K = bulk modulus

AP = pressure drop in pipe

Fd =drag force A = area normal to flow e, = friction loss

(energy/mass) rw = wall stress yv2



(Buoyant x inertial)/ (viscous) forces

(Yield/viscous) stresses

(Gravity/surface tension) forces

(Inertial/compressible) forces

(Pressure energy)/ (kinetic energy)

(Drag stress)/ momentum flux)

(Wall stress)/ (momentum flux)

Settling particles, fluidization

Flow of Bingham plastics

Rise or fall of drops or bubbles

Compressible flow

Flow in closed conduits

External flows

Flow in pipes, channels, fittings, etc.

Fraude number /VFr /VFr = V2/gL

Hedstrom number NHe T0D2p

Reynolds number /VRe flows /VRe =


4Qp TzDpL

Mach number NUa V

L= characteristic length (Inertial/gravity) forces

Free surface flows t0 = yield stress fj,^ = limiting viscosity

Pipe flow: rw =wall stress

(Yield x inertia)/ viscous stresses

(inertial momentum flux)/ (viscous momentum flux)

Flow of Bingham plastics

Pipe/internal flows (Equivalent forms for external flows)

c = speed of sound (Gas velocity)/(speed of sound)

High speed compressible flow quantities, it is important that we understand and appreciate which of the numbers we deal with are useful and which are not.

First of all, we should make a clear distinction between accuracy and precision. Accuracy is a measure of how close a given value is to the ''true'' value, whereas precision is a measure of the uncertainty in the value or how ''reproducible'' the value is. For example, if we were to measure the width of a standard piece of paper using a ruler, we might find that it is 21.5 cm, give or take 0.1 cm. The ''give or take'' (i.e., the uncertainty) value of 0.1 cm is the precision of the measurement, which is determined by how close we are able to reproduce the measurement with the ruler. However, it is possible that when the ruler is compared with a ''standard'' unit of measure it is found to be in error by, say, 0.2 cm. Thus the ''accuracy'' of the ruler is limited, which contributes to the uncertainty of the measurement, although we may not know what this limitation is unless we can compare our ''instrument'' to one we know to be true.

Thus, the accuracy of a given value may be difficult to determine, but the precision of a measurement can be determined by the evaluation of reproducibility if multiple repetitions of the measurement are made. Unfortunately, when using values or data provided by others from handbooks, textbooks, journals, and so on, we do not usually have access to either the ''true'' value or information on the reproducibility of the measured values. However, we can make use of both common sense (i.e., reasonable judgment) and convention to estimate the implied precision of a given value. The number of decimal places when the value is represented in scientific notation, or the number of digits, should be indicative of its precision. For example, if the distance from Dallas to Houston is stated as being 250 miles and we drive at 60 miles/hr, should we say that it would take us 4.166667 (=250/60) hours for the trip? This number implies that we can determine the answer to a precision of 0.0000005 hr, which is one part in 107, or less than 2 milliseconds! This is obviously ludicrous, because the mileage value is nowhere near that precise (is it ±1 mile, ±5 miles?—exactly where did we start and end?), nor can we expect to drive at a speed having this degree of precision (e.g., 60 ± 0.000005 mph, or about ±20 mm/s!). It is conventional to assume that the precision of a given number is comparable to the magnitude of the last digit to the right in that number. That is, we assume that the value of 250 miles implies 250 ± 1 mile (or perhaps ± 0.5 mile). However, unless the numbers are always given in scientific notation, so that the least significant digit can be associated with a specific decimal place, there will be some uncertainty, in which case common sense (judgment) should prevail.

For example, if the diameter of a tank is specified to be 10.32 ft, we could assume that this value has a precision (or uncertainty) of about

0.005 ft (or 0.06 in., or 1.5 mm). However, if the diameter is said to be 10 ft, the number of digits cannot provide an accurate guide to the precision of the number. It is unlikely that a tank of that size would be constructed to the precision of 1.5 mm, so we would probably assume (optimistically!) that the uncertainty is about 0.5 in or that the measurement is "roughly 10.0 ft.'' However, if I say that I have five fingers on my hand, this means exactly five, no more, no less (i.e., an "infinite" number of "significant digits").

In general, the number of decimal digits that are included in reported data, or the precision to which values can be read from graphs or plots, should be consistent with the precision of the data. Therefore, answers calculated from data with limited precision will likewise be limited in precision (computer people have an acronym for this—"GIGO," which stands for ''garbage in, garbage out"). When the actual precision of data or other information is uncertain, a general rule of thumb is to report numbers to no more than three ''significant digits,'' this corresponds to an uncertainty of somewhere between 0.05% and 0.5 % (which is actually much greater precision than can be justified by most engineering data). Inclusion of more that three digits in your answer implies a greater precision than this and should be justified. Those who report values with a large number of digits that cannot be justified are usually making the implied statement ''I just wrote down the numbers—I really didn't think about it.'' This is most unfortunate, because if these people don't think about the numbers they write down, how can we be sure that they are thinking about other critical aspects of the problem?

Example 2-4: Our vacation time accrues by the hour, a certain number of hours of vacation time being credited per month worked. When we request leave or vacation, we are likewise expected to report it in increments of 1 hr. We received a statement from the accountants that we have accrued "128.00 hours of vacation time." What is the precision of this number?

The precision is implied by half of the digit furtherest to the right of the decimal point, i.e., 0.005 hr, or 18 s. Does this imply that we must report leave taken to the closest 18 s? (We think not. It takes at least a minute to fill out the leave request form—would this time be charged against our accrued leave? The accountant just ''wasn't thinking'' when the numbers were reported.)

When combining values, each of which has a finite precision or uncertainty, it is important to be able to estimate the corresponding uncertainty of the result. Although there are various "rigorous" ways of doing this, a very simple method that gives good results as long as the relative uncertainty is a small fraction of the value is to use the approximation (which is really just the first term of a Taylor series expansion)

which is valid for any value of x if a < 0.1 (about). This assumes that the relative uncertainty of each quantity is expressed as a fraction of the given value, e.g., the fractional uncertainty in the value A is a or, equivalently, the percentage error in A is 100a.

Example 2-5: Suppose we wish to calculate the shear stress on the bob surface in a cup-and-bob viscometer from a measured value of the torque or moment on the bob. The equation for this is

0 0

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