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_C4H10≔Chemicals:-PropertyHeatOfFormation,C4H10(l),n-buta,useunits
−150.66kJmol
Enthalpies of the combustion products at a temperature T
h_N2≔Chemicals:-PropertyHmolar,N2(g),temperature=T:h_O2≔Chemicals:-PropertyHmolar,O2(g),temperature=T:h_H2O≔Chemicals:-PropertyHmolar,H2O(g),temperature=T:h_CO2≔Chemicals:-PropertyHmolar,CO2(g),temperature=T:
Enthalpy of the reactants
H_reactants≔1mol⋅ h_f_C4H10
−150.66kJ
Total enthalpy of the combustion products
H_products≔4 mol⋅ h_CO2+ 5 mol ⋅ h_H2O+ 24.44 mol ⋅ h_N2
H_products≔4Chemicals:−PropertyHmolar,CO2(g),temperature=Tmol+5Chemicals:−PropertyHmolar,H2O(g),temperature=Tmol+24.44Chemicals:−PropertyHmolar,N2(g),temperature=Tmol
Equating the enthalpy of the reactants and the enthalpy of the combustion products gives the adiabatic flame temperature
fsolveH_reactants=H_products,T=2000K
2379.85K
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
obj≔12⋅4⋅π⋅R2+2⋅π⋅R⋅L+π⋅R⋅H2+R2:
Constraint on the volume area of pod
cons1≔1243π⋅R3+π R2⋅L+13⋅π⋅R2⋅H=3m3:
All dimensions must be greater than 0
cons2≔0≤R,0≤L,0≤H:
Hence the optimized dimensions are
dimensions≔Optimization:-Minimizeobj,cons1,cons2,initialpoint=H=1m,L=1m,R=1m
10.25m2,H=0.79m,L=0.39m,R=0.88m
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.
P≔ThermophysicalData:-Propertypressure,methane,temperature=350K,density=1V:
We then perform the numeric integration ∫1.0 m3 kg−10.5 m3 kg−1P ⅆV to calculate the work done.
−intP,V=1.0 m3 kg−1.. 0.5 m3 kg−1,numeric
125.43kJkg
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
...to...
PropertyHmolar,C2H5OH(L),temperature=320K
−274.96kJmol
...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
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