What are the uses of differential equations in engineering?

Answer 1

You'll find ordinary differential equations (ODEs) being used in chemical engineering for many things, such as determining reaction rates, activation energies, mass transfer operations, heat transfer operations, and momentum transfer operations.

Answer 2

A differential equation is just an equation that relates the value of the function to its derivatives. What that means in English is that the function's value depends on how it's changing. If we take kinematics as an example (motion), the original function describes position. The function's first derivative with respect to time gives an equation for velocity, aka the change in position over time. The function's second time derivative yields an equation for acceleration, the change of velocity over time, or the change of the change of position over time.

So now if we say position is dependent on its current velocity and/or its current acceleration, we get a differential equation.

So what does this mean for engineering? It means we have a huge mathematical tool for modeling dynamic scenarios. For instance, let's model throwing a ball on earth. Earth has a gravitational field. For simplicity, let's assume that the acceleration due to the gravitational field is constant (represented as g). So our force on a ball with mass m, due to gravity, is F(t) = -mg (the negative means down, the force is pointed down into the earth).

So where do we go from here? Newton's equation of motion says F = ma:

Force = mass times acceleration.

We'll use the following notation from here on out: a(t) is acceleration, v(t) is velocity, x(t) is position, ' (prime) denotes the derivative with respect to time, and bolded values mean they're vectors.

This gives us:

F(t) = ma(t) = mv'(t) = mx''(t)

So this is our differential equation; force is mass times the change of the change of position (aka acceleration). Whats the next step? Simplify:

mx''(t) = F(t) = -mg

x''(t) = -g

Next step? Integration:

x'(t) = -gt + v0, where v0 is an integration constant which is solved by initial conditions (plug in a value for x'(t) or v(t) that is known, and solve; e.g. if v(0) = 0 then v0 = 0).

Then integrate again:

x(t) = -1/2 gt2 + v0t + x0, where x0 is also an integration constant and, like before, found through initial conditions (If x(0) = 0 then x0=0).

So what does this all mean? We started with a differential equation relating force to the change of the change of position of the ball with respect to time and in the end, we found an equation that completely describes the position of the ball as a function of time from the effects of gravity. We made an assumption and figured out the implications of this assumption.

This was a simple example. There are hundreds of thousands of situations out there waiting to be modeled and solved. For instance, we can take that last example a step further in the following way. Since we know that the acceleration due to gravity actually depends on the distance between the two masses in question, we could model the situation using the function g(r), instead of constant g, where r is the distance between the two objects. However, solving this equation is quite a bit harder. Another situation we could model with a differential equation could be time effects, such as if gravity got stronger over time. There are also partial derivatives and other complicated things that could be added, such as if the x component of the force vector depended on the position of y. All of this takes years of practice and a lot of headaches, even to get through the "simple" stuff, but it is extremely satisfying when you get past that point. Then, of course, the process repeats all over again for harder problems.

Answer 3

As useless as partial derivatives may seem during your undergraduate studies, they do serve very important applications. For example, the energy balance of a differential volume is a partial derivative equation, being very difficult to integrate without having boundary conditions. A lot of the equations you see can be derived from first principle balances on differential units, so they are extremely useful.

Also, another very important application is related to transport phenomena, specifically fluid flow. As a chemical engineer, this is core to your knowledge database, but it is often forgotten (or never taught) that fundamental equations are derived from the Navier-Stokes equations related to transport phenomena!

It is unlikely that you will be required to solve such fundamental equations unless you pursue a masters in say computation fluid dynamics (CFD) or are required by your employer to do such R&D work that requires such high amount of knowledge.

A great chemical engineer should be able to derive all their necessary knowledge from first principles, so study hard.