Example of mixing problems, Mathematics

Assignment Help:

A 1500 gallon tank primarily holds 600 gallons of water along with 5 lbs of salt dissolved into it. Water enters the tank at a rate of 9 gal/hr and the water entering the tank has a salt concentration of 1/5 (1 + cos (t)) lbs/gal. If a well mixed solution goes away the tank at a rate of 6 gal/hr, how much salt is in the tank while it overflows?

Solution

Firstly, let's address the "well mixed solution" bit. It is the assumption that was mentioned earlier. We are going to suppose that the instant the water enters the tank this somehow immediately disperses evenly throughout the tank to provide a uniform concentration of salt into the tank at every point.  Again, it will evidently not be the case in actuality, but it will permit us to do the problem.

Now, to set up the Initial Value Problem that we'll require to solve to get Q(t) we'll require the flow rate of the water entering as we've got that the concentration of the salt into the water entering when we've got that, the flow rate of the water leaving and the concentration of the salt into the water exiting but we don't have this yet.

Thus, we first require determining the concentration of the salt in the water exiting the tank. As we are assuming a uniform concentration of salt in the tank the concentration at some point into the tank and thus in the water exiting is specified by,

Concentration = Amount of salt in the tank at any time, t/Volume of water in the tank at any time, t

 The amount at any time t is simple it's just Q(t). The volume is also pretty simple. We begin with 600 gallons and each hour 9 gallons enters and 6 gallons leave. Thus, if we use t in hours, each hour 3 gallons enters the tank, or at any time t there as 600 + 3t gallons of water into the tank.

Thus the Initial Value Problem for this condition is:

Q'(t) = 9 ((1/5)(1 + cos(t))) - 6 (Q(t)/(600 + 3t)),                   Q(0) = 5

Q'(t) = 9/5 ( 1 + cos (t)) - (2Q(t))/(200 + t),                           Q(0) = 5

It is a linear differential equation and this isn't too hard to solve hopefully. We will demonstrate most of the details, although leave the explanation of the solution process out.  If you require a refresher on solving linear first order differential equations go back and see that section.

Q'(t) + ((2Q(t))/(200 + t)) = 9/5(1 + cos(t))

µ(t) =  e∫(2/(200 + t)) dt = e2In(200 + t)) =(200 + t)2

∫((200 + t)2 Q(t))' dt = ∫(9/5(200+ t)2 (1 + cos(t))dt

 (200 + t)2 Q(t) = 9/5((1/3 (200 + t)3) + ((200 + t)2 sin(t)) + (2 (200 + t) cos(t)) - (2 sin(t))) + c

Q(t) = 9/5((1/3 (200 + t)) + sin(t) + ((2cos (t))/(200 + t)) - ((2sin(t))/(200 + t)2)) +(c/(200 + t)2)

Thus, here's the general solution. Here, apply the initial condition to find the value of the constant, c.

5 = Q(0) =  9/5((1/3 (200) + (2/200)) + c/(200)2

C= - 4600720

Hence, the amount of salt into the tank at any time t as:

Q(t) = 9/5((1/3 (200 + t)) + sin(t) + ((2cos (t))/(200 + t)) - ((2sin(t))/(200 + t)2))-(4600720/(200 + t)2)

Now, the tank will overflow at t = 300 hrs. The amount of salt in the tank at that time is.

Q (300) = 279.797 lbs

There is a graph of the salt into the tank before it overflows.

1351_Example of Mixing Problems.png

Remember that the complete graph must have small oscillations in it as you can notice in the range from 200 to 250. The scale of the oscillations though was small adequate that the program used to produce the image had trouble demonstrating all of them.

The work was a little messy along with that one, but they will frequently be that way so don't get excited regarding it. This first illustration also assumed that nothing would change during the life of the process. That, of course will generally not be the case.


Related Discussions:- Example of mixing problems

Eigenvalues and eigenvectors, Review: Systems of Equations - The tradition...

Review: Systems of Equations - The traditional initial point for a linear algebra class. We will utilize linear algebra techniques to solve a system of equations. Review: Matr

Linear independence and dependence, It is not the first time that we've loo...

It is not the first time that we've looked this topic. We also considered linear independence and linear dependence back while we were looking at second order differential equation

Differential equation, Find the normalized differential equation which has ...

Find the normalized differential equation which has {x, xex} as its fundamental set

How much money did carlie have after she had paid her friend, Carlie receiv...

Carlie received x dollars every hour she spent babysitting. She babysat a total of h hours. She then gave half of the money to a friend who had stopped through to help her. How muc

Equation, how do you slove 4u-5=2u-13

how do you slove 4u-5=2u-13

Expected value, Expected Value For taking decisions under conditions of...

Expected Value For taking decisions under conditions of uncertainty, the concept of expected value of a random variable is used. The expected value is the mean of a probability

What are the average total repair costs per month, An automobile manufactur...

An automobile manufacturer needs to build a data warehouse to store and analyze data about repairs of vehicles. Among other information, the date of repair, properties of the vehic

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd