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Chemical Engineering 150A February 29, 2008
Midterm #
(155) 1. In the system below, a free liquid jet flows downwards at a known volumetric flow rate Qo against a suspended inverted solid cone of mass m , as shown in Figure 1. A suspending force Fo is applied to the cone to keep it stationary, as illustrated. The liquid deflects around the cone (dimensions are given) with a local x -velocity profile given by
2 2 x^1
y v ( y ) a [ ]
= − ^
where δ is the thickness of the fluid layer, μ is viscosity of the liquid, and
a = ρ g cos β / 2 ( ρ is the liquid density and g is the acceleration of gravity).
Figure 1. A Free Liquid Jet Impinging on a Suspended Inverted Cone
x
y
s
Qo
δ( x)
patm
patm
patm
m
2β
h
R
Fo
Chemical Engineering 150A Midterm #1 continued
(40) a. Since the circumference of the cone increasing as the fluid travels along its length s , the the liquid film occupies larger cross sectional areas. Hence the film thickness δ decreases with increasing x. Perform a mass balance on the fluid to find an expression for the film
thickness, δ , as a function of x and the parameters as a function of x, Qo ρ , β , g, and μ..
The volumetric flow of liquid at any point x along the cone can be written approximately as
Q = 2 π δ x sin β vx
where vx is the average x -velocity at position x.
(15) b. What role does pressure play in keeping the cone stationary?
(15) c. Which components of the shear stress tensor act on the cone (i.e., which of the various
τ ij )?
(25) d. Find the expression for the vertical component of the drag force exerted by the liquid
on the cone in terms of x, Qo ρ , β , g, and μ. Please note that because the liquid film
thickness varies in the x -direction along the cone, so does the shear stress.
(40) e. Perform a vertical momentum balance on the cone to establish an expression for the
force Fo in terms of the parameters Qo ρ , β , g, and μ.
(10) f. What will happen if Fo is held constant and the volumetric flow rate of the jet, Qo , is increased?
(10) g. What happens to the cone if it tips slightly off axis?
