Differential Equations - Mixture in an overflowing tank


by proctortom
Tags: application problems, concentration, differential eqn
proctortom
proctortom is offline
#1
Sep28-12, 01:19 AM
P: 12
HOW IS THE FOLLOWING QUESTION DONE?


Milk chocolate is being produced in a 300 litre tank, which initially contains 100 litres of milk. The following things then occur simultaneously:

1. Liquid cocoa (made up of equal parts cocoa solids and cocoa butter, both in liquid form) is added at a rate of 6 litres per minute.

2. Milk is added at a rate of 3 litres per minute.  The well-stirred mixture leaves the tank via a tube, at a rate of 6 litres per minute.


Let x(t) be the amount of cocoa solids in the mixture


The differential equation is dx/dt + 6x / (100 + 3t) = 3
......Therefore......
x(t) = (9t^3 + 900t^2 + 30,000t) / (3t + 100)^2


After the tank is full, the process continues as above. However, in addition to the well-stirred mixture leaving via the tube, it also flows over the edges of the tank and is collected by overflow tubing which takes it to the cooling process. Let y(t) be the number of litres of cocoa solids present in the tank t minutes after it is full.

Find the differential equation satisfied by y(t)
Phys.Org News Partner Science news on Phys.org
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered
haruspex
haruspex is offline
#2
Sep28-12, 10:12 AM
Homework
Sci Advisor
HW Helper
Thanks ∞
P: 9,151
Not sure about the = 3 in your first eqn. Shouldn't it be 6?
The soln doesn't look quite right to me. Did you check it satisfied the differential eqn?
Please show your attempt at the last part.
sad_song009
sad_song009 is offline
#3
Sep29-12, 09:47 AM
P: 2
It is 3 because proctortom is asking for cocoa solids. dx/dt = rate of cocoa solids in - rate of cocoa solids out. Which is dx/dt = 3 [Liquid cocoa (made up of equal parts cocoa solids and cocoa butter, both in liquid form) is added at a rate of 6 litres per minute] - (x/100+3t)*6 = dx/dt + 6x/100+3t = 3

I also get stuck in finding the differential equation satisfied by y(t). The eqn should be dy/dt (3/100)*(100-y). But I don't know how to get to this !

nishcha7
nishcha7 is offline
#4
Oct1-12, 09:56 PM
P: 2

Differential Equations - Mixture in an overflowing tank


dy/dt = rate in - rate out

rate in = 3L/m (cocoa solids going into mixture)

rate out = concentration x flow rate out
= amount/volume x flow rate out

where the flow rate out must equal the flow rate in as the tank is remaining full! (6L/m via tube and 3L/m via overflow tubing)

= y/300 * 9
= 3y/100

so dy/dt = 3 - 3y/100
= 3/100 (100 - y)


Register to reply

Related Discussions
Mixing Tank, Differential Equations Problem Calculus & Beyond Homework 11
United States Elementary Differential Equations - 1st Order Differential Equations Calculus & Beyond Homework 1
Differential equations capacity tank problem (chemical solutions) mixture Calculus & Beyond Homework 1
Differential Equations Tank problem Calculus & Beyond Homework 7
Mixture Problem (differential equations) Introductory Physics Homework 9