Modeling Salt Mixing in a Chemical Plant: Finding Eigenvalues and Equilibria

[DeTroi1616]

A chemical plant with 3 tanks in succession with mizing of reactants and products. Brine tanks = fresh water flows into tank, mixed brine from tank 1 to tank 2, then tank 2 into tank 3, and out of tank 3.

Let xi(t) = lbs of salt in tank i at time t for i=1,2,3

Assume each flow rate is in gal/min

dx1/dt = (rate in) - (rate out) = 0 lbs/min - (r gal/min)*(x1 lbs / V1 gal)

dx1/dt = (-r / V1)*x1

dx2/dt = (r / V1)*x1 - (r / V2)*x2

dx3/dt = (r / V2)*x2 - (r / V3)*x3

To simplify the variables:

x1(t) = x x2(t) = y x3(t) = z

Let x1(t),y1(t),z1(t) = lbs of salt in tank 1 after t minutes

Suppose V1 = 20 gal, V2= 40 gal, and V3 = 50 gal and r = 10 gal/min with initial amounts of salt in each tank:

x(0) = 15 lbs, y(0) = 0 lbs, z(0) = 0 lbs

(a) Write the mathematical model in the form of dŷ/dt = Aŷ and find the 3x3 coefficent matrix A.

ŷ = vector y notation

I did this part and got dx/dt = -1/2x dy/dt = 1/2x-1/4y dz/dt = 1/4y - 1/5z

coefficent matrix: | 1/2 - λ 0 0 |

| 1/2 -1/4-λ 0 |

| 0 1/4 -1/5-λ|

b) Show hand work in finding the eigen values, by solving det(A-λI)=0, and eigenvectors of A.

I got Eigenvalues: λ1 = -1/2 λ2 = -1/4 λ3 = -1/5, but I am having trouble getting the eigen vectors



E = NOATION FOR EIGENVECTOR

c) Fidn the general solution as a linear combination of eigen solutions:

ŷ(t) = c1e^(λ1t)*E1 + c2e^(λ2t)*E2 + c3e^(λ3t)*E3



(d) find the formulas for x(t), y(t), and z(t), the maountf of salt in each tank after t minutes using theinitial data



(e) Find all equilibrium of the DE system dŷ/dt = Aŷ and describe their type (spiral or sink or real source, etc)

 

[HallsofIvy]

I am puzzled as to what happens in tanks 2 and 3. If solution simple flows in and then out, with no additional water or solution, the whole problem is just what comes out of tank1. Tanks 2 and 3 might as well be a single outflow pipe.

 

[chogg]

I'm also confused, but for a different reason.

This sure sounds to me like a "homework style" question. I'm not a regular here, but shouldn't it be redirected to one of the homework forums?

 

[berkeman]

I'm also confused, but for a different reason.

This sure sounds to me like a "homework style" question. I'm not a regular here, but shouldn't it be redirected to one of the homework forums?

Done.