The two ponds in the figure below can be modeled as completely-stirred tank reactors (CSTRs) in series. The first-order decay rates for each pond are different but the inflow and outflow rates and volumes for the two ponds are the same.
a. Write a mass-balance (differential) equation describing dC2/dt.
b. Solve your mass balance to develop an expression for C2(t). Assume a spill results in an instantaneous input of mass M into the first pond. Assume the upstream concentration C(0)=0 and the initial concentration of the second pond, C2(t=0)=0.
c. For parameter values k1=0.1/day, k2=0.15/day, Q=10 m3/day, V=1000 m3, and M=1 kg, prepare a figure showing C2 as a function of time for 0<=t(days)<=50.
d. What is the maximum concentration in the second pond and when does it occur?
Q, C(0) Q, C1
------>Pond 1: V, C1, k1------>
Q, C2
Pond 2: V, C2, k2------>
To clarify the 'figure':
Q, C(0) are the inflow parameters into the first pond.
Q, C1 are parameters from pond 1 as it is about to enter pond 2.
Q, C2 are outflow parameters from the second pond.
Pond 1 has parameters V, C1, k1.
Pond 2 has parameters V, C2, k2.
Thanks, Christine
a. The mass-balance equation for the second pond can be written as:
dC2/dt = (Q/V)(C1 - C2) - k2C2
where Q is the flow rate, V is the volume of each pond, C1 is the concentration in Pond 1, and k2 is the decay rate for Pond 2.
b. To solve the mass balance, we need to separate the variables and integrate:
dC2/(C1 - C2) = (Q/V)dt - k2 dt
Integrating both sides:
-ln(C1 - C2) = (Q/V)t - k2t + constant
Simplifying and solving for C2:
C1 - C2 = Ce^(-(Q/V + k2)t)
Since the initial concentration of Pond 2, C2(t=0) = 0:
C1 - 0 = Ce^(-0) ==> C1 = C
Therefore, the expression for C2(t) is:
C2(t) = C - Ce^(-(Q/V + k2)t)
c. To plot C2 as a function of time, we can substitute the given parameter values into the expression for C2(t). Using k1=0.1/day, k2=0.15/day, Q=10 m3/day, V=1000 m3, and M=1 kg, we can calculate the concentration at different time points.
d. The maximum concentration in the second pond can be found by differentiating C2(t) with respect to time and setting it equal to zero. Solving for t will give us the time at which the maximum concentration occurs.
To answer these questions, we need to use the concept of mass balance for each pond.
a. The mass balance equation for the second pond (C2) can be written as:
dC2/dt = (Q/V) * (C1 - C2) - k2 * C2
This equation represents the rate of change of concentration in the second pond with respect to time (dC2/dt). The first term on the right side represents the inflow and outflow of mass from the second pond, while the second term represents the decay of mass within the second pond.
b. To solve the mass balance equation, we need to write an expression for C2(t). Given the initial conditions C(0) = 0 and C2(t=0) = 0, we can solve the differential equation using standard techniques (such as separation of variables or integrating factors). The result would be an expression for C2 as a function of time (t).
c. Using the given parameter values:
k1 = 0.1/day, k2 = 0.15/day, Q = 10 m3/day, V = 1000 m3, and M = 1 kg, we can plot C2 as a function of time (0 <= t <= 50 days). To do this, we need to numerically integrate the differential equation from part a using a numerical method like Euler's method or a more accurate method like Runge-Kutta.
d. The maximum concentration in the second pond (C2) can be determined by observing the plot obtained in part c. The point where C2 is at its highest value corresponds to the maximum concentration. The time at which this maximum occurs can be read from the plot.