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 temperature of its surroundings.
Thus if T is the temperature of an object at time t, we have:
Where is the temperature of the surroundings.
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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.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).
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Example 1: The Big Pot of Soup As part of his summer job at a resturant, 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 accomodate 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 tht 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 ? Solution: Let us summarize the information briefly and define notation for this problem. 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: where is a positive constant. Note that if we take a derivative of , and use the Newton's law of cooling, we arrive at (We have used the fact that is constant to eliminate its derivative, and we plugged in for in the last step.) What a nice surprize ! By defining this new variable, we have arrived once more at the familiar equation whose solution is well known to us, namely: We can use this result to conclude (by plugging in and ) that It follows that We found the solution in general form, but it looks quite complicated. Let's try to understand this expression and its predictions in the case of the problem described above. so that, We also know that after 10 minutes, the soup cools to 60 degrees, so that . Plugging into the last equation, we find that Rearranging, (The steps are much the same as in our previous work in the example on radioactive decay. In the last step we took a reciprocal of both sides of the equation. This just makes all the quantities come out to be positive in the next step, so it is done for convenience, though it is not an essential step). We have found that Taking the natural logarithm of both sides, and solving for , we find that Thus, So we see that the constant which governs the rate of cooling is per minute. Now we can specify the solution fully, since all constants have been determined from the information in the problem. The prediction is that the temperature of the pot of soup at time t will be This equation can be solved for in much the same way as before. Subtracting 5 from both sides and dividing by 95 we get: Taking logarithms of both sides, we find that Thus, using the fact that we have Thus, it will take a little over half an hour for Jim's soup to cool off enough to be put into the refrigerator.
Let = Temperature of the soup at time t (in min). = Initial Temperature of the soup =100 deg. = Ambient temperature (temp of water in sink) = 5 deg .
Given: The rate of change of the temperature , is (by Newton's Law of Cooling) proportional to the difference between the temperature of the soup and the ambient temperature This means that:
This equation is another example of a differential equation. The independent variable is for time, the function we want to find is , and the quantities are constants. In fact, from Jim's measurements, we know that , but we still don't know what value to put in for the constant . We will discuss this further below. For your consideration:
Back to the same old equation
The equation we arrived at above looks different from the ones we have just investigated, but as we shall soon see, the difference is rather superficial. Indeed, by defining a new variable, we will show that the equation is really completely related to the exponential decay seen previously. To see this, define = Temperature difference between soup and water in sink at time t. = Initial temperature difference at time t=0
How the soup will cool
From the information in the problem, we know that
The behaviour of this solution is shown in the diagram.
For your consideration:
Solving Jim's Soup problem
To finish our work, let us determine how long it takes for the soup to be cool enough to put into the refrigerator. We need to wait until , so at that time:
Summary:
We have just seen yet another example of a simple differential equation and how it can be used to make predictions. To summarize what we found, here is the connection between the differential equation of Newton's Law of Cooling and its solution:Newton's Law of Cooling Solution
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When hot bodies are left in the open they are found to cool gradually. Newton found that the rate of cooling was proportional to the excess of temperature of the body over that of the surroundings. This observation is what is called Newton’s law of Cooling. It is not known if Newton attempted any theoretical explanation of this phenomenon. But it is unlikely because the concepts about heat were not clear in those times. But what is important is that the original statement of the conditions for the validity of Newton’s law included the presence of a draught
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In the course of time Stefan’s law of
radiation was discovered and it explained how
a hot body loses heat by radiation. According
to this law every body radiates energy at a rate
proportional to the fourth power of its absolute
temperature T. Since the body absorbs energy
radiated from the surroundings which is
proportional to the fourth power of the absolute
temperature T0 of the environment and the
constant of proportionality is the same, the net
loss of heat by the body from radiation is
proportional to the difference (T4 –T0)
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