2.6: Untitled Page 18

Chapter 2

is zero (

0 ) the drop size is not infinite as predicted by Eq. 2‐13, but instead it a

V

is on the order of the diameter of the atomizer jet. Once again, this indicates that Eq. 2‐13 is only valid for some range of values of X vs ; however, the range is known only to those persons who examine the original experimental data.

Clearly a dimensionally incorrect empiricism carries its own warning: Beware!

The dimensionally incorrect result given by Eq. 2‐13 can be used to construct a dimensionally correct equation by finding the units that should be associated with the coefficients 1920 and 597. For example, the correct form of the first term should be expressed as

 correct form needed

1920 (units) 

(2‐14)

 to produce micrometers

a

V

in which the correct units are determined by

3 

3 

(ft/s) g/cm

(units) 

(micrometers)

4

(ft/s) g/cm 

10 cm 

dyne/cm 

g s

(2‐15)

 30.48

4

10

cm

Thus the dimensionally correct form of the first term in Eq. 2‐13 is given by 1920 

 5.85 cm 

(2‐16)

a

V

a

V

and it is an exercise for the student to determine the correct form of the second term in Eq. 2‐13.

2.4 Convenience Units

Occasionally one finds that certain quantities are represented by terms that do not have the appropriate units. The classic example of this situation is associated with the use of mercury barometers to measure the pressure. It is a straightforward matter to use the laws of hydrostatics to show that the atmospheric pressure measured by the barometer shown in Figure 2‐1 is given by

o

 

o

p

g h

p

(2‐17)

Hg

Hg

Here o

p

represents the vapor pressure of mercury and under normal

Hg

Units

23

Figure 2‐1. Mercury barometer

circumstances this pressure is extremely small compared to

. This allows us to

o

p

write Eq. 2‐17 as

 

o

p

gh

(2‐18)

Hg

Since the density of mercury and the gravitational constant are essentially constant, the atmospheric pressure is often reported in terms of h, i.e. millimeters of mercury. While this is convenient, it can lead to errors if units are not used carefully. The message here should be clear: Beware of convenience units!

The pressure over and above the constant ambient pressure is often a convenient quantity to use in engineering calculations. For example, if one is concerned about the possibility that a tank might rupture because of an excessively high pressure, it would be the pressure difference between the inside and outside that one would want to know. The pressure over and above the surrounding ambient pressure is usually known as the gauge pressure and is identified as pg . The gauge pressure is defined by p

g

p

o

p

(2‐19)

24