Introduction
Basically this whole lecture introduces dynamic system to us rather than the steady state ones we have been working with.
Here the goal is to show us how we can take dynamic equations and use the laplace transformation inorder to derive important information in an easier and faster way than solving the atually equations.
Here the transformation is only ment to be used as a tool for further much more diffuclt topics to come. it is really a buliding block for the future
What assumptions must apply for equations to approximate process dynamics?
Process dynamics can be very well approximated by finite order, linear, time-invariant differential equations when each of the assumptions apply:
- the content of a relevant reactor volume can be considered homogeneous
- the conditions and properties don’t change as function of time
- the process is operated in a well-defined operating point
In what domain do we usually analyse data, and which one is most convenient for dynamic systems
We as humans are most comfortable using the time domain; however, for dynamic processes, it is much more convenient to use the frequency domain (which will be explained later)
What is the Laplace transformation?
It is a transformation that takes ordinary, linear, time-invariant, finite order differential equations representing dynamic processes in the time domain to
algebraic equations in the frequency domain.
Mathatically, Let f(t) be an integrable function defined on [0, +∞). The Laplace transform of f(t) is defined as
digital notes
where the parameter s in the integral is understood to be a complex number, therefore the integral defines a function of a complex variable s.
What are some notations to keep in mind off?
The notation L[f(t)] is also used to denote the Laplace transform of the function f(t).
Note that the lower-case symbol f is normally used to denote a function of t and the corresponding upper-case symbol F is used to denote its Laplace transform, function of s. In dynamics applications, the variable t represents time. The variable s does not have a physical significance; it is a mathematical entity, with units of inverse time (#Frequency).
What are the conditions and or assumptions of the Laplace equation? Note last condition is just understanding, really not memorising at all
- It is assumed that f(t) = 0 for all values of t <= 0; this will be clearer with the example of a step function
- f(t) must be integrable
- ROC: Region of conversion. Where the improper integral must converge (not blow up). This happens only when the exponential decay term e^−σt dominates any growth of f(t). Y3ani, in mathematical terms, there exist constants M>0, α∈R, and t0 > 0 such that: ∣f(t)∣ ≤ M*e^(αt), ∀t≥t0
Then the Laplace transform converges for all s∈C with Re(s)>α. In that region, F(s) is analytic (smooth and well-behaved). This will also become clear with examples
Break moment
Okay, we are almost done, but I want to remind you that the purpose of all of this is to show the value and the benefit of what information we get from laplace transformation. Now we will proceed to examples
Example 1: Step function
Digital notes
Example 2: e^-at
Digital notes
Example 3: Find the Laplace transform of the impulse (Dirac) function? (Exact understanding of the math is not needed
The Dirac function is super important in engineering processes where it is used to monotior the progress of a reaction by dropping (for example) a bit of paint in a reactor and watch how it progresses through. Area under the curve is always 1
Example in digital notes
Okay, finally, what is the big thing that the transformation tells us?
Okay, so all of this is in relation to the digital notes:
- System stability in the s-plane: The system is stable as long as all poles lie on the left-hand side of the complex plane, which corresponds to Re(s)<0.
- Further left = faster response: The farther to the left the poles are, the faster the system converges (the faster the reaction settles).
- Right-hand side = instability: If poles end up on the right-hand side of the real axis, the system (or reaction) is unstable and usually cannot be solved numerically.
*Controllers in chemical reactions: In chemical processes, poles may naturally lie on the right-hand side, which would make the reaction unstable. In such cases, a controller is used to shift the poles to the left-hand side, ensuring stability and preventing the reaction from blowing up.
Okay final note
Again, Laplace should just be a tool; in the book and in the digital notes, there is a slide showing a table of the most relevant Laplace transformations of different functions; however, next lecture, we will still expand on the rules of Laplace transformations
