GNG1506 – Fundamentals of Engineering in information processing
During Project
chemical Engineering
The transient response of coupled chemical reactors
The study of chemical reactors plays an important role in chemical engineering. The transient response of a reactor is the change in the concentration of a chemical in the reactor when the concentration in the input stream is changed.
The majority of applications in chemical engineering pays attention on mass balance for reactors. The balance of the mass is derived from the conservation of mass and the said inputs to the system minus the attractions System enables the accumulation of mass in the system.
Accumulation = input – output (Equation 1)
The balance of mass is used in the engineering problem solving by expressing the input and output variables and measurable parameters. For example, to apply the mass balance for a conservative substance (ie, d. A substance that does not increase or does not decrease by chemical transformations) in a reactor (Figure 1), the flow rates of the mass enters the reactor through the inlet pipe and out of the reactor through the outlet pipe must be quantified. This quantification is made with the product of the flow rate Q (in cubic meters per minute) and the concentration c of the substance (in milligrams per cubic meter). So the system input is Qcin and the system output is Qc.
Figure 1 – Single reactor with an inlet and an outlet
We are interested in the transient response (how the concentration of the substance in the reactor changes over time) of a completely mixed reactor (i.e., the concentration is similar across the reactor). First, a mathematical expression of mass accumulation in the reactor is necessary. For the reactor with a constant volume (the volume of the liquid does not change), the formula is simply:
Accumulation (Equation 2)
where V = constant volume and c = concentration in the reactor. So the mathematical formula for accumulation is the volume times the derivative of C versus t; The equation represents how the accumulation of the chemical changes in relation to the change in concentration.
where V = constant volume and c = concentration in the reactor. So the mathematical formula for accumulation is the volume times the derivative of C versus t; The equation represents how the accumulation of the chemical changes in relation to the change in concentration.
(Equation 3)
Equation 3 is used to find the transient response of the reactor, i.e. how the concentration of the reactor substance changes relative to time, given the initial concentration value, C0 (at time t = 0) and the CIN input concentration (in Inlet flow) and C (the concentration in the reactor).
A system of several reactors are connected to form a system of coupled reactors. In this project, we will work with a system of 3 coupled reactors shown in Figure 2. Each reactor contains constant volumes given by V1 (reactor 1), V2 (reactor 2) et V3 (reactor 3).
Figure 2 – System of 3 coupled reactors
For this system of coupled reactors, three equations are necessary to represent the transient change of concentrations in the reactors.
(Equation 3)
In order for the volumes in each reactor to remain constant, flow rates must respect the following relationships:
(Équation 4)
Although the analytical solution is often possible, it is sometimes difficult or impossible to find it in several problems. Numerical methods can be used to find a solution directly with the differential equation. A simple method that gives a solution with a differential equation and the Euler method.
Given a normal form of the differential equation where the right expression contains no derivatives,
, (Équation 5)
The Euler method gives the difference equation in the following form:
(Équation 6)
By applying equation 6 to Equation 3a, the difference equation for the change in concentration of reactor 1 becomes:
(Équation 7)
Where concentration C1, I, and C1, I-1 represents the concentration of the chemical in reactor 1 at time Ti and TI-1, respectively, and the concentration C3, I-1 represents the concentration in reactor 3 at time TI-1. Two other equations of difference must be developed from the differential equations 3b and 3c in order to be able to determine the changes in the C2 and C3 concentrations. Therefore, given the values of C1, 0, C2, 0, and C3, 0, at time t0, it is possible to compute all the values of C1, I, C2, I, C3, I as the time is incremented by Δt (which also gives the values T1 , T2, T3,…).
Develop software that allows the user to study the transient response of the three reactors shown in Figure 2. The user will provide the following values:
- The volumes in each reactor V1, V2 and V3,
- The values of flow rates Q01, Q03, Q12, Q23, Q31, and Q33,
- The input concentrations C01 and c03, the initial concentration values in the reactors at time t = 0, C1, 0, C2, 0, and C3, 0, as well as the final time TF, to define the time span to study the response (i.e., between t = 0 and t = TF).
The software must verify that the input values meet the constraints of Equation 4. If the constraints are met, the software will have to display 3 paths on a chart. Each plot should show the change of substance concentration in one of the reactors. Used colors and different styles of lines in order to distinguish paths well.
When new data are entered, the user is given the option to save it to a file; up to five sets of input values can be saved. When the program starts, the user has the option to choose one of the saved sets, or to give a new set of input values.