Use of the unit impulse function

BAE 3023 Use of the unit impulse function The dump variation of the Dilution Problem

Problem Description BAE 3023 • Remember the following Dilution problem: • The inflow concentration cin [kg/m3] of A is time varying • The flowrate F [m3/s] is constant • The material A in the fluid is not reduced by reaction or other means in the tank • The Tank is well mixed • Find the outflow concentration cout [kg/m3] of A

An Impulse as a rate BAE 3023 How can we deal with the situation where a bucket of component A were dumped into the tank at the inflow point? Where IA is the inflow rate of A and FBCB is the inflow rate of A from the bucket. The bucket of A dumped in would be so many kg of A but can be thought of as a rate of A. The unit impulse function is defined as follows and has a non-zero value only at d(0): and: Note that if d(t) has units of [kg A/s] that the integral of d(t) has units of [kg A]. The magnitude of the impulse is the mass of A.

Impulse Input BAE 3023 For our tank, Md(t) [kgA/s] is the rate of A dumped into the tank from the bucket where M is the Mass of A in the bucket.

Basic Equation BAE 3023 A mass balance for component A can be written Inflow rate of A - Outflow rate of A = Accumulation rate of A Subtracting this from the unsteady state equation above:

Solution BAE 3023 Letting cin be at steady state (Cin=0) and rearranging, we get: Transforming:

Graphical Response BAE 3023

Use of the unit impulse function