1. Chp3 Tutorial: Prob 3.14 Solution
Engineering 25
Chp3 Tutorial: Prob 3.14 Solution
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. P3.14 Water Reservoir Vol
Using estimates of rainfall, evaporation, and water consumption, the town engineer developed the following model of the water volume in the reservoir as a function of time
where V is the water volume in liters, t is time in days, and r is the town's consumption rate in liters per day. Write TWO user-defined functions. The first function should define the function V(t) for use with the fzero function. The second function should use fzero to compute how long it will take for the water volume to decrease to x percent of its initial value of 109 L. The inputs to the second function should be x and r. Test your functions for the case where x = 50 percent and r = 107 L/day.

3. A NearBy Reservoir

4. A NearBy Reservoir
This 315-acre anglers oasis sits in the heart of Suburban Castro Valley but, you'd never know it from your surroundings once inside the park. Miles of parkland and trails make Chabot a haven for outdoor enthusiasts. Hikers can enjoy scenic walks on the 280-acre Fairmont Ridge. Park trails are designed for shared useage. Lake Chabot is well known among the running community. The park hosts Half Marathons, 5K's and the Kids Fun Run. In fact, Trail Runner magazine chose this setting as one of America's Most Scenic Races.

5. Water Reservoir Vol Problem 3-14 appears to have an inconsistency
The fzero built-in function operates on any SINGLE VARIABLE function as described in Tab of the text book.
In this case want to input a V(t), function, NOT a V(t,x,r) function to determine the %-decrease time.
The problem, however, says that TWO parameters, x & r, must also be sent to the V(t) function before using fzero

6. Use GLOBALS for x&r
We can “work around” the fzero single-varialbe requirement for V(t) by declaring x & r as GLOBAL Variables as described on pg 124 of the Text
Declaring x & r as globals in BOTH the calling function and the V(t) function gives the single-Var function, V(t), access to r & x withOUT listing them in the V-function argument

7. Physical Analysis The Reservoir Volume vs t Function
Now need to find the time, td, for the volume to decrease TO x-percent of the initial Volume for a given water consumption rate, r
i.e, Find td such that

8. Physical Analysis Now need to define a function for which zeros exist
In this case define a DIFFERENCE Function
Want to ZERO Vguess - Vcalc
Now ΔV will be ZERO when t = td so that V(td) = Vfinal
Thus the function to be zeroed is the DIFFERENCE between Vfinal (the Goal) and V(t) (the Guess)
Want: Vgoal − Vguess = 0

9. Guess&Check Demo Make a Time Guess, tGuess Calc VGuess = V(tGuess)
Calc % Full for the Guess
Calc Difference
Guess AGAIN

10. Guess&Check Demo First Trial Run (Cmd Window) >> r = 1e7 r =
>>
>> VtoverVi (1e9 + 1e8*(1 - exp(-t/100)) - r*t)/1e9
VtoverVi =
@(t)(1e9+1e8*(1-exp(-t/100))-r*t)/1e9
>> Goal = 0.55
Goal =
0.5500
>> Guess31 = VtoverVi(31)
Guess31 =
0.7167
>> del31 = Goal - Guess31
del31 =

11. Guess & Check Ans >> P3_14_ReservoirVolume_1302 r = 10000000
VtoverVi =
@(t)(1e9+1e8*(1-exp(-t/100))-r*t)/1e9
Goal =
0.5500
Guess31 =
0.7167
del31 =
Guess62 =
0.4262
del62 =
0.1238
Guess47 =
0.5675
del47 =
Guess48 =
0.5581
del48 =
Guess49 =
0.5487
del49 =
0.0013

12. Guess&Check .m-file % Bruce Mayer, PE % ENGR25 * 12Feb13
% Problem 3_14
% P3_14_ReservoirVolume_1302.m
r = 1e7 % consumption rate
% Create ANON fcn to return % of Initial Vol
VtoverVi (1e9 + 1e8*(1 - exp(-t/100)) - r*t)/1e9
Goal = 0.55 % to 55% of initial Volume
Guess31 = VtoverVi(31)
del31 = Goal - Guess31
Guess62 = VtoverVi(62)
del62 = Goal - Guess62
Guess47 = VtoverVi(47)
del47 = Goal - Guess47
Guess48 = VtoverVi(48)
del48 = Goal - Guess48
Guess49 = VtoverVi(49)
del49 = Goal - Guess49

13. MATLAB GamePlan
Write the single variable (var = t) function deltaV(t) for use in fzero
This function receives parameters r & x as GLOBAL variables rd & xd
Write the calling function, time_to_decrease(x,r), that calls into action fzero with deltaV as its argument
This calling function sends rd & xd to deltaV by way of the GLOBAL declaration

14. Structure Chart
Note How GLOBALS ByPass the Normal Communication Path

15. Fcn deltaV.m

16. time_to_decrease(x,r).m
Only 4 active commands

17. The Calling Program

18. Example Call User Inputs
x = 50 & r = 1e7
Takes 54.2 days to drop to 50% initial for use rate of 1e7 Liters per Day
Percent of Initial Storage, x = 50
Consumption Rate in L/day, r = 1e7
for x = 50
for r =
The time to Decrease in Days =

19. Cmd Window Session For x = 50%, and r = 1e7 liters per day
>> td50_1e7 = time_to_decrease(50,1e7)
Rd =
Xd = 50
t_dec =
td50_1e7 =
Thus for a 1e7 liter/day consumption rate, a 50% decrease in volume takes about 54 days

20. Cmd Window Session For x = 27%, and r = 4e6 liters/day
>> td27_4e6 = time_to_decrease(27,4e6)
Rd =
Xd = 27
t_dec =
td27_4e6 =
Thus for a 4e6 liter/day consumption rate, a decrease to 27% of the initial volume takes about 204 days

21. DeltaV Plot Anonymous fcn For x = 27%, and r = 4e6 liters/day
dV = Goal – Guess
Goal = 1e9*(27/100)
Guess = (1e9 + 1e8*(1-exp(-tg/100)) - 4e6*tg);
>> dV27_4e6 (1e9*(1-73/100 )) - (1e9 + 1e8*(1-exp(-tg/100)) - 4e6*tg); % Vfinal - Vguess
>> tplot = linspace(-350,350,700);
>> VolDiff = dV27_4e6(tplot);
>> plot(tplot, VolDiff, 'LineWidth', 3), grid, xlabel('t (days)'), ylabel('DeltaV (L)'), title('P3-14: Need to ZERO‘)

22. DeltaV Plot
Note that fzero can Return ERRONEOUS (negative) Results if Start-Pt is Off

23. Erroneous result For x = 65%, and r = 1e6 liters/day
>> td65_1e6 = time_to_decrease(65,1e6)
Rd =
Xd = 65
t_dec =
td65_1e6 =
For THIS SET of Parameters fzero found a NEGTIVE (erroneous Root)

24. Erroneous DeltaV Plot fzero found the negative root
When in Doubt PLOT it
fzero found the negative root
The can be fixed by changing the starting point to 200

25. Erroneous result For x = 65%, and r = 1e6 liters/day
Change This Line in the time_to_decrease code
t_dec = fzero('deltaV', 200)
The new (Mathimatically Correct) result
>> td65_1e6 = time_to_decrease(65,1e6)
Rd =
Xd = 65
t_dec =
td65_1e6 =

26. Erroneous Comment
Even though Days is correct by the Equation, it is NOT physically reasonable
Since Weather is CYCICAL on a 365 day basis, anything over a Year violates nature and is thus NOT Applicable
Some Advice
Always do a Reality Check on the Results returned by ANY computer program
When in Doubt PLOT