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MIXING_SOLUTION

# MIXING_SOLUTION - ENV 6015 Solution to Mixing Problem Set 1...

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1 ENV 6015 Solution to Mixing Problem Set 1. A slug of dye (100 mg) is injected into a single tank with continuous mixing. The flow in and out of the tank is 0.5 gpm. The tank volume is 500 gallons. When will the dye concentration equal 26.8 ug/L? Solution: Q, C o Q, C 2 Q, C 1 M V 1 V 2 Mass balance : in - out = acc where : in = 0 because there is no more input () dt V C d QC 0 m m = where V M C m = at 0 t = at all points and volume V is constant. m m C dC dt V Q = ; and T 1 V Q = (by definition T is hydraulic residence time, HRT) = = t C V M C m m t 0 C dC dt T 1 Æ [] t m C V M C m C ln T t = = = V M ln C ln T t t = M V C ln T t t Where V = 500 gallons M = 100 mg

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2 C t = 26.8 L g μ Q = 0.5 min g T = gal/min 0.5 gal 500 = 100 min × × × = × × = μ g 100,000 L/gal 3.7854 gal 500 μ g/L) (26.8 ln gal/min 0.5 gal 500 M V C ln T t t t = 679 min. =11.3 hr. for μg/L 26.8 C C t m = =
3 2. Is the differential equation shown below the correct differential equation for the system? 1 T t 1 2 2 2 2 e V M Τ 1 C Τ 1 dt dc = + Solution: For reactor 1 1 t 1 M/V C ln T t = ------------------------------------------------- (1) where T 1 = 1 V Q For reactor 2 : Mass Balance (No generation) ; In – Out + generation = accumulation dt ] V d[C QC QC 2 2 2 1 = but V 2 = constant dt dc C V Q C V Q 2 2 2 1 2 = ; dt dc C T 1 C T 1 2 2 2 1 2 = where 2 2 V Q T = Rearranging : we get 1 2 2 2 2 C T 1 C T 1 dt dc = + -------------- (2) Q, C o Q, C 2 Q, C 1 M V 1 V 2

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4 solving equation (1) for 1 t C C = we get 1 t/T 1 e V M C = --------------------------------------------------------- (3) Substituting (3) in (2) 1 t/T 2 2 2 2 e V M T 1 C T 1 dt dc = + Which is the same equation, and it is the correct differential equation for the system.
5 3. The data shown in the succeeding table represent concentration of dye following time of addition. Assume the data points can be connected by straight lines to provide a continuous representation of mass flow out of the tank. This data is typical of data generated for a resident time distribution study. How many pounds of dye were added to the tank if the tank volume was 53,000 gallons and the flow was 26,500 gallons/hr? Draw a concentration verses time curve, and a cumulative mass verses time curve for this data. Report the times when 10 %, 40 % and 90 % of the mass has exited the tank. Do

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MIXING_SOLUTION - ENV 6015 Solution to Mixing Problem Set 1...

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