~jdpipe/ascend/trunk-old

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\begin_body

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

\begin_layout Plain Layout
Chemical Engineering Handbook
\end_layout

\end_inset

 by Perry
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\begin_layout Plain Layout
Perry
\end_layout

\end_inset

 and Chilton
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\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