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Units

Introduction

Maple 2018 offers enhanced tools for using units throughout your analysis.

Units in Solving, Integration, and Optimization

You can now use use units in numeric equation solving, integration, and optimization. Several functions support units, including fsolve and the optimizers in the Optimization package. In addition, units handling has been improved in the solve command and the int command for symbolic and numeric inputs.

Example: Adiabatic Flame Temperature of Butane

Liquid butane is burnt with 100% theoretical air at an initial temperature of 298.15 K. The combustion reaction is

 

C4H10 (l) + 6.5 O2 (g)+ 24.44 N2 (g) → 4 CO2 (g) + 5 H2O (g) + 24.44 N2 (g)

 

Here, we will calculate the adiabatic flame temperature of the combustion products.

 

withThermophysicalData:

 

Heat of formation of butane

h_f_C4H10Chemicals:-PropertyHeatOfFormation,C4H10(l),n-buta,useunits

150.6640000kJmol

(2.1.1)

Enthalpies of the combustion products at a temperature T

h_N2Chemicals:-PropertyHmolar,N2(g),temperature=T:h_O2Chemicals:-PropertyHmolar,O2(g),temperature=T:h_H2OChemicals:-PropertyHmolar,H2O(g),temperature=T:h_CO2Chemicals:-PropertyHmolar,CO2(g),temperature=T:

 

Enthalpy of the reactants

H_reactants1mol h_f_C4H10

150.6640000kJ

(2.1.2)

Total enthalpy of the combustion products

H_products4 mol h_CO2+ 5 mol  h_H2O+ 24.44 mol  h_N2

H_products4Chemicals:−PropertyHmolar,CO2(g),temperature=Tmol+5Chemicals:−PropertyHmolar,H2O(g),temperature=Tmol+24.44Chemicals:−PropertyHmolar,N2(g),temperature=Tmol

(2.1.3)

 

Equating the enthalpy of the reactants and the enthalpy of the combustion products gives the adiabatic flame temperature

fsolveH_reactants=H_products,T=2000K

2379.853026K

(2.1.4)

Example: Optimizing the Design of a Fuel Pod

You are designing a fuel pod with a hemispherical cap, cylindrical mid-section and conical cap.

 

 

What are values of L, H and R that minimize the surface area while maintaining the volume V at 3 m3?

 

restart:

 

Objective function - surface area of pod

obj124πR2+2πRL+πRH2+R2:

 

Constraint on the volume area of pod

cons11243πR3+π R2L+13πR2H=3m3:


All dimensions must be greater than 0

cons20R,0L,0H:


Hence the optimized dimensions are

dimensionsOptimization:-Minimizeobj,cons1,cons2,initialpoint=H=1m,L=1m,R=1m

10.2533536615869920m2,H=0.785093823049978m,L=0.392546902492684m,R=0.877761593519080m

(2.2.1)

Example: Work Done in Isothermal Compression of Methane

Methane at 350 K is isothermally compressed from a specific volume of 1.0 m3 kg-1 to 0.5 m3 kg-1. Here, we will calculate the work done.

 

First, we define an expression that gives the pressure of methane as a function of the specific volume V.

PThermophysicalData:-Propertypressure,methane,temperature=350K,density=1V:

 

We then perform the numeric integration 1.0 m3 kg10.5 m3 kg1P ⅆV to calculate the work done.

intP,V=1.0 m3 kg1..  0.5  m3 kg1,numeric

125.4345869kJkg

(2.3.1)

Unit Conversion

Maple 2018 moves the units formatting dialog into the new Context Panel, and enhances its functionality.

 

 

Changing units takes fewer mouse clicks, and can be used in tandem with numeric formatting to quickly create presentable, readable results. For example, you can reformat this result...

 

withThermophysicalData:-Chemicals:PropertyHmolar,C2H5OH(L),temperature=320K

274957.7153Jmol

(3.1)

...to...

PropertyHmolar,C2H5OH(L),temperature=320K

274.9577153kJmol

(3.2)

 

...with a few mouse clicks, and without navigating between different dialogs.


The list of suggested unit conversions now gives you

 

• 

typeset Greek letters, e.g. micro Joules are typeset as μJ

• 

typeset math, e.g. Jmol K instead of J/mol/K

• 

sensible suggestions for unit conversions, e.g. selecting a result with units of kg m2s2 mol offers Jmol and kJmol as potential conversions

• 

a cleaner, more streamlined experience by removing little used unit systems (although all unit systems can still be set via the command line)

Applications

Three Reservoir Problem

Ultimate Strength of a Bolt Group

Single Stub Matching

Spring Design Optimization