sinking speed of particle in water

Question

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

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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

Answer 1

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.