1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

3 Cascading Brine Tanks

  1. Apr 14, 2014 #1
    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)
     
  2. jcsd
  3. Apr 15, 2014 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
     
  4. Apr 15, 2014 #3
    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?
     
  5. Apr 15, 2014 #4

    berkeman

    User Avatar

    Staff: Mentor

    Done.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: 3 Cascading Brine Tanks
  1. Draining a Tank (Replies: 5)

  2. Cascade of LTI (Replies: 0)

Loading...