10.6 二個桶子間之水位問題

不同水位之兩個桶子,其下用管路相通。當其開關打開後，兩桶之水位隨時間而變化，水位高著向水位低者流動，其流動速率與兩桶水位差成正比。設每公分之水位差之流動量為Ｃ[cm³]，兩桶之斷面直徑分別為[D₁、D₂]cm，水位高度分別為[U₁、U_２]cm，則在管路之流動量Ｖ為：

  V=C*H*(U1-U2)
　U1=U1-V/A1
  U2=U2+V/A2

式中，H為每次計算時之時段，A₁A₂分別為水桶之斷面積，U₁、U₂則隨著時間而變化。

程式設計

下面之程式係以上式為計算基礎，其輸入分三種方式，即使用預設資料、使用參數輸入與逐項輸入之方式：

預設資料：pail2pail
使用參數輸入：pail2pail(U,D,C,Delt)
逐項輸入：pail2pail(1)

程式內容

function pail2pail(U,D,C,Delt)
% pail2pail.m
% program to calculate waterflows between two pails.
% U1,U2:water levels for each pails, cm
% D1,D2:diameters of both pails, cm
% T:time elapsed through the experiment.
% C:flow coefficent, in cm^3/cm of water level difference.
% designed by D.S. Fon, BIME, NTU. Date:Nov. 26,2006
if nargin==0,
U1=20;U2=10;D1=5;D2=5;C=1;delt=1;
elseif U==1,
U1=inopt1('A. Input water level in Pail No.1,cm',[50 0 500]);
U2=inopt1('B. Input water level in Pail No.2,cm',[0 0 500]);
D1=inopt1('C. Input diameter of pail 1,cm',[5 0 500]);
D2=inopt1('D. Input diameter of pail 2,cm',[5 0 500]);
C=inopt1('E. Input flow coefficient,cm^3/cm',[1 0 10]);
delt=inopt1('F. Input time interval,seconds',[1 0.1 10]);
else
U1=U(1);U2=U(2);D1=D(1);D2=D(2);
end
A1=pi*D1^2/4;A2=pi*D2^2/4;
T=0;i=0;
clf;
disp('  Time      Pail 1 Level  Pail 2 Level');
disp('========    ============  ============');
while abs(U2-U1)>delt,
fprintf(1,'%8.3f    %10.4f  %10.4f\n',T,U1,U2);
T=T+delt;i=i+1;
vel=C*delt*(U1-U2);
U1=U1-vel/A1;U2=U2+vel/A2;
f1(i)=U1;f2(i)=U2;Tx(i)=T;
line(Tx(i),f1(i),'Marker','o');
line(Tx(i),f2(i),'Marker','s');
end
j=fix(i/3);
line(Tx,f1,'color','red');
line(Tx,f2,'color','blue');
line([0 Tx(end)],[(f1(1)+f2(1))/2 (f1(1)+f2(1))/2]);
text(Tx(j),f1(j),'\leftarrow---U1');
text(Tx(j),f2(j),'\leftarrow---U2');
xlabel('Time, seconds');
ylabel('Water levels, U1 & U2');

function [kk]=inopt1(A,s)
while 1
      mtxt=['(',num2str(s(2)),'-',num2str(s(3)),')[',...
          num2str(s(1)),']=='];
      kk=input([A,mtxt]);
      if isempty(kk), kk=s(1);end
      if kk>=s(2)|kk<=s(3), return;end
      disp('*****Exceeding the limit, please reinput*****')
end

執行範例

>> pail2pail(1)
A. Input water level in Pail No.1,cm(0-500)[50]==>
B. Input water level in Pail No.2,cm(0-500)[0]==>
C. Input diameter of pail 1,cm(0-500)[5]==>
D. Input diameter of pail 2,cm(0-500)[5]==>
E. Input flow coefficient,cm^3/cm(0-10)[1]==>
F. Input time interval,seconds(0.1-10)[1]==>
Time      Pail 1 Level  Pail 2 Level
========    ============  ============
0.000       50.0000      0.0000
1.000       47.4535      2.5465
2.000       45.1664      4.8336
3.000       43.1123      6.8877
4.000       41.2674      8.7326
5.000       39.6104     10.3896
6.000       38.1222     11.8778
7.000       36.7856     13.2144
8.000       35.5851     14.4149
9.000       34.5069     15.4931
10.000       33.5386     16.4614
11.000       32.6688     17.3312
12.000       31.8877     18.1123
13.000       31.1861     18.8139
14.000       30.5560     19.4440
15.000       29.9901     20.0099
16.000       29.4818     20.5182
17.000       29.0253     20.9747
18.000       28.6153     21.3847
19.000       28.2470     21.7530
20.000       27.9163     22.0837
21.000       27.6192     22.3808
22.000       27.3524     22.6476
23.000       27.1128     22.8872
24.000       26.8976     23.1024
25.000       26.7043     23.2957
26.000       26.5307     23.4693
27.000       26.3748     23.6252
28.000       26.2348     23.7652
29.000       26.1090     23.8910
30.000       25.9960     24.0040
31.000       25.8946     24.1054
32.000       25.8035     24.1965
33.000       25.7216     24.2784
34.000       25.6481     24.3519
35.000       25.5821     24.4179
36.000       25.5228     24.4772

這是產生的兩桶之水位變化圖