Newton's Law Of Cooling Equation Newton's Law Of Cooling Equation 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 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. Know More About Rational Numbers Properties
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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. Let T(t) = Temperature of the soup at time t (in min). T(0)=Tv0 = Initial Temperature of the soup =100 deg. Tva = 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: 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. Read More About Implicit Differentiation XY
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(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. 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. This solution assumes that Kb repeats itself (ie high bond rupture have an equal likelihood of occuring at any time in the time history), because it should be obvious from the first equation that high stresses will cause more damage (bigger increase in n) if they occur at the start than at the end of the polymers life. I'd like to improve the code so that it can be used to analyse time histories that do not repeat themselves. Kb cannot be described mathmatically (it is a piecewise linear time history), and I cannot solve the equation by time stepping the first equation as the time penalty isn't acceptable (I'm currently using MatLab so if it's possible to carry on using vectorised operations I'd like to, although I do have a simple time stepping equations for checking my answers).
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Thank You
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