Quesition below

Question

A coolant is injected into engine fluid, causing the temperature of the engine fluid to decrease with respect to time at a rate that is proportional to the difference of the fluid's instantaneous temperature, T, and the fluid's original temperature, T_o. Select the differential equation that represents the relationship.

Answer 1

Given:

The engine fluid temperature rate of change, dT/dt

- dT/dt ∝ ( T - T₀ ) [...proportional to the difference of the fluid's instantaneous temperature, T, and the fluid's original temperature, T₀.]

Find:

The Differential Equation that represents the relationship.

Solution:

The framework is already set, there is only one thing that changes the proportionality to an equation. And that is a proportionality constant.

Let's call it 'k'

Since this is a directly proportional set up, the constant 'k' will be multiplied to the term of ( T - T₀ )

- dT/dt = k * ( T - T₀ )

And to make this look like a typical equation, multiply both sides by a -1 to get:

∴ dT/dt = - k * ( T - T₀ )

As expected, this comes to represent the same thermodynamic effect you'd see in Newton's Law of Cooling; it is expected because most of all fluids exhibit this cooling effect when interacting with other fluids of differing temperatures.

I hope this helps! Message me in the comments if you have any questions, comments, or concerns!