<html><body><div style="color:#000; background-color:#fff; font-family:lucida console, sans-serif;font-size:14pt"><div><span><br>Hiall,</span></div><div style="color: rgb(0, 0, 0); font-size: 18.6667px; font-family: lucida console,sans-serif; background-color: transparent; font-style: normal;"><span><br></span></div><div><!--[if gte mso 9]><xml>
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<div class="MsoNormal" style="text-align:justify">I humbly seek kind <span style="mso-spacerun:yes"> </span>assistance in solving a Simulation Problem (GRAVITY
FLOW TANK) that was initially written <span style="mso-spacerun:yes"> </span>with FORTRAN to numerically integrate two ODEs
(to be given below, eqns 1&2).</div>
<div class="MsoNoSpacing"><b style="mso-bidi-font-weight:normal">Gravity-flow tank
data:</b></div>
<div class="MsoNoSpacing"><i style="mso-bidi-font-style:normal">Pipe:</i></div>
<div class="MsoNoSpacing">Internal Diameter = 3 ft<span style="mso-tab-count:2"> </span>Area
= 7.06 ft<sup>2<span style="mso-tab-count:2"> </span></sup>Length
= 3000 ft</div>
<div class="MsoNoSpacing"><i style="mso-bidi-font-style:normal">Tank:</i></div>
<div class="MsoNoSpacing">Internal Diameter = 12ft<span style="mso-tab-count:2"> </span>Area
= 113 ft<sup>2<span style="mso-tab-count:2"> </span></sup>Height
= 7 ft</div>
<div class="MsoNoSpacing"><b style="mso-bidi-font-weight:normal"> </b></div>
<div class="MsoNoSpacing"><b style="mso-bidi-font-weight:normal">Steady State
values:</b></div>
<div class="MsoNoSpacing">F = 35.1 ft<sup>3</sup>/s (15, 700 gpm)</div>
<div class="MsoNoSpacing">h = 4.72 ft</div>
<div class="MsoNoSpacing"><span style="font-family:"Times New Roman","serif"">υ</span>
= 4.97 ft/s</div>
<div class="MsoNoSpacing"> </div>
<div class="MsoNormal" style="text-align:justify">d<span style="font-family:"Times New Roman","serif"">υ/dt
= 0.0107h – 0.00205υ<sup>2 </sup>…………………eqn(1)</span></div><div class="MsoNormal" style="text-align: justify; color: rgb(0, 0, 0); font-size: 16.2167px; font-family: "Times New Roman","serif"; background-color: transparent; font-style: normal;"><span style="font-family:"Times New Roman","serif""><br></span></div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif"">dh/dt
= 0.311 – 0.0624υ………………………eqn(2)</span></div><div class="MsoNormal" style="text-align: justify; color: rgb(0, 0, 0); font-size: 18.6667px; font-family: "Times New Roman","serif"; background-color: transparent; font-style: normal;"><span style="font-family:"Times New Roman","serif""><br></span></div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif"">The
following FORTRAN program numerically integrates the two ODEs describing the
system for two different initial conditions of flow and liquid level in the
tank: (1) when the initial flow rate is 50 percent of the design rate, and (2)
when the initial flow rate is 67 percent of the design flow rate. At time equal
zero, the flow rate into the tank is increased to the maximum design flow rate
of 35.1 ft<sup>3</sup>/s.</span></div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif"">Gravity-flow
tank simulation (FORTRAN)</span></div>
<div class="MsoNoSpacing">C<span style="mso-tab-count:1"> </span><span style="mso-tab-count:1"> </span>(TIME IN SECONDS)</div>
<div class="MsoNoSpacing">C TWO CASES ARE RUN. AT TIME EQUAL ZERO THE FEED FLOW
RATE IS</div>
<div class="MsoNoSpacing">C<span style="mso-tab-count:1"> </span><span style="mso-spacerun:yes"> </span>INCREASED TO 100% OF THE DESIGN FLOW RATE
(15, 700 GPM)</div>
<div class="MsoNoSpacing">C<span style="mso-tab-count:1"> </span><span style="mso-spacerun:yes"> </span>CASE NO.1 STARTS FROM 67% OF DESIGN FLOW
RATE</div>
<div class="MsoNoSpacing">C<span style="mso-tab-count:1"> </span><span style="mso-spacerun:yes"> </span>CASE NO.2 STARTS FROM 50% OF DESIGN FLOW
RATE</div>
<div class="MsoNoSpacing"><span style="mso-tab-count:1"> </span>DATA
V1,H1/3.40,2.05/</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>DATA V2,H2/2.50,1.2/</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>DO 200 N=1,2</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>IF(N.=EQ.1)THEN</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>V=V1</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>H=H1</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>ELSE</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>V=V2</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>H=H2</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"></span><span class="tab"></span><span class="tab"></span>ENDIF</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>TIME=0.</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>DELTA=1.</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>TPRINT=0.</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>WRITE(6,1)V,H</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes"> </span>1
FORMAT(‘INITIAL CONDITIONS: V= ‘,F6.3,’ H=’,F8.3)</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span><span class="tab"> </span>WRITE(6,2)</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes"> </span>2 FORMAT(‘<span style="mso-tab-count:1"> </span>TIME<span style="mso-tab-count:
1"> </span>V<span style="mso-tab-count:1"> </span>H’)</div>
<div class="MsoNoSpacing">C</div>
<div class="MsoNoSpacing">C EVALUATE DERIVATIVES</div>
<div class="MsoNoSpacing">C</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes"> </span>100
VDOT=0.0107*H-0.00205*V**2</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes">
</span>HDOT=0.311-0.0624*V</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes">
</span>IF(TIME.LT.TPRINT)GO TO 10</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes"> </span>WRITE(6,3)TIME,V,H</div>
<div class="MsoNoSpacing"><span class="tab"> </span><span class="tab"> </span>3 FORMAT(3X,3F8.2)</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes">
</span>TPRINT=TPRINT+20.</div>
<div class="MsoNoSpacing">C</div>
<div class="MsoNoSpacing">C INTEGRATE USING EULER</div>
<div class="MsoNoSpacing">C</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes"> </span>10 V = V+
VDOT+VDELTA</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes"> </span>H=
H+HDOT+DELTA</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes"> </span>TIME=TIME+DELTA</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes"> </span>IF(TIME.LE.200.)GO
TO 100</div>
<div class="MsoNoSpacing">200 CONTINUE</div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes"> </span>STOP </div>
<div class="MsoNoSpacing"><span style="mso-spacerun:yes"> </span>END</div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif""> </span></div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif"">The
explicit first-order Euler algorithm is used. The variables that we are solving
for as functions of time are V and H. the right-hand sides of eqns (1) and (3)
are the derivative functions. These are called VDOT and HDOT in the program. At
the nth step in time</span></div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif"">(VDOT)<sub>n
</sub>= 0.0107(H)<sub>n </sub>– 0.00205[(V)<sub>n</sub>]<sup>2 </sup>…………………………….eqn(3)</span></div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif"">(HDOT)<sub>n
</sub>= 0.311 – 0.0624(V)<sub>n</sub>…………………………………….eqn(4)</span></div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif"">The
new values of H and V at the (n+1)st step are calculated from the Euler
algorithm with a step size of DELTA.</span></div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif"">(H)<sub>n+1
</sub>= (H)<sub>n </sub>+ DELTA(HDOT)<sub>n</sub>……………………………eqn(5)</span></div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif"">(V)<sub>n+1
</sub>= (V)<sub>n</sub> +DELTA(VDOT)<sub>n</sub>……………………………eqn(6)</span></div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif"">A
plot was equally generated to illustrate the relationship between h, ft and
process time, s.</span></div><div class="MsoNormal" style="text-align: justify; color: rgb(0, 0, 0); font-size: 18.6667px; font-family: "Times New Roman","serif"; background-color: transparent; font-style: normal;"><span style="font-family:"Times New Roman","serif""><br></span></div>
<div class="MsoNormal" style="text-align:justify"><span style="font-family:"Times New Roman","serif"">Thanks
as I anticipate various professional inputs.</span></div>
<div class="MsoNormal" style="text-align:justify"> </div>
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<![endif]--></div><div>ABDULJELEEL A. OSUNKUNLE<br>SCIENTIFIC OFFICER<br>PROJECTS DEVELOPMENT INSTITUTE, P.M.B. 01609, EMENE INDUSTRIAL LAYOUT, ENUGU, NIGERIA.<br>+2347058408440.</div></div></body></html>