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\shortcut idx |
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\end_index |
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\end_header |
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|
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\begin_body |
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|
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\begin_layout Chapter |
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Entering Dimensional Equations |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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equation, dimensional |
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\end_layout |
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|
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\end_inset |
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|
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from Handbooks |
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\begin_inset CommandInset label |
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LatexCommand label |
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name "cha:dimeqns" |
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|
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\end_inset |
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|
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|
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\end_layout |
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|
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\begin_layout Standard |
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Often in creating an ASCEND model one needs to enter a correlation |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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correlation |
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\end_layout |
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|
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\end_inset |
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|
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given in a handbook that is written in terms of variables expressed in |
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specific units. |
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In this chapter, we examine how to do this easily and correctly in a system |
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like ASCEND where all equations must be dimensionally correct. |
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\end_layout |
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|
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\begin_layout Section |
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Example 1-- vapor pressure |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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pressure, vapor |
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\end_layout |
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|
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\end_inset |
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|
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|
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\end_layout |
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|
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\begin_layout Standard |
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Our first example is the equation to express vapor pressure using an Antoine |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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Antoine |
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\end_layout |
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|
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\end_inset |
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|
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-like equation of the form: |
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\end_layout |
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|
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\begin_layout Standard |
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\begin_inset Formula |
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\begin{equation} |
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\ln(P_{sat})=A-\frac{B}{T+C}\label{eqn:dimeqns.lnPsat} |
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\end{equation} |
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|
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\end_inset |
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|
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where |
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\begin_inset Formula $P_{sat}$ |
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\end_inset |
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|
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is in {atm} and |
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\begin_inset Formula $T$ |
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\end_inset |
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|
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in {R}. |
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When one encounters this equation in a handbook, one then finds tabulated |
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values for |
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\begin_inset Formula $A$ |
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\end_inset |
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|
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, |
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\begin_inset Formula $B$ |
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\end_inset |
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|
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and |
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\begin_inset Formula $C$ |
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\end_inset |
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|
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for different chemical species. |
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The question we are addressing is: |
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\end_layout |
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|
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\begin_layout Quote |
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How should one enter this equation into ASCEND so one can then enter the |
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constants A, B, and C with the exact values given in the handbook? |
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\end_layout |
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|
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\begin_layout Standard |
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ASCEND uses SI |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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SI |
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\end_layout |
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|
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\end_inset |
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|
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units internally. |
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Therefore, P would have the units {kg/m/s^2}, and T would have the units |
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{K}. |
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\end_layout |
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|
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\begin_layout Standard |
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Eqn |
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\begin_inset CommandInset ref |
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LatexCommand ref |
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reference "eqn:dimeqns.lnPsat" |
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|
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\end_inset |
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|
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|
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\noun off |
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is, in fact, dimensionally incorrect as written. |
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We know how to use this equation, but ASCEND does not as ASCEND requires |
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that we write dimensionally correct equations. |
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For one thing, we can legitimately take the natural log (ln) only of unitless |
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quantities. |
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Also, the handbook will tabulate the values for A, B and C without units. |
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If A is dimensionless, then B and C would require the dimensions of temperature. |
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\end_layout |
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|
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\begin_layout Standard |
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The mindset we describe in this chapter is to enter such equations is to |
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make all quantities that must be expressed in particular units into dimensionle |
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ss quantities that have the correct numerical value. |
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\end_layout |
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|
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\begin_layout Standard |
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We illustrate in the following subsections just how to do this conversion. |
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It proves to be very straight forward to do. |
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\end_layout |
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|
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\begin_layout Subsection |
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Converting the ln term |
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\end_layout |
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|
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\begin_layout Standard |
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Convert the quantity within the ln() term into a dimensionless number that |
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has the value of pressure when pressure is expressed in {atm}. |
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\end_layout |
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|
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\begin_layout Standard |
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Very simply, we write |
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\end_layout |
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|
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\begin_layout LyX-Code |
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P_atm = P/1{atm}; |
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\end_layout |
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|
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\begin_layout Standard |
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Note that P_atm has to be a dimensionless quantity here. |
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\end_layout |
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|
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\begin_layout Standard |
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We then rewrite the LHS of Equation |
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\begin_inset CommandInset ref |
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LatexCommand ref |
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reference "eqn:dimeqns.lnPsat" |
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|
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\end_inset |
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|
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|
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\noun off |
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as |
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\end_layout |
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|
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\begin_layout LyX-Code |
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ln(P_atm) |
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\end_layout |
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|
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\begin_layout Standard |
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Suppose P = 2 {atm}. |
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In SI units P= 202,650 {kg/m/s^2}. |
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In SI units, the dimensional constant 1{atm} is about 101,325 {kg/m/s^2}. |
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Using this definition, P_atm has the value 2 and is dimensionless. |
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ASCEND will not complain with P_atm as the argument of the ln |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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ln |
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\end_layout |
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|
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\end_inset |
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|
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() function, as it can take the natural log of the dimensionless |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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dimensionless |
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\end_layout |
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|
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\end_inset |
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|
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quantity 2 without any difficulty. |
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\end_layout |
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|
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\begin_layout Subsection |
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Converting the RHS |
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\end_layout |
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|
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\begin_layout Standard |
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We next convert the RHS of Equation |
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\begin_inset CommandInset ref |
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LatexCommand ref |
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reference "eqn:dimeqns.lnPsat" |
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|
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\end_inset |
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|
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|
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\noun off |
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, and it is equally as simple. |
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Again, convert the temperature used in the RHS into: |
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\end_layout |
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|
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\begin_layout LyX-Code |
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T_R = T/1{R}; |
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\end_layout |
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|
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\begin_layout Standard |
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ASCEND converts the dimensional constant 1{R} into 0.55555555...{K}. |
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Thus T_R is dimensionless but has the value that T would have if expressed |
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in {R}. |
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\end_layout |
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|
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\begin_layout Subsection |
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In summary for example 1 |
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\end_layout |
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|
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\begin_layout Standard |
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We do not need to introduce the intermediate dimensionless variables. |
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Rather we can write: |
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\end_layout |
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|
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\begin_layout LyX-Code |
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ln(P/1{atm}) = A - B/(T/1{R} + C); |
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\end_layout |
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|
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\begin_layout Standard |
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as a correct form for the dimensional equation. |
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When we do it in this way, we can enter A, B and C as dimensionless quantities |
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with the values exactly as tabulated. |
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\end_layout |
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|
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\begin_layout Section |
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Fahrenheit |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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Fahrenheit |
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\end_layout |
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|
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\end_inset |
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|
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-- variables with offset |
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\begin_inset CommandInset label |
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LatexCommand label |
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name "sec:dimeqns.Fahrenheit" |
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|
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\end_inset |
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|
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|
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\end_layout |
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|
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\begin_layout Standard |
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What if we write Equation |
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\begin_inset CommandInset ref |
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LatexCommand ref |
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reference "eqn:dimeqns.lnPsat" |
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|
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\end_inset |
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|
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|
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\noun off |
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but the handbook says that T is in degrees Fahrenheit, i.e., in {F}? The |
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conversion from {K} to {F} is |
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\end_layout |
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|
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\begin_layout LyX-Code |
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T{F} = T{K}*1.8 - 459.67 |
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\end_layout |
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|
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\begin_layout Standard |
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and the 459.67 is an offset. |
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ASCEND does not support an offset for units conversion. |
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We shall discuss the reasons for this apparent limitation in Section |
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\begin_inset CommandInset ref |
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LatexCommand ref |
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reference "ssec:dimeqns.handlingUnitConv" |
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|
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\end_inset |
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|
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. |
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\end_layout |
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|
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\begin_layout Standard |
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You can readily handle temperatures in {F} if you again think as we did |
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above. |
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The rule, even for units requiring an offset for conversion, remains: convert |
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a dimensional variable into dimensionless one such that the dimensionless |
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one has the proper value. |
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\end_layout |
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|
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\begin_layout Standard |
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Define a new variable |
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\end_layout |
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|
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\begin_layout LyX-Code |
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T_degF = T/1{R} - 459.67; |
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\end_layout |
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|
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\begin_layout Standard |
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Then code |
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\begin_inset CommandInset ref |
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LatexCommand ref |
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reference "eqn:dimeqns.lnPsat" |
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|
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\end_inset |
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|
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|
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\noun on |
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Equation 7.1 |
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\noun off |
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as |
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\end_layout |
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|
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\begin_layout LyX-Code |
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ln(P/1{atm}) = A - B/(T_degF + C); |
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\end_layout |
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|
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\begin_layout Standard |
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when entering it into ASCEND. |
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You will then enter constants A, B, and C as dimensionless quantities having |
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the values exactly as tabulated. |
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In this example we must create the intermediate variable T_degF. |
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\end_layout |
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|
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\begin_layout Section |
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Example 3-- pressure drop |
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\begin_inset CommandInset label |
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LatexCommand label |
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name "ssec:dimeqns.pressure drop" |
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|
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\end_inset |
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|
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|
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\end_layout |
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|
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\begin_layout Standard |
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From the Chemical Engineering Handbook |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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Chemical Engineering Handbook |
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\end_layout |
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|
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\end_inset |
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|
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by Perry |
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\begin_inset Index idx |
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status collapsed |
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\begin_layout Plain Layout |
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Perry |
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\end_layout |
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\end_inset |
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|
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and Chilton |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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Chilton |
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\end_layout |
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|
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\end_inset |
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|
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, Fifth Edition, McGraw-Hill, p10-33, we find the following correlation: |
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\end_layout |
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|
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\begin_layout Standard |
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\begin_inset Formula |
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\[ |
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\Delta P_{a}^{\prime}=\frac{y(V_{g}-V_{l})G^{2}}{144g} |
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\] |
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|
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\end_inset |
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|
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where the pressure drop on the LHS is in psi, y is the fraction vapor by |
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weight (i.e., dimensionless), Vg and Vl are the specific volumes of gas and |
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liquid respectively in ft3/lbm, G is the mass velocity in lbm/hr/ft2 and |
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g is the acceleration by gravity and equal to 4.18x108 ft/hr2. |
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\end_layout |
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|
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\begin_layout Standard |
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We proceed by making each term dimensionless and with the right numerical |
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value for the units in which it is to be expressed. |
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The following is the result. |
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We do this by simply dividing each dimensional variable by the correct |
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unit conversion factor. |
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\end_layout |
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|
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\begin_layout LyX-Code |
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delPa/1{psi} = y*(Vg-Vl)/1{ft^3/lbm}* |
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\end_layout |
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|
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\begin_layout LyX-Code |
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(G/1{lbm/hr/ft^2})^2/(144*4.18e8); |
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\end_layout |
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|
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\begin_layout Section |
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The difficulty of handling unit conversions defined with offset |
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\begin_inset CommandInset label |
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LatexCommand label |
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name "ssec:dimeqns.handlingUnitConv" |
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|
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\end_inset |
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|
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|
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\end_layout |
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|
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\begin_layout Standard |
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How do you convert temperature from Kelvin to centigrade? The ASCEND compiler |
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encounters the following ASCEND statement: |
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\end_layout |
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|
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\begin_layout LyX-Code |
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d1T1 = d1T2 + a.Td[4]; |
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\end_layout |
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|
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\begin_layout Standard |
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and d1T1 is supposed to be reported in centigrade. |
| 527 |
We know that ASCEND stores termperatures in Kelvin {K}. |
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We also know that one converts {K} to {C} with the following relationshipT{C} |
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= T{K} - 273.15. |
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\end_layout |
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|
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\begin_layout Standard |
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Now suppose d1T2 has the value 173.15 {K} and a.Td{4} has the value 500 {K}. |
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What is d1T1 in {C}? It would appear to have the value 173.15+500-273.15 |
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= 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 |
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{dC} to be the units for temperature difference in centigrade and {dK} |
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for differences in Kelvin. |
| 540 |
Then the correct answer is 173.15+500=673.15 {dC}. |
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|
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\end_layout |
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|
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\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). |
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It turns out that a.Td[4] is then required to be a temperature and not a |
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temperature difference for this equation to make sense. |
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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 |
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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 |
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"fits" the data without requiring the equation to be dimensionally (and |
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physically) reasonable. |
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\end_layout |
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|
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\begin_layout Standard |
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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. |
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|
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\end_layout |
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|
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\begin_layout Subsection |
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General offset |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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offset |
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\end_layout |
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|
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\end_inset |
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|
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and difference units |
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\begin_inset Index idx |
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status collapsed |
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|
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\begin_layout Plain Layout |
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difference units |
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\end_layout |
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|
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\end_inset |
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|
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|
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\end_layout |
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|
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\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 |
|
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\begin_layout Standard |
| 601 |
\begin_inset Formula |
| 602 |
\begin{equation} |
| 603 |
t_{1}+t_{2}-(t+t_{4})=0\label{eqn:t1+t2} |
| 604 |
\end{equation} |
| 605 |
|
| 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 |
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