# sinking speed of particle in water

For a Newtonian fluid the sinking final speed of a metal particle is given by this equation :

while g is Gravitational acceleration , $$\rho_P$$ = is particle density , $$\rho_F$$ = is fluid density , $$D_p$$ = is particle diameter , and $$C_D$$ is given by Reynolds number function with the given values :

Find the sinking speed of a metal particle with density=(7,850 $$kg/m^3$$) , that is sinking in water with density = (1,000 $$kg/m^3$$) , and viscosity = (0.001$$PA*s$$) as a function of particle diameter (in meters) in the range of 0.1mm and 0.15cm with a growth = 0.2mm . I need to display the answer with a graph using MATLAB.

NOTE

it's a previous exam question, and I have no clue how to solve the question

My attempt

function [Ut]=myfunc(D_p)
%calculate the violecity Ut [m/s] of sphere in water
%Cd drag coefficient
%D_p the diameter of the sphere
%ru_p practicle density
%ru_f fluid density
%g acceleration of gravity [m/s^2]

ru_f=1000;%[kg/m^3]
meu=0.001;%[Pa*s]
ru_p=7850;%[kg/m^3]
g=9.8;%[m/s^2]


for D_p=0.0001:0.0002:0.15;%[m] disp(D_p) end

function [Cd]=myfunc(Re)
%calculate the drag coefficient Cd
%Re reynpld number
if Re<0.1
disp('Cd=24/Re')
elseif 0.1<Re<10^3
disp('Cd=[24/Re][1+0.14(Re)^0.7]')
elseif 10^3<Re<3.5*10^5
disp('Cd=0.445')
elseif Re==350000
disp('Cd=0.396')
elseif Re==400000
disp('Cd=0.0891')
elseif Re==500000
disp('Cd=0.0799')
elseif Re==700000
disp ('Cd=0.00945')
elseif Re==1000000
disp ('Cd=0.110')
end
Ut=squart((4*g*(ru_p*ru_f)*D_p)/3*Cd*ru_f)
end
plot(plotdata_x,plotdata_y)
xlabel('D(m)')
ylabel('Vt(m/s)')
end

• What have you tried so far? The exam question appears to be more about applying matlab than about the engineering/physics. (cryptic hint: Anything to with flow has to be soverd iterativly, outside a very few special cases) – mart Jun 15 '20 at 8:13
• @mart I'm sorry , I added my attempt , i'm new in matlab and not sure about anything iv'e wrote in this code. – Mahajna Jun 15 '20 at 13:53

I think I understand where you are stuck - it appears (I don't "speak" Matlab) you're trying to solve u directly, without adressing that u goes into Re.

Here's what I'd do:

• Guess $$u$$ for the smallest diameter, calculate $$Re$$ from this $$u$$ to arrive at a new $$u'$$. Repeat/Reiterate by calculating $$Re$$ and a new $$u'$$ until the difference between $$u$$ and $$u'$$ is negliblge (<1% or even <10% depending on what accuracy you need)
• Repeat for all other diameters (I'd use the $$u$$ value from an adjectant diameter as a starting point to keep the iteration loop short)

This is a general theme in flow calculations: to compute flow velocity you need some sort of friction factor or $$C_d$$ value, this depends on $$Re$$ so it's also velocity dependent. I guess this is one reason, why charts for sinking speeds (or Moody diagrams for pipe friction factors) are so common, iterating through these calculations must be really tiring without a programmable computer.

• I believe you exactly know what is my problem , 1) can't "speak MATLAB" , and the need to translate the question from my second language to my third language which is English . – Mahajna Jun 15 '20 at 14:41
• Is my answer clear enough, at least, so you can try this approach? – mart Jun 15 '20 at 14:42
• I've updated my attempt I believe that I have implemented your approach , but still i'm having errors which I couldn't solve . – Mahajna Jun 15 '20 at 14:42
• I don't see the iteration loop in the update. – mart Jun 15 '20 at 14:44
• the first iteration should start with the smallest diameter which is 0.1mm to 0.15cm and I need to increase by 0.2mm right ? – Mahajna Jun 15 '20 at 14:52