Solving complex equation.

I want to create an equation in matlab and for each one variable that is unknown it can solve it directly.

For example I sign every variable to equal a number and solve for Cj.

Below is my coding to solve for Cj.

However I want to create an equation that let say if I have Cj known and I want to find H or y it can solve it without me having to creat another equation to solve for it

mdot = 115; % Source Strength g/s 
U = 6.82; % Wind speed m/s

h = 62.6; % Stack Height (m)
Sh = 25.5; % Plume Rise Height (m)
H = h + Sh; % Effective Stack Height (m)

x = 2; % X location (m)
y = 0; % Y location (m)
z = 1.49; % Z location (m)


% Dispersion Coefficients
a = 68;
b = 0.894;
c = 44.5;
d = 0.516;
f = -13;

% Dispersion Coefficients
Sigma_Y = a * (x^b);
Sigma_Z = c * (x^d) + f;

% Exp
e1 = exp((-1/2)*((y/Sigma_Y)^2 + ((z-H)/Sigma_Z)^2));
e2 = exp((-1/2)*((y/Sigma_Y)^2 + ((z+H)/Sigma_Z)^2));

% Equation for Gaussian concentration for absorbing ground
Cj1 = (mdot * e1) / (2*pi*U*Sigma_Y*Sigma_Z);
Cj_Absorbing = Cj1 * 10^6;

% Equation for Gaussian concentration for reflective ground
Cj1 = (mdot * (e1 + e2)) / (2*pi*U*Sigma_Y*Sigma_Z);
Cj_Reflecting = Cj1 * 10^6;

[Benjamin Thompson]

You could use the Symbolic Math Toolbox. There are a lot of examples here:

https://www.mathworks.com/help/symbolic/examples.html?s_tid=CRUX_topnav

[Star Strider]

Function Basics

[Torsten]

Example:

syms x y
expr = 3*x^2 + sin(y) == 0;
solx = solve(expr,x)
soly = solve(expr,y)

[Abdullah Alsayegh]

its will solve as an equation can it solve it as a number if I give all the variables = number and find one variable?

[Torsten]

syms x y
expr = 3*x^2 + sin(y) == 0;
solx = solve(expr,x);
solx = double(subs(solx,y,0.3))
soly = solve(expr,y);
soly = double(subs(soly,x,0.2))

Answer 1

[David Hill]

function Output = solveE(Input,guess)
%Input is an array [U,mdot,H,z,y,sigmaY,sigmaZ,Cj] of all values except nan
%for the variable you want to solve for.
syms U mdot H y z sigmaY sigmaZ Cj
eqn=mdot/(2*pi*U*sigmaY*sigmaZ)*exp(-0.5*((y/sigmaY)^2+((z-H)/sigmaZ)^2))-Cj==0;
switch find(isnan(Input))
    case 1
        Output=vpasolve(subs(eqn,[mdot,H,z,y,sigmaY,sigmaZ,Cj],Input(2:8)),U,guess);
    case 2
        Output=vpasolve(subs(eqn,[U,H,z,y,sigmaY,sigmaZ,Cj],Input([1,3:8])),mdot,guess);
    case 3
        Output=vpasolve(subs(eqn,[U,mdot,z,y,sigmaY,sigmaZ,Cj],Input([1:2,4:8])),H,guess);
    case 4
        Output=vpasolve(subs(eqn,[U,mdot,H,y,sigmaY,sigmaZ,Cj],Input([1:3,5:8])),z,guess);
    case 5
        Output=vpasolve(subs(eqn,[U,mdot,H,z,sigmaY,sigmaZ,Cj],Input([1:4,6:8])),y,guess);
    case 6
        Output=vpasolve(subs(eqn,[U,mdot,H,z,y,sigmaZ,Cj],Input([1:5,7:8])),sigmaY,guess);
    case 7
        Output=vpasolve(subs(eqn,[U,mdot,H,z,y,sigmaY,Cj],Input([1:6,8])),sigmaZ,guess);
    case 8
        Output=vpasolve(subs(eqn,[U,mdot,H,z,y,sigmaY,sigmaZ],Input(1:7)),Cj,guess);
end

[Abdullah Alsayegh]

Thankk you so much for taking your time.I am new to matlab and I just want to know now how I can use it if let say i have all the variables and I want to solve for y ?

[David Hill]

Added a guess for the value you are trying to find.

mdot = 115; % Source Strength g/s 
U = 6.82; % Wind speed m/s
h = 62.6; % Stack Height (m)
Sh = 25.5; % Plume Rise Height (m)
H = h + Sh; % Effective Stack Height (m)
x = 2; % X location (m)
y = 0; % Y location (m)
z = 1.49; % Z location (m)
% Dispersion Coefficients
a = 68;
b = 0.894;
c = 44.5;
d = 0.516;
f = -13;
% Dispersion Coefficients
Sigma_Y = a * (x^b);
Sigma_Z = c * (x^d) + f;

k=[U,mdot,H,z,y,Sigma_Y,Sigma_Z,nan];%move the nan to different spots to get other results

Cj=solveE(k,1)%function call

Cj =

   9.7124e-05

[Abdullah Alsayegh]

I am getting a error after running the code.

[David Hill]

Save the function, then run the above script in the command line.

[Abdullah Alsayegh]

Thank you so much for helping me on this it worked so perfect.

One quick question, if i need to set up my variable x to be x = 1:1:10 can it work on this function? because I am planning to plot this to >> plot (x,y) however the answer showing for y is >> Empty sym: 0-by-1

the plot should be similar to this picture

[Walter Roberson]

? your equation has no x, and your code has no x, so I do not see where you would put x=1:1:10 ?

If you are doing symbolic work and you put a vector into vpasolve() or solve() then those functions need to solve the set of equations simultaneously . For example if you had 3*x == z and you wanted to know z and you put in x = [1, 2] then it would be trying to solve [3 == z, 6 == z] simultaneously , which would fail. vpasolve() and solve() are for simultaneous equations .