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<div class="moz-cite-prefix">Hello,<br>
<br>
To be honest, I do not understand your problem, sorry. There are
two ordinary differential equations<br>
<div style="text-align: justify;" class="yiv7093776338MsoNormal">d<span>υ/dt
=
0.0107h – 0.00205υ<sup>2 </sup>…………………eqn(1)</span></div>
<div style="text-align: justify;" class="yiv7093776338MsoNormal"><span>dh/dt
=
0.311 – 0.0624υ………………………...eqn(2)</span></div>
<br>
what is the interval of integration t=[0,?] ?<br>
what are the initial conditions v, h for t=0 ?<br>
I suppose that v,h are changed somehow in t=[0,?]. In what way?<br>
<br>
Regards,<br>
Radovan<br>
<br>
On 8/9/2013 7:00 PM, Debola Abduljeleel wrote:<br>
</div>
<blockquote
cite="mid:1376067655.31907.YahooMailNeo@web160304.mail.bf1.yahoo.com"
type="cite">
<div style="color:#000; background-color:#fff; font-family:lucida
console, sans-serif;font-size:14pt">
<div><font size="2" face="Arial"><br>
</font><span><var id="yui-ie-cursor"></var>Hi All,</span></div>
<div style="font-family: lucida console, sans-serif; font-size:
14pt;">
<div style="font-family: times new roman, new york, times,
serif; font-size: 12pt;">
<div class="y_msg_container">
<div id="yiv7093776338">
<div>
<div style="color: rgb(0, 0, 0); font-family: lucida
console, sans-serif; font-size: 14pt;
background-color: rgb(255, 255, 255);">
<div style="color: rgb(0, 0, 0); font-family: lucida
console, sans-serif; font-size: 18.66px;
font-style: normal; background-color:
transparent;"><span><br>
</span></div>
<div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal">I humbly seek
kind <span> </span>assistance in solving a
Simulation Problem (GRAVITY
FLOW TANK) that was initially written <span> </span>with
FORTRAN to numerically integrate two ODEs
(to be given below, eqns 1&2).</div>
<div class="yiv7093776338MsoNoSpacing"><b>Gravity-flow
tank
data:</b></div>
<div class="yiv7093776338MsoNoSpacing"><i>Pipe:</i></div>
<div class="yiv7093776338MsoNoSpacing">Internal
Diameter = 3 ft<span> </span>Area
= 7.06 ft<sup>2<span> </span></sup>Length
= 3000 ft</div>
<div class="yiv7093776338MsoNoSpacing"><i>Tank:</i></div>
<div class="yiv7093776338MsoNoSpacing">Internal
Diameter = 12ft<span> </span>Area
= 113 ft<sup>2<span> </span></sup>Height
=
7 ft</div>
<div class="yiv7093776338MsoNoSpacing"><b> </b></div>
<div class="yiv7093776338MsoNoSpacing"><b>Steady
State
values:</b></div>
<div class="yiv7093776338MsoNoSpacing">F = 35.1 ft<sup>3</sup>/s
(15, 700 gpm)</div>
<div class="yiv7093776338MsoNoSpacing">h = 4.72 ft</div>
<div class="yiv7093776338MsoNoSpacing"><span>υ</span>
= 4.97 ft/s</div>
<div class="yiv7093776338MsoNoSpacing"> </div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal">d<span>υ/dt
= 0.0107h – 0.00205υ<sup>2 </sup>…………………eqn(1)</span></div>
<div style="text-align: justify; color: rgb(0, 0,
0); font-size: 16.21px; font-style: normal;
background-color: transparent;"
class="yiv7093776338MsoNormal"><span><br>
</span></div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal"><span>dh/dt
= 0.311 – 0.0624υ………………………eqn(2)</span></div>
<div style="text-align: justify; color: rgb(0, 0,
0); font-size: 18.66px; font-style: normal;
background-color: transparent;"
class="yiv7093776338MsoNormal"><span><br>
</span></div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal"><span>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 style="text-align: justify;"
class="yiv7093776338MsoNormal"><span>Gravity-flow
tank
simulation (FORTRAN)</span></div>
<div class="yiv7093776338MsoNoSpacing">C<span>
</span><span> </span>(TIME IN
SECONDS)</div>
<div class="yiv7093776338MsoNoSpacing">C TWO CASES
ARE RUN. AT TIME EQUAL ZERO THE FEED FLOW
RATE IS</div>
<div class="yiv7093776338MsoNoSpacing">C<span>
</span><span> </span>INCREASED TO 100% OF THE
DESIGN FLOW RATE
(15, 700 GPM)</div>
<div class="yiv7093776338MsoNoSpacing">C<span>
</span><span> </span>CASE NO.1 STARTS FROM
67% OF DESIGN FLOW
RATE</div>
<div class="yiv7093776338MsoNoSpacing">C<span>
</span><span> </span>CASE NO.2 STARTS FROM
50% OF DESIGN FLOW
RATE</div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>DATA
V1,H1/3.40,2.05/</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>DATA
V2,H2/2.50,1.2/</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>DO 200
N=1,2</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>IF(N.=EQ.1)THEN</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>V=V1</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>H=H1</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>ELSE</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>V=V2</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>H=H2</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"></span><span
class="yiv7093776338tab"></span><span
class="yiv7093776338tab"></span>ENDIF</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>TIME=0.</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>DELTA=1.</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>TPRINT=0.</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>WRITE(6,1)V,H</div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>1
FORMAT(‘INITIAL CONDITIONS: V= ‘,F6.3,’
H=’,F8.3)</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>WRITE(6,2)</div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>2 FORMAT(‘<span> </span>TIME<span>
</span>V<span> </span>H’)</div>
<div class="yiv7093776338MsoNoSpacing">C</div>
<div class="yiv7093776338MsoNoSpacing">C EVALUATE
DERIVATIVES</div>
<div class="yiv7093776338MsoNoSpacing">C</div>
<div class="yiv7093776338MsoNoSpacing"><span> </span>100
VDOT=0.0107*H-0.00205*V**2</div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>HDOT=0.311-0.0624*V</div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>IF(<a moz-do-not-send="true"
href="http://time.lt.tp/" target="_blank">TIME.LT</a>.TPRINT)GO
TO 10</div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>WRITE(6,3)TIME,V,H</div>
<div class="yiv7093776338MsoNoSpacing"><span
class="yiv7093776338tab"> </span><span
class="yiv7093776338tab"> </span>3
FORMAT(3X,3F8.2)</div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>TPRINT=TPRINT+20.</div>
<div class="yiv7093776338MsoNoSpacing">C</div>
<div class="yiv7093776338MsoNoSpacing">C INTEGRATE
USING EULER</div>
<div class="yiv7093776338MsoNoSpacing">C</div>
<div class="yiv7093776338MsoNoSpacing"><span> </span>10
V = V+
VDOT+VDELTA</div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>H=
H+HDOT+DELTA</div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>TIME=TIME+DELTA</div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>IF(TIME.LE.200.)GO
TO 100</div>
<div class="yiv7093776338MsoNoSpacing">200
CONTINUE</div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>STOP </div>
<div class="yiv7093776338MsoNoSpacing"><span>
</span>END</div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal"><span> </span></div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal"><span>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 style="text-align: justify;"
class="yiv7093776338MsoNormal"><span>(VDOT)<sub>n
</sub>= 0.0107(H)<sub>n </sub>– 0.00205[(V)<sub>n</sub>]<sup>2
</sup>…………………………….eqn(3)</span></div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal"><span>(HDOT)<sub>n
</sub>= 0.311 – 0.0624(V)<sub>n</sub>…………………………………….eqn(4)</span></div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal"><span>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 style="text-align: justify;"
class="yiv7093776338MsoNormal"><span>(H)<sub>n+1
</sub>= (H)<sub>n </sub>+ DELTA(HDOT)<sub>n</sub>……………………………eqn(5)</span></div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal"><span>(V)<sub>n+1
</sub>= (V)<sub>n</sub> +DELTA(VDOT)<sub>n</sub>……………………………eqn(6)</span></div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal"><span>A
plot was equally generated to illustrate the
relationship between h, ft and
process time, s.</span></div>
<div style="text-align: justify; color: rgb(0, 0,
0); font-size: 18.66px; font-style: normal;
background-color: transparent;"
class="yiv7093776338MsoNormal"><span><br>
</span></div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal"><span>Thanks
as I anticipate various professional inputs.</span></div>
<div style="text-align: justify;"
class="yiv7093776338MsoNormal"> </div>
</div>
<div>ABDULJELEEL A. OSUNKUNLE<br>
SCIENTIFIC OFFICER<br>
PROJECTS DEVELOPMENT INSTITUTE, P.M.B. 01609,
EMENE INDUSTRIAL LAYOUT, ENUGU, NIGERIA.<br>
+2347058408440.</div>
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