Dimensional Analysis

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My fictitious vehicle's fuel consumption rate is dependent on several factors:

• The current acceleration (how far I've pushed the gas pedal)
• The speed (due to increased wind resistance)
• The weight of my vehicle and it's contents (increased tire/road friction)
• A constant term representing my engine idling

From these terms, we can conclude that the fuel consumption (\(Q\)) is directly proportional to acceleration (\(a\)) and speed (\(v\)), and inversely proportional to weight (\(f\)), plus a constant term (\(C\)). We want to know two things:

1. What are the dimensions on the constant term? \(\dim(C)\)
2. What are the dimensions on the proportionaly constant? \(\dim(x)\)

Let's set up our equation and substitute in our variables:

$$\text{Fuel consumption} = x \times \left[ \frac{\text{Acceleration} \times \text{Speed}}{\text{Weight}} + \text{C} \right]$$

$$Q = x \times \left[ \frac{a \times v}{f} + C \right]$$

From our dimensional analysis or physics class that we paid attention in, we know that we can only add terms of like dimensions, so that means that:

$$\dim\left(C\right) = \dim\left(\frac{a \times v}{f}\right)$$

We can also isolate \(x\) to determine its dimensionality as well:

$$\dim\left(x\right) = \dim\left(\frac{\frac{a \times v}{f} + C}{Q}\right) = \dim\left(\frac{\text{C}}{\text{Q}}\right)$$

Use Dimensional to define each one of the known dimensions and calculate the unknown dimensions for \(C\) and \(x\).

Installation

Follow these steps to create a new project workspace and install the dimensional dependency to run this example.

# Create and open project folder
mkdir Dimensional_Analysis_demo
cd Dimensional_Analysis_demo
# Initialize project and install dependencies
npm init -y
npm i dimensional@1.3.1
# Create and open source file
touch "Dimensional Analysis.mjs"
open "Dimensional Analysis.mjs"
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Source

Copy and paste this source code into Dimensional Analysis.mjs.

import { dimensions } from 'dimensional';

// Define our dimensions
const a = dimensions.acceleration; // Acceleration
const v = dimensions.velocity; // Speed
const f = dimensions.force; // Weight
const Q = dimensions.volume.over(dimensions.Time); // Fuel consumption, e.g. gal/min - not a default dimension

// Determine dimensions on C (idling constant term)
const C = a.times(v).over(f);
console.log(C.toString());

// Determine dimensions on x (proportionality constant)
const x = C.over(Q);
console.log(x.toString());
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Output

In Dimensional_Analysis_demo/, execute Dimensional Analysis.mjs with NodeJS to generate an output.

node "Dimensional Analysis.mjs"
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You should expect to see an output similar to the one below.

\frac{{\textbf{L}}}{{\textbf{T}} \cdot {\textbf{M}}}
\frac{1}{{\textbf{L}}^{2} \cdot {\textbf{M}}}
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