Differential Equations - Mixture in an overflowing tank

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Discussion Overview

The discussion revolves around a differential equations problem involving a mixture in an overflowing tank, specifically focusing on the dynamics of cocoa solids in a milk chocolate production process. Participants explore the formulation of differential equations related to the rates of addition and removal of materials in the tank.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Post 1 presents a differential equation for the amount of cocoa solids, suggesting that the equation is dx/dt + 6x / (100 + 3t) = 3, with a proposed solution for x(t).
  • Post 2 questions the validity of the equation, suggesting that the right-hand side should be 6 instead of 3, and challenges the correctness of the proposed solution.
  • Post 3 argues that the equation is correct as it pertains to cocoa solids, explaining the rates of cocoa solids in and out, and expresses uncertainty about deriving the differential equation for y(t).
  • Post 4 provides a formulation for dy/dt based on the rates of cocoa solids entering and leaving the tank, concluding with dy/dt = 3/100 (100 - y).

Areas of Agreement / Disagreement

Participants express disagreement regarding the formulation of the initial differential equation, particularly the right-hand side value. There is also uncertainty about the derivation of the differential equation for y(t), with differing approaches presented.

Contextual Notes

Participants have not reached a consensus on the correct formulation of the initial differential equation or the subsequent equation for y(t). The discussion includes various assumptions about the rates of flow and concentration that have not been fully resolved.

proctortom
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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)
 
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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.
 
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 !
 
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)
 

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