Tank with Varying Concentration and Volume, Problem 2

Outline

These problems have a typical solution outline

1) Perform material balance on tank

2) Get 1st-order Differential Equation in terms of mass

3) Get volume as a function of time (V decreases in this example)

4) Solve differential Equation

Material Balance Notation

Change in NaOH = NaOH in - NaOH Out [lb/hr] C -Concentration [lb/lb]

D - density [lb/gal]

t - time

V - Volume [gallons]

Q - Flow [gal/hr]

M - Mass [lbs]

Since, MC = DC(t)V(t) and Co = C(t)

subscripts

i - in

o - out

c - caustic

Notice both V and C vary with time.

Get an V(t) from the overall material balance for the tank.

V(t) = 8000 - (1500-1000).t = 8000 - 500

.t

V'(t) = -500 by differentiation

Substituting for the variables in the differential equation above

A solution of 10% by weight caustic is being used for a caustic washing process. A 10,000 gallon

surge tank is being used to equalize concentrations. The discharge rate from the tank is a constant

1,500 gallons per hour. The tank initially contains 8,000 gallons at 10% caustic, when an upset occurs,

and the inlet concentration drops to 5% by weight at an inlet flow rate of 1,000 gallons per hour.

Assume constant density for the mixtures. The mixture is considered outside the specification below

9.2% by weight caustic. When does this occur?

Assuming constant density, D cancels out. Using the

product rule, we get:

Vt=0=8,000 gal Ct=0=0.10 C(t)

1,000 gal/hr 5% Caustic

oooiii

c CQCQdt

dM

1,500 gal/hr C(t)

CQCQdt

VCdooiiio

)(

CQCQdt

tdCV

dt

tdVC oii

)()(

)(1500)05.0)(1000()(

)5008000()(500 tCdt

tdCttC

The differential equation is solved by Runge-Kutta on the next page

)5008000(

)100050(

t

C

dt

dC

Runge Kutta Method

click here to insert a different function

Activates the change by using the function once

Initial Condition

at t= 0 , C(t) = 10.00%

h= 0.083333 increment t by these hours (5 minutes in this example)

n t C(t) dC(t)/dt k1 k2 k3 k4

(hours) (%)

0 0.000 10.00% -0.00625 -0.00052 -0.00052 -0.00052 -0.00052

1 0.083 9.95% -0.00622 -0.00052 -0.00052 -0.00052 -0.00052

2 0.167 9.90% -0.00618 -0.00052 -0.00051 -0.00051 -0.00051

3 0.250 9.84% -0.00615 -0.00051 -0.00051 -0.00051 -0.00051

4 0.333 9.79% -0.00612 -0.00051 -0.00051 -0.00051 -0.00051

5 0.417 9.74% -0.00609 -0.00051 -0.00051 -0.00051 -0.00050

6 0.500 9.69% -0.00605 -0.00050 -0.00050 -0.00050 -0.00050

7 0.583 9.64% -0.00602 -0.00050 -0.00050 -0.00050 -0.00050

8 0.667 9.59% -0.00599 -0.00050 -0.00050 -0.00050 -0.00050

9 0.750 9.54% -0.00596 -0.00050 -0.00050 -0.00050 -0.00049

10 0.833 9.49% -0.00592 -0.00049 -0.00049 -0.00049 -0.00049

11 0.917 9.44% -0.00589 -0.00049 -0.00049 -0.00049 -0.00049

12 1.000 9.39% -0.00586 -0.00049 -0.00049 -0.00049 -0.00049

13 1.083 9.35% -0.00583 -0.00049 -0.00048 -0.00048 -0.00048

14 1.167 9.30% -0.00579 -0.00048 -0.00048 -0.00048 -0.00048

15 1.250 9.25% -0.00576 -0.00048 -0.00048 -0.00048 -0.00048

16 1.333 9.20% -0.00573 -0.00048 -0.00048 -0.00048 -0.00047

17 1.417 9.15% -0.00570 -0.00047 -0.00047 -0.00047 -0.00047

18 1.500 9.11% -0.00566 -0.00047 -0.00047 -0.00047 -0.00047

19 1.583 9.06% -0.00563 -0.00047 -0.00047 -0.00047 -0.00047

20 1.667 9.01% -0.00560 -0.00047 -0.00047 -0.00047 -0.00046

Change Function

8.80%9.00%9.20%9.40%9.60%9.80%

10.00%10.20%

0.000 0.500 1.000 1.500 2.000

Co

nc.

by w

t.

time (hours)

Runge Kutta Results

Activate Function

Solution: The concentration of 9.2% occurs at 1 hour and 20 minutes.