trunk/doc/howto-dimeqns.lyx

#LyX 1.4.1 created this file. For more info see http://www.lyx.org/
\lyxformat 245
\begin_document
\begin_header
\textclass book
\language english
\inputencoding auto
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\graphics default
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\spacing single
\papersize a4paper
\use_geometry false
\use_amsmath 2
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\papercolumns 1
\papersides 2
\paperpagestyle default
\tracking_changes false
\output_changes true
\end_header

\begin_body

\begin_layout Chapter
Entering Dimensional Equations
\begin_inset LatexCommand \index{equation, dimensional}

\end_inset

 from Handbooks
\begin_inset LatexCommand \label{cha:dimeqns}

\end_inset


\end_layout

\begin_layout Standard
Often in creating an ASCEND model one needs to enter a correlation
\begin_inset LatexCommand \index{correlation}

\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 LatexCommand \index{pressure, vapor}

\end_inset


\end_layout

\begin_layout Standard
Our first example is the equation to express vapor pressure using an Antoine
\begin_inset LatexCommand \index{Antoine}

\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 LatexCommand \index{SI}

\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 LatexCommand \ref{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 LatexCommand \ref{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 LatexCommand \index{ln}

\end_inset

 () function, as it can take the natural log of the dimensionless
\begin_inset LatexCommand \index{dimensionless}

\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 LatexCommand \ref{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 LatexCommand \index{Fahrenheit}

\end_inset

-- variables with offset
\begin_inset LatexCommand \label{sec:dimeqns.Fahrenheit}

\end_inset


\end_layout

\begin_layout Standard
What if we write Equation 
\begin_inset LatexCommand \ref{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 LatexCommand \ref{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 LatexCommand \ref{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 LatexCommand \label{ssec:dimeqns.pressure drop}

\end_inset


\end_layout

\begin_layout Standard
From the Chemical Engineering Handbook
\begin_inset LatexCommand \index{Chemical Engineering Handbook}

\end_inset

 by Perry
\begin_inset LatexCommand \index{Perry}

\end_inset

 and Chilton
\begin_inset LatexCommand \index{Chilton}

\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 LatexCommand \label{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 LatexCommand \index{offset}

\end_inset

 and difference units
\begin_inset LatexCommand \index{difference units}

\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	#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