties should be evaluated at the mean or bulk liquid temperature.

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