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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">
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                <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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