- #1
DeTroi1616
- 1
- 0
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)
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)