% ****** Start of file aipsamp.tex ****** % % This file is part of the AIP files in the AIP distribution for REVTeX 4. % Version 4.1 of REVTeX, October 2009 % % Copyright (c) 2009 American Institute of Physics. % % See the AIP README file for restrictions and more information. % % TeX'ing this file requires that you have AMS-LaTeX 2.0 installed % as well as the rest of the prerequisites for REVTeX 4.1 % % It also requires running BibTeX. The commands are as follows: % % 1) latex aipsamp % 2) bibtex aipsamp % 3) latex aipsamp % 4) latex aipsamp % % Use this file as a source of example code for your aip document. % Use the file aiptemplate.tex as a template for your document. \documentclass[% aip, %jmp,% %bmf,% sd,% %rsi,% amsmath,amssymb, %preprint,% reprint,% %author-year,% author-numerical,% twoside ]{revtex4-1} \usepackage{graphicx}% Include figure files \usepackage{dcolumn}% Align table columns on decimal point \usepackage{bm}% bold math \usepackage[mathlines]{lineno}% Enable numbering of text and display math %\linenumbers\relax % Commence numbering lines \usepackage{hyperref} \usepackage{fancyhdr} \usepackage{appendix} \usepackage{ifthen} \pagestyle{fancy} \fancyhf{} \lhead{\textit{Metal CANDU Fuel Heat Transfer}} \fancyfoot[RO]{\textit{\small{McMaster Journal of Engineering Physics}, 2017, \bf{[vol]}, \thepage}} \fancyfoot[LE]{\textit{\small{McMaster Journal of Engineering Physics}, 2017, \bf{[vol]}, \thepage}} \renewcommand{\footrulewidth}{0.4pt} %\fancyfoot{\ifthenelse{\value{page}=1}{\small{\textit{McMaster Journal of Engineering Physics}, 2016, \bf{[vol]}, \thepage}}} \begin{document} \preprint{AIP/123-QED} \title[\textit{Metal CANDU Fuel Heat Transfer}]{Simulated Heat Transfer out of a Metallic Cruciform CANDU Fuel Element} \author{L.P. Palumbo} %only include a), b) etc if there are footnotes on the authors (see section III.3 Footnotes, and Notes and References at end \affiliation{ McMaster University, 1280 Main Street West, Hamilton, Canada. %\\This line break forced with \textbackslash\textbackslash } \date{\textbf{\today}}% It is always \today, today, % but any date may be explicitly specified \begin{abstract} FlexPDE was used to simulate the temperature distribution in a theoretical Uranium-Zirconium alloy, helical, cruciform shaped fuel element, based on calculated thermohydraulic conditions inside a metal fuel bundle loaded CANDU-6 pressure tube. At the conventional CANDU fuel melting power level of 70 kW/m, the metal fuel had a simulated peak temperature of 602$^\circ$C, which is 1123$^\circ$C below its melting temperature. Furthermore the metal CANDU fuel can thermally sustain 40\% higher fission power than conventional fuel before it is limited by sustained subcooled nucleate boiling, therefore fuel melting would be eliminated as a power constraint. \end{abstract} \pacs{28.41.Fr, 28.41.Bm, 28.41.Ak}% PACS, the Physics and Astronomy % Classification Scheme. \keywords{CANDU fuel bundle, metallic nuclear fuel, U-Zr}%Use showkeys class option if keyword %display desired \maketitle \thispagestyle{fancy} \section{\label{sec:intro}INTRODUCTION} CANDU-6 is a commercial nuclear power reactor design powered by nuclear fuel arranged inside an array of 380 horizontal tubes. Each pressurized tube contains 13 cylindrical bundles of fuel elements being actively cooled by pumping heavy water at high pressure through the fuel channel\cite{CANDU-6}. Each fuel bundle is a 0.5 m long, 10.24 cm diameter, array of 37 hollow tubes 1.308 cm in diameter, with end plates to hold the bundle together. The fuel bundle is made out of a corrosion resistant and relatively neutron transparent alloy known as Zircaloy-4\cite{canteachBundle}. Each fuel element is filled with 19 mm long pellets of the dense Uranium containing ceramic, UO$_2$. As the Uranium fissions inside the reactor, the pellets undergo almost uniform volumetric heating. UO$_2$ has a thermal conductivity of 2-4 W/mK in its normal, full power, temperature operating range, and as its temperature is increased, the thermal conductivity decreases\cite{UO2thermal}. Because of its low, and inversely temperature dependent thermal conductivity, and air/helium gap thermal resistance between a fuel pellet and the inside cladding tube wall, a fuel pellet can experience a 1500$^{\circ}$C temperature differential between its centerline and surface\cite{canteachPellet1}. The highest pellet centerline temperatures are found in the outer fuel element of a bundle near the axial and azimuthal center of the reactor core at an intermediate burnup of 40 MWh/kgU, these elements have a maximum linear power of 57 kW/m\cite{Manzer}. The license limit for linear element power is is 65 kW/m in a CANDU-6 reactor because high power experiments found fuel pellets begin melting at 70 kW/m\cite{song}. Commercial entities in the United States are currently developing twisted, non-circular, solid metal nuclear fuel elements for pressurized water reactors\cite{Lightbridge}, that enhance heat transfer, and have substantially higher thermal conductivity then UO$_2$. If adapted to CANDU, a fuel bundle of metal fuel elements may remove the power constraint caused by pellet centerline melting. \begin{figure}[h] \includegraphics[scale=1.2]{bundle.png}% Here is how to import PDF art \caption{\label{fig:bundle} A CANDU 37 element fuel bundle visualized with helical cruciform fuel elements} \end{figure} \subsection{\label{sec:subsect1}Metal Nuclear Fuel} Pure Uranium metal would be an obvious choice for metal fuel due to its high thermal conductivity and maximum nuclear fuel density, however it has a low melting temperature and swells substantially at low burnup levels\cite{UZrSystem}. Historically Uranium metal has been alloyed with 10\% Zirconium to increase the melting temperature and reduce the burnup swelling, although it is still found to axially swell 8\% at 1\% atom burnup\cite{Hofman1990}, an unacceptable amount for a CANDU pressure tube designed to hold bundles of a fixed length. Irradiation swelling in nuclear fuel is caused by the build up of Xe and Kr fission gas bubbles\cite{Hofman1990}. Uranium metal and low Zr alloy U-Zr form an $\alpha$-phase crystal structure that is highly susceptible to swelling\cite{UZrSystem}. A U-Zr alloy at 70at\% Zr has a solidus melting temperature of 1725$^{\circ}$C and forms the $\delta$-phase crystal structure below 616$^{\circ}$C\cite{UZrSystem}, this structure is much more resistant to irradiation swelling\cite{Lightbridge}, this alloy is assumed for this study. Cladding metallic nuclear fuel can be achieved by co-extrusion of the fuel inside a layer of cladding\cite{Lightbridge}, the cladding is assumed to be Zircaloy-4 due to its successful deployment in existing reactors and preexisting supply chain\cite{canteachBundle}. Co-extrusion benefits heat transfer by eliminating gap thermal resistance between layers because they are welded together during hot extrusion. The thermal conductivity of the fuel alloy is modeled by equation \ref{eq:kfuel}\cite{thermalconductivity}. \begin{equation} \\k_f = {-}9\times10^{-11}T^3+4\times10^{-7}T^2 {-} 0.0002T+0.114 \label{eq:kfuel} \end{equation} The thermal conductivity of the Zircaloy-4 cladding is given by equation \ref{eq:kclad}\cite{Zirc4}. \begin{equation} \\k_c = 0.113+2.25\times10^{-5}T+0.725\times10^{-7}T^2\\ \label{eq:kclad} \end{equation} Where: $T$ = temperature in K $k$ = thermal conductivity in W/cmK \begin{figure}[h] \includegraphics[scale=0.4]{kgraph.png}% Here is how to import PDF art \caption{\label{fig:kgraph} Thermal Conductivity of traditional UO$_2$ fuel and Zircaloy-4 cladding compared to the proposed U-70\%Zr alloy} \end{figure} \subsection{\label{sec:subsect2}Heat Transport Mechanisms} \subsubsection{Surface Convection} Nuclear heat is transported away from the fuel bundles by pumping 24 Kg/s of D$_2$O through each pressure tube at an average static pressure of 10.875 MPa(a) and an average temperature of 288$^\circ$C \cite{CANDU-6}. This analysis is considering the subcooled liquid flow regime where the primary heat transport phenomenon is one phase, forced convection by highly turbulent flow\cite{2Phase}. As the wall temperature passes the saturation temperature associated with the coolant's pressure, subcooled nucleate boiling emerges as a secondary heat transfer mechanism\cite{SubcooledBoiling}. Finally, as the wall temperature continues to rise, the localized subcooled boiling forms a layer of bubbles that begins insulating the fuel element surface, this is known as departure from nucleate boiling (DNB)\cite{canteachPellet1}, the heat flux as a function of wall temperature reaches a local maximum at this point before it begins falling towards the Leidenfrost point\cite{2Phase}. \subsubsection{Internal Conduction} Inside the fuel element, heat is transported from the fission heated fuel region, through the cladding, to the fuel element wall. Conduction heat transport is dependent on the divergence of the temperature gradient multiplied by the local thermal conductivity. The balance of volumetric thermal power generation in the fuel, $s$ W/cm$^3$, and heat transport inside the fuel element is given by the heat equation \ref{eq:heat}. \begin{equation} \\-\nabla\cdot (k\nabla{T})+s=0 \label{eq:heat} \end{equation} \subsection{\label{sec:subsect3}CANDU Hydraulic Diameter} Conventional, and well documented thermohydraulic relationships for internal turbulent flows can be used for non circular pipes if the equivalent hydraulic diameter is substituted instead\cite{ThermalFluids}. The hydraulic diameter of flow through a fueled CANDU pressure tube is given by equation \ref{eq:Dh}\cite{1Phase}. \begin{equation} \\ D_H=\frac{4[\frac{\pi D_{PT}}{4}-37A_{element}]}{37P_{element}+\pi D_{PT}} \label{eq:Dh} \end{equation} \section{\label{sec:procedure}Modeling Approach} \subsection{Hydraulic Modeling} The apparent friction factor in a conventional CANDU-6 pressure tube was first calculated using equation \ref{eq:pressuredrop}, to base an estimate on for a metal fueled channel. \begin{equation} \\\Delta p=\frac{fL}{D_H2}\rho v^2 \label{eq:pressuredrop} \end{equation} Given that the pressure drop across a conventional 5.94 m fuel channel is 838 kPa\cite{CANDU-6}, the friction factor must be 0.0314. M.M.K. Bhuiya et.al. found comparing a twisted tape with an array of linear aligned perforations, to a plain tube, showed increases in friction factor at all Reynolds numbers tested\cite{TwistedTape}. The combination of helically twisting flow, and of flow through aligned holes is assumed analogous to the combination of subchannel linear flow and induced rotational flow across the helical fuel elements encountered in the Metal CANDU fuel bundle. The enhancement did however show diminishing improvement as Reynolds number was increased, so an empirical relation \ref{eq:twistedf} was created here to extrapolate the increase in friction factor to higher Reynolds numbers in this analysis. \begin{equation} \ \Delta f = 151.87Re^{-0.817} \label{eq:twistedf} \end{equation} The friction factor in the theoretical metal CANDU fuel channel is estimated to be 0.0365. The larger flow area of the metal fuel channel produces a lower pressure drop for any given mass flow rate compared to the conventional fuel channel. The flow rate in the metal fuel bundle channel was modeled at 29.9 kg/s to match the pressure drop of the conventional channel, because the pumping power would remain the same in a retro-fitted plant. {\setlength{\extrarowheight}{3pt}% \begin{table}[h] \caption{\label{tab:table1}The hydraulic parameters of a conventional and metal CANDU fuel channel with equivalent pressure drops} \begin{ruledtabular} \begin{tabular}{lllll} Fuel & Flow Area ($m^2$)&$D_H$ ($cm$) & Re & $\dot{M}$ ($kg/s$) \\ \hline UO$_2$& $3.43\times 10^{-3}$ & 0.743 & 525000 & 24.0\\ U-Zr & 0.0057 & 1.19 & 1040000&29.6\\ \end{tabular} \end{ruledtabular} \end{table} \subsection{Convection Modeling} The single phase convection heat flux out of the fuel element surface is modeled as a function of the temperature difference between the surface and the bulk coolant temperature by equation \ref{eq:convection}. \begin{equation} \\q''=h(T_w-T_b) \label{eq:convection} \end{equation} Where $h$ is the convection coefficient evaluated in equation \ref{eq:nusselt} as a function of the coolants thermal conductivity and the system's dimensionless Nusselt number\cite{ThermalFluids}. \begin{equation} \\h=\frac{(Nu)(k)}{D_{h,thermal}} \label{eq:nusselt} \end{equation} $D_{H,thermal}$ is the hydraulic diameter from equation \ref{eq:Dh} with the denominator only considering the heated perimeter not the total wetted perimeter. The Nusselt number can be calculated using the Gnielinski equation \ref{eq:gnielinski}\cite{ThermalFluids}, given the Prandtl number of D$_2$O is 1.046 at 288$^\circ$C and 10.875 MPa\cite{D2O}. \begin{equation} \\Nu=\frac{(\frac{f}{8})(Re-1000)Pr}{1+12.7(\frac{f}{8})^{0.5}(Pr^{\frac{2}{3}}-1)} \label{eq:gnielinski} \end{equation} A correction is applied to the Nusselt number for the case of a narrow annulus and an adiabatic outer wall in equation \ref{eq:nusseltprime}\cite{annulus}. \begin{equation} \\ \frac{Nu'}{Nu}=0.86\bigg(\frac{D_{PT}-D_H}{D_{PT}}\bigg)^{-0.16} \label{eq:nusseltprime} \end{equation} \begin{table}[h] \caption{\label{tab:table2}Convection parameters compared between conventional and the proposed Metal fuel bundles} \begin{ruledtabular} \begin{tabular}{lllll} Fuel & $f$ & $D_{H,thermal}$ ($cm$) & Nu & $h$ ($W/cm^2K$)\\ \hline UO$_2$& 0.0314& 0.901 & 1830 & 9.96\\ U-Zr & 0.0365 &1.43 & 4230 & 14.5\\ \end{tabular} \end{ruledtabular} \end{table} \section{\label{sec:Sections}RESULTS AND DISCUSSION} The steady state heat equation, given in equation \ref{eq:heat}, was solved with the surface boundary given by equation \ref{eq:convection}, for a cross section of a metal CANDU fuel element using the multi-physics, finite element simulation package, flexPDE. \begin{table}[h] \caption{\label{tab:table3}Simulated peak surface and centerline temperatures in high power fuel elements} \begin{ruledtabular} \begin{tabular}{llll} Linear Power & Max $T$ & Max Surface $T$ & Fission Power\\ $kW/m$ & $^\circ C$ & $^\circ C$ & $W/cm^3$\\ \hline 27.5& 429& 297 & 500 \\ 40 & 485 &301 & 730 \\ 57 & 554 &306 & 1042 \\ 65 & 584 &309 & 1188 \\ 70 & 602 &310 &1279\\ 75 & 619 &312 & 1370 \\ 80 & 637 &314 & 1462 \\ \end{tabular} \end{ruledtabular} \end{table} \begin{figure}[h] \includegraphics[scale=0.45]{composite2.png} \caption{\label{fig:4view} A comparison of the steady-state temperature distribution in 4 cruciform fuel element cross sections. 57 $kW/m$ is the maximum linear power in operating CANDU-6 reactors, 70 $kW/m$ would begin melting a conventional fuel element} \end{figure} The Metal CANDU fuel element was over 1000$^\circ$C below its solidus melting temperature at all power levels tested, including up to 40\% higher than current max operating power levels. The centerline temperature of the fuel element passed the $\gamma$-phase transition temperature of 616$^\circ$C at a linear power of 74 kW/m. The $\gamma$-phase of U-Zr has low irradiation swelling properties\cite{UZrSystem}, however it is unknown if repeated operational cycling between $\gamma$ and $\delta$-phase crystal structure is undesirable from a thermo-mechanical or nuclear properties perspective. 80 kW/m is the highest power level simulated that did not generate a surface temperature over the subcooled nucleate boiling temperature of 315$^\circ$C, meaning it is the highest theoretical power level that could be used and still maintain a near 0\% exit steam quality at 10.875 MPa. Many considerations would go into selecting metal CANDU fuel, including enrichment requirements, reactivity implications based on differnt fuel absorption and scattering cross sections, lower resonance absorption due to lower average temperatures, and changes in burnup dynamics due to the change in $^{235}$U fission versus $^{239}$Pu breeding ratios. From a thermohydraulic perspective this analysis suggests it is viable and could potentially be studied as a method to increase thermal margins and up-rate existing CANDU reactor's power output. \section{\label{sec:conc}Conclusions} Simulation of the heat transport out of a cruciform Uranium-Zirconium CANDU fuel element has shown that the power limiting factor of centerline melting at 70 kW/m would be completely eliminated compared to conventional UO$_2$ fuel. Furthermore, increases in mass flow with the same pumping power mean more enthalpy could be transported away from the fuel without significant boiling, this combined with higher max linear power suggests an existing CANDU-6 reactor could be fueled with metal fuel bundles, similar to figure \ref*{fig:bundle}, and could, conservatively generate 10-20\% more thermal power, while maintaining +1000$^\circ$C thermal margins. \begin{acknowledgments} The author would like to thank Dr. Turak and K. Groves for their feedback and guidance throughout the research and writing of this paper, and Dr. Minnick for his enthusiastic and expert help with flexPDE and heat transfer physics. \end{acknowledgments} \subsubsection*{\label{sec:notes}Notes and References} %\nocite{*} \bibliographystyle{unsrtnat} \bibliography{MetalFuel.bib}% Produces the bibliography via BibTeX. \end{document} % % ****** End of file aipsamp.tex ******