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A comprehensive guide to solving mixing problems, a common type of compartmental system in engineering and related fields. It explores the concept of compartmental systems, their graphical representation using block diagrams, and the derivation of differential equations to model the mixing process. Detailed examples and step-by-step solutions to illustrate the application of these principles. It is particularly useful for students studying chemical engineering, environmental engineering, or other disciplines involving mass transfer and material balance.
Typology: Exercises
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(^1) T
Problems Exercises 1.7^1
broken (^) down into distinct
entire system modeled
interactions between (^) the various
Such
and (^) are graphically depicted^ by^ block (^) diagrams
rate
rate It (^) represents the (^) amount of^ a substance at^ time t The (^) substance enters^ the (^) compartment
output rate The rate of^
in (^) Xlt is given (^) by dog input rate^ output rate
Mixing
Example consider^ a
holding 1000L of^ pure water into which a^ brine solution of salt
of (^6) 4m The solution^ inside (^) the tank
flowing
If (^) the concentration of salt in^ the brine^ entering
concentration of^ salt in^ the will
64min Is
X It^ kg 1000 L 101 0
E (^6) 4min Solution 8 rate (^) in rate^ out
in
X t^006 x 6 multiply by^
e 00Gt x t^006 e^ 00Gt
00Gt e out x 6 e 00Gt
tx f (^600006) tdt (^) t
e 00Gt
gl qje^
fektdt fekttcx E.at gge
x ogE 00bte out
oobt
00Gt
0067 We know that^ x^07 0 so (^) c
E O 100 C C 100 X
00Gt J t what (^) happens to (^) x as t (^) becomes large Find 7
(^006) t 100 too Iott O
Example For (^) the mixing problem from the last^ example assume that the brine^ leaves^ the^ tank^ at^ a^ rate
being
the amount of^ salt^ at^ time t and the^ concentration^ at^ time^ t 64min (^) so
X It^ kg o 1000L^ 54min X IO O Kg Solution
Rate
in volume^ dot 64min^ 54min V tt^ C We (^) have Vio^ 7000kg a (^) volume at (^) to 1000 0 C^1000 t 1000
equation to (^) find X
X rate^ in^ rate^ out
1kg L
115 4min t t^1000 T (^) T concentration
out at time^ t It
IW X
of
9
X t^ 5
Integrating
I (^) e Spit^
Sfo d (^) t I e^ stilettos S (^) Edu
en Ctt 1000 5
5 I
x III I ooo^ s Concentration Cit II tT^ Etops of t 11ft kg L A s^ t^ o we (^) have l im^ cuts^ II to so YES to to kg L I kg L
