**6. Hydrodynamic structure of fractal aggregates**

As discussed earlier, the ratio of the internal permeability and the square of aggregate radius is expected to be constant for aggregates of the same fractal dimension. Consider an early stage of aggregate growth in which the constancy of the normalized permeability is attained. At the beginning the aggregate consists of two and then several monomers. The number of pores and their size are of the order of aggregation number and monomer size, respectively. At a certain aggregation number, however, the size of new pores formed starts to be much larger than that formerly created. This means that the hydrodynamic structure building has been finished and the smaller pores become not active in the flow and can be regarded as connected to the interior of hydrodynamic blobs.

A part of the aggregate interior is effectively excluded from the fluid flow, so one can consider this part as the place of existence of impermeable objects greater than the monomers. Since both the impermeable object size and the pore size are greater than formerly, the real permeability is bigger than that calculated by a formula valid for a

Hydrodynamic Properties of Aggregates with Complex Structure 263

**1.75 2.00 2.25 2.50**

*D*

Due to self-similarity, the number of monomers deduced from Fig. 8 is the number of hydrodynamic blobs which are the fractal aggregates similar to the whole aggregate. Hydrodynamic picture of a growing aggregate is such that after receiving a given number of monomers the number of hydrodynamic blobs becomes constant and further growth causes

As this estimation shows, the number of hydrodynamic blobs rises with the aggregate fractal dimension. The knowledge of this number makes it possible to estimate the aggregate permeability in the slip regime where the free molecular way of the molecules of the dispersing medium becomes longer than the aggregate size. In this region the dynamics

The permeability of a homogeneous arrangement of solid particles of radius *a*, present at

2 6 9 *packing k a a f*

The friction factor of a particle in a packing can be presented as the friction factor of

*packing f fS*

<sup>0</sup> 6 *f f a continuum*

<sup>0</sup> 6 / 1 1.612 *slip ff a*

0

*a* 

(51)

(48)

(49)

(50)

Fig. 8. Number of hydrodynamic blobs as dependent on fractal dimension.

, can be calculated (Brinkman, 1947) as

individual particle multiplied by a function of volume fraction of particles

2

**5**

the increase in blob mass not their number.

of the continuum media is no longer valid.

whereas in the slip one (Sorensen & Wang, 2000)

volume fraction

In the continuum regime

**10**

*I*

**15**

**20**

uniform packing of monomers. So this point can be considered as manifested by the beginning of the decrease of the normalized aggregate permeability calculated.

During the aggregate growth the number of large pores tends to a value which remains unchanged during the further aggregation. The self-similar structure exists, which can be described by an arrangement of pores and effective impermeable monomers (hydrodynamic blobs) of the size growing proportional to the pore size.

According to the above considerations one can expect effective aggregate structure such that the normalized aggregate permeability <sup>2</sup> *k R*/ attains maximum. To determine the hydrodynamic structure of fractal aggregate the aggregate permeability is estimated by the Happel formula

$$\frac{k}{a^2} = \frac{2}{9\rho} \cdot \frac{3 - 4.5\rho^{1/3} + 4.5\rho^{5/3} - 3\rho^2}{3 + 2\rho^{5/3}}\tag{45}$$

where the volume fraction of solid particles in an aggregate is described as

$$
\varphi = \mathbf{i} \cdot \left(\frac{a}{R}\right)^3 \tag{46}
$$

The normalized aggregate permeability is calculated as

$$\frac{k}{R^2} = \frac{k}{a^2} \frac{a^2}{R^2} = \frac{k}{a^2} \left(\frac{i}{\varphi}\right)^{-2/3} \tag{47}$$

The results are presented in Fig. 7.

uniform packing of monomers. So this point can be considered as manifested by the

During the aggregate growth the number of large pores tends to a value which remains unchanged during the further aggregation. The self-similar structure exists, which can be described by an arrangement of pores and effective impermeable monomers (hydrodynamic

According to the above considerations one can expect effective aggregate structure such that the normalized aggregate permeability <sup>2</sup> *k R*/ attains maximum. To determine the hydrodynamic structure of fractal aggregate the aggregate permeability is estimated by the

2 3 4.5 4.5 3

2 5/3

9 3 2

2 22 2 *k ka k i R aR a*

1/3 5/3 2

 

(45)

(46)

(47)

1.75

2.00

2.25

2.50

*D*

3 *<sup>a</sup> <sup>i</sup> R*

2/3 2

**1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20**

*i*

Fig. 7. Normalized aggregate permeability calculated by the Happel formula for different fractal dimensions. The maxima (indicated) determine the number of hydrodynamic blobs

beginning of the decrease of the normalized aggregate permeability calculated.

blobs) of the size growing proportional to the pore size.

*k a*

The normalized aggregate permeability is calculated as

The results are presented in Fig. 7.

**1.e-1**

**1.e-4**

**1.e-3**

*k/R*

2

in aggregate.

**1.e-2**

where the volume fraction of solid particles in an aggregate is described as

Happel formula

Fig. 8. Number of hydrodynamic blobs as dependent on fractal dimension.

Due to self-similarity, the number of monomers deduced from Fig. 8 is the number of hydrodynamic blobs which are the fractal aggregates similar to the whole aggregate. Hydrodynamic picture of a growing aggregate is such that after receiving a given number of monomers the number of hydrodynamic blobs becomes constant and further growth causes the increase in blob mass not their number.

As this estimation shows, the number of hydrodynamic blobs rises with the aggregate fractal dimension. The knowledge of this number makes it possible to estimate the aggregate permeability in the slip regime where the free molecular way of the molecules of the dispersing medium becomes longer than the aggregate size. In this region the dynamics of the continuum media is no longer valid.

The permeability of a homogeneous arrangement of solid particles of radius *a*, present at volume fraction , can be calculated (Brinkman, 1947) as

$$\frac{k}{a^2} = \frac{2}{9\rho} \cdot \frac{6\pi\eta\_0 a}{f\_{\text{pucking}}} \tag{48}$$

The friction factor of a particle in a packing can be presented as the friction factor of individual particle multiplied by a function of volume fraction of particles

$$f\_{\text{pucking}} = f \cdot S(\boldsymbol{\sigma}) \tag{49}$$

In the continuum regime

$$f = f\_{\text{continuum}} = \text{€}\pi\eta\_0 a \tag{50}$$

whereas in the slip one (Sorensen & Wang, 2000)

$$f = f\_{slip} = 6\pi\eta\_0 a \left(1 + 1.612\frac{\lambda}{a}\right) \tag{51}$$

Hydrodynamic Properties of Aggregates with Complex Structure 265

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where is the gas mean free path.

For a given structure of arrangement *a*, it possible to calculate the permeability coefficient in the slip regime from that valid in the continuum regime (Gmachowski, 2010)

$$k\_{slip} = k \frac{f\_{continuum}}{f\_{slip}} = \left(1 + 1.612 \frac{\lambda}{a}\right) \cdot k \tag{52}$$

in which the monomer size should be replaced by the hydrodynamic blob radius rising such as the growing aggregate. So large differences in permeabilities at the beginning diminish when the aggregate mass increases and disappear when the aggregate size greatly exceeds the gas mean free path.

Calculated mobility radius *rm* , representing impermeable aggregate in the slip regime, is smaller than the hydrodynamic one because of higher permeability and tends to the hydrodynamic size when the difference in permeabilities becomes negligible. At an early stage of the growth of aerosol aggregates it can be approximated as a power of mass (Cai & Sorensen, 1994)

$$r\_m = a \cdot i^{1/2.3} \tag{53}$$

in which the number 2.3 greatly differs from the fractal dimension equal to 1.8.
