Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

[email protected] • ENGR-25_HW-01_Solution.ppt1

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

ENGR 25

Chp3Chp3Tutorial: Tutorial: Prob 3.14Prob 3.14SolutionSolution



Water Reservoir VolWater Reservoir Vol

Problem 3-14 appears to have an inconsistency• The fzero built-in function

operates on any SINGLE variable function as described on pg 156-157 of the text book.– In this case want to input a V(t)

function to determine the %-decrease time.

• The problem, however says that TWO parameters, x & r, must also be sent to the V(t) before using fzero



Use GLOBALS for x&rUse GLOBALS for x&r

We can “work around” the fzero single-var requirement for V(t) by declaring x & r as GLOBAL Variables as described on pgs 153-155 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 them being in the V-function argument



From MATLAB HelpFrom MATLAB Help global X Y Z defines X, Y,

and Z as global in scope. Ordinarily, each MATLAB

function, defined by an M-file, has its own local variables, which are separate from those of other functions, and from those of the base workspace.

However, if several functions, and possibly the base workspace, ALL declare a particular name as global, they all share a single copy of that variable. Any assignment to that variable, in any function, is available to ALL the functions declaring it global.



Physical AnalysisPhysical Analysis

The Reservoir Volume vs time Function

rtetV t 10089 11010

Now need to find the time, td, for the volume to decrease to x-percent of the initial value for a given drinking rate, r• i.e., Find td such that

initialfinald Vx

VtV100



Physical AnalysisPhysical Analysis

Now need to define a function for which zeros exist

In this case define a DIFFERENCE Function

GuessGuess

final

VVV

ortVVtV

target

Now ΔV will be ZERO whent = td so that V(td) = Vfinal

Thus the function to be ‘zeroed’ is the DIFFERENCE between Vfinal and V(t)



MATLAB GamePlanMATLAB GamePlan

Write the single variable (var = t) function deltaV(t) for use in fzero • This function receives

PARAMETERS r & x as GLOBAL vars rd & xd

Write the calling function, time_to_decrease(x,r), that calls into action fzero with deltaV as the fzero argument• The calling function sends

rd & xd to deltaV by way of the GLOBAL declaration



Fcn deltaV.mFcn deltaV.mfunction diffV = deltaV(t);% Bruce Mayer, PE * 25Sep07% ENGR25 * Problem 3-14%% Function to find Vfinal - V(t) where%% Vfinal = 1e9*(x/100)%%% x = the percent decrease relative to the 1e9 starting vol%% V(t) = 1e9+1e8(1-exp(-t/100) - r*t per eqn in P3-14%%% r = water consumption rate%% This function is to be used in the fzero built-in function% fzero accepts ONLY SINGLE variable functions%% Thus Parameters x & r MUST be passed into this fcn, deltaV.m, as GLOBALS%global rd xdV_of_t = 1e9 + 1e8*(1-exp(-t/100)) - rd*t;Vfinal = 1e9*(xd/100); %% calc DIFFERENCE between the final and V(t) valuesdiffV = Vfinal - V_of_t;



time_to_decrease(x,r).mfunction t_dec = time_to_decrease(x,r);% Bruce Mayer, PE * 25Sep07% ENGR25 * Problem 3-14%% Function to find: diffV = Vfinal - V(td) = 0 % calc diffV using Function deltaV%% Now function deltaV needs parameters x & r passed into % it as GLOBALS as discussed within deltaV.m%% need to change the names of the Globalsso that they are not%% confused with local variables x & rglobal rd xdrd = rxd = x%% calc time to decrease stored volume by x-percent using% fzero on diffV caluation using fcn deltaV%% use general guess of 10 days for the diffv(t_dec) = 0t_dec = fzero('deltaV', 10);



Cmd Window SessionCmd Window Session

For x = 50%, and r = 1e7 liters per day>> td = time_to_decrease(50, 1e7)rd = 10000000xd = 50t_dec = 54.1832td = 54.1832

Thus for a 1e7 liter/day consumption rate, a decrease to 50% of the initial volume takes about 54 days



Cmd Window SessionCmd Window Session

For x = 73%, and r = 4e6 liters/day>> td = time_to_decrease(73, 4e6)rd = 4000000xd = 73t_dec = 81.4258td = 81.4258

Thus for a 4e6 liter/day consumption rate, a decrease to 73% of the initial volume volume takes about 81 days



Plot for r = 1e7Plot for r = 1e7

0 20 40 60 80 100 120-20

0

20

40

60

80

100

Time (days)

Re

srv

oir

Ca

pa

cit

y (

%)

* r

- 1

e7

50 %

54d



Appendix – UseFul PlotsAppendix – UseFul Plots% Bruce Mayer, PE * 22Feb09 * Rev 13Sep09 * Rev 14Sep10% ENGR25 * Problem 3-14% Prob3_14_Plot_1009.m%r = 1e7; % set CONSUMPTION RATE HERE %% the Reservoir volume function - u is dummy variable for tVres = @(u) 1e9 + 1e8*(1 - exp(-u/100)) - r*u%t = linspace(0, 200);Voft = Vres(t);plot(t, Voft), grid, title('Unit-filled plot')display('showing unit plot - hit any key to continue')pause%Vpct = 100*Voft/Vres(0);plot(t, Vpct), grid, title('Normalized Plot')%display('showing NORMALIZED plot - hit any key to continue')pause% Compare Exponential and Linear TermseTerm = 1e8*(1 - exp(-t/100));LinTerm = r*t;plot(t, eTerm, t, LinTerm), grid, legend('eTerm, LinTerm')%display('showing unit-filled comparison plot - hit any key to continue')pause%%Compare Exponential and Linear Terms by RATIOLToverET = LinTerm./eTerm;plot(t, LToverET), grid, title('LinTerm/eTerm Plot')%display('showing linTerm/eTerm RATIO comparison plot - hit any key to continue‘)

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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