trunk/doc/howto-dimeqns.lyx

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\begin_body

\begin_layout Chapter
Entering Dimensional Equations
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
equation, dimensional
\end_layout

\end_inset

 from Handbooks
\begin_inset CommandInset label
LatexCommand label
name "cha:dimeqns"

\end_inset


\end_layout

\begin_layout Standard
Often in creating an ASCEND model one needs to enter a correlation
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
correlation
\end_layout

\end_inset

 given in a handbook that is written in terms of variables expressed in
 specific units.
 In this chapter, we examine how to do this easily and correctly in a system
 like ASCEND where all equations must be dimensionally correct.
\end_layout

\begin_layout Section
Example 1-- vapor pressure
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
pressure, vapor
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Our first example is the equation to express vapor pressure using an Antoine
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
Antoine
\end_layout

\end_inset

-like equation of the form:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\ln(P_{sat})=A-\frac{B}{T+C}\label{eqn:dimeqns.lnPsat}
\end{equation}

\end_inset

where 
\begin_inset Formula $P_{sat}$
\end_inset

 is in {atm} and 
\begin_inset Formula $T$
\end_inset

 in {R}.
 When one encounters this equation in a handbook, one then finds tabulated
 values for 
\begin_inset Formula $A$
\end_inset

, 
\begin_inset Formula $B$
\end_inset

 and 
\begin_inset Formula $C$
\end_inset

 for different chemical species.
 The question we are addressing is:
\end_layout

\begin_layout Quote
How should one enter this equation into ASCEND so one can then enter the
 constants A, B, and C with the exact values given in the handbook?
\end_layout

\begin_layout Standard
ASCEND uses SI
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
SI
\end_layout

\end_inset

 units internally.
 Therefore, P would have the units {kg/m/s^2}, and T would have the units
 {K}.
\end_layout

\begin_layout Standard
Eqn 
\begin_inset CommandInset ref
LatexCommand ref
reference "eqn:dimeqns.lnPsat"

\end_inset


\noun off
 is, in fact, dimensionally incorrect as written.
 We know how to use this equation, but ASCEND does not as ASCEND requires
 that we write dimensionally correct equations.
 For one thing, we can legitimately take the natural log (ln) only of unitless
 quantities.
 Also, the handbook will tabulate the values for A, B and C without units.
 If A is dimensionless, then B and C would require the dimensions of temperature.
\end_layout

\begin_layout Standard
The mindset we describe in this chapter is to enter such equations is to
 make all quantities that must be expressed in particular units into dimensionle
ss quantities that have the correct numerical value.
\end_layout

\begin_layout Standard
We illustrate in the following subsections just how to do this conversion.
 It proves to be very straight forward to do.
\end_layout

\begin_layout Subsection
Converting the ln term
\end_layout

\begin_layout Standard
Convert the quantity within the ln() term into a dimensionless number that
 has the value of pressure when pressure is expressed in {atm}.
\end_layout

\begin_layout Standard
Very simply, we write
\end_layout

\begin_layout LyX-Code
P_atm = P/1{atm};
\end_layout

\begin_layout Standard
Note that P_atm has to be a dimensionless quantity here.
\end_layout

\begin_layout Standard
We then rewrite the LHS of Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eqn:dimeqns.lnPsat"

\end_inset


\noun off
 as
\end_layout

\begin_layout LyX-Code
ln(P_atm)
\end_layout

\begin_layout Standard
Suppose P = 2 {atm}.
 In SI units P= 202,650 {kg/m/s^2}.
 In SI units, the dimensional constant 1{atm} is about 101,325 {kg/m/s^2}.
 Using this definition, P_atm has the value 2 and is dimensionless.
 ASCEND will not complain with P_atm as the argument of the ln
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
ln
\end_layout

\end_inset

 () function, as it can take the natural log of the dimensionless
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
dimensionless
\end_layout

\end_inset

 quantity 2 without any difficulty.
\end_layout

\begin_layout Subsection
Converting the RHS
\end_layout

\begin_layout Standard
We next convert the RHS of Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eqn:dimeqns.lnPsat"

\end_inset


\noun off
, and it is equally as simple.
 Again, convert the temperature used in the RHS into:
\end_layout

\begin_layout LyX-Code
T_R = T/1{R};
\end_layout

\begin_layout Standard
ASCEND converts the dimensional constant 1{R} into 0.55555555...{K}.
 Thus T_R is dimensionless but has the value that T would have if expressed
 in {R}.
\end_layout

\begin_layout Subsection
In summary for example 1
\end_layout

\begin_layout Standard
We do not need to introduce the intermediate dimensionless variables.
 Rather we can write:
\end_layout

\begin_layout LyX-Code
ln(P/1{atm}) = A - B/(T/1{R} + C);
\end_layout

\begin_layout Standard
as a correct form for the dimensional equation.
 When we do it in this way, we can enter A, B and C as dimensionless quantities
 with the values exactly as tabulated.
\end_layout

\begin_layout Section
Fahrenheit
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
Fahrenheit
\end_layout

\end_inset

-- variables with offset
\begin_inset CommandInset label
LatexCommand label
name "sec:dimeqns.Fahrenheit"

\end_inset


\end_layout

\begin_layout Standard
What if we write Equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eqn:dimeqns.lnPsat"

\end_inset


\noun off
 but the handbook says that T is in degrees Fahrenheit, i.e., in {F}? The
 conversion from {K} to {F} is
\end_layout

\begin_layout LyX-Code
T{F} = T{K}*1.8 - 459.67
\end_layout

\begin_layout Standard
and the 459.67 is an offset.
 ASCEND does not support an offset for units conversion.
 We shall discuss the reasons for this apparent limitation in Section 
\begin_inset CommandInset ref
LatexCommand ref
reference "ssec:dimeqns.handlingUnitConv"

\end_inset

.
\end_layout

\begin_layout Standard
You can readily handle temperatures in {F} if you again think as we did
 above.
 The rule, even for units requiring an offset for conversion, remains: convert
 a dimensional variable into dimensionless one such that the dimensionless
 one has the proper value.
\end_layout

\begin_layout Standard
Define a new variable
\end_layout

\begin_layout LyX-Code
T_degF = T/1{R} - 459.67;
\end_layout

\begin_layout Standard
Then code 
\begin_inset CommandInset ref
LatexCommand ref
reference "eqn:dimeqns.lnPsat"

\end_inset


\noun on
Equation 7.1
\noun off
 as
\end_layout

\begin_layout LyX-Code
ln(P/1{atm}) = A - B/(T_degF + C);
\end_layout

\begin_layout Standard
when entering it into ASCEND.
 You will then enter constants A, B, and C as dimensionless quantities having
 the values exactly as tabulated.
 In this example we must create the intermediate variable T_degF.
\end_layout

\begin_layout Section
Example 3-- pressure drop
\begin_inset CommandInset label
LatexCommand label
name "ssec:dimeqns.pressure drop"

\end_inset


\end_layout

\begin_layout Standard
From the Chemical Engineering Handbook
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
Chemical Engineering Handbook
\end_layout

\end_inset

 by Perry
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
Perry
\end_layout

\end_inset

 and Chilton
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
Chilton
\end_layout

\end_inset

, Fifth Edition, McGraw-Hill, p10-33, we find the following correlation:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\Delta P_{a}^{\prime}=\frac{y(V_{g}-V_{l})G^{2}}{144g}
\]

\end_inset

where the pressure drop on the LHS is in psi, y is the fraction vapor by
 weight (i.e., dimensionless), Vg and Vl are the specific volumes of gas and
 liquid respectively in ft3/lbm, G is the mass velocity in lbm/hr/ft2 and
 g is the acceleration by gravity and equal to 4.18x108 ft/hr2.
\end_layout

\begin_layout Standard
We proceed by making each term dimensionless and with the right numerical
 value for the units in which it is to be expressed.
 The following is the result.
 We do this by simply dividing each dimensional variable by the correct
 unit conversion factor.
\end_layout

\begin_layout LyX-Code
delPa/1{psi} = y*(Vg-Vl)/1{ft^3/lbm}*
\end_layout

\begin_layout LyX-Code
              (G/1{lbm/hr/ft^2})^2/(144*4.18e8);
\end_layout

\begin_layout Section
The difficulty of handling unit conversions defined with offset
\begin_inset CommandInset label
LatexCommand label
name "ssec:dimeqns.handlingUnitConv"

\end_inset


\end_layout

\begin_layout Standard
How do you convert temperature from Kelvin to centigrade? The ASCEND compiler
 encounters the following ASCEND statement:
\end_layout

\begin_layout LyX-Code
d1T1 = d1T2 + a.Td[4];
\end_layout

\begin_layout Standard
and d1T1 is supposed to be reported in centigrade.
 We know that ASCEND stores termperatures in Kelvin {K}.
 We also know that one converts {K} to {C} with the following relationshipT{C}
 = T{K} - 273.15.
\end_layout

\begin_layout Standard
Now suppose d1T2 has the value 173.15 {K} and a.Td{4} has the value 500 {K}.
 What is d1T1 in {C}? It would appear to have the value 173.15+500-273.15
 = 400 {C}.
 But what if the three variables here are really temperature differences?
 Then the conversion should be T{dC} = T{dK}, where we use the notation
 {dC} to be the units for temperature difference in centigrade and {dK}
 for differences in Kelvin.
 Then the correct answer is 173.15+500=673.15 {dC}.
 
\end_layout

\begin_layout Standard
Suppose d1T1 is a temperature and d1T2 is a temperature difference (which
 would indicate an unfortunate but allowable naming scheme by the creator
 of this statement).
 It turns out that a.Td[4] is then required to be a temperature and not a
 temperature difference for this equation to make sense.
 We discover that an equation written to have a right-hand-side of zero
 and that involves the sums and differences of temperature and temperature
 difference variables will have to have an equal number of positive and
 negative temperatures in it to make sense, with the remaining having to
 be temperature differences.
 Of course if the equation is a correlation, such may not be the case, as
 the person deriving the correlation is free to create an equation that
 "fits" the data without requiring the equation to be dimensionally (and
 physically) reasonable.
\end_layout

\begin_layout Standard
We could create the above discussion just as easily in terms of pressure
 where we distinguish absolute from gauge pressures (e.g., {psia} vs.
 {psig}).
 We would find the need to introduce units {dpisa} and {dpsig} also.
 
\end_layout

\begin_layout Subsection
General offset
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
offset
\end_layout

\end_inset

 and difference units
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
difference units
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Unfortunately, we find we have to think much more generally than the above.
 Any unit conversion can be introduced both with and without offset.
 Suppose we have an equation which involves the sums and diffences of terms
 t1 to t4:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
t_{1}+t_{2}-(t+t_{4})=0\label{eqn:t1+t2}
\end{equation}

\end_inset

where the units for each term is some combination of basic units, e.g., {ft/s^2/R}.
 Let us call this combination {X} and add it to our set of allowable units,
 i.e., we define
\emph on
{X} = {ft/s^2/R}.

\emph default
 
\end_layout

\begin_layout Standard
Suppose we define units {Xoffset} to satisfy: {Xoffset} = {X} - 10 as another
 set of units for our system.
 We will also have to introduce the concept of {dX} and and should probably
 introduce also {dXoffset} to our system, with these two obeying{dXoffset}
 = {Xoffset}.
 
\end_layout

\begin_layout Standard
For what we might call a "well-posed" equation, we can argue that the coefficien
t of variables in units such as {Xoffset} have to add to zero with the remaining
 being in units of {dX} and {dXoffset}.
 Unfortunately, the authors of correlation equations are not forced to follow
 any such rule, so you can find many published correlations that make the
 most awful (and often unstated) assumptions about the units of the variables
 being correlated.
\end_layout

\begin_layout Standard
Will the typical modeler get this right? We suspect not.
 We would need a very large number of unit conversion combinations in both
 absolute, offset and relative units to accomodate this approach.
\end_layout

\begin_layout Standard
We suggest that our approach to use only absolute units with no offset is
 the least confusing for a user.
 Units conversion is then just multiplication by a factor both for absolute
 {X} and difference {dX} units-- we do not have to introduce difference
 variables because we do not introduce offset units.
 
\end_layout

\begin_layout Standard
When users want offset units such as gauge pressure or Fahrenheit for temperatur
e, they can use the conversion to dimensionless variables having the right
 value, using the style we introduced above, i.e., T_defF = T/1{R} - 459.67
 and P_psig = P/1{psi} - 14.696 as needed.
\end_layout

\begin_layout Standard
Both approaches to handling offset introduce undesirable and desirable character
istics to a modeling system.
 Neither allow the user to use units without thinking carefully.
 We voted for this form because of its much lower complexity.
\end_layout

\end_body
\end_document
1	jpye	2785	#LyX 2.1 created this file. For more info see http://www.lyx.org/
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68	johnpye	825	\end_header
69
70			\begin_body
71
72			\begin_layout Chapter
73			Entering Dimensional Equations
74	jpye	2785	\begin_inset Index idx
75			status collapsed
76	johnpye	825
77	jpye	2785	\begin_layout Plain Layout
78			equation, dimensional
79			\end_layout
80
81	johnpye	825	\end_inset
82
83			from Handbooks
84	jpye	2785	\begin_inset CommandInset label
85			LatexCommand label
86			name "cha:dimeqns"
87	johnpye	825
88			\end_inset
89
90
91			\end_layout
92
93			\begin_layout Standard
94			Often in creating an ASCEND model one needs to enter a correlation
95	jpye	2785	\begin_inset Index idx
96			status collapsed
97	johnpye	825
98	jpye	2785	\begin_layout Plain Layout
99			correlation
100			\end_layout
101
102	johnpye	825	\end_inset
103
104			given in a handbook that is written in terms of variables expressed in
105			specific units.
106			In this chapter, we examine how to do this easily and correctly in a system
107			like ASCEND where all equations must be dimensionally correct.
108			\end_layout
109
110			\begin_layout Section
111			Example 1-- vapor pressure
112	jpye	2785	\begin_inset Index idx
113			status collapsed
114	johnpye	825
115	jpye	2785	\begin_layout Plain Layout
116			pressure, vapor
117			\end_layout
118
119	johnpye	825	\end_inset
120
121
122			\end_layout
123
124			\begin_layout Standard
125			Our first example is the equation to express vapor pressure using an Antoine
126	jpye	2785	\begin_inset Index idx
127			status collapsed
128	johnpye	825
129	jpye	2785	\begin_layout Plain Layout
130			Antoine
131			\end_layout
132
133	johnpye	825	\end_inset
134
135			-like equation of the form:
136			\end_layout
137
138			\begin_layout Standard
139	jpye	2785	\begin_inset Formula
140			\begin{equation}
141			\ln(P_{sat})=A-\frac{B}{T+C}\label{eqn:dimeqns.lnPsat}
142			\end{equation}
143	johnpye	825
144			\end_inset
145
146			where
147			\begin_inset Formula $P_{sat}$
148			\end_inset
149
150			is in {atm} and
151			\begin_inset Formula $T$
152			\end_inset
153
154			in {R}.
155			When one encounters this equation in a handbook, one then finds tabulated
156			values for
157			\begin_inset Formula $A$
158			\end_inset
159
160			,
161			\begin_inset Formula $B$
162			\end_inset
163
164			and
165			\begin_inset Formula $C$
166			\end_inset
167
168			for different chemical species.
169			The question we are addressing is:
170			\end_layout
171
172			\begin_layout Quote
173			How should one enter this equation into ASCEND so one can then enter the
174			constants A, B, and C with the exact values given in the handbook?
175			\end_layout
176
177			\begin_layout Standard
178			ASCEND uses SI
179	jpye	2785	\begin_inset Index idx
180			status collapsed
181	johnpye	825
182	jpye	2785	\begin_layout Plain Layout
183			SI
184			\end_layout
185
186	johnpye	825	\end_inset
187
188			units internally.
189			Therefore, P would have the units {kg/m/s^2}, and T would have the units
190			{K}.
191			\end_layout
192
193			\begin_layout Standard
194			Eqn
195	jpye	2785	\begin_inset CommandInset ref
196			LatexCommand ref
197			reference "eqn:dimeqns.lnPsat"
198	johnpye	825
199			\end_inset
200
201
202			\noun off
203			is, in fact, dimensionally incorrect as written.
204			We know how to use this equation, but ASCEND does not as ASCEND requires
205			that we write dimensionally correct equations.
206			For one thing, we can legitimately take the natural log (ln) only of unitless
207			quantities.
208			Also, the handbook will tabulate the values for A, B and C without units.
209			If A is dimensionless, then B and C would require the dimensions of temperature.
210			\end_layout
211
212			\begin_layout Standard
213			The mindset we describe in this chapter is to enter such equations is to
214			make all quantities that must be expressed in particular units into dimensionle
215			ss quantities that have the correct numerical value.
216			\end_layout
217
218			\begin_layout Standard
219			We illustrate in the following subsections just how to do this conversion.
220			It proves to be very straight forward to do.
221			\end_layout
222
223			\begin_layout Subsection
224			Converting the ln term
225			\end_layout
226
227			\begin_layout Standard
228			Convert the quantity within the ln() term into a dimensionless number that
229			has the value of pressure when pressure is expressed in {atm}.
230			\end_layout
231
232			\begin_layout Standard
233			Very simply, we write
234			\end_layout
235
236			\begin_layout LyX-Code
237			P_atm = P/1{atm};
238			\end_layout
239
240			\begin_layout Standard
241			Note that P_atm has to be a dimensionless quantity here.
242			\end_layout
243
244			\begin_layout Standard
245			We then rewrite the LHS of Equation
246	jpye	2785	\begin_inset CommandInset ref
247			LatexCommand ref
248			reference "eqn:dimeqns.lnPsat"
249	johnpye	825
250			\end_inset
251
252
253			\noun off
254			as
255			\end_layout
256
257			\begin_layout LyX-Code
258			ln(P_atm)
259			\end_layout
260
261			\begin_layout Standard
262			Suppose P = 2 {atm}.
263			In SI units P= 202,650 {kg/m/s^2}.
264			In SI units, the dimensional constant 1{atm} is about 101,325 {kg/m/s^2}.
265			Using this definition, P_atm has the value 2 and is dimensionless.
266			ASCEND will not complain with P_atm as the argument of the ln
267	jpye	2785	\begin_inset Index idx
268			status collapsed
269	johnpye	825
270	jpye	2785	\begin_layout Plain Layout
271			ln
272			\end_layout
273
274	johnpye	825	\end_inset
275
276			() function, as it can take the natural log of the dimensionless
277	jpye	2785	\begin_inset Index idx
278			status collapsed
279	johnpye	825
280	jpye	2785	\begin_layout Plain Layout
281			dimensionless
282			\end_layout
283
284	johnpye	825	\end_inset
285
286			quantity 2 without any difficulty.
287			\end_layout
288
289			\begin_layout Subsection
290			Converting the RHS
291			\end_layout
292
293			\begin_layout Standard
294			We next convert the RHS of Equation
295	jpye	2785	\begin_inset CommandInset ref
296			LatexCommand ref
297			reference "eqn:dimeqns.lnPsat"
298	johnpye	825
299			\end_inset
300
301
302			\noun off
303			, and it is equally as simple.
304			Again, convert the temperature used in the RHS into:
305			\end_layout
306
307			\begin_layout LyX-Code
308			T_R = T/1{R};
309			\end_layout
310
311			\begin_layout Standard
312			ASCEND converts the dimensional constant 1{R} into 0.55555555...{K}.
313			Thus T_R is dimensionless but has the value that T would have if expressed
314			in {R}.
315			\end_layout
316
317			\begin_layout Subsection
318			In summary for example 1
319			\end_layout
320
321			\begin_layout Standard
322			We do not need to introduce the intermediate dimensionless variables.
323			Rather we can write:
324			\end_layout
325
326			\begin_layout LyX-Code
327			ln(P/1{atm}) = A - B/(T/1{R} + C);
328			\end_layout
329
330			\begin_layout Standard
331			as a correct form for the dimensional equation.
332			When we do it in this way, we can enter A, B and C as dimensionless quantities
333			with the values exactly as tabulated.
334			\end_layout
335
336			\begin_layout Section
337			Fahrenheit
338	jpye	2785	\begin_inset Index idx
339			status collapsed
340	johnpye	825
341	jpye	2785	\begin_layout Plain Layout
342			Fahrenheit
343			\end_layout
344
345	johnpye	825	\end_inset
346
347			-- variables with offset
348	jpye	2785	\begin_inset CommandInset label
349			LatexCommand label
350			name "sec:dimeqns.Fahrenheit"
351	johnpye	825
352			\end_inset
353
354
355			\end_layout
356
357			\begin_layout Standard
358			What if we write Equation
359	jpye	2785	\begin_inset CommandInset ref
360			LatexCommand ref
361			reference "eqn:dimeqns.lnPsat"
362	johnpye	825
363			\end_inset
364
365
366			\noun off
367			but the handbook says that T is in degrees Fahrenheit, i.e., in {F}? The
368			conversion from {K} to {F} is
369			\end_layout
370
371			\begin_layout LyX-Code
372			T{F} = T{K}*1.8 - 459.67
373			\end_layout
374
375			\begin_layout Standard
376			and the 459.67 is an offset.
377			ASCEND does not support an offset for units conversion.
378			We shall discuss the reasons for this apparent limitation in Section
379	jpye	2785	\begin_inset CommandInset ref
380			LatexCommand ref
381			reference "ssec:dimeqns.handlingUnitConv"
382	johnpye	825
383			\end_inset
384
385			.
386			\end_layout
387
388			\begin_layout Standard
389			You can readily handle temperatures in {F} if you again think as we did
390			above.
391			The rule, even for units requiring an offset for conversion, remains: convert
392			a dimensional variable into dimensionless one such that the dimensionless
393			one has the proper value.
394			\end_layout
395
396			\begin_layout Standard
397			Define a new variable
398			\end_layout
399
400			\begin_layout LyX-Code
401			T_degF = T/1{R} - 459.67;
402			\end_layout
403
404			\begin_layout Standard
405			Then code
406	jpye	2785	\begin_inset CommandInset ref
407			LatexCommand ref
408			reference "eqn:dimeqns.lnPsat"
409	johnpye	825
410			\end_inset
411
412
413			\noun on
414			Equation 7.1
415			\noun off
416			as
417			\end_layout
418
419			\begin_layout LyX-Code
420			ln(P/1{atm}) = A - B/(T_degF + C);
421			\end_layout
422
423			\begin_layout Standard
424			when entering it into ASCEND.
425			You will then enter constants A, B, and C as dimensionless quantities having
426			the values exactly as tabulated.
427			In this example we must create the intermediate variable T_degF.
428			\end_layout
429
430			\begin_layout Section
431			Example 3-- pressure drop
432	jpye	2785	\begin_inset CommandInset label
433			LatexCommand label
434			name "ssec:dimeqns.pressure drop"
435	johnpye	825
436			\end_inset
437
438
439			\end_layout
440
441			\begin_layout Standard
442			From the Chemical Engineering Handbook
443	jpye	2785	\begin_inset Index idx
444			status collapsed
445	johnpye	825
446	jpye	2785	\begin_layout Plain Layout
447			Chemical Engineering Handbook
448			\end_layout
449
450	johnpye	825	\end_inset
451
452			by Perry
453	jpye	2785	\begin_inset Index idx
454			status collapsed
455	johnpye	825
456	jpye	2785	\begin_layout Plain Layout
457			Perry
458			\end_layout
459
460	johnpye	825	\end_inset
461
462			and Chilton
463	jpye	2785	\begin_inset Index idx
464			status collapsed
465	johnpye	825
466	jpye	2785	\begin_layout Plain Layout
467			Chilton
468			\end_layout
469
470	johnpye	825	\end_inset
471
472			, Fifth Edition, McGraw-Hill, p10-33, we find the following correlation:
473			\end_layout
474
475			\begin_layout Standard
476	jpye	2785	\begin_inset Formula
477			\[
478			\Delta P_{a}^{\prime}=\frac{y(V_{g}-V_{l})G^{2}}{144g}
479			\]
480	johnpye	825
481			\end_inset
482
483			where the pressure drop on the LHS is in psi, y is the fraction vapor by
484			weight (i.e., dimensionless), Vg and Vl are the specific volumes of gas and
485			liquid respectively in ft3/lbm, G is the mass velocity in lbm/hr/ft2 and
486			g is the acceleration by gravity and equal to 4.18x108 ft/hr2.
487			\end_layout
488
489			\begin_layout Standard
490			We proceed by making each term dimensionless and with the right numerical
491			value for the units in which it is to be expressed.
492			The following is the result.
493			We do this by simply dividing each dimensional variable by the correct
494			unit conversion factor.
495			\end_layout
496
497			\begin_layout LyX-Code
498			delPa/1{psi} = y(Vg-Vl)/1{ft^3/lbm}
499			\end_layout
500
501			\begin_layout LyX-Code
502			(G/1{lbm/hr/ft^2})^2/(144*4.18e8);
503			\end_layout
504
505			\begin_layout Section
506			The difficulty of handling unit conversions defined with offset
507	jpye	2785	\begin_inset CommandInset label
508			LatexCommand label
509			name "ssec:dimeqns.handlingUnitConv"
510	johnpye	825
511			\end_inset
512
513
514			\end_layout
515
516			\begin_layout Standard
517			How do you convert temperature from Kelvin to centigrade? The ASCEND compiler
518			encounters the following ASCEND statement:
519			\end_layout
520
521			\begin_layout LyX-Code
522			d1T1 = d1T2 + a.Td[4];
523			\end_layout
524
525			\begin_layout Standard
526			and d1T1 is supposed to be reported in centigrade.
527			We know that ASCEND stores termperatures in Kelvin {K}.
528			We also know that one converts {K} to {C} with the following relationshipT{C}
529			= T{K} - 273.15.
530			\end_layout
531
532			\begin_layout Standard
533			Now suppose d1T2 has the value 173.15 {K} and a.Td{4} has the value 500 {K}.
534			What is d1T1 in {C}? It would appear to have the value 173.15+500-273.15
535			= 400 {C}.
536			But what if the three variables here are really temperature differences?
537			Then the conversion should be T{dC} = T{dK}, where we use the notation
538			{dC} to be the units for temperature difference in centigrade and {dK}
539			for differences in Kelvin.
540			Then the correct answer is 173.15+500=673.15 {dC}.
541
542			\end_layout
543
544			\begin_layout Standard
545			Suppose d1T1 is a temperature and d1T2 is a temperature difference (which
546			would indicate an unfortunate but allowable naming scheme by the creator
547			of this statement).
548			It turns out that a.Td[4] is then required to be a temperature and not a
549			temperature difference for this equation to make sense.
550			We discover that an equation written to have a right-hand-side of zero
551			and that involves the sums and differences of temperature and temperature
552			difference variables will have to have an equal number of positive and
553			negative temperatures in it to make sense, with the remaining having to
554			be temperature differences.
555			Of course if the equation is a correlation, such may not be the case, as
556			the person deriving the correlation is free to create an equation that
557			"fits" the data without requiring the equation to be dimensionally (and
558			physically) reasonable.
559			\end_layout
560
561			\begin_layout Standard
562			We could create the above discussion just as easily in terms of pressure
563			where we distinguish absolute from gauge pressures (e.g., {psia} vs.
564			{psig}).
565			We would find the need to introduce units {dpisa} and {dpsig} also.
566
567			\end_layout
568
569			\begin_layout Subsection
570			General offset
571	jpye	2785	\begin_inset Index idx
572			status collapsed
573	johnpye	825
574	jpye	2785	\begin_layout Plain Layout
575			offset
576			\end_layout
577
578	johnpye	825	\end_inset
579
580			and difference units
581	jpye	2785	\begin_inset Index idx
582			status collapsed
583	johnpye	825
584	jpye	2785	\begin_layout Plain Layout
585			difference units
586			\end_layout
587
588	johnpye	825	\end_inset
589
590
591			\end_layout
592
593			\begin_layout Standard
594			Unfortunately, we find we have to think much more generally than the above.
595			Any unit conversion can be introduced both with and without offset.
596			Suppose we have an equation which involves the sums and diffences of terms
597			t1 to t4:
598			\end_layout
599
600			\begin_layout Standard
601	jpye	2785	\begin_inset Formula
602			\begin{equation}
603			t_{1}+t_{2}-(t+t_{4})=0\label{eqn:t1+t2}
604			\end{equation}
605	johnpye	825
606			\end_inset
607
608			where the units for each term is some combination of basic units, e.g., {ft/s^2/R}.
609			Let us call this combination {X} and add it to our set of allowable units,
610			i.e., we define
611			\emph on
612			{X} = {ft/s^2/R}.
613
614			\emph default
615
616			\end_layout
617
618			\begin_layout Standard
619			Suppose we define units {Xoffset} to satisfy: {Xoffset} = {X} - 10 as another
620			set of units for our system.
621			We will also have to introduce the concept of {dX} and and should probably
622			introduce also {dXoffset} to our system, with these two obeying{dXoffset}
623			= {Xoffset}.
624
625			\end_layout
626
627			\begin_layout Standard
628			For what we might call a "well-posed" equation, we can argue that the coefficien
629			t of variables in units such as {Xoffset} have to add to zero with the remaining
630			being in units of {dX} and {dXoffset}.
631			Unfortunately, the authors of correlation equations are not forced to follow
632			any such rule, so you can find many published correlations that make the
633			most awful (and often unstated) assumptions about the units of the variables
634			being correlated.
635			\end_layout
636
637			\begin_layout Standard
638			Will the typical modeler get this right? We suspect not.
639			We would need a very large number of unit conversion combinations in both
640			absolute, offset and relative units to accomodate this approach.
641			\end_layout
642
643			\begin_layout Standard
644			We suggest that our approach to use only absolute units with no offset is
645			the least confusing for a user.
646			Units conversion is then just multiplication by a factor both for absolute
647			{X} and difference {dX} units-- we do not have to introduce difference
648			variables because we do not introduce offset units.
649
650			\end_layout
651
652			\begin_layout Standard
653			When users want offset units such as gauge pressure or Fahrenheit for temperatur
654			e, they can use the conversion to dimensionless variables having the right
655			value, using the style we introduced above, i.e., T_defF = T/1{R} - 459.67
656			and P_psig = P/1{psi} - 14.696 as needed.
657			\end_layout
658
659			\begin_layout Standard
660			Both approaches to handling offset introduce undesirable and desirable character
661			istics to a modeling system.
662			Neither allow the user to use units without thinking carefully.
663			We voted for this form because of its much lower complexity.
664			\end_layout
665
666			\end_body
667			\end_document