Lab Session $11$: Numerical Differentiation

Use of Splines to estimate heat transfer $($Numerical Differentiation$)$

Lakes in the temperate zone become thermally stratified during the summer. Such

stratification effectively divides the lake vertically into two layers: the epilimnion and the

hypolimnion, separated by a plane called the thermocline.

The location of the Thermocline has great significance for environmental engineers. In

particular, thermocline greatly diminishes mixing between the two layers, as a result, the

decomposition of organic matter can lead to severe depletion of oxygen in the isolated

bottom waters.

The location of the thermocline can be defined as the inflection point of the temperaturedepth curve; that is$,$ the point at which $($d

$2$T$/$dz$2$

$)=0.$ It is also the point at which the

absolute value of the first derivative or gradient is a maximum.

Use cubic splines to determine the thermocline depth for a Lake for which the

Temperature vs Depth data is given as follows

z$,[$m$]-0-2\mathrm{.}3-4\mathrm{.}6-6\mathrm{.}9-9\mathrm{.}2-11\mathrm{.}5-13\mathrm{.}8-16\mathrm{.}1$

T$,[$oC$]22\mathrm{.}822\mathrm{.}722\mathrm{.}520\mathrm{.}613\mathrm{.}911\mathrm{.}711\mathrm{.}211\mathrm{.}1$

The purpose of this task is to analyze the data with the use of MATLAB spline built$-$in

function.

a$)$ First, plot the data points with MATLAB, $($notice that x$-$axis values are Temperatures,

while y$-$axis values are depth$)$

$>$ plot$($z$,$T$,$r$*)$

$($don$$t forget your axis labels and other details in your plot$)$

b$)$ on top of this plot, also plot the results of MATLAB spline interpolant using built$-$in

function $$spline$($use help to learn more on spline command$)$

zz $=$ linspace$(0,-16\mathrm{.}1)$;

TS $=$ spline$($z$,$T$,$zz$)$;

plot$($zz$,$TS$)\%$select a different color if you like

c$)$ Now use FDD$,$ BDD$,$ or CDD $($whichever is appropriate$)$ to find the $1$st and $2$nd

derivative of Temperature $($T$)$ with respect to depth $($z$)$ with $($

$$

$).$

Plot these results separately, and locate the thermocline from $$

$2/$

$2$ plot. $($Formulations

for FDD$,$ BDD$,$ and CDD are given in Figures $21\mathrm{.}3,21\mathrm{.}4,$ and $21\mathrm{.}5,$ respectively$).$

Remember Thermocline is at a depth where $($

$2/$

$2$

$)=0.$

d$)$ If the heat flux from the surface to the bottom layer can be computed with Fourier$$s

LAW

$=$

$$

$$

Compute the flux at this thermocline interface $(=0\mathrm{.}01$ cal$/($s$.$cm$.\backslash$deg C$))$

NOTE: Please check $$help spline$$ and $$help diff$$ to get more information on using built in

function