Despite the unfortunate use of the pound as both unit of mass and unit of force-a traditional source of frustration to engineers-the
pitfalls can be avoided
The American Engineering System
of Units and Its Dimensional Constant gC ADRIAN KLINKENBERG
he engineering systems of units, having four funda-
Tmental units (force, mass, length, and time), need to
be clearly distinguished from the gravitational and dynamic systems, having three fundamental units (force, length, and time or mass, length, and time), so that a brief survey is in place : Units of Mass Length
System
Force
Time
“Old version” English engineering “New version” English engineering English gravitational = static English dynamic = absolute
Lb
Lb
Ft
Lbf
Lb,
Ft
Lb
Slug
Ft
Sec or hr Sec or hr Sec
Poundal
Lb
Ft
Sec
The name, “English Engineering System,” is used, for example, by Walker et al. (48,p 28) and corresponds to the “Old version” in the table. The name, “American Engineering System,” is employed by Chilton ( 5 ) and Himmelblau (27) for the “New version,” but also by Hall (79) for the “Old version.” However, the name, “British Engineering System,” as used by Coulson and Richardson (9), refers to the English gravitational system of Walker et al. (48). The engineering systems with four fundamental units came into being as a compromise between the desire of mechanical engineers to use the “pound” as a unit of force and the desire of chemical engineers to employ it as a unit of mass. The “New version” of the English engineering system was proposed when the “Old version” was found to be too confusing for anyone not
~~~~
Examples of common errors and inconsistencies in the literature are classified in this article as follows:
l i
1.
Using faulty numerical values or units for g,
c.
Using poundal and Ibf, or Ib, together in the same treatment
2.
Swerving between old and new engineering systems
d.
Writing go in a system based on FLT or MLT
a.
I, Using Ib for Ibf and also for b with go
b.
Confusing g and g,
c.
In discussing - Bernoulli’s theorem
together
3.
Using various names for gc
4.
Swerving between engineering systems and corresponding -~dynamic and static systems a.
Using FMLT units together with MLT dimensions
b.
Arriving at the paradox: g, = 1
5.
and slug,
e.
Omitting goor go (g) in an FMLT system
f.
Various
Failing to note dualism in deriving units from either Ib, or Ibr
6. Denying go its existence When an author gives two conflicting statements without having explained his intentions, it may well be that each statement can be b y itself correct, but only i f the other is wrong. The allocation of the examples to the section headings may therefore be somewhat arbitrary.
VOL. 6 1
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APRIL 1969
53
much familiar with it, and sometimes too confusing even for proponents of the “Old version.” Examples of this confusion are given later in this article. For an introduction to the “New version,” see Chilton ( 5 ); for a general survey of systems, both English and metric, see Klinkenberg (25). There have been some corresponding developments in the metric countries when a kgf, kg,, m, sec system has been proposed. However, the dynamic system based on kg,, m, sec has an inherently stronger position than the lb,, ft, see system, because of its coherence with practical electrical units and its adoption as the base for the International System (the “SI”). Consequently, a compromise similar to that which created the English engineering system makes less sense. Klinkenberg (26) therefore considered a logical choice of systems to be:
either or
Metric: The SI Pounds and feet: The “New version” English engineering system
E. Schmidt, in his work on engineering thermodynamics, abandoned the kgf, kg,, m, sec system for the SI. Changeover from various metric systems into SI is discussed by Hahnemann (78). Hall (79) and several other authors in the same volume discuss the SI. An introduction to the SI and an encouragement to use this system have further been given by the American Chemical Society (7). I n this article, the symbols lbf and lb, are used for conciseness and clarity. Equivalent expressions are : lb (force) and Ib (mass); lbf and lb-see McAdams ( 3 4 , Chilton ( 5 ) , and McCabe and Smith (35), p 969; lb and lb,-see Badger and Banchero (2); Lbf and Lbm-see Maker (36). (Maker, in fact, says: “Joe” is not enough description of a man if both “Joe Smith” and “Joe Doakes” are working on a job.)
“New version” engineering system, g p = mu
(4)
with go = 32.174 lb, ft/lbf sec2
(5)
Gravitational and dynamic systems, F = mu
(6)
The value of go or go is standardized (45” latitude a t sea level) to avoid the variability in lbf that would be introduced by defining this at the local g [see, e.g., Comings (7) I. Quite often, in the “Old version” system these differences are ignored when one writes the local g instead of go. This gives rise to the confusion of seeing g appear at the “wrong places.” I n view of the approximately 0.5y0variation i n g with latitude, a statement that the acceleration due to gravity at sea level equals 32.1739 ft/sec2 [Giedt (77, p 363); Thatcher (45, p 417)] holds at 45” latitude only. An effort to distinguish the dimensional constant from the local acceleration due to gravity is made by Lockhart and Martinelli (32) by writing for the dimensional constant
g
=
32.2 lb ft/lb sec2
Here the reader is expected to remember the difference of the two “pounds.” I n Maker’s paper (36), to clarify the situation, the starting point for the discussion on units is
1 Lbf = (1/32.1’74) Lbm X 32.174 ft/sec2 This equation is stated to have been experimentally found. Maker prefers to use k = 1/32.174 instead of go to avoid confusion with g, and he also prefers not to assign a unit to k. If one does not give k a unit, the above equation simplifies to
1 Lbf = 1 Lb, X 1 ft/sec2 Definitions-Newton’s
Second Law
The starting point is, of course, the application of Newton’s second law to free fall, when 1 lbf imparts to 1 lb, an acceleration of 32 ft/sec2. In the interest of using a coherent (consistent) system of units, one does not want to write this in the form of:
1 lbf
=
(11
1 lb, X 32 ft/sec2
[This equation, though, can be met in literature-see Kern (24, p 28).] Numerical equality is obtained by choosing such units that free fall of a pound is described: in engineering systems, by: 32 X 1 = 1 X 32 in the gravitational system, by: 1 = (1/32) X 32 in the dynamic system, by: 32 = 1 X 32 Accordingly, Newton’s second law is made to read: “Old version” engineering system,
g,F
=
mu
with go = 32.174 ft/sec2 54
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
(2) (3)
which is extremely confusing, especially in view of Equation 1. If one does give k the units of 1/go, the equation indeed describes free fall of a pound weight at 45” latitude and at sea level (Equation 4). 1.
Using Faulty Numerical Values or Units for g,
Some of the errors or misprints Klinkenberg (26) has collected from books [his number A3 from McAdams (33) and his A1 from Lapple (29) in Perry’s Handbook] have been corrected in later editions. Hershey and Song (20) expressed g, in Ibf ft/lb, sec2. Further discussed below are statements : gc = g (in Section 2) go = 1 (in Sections 4a, 4d) g, = a numerical factor (in Sections 2b and 4a).
Nissan and George (38), in their Equation 19, use a wrongly placed go.
2.
Swerving between Old and N e w Engineering Systems
U
I n consequence, the resultant equation :
a. Using lb for lbf and also for lb,, together with g, ( L e . , not g). I n this case a check on homogeneity of units leaves one with a product such as lb-lbf/lb-lb,. The reader is expected to remember which pound is which. The “New version” was, however, specifically meant to eradicate the habit of using the same symbol for the two kinds of pound to create more clarity in the concepts. Such clarity, which we owe to Newton, tended to be jeopardized in the “Old version.’’ Sherwood and Pigford (42, p 2391, use g, together with a pressure P in lb/ft2 and a density p in lb/ft*. Leva (37) and Fan and Wen (73) use g, together with both pounds. Susskind and Becker (44) use g, together with a pressure drop, Ap, in lb/ft2 and a mass flow density G in lb/sec-ft2. The interpretation of texts is made more difficult, in general, if there is too much looseness in writing lb or lb, and lb or lbf. Badger and Banchero ( 2 ) announce on p 11, lines 1-2: “ I n this book the mass pound will be denoted by lb mass and the force pound by lb without qualification”; and continue in the very next sentence: ‘‘Pressures in the fps system are expressed by lb force/sq ft or lb force/sq in.” Thatcher (45) says on p 82 : “ . , . . . . if P is expressed in lb/ft2 and V is in ft3, . . . . . . the PV product will then be in lbf ft.” b. Confusing g and g,. If one defines g, on the basis of the “New” FMLT system, one cannot later equate it to the go or g of the “Old” system-g, and g or go are different dimensionally and have different units. Badger and Banchero (2, pp 8-9) write a dimensional constant, k, in Newton’s second law, eliminate this constant to derive the poundal, introduce gc as a numerical constant to convert poundals to pound weight, and reintroduce g, as a dimensional constant in an FMLT system. Finally, g/g, is taken as unity. Confusing g and g, has resulted in an incorrect treatment of the derivation of Bernoulli’s theorem [see warning by Comings (7)]. Because of the fundamental importance of this theorem, this subject will be analyzed. c. In discussing Bernoulli’s theorem. I n the following discussion we shall leave out heat effects and external work and consider the fluid incompressible. Authors will be quoted for such simplified versions of their formulas only. Drew et al. (72), in the 3rd edition of Perry’s Handbook, used the “New” engineering system. They derived a total energy balance per “unit weight of fluid,” in which all terms are expressed in ft Ibf/lb,. This balance would read : @/P
+ Zg/g, 4- v2/2g, = constant
(7)
However, even before making the above simplification, the ratio g/gc was regarded as unity “since the numerical value of g rarely differs from 32.1740 by as much as 3 parts per 1000.”
p/p
+ Z + v2/2gc = constant
(8)
was dimensionally inconsistent. Badger and Banchero (2, p 38), follow this reasoning and observe that such inconsistency appears to be necessary to arrive at a practical and simple form of Bernoulli’s theorem in which all terms numerically (though not dimensionally) are heights. Also, Hughmark and Pressburg (22) and Larian (30) at this point “assume” g c = g. Thatcher (45, p 244), notes that the units of ft lbf/lb, “are commonly reduced to feet, even though the cancellation of lbf by lb, is not quite proper. . ,” and asks the student, “to keep the true significance of the term head and its pseudodimension of length well in mind.” I t is, however, sufficient to divide Equation 7 as a whole by g/gc to obtain a dimensionally correct equation in which the terms are true heads [Chilton ( 5 ) ]: gg/pg
+ Z + v2/2g = constant
(9)
This then is Bernoulli’s theorem in terms of heads, for use in the “New” engineering system. The equivalent form employing specific weight y (weight per volume) : p/y
+ Z + v2/2g = constant
(10)
though correct, is not to be suggested for use in the “New” FMLT system. For the other systems, accordingly: “Old” FMLT: Replace g, by go (or g) in Equation 9. De Rienzo (70) and Giedt ( I 7, p 82), use p/p
MLT :
+ 2 + v2/2g
=
constant
(11)
Note that the consequence of identifying g and g, is also identifying y and p (compare Equations 10 and 11) Omit g, from Equation 9 to give p/pg
+ Z + v2/2g
=
constant (12)
FLT :
Equation 10, though correct, is impractical Use Equation 10 or-if one is prepared to use the density expressed in slugs/ft3-Equation 12
Silberberg and McKetta (43) gave a unified treatment that is valid for the “New” FMLT, the MLT, and FLT systems and which exposes the confusion quite clearly. Unfortunately, there is an error at the crucial spot. I n the final equation for Bernoulli’s law in terms of heads, valid for all systems, a specific volume u (units = volume/mass) should be deleted in the pressure term. Thereupon it becomes our Equation 10. Silberberg and McKetta, however, do not discuss the ‘‘Old” FMLT VOL. 6 1
NO. 4
APRIL 1969
55
system. They mention that Equation 11 is often used, but consider it dimensionally incorrect. The present author, though disliking the method of equating the dimensions and the units of mass and force, feels that he cannot logically forbid it. But once this procedure is accepted, thereby giving the “Old” F M L T system the status of a coherent system-as has been done in the present paper-then, because of the resultant identity of p and 7,Equation 11 becomes just another way of writing Equation 10. McCabe and Smith (35)) after deriving Equation 7, say (footnote, p 74) that in “hydraulic practice” it is replaced by Equation 10. Boucher and Alves ( 3 ) , in the 4th edition of Perry’s Handbook, present the general form corresponding to Equation 7. Brown et al. ( 4 ) observe that the terms of Equation 7 are frequently and erroneously called liquid heads.
3. Using Various Names for g, Lockhart and Martinelli (32) as discussed in Section 2, refer to g (= 32.2 lb ft/lb secz) as the gravitational force per unit mass. I t is inviting to interpret this as an acceleration. McCabe and Smith (35) and Shenvood and Pigford (42) refer to g , as, respectively, Newton’s law conversion factor and the gravitational conversion factor. Drew (72, p 360)) and Klinkenberg (25-27) refer to g , as a dimensional constant. The relative merits of the latter expressions will be discussed in the next section. 4. Swerving between Engineering Systems and Corresponding Dynamic (MLT) and Static (FLT) Systems a. Using FMLT units together with MLT dimensions. Thatcher (45), while preparing the way for use of the lbf, lb,, ft, sec system, continues to consider the dimensional formula IF] for force to be related to [MI, [L], and [TI-see, e.g., p 417. McCabe and Smith (35))on the other hand, use the identical F M L T structure for both systems-i.e., for any quantity, one finds the unit by substituting the fundamental units in the dimensional formula. This approach is highly recommended. I t is unfortunate that McCabe and Smith depart from this principle in one aspect, viz., by identifying [MI and [Mole] in the dimensions, while keeping lb and lb mol distinguished as units. I n consequence, the molecular weight in the column “Units” is said to be dimensionless, whereas for converting, say, a specific energy (Btu/lb) into a molar energy (Btu/lb mol) it must have the unit lb/lb mol. b. Arriving at the paradox g, = I. McCabe and Smith (35, Appendix 2) have listed g , together with “other” conversion factors, such as 1 “C/1 OF = 1.8, and write in the column “Factors” :
1 lbf sec2/1 lb, ft = 32.174
56
g,
=
32.174 lb, ft/lbr sec2
(5)
the result is
The solution to this paradox is that definitions of derived units, to show how these depend on fundamental units, are as good as pure algebra, only if one stays within a single dimensional system. One cannot really multiply a lb, by a ft and divide by the square of a second to obtain a poundal. But since the derived unitpoundal-depends on these fundamental units in the way Equation 14 indicates, there is no harm done if we do call this a multiplication, provided we remember the basic convention of the M L T systems, which is not to have such a thing as g,. It is because of this violation of basic conventions that Equation 13 has no place in a treatise on the FMLT system. A strong warning: The combination of two statements, each correct within its own philosophy, will produce nonsense if the philosophies are in conflict. The conclusion reached is that the users of the F M L T system should not call the dimensional constant g , a < C conversion factor,” because this is a concept derived outside their own world. They have, however, no right to stop others from doing this. But, of course, these other people have no business to meddle with g,. Similar paradoxes arise in connection with other dimensional constants, such as J, eo, and po, if one does not stay in one’s system: MECHANICAL EQUIVALENT OF HEAT. McCabe and Smith (35, Appendix l ) , make J a secondary quantity so that :
J
= 778.26 ft lbf/Btu
but in Appendix 2 they introduce as a conversion factor
1 Btu/l ft lbf = 778.26 whence one would conclude that J = 1, an obvious falsity in the “new” engineering system. PERMITTIVITY, eo, AND PERMEABILITY, PO, OF VACUUM. Pohl (40)revealed and Klinkenberg (27) discussed a paradox in which the cgs electrostatic and electromagnetic units are related to one another, each being expressed in terms of cm, g, and sec, with the respective conventions e,, = 1 and po = 1 .
A . Klinkenberg, now retired, w a s Visiting Professor
(13)
AUTHOR
(14)
of Chemical Engineering at the University of California, Berkeley, at the time of writing this paper. H i s present address in Glipperweg 28, Heemstede, The Netherlands.
Since
1 Ibf = 32.174 poundals = 32.174 lb, ft/sec2
Equation 13 embodies the lbr/poundal conversion, together with the definition of the poundal. As such, it is a correct statement. Also, Brown et al. (4, p 132), define g, = conversion factor (32.17 poundals per pound). However, if the unwary reader combines Equation 13 with the definition of g , as the dimensional constant belonging to the F M L T system, viz,:
INDUSTRIAL AND ENGINEERING CHEMISTRY
For the units of electric charge this would give
3.1010 cms/2 gl” sec-1
1 cm’/2
gl/z
or 3.1010 cm = 1 sec I n view of the fact that copo = 1/c2, a result from Maxwell’s theory of light, the conventions can be reconciled only if the units are such that the velocity of light, c, is unity. This is indeed what one finds. The existence of confusion of the kind described above is a powerful argument for standardization in the field of dimensional systems. I n contrast, the ordinary conversion factors for similar quantities, though a bother, constitute no hazard in thinking. The mix-up between the concepts of ‘‘dimensional constant” and “conversion factor” has also caused the mishap mentioned by Klinkenberg (26, Example A8). Here Van Heerden et al. (47) used a metric M L T equation, which they therefore wrote without the g,. Conversion to English engineering units was meant to be achieved by introducing g, in the conversion factor. I n their nomenclature section, this led to the statement:
Afi = pressure drop N/m2 (= 8.7 X lo6 lb force/ft2) The factor, however, is not the conversion factor, as the equals sign would suggest, but the product of gc and the conversion factor, using hours. c. Using poundal and lbf, or lb, and slug, together in the same treatment. Kern (24, p 33)) states: “The force-pound is the poundal,” and analogously, “The force-gram is the dyne.” It is clear that an FMLT system is used (Kern, though, uses 0 for time and T for temperature), with lbf and lb as units of F and M (compare, however: 1 Btu = 778 ft-lb). Thus, the “forcepound” and the “force-gram’) indeed are the lbf and the gf,and the statements about the poundal and the dyne are erroneous. I n the lbf-lb-ft-hr system, shear stress is expressed in poundal/ft2 = lb,/ft sec2. This violates the consistent use of hr, and also violates the convention that the unit for is better derived from lbt (Section 5). Giedt (77, p 353 ff), on the whole, uses the FLT system with viscosity p in lb sec/ft2 and mass density p in slugs/ ft*. But the mass velocity G is in lb/hr ft2, and this cannot serve in the definition Re = dG/p which would give R e units (ft/hr sec). d. Writing g, in a system based on FLT or MLT. I n F L T or M L T equations, one must not write go. If FMLT equations are to be used in FLT or M L T systems, the g, must be left out. This last rule probably perpetuates the unfortunate belief that, fundamentally, g, = 1. This, however, one cannot say within the FMLT system. Foust et al. (75, p. 517)) use g, in FLT (and MLT) systems, stating that g, = 1 slug ft/lbf sec2, and saying that this g, must be included for dimensional reasons. Dent (77) does not use a fundamental unit of force, yet he writes g,r = pa,where r = shear stress [M/ LT2J, and g, is dimensionless. Lapple (29) lists units in two columns headed: “Sys-
tem of consistent units,” subtitled ‘‘Metric (cgs)’) and “English,” while explaining ‘‘that any system of consistent units may be employed,” and “that cgs and English units are merely given by way of example because they represent the most common such systems used.” But by including in each column units from more than one consistent system (e.g., poundal as well as lbf and go and, also, dyne, as well as gf and a metric g,) , he violates both the general heading and the cgs subtitle. A warning must, however, be issued that the name ‘‘cgs system” is sometimes [Treybal (46,p 31)] used for the gf, g,, cm, sec system, and this of course uses a g,. From the author’s point of view this has the advantage that his equations can be used with both metric and English FMLT units, but the regular users of the cgs system will object. e. Omitting g, or go(g)when definitions make clear that an FMLT system is used. Omitting gc leads to numerical mistakes by author or reader, unless the matter is spotted in time by a dimensional (units) check. Coons et al. (8))while using the “Old” FMLT system omitted g in a friction factor. Accordingly their graph 9 does not show the usual dimensionless friction factor plotted against the Reynolds number, Nissan and Bresan (37) omitted g, in their notation in a drag coefficient c p (stated to be dimensionless), and omitted it again in their text, in an equation relating pressure to kinetic energy. Fan and Wen (73) omitted g in equations for pressure drop in fluidized beds, for which lb-lb-ft-hr units are specified. Samuels and Churchill (47), in defining a dimensionless pressure P, use a pressure in lbf/ft2 and a density in lb,/ft3, but not g,. Also, with the FMLT units employed, there should have been a g, in their NavierStokes equations. Kern (24, p 27)) relates shear stress r and viscosity p by du
7
= p-
dY though (p 33) r is derived from lbf and p from lb,. Galloway and Sage (76))in their notation, omitted g 1 in the definition of a pressure coefficient Ap/--uVZ,
2
where Ap was expressed in lb/ft2, u in lb/ft3, and U in ft/sec. f. Various. Giedt (77, pp 94-95) successively discusses: (1) the four primary dimensions of F, M, L, and T ; (2) their relationship by Newton’s second law; (3) viscosity expressed as FT/L2 or M/LT; ( 4 ) the units of viscosity lb sec/ft2 and slug/ft sec. Items 1 and 3 belong to the FMLT system, while items 2 and 4 belong to the FLT system. The two units for viscosity, in view of: 1 slug = 1 lb sec2/ft, are, in fact, identical. It should be observed that the slug/ft sec is not an M / L T unit, since the slug itself is a derived unit. For a further example of the ways in which different reasonings can interfere, see Pankhurst (39, p 35). H e introduces the FMLT system correctly as one based on four dimensions. But his earlier chapter on “Fundamental and Derived Units and Dimensions in MechanVOL. 6 1
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57
ics” dealt with FLT and M L T systems only, and thus put [MI = [FL-’T21 or [F] = [MLTW2]. O n p 23, Newton’s relation, “ F proportional to ma” was used to equate the dimensions of F to those of ma-this shuts the door to the FMLT system. In consequence, Pankhurst also equates the two dimensional formulas for work: [FL] = [ML2T-2]. Again, in the FMLT system these are essentially different. In fact, a choice by convention is required (Section 5). 5.
Failing to Note Dualism in Deriving Units from Either Ib, or Ibf Having defined go from an equation that directly
relates a force to a mass (Newton’s second law), let us now consider a more devious route from force to mass. This is by way of the following concepts, all related by well known equations : Force,
F
Pressure, p
dv
gcr, = p dx
(7,
= molecular r )
and here the T, and p are specified as r Fand p a , so that the goshould not be present. Their Poiseuille equation (Equation 512) likewise either uses the wrong p or uses gowhere it should not. Kozicki et al. (28) define or use j F , T F , p M , and pM for Newtonian fluids and yet use p F and p F for nonNewtonian fluids. Fishenden and Saunders (74, p 79), relate p F to r M ,so that here Equation 13 contains the g, (here a s ) . The same authors, p 85, relate$, to rw Jakob (23) defines the Prandtl group in the usual manner as pc,/k, but gives his data for it in inconsistent units, viz.,p in kg, hr/m2; c p in kcal/kg “C and k in kcal/ hr m “C, so that either a metric goshould have been used or a p M . I t is not suggested that Jakob consequently
Shear Stress, T
Viscosity,
Density, p
Mass, M
Lbf sec2/ft (= slug) Lb,
FLT units
Lbr
Lb,/ft
Lbr/ft
Lbf sec/ft2
Lbt sec2/ft4
MLT units
Lb, ft/sec2 ( = poundal)
Lbm/ft sec2
Lbm/ft sec2
Lbm/ft sec
Lbm/ft3
The FMLT system uses lbf in the units at the left of the table and lb, in the units at the right (iz.,never the poundal or the slug), but precisely where the crossover is to be made must be the subject of a convention. At this crossover there is an equation involving g,: McAdams (34, p 128), uses p M for viscosity expressed in lb,/ft sec and p for viscosity in lb, sec/ft2. McCabe and Smith (35, p 971), write p and kF; Comings (6, 7) uses p’ and p . McCabe and Smith refer to absolute and gravitational viscosity. Comings, however, refers to both concepts as absolute viscosity. McAdams (34) expresses a preference for pw McCabe and Smith say (p 51) that “another viscosity is sometimes used.” Dimensional analysis, e.g., is carried out (p 19) employing p F so that the Reynolds group contains a g,. However, they mostly use p (i.e., p M ) . If, for the sole purpose of the present discussion, we use both subscripts M and F for all properties concerned (except force and mass themselves), then it can be stated : McAdams would use: j F ,T F ,p M , and pM McCabe and Smith would prefer: the same set, but sometimes use: P F , T F , p F , and pM For the preferred route, accordingly : d
Shear stress at wall of pipe : r F = AbF . 4L Definition of Relation between
p
and
p:
(13)
dv
gcrF = p M dY
Y: pM
=
v ~ M
(15 )
Brown et al. (4, p 6 7 ) say that viscosity has dimensions M / L T or FT/L2 and later (p 67), without explanation, narrow this down to the unit lb,/ft sec (i.e., to p M ) . 58
But on page 520, their Equation 511 reads:
INDUSTRIAL AND ENGINEERING CHEMISTRY
calculated faulty Pr values-his values in fact are correct. However, his readers, upon following his prescriptions, will obtain incorrect numerical values. I n addition, some of the inconsistencies discussed in earlier sections are related to these same problems. I t clearly results from the above that a very stringent rule is required to end the confusion concerning the use of gc. The rule employingpF, T F , p M , and p M , as used by McAdams, seems most suited. Similar problems arise in connection with work, which is commonly taken to be W Fin ft-lbf (included for kinetic energy), and with interfacial tension (commonly lbf/ft). 6.
Denying g, Its Existence
Coulson and Richardson (9, pp 1-2) use Newton’s second law only without the dimensional constant, i.e., they define: the MLT system, called the Absolute FPS System the FLT system, called the British Engineering System or Gravitational FPS System. They state, pp 2-3 : “It will be noted that the units of 1 lb mass and 1 lb weight belong to two different sets of units, and therefore these two quantities must not be used together in any dimensionally consistent equation.” The philosophy for the American (“New version”) Engineering System must, however, be that its equations are, in fact, dimensionally consistent; this is achieved by the inclusion of the dimensional constant g,. The inclusion of a dimensional constant cannot be held against the consistency of a system. I t is recalled that the cgs system and the International System (SI) also employ dimensional constants (eo or po, respectively,
The criterion for consistency should be: € 0 and PO). “Homogeneity of the equation in its units.” Conclusions-Fundamental
1. Systematic application of the FMLT system based on lb,, lbf, ft, sec requires that gobe considered a dimensional constant. T o say that it is a conversion factor is foreign to the FMLT system. The two views combined lead to a paradox. 2. Much confusion in thinking is caused by swerving between systems that are differently based and by uncertainty concerning the fundamental quantities of the system. 3. This points to the need to reduce the number of systems, ideally to reduction to a single system. 4. I t is advisable to use a dimensional system and a units system of the same construction, so that derived units are found by substituting fundamental units into dimensional formulas. Conclusions-Practical
The fact that brilliant chemical engineers sometimes have a weak moment regarding units lends extra weight to the following recommendations :
5 . All of the confusion described above could be prevented if authors would: (a) Decide at an early stage what system to use, and state this clearly in their publications (b) Use that system consistently (c) Apply dimensional (units) checks to see that they have indeed been consistent
d
‘c
6 . Of the four systems discussed, no more than two need remain in use. The FLT system could be dispensed with; its adherents have no reason to complain if they can keep the lbf, lbf/ft2, and ft lbf, as in the “New” FMLT system. The “Old” FMLT system changes into the “New” by using lbf, lb,, and go instead of lb, lb, and g, and by reviving a factor g/g0 sometimes seen in formulas. All this works toward better clarification, so that no one can, within reason, object. The M L T system holds its ground. I t is widely used in Britain, is analogous in structure to some important metric systems (cgs and the International System), and avoids the ambiguities of the FMLT systems which were discussed in Section 5. The “New” FMLT system needs stringent specifications to avoid such ambiguities. 7. I t is not correct to put the blame for the present situation on the “New” FMLT system. There would have been considerably more confusion if only the “Old” FMLT system were used. 8. Journals cannot ignore these problems, since these have a direct bearing on the quality of the work. They can exercise a beneficial influence : (a) By insisting and checking that authors of journal articles are consistent in their calculations
(b) By stating what systems are allowed in their columns (c) By having the authors of book reviews, wherever appropriate, include a statement on the consistent use of units.
The Editor of I&EC invites correspondence from readers concerning this article and the use of consistent units by scientists and engineers.
REFERENCES (1) American Chemical Society, “Handbook for Authors;’ ACS, Washington, D. C., 1967. (2) Badger, W. L., and Banchero, J. T., “Introduction to Chemical Engineering,” McGraw-Hill, New York, N. Y., 1955. (3) Boucher, D. F., and Alves, C. D., in “Chemical Engineers’ Handbook,” 4th ed., J. H.Perry, Ed., McGraw-Hill, NewYork, N. Y., 1963, pp 5-17. (4) Brown G G., et al., “Unit Operations,” John Wiley and Sons, New York, N. Y., 1650.‘ (5) Chilton T. H (1959) in “Systems of Units,” C. F. Kayan, Ed., Publ. 57 of AAAS, dashing;)on, D. C., 1959, pp 87-100. (6) Comin s, E. W., “High Pressure Technology,” McGraw-Hill, New York, N. Y., 1856. (7) Comings, E. W., IND.ENO.CHEM.,32, 984 (1950). (8) Coons, K. W., Hargis, A. M., Hewes, P. Q., and Weems, F. T., Cham. Eng. Progr., 43, 405 (1947). (9) Coulson, J. M., and Richardson, J. F., “Chemical Engineering,” Vol. I, Pergamon Press, Oxford, England, 1961. (10) De Rienzo, P. E., “Chemical Engineering,” The MacMillan Co., New York, N. Y., 1964. (11) Dent, J. C., A.I.Ch.E. J., 13, 1114 (1967). (12) Drew, ‘I’;B Dunkle H. H and Generaux R. P., in “Chemical Engineers’ Handbook, i;d ed., H. Pe‘rry, Ed., McGiaw-Hill, New York, N. Y., 1950, p 375 ff. (13) Fan, L. T., and Wen, C. Y., A.I.Ch.E. J.,7,610 (1961). (14) Fishenden, M., and Saunders 0. A “Introduction to Heat Transfer,” Oxford Univ. Press, London, EnglHnd, 19;O. (15) Foust, A. S., Wendel, L. A Glum , C . W., Maus, L., and Andersen, L. B., Principles of Unit Operations;:’ Johnkiley and Sons, New York, N. Y., 1960. (16) Galloway,T. R., and Sage, B. H., A.I.Ch.E. J., 13,563 (1967). (17) Giedt W. H “Principles of Engineering Heat Transfer,” Van Nostrand, Princetoh, N. J.,”1957. (18) Hahnemann H. W., “Die Umstellun auf das Internationale Einheiten System in Mecianik and WPrmetechnik,” q.D.1. Verlag, Diisseldorf, 1964. (19) Hall N. A (1959) in “S stems of Units,” C. F. Kayan, Ed., Publ. 57 of AAAS, kashiI;)gton, D. C., 19l9, pp 101-8. (20) Hershey, D., and Song, G., A.I.Ch.E. J., 13, 491 (1967). (21) Himmelblau, D. M., “Basic Principles and Calculations in:Chemical Engineering,’’ Prentice Hall, Englewood Cliffs, N. J., 1962. (22) Hughmark, G. A., and Pressburg, B. S., A.I.Ch.E. J., 7,677 (1961). (23) Jakob, M., “Heat Transfer,” John Wiley and Sons, New York, N. Y., Vol. I, 1949, p 638. (24) Kern, D. Q., “Process Heat Transfer,” 1st ed., McGraw-Hill, New York, N. Y., 1950. (25) Klinkenberg, A., Chem. Eng. Sci., 4, 130 (1955). (26) Klinkenberg, A., ibid., p 167. (27) Klinkenberg, A., Trans. Inst. Chem. Eng., 37, 335 (1959). andTiu, C., Chbm. Eng. Sci., 21, 665 (1966). (28) Kozicki, W., Chou, C. H., (29) Lapple, C. E., in “Chemical Engineers’ Handbook,” 3rd ed., J. H. Perry, Ed., McGraw-Hill, New York, N. Y., 1949, p 1013 ff. (30) Larian, M. G., “Fundamentals of Chemical Engineering Operations,” Prentice-Hall, Englewood Cliff 8, N. J., 1958. (31) Leva, M., “Fluidization,” McGraw-Hill, New York, N. Y., 1959. (32) Lockhart, R. W., and Martinelli, R . C., Chcm. Eng. Progr., 45,44 (1949). (33) McAdams, W. H., “Heat Transfer,” 2nd ed., McGraw-Hill, New York, N. Y., 1942, pp 97 and 256. (34) McAdams, W. H., “Heat Transmission,” 3rd ed., McGraw-Hill, New York, N. Y., i954. (35) McCabe W. L and Smith J. C “Unit 0 erations of Chemical Engineering,’’ 2nd ed., Mcdlraw-Hill, NAw Yo;k, N. Y., lt67. (36). Maker, F. L., “Boelter Anniversary Volume,” H. A. Johnson, Ed., McGrawHill, New York, N. Y., 1964, p 450. (37) Nissan, A. H., and Bresan, V. P., A.I.Ch.E. J., 7,543 (1961). (38) Nissan, A. H., and George, H. H., ibid., p 635. (39) Pankhurst R. C “Dimensional Anal sis and Scale Factors ” Chapman and Hall, Londoi, E n g l a d , and Reinhold Puhshing, New York, N: Y., 1964. (40) Pohl, R. W., “Einfiihrung in die Physik,” 6-7th ed., Vol. 11, “Elektrizitltslehre,” Springer Verlag, Berlin, Germany, 1941. (41) Samuels, M. R., and Churchill, S . W., A.I.Ch.E. J.,13,77 (1967). (42) Sherwood T. K and Pigford R. L., “Absorption and Extraction,” 2nd ed., McGraw-Hili, New’jlork, N. Y., {952. (43) Silberberg, 1. H., and McKetta Jr., J. J., Petrol. Rejnar, May 1953. (44) Susskind, H., andBecker, W., A.I.Ch.E. J., 13,1155 (1967). (45) Thatcher, C. M “Fundamentals of Chemical Engineering,’’ Merrill Books, Columbus, 0 2 0 , 1 9 6 2 . (46) Treybal, R. E., “Mass Transfer Operations,” McGraw-Hill, New York, N. Y., 1955. (47) Van Heerden, C., Nobel, A. P. P., and van Krevelen, D:W.,’Chcm. Eng. Sci., 1, 37 (1951). (48) Walker W. H Lewis W. K McAdams W. H a n d Gilliland E. R., “Principles of ChemicayEnginkering,);’3rd ed., MbGraw&ill, New Yord, N. Y., 1937.
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