\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 181, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/181\hfil Optimal design of a bar]
{Optimal design of a bar with an attached mass for maximizing the heat transfer}
\author[B. P. Belinskiy, J. W. Hiestand, M. L. McCarthy \hfil EJDE-2012/181\hfilneg]
{Boris P. Belinskiy, James W. Hiestand, Maeve L. McCarthy } % in alphabetical order
\address{Boris P. Belinskiy \newline
Department of Mathematics\\
University of Tennessee at Chattanooga\\
615 McCallie Avenue\\
Chattanooga, TN 37403-2598, USA}
\email{Boris-Belinskiy@utc.edu}
\address{James W. Hiestand \newline
College of Engineering\\
University of Tennessee at Chattanooga\\
615 McCallie Avenue\\
Chattanooga, TN 37403-2598, USA}
\email{James-Hiestand@utc.edu}
\address{Maeve L. McCarthy \newline
Department of Mathematics \& Statistics\\
Murray State University\\
6C Faculty Hall\\
Murray, KY 42071-334, USA}
\email{mmccarthy@murraystate.edu}
\thanks{Submitted August 15, 2012. Published October 19, 2012.}
\subjclass[2000]{74A15, 74P10, 49K15, 34B24}
\keywords{Optimal design; heat transfer; heat equation; least eigenvalue;
\hfill\break\indent Sturm-Liouville problem;
Helly's principle; calculus of variations}
\begin{abstract}
We maximize, with respect to the cross sectional area, the rate of heat transfer
through a bar of given mass. The bar serves as an extended surface to enhance
the heat transfer surface of a larger heated known mass to which the bar is
attached. In this paper we neglect heat transfer from the sides of the bar and
consider only conduction through its length. The rate of cooling is defined by
the first eigenvalue of the corresponding Sturm-Liouville problem.
We establish existence of an optimal design via rearrangement techniques.
The necessary conditions of optimality admit a unique optimal design.
We compare the rate of heat transfer for that bar with the rate for the bar
of the same mass but of a constant cross-section area.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\def \pd#1#2{\frac{\partial#1}{\partial#2}}
\section{Introduction}\label{intro_sect}
Materials are often cooled by convection to a surrounding ambient medium such as the atmosphere. For example, the heat generated in an automobile engine is transferred first to the cooling water that circulates through the engine and then to the atmosphere through the radiator. Convective heat transfer is described by the equation
\begin{equation}
\dot Q=h A_s (T-T_\infty)
\label{1.1}
\end{equation}
where $\dot Q$ is the heat transfer rate, $h$ is an empirical heat transfer
coefficient, $A_s$ is the surface area, $T$ is the temperature of the surface
and $T_\infty$ is the temperature of the surrounding medium.
We want to maximize the surface area since the convective heat transfer
rate is proportional to this area. On the other hand, we want to minimize
the volume of the heat transfer region, in order to keep its weight and hence
material cost as low as possible. Thus we seek to maximize the surface to volume
ratio of the heat transfer surface.
Extended surfaces attached to a given base mass, $M_0$, are frequently used in
commercial applications to increase the heat transfer surface area without
significantly increasing the associated mass and hence the material cost of
the device. The additional surface might be in the form of thin donuts around a
central pipe, parallel plates attached to the surface as in small engines or
automobile radiators, or fins extending outward like hairs from a surface.
The mass of the added extended surface is small compared to the base mass $M_0$.
For this reason a high surface-to-volume ratio for the extended surface is sought.
Such mass additions to enhance heat transfer are referred to as fins.
Nature also ``designs" according to this criterion. The ears of an elephant have
large surface area compared to their volume which allows the blood passing through
them to be efficiently cooled. Likewise members of species like deer that live
near the equator must be able to dissipate heat efficiently. They achieve this
desirable ratio by being smaller than their counterparts that live towards the
poles (think of a sphere where the surface to volume is inversely proportional to $r$).
Engineering heat transfer texts sometimes consider fins of variable
cross-section \cite[pp. 124-126]{Incropera} but the cross-section is assumed
to be for a regular shape, e.g. a cylinder with variable radius.
Or the optimization of a fin with a given cross-section (e.g. a rectangle)
is optimized with respect to the length and thickness \cite[pp. 74]{Schneider}.
However, a general variation of shape to maximize the heat transfer from the fin
is not considered.
Heat transfer within the surface is by conduction and the rate is given by
the equation
\begin{equation}
\dot Q=-{{kA \Delta T}\over l}
\end{equation}
where $k$ is the thermal conductivity of the material, $A$ is the cross-sectional
area, and $l$ is the length of the material. Here $\Delta T$ is the temperature
difference between the end points of the heat transfer. The equations above,
along with the corresponding physical background, may be found in
\cite[pp. 110-114]{Incropera}.
If an energy balance is performed for the region of the bar between $x$ and
$x+\Delta x$, energy enters by conduction at $x$ and leaves by conduction
at $x+\Delta x$ and also from the side by convection (see Equation \eqref{1.1}).
The difference is the rate of change of the energy content of that region of the bar
$$
\text{rate of change of energy}=\text{energy in}-\text{energy out}\quad \text{or}
$$
\begin{equation}
\rho c A \Delta x {{\Delta T}\over {\Delta t}}=-kA{{\Delta T}\over {\Delta x}}
\Big \vert_x+kA{{\Delta T}\over {\Delta x}}\Big \vert_{x+\Delta x}-hA_s(T-T_{\infty}).
\label{1.3}
\end{equation}
Here $T(x,t)$ is the temperature distribution. The surface area is
$A_s=P\Delta x$, where $P(x)$ is the perimeter of the cross-section at the point $x$.
The ratio ${{\Delta T}\over {\Delta t}}$ is the rate of change of temperature with
time and ${{\Delta T}\over {\Delta x}}$ is the local temperature gradient.
The following bar material parameters are introduced, the density $\rho$,
the specific heat capacity $c$, the thermal conductivity $k$, and the convective
heat transfer coefficient $h$. It is assumed that $\rho,c,k,h$ are positive constants.
Dividing by $\Delta x$, and taking the limit as $\Delta x$ and $\Delta t$ approach
zero, yields the partial differential equation
\begin{equation}
A{\pd {T}{t}}={k\over {\rho c}}{\pd {}{x}}\Big(A{\pd {T}{x}}\Big)
-{{hP}\over {\rho c}}(T-T_{\infty}), (x,t)\in (0,l)\times (0,\infty).
\label{1.4}
\end{equation}
We discuss a particular case of this general equation when convective heat transfer
from the side of the bar is neglected, i.e., the limiting case $h\to 0$ is considered.
It is the purpose of this paper to find the optimal distribution of the
cross-section area, $A$, of a surface of revolution of a given mass such that
the heat transfer rate is a maximum. This will produce a maximum cooling per
unit mass and may be considered the optimum.
Detailed discussion of techniques and results in structural optimization can
be found in \cite{Dym,Atanackovic} and the references therein.
We mention in particular the maximization of a column's buckling load
\cite{Taylor,TaylorLiu}, the minimization of the mass of an oscillating
bar \cite{Taylor2,Turner} or a rotating rod \cite{Atanackovic2001,Atanackovic2004},
the maximization of a column's height \cite{KellerNiordson} and the minimization
of the moment of inertia of an oscillating turbine \cite{BelinskiyMcCarthy}.
If the design variable tapers too rapidly, the eigenvalue being optimized is not
isolated from the remainder of the spectrum
\cite{KellerNiordson,CoxMcCarthy,Atanackovic2001}. In these cases optimality
conditions can be derived using non-smooth analysis in conjunction with the more
classical Calculus of Variations techniques \cite{McCarthy,McCarthy2}.
The complexity of the Sturm-Liouville problem also increases if the boundary
conditions contain an eigenparameter.
This is due to the fact the the Sturm-Liouville operator is not self-adjoint with
respect to the usual $L^2(0,l)$ inner product.
The spectral properties of the Sturm-Liouville problems that arise from diverse
mechanical models and contain the spectral parameter in the boundary condition(s)
have been studied in \cite{Walter,Fulton,Hinton2,BBS,Belinskiy}. Numerical schemes for the inverse problem were developed by \cite{McCarthyRundell}. Design problems of this type have been considered by \cite{Turner,BelinskiyMcCarthy}.
In the latter, existence of an optimal design was treated seriously,
as it will be here.
In the design problem considered here, we encounter a spectral parameter in a
boundary condition.
In Section \ref{sl_sect} we give the mathematical description of the model,
apply separation of variables and formulate the spectral properties of the
corresponding Sturm-Liouville problem. We also give the solution for an elementary
case of the problem when the cross-section area is constant, which we need later
for comparison with the solution in case of variable cross-section area.
In Section \ref{cov_sect} we derive the necessary conditions of optimality and
hence find an optimal form of the bar.
In Section \ref{rearr_sect} we use a rearrangement technique to prove that the
optimal design is increasing and maximizes the first eigenvalue of
the Sturm-Liouville problem.
In Section \ref{num_sect} we give the numerical comparison of cooling properties
for the bar of optimal shape and a bar having the same mass but with the
constant cross-section area. In the Appendix, we prove that the rate of cooling
for the bar with the optimal cross-section area is greater than for a bar of the
same mass but with constant cross-section area.
\section{Heat transfer of a bar of a variable cross-section area: separation
of variables and the Sturm-Liouville problem}
\label{sl_sect}
We consider the heat transfer in a bar $\{0z
\label{6.20}
\end{equation}
holds for any positive ratio $M/M_0$. This is confirmed by our
numerical results in Section \ref{num_sect}.
\section{The Optimal Design is Increasing}
\label{rearr_sect}
We prove here that the heat transfer rate $\lambda_1(A)$ can be increased
through the use of increasing rearrangements of the cross-sectional
area $A(x)$. This is achieved through the use of an alternative characterization
of $\lambda_1(A)$. We begin by defining decreasing and increasing rearrangements,
and stating some of their relevant properties.
\begin{definition} \rm
The decreasing rearrangement of a nonnegative function, $f$, on $(a,b)$ is simply
$$
f^*(x) \equiv \sup \{ t > 0 : \mu_f(t) > x \},
$$
where $\mu_f$ is the distribution function of $f$,
$$\mu_f(t) = |\{x\in (a,b):f(x)>t\}| \quad t \ge 0.
$$
The increasing rearrangement of $f$ is $f_*(x) \equiv f^*(b-x)$.
\end{definition}
If $g$ and $h$ are nonnegative functions on $(a,b)$, with $g$ increasing
and $h$ decreasing, then
\begin{equation}
\int_a^b f dx = \int_a^b f^* dx = \int_a^b f_* dx ,
\label{5.7}
\end{equation}
By \eqref{5.7}, if we replace a particular design $A\in ad$ by either its increasing
or decreasing rearrangements $A_*$ or $A^*$ then the new design has the same integral.
Furthermore,
\begin{equation}
\int_a^b f^* g dx \leq \int_a^b fg dx , \qquad \int_a^b f_* h dx
\leq \int_a^b fh dx.
\label{5.8}
\end{equation}
These results are a special case of those established in \cite[pp. 153]{PolyaSzego}.
\begin{theorem} \label{rearr_thm}
For any cross-sectional area $A$ satisfying
$$
00$. Since $\lambda_1(A_*), M_0, \rho, u_1(x), A(x) >0$,
this implies that $u_1$ is decreasing.
The second part of \eqref{5.8} implies that
$\int_0^{x} Au_1 dy \geq \int_0^{x} A_* u_1 dy $ and so
$$
{{1}\over{\lambda_1(A)}} \geq {{M_0}\over{\rho}} \int_0^l
\left( \int_0^x A_*(y) u_1(y) dy \right) {{dx}\over{A(x)}}
+\int_0^l \left( \int_0^x A_*(y) u_1(y) dy \right)^2 {{dx}\over{A(x)}}.
$$
Clearly, the functions $( \int_0^x A_* u_1 dy)$ and
$( \int_0^x A_*u_1 dy)^2$
are nonnegative increasing functions of $x$. Once again \eqref{5.8} yields
\begin{align*}
{{1}\over{\lambda_1(A)}}
& \geq {{M_0}\over{\rho}} \int_0^l \Big( \int_0^x A_*(y) u_1(y) dy \Big)
\Big( {{1}\over{A(x)}} \Big)^* dx \\
&\quad +\int_0^l \Big( \int_0^x A_*(y) u_1(y) dy \Big)^2
\Big( {{1}\over{A(x)}} \Big)^* dx.
\end{align*}
If $f$ is decreasing on the range of $g$ then the composition
$(f\circ g)^*=f\circ g_*$, see \cite{Cox}, which implies that
\begin{align*}
{{1}\over{\lambda_1(A)}}
&\geq {{M_0}\over{\rho}} \int_0^l \Big( \int_0^x A_*(y) u_1(y) dy \Big)
{{dx}\over{A_*(x)}}
+ \int_0^l \Big( \int_0^x A_*(y) u_1(y) dy \Big)^2 {{dx}\over{A_*(x)}}\\
& = {{1}\over{\lambda_1(A_*)}}. \qedhere
\end{align*}
\end{proof}
\begin{theorem} \label{thm4.3}
The design satisfying the first order optimality conditions
\eqref{XXX}-\eqref{var-A}
$$
A(x)=\frac{M\sqrt{\lambda_1}\coth(\sqrt{\lambda_1}l)}{\rho \cosh^2 (\sqrt{\lambda_1}(x-l))}
$$
maximizes the functional $A\mapsto\lambda_1(A)$ on the set
$$
ad = \big\{ A : 01$ in the graph are
shown for illustrative purposes only.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\end{center}
\caption{Comparision of optimal design and constant area design}
\label{bhm}
\end{figure}
Numerical results show that the advantage of the optimum cross-section over
the constant cross-section is small and becomes less so as the base mass,
$M_0$, increases. This is physically reasonable. Indeed, recall that
convective heat transfer from the side of the area has been neglected.
Hence addition of the extended surface does little but move the boundary
condition at $x=l$ that distance from $M_0$. Furthermore, as $M$ becomes
small compared to $M_0$, its very presence becomes negligible and hence
its shape does not matter.
In each case $z^{\rm opt}\ge z$, as shown in the Appendix.
This numerical observation is certainly in agreement with the general
results of Sections \ref{cov_sect} and \ref{rearr_sect}. However,
the effect is not large because of the physical reasons explained above.
Moreover, for $M/M_0 \to 0$ the optimum and constant area results merge.
This numerical observation is in agreement with the asymptotic formula \eqref{B.5}.
\section{Conclusion}
We have found the optimal distribution of the cross-section area of a bar in
the form of a surface of revolution of a given total mass with a point mass
attached at the end such that the heat transfer rate is a maximum.
That rate is defined by the least eigenvalue of the corresponding
Sturm-Liouville problem. This is of independent interest because the spectral
parameter appears not only in the differential equation but also in the
boundary condition. The bar will produce the maximum cooling per unit mass
and may be considered the optimum. The optimal distribution coincides with
one found by Taylor \cite{Taylor} and M.J. Turner \cite{Turner} for the
design of a bar having a maximum lowest eigenfrequency with the given mass.
Numerical results show that the advantage of the optimal design over the
constant cross-section is small and decreases as the base mass increases.
We believe this to be a result of the fact that our model neglects heat
transfer from the side of the bar.
We should emphasize that we have considered a special case of the heat
transfer assuming that convective heat transfer from the side of the bar is neglected
and only conduction through the length of the bar is considered
(see Section \ref{intro_sect}).
We expect that the solution of the optimal design problem for
the more general problem will show a more noticeable difference between the
optimal design and the constant case.
If we were to include the heat transfer phenomenon from the sides of the bar,
we would have to consider the partial differential equation
\begin{equation}
a^2(x){\pd {T}{t}}={k\over {\rho c}}{\pd {}{x}}\Big(a^2(x){\pd {T}{x}}\Big)
-{{ha(x)\sqrt{1+(a'(x)^2}}\over {\rho c}}(T-T_{\infty}),
\end{equation}
where $ (x,t)\in (0,l)\times (0,\infty)$, and $a(x)$ would be the radius of the body
of revolution that represents the bar.
The corresponding Sturm-Liouville problem has a discrete spectrum and a complete
set of eigenfunctions. We could derive a Rayleigh-Ritz ratio for the least
eigenvalue similar to expression \eqref{2.14}.
But the technique of the Calculus of Variations used in Section \ref{cov_sect}
will not lead to an explicit form of the cross-section area and the least eigenvalue.
For that problem, we had hoped to use a numerical approach based on
the discretization of our bar that would reduce the problem of the optimal
design to the problem of optimization for a function of several variables
(this idea was developed for the optimal design of mechanical systems
in \cite{Cardou}). We have recently learned that an equation with similar
appearance of the function $a(x)$ is optimized in \cite{Henrot-Privat}.
The techniques used there may be applicable when we consider the more general
heat transfer model.
This will be will be considered in a future paper.
\section{Appendix}
Having derived an explicit formula for the optimal rate of cooling and
an equation for the rate for the bar with the constant cross-section area,
we may compare them directly. We prove below that the optimal rate is greater
than the rate for the bar of the same mass but with the constant cross-section area.
This inequality clearly is demonstrated in our numerical results above,
but is proven here for completeness.
\begin{lemma}
The inequality
\begin{equation}
z^{\rm opt} >z \label{B.1}
\end{equation}
where
\begin{equation}
z^{\rm opt}=\ln\Big(\mu+\sqrt{\mu^2+1} \Big) \label{B.2}
\end{equation}
and $z$ is the minimal positive root of the equation
\begin{equation}
z \tan z= \mu^2 \label{B.3}
\end{equation}
holds for any positive quantity
\begin{equation}
\mu\equiv \sqrt{{M\over {M_0}}}. \label{B.4}
\end{equation}
\end{lemma}
\begin{proof}
We consider both $z^{\rm opt}$ and $z$ as functions of $\mu> 0$.
Note first that
$z^{\rm opt}(\mu)\asymp \mu, z(\mu)\asymp \mu$, and hence
\begin{equation}
\lim_{\mu\to 0} z^{\rm opt}(\mu)=\lim_{\mu\to 0} z(\mu)=0. \label{B.5}
\end{equation}
Hence the inequality \eqref{B.1} holds if the derivative of $z^{\rm opt}(\mu)$
is greater or equal to the derivative of $z(\mu)$.
It is easy to prove that the first positive solution of \eqref{B.3} is a
uniquely defined function on $z\in (0,\pi/2)$ with the derivative
\begin{equation}
z'(\mu)={{2\mu}\over {\tan z+{z\over {\cos^2 z}}}}={{2\mu}\over {{{\mu^2}\over z}
+z+{{\mu^4}\over z}}}
\label{B.6}
\end{equation}
where we used \eqref{B.3} to get the final form. The derivative of $z^{\rm opt}$
can be easily found from \eqref{B.2}. We finally come up with the
necessity to prove the following inequality
\begin{equation}
F(\mu)\equiv {1\over {\sqrt{1+\mu^2}}}-{{2\mu z}\over {\mu^2+\mu^4+z^2}}\ge 0.
\label{B.7}
\end{equation}
We find first
\begin{equation}
\mu^2+\mu^4+z^2\ge 2\sqrt{\mu^2+\mu^4} z=2\mu \sqrt{1+\mu^2} z.
\label{B.8}
\end{equation}
Hence
\begin{equation}
F(\mu)\ge {1\over {\sqrt{1+\mu^2}}}-{{2\mu z}\over {2\mu \sqrt{1+\mu^2} z}}=0
\label{B.9}
\end{equation}
which proves \eqref{B.7} and, along with \eqref{B.5}, proves \eqref{B.1}.
\end{proof}
\subsection*{Acknowledgments}
The first author was supported in part by a University of Tennessee
at Chattanooga Faculty Research Grant. The third author was supported
in part by NSF DMS \# 0209562 and \# 0531865.
The authors are grateful to the referee for pointing out the
work of Henrot and Privat.
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\end{document}