The more frequent the stew is stirred, more particles will be in contact with the inner wall and hence more particles will be cooled down. Figure 1: A pot of stew in a sink full of ice If Eddie did not stir the stew occasionally, it will take a longer time for the stew to cool down. This is because less liquid particles would be in contact with the inner wall of the container adjacent to the ice water. Therefore, less liquid particles would be cooled down, and hence it would take a longer time for the whole liquid particles to cool down.
However, it was only stirred occasionally, and not all the time. This is because if the stew is stirred too frequently, the stews temperature will drop too quickly and once it reaches the customer, it would be lower than room temperature, hence not able to meet the customers request. Furthermore, it is important that the rate of stirring is constant throughout in order to make sure that the stew will be cooled down in a constant speed the whole time.
Besides the assumption stated in Q above, list two other assumptions that need to be made in order for your answer in Q to be valid. (5 marks) 1. The value of k is always positive. . The rate at which the stew is stirred is the same throughout.
Q: Write the final mathematical model that you have verified in Q. Compare and discuss your model with the mathematical model that can be obtained by using a software tool such as Excel that is able to give a reasonably good approximating function with its graph fitting capability.
Refer to Appendix C . (5 marks) Final mathematical model verified: Mathematical model obtained from Excel: Comparing the two equations, The value of was previously calculated in CNN. Given that the stews temperature drops from ICC to ICC in 12 minutes, Calculated On the other hand, using Excel, from the equation given above, Therefore, if we calculate the percentage discrepancy, The trending option chosen for this particular graph is linear. Since the percentage discrepancy is only 2. 1%, this proves that Excel is able to give a reasonably good approximating function with its graph-fitting capability when the linear option is chosen for the graph. (Explore and Further Insights)
Q: Read about Newton’s Law of Cooling and indicate how it can be used to solve this problem. Please cite your reference/source. (5 marks) 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).
Given that the rate of change of the temperature, is (by Newton’s Law of Cooling) proportional to the difference between the temperature of the soup T(t) and the ambient temperature Ta. This means that is directly proportional to 0 However, since the soup is cooling down, this means that should be negative. This means that the equation should be . If the derivative of T(t) is taken, and use Newton’s law of cooling concept, we arrive at the following equation: Therefore, Using differentiation techniques, Since: is the temperature difference between the stew and ice water at .
Imperative difference at , is the initial In order to solve the problem using this equation, we need to assume that the temperature of the ice water is constant, hence, as an example, let be ICC. Hence, Percentage discrepancy: The reason for the huge percentage of discrepancy is that for the calculations regarding Newton’s law of cooling, we assumed that the temperature of the ice water is constant at ICC, while in the calculation using the exponential decay ormolu, there were no assumptions or values made or given.