Newton's law of cooling with multiple containers DE's

In summary, we need to make a substitution by solving for the derivative of (y - x) and substituting it into one of the equations to reduce the system to two variables, x and y. Then we can solve for the temperature of the metal bar at time t.
  • #1
asantas93
3
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I've been trying to figure out how to solve the following problem:

"A metal bar is placed in a container (call it container A) which is inside of a much larger container (call it container B), whose temperature can be assumed to be constant. Find the function for the temperature of the metal bar at time t."

First, I represent the temperature of container A as x(t) and the metal bar as y(t). The constant temperature of container B is TB. I assume that the temperature of the bar is proportional to the temperature of container A but the temperature of container A is proportional to both TB and the temperature of the bar. Then

dx/dt = k1(TB - x) + k2(y - x) and
dy/dt = k3(x - y)

I was told that I need to make a substitution to solve this, but I can't seem to take this DE down to two variables. Any help would be appreciated.
 
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  • #2
Hint: Try solving for the derivative of (y - x) and substituting it into one of the equations. Substituting dy/dt = k3(x - y) into dx/dt = k1(TB - x) + k2(y - x) gives us: dx/dt = k1(TB - x) + k2(dy/dt - k3(x - y))Rearranging this and solving for dy/dt yields: dy/dt = (k1/k2)*(TB - x) + (k3/k2)*(y - x)Now we have two equations with two unknown variables, x and y. We can now proceed to solve the system of differential equations.
 

FAQ: Newton's law of cooling with multiple containers DE's

1. What is Newton's law of cooling with multiple containers DE's?

Newton's law of cooling with multiple containers DE's is a mathematical equation that describes the rate at which the temperature of an object decreases over time when placed in an environment with a different temperature. It takes into account the number of containers and their respective temperatures.

2. How is Newton's law of cooling with multiple containers DE's used in science?

This law is used in many fields of science, such as physics, chemistry, and engineering, to model and predict the cooling behavior of objects in different environments. It is particularly useful in studying heat transfer and thermodynamics.

3. What are the variables in the equation for Newton's law of cooling with multiple containers DE's?

The variables in the equation are the initial temperatures of the containers, the ambient temperature, the number of containers, and a constant value that represents the rate of cooling for each container.

4. How accurate is Newton's law of cooling with multiple containers DE's?

While Newton's law of cooling with multiple containers DE's provides a good approximation of the cooling behavior, it is not always accurate in real-world scenarios. Factors such as air flow, humidity, and the shape of the containers can affect the accuracy of the equation.

5. Can Newton's law of cooling with multiple containers DE's be applied to other situations besides cooling?

While this law is commonly used to describe cooling, it can also be applied to situations where an object's temperature is changing due to other factors, such as heating or chemical reactions. However, the equation may need to be modified to account for these different scenarios.

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