# 3.22: Untitled Page 48

## Chapter 3

and apply Eq. 6 of Example 3.1.

(b) Construct an appropriate control volume and develop a macroscopic balance that will allow you to determine the thickness of the liquid film, b, on the moving web during steady operation.

3‐6. One method for continuously depositing a thin liquid coating on a moving web is known as slot die coating and the process is illustrated in Figure 3.6. In slot Figure 3.6. Slot die coating

die coating, the liquid is forced through a slot of thickness d and flows onto a vertical web moving at a constant speed, U . The velocity profile in the slot is o

given by

 2

v

6 v

 

x

x

y d

y d

(1)

and this profile is illustrated in Figure 3.6. Variations of the velocity, v , across x

the width, w , of the slot can be ignored. In a steady operation all the feed liquid to the slot die is picked up by the web. As in the case of slide coating (see Problem 3‐5), the fluid velocity on the moving web can be approximated by

Single component systems

81

v

,

far downstream on the moving web

(2)

x

U o

(a) Select an appropriate control volume and construct the macroscopic balance that will allow you to determine the thickness, b, of the liquid film on the moving web during steady operation.

(b) If far downstream on the moving web all the liquid in the coated film of thickness, b, moves at the velocity of the web, U , determine the thickness of the o

coated film.

(c) If the gap, d, of the die slot changes by 10% and the average velocity remains constant, how much will the final thickness, b, of the coated film change?

3‐7. If the delivery charge for the propane tank described in Example 3.3 is $37.50, and the cost of the next largest available tank is$2500 (for a 2.2 cubic meter tank), how long will it take to recover the cost of a larger tank?

3‐8. The steady‐state average residence time of a liquid inside a holding tank is determined by the ratio of the volume of the tank to the volumetric flow rate of liquid into and out of the tank,   V Q . Consider a cylindrical tank with volume

3

V  3 m and an input mass flow rate of water of 250 kg/minute. At steady‐state, the output flow rate is equal to the input flow rate. What is the average residence time of water in the tank? What would be the average residence time if the mass flow rate of water is increased to 300 kg/minute?

What would be the average residence time if a load of 1.2 m3 of stones is dropped into the holding tank?

3‐9. Mono Lake is located at about 6,000 ft above sea level on the eastern side of the Sierra Nevada mountains, and a simple model of the lake is given in Figure 3.9a. The environment is that of a high, cold desert during the Winter, a thirsty well during the Spring runoff, and an cornucopia of organic and avian life during the Summer. Mono Lake is an important resting place for a variety of birds traveling the flyway between Canada and Mexico and was once the nesting place of one‐fourth of the worldʹs population of the California gull.

The decline of Mono Lake began in 1941 when Los Angeles diverted all the water from four of the five creeks flowing into the lake. This sent 56,000 acre‐feet per year into the Owens River and on to the Los Angeles aqueduct. Details concerning the fight to save Mono Lake from a diminishing influx of water are

82