solve with matlab code
Lab Session 12: Ordinary Differential Equations
Problem Statement (Please read the lab manual and work on your own to complete the lab
report)
Part 1)
Accumulation represents the change in mass in the reactor per change in time. For a constant-
volume system, it can be formulated as
Accumulation =
where V is the volume, c is the concentration. Thus, a mathematical formulation for
accumulation is volume times the derivative of concentration with respect to time.
Oc
Or
3
Figure 1. A single, completely mixed reactor with an inflow and an outflow.
Since Accumulation . inputs - outputs, one can write the mass balance for a single reactor as
shown in Figure 1:
This equation can be used to determine transient or time-variable solutions for the reactor.
For example, if c =catt=0,calculus can be employed to solve the equation as follows
analytically
If Cin=50 mg/m^3, Q = 5 m^3/min, V = 100 m^3, and Co = 10 mg/m^3, the equation is
Cexact =50(1-e-005t)+10e-0.05
THIS IS THE ANALYTICAL SOLUTION
this plot in your report

Question

Accepted Answer

```matlab % Define the function to calculate the concentration over time function Cexact = calculate_concentration(t) % Given values for Cin, Q, V, and Co Cin = 50; % mg/m^3 Q = 5; % m^3/min V = 100; % m^3 Co = 10; % mg/m^3 k = Q / V; % 1/min, the rate constant % Calculate the exact concentration using the provided formula Cexact = Cin * (1 - exp(-k * t)) + Co * exp(-k * t); end ``` ```matlab % Main function to test the calculate_concentration function function main_test() % Test the calculate_concentration function % Use case 1: At time t = 0 assert(abs(calculate_concentration(0) - 10) < 1e-4); % Use case 2: At time t = 10 assert(abs(calculate_concentration(10) - (50 * (1 - exp(-0.05 * 10)) + 10 * exp(-0.05 * 10))) < 1e-4); end ``` ∻Explanation∻ ⚹The MATLAB code defines a function `calculate_concentration` that computes the concentration of the substance in the reactor at a given time `t` using the provided analytical solution. The main function `main_test` tests this function with two use cases: at the initial time (t=0) and at t=10 minutes. The `assert` statements check if the calculated concentration is within a small error tolerance (1e-4) of the expected values.⚹ ∻Step-by-step Instruction∻ ‖ ▪ ⋮ Define the function `calculate_concentration` to compute the concentration at time `t`. ‖ ‖ ▪ ⋮ Assign given values for `Cin`, `Q`, `V`, and `Co` within the function. ‖ ‖ ▪ ⋮ Calculate the rate constant `k` as `Q / V`. ‖ ‖ ▪ ⋮ Use the provided analytical solution formula to calculate `Cexact`. ‖ ‖ ▪ ⋮ Define the main function `main_test` to test the `calculate_concentration` function. ‖ ‖ ▪ ⋮ Use `assert` statements to check the correctness of the function with two use cases. ‖ ∻Time Complexity∻ ⚹The time complexity of the function `calculate_concentration` is O(1) because it performs a constant number of operations regardless of the input size.⚹ ∻Space Complexity∻ ⚹The space complexity of the function is also O(1) as it uses a fixed amount of space for the variables and does not depend on the input size.⚹

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