Three CSTRs are used in series. The second reactor has a volume twice that of the first and third reactors. The influent flow has a concentration of 150 mg/l of A, and the flowrate is 380 l/min. The reaction is first order, the rate equation is -dCa/dt= KCa= -ra and the rate constant is 0.4/hr. Determine:
a) A graph of -ra on the y-axis versus Ca on the x-axis on arithmetic paprr.
b) The mean residence time and volume of each reactor if the removal or conversion of A is 90%. What are the concentrations of A and the rate of conversion in each reactor? Use a graphic solution using the graph from part(a). Express volume as litres.
a) To plot the graph of -ra versus Ca, we need to determine the rate of reaction (-ra) at various concentrations of A (Ca).
Given the rate equation, -dCa/dt = KCa = -ra, we know that the rate of reaction is proportional to the concentration of A.
Since the reaction is first order, we can rearrange the rate equation to solve for -ra:
-ra = KCa = 0.4Ca
Now, we can plot the graph with -ra on the y-axis and Ca on the x-axis.
b) To determine the mean residence time and volume of each reactor, we need to consider the stoichiometry of the reaction and the conversion of A.
Let's denote the volumes of the first, second, and third reactors as V1, V2, and V3, respectively. Given that the second reactor has a volume twice that of the first and third reactors, we can write:
V2 = 2V1
V3 = V1
The mean residence time (θ) is calculated by dividing the reactor volume (V) by the flow rate (Q):
θ = V / Q
Considering the first reactor, we can write the mass balance equation for A as follows:
V1 * dCa1/dt = Qin * Ca0 - Qout * Ca1 - ra1 * V1
Since the influent flow has a concentration of 150 mg/L of A and the flow rate is 380 L/min, we substitute the values for Qin and Ca0:
V1 * dCa1/dt = (380 L/min) * (150 mg/L) - 380 L/min * Ca1 - 0.4 (1/hr) * Ca1 * V1
We can solve this equation graphically to determine the concentration of A and conversion in each reactor by finding the intersection of the -ra versus Ca graph from part (a) with the mass balance equation.
Using this graphical solution, we can determine the concentrations of A and the rate of conversion in each reactor.
a) To plot the graph of -ra versus Ca, we can first determine the values of -ra at different concentrations of Ca.
Given:
Concentration of A in influent flow (Ca0) = 150 mg/l
Flowrate (Q) = 380 l/min
Rate constant (K) = 0.4/hr
We can use the first-order rate equation to calculate the rate of reaction (-ra) at different concentrations of A (Ca).
The rate equation is: -dCa/dt = KCa = -ra
Since it is a first-order reaction, the rate of reaction (-ra) is proportional to the concentration of A (Ca).
At steady-state conditions, the rate of reaction in each reactor is constant. Therefore, we can assume that the concentration of A at the outlet of each reactor is the same as that at the inlet.
We can calculate the concentration of A (Ca) in each reactor using the following formula:
Ca = (Ca0 * e^(-ra*t))/Q
where:
- Ca is the concentration of A in each reactor.
- Ca0 is the concentration of A in the influent flow.
- ra is the rate of reaction.
- t is the residence time in each reactor (t = V/Q, where V is the volume of each reactor).
- Q is the flowrate.
Let's assume the volume of the first reactor is V1, the volume of the second reactor is V2 (2 * V1), and the volume of the third reactor is V3 (also 2 * V1).
Now we can calculate the concentration of A (Ca) and the rate of conversion (-ra) in each reactor using the formulas above and plot the graph.
b) To determine the mean residence time and volume of each reactor when the removal or conversion of A is 90%, we can use the graph from part (a). We need to find the point on the graph where the -ra value corresponds to 90% conversion.
1. Locate the point on the graph where the -ra value corresponds to 90% conversion.
2. Determine the corresponding value of Ca at this point.
3. Use the formula Ca = (Ca0 * e^(-ra*t))/Q to calculate the residence time (t) for this conversion.
4. Calculate the volume of each reactor using the formula V = Q * t.
Note: The units given for flowrate (l/min) and rate constant (1/hr) need to be consistent for accurate calculations.
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