Help me convert this Equation

i converted this equation before and got this figure.

I forgot how but i lost the file that i wrote the code to get the figure above.

below i am trying to rewrtie it but it is not working. the exponential function is zero or infinity.

Edit!

this is the original equation for the R_e.

i checked the parameters with my report and they are the same.

maybe there is a problem here too?

i did what you asked about the final denominator, still the same results.

clc
Contact_angle = 2.4783666516568;
Volume = 5;
ST_L = 72;
Density = 0.973;
Gravity = 9.807;
Shape_factor = 37.1;
T_zero = 0;
Dynamic_viscosity = 8.9e-4;
R_e = ( 4*Volume / (pi*Contact_angle) )^1/3;
T = 0:0.1:6;
R = R_e*(1 - exp(-(2*ST_L/R_e^12 + Density*Gravity/(9*R_e^10))*(24*Shape_factor*Volume^4*(T+T_zero)/(pi^2*Dynamic_viscosity)))).^(1/6);
plot(T,R);

Thank you All for the help

[Dyuman Joshi]

You need to club the pi^2*neta together in the denominator to correct the definition.

But even after the correction, the result is not similar to the figure above. Are you sure the values are correct?

%Value edited
Contact_angle = 2.4783666516568;%142
Volume = 5;
ST_L = 72;
Density = 0.973;
Gravity = 9.807;
Shape_factor = 37.1;
T_zero = 0;
Dynamic_viscosity = 8.9e-4;
R_e = ( 400*Volume / (pi*Contact_angle))^(1/3);
%T according to the figure
T = 0:6;
R = R_e*(1 - exp(-(2*ST_L/R_e^12 + Density*Gravity/(9*R_e^10))*24*Shape_factor*Volume^4*(T+T_zero)/(pi^2*Dynamic_viscosity))).^(1/6);
plot(T,R)

[shammas mohamed]

the resutls should be in mm or m. i tried converting all of them but i am still facing the same problem

[Dyuman Joshi]

What are the units of each parameter used?

[shammas mohamed]

Volume = mm^3;

ST_L = mN/m;

Density = 0.973g/cm^3;

Gravity = m/s^2;

Shape_factor = 1/m;

Dynamic_Viscosity = Pa s

[Dyuman Joshi]

After correcting the units, you get a similar graph -

%Converting to SI units
Contact_angle = 142*pi/180;
Volume = 5 * 1e-9; %m^3;
ST_L = 72 * 1e-3; %N.m
Density = 0.973 * 1e3; %kg/m^3
Gravity = 9.807;
Shape_factor = 37.1;
T_zero = 0;
Dynamic_viscosity = 8.9e-4;
R_e = ( 400*Volume / (pi*Contact_angle))^(1/3);
%T according to the figure
T = 0:6; %seconds
R = R_e*(1 - exp(-(2*ST_L/R_e^12 + Density*Gravity/(9*R_e^10))*24*Shape_factor*Volume^4*(T+T_zero)/(pi^2*Dynamic_viscosity))).^(1/6);
%Convert R to mm
plot(T,R*1e3)

[shammas mohamed]

thank you so much. its working.

However,

why are you multiplying v by 400 in the R_e equation?

also in my experiment, the final radius reaches a value around 5 just like the original figure

[Cris LaPierre]

The 400 is likely because that is what was in your original post before it was edited.

[shammas mohamed]

yeah sorry its not as the original equation. check the updated question

Answer 1

[Sam Chak]

Hi @shammas mohamed

Edit: There are two correction in the equation. Thanks @Walter Roberson for pointing out.

% Parameters
Contact_angle     = 2.4783666516568;
Volume            = 5;
ST_L              = 72;
Density           = 0.973;
Gravity           = 9.807;
Shape_factor      = 37.1;
T_zero            = 0;
Dynamic_viscosity = 8.9e-4;
R_e = ( 4*Volume / (pi*Contact_angle) )^1/3;

% suggest to split up the terms 
T   = linspace(0,6e-12, 601);
f5  = 24*Shape_factor*Volume^4*(T + T_zero)/((pi^2)*Dynamic_viscosity); % <--- correction
f4  = (Density*Gravity)/(9*(R_e^10));
f3  = (2*ST_L)/(R_e^12);
f2  = f3 + f4;
f1  = - f2*f5;
R   = R_e*(1 - exp(f1)).^(1/6);     % <--- correction
plot(T, R), grid on, xlabel('T')

[shammas mohamed]

Thank you for the correction. it gave me a good start on the initial rate that the radius will start increasing exponentially from. however there is still problem that the function is not increasing. (the exponent part is 0).

I dont understand why this is happening. i edited my problem with more information if you need it.

Answer 2

[Walter Roberson]

x^1/3 means (x^1) divided by 3 which is x/3 . If you want to raise to the power of 1/3 you need x^(1/3)

Watch out for ^1/6 for the same issue.

[Dyuman Joshi]

Walter, I did implement this in the code in my comment, and it still doesn't give any similar result to what OP achieved earlier.

Answer 3

[Cris LaPierre]

As written, you need to either include your final denominator in parentheses, or divide by Dynamic_Viscocity.

We have to take the rest of the parameters you have given as correct. One potential error to look into - should your angles be in radians instead of degrees?

Contact_angle = 142;
Volume = 5;
ST_L = 72;
Density = 0.973;
Gravity = 9.807;
Shape_factor = 37.1;
T_zero = 0;
Dynamic_viscosity = 8.9e-4;
R_e = ( 400*Volume / (pi*Contact_angle))^1/3;
T = 0:300;
R = R_e * ( 1 - exp(-(2*ST_L/R_e^12 + (Density*Gravity) / (9*R_e^10))*24*Shape_factor*Volume^4*(T+T_zero)/(pi^2*Dynamic_viscosity))).^(1/6);
plot(T,R);

[shammas mohamed]

this is the original equation for the R_e.

i checked the parameters with my report and they are the same.

maybe there is a problem here too?

i did what you asked about the final denominator, still the same results.

clc
Contact_angle = 2.4783666516568;
Volume = 5;
ST_L = 72;
Density = 0.973;
Gravity = 9.807;
Shape_factor = 37.1;
T_zero = 0;
Dynamic_viscosity = 8.9e-4;
R_e = ( 4*Volume / (pi*Contact_angle) )^1/3;
T = 0:0.1:6;
R = R_e*(1 - exp(-(2*ST_L/R_e^12 + Density*Gravity/(9*R_e^10))*(24*Shape_factor*Volume^4*(T+T_zero)/(pi^2*Dynamic_viscosity)))).^(1/6);
plot(T,R);

the value of R_e here is the most accurate to my experiment

[Cris LaPierre]

It's likely an issue with inconsistent units. It might be worth doing a dimensional analysis to be sure you are consistent.

[shammas mohamed]

But why the exponential function becomes zero?

[Cris LaPierre]

I also suspect you are using the wrong equation to reproduce the plot you have shared. The equation you have shared is for the partial wetting condition. I wonder if you shouldn't be using the complete wetting situation equation.

See here.

Using that equation, I produced the following curve. I did have to make some assumptions about units, and adjusted the time scale to match the plot you shared.

[Cris LaPierre]

The exponential function becomes 0 because that's what happens when you raise e to a larger and larger negative number.

exp(0)

ans = 1

exp(-0.1)

ans = 0.9048

exp(-1)

ans = 0.3679

exp(-10)

ans = 4.5400e-05

Because the entire expression 1-exp(...) is multiplied by , as time increases, your result will approach .

If it happens to be doing that quicker or slower than expected, check your units.

[shammas mohamed]

i understand thanks