JakobCHR.com

Quick Navigation:

Personal:

Go to Home
MS Research
PhD Research
Curriculum Vitae

General:

Linux

Soon to come:

Matlab
On-line Stores
Cycling
Medicine & Health
LaTeX
OOP & C++
Sony PCM-R500 DAT

Next: Derivation of curve fits Up: Coupled model theory and Previous: Conclusion Contents Index

Importance of kinetic and potential energies

This appendix describes the importance of the kinetic and potential energies in the mixture energy equation (6.45) described in section 6.6.

To analyze the importance we write a control volume formulation of the energy equation for the whole of one fuel element. The resulting equation can be written as

$\begin{eqnarray*} \lefteqn{\left[ (\mbox{$<\!{h_\ell}\!>$}_\ell + \frac{1}{2}\m... ...ox{$\dot{m}$}_g \right]_{\mbox{\protect\scriptsize inlet}} = q \end{eqnarray*}$

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

where q is the total generated power within the fuel element [W].

Upon moving the terms which describe the change in kinetic and potential energies to the RHS reveals

$\begin{displaymath} \mbox{$<\!{h_g}\!>$}_{g{\mbox{\protect,exit}}} \mbox{$\dot{... ...t\scriptsize kin}} - \Delta E_{\mbox{\protect\scriptsize pot}} \end{displaymath}$

(A1)

where $\mbox{$\dot{m}$}$ is the total mass flow rate defined by

$\begin{displaymath} \mbox{$\dot{m}$}\;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$... ...tstyle\triangle$}}\;\mbox{$\dot{m}$}_\ell + \mbox{$\dot{m}$}_g \end{displaymath}$

(A2)

and $\Delta E_{\mbox{\protect\scriptsize kin}}$ and $\Delta E_{\mbox{\protect\scriptsize pot}}$ are the changes in kinetic and potential energies (strictly speaking energy-rates) defined by

$\begin{displaymath} \Delta E_{\mbox{\protect\scriptsize kin}} \;\hbox{$=$\kern-... ...}\!>$}_{\ell{\mbox{\protect,inlet}}}^2 \mbox{$\dot{m}$}\right) \end{displaymath}$

(A3)

and

$\begin{displaymath} \Delta E_{\mbox{\protect\scriptsize pot}} \;\hbox{$=$\kern-... ...tsize core}} \mbox{$\dot{m}$}_{\mbox{\protect\scriptsize tot}} \end{displaymath}$

(A4)

where $L_{\mbox{\protect\scriptsize core}}$ is the length of the fuel element [m].

From calculations in [20] we have the following results in regard to a General Electric Co. SBWR design

$\begin{eqnarray*} \dot{Q} &=& 2.38 {\mbox{ MW}} \\ S &=& 1.9 {\mbox{ (Constan... ...}}\\ L_{\mbox{\protect\scriptsize core}}&=& 2.44 {\mbox{ m}} \end{eqnarray*}$

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

These results reveals

$\begin{displaymath} \Delta E_{\mbox{\protect\scriptsize kin}} \approx 52.1 {\mbox{ W}} \end{displaymath}$

(A5)

and

$\begin{displaymath} \Delta E_{\mbox{\protect\scriptsize pot}} \approx 217.3 {\mbox{ W}} \end{displaymath}$

(A6)

or in percent of the heat input

$\begin{displaymath} \frac{\Delta E_{\mbox{\protect\scriptsize kin}}}{q} \approx 0.002 \end{displaymath}$

(A7)

and

$\begin{displaymath} \frac{\Delta E_{\mbox{\protect\scriptsize pot}}}{q} \approx 0.009 \end{displaymath}$

(A8)

Therefore we conclude that the changes in the kinetic and potential energies are in the order of 1 hundredth of a percent of the heat input and that it is justified to neglect these terms in the energy equation. Note that the importance of the kinetic and potential energies increases with decreasing heat input, but even at a very low power level (in the order of 100 kW) the assumption seem valid.

Next: Derivation of curve fits Up: Coupled model theory and Previous: Conclusion Contents Index

Top Quality
Developed with
Danish
Brain Power