CHEE 3363: Fluid Mechanics for Chemical Engineers

Homework #2Due date: Wednesday, January 29, at 4:00pm.

Recommended reading: Fox, sections 1.41.6.Recommendd viewing: the NCFM movie Surface Tension.

1. Plan an experiment to measure the surface tension of a liquid similar to water. If necessary,review the NCFMF video Surface Tension for ideas. Which method would be most suitablefor use in an undergraduate laboratory? What experimental precision could be expected?

2. Honey with viscosity 2,000 Poise and density = 1300 kg/m3 (recall: 1 cP = 103 Pa-s)flows down the side of a storage container whose surface makes a 15 degree angle with thehorizontal. The thickness of the honey is h = 1.5 cm. The velocity profile is given by

u =g

(hy y

2

2

)sin

with coordinate x oriented along the surface of the container and coordinate y normal to thesurface. Determine the magnitude and direction of the shear stress that acts on the surfacedue to the flow of the honey.

3. An air-hockey puck of diameter D = 8 cm has mass m = 50 g. When the puck is placed onan air table at 20C, a film of thickness h = 0.10 mm forms underneath the puck. The puckis struck such that its initial velocity is v0= 15 m/s.

(a) How long will it take for the puck to slow down to 2 m/s?

(b) How far has the puck traveled when it slows down to 2 m/s?

(c) How long will it take for the puck to stop completely?

4. The belt in the figure moves at steady velocity V and skims the top of a tank of oil of viscosity. Assuming that the film of oil between the belt and the surface is thin, develop a simpleformula for the belt-drive power P required as a function of h, L, V,B, . Neglect airdrag.What power P in watts is required if the belt moves at 2 m/s over SAE 30W oil at 20C,with L = 4 m, b = 50 cm, and h = 2 cm?

Chapter 1 Introduction 29

The total error is dominated by the 8% error in the estimate of clearance, (Ro Ri). We might state the experimental result for viscosity as

exp 0.29 8.2% Ans = .kg0.29 0.024

m s

1.52 The belt in Fig. P1.52 moves at steady velocity V and skims the top of a tank of oil of viscosity . Assuming a linear velocity profile, develop a simple formula for the belt-drive power P required as a function of (h, L, V, B, ). Neglect air drag. What power P in watts is required if the belt moves at 2.5 m/s over SAE 30W oil at 20C, with L = 2 m, b = 60 cm, and h = 3 cm?

Fig. P1.52

Solution: The power is the viscous resisting force times the belt velocity:

oil belt beltVP A V (bL)V .h

Ans ! "= =# $% &

2 LV bh

(b) For SAE 30W oil, 0.29 kg/m s. Then, for the given belt parameters, 2 2

23

kg m 2.0 m kg mP V bL/h 0.29 2.5 (0.6 m) 73 . (b)m s s 0.03 m s

Ans ! " ! "= = =# $# $

% &% &

73 W

1.53* A solid cone of base ro and initial angular velocity o is rotating inside a conical seat. Neglect air drag and derive a formula for the cones angular velocity (t) if there is no applied torque.

Solution: At any radial position r < ro on the cone surface and instantaneous rate ,

Fig. P1.53

w

r drd(Torque) r dA r 2 r ,h sin

! "! "= = # $# $

% &% &

5. A block of mass M and contact area A slides on a thin film of oil with thickness h as shownin the figure. When released, a mass m exerts tension on the cord, causing the block toaccelerate.

1

(a) Neglect friction in the pulley and air resistance, and develop an algebraic expression forthe viscous force that acts on the block when it moves at a speed V .

(b) Derive a differential equation for the block speed as a function of time, and obtain anexpression for the block speed as a function of time.

(c) For the values = 0.04 Pa-s, m = 2 kg, A = 10 cm2, and h = 0.1 mm, determine theblock mass M if it takes 10 s for the block speed to reach 1 m/s.

Mass M, area A

Mass m

Oil film thickness h, viscosity g

CordV

h

6. Under the right conditions, metal objects can float on water due to surface tension. What isthe maximum length of a steel rod (density 490 lb/ft3) that can float on water if the rod has adiameter of 0.036 in (i.e. a paper clip)? Assume that the surface tension force acts verticallyupwards.

2