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1 johnpye 825 #LyX 1.4.1 created this file. For more info see http://www.lyx.org/
2     \lyxformat 245
3     \begin_document
4     \begin_header
5     \textclass book
6     \language english
7     \inputencoding auto
8     \fontscheme default
9     \graphics default
10     \paperfontsize default
11     \spacing single
12     \papersize a4paper
13     \use_geometry false
14     \use_amsmath 2
15     \cite_engine basic
16     \use_bibtopic false
17     \paperorientation portrait
18     \secnumdepth 3
19     \tocdepth 3
20     \paragraph_separation indent
21     \defskip medskip
22     \quotes_language english
23     \papercolumns 1
24     \papersides 2
25     \paperpagestyle default
26     \tracking_changes false
27     \output_changes true
28     \end_header
29    
30     \begin_body
31    
32     \begin_layout Chapter
33     Entering Dimensional Equations
34     \begin_inset LatexCommand \index{equation, dimensional}
35    
36     \end_inset
37    
38     from Handbooks
39     \begin_inset LatexCommand \label{cha:dimeqns}
40    
41     \end_inset
42    
43    
44     \end_layout
45    
46     \begin_layout Standard
47     Often in creating an ASCEND model one needs to enter a correlation
48     \begin_inset LatexCommand \index{correlation}
49    
50     \end_inset
51    
52     given in a handbook that is written in terms of variables expressed in
53     specific units.
54     In this chapter, we examine how to do this easily and correctly in a system
55     like ASCEND where all equations must be dimensionally correct.
56     \end_layout
57    
58     \begin_layout Section
59     Example 1-- vapor pressure
60     \begin_inset LatexCommand \index{pressure, vapor}
61    
62     \end_inset
63    
64    
65     \end_layout
66    
67     \begin_layout Standard
68     Our first example is the equation to express vapor pressure using an Antoine
69     \begin_inset LatexCommand \index{Antoine}
70    
71     \end_inset
72    
73     -like equation of the form:
74     \end_layout
75    
76     \begin_layout Standard
77     \begin_inset Formula \begin{equation}
78     \ln(P_{sat})=A-\frac{B}{T+C}\label{eqn:dimeqns.lnPsat}\end{equation}
79    
80     \end_inset
81    
82     where
83     \begin_inset Formula $P_{sat}$
84     \end_inset
85    
86     is in {atm} and
87     \begin_inset Formula $T$
88     \end_inset
89    
90     in {R}.
91     When one encounters this equation in a handbook, one then finds tabulated
92     values for
93     \begin_inset Formula $A$
94     \end_inset
95    
96     ,
97     \begin_inset Formula $B$
98     \end_inset
99    
100     and
101     \begin_inset Formula $C$
102     \end_inset
103    
104     for different chemical species.
105     The question we are addressing is:
106     \end_layout
107    
108     \begin_layout Quote
109     How should one enter this equation into ASCEND so one can then enter the
110     constants A, B, and C with the exact values given in the handbook?
111     \end_layout
112    
113     \begin_layout Standard
114     ASCEND uses SI
115     \begin_inset LatexCommand \index{SI}
116    
117     \end_inset
118    
119     units internally.
120     Therefore, P would have the units {kg/m/s^2}, and T would have the units
121     {K}.
122     \end_layout
123    
124     \begin_layout Standard
125     Eqn
126     \begin_inset LatexCommand \ref{eqn:dimeqns.lnPsat}
127    
128     \end_inset
129    
130    
131     \noun off
132     is, in fact, dimensionally incorrect as written.
133     We know how to use this equation, but ASCEND does not as ASCEND requires
134     that we write dimensionally correct equations.
135     For one thing, we can legitimately take the natural log (ln) only of unitless
136     quantities.
137     Also, the handbook will tabulate the values for A, B and C without units.
138     If A is dimensionless, then B and C would require the dimensions of temperature.
139     \end_layout
140    
141     \begin_layout Standard
142     The mindset we describe in this chapter is to enter such equations is to
143     make all quantities that must be expressed in particular units into dimensionle
144     ss quantities that have the correct numerical value.
145     \end_layout
146    
147     \begin_layout Standard
148     We illustrate in the following subsections just how to do this conversion.
149     It proves to be very straight forward to do.
150     \end_layout
151    
152     \begin_layout Subsection
153     Converting the ln term
154     \end_layout
155    
156     \begin_layout Standard
157     Convert the quantity within the ln() term into a dimensionless number that
158     has the value of pressure when pressure is expressed in {atm}.
159     \end_layout
160    
161     \begin_layout Standard
162     Very simply, we write
163     \end_layout
164    
165     \begin_layout LyX-Code
166     P_atm = P/1{atm};
167     \end_layout
168    
169     \begin_layout Standard
170     Note that P_atm has to be a dimensionless quantity here.
171     \end_layout
172    
173     \begin_layout Standard
174     We then rewrite the LHS of Equation
175     \begin_inset LatexCommand \ref{eqn:dimeqns.lnPsat}
176    
177     \end_inset
178    
179    
180     \noun off
181     as
182     \end_layout
183    
184     \begin_layout LyX-Code
185     ln(P_atm)
186     \end_layout
187    
188     \begin_layout Standard
189     Suppose P = 2 {atm}.
190     In SI units P= 202,650 {kg/m/s^2}.
191     In SI units, the dimensional constant 1{atm} is about 101,325 {kg/m/s^2}.
192     Using this definition, P_atm has the value 2 and is dimensionless.
193     ASCEND will not complain with P_atm as the argument of the ln
194     \begin_inset LatexCommand \index{ln}
195    
196     \end_inset
197    
198     () function, as it can take the natural log of the dimensionless
199     \begin_inset LatexCommand \index{dimensionless}
200    
201     \end_inset
202    
203     quantity 2 without any difficulty.
204     \end_layout
205    
206     \begin_layout Subsection
207     Converting the RHS
208     \end_layout
209    
210     \begin_layout Standard
211     We next convert the RHS of Equation
212     \begin_inset LatexCommand \ref{eqn:dimeqns.lnPsat}
213    
214     \end_inset
215    
216    
217     \noun off
218     , and it is equally as simple.
219     Again, convert the temperature used in the RHS into:
220     \end_layout
221    
222     \begin_layout LyX-Code
223     T_R = T/1{R};
224     \end_layout
225    
226     \begin_layout Standard
227     ASCEND converts the dimensional constant 1{R} into 0.55555555...{K}.
228     Thus T_R is dimensionless but has the value that T would have if expressed
229     in {R}.
230     \end_layout
231    
232     \begin_layout Subsection
233     In summary for example 1
234     \end_layout
235    
236     \begin_layout Standard
237     We do not need to introduce the intermediate dimensionless variables.
238     Rather we can write:
239     \end_layout
240    
241     \begin_layout LyX-Code
242     ln(P/1{atm}) = A - B/(T/1{R} + C);
243     \end_layout
244    
245     \begin_layout Standard
246     as a correct form for the dimensional equation.
247     When we do it in this way, we can enter A, B and C as dimensionless quantities
248     with the values exactly as tabulated.
249     \end_layout
250    
251     \begin_layout Section
252     Fahrenheit
253     \begin_inset LatexCommand \index{Fahrenheit}
254    
255     \end_inset
256    
257     -- variables with offset
258     \begin_inset LatexCommand \label{sec:dimeqns.Fahrenheit}
259    
260     \end_inset
261    
262    
263     \end_layout
264    
265     \begin_layout Standard
266     What if we write Equation
267     \begin_inset LatexCommand \ref{eqn:dimeqns.lnPsat}
268    
269     \end_inset
270    
271    
272     \noun off
273     but the handbook says that T is in degrees Fahrenheit, i.e., in {F}? The
274     conversion from {K} to {F} is
275     \end_layout
276    
277     \begin_layout LyX-Code
278     T{F} = T{K}*1.8 - 459.67
279     \end_layout
280    
281     \begin_layout Standard
282     and the 459.67 is an offset.
283     ASCEND does not support an offset for units conversion.
284     We shall discuss the reasons for this apparent limitation in Section
285     \begin_inset LatexCommand \ref{ssec:dimeqns.handlingUnitConv}
286    
287     \end_inset
288    
289     .
290     \end_layout
291    
292     \begin_layout Standard
293     You can readily handle temperatures in {F} if you again think as we did
294     above.
295     The rule, even for units requiring an offset for conversion, remains: convert
296     a dimensional variable into dimensionless one such that the dimensionless
297     one has the proper value.
298     \end_layout
299    
300     \begin_layout Standard
301     Define a new variable
302     \end_layout
303    
304     \begin_layout LyX-Code
305     T_degF = T/1{R} - 459.67;
306     \end_layout
307    
308     \begin_layout Standard
309     Then code
310     \begin_inset LatexCommand \ref{eqn:dimeqns.lnPsat}
311    
312     \end_inset
313    
314    
315     \noun on
316     Equation 7.1
317     \noun off
318     as
319     \end_layout
320    
321     \begin_layout LyX-Code
322     ln(P/1{atm}) = A - B/(T_degF + C);
323     \end_layout
324    
325     \begin_layout Standard
326     when entering it into ASCEND.
327     You will then enter constants A, B, and C as dimensionless quantities having
328     the values exactly as tabulated.
329     In this example we must create the intermediate variable T_degF.
330     \end_layout
331    
332     \begin_layout Section
333     Example 3-- pressure drop
334     \begin_inset LatexCommand \label{ssec:dimeqns.pressure drop}
335    
336     \end_inset
337    
338    
339     \end_layout
340    
341     \begin_layout Standard
342     From the Chemical Engineering Handbook
343     \begin_inset LatexCommand \index{Chemical Engineering Handbook}
344    
345     \end_inset
346    
347     by Perry
348     \begin_inset LatexCommand \index{Perry}
349    
350     \end_inset
351    
352     and Chilton
353     \begin_inset LatexCommand \index{Chilton}
354    
355     \end_inset
356    
357     , Fifth Edition, McGraw-Hill, p10-33, we find the following correlation:
358     \end_layout
359    
360     \begin_layout Standard
361     \begin_inset Formula \[
362     \Delta P_{a}^{\prime}=\frac{y(V_{g}-V_{l})G^{2}}{144g}\]
363    
364     \end_inset
365    
366     where the pressure drop on the LHS is in psi, y is the fraction vapor by
367     weight (i.e., dimensionless), Vg and Vl are the specific volumes of gas and
368     liquid respectively in ft3/lbm, G is the mass velocity in lbm/hr/ft2 and
369     g is the acceleration by gravity and equal to 4.18x108 ft/hr2.
370     \end_layout
371    
372     \begin_layout Standard
373     We proceed by making each term dimensionless and with the right numerical
374     value for the units in which it is to be expressed.
375     The following is the result.
376     We do this by simply dividing each dimensional variable by the correct
377     unit conversion factor.
378     \end_layout
379    
380     \begin_layout LyX-Code
381     delPa/1{psi} = y*(Vg-Vl)/1{ft^3/lbm}*
382     \end_layout
383    
384     \begin_layout LyX-Code
385     (G/1{lbm/hr/ft^2})^2/(144*4.18e8);
386     \end_layout
387    
388     \begin_layout Section
389     The difficulty of handling unit conversions defined with offset
390     \begin_inset LatexCommand \label{ssec:dimeqns.handlingUnitConv}
391    
392     \end_inset
393    
394    
395     \end_layout
396    
397     \begin_layout Standard
398     How do you convert temperature from Kelvin to centigrade? The ASCEND compiler
399     encounters the following ASCEND statement:
400     \end_layout
401    
402     \begin_layout LyX-Code
403     d1T1 = d1T2 + a.Td[4];
404     \end_layout
405    
406     \begin_layout Standard
407     and d1T1 is supposed to be reported in centigrade.
408     We know that ASCEND stores termperatures in Kelvin {K}.
409     We also know that one converts {K} to {C} with the following relationshipT{C}
410     = T{K} - 273.15.
411     \end_layout
412    
413     \begin_layout Standard
414     Now suppose d1T2 has the value 173.15 {K} and a.Td{4} has the value 500 {K}.
415     What is d1T1 in {C}? It would appear to have the value 173.15+500-273.15
416     = 400 {C}.
417     But what if the three variables here are really temperature differences?
418     Then the conversion should be T{dC} = T{dK}, where we use the notation
419     {dC} to be the units for temperature difference in centigrade and {dK}
420     for differences in Kelvin.
421     Then the correct answer is 173.15+500=673.15 {dC}.
422    
423     \end_layout
424    
425     \begin_layout Standard
426     Suppose d1T1 is a temperature and d1T2 is a temperature difference (which
427     would indicate an unfortunate but allowable naming scheme by the creator
428     of this statement).
429     It turns out that a.Td[4] is then required to be a temperature and not a
430     temperature difference for this equation to make sense.
431     We discover that an equation written to have a right-hand-side of zero
432     and that involves the sums and differences of temperature and temperature
433     difference variables will have to have an equal number of positive and
434     negative temperatures in it to make sense, with the remaining having to
435     be temperature differences.
436     Of course if the equation is a correlation, such may not be the case, as
437     the person deriving the correlation is free to create an equation that
438     "fits" the data without requiring the equation to be dimensionally (and
439     physically) reasonable.
440     \end_layout
441    
442     \begin_layout Standard
443     We could create the above discussion just as easily in terms of pressure
444     where we distinguish absolute from gauge pressures (e.g., {psia} vs.
445     {psig}).
446     We would find the need to introduce units {dpisa} and {dpsig} also.
447    
448     \end_layout
449    
450     \begin_layout Subsection
451     General offset
452     \begin_inset LatexCommand \index{offset}
453    
454     \end_inset
455    
456     and difference units
457     \begin_inset LatexCommand \index{difference units}
458    
459     \end_inset
460    
461    
462     \end_layout
463    
464     \begin_layout Standard
465     Unfortunately, we find we have to think much more generally than the above.
466     Any unit conversion can be introduced both with and without offset.
467     Suppose we have an equation which involves the sums and diffences of terms
468     t1 to t4:
469     \end_layout
470    
471     \begin_layout Standard
472     \begin_inset Formula \begin{equation}
473     t_{1}+t_{2}-(t+t_{4})=0\label{eqn:t1+t2}\end{equation}
474    
475     \end_inset
476    
477     where the units for each term is some combination of basic units, e.g., {ft/s^2/R}.
478     Let us call this combination {X} and add it to our set of allowable units,
479     i.e., we define
480     \emph on
481     {X} = {ft/s^2/R}.
482    
483     \emph default
484    
485     \end_layout
486    
487     \begin_layout Standard
488     Suppose we define units {Xoffset} to satisfy: {Xoffset} = {X} - 10 as another
489     set of units for our system.
490     We will also have to introduce the concept of {dX} and and should probably
491     introduce also {dXoffset} to our system, with these two obeying{dXoffset}
492     = {Xoffset}.
493    
494     \end_layout
495    
496     \begin_layout Standard
497     For what we might call a "well-posed" equation, we can argue that the coefficien
498     t of variables in units such as {Xoffset} have to add to zero with the remaining
499     being in units of {dX} and {dXoffset}.
500     Unfortunately, the authors of correlation equations are not forced to follow
501     any such rule, so you can find many published correlations that make the
502     most awful (and often unstated) assumptions about the units of the variables
503     being correlated.
504     \end_layout
505    
506     \begin_layout Standard
507     Will the typical modeler get this right? We suspect not.
508     We would need a very large number of unit conversion combinations in both
509     absolute, offset and relative units to accomodate this approach.
510     \end_layout
511    
512     \begin_layout Standard
513     We suggest that our approach to use only absolute units with no offset is
514     the least confusing for a user.
515     Units conversion is then just multiplication by a factor both for absolute
516     {X} and difference {dX} units-- we do not have to introduce difference
517     variables because we do not introduce offset units.
518    
519     \end_layout
520    
521     \begin_layout Standard
522     When users want offset units such as gauge pressure or Fahrenheit for temperatur
523     e, they can use the conversion to dimensionless variables having the right
524     value, using the style we introduced above, i.e., T_defF = T/1{R} - 459.67
525     and P_psig = P/1{psi} - 14.696 as needed.
526     \end_layout
527    
528     \begin_layout Standard
529     Both approaches to handling offset introduce undesirable and desirable character
530     istics to a modeling system.
531     Neither allow the user to use units without thinking carefully.
532     We voted for this form because of its much lower complexity.
533     \end_layout
534    
535     \end_body
536     \end_document

john.pye@anu.edu.au
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