**4. Free settling velocity**

Free settling velocity of an aggregate with mixed statistics can be determined by equating the gravitational force allowing for the buoyancy of the surrounding fluid with the opposing

Hydrodynamic Properties of Aggregates with Complex Structure 257

0 1 1/*D*

1

*D*

(18)

(19)

(20)

*u r* (21)

(22)

(23)

(25)

(24)

*r* . This

gives the

*a <sup>u</sup> <sup>i</sup> u*

*a u r u a*

0

describe the free settling velocity of aggregates with mixed statistics.

mass *Ma* . To calculate the intrinsic viscosity

overall polymer mass concentration in the solution.

their mass <sup>3</sup> 4/3 *<sup>s</sup>*

 *a* 

of aggregate equivalent to coil

0 0

characteristic for fractal aggregates with one-level structure. Hence the following formulae

*B <sup>u</sup> <sup>i</sup> u*

> 0 *u r*

**5. Intrinsic viscosity of macromolecular coils and the thermal blob mass** 

A macromolecular coil in a solution is modeled as an aggregate with mixed statistics consisting of *I* thermal blobs of 2 *DB* , each containing *Bi* solid monomers of radius *a* and

> <sup>0</sup> <sup>0</sup> <sup>0</sup> lim *<sup>c</sup> c*

one has to define the mass concentration *c* of a macromolecular solution analyzed. The mass concentration in the coil, represented by the equivalent impermeable sphere, can be calculated as the product of the total number of non-porous monomers *<sup>B</sup> Ii* multiplied by

and divided by the hydrodynamic volume of the coil <sup>3</sup> 4/3

3 3 *<sup>B</sup> <sup>s</sup> Ii a*

2

*D*

 

concentration multiplied by the volume fraction of equivalent aggregates

*c*

*B*

nonporous monomer is the aggregation number of the thermal blob

*<sup>r</sup>* 

Mass-radius relations are then employed. The thermal blob mass related to that of

*B B*

*a M r <sup>i</sup> M a*

whereas the macromolecular mass related to that of thermal blob is the aggregation number

*B B M r <sup>I</sup> M r* 

1/ 1/

0

*D DB*

hydrodynamic force which depends on the aggregate size and its permeability. The use of hydrodynamic radius which is the radius of an impermeable sphere of the same mass

Fig. 4. Graphical representation of the mass-radius relation for asphaltene aggregates.

having the same dynamic properties, instead of the aggregate radius, makes it possible to neglect the internal permeability. For an aggregate of hydrodynamic radius *r* composed of *<sup>B</sup> i Ii* primary particles of radius *a* the force balance is

$$\frac{4}{3}\pi a^3 \text{li}\_{\mathcal{B}}\left(\rho\_s - \rho\_f\right) \mathbf{g} = 6\pi \eta\_0 ru \tag{14}$$

Using the mass-hydrodynamic radius relations for blob and aggregate (Eqs. 9,10), one gets

$$\frac{\mu}{\mu\_a} = I^{1 - 1/D} \dot{\mathbf{i}}\_B^{1 - 1/D\_B} \tag{15}$$

where

$$
\mu\_a = \frac{2}{9\eta\_0} (\rho\_s - \rho\_f) g a^2 \tag{16}
$$

is the Stokes falling velocity of primary particle. Alternatively, using the expression for the hydrodynamic radius changed by the presence of blobs (Eq. 12), one obtains

$$\frac{\mu}{\mu\_a} = \left(\frac{r\_0}{a}\right)^{D-1} \Big/ \frac{r\_0}{r} \tag{17}$$

If the blobs of the fractal dimension different from that of the aggregate are not present ( *D D <sup>B</sup>* and 0*r r* ), the corresponding dependences reduce to the following relations

hydrodynamic force which depends on the aggregate size and its permeability. The use of hydrodynamic radius which is the radius of an impermeable sphere of the same mass

*rB*=2<sup>m</sup>

**10 100 1000**

*rB*=4<sup>m</sup>

*D*=1.5

**2***R***[m]**

0

*a Ii g ru* (14)

(15)

(16)

(17)

 

Fig. 4. Graphical representation of the mass-radius relation for asphaltene aggregates.

<sup>3</sup>

*<sup>u</sup> I i*

0 2 <sup>9</sup> *u g a sf a*

Alternatively, using the expression for the hydrodynamic radius changed by the presence of

*u r r ua r* 

If the blobs of the fractal dimension different from that of the aggregate are not present ( *D D <sup>B</sup>* and 0*r r* ), the corresponding dependences reduce to the following relations

*a*

 

*a*

*u*

<sup>4</sup> <sup>6</sup> 3 *Bs f*

Using the mass-hydrodynamic radius relations for blob and aggregate (Eqs. 9,10), one gets

1 1/*D* 1 1/*DB B*

<sup>2</sup>

1 0 0 / *D*

*<sup>B</sup> i Ii* primary particles of radius *a* the force balance is

is the Stokes falling velocity of primary particle.

blobs (Eq. 12), one obtains

having the same dynamic properties, instead of the aggregate radius, makes it possible to neglect the internal permeability. For an aggregate of hydrodynamic radius *r* composed of

**1**

**10**

*I*

where

**100**

$$\frac{\mu\_0}{\mu\_a} = i^{1 - 1/D} \tag{18}$$

$$\frac{\mu\_0}{\mu\_a} = \left(\frac{r\_0}{a}\right)^{D-1} \tag{19}$$

characteristic for fractal aggregates with one-level structure. Hence the following formulae

$$\frac{\mu}{\mu\_0} = i\_B \mathbf{1}^{1/D - 1/D\_B} \tag{20}$$

$$\frac{\mu}{\mu\_0} = \frac{r\_0}{r} \tag{21}$$

describe the free settling velocity of aggregates with mixed statistics.
