The Three Reservoir Problem - Maple Programming Help

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The Three Reservoir Problem

Introduction

Three reservoirs at different elevations are connected through a piping network at a single point, with an outflow from the common junction.  This application will calculate the flow rates, flow directions, and head at the common junction.

The Bernoulli equation (ignoring the losses associated with pipe fittings) for Reservoir 1 is

where z1 is the reservoir elevation, f1 is the friction factor, L1 is the pipe length, D1 is the pipe diameter, V1 is the liquid velocity, and H is the head at the common junction.  However, since flow can be either into or out of a reservoir (that is, a positive or negative velocity), we rewrite the Bernoulli equation thus

Similar equations can be defined for the other reservoirs.  These equations, along with the continuity equation, 

${\mathrm{Q}}_{1}+{\mathrm{Q}}_{2}+{\mathrm{Q}}_{3}={\mathrm{Q}}_{\mathrm{outflow}}$

and a correlation for the friction factor, can be used to find the flow rates, flow directions, and the head at the junction.

 > $\mathrm{restart}:$$\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{with}\left(\mathrm{ThermophysicalData}\right):$

Parameters

Liquid viscosity and density

 > $\mathrm{μ}≔\mathrm{Property}\left("viscosity","water","pressure"=1⟦\mathrm{atm}⟧,"temperature"=298⟦\mathrm{K}⟧\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$$\mathrm{ρ}≔\mathrm{Property}\left("density","water","pressure"=1⟦\mathrm{atm}⟧,"temperature"=298⟦\mathrm{K}⟧\right)$
 ${0.0008930728887}{}⟦{\mathrm{Pa}}{}{\mathrm{s}}⟧$
 ${997.0860088}{}⟦\frac{{\mathrm{kg}}}{{{\mathrm{m}}}^{{3}}}⟧$ (2.1)

Reservoir elevations

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Lengths, diameters, and cross-sectional areas of the pipes connecting the reservoirs to the junction

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Roughness of the pipe

 > $\mathrm{e}≔0.0005⟦\mathrm{m}⟧:$

Outflow from the junction

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Gravitational constant

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Bernoulli Equations for the Three Reservoir System

Bernoulli analysis of the system

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Calculation of Flow Rates, Flow Directions, and Junction Head

Positive flow rates indicate flow out of a reservoir, while negative flow rates indicate flow into a reservoir

 > $\mathrm{Digits}≔20:$
 > $\mathrm{results}≔\mathrm{fsolve}\left(\left\{\mathrm{sys}\right\},\left\{\mathrm{estimates}\right\}\right)$
 $\left\{{\mathrm{H}}{=}{83.061763740932816446}{}⟦{\mathrm{m}}⟧{,}\mathrm{Q__1}{=}{0.066687156331014103433}{}⟦\frac{{{\mathrm{m}}}^{{3}}}{{\mathrm{s}}}⟧{,}\mathrm{Q__2}{=}{0.024938188343994482394}{}⟦\frac{{{\mathrm{m}}}^{{3}}}{{\mathrm{s}}}⟧{,}\mathrm{Q__3}{=}{-}{0.061625344675008585827}{}⟦\frac{{{\mathrm{m}}}^{{3}}}{{\mathrm{s}}}⟧{,}\mathrm{Rey__1}{=}{3.1599250299840063132}{}{{10}}^{{5}}{,}\mathrm{Rey__2}{=}{1.4180146816355551248}{}{{10}}^{{5}}{,}\mathrm{Rey__3}{=}{3.5040894833508434552}{}{{10}}^{{5}}{,}\mathrm{fric__1}{=}{0.022941390153041356289}{,}\mathrm{fric__2}{=}{0.024556386647882404062}{,}\mathrm{fric__3}{=}{0.023923865415019612453}\right\}$ (4.1)
 >