Particle trajectory in an electric dust collector The partic

Particle trajectory in an electric dust collector. The particle that begins at z = 0 and ends at x = + B may not necessarily travel the longest distance in the z direction. Fully-developed laminar flow has been established by means of a pressure gradient (p_0 - p_L)/L between a pair of horizontal plates, as pictured in, Fig. 2C.1, BSL, p. 69. The pressure gradient has a magnitude of 50 dyne/cm^3. In this system, B = 0.30 cm, and the fluid viscosity and density are mu = 5.0 cP and rho = 1.0 g/cm^3, resp. A small spherical particle of radius R = 0.01 cm and density rho_s = 2.4 g/cm^3 is injected just below the top plate at a position z = 0, and instantly begins to settle at its terminal velocity. While it is settling, it is carried along in the z-direction at a velocity equal to the local fluid velocity. Derive and plot the expression for the particle trajectory x_s(z_s), and compute the total horizontal distance traveled by the particle.

Solution

FOR LAMINAR FLOW, GIVEN:- Pressure gradient P = (Po-PL) / L = 50 dyne/cm^3, B = 0.3cm (2B=D=0.6cm),

Viscocity u = 0.05 dyne.s/cm2 = 5cP, Density d = 1g/cm^3, Sphere radius R = 0.01cm and density ds = 2.4gm/cm^3

TO FIND:- Derive & plot expression for sphere particle trajectory = ? and Horizontal distance Ls = ?

SOLUTION:-

Reynolds number Re for laminar flow = 2000 and by formula Re = d x v x D / u

so 2000 = 1 x v x 0.6/ 0.05, gives (Velocity of fluid) v = 166.66 cm/s

Correlation between Re, diameter (D) and Length L for laminar flow is given by L = 0.06.Re.D = 0.06x2000x0.6

Horizontal distance L = 72cm (ANSWER)

The sin component of the drag force or buoyant force acting on the sphere is balanced or exceeded by the gravity force acting on the sphere, so Fd Sin(o) = Fg

That is 6(3.142)xuxRxV = mxg (ANSWER) is the expression