Water with a dissolved oxygen concentration of 1.0 mg/L enters a well-mixed tank

Question

Water with a dissolved oxygen concentration of 1.0 mg/L enters a well-mixed tank at a rate of Q = 1000 L/min. The tank volume V = 1.6 x 105 L. In the tank, bacteria consume oxygen for metabolism in proportion to the amount of oxygen present; i.e., at a rate r1 = k1C where k1 = 2x10-3 min-1 is the first order rate constant and C is the concentration of oxygen. Air is supplied to the tank at a rate proportional to the degree of undersaturation; i.e., r2 = k2(Cs – C) where k2 = 4x10-3 min-1 is the rate constant and Cs = 10 mg/L is the oxygen saturation. Write a mass balance for the oxygen concentration in the system. What are the independent variables, the dependent variables, and the parameters in the system? Find the steady-state concentration of oxygen.

Explanation / Answer

The equation for mass balance is

(mass accumulation rate)= (mass flux input) - (mass flux out)+ (mass flux net rate)

dm/dt = minput -moutput+ mreaction

dm/dt =V dC/dt      C=concentration

V= volume

minput = Q x C =1000 x 10 =104

moutput = Q x C =104

mreaction = -VkC = - 1.6 x 105 x 2 x 10-3 x 10 = - 336 x 10-2

For steady state dm/dt=0

dm/dt = 104 - 104 +(- 336 x 10-2)

The independent variables are volume as it is constant

The dependent variables are mass flux , conconcentration , rate constant.

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