• Author / Uploaded
• C. M

• Categories
• Esfuerzo cortante
• Ciencia de los Materiales
• Mecánica de fluidos
• Física y matemáticas
• Física

Citation preview

The Ellis Model

By considering a force balance across the small element ABCD in the pipe at steady state conditions it can be shown that, in general the volumetric flow rate is given by; w

R 3 Q  3   rz2 f  rz d rz w 0 Where f  rz   

(1)

du z dr

The viscosity of a fluid that follows the Ellis model is given by;



0

1   rz  0.5 

 1

(2)

The shear stress is related to the shear rate according to;

 du z    dr 

 rz    

( 3)

Substituting (2) into (1) gives;

 rz 

0    1   rz     0.5  

 1

 du z      dr    

(4)

Re-arranging Eq. (4) gives;

    1   du   rz 1   rz     0   z     0.5    dr   

(5)

 1  du z   rz    rz    1      dr   0    0.5  

(6)

   du z  1   rz  rz1     0.5   dr   0 

(7)

Therefore

du z  rz  1   f  rz      rz   1  dr  0   0.5 

(8)

Substituting (8) into (1) gives; w

  R 3 1   rz  rz1 d rz Q  3   rz2  w 0 0   0.5 

(9)

Bringing out constants out of integration sign

Q

w

 rz  R 3 1 2  d rz    rz rz  w3  0 0   0.51 

(10)

Expanding the bracket in the integration

R 3 1 Q 3  w 0

w

 3  rz  2  0  rz   0.51 d rz

(11) w

R 3 1  rz4 1  rz 3  Q 3     w  0  4   3  0.51  0

(12)

Q

R 3 1  w4 1  w 3      w3  0  4   3  0.51 

(13)

Q

R 3 1  w4  4  w 1  1     w3  0 4    3  0.51 

(14)

R 3  w  4   w  Q 1 4  0    3   0,5 

 1



   

(15)

It is also known that

Q  uA  R 2 u

(16)

Where u is the average velocity of the fluid in the pipe. Hence, the average velocity of the fluid can be expressed as;  1 R w  4   w   1   u 4  0    3   0,5    

(17)