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410 13 Nanoaerosol

Equation (13.28) becomes

" #
2
2 v cr 4 v cr 4 v cr
g ¼ erf p ffiffiffi exp ð13:30Þ
ad
p v im p v im p v im
For the ease of presentation, we define an interim term
2 v cr
z ¼ p ffiffiffi ð13:31Þ
p v im

The equation for adhesion energy can be simply presented as
2z 2
g ¼ erf zðÞ p exp z ð13:32Þ
ad
ffiffiffi
p
For the ease of calculation without software, the error function can be approx-
imated with

1
erf zðÞ 1 ð13:33Þ
2 3 4 4
ð 1 þ a 1 z þ a 2 z þ a 3 z þ a 4 z Þ
where a 1 = 0.278393, a 2 = 0.230389, a 3 = 0.000972, and a 4 = 0.078108. The
−4
maximum error is 5 × 10 (Fortran 77 manual).

13.4.5 Adhesion Energy

Several models of adhesion energy (E ad ) were developed before and they were
summarized by Givehchi and Tan [16]. As guidance, we will introduce only two of
them, the JKR model [24] and the DMT model [11]. These two models complement
each other because they represent two extremes in the Tabor parameter spectrum.
JKR model is applicable to soft material, large radius, compliant spheres, and large
adhesion energy and DMT model is for hard material, small radius with low
adhesion energy [33]. The effectiveness of the DMT model has been proven for
smaller and stiffer contact solids [44]. However, the main defect in this theory is
that it neglects deformations outside the contact area [33].
Consider a nanoparticle deformed on the solid surface in Fig. 13.6. The adhesion
energy between the particles and the surface of the filter material can be mathe-
matically calculated based on consideration of elastic or plastic impaction.

E ad ¼ Dcpa 2 ð13:34Þ