Movement of Al2O3 particulate in water

Abstract:

In the experiment analysis of sedimentation (Physical Chemistry, Peking University Press ) there is a basic assumption without any proof that the Al₂O₃ particulate reaches its maximum velocity from stillness in no time and then moves in the maximum velocity. By deducing the velocity-time (v-t) equation theoretically such an assumption is found incorrect. Under the correct v-t equation the P-t equation (see below) is developed.

Key words: Al₂O₃ particulates

Introduction:

To get the radius distribution of Al₂O₃ particulates, there is an experiment analysis of sedimentation in the textbook of Peking University. By letting a lot of Al₂O₃ particulates fall from stillness at the same time and at the same point in water, and measuring the time(t) and at which the mass of Al₂O₃ particulates (P) which have fallen to the bottom of the water, we can get a P-t curve, by which we can get the distribution of radius of the particulates.

In the beginning the particulates accelerate from stillness because the gravity is greater than buoyancy and resistance of water, and then get the maximum velocity because the gravity is equal to buoyancy and resistance of water. Under its own maximum velocity, each particulate falls to the bottom. Of course the ones with larger radiuses will get the bottom first and then those with smaller radiuses.

The textbook makes an assumption that the time from stillness to the maximum velocity of each particulate is less than 0.01 second and thus can be omitted. Under such an assumption the textbook deduces that the derivative of P-t curve is the range of the radius.

In this paper I will prove that the assumption is incorrect and deduced the correct P-t equation.

Movement of Al₂O₃ particulate in water:

Each particulate is driven by three forces:

(1)gravity G=mg=d·V·g=(4/3)·Pi·r³d·g

(2) buoyancy F_l=d₀·V·g

The sum of gravity and buoyancy is F＝k₁·r³ where k₁=(4/3) ·Pi·(d-d₀) ·g

(3)resistance of H₂O: f=6Pi·n·r·v=k₀·v

where k₀=6Pi·n·r

According to the second law of Newton:

dv/dt=(F-f)/m

and v=0 where t=0

so v=(F/k0)·(1-e^-k0·t) (1)

so v_max= F/k0

One of the basic assumptions is that the time from stillness to the maximum velocity of each particulate can be omitted. The following is the time.

Table 1: Time for particulates with different radiuses to reach their 0.9 maximum velocity

radius,r		time,t
meter	second		minute	hour
10^-7	10⁹		2·10⁷	4·10⁵
10^-6	10⁸		2·10⁶	4·10⁴
10^-5	10⁷		2·10⁵	4·10³
10^-4	10⁶		2·10⁴	4·10²
10^-3	10⁵		2·10³	2·10¹

The radius of the particulates is about 10^-5metre, and the time is long and far from omitted. Table 1 reveals that the assumption is incorrect.

Ddeducing of the h-t equation:

The calculus of equation (1) leads the h-t equation:

h=h/k₀[t-(1/k₀)·(1-e^-k0t)] (2)

Figure 2: h-t Curve

According the results draw from the experiment using the method based on the textbook, r is about 10^-5meter. The particulates with such a radius will spend 20 hour to reach the bottom. That’s to say, in 30 minutes, the particulates can’t reach the bottom.

Table 2: Time for particulates with different radiuses to reach their 0.9 maximum velocity

radius,r	time, t
meter	second	minute	hour
10^-7	7·10⁷	1.2·10⁶	2.0·10⁴
10^-6	2·10⁶	3.7·10⁴	621.8
10^-5	7·10⁴	1.2·10³	19.58
10^-4	2·10³	37	0.62
10^-3	7·10¹	1.2	0.02

Deducing of the P-t equation:

Suppose that at the point of t₀ the particulates with the radius r₀ and greater than r₀ have completely reach the bottom, we can divide the particulates in the bottom into two group:

a. with radius not less than r₀, mass:Wa

b.with radius less than r₀, mass:Wb

Suppose that the range of r conforms to standard distribution.

f(r)=

Wa=

Wb=

P(t)=Wa+Wb

Conclusion/discussion:

From the data shown above, we can see that the basic assumption in the textbook is wrong. However, though the correct P-t equation is deduced, the curve of P-t is not plotted because of the computer’s ability.

New methods such as direct observation should used to get the distribution of radius.