# 3.24: Untitled Page 50

- Page ID
- 18183

## Chapter 3

*Figure 3.10*. Water storage tank for distribution

3‐11. The 7 *th* Edition of Perry’s Chemical Engineering Handbook (Perry, *et al*.

1997) gives the following formula to compute the volume of liquid inside a partially filled horizontal cylinder:

2

*L R*

2

*V*

*LR *

sin

(1)

2

Here *L* is the length of the cylinder and *R* is the radius of the cylinder. The angle,

, illustrated in Figure 3.11a, is measured in radians. In this problem we wish to *Figure 3.11a. * Definition of geometric variables in horizontal cylindrical tank.

determine the depth of liquid in the tank, *h *, as a function of time when the *net* *flow* into the tank is *Q *. This flow rate is positive when the tank is being filled and negative when the tank is being emptied. The depth of the liquid in the tank is given in terms of the angle by the trigonometric relation *h * *R * *R * cos 2

(2)

85

It follows that *h * 0 when 2 and *h * 2 *R * *H * when 0 .

Part I. Choose an appropriate control volume and show that the macroscopic mass balance for a constant density fluid leads to

*d*

2 *Q*

*, *

*Q * *Q*

*Q *

(3)

2

*dt*

*LR *

*in*

*out*

cos

1

Part II. Given an initial condition of the form

IC

*, *

*t *

(4)

o

0

show that the implicit solution for (

*t*) is given by

2 *Q t*

sin (

*t*) (

*t*) sin

(5)

o

o

2

*L R*

This equation can be solved using the methods described in Appendix B in order to determine ( *t*) which can then be used in Eq. 2 to determine the fluid depth, (

*h t*) .

Part III. In Figure 3.11b values of

(

*t*) are shown as a function of the

dimensionless time,

2

*Q t / * *L R * for

o

0 . The curve shown in Figure 3.11b

represents values of (

*t*) when the tank is being drained, while the curve shown in Figure 3.11c represents the values of (

*t*) when the tank is being filled.

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