Newton's law of cooling

Newton’s law of cooling states that “the rate of change of the temperature of an object is proportional to the difference in temperatures between the body and its surroundings.”

The law is given as the differential equation:

Where, Q = Thermal energy in joules
 h = Convection Heat Transfer Coefficient
A = Surface area of the heat being transferred
T = Temperature of the object's surface and interior (since these are the same in this approximation)
T_env = Temperature of the environment
ΔT(t) = T(t) − T_env is the time-dependent thermal gradient between environment and object

Example: ("coffee cooling problem”)

Suppose, You are having lunch at a restaurant. You place your order, and the waitress brings you your coffee much earlier than the rest of your meal. You want the coffee to stay warm until your meal arrives so you can have them at the same time. You always add cream to your coffee, but know that from Newton’s Law of Cooling equation that a hot object transfers heat to its surroundings at a rate proportional to the difference in temperature between the two. So your choice is to either add the cream to your coffee now, or add the cream to your coffee once your meal arrives. You think about the problem for a moment and come to a conclusion.

If you add the cream right away the temperature difference between the coffee and its surrounding air is brought closer together than between just the hot coffee without cream and restaurant air. A hot object cools at a rate that is faster when the difference between the temperatures of liquid and the surrounding air and cup is the greatest. Adding cool cream at the beginning slows down the cooling speed because it decreases the difference in temperature between the hot coffee and its surroundings. If you did not add cream right away the difference in temperatures of the hot coffee and restaurant air and cup is the greatest, so it would cool more rapidly and then when the cream would be added, it would cool even further. You add your cream to your coffee as soon you got it,and enjoy a nice hot cup of coffee when your meal arrives all thanks to Newton’s Law of Cooling to help you out.

Newton's Law of Cooling

Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings).

Newton's Law makes a statement about an instantaneous rate of change of the temperature. We will see that when we translate this verbal statement into a differential equation, we arrive at a differential equation. The solution to this equation will then be a function that tracks the complete record of the temperature over time. Newton's Law would enable us to solve the following problem.

Example 1: The Big Pot of Soup As part of his summer job at a restaurant, Jim learned to cook up a big pot of soup late at night, just before closing time, so that there would be plenty of soup to feed customers the next day. He also found out that, while refrigeration was essential to preserve the soup overnight, the soup was too hot to be put directly into the fridge when it was ready. (The soup had just boiled at 100 degrees C, and the fridge was not powerful enough to accommodate a big pot of soup if it was any warmer than 20 degrees C). Jim discovered that by cooling the pot in a sink full of cold water, (kept running, so that its temperature was roughly constant at 5 degrees C) and stirring occasionally, he could bring that temperature of the soup to 60 degrees C in ten minutes. How long before closing time should the soup be ready so that Jim could put it in the fridge and leave on time

Here a bit of care is needed: Clearly if the soup is hotter than the water in the sink  , then the soup is cooling down which means that the derivative  should be negative. (Remember the connection between a decreasing function and the sign of the derivative ?). This means that the equation we need has to have the following sign pattern: