# 3.26: Untitled Page 52

## Chapter 3

where h is the depth of the fluid in the tank. This leads to Torricelli’s law in the form

Q C A o 2 gh ,

hydrostatic conditions

d

(3)

Use this information to derive an equation for the depth of the fluid as a function of time. For a tank filled with water to a depth of 1.6 m having a diameter of 20

cm, how long will it take to lower the depth to 1 cm if the diameter of the orifice is 3 mm?

3‐13. The system illustrated in Figure 3.13 was analyzed in Example 3.5 and the Figure 3.13. Tank filling process

depth at a single specified time was determined using the bisection method. In this problem you are asked to repeat the type of calculation presented in Example 3.5 applying methods described in Appendix B. Determine a sufficient number of dimensionless times so that a curve of x   h  versus 2

   t 2

can be constructed.

Part (a). The bi‐section method

Part (b). The false position method

Part (c). Newton’s method Single component systems

89

Part (d). Picard’s method

Part (e). Wegstein’s method

3‐14a. When full, a bathtub contains 25 gallons of water and the depth of the water is one foot. If the empty bathtub is filled with water from a faucet at a flow rate of 10 liters per minute, how long will it take to fill the bathtub?

3‐14b. Suppose the bathtub has a leak and water drains out of the bathtub at a rate given by Torricelli’s law (see Problem 3‐12)

Q

C A o 2

leak

d

gh

(1)

Here h is the depth of water in the bathtub, and represents the discharge

d

C

coefficient associated with the area of the leak in the bathtub, A o . Since neither nor

d

C

A o are known, we express Eq. 1 as

le

Q ak

k h

(2)

in which k C A o 2

d

g . To find the value of k we have a single experimental condition given by

Experiment:

5

3

Q

 3 1

. 6  10 m / s ,

h  0 1

. 0m

leak

(3)

Given the experimental value of k , assume that the cross section of the bathtub is constant and determine how long it will take to fill the leaky bathtub.

3‐15. The flow of blood in veins and arteries is a transient process in which the elastic conduits expand and contract. As a simplified example, consider the artery shown in Figure 3.15. At some instant in time, the inner radius has a Figure 3.15. Expanding artery

radial velocity of 0.012 cm/s. The length of the artery is 13 cm and the volumetric flow rate at the entrance of the artery is 0.3 cm3/s. If the inner radius of the artery  90