# 3.21: Untitled Page 47

- Page ID
- 18180

## Chapter 3

*Section* 3.2

3‐3. To describe flow of natural gas in a pipeline, a utility company uses *mass* flow rates. In a 10 inch internal diameter pipeline, the flow is 20,000 lb */ * h . The m

average density of the gas is estimated to be 10 kg/m3. What is the volumetric flow rate in ft3/s ? What is the average velocity inside the pipe in m/s?

3‐4. For the coating operation described in Example 3.2, we have produced an optical fiber having a diameter of 125 micrometers. The speed of the coated fiber at the take‐up wheel is 4.5 meters per second and the desired thickness of the polymer coating is 40 micrometers. Assume that there is no change in the polymer density and determine the volumetric flow rate of the coating polymer that is required to achieve a thickness of 40 micrometers.

3‐5. Slide coating is one of several methods for continuously depositing a thin liquid coating on a moving web. A schematic of the process is shown in Figure 3.5. In slide coating, a liquid film flows down an inclined plate (called the slide) owing to a gravitational force that is balanced by a viscous force. In a subsequent course in fluid mechanics, it will be shown that the velocity profile upstream on the slide is given by

2

*g h * sin

2

1

v

*y h *

*y h*

*, *

o

n the slide

*x*

(1)

2

Here *y* is the distance perpendicular to the slide surface and *h* is the thickness of the liquid film. Variations of the velocity, v , across the width of the slide can be *x*

ignored. In a steady operation all the liquid flowing down the slide is picked up by a vertical web moving at a constant speed, *U *. Far downstream on the o

moving web, the velocity profile is given by

2

*g b *

2

1

v

*U *

*y b *

o

*y b*

*, *

o

n the moving web

*x*

(2)

2

and this velocity profile is illustrated in Figure 3.5. Here one must note that the coordinate system used in Eq. 2 is different from that associated with Eq. 1. In slide coating operations, the system is operated in a manner such that 2

*g b*

*U* o

(3)

*Single component systems *

79

and Eq. 2 and be replaced with the approximation given by

v

*, *

far downstream on the moving web

(4)

*x*

*U* o

In this problem you are asked to carry out the following steps in the analysis of the slide coating process.

*Figure 3.5*. Slide coating

(a) Demonstrate that the flow on the slide can be expressed as v

3 v

*x*

*x*

*y h*

*y h*2

1

2

(5)

To accomplish this, make use of Eq. 3‐35 in the form

*Q*

*dA , *

*entrance*

**v n**

**v n**

v *x *

(6)

*A entrance*

80