Newton's Law of Cooling and Fuzzy Method for Decision Making

Following to article of “Fuzzy Method for Decision Making:  A Case of Asset Pricing Model”, the purpose of this article is more practice of this method by presenting a case in physics which is about Newton’s law of cooling. At the first, a case has been explained then reviewing Newton’s law of cooling and finally the application of method mentioned in above article to analyze the case is demonstrated.  Of course, there are many cases to analyze by this method such as the circuit electric of R – L in which we have:  i = V/ R [1- (e^ (-Rt/L))] in the field of electrical engineering, Benjamin Franklin (1706-1790)’s devise about composite interest rate in finance, Compressor Performance Control to track the distance between the operating point and set point, Head Generated by the Circulation Pumps to find out the operating point which is the intersection of pump head and circuit losses in mechanical engineering and so on.

I have chosen Newton’s law of cooling because it is the general case for all engineers.

Case Study: Predictions in temperature transferring 

Mr. X is willing to decrease the temperature of a pot filled of water boiled at 100 degrees C. He puts this pot into a sink full of water at 5 degrees C in which water temperature into sink will be stayed the constant at 5 degrees C during the period of the time. He would like to predict the temperature of water into pot after one hour. How?

We as well as know that there are three mechanisms to transfer the temperature which are Conduction, Convection and Radiation. This case is referred to only Conduction.

Definitely he should use Newton’s law of cooling.

Newton’s law of cooling

In early 1701, Isaac Newton observed that the rate of heat loss a hot area during the period of the time has directly the relationship with the difference temperature between this area and adjacent area where the temperature of adjacent area (surrounding) will be the constant during the period of the time. Where we have:

T0 = the temperature of hot area in time zero

Ta = the temperature of adjacent area (the constant)

T (t) = the temperature of hot area at time “t”

T (t) > Ta, T = T (t)

We can replace above proportional relationship to a differential equation, if we include a heat transfer coefficient (k) which is positive. Since we have: (T – Ta) > 0, the trend of the curve will fall down. Therefore, we should consider a negative mark for this differential equation as follows:

Solution of differential equation:

We have:

In the reference with data included in the case study, the rate of heat loss will be as follows:

As we can see, Mr. “X” will be able to predict the temperature of water into pot after one hour, if he finds out a constant amount for “k”.

One the best ways to obtain “k” is to utilize Fuzzy set theory.

Fuzzy Method for Decision Making

Here, I am willing to apply fuzzy method mentioned in my previous article to calculate “k” step by step as follows:

Ø  On the spreadsheet of excel, we enter a algorithm to calculate T (t)

Ø  We use of the formula of µ = T (t) / T0 to change rate of temperature loss to Fuzzy membership function assigned to “t”

Ø  If we have the heat loss by falling down the trend ((T – Ta) > 0), it can be also considered vice versa. It is means, if the temperature of pot water is 5 degrees C and the temperature of sink water is 100 degrees C (the constant), we will have the increase heat into pot where the trend of the curve will go up ((T – Ta) < 0). Therefore, the differential equation will be the same and we can do above two steps for this situation (Please see below algorithms on my spreadsheet of excel).

Ø  The most important step is to find out the domain for rectangular k – t. As the first try, I start by method of try and error in which I reach a constant “µ” or where the rate of changes for “µ” will be negligible.

Ø  By Using of “µ” discovered from previous try, I utilize PLUG order in excel to limit “µ” to zero in which it is named α – cut.

Ø  As the second try, I use from the table of two way sensitivity analysis for rectangular k – t where the domain has been extracted from previous step.

(Please see Appendix I)

If we review Appendix I, we will see that the range of “k” is between 0.0106 and 0.1 (per min) while the range of “t” is between 1 (min) and 45 (min)

Ø  As the final try, I repeat to make the table of two way sensitivity analysis for rectangular k – t in accordance with above ranges. (Please see Appendix II and III)

Appendix II is for status of warming to cooling and Appendix III is for status of cooling to warming.

The final result has been presented in below diagram:

As we can see, the most density has been included into triangular ABC. Therefore, gravity centre of this triangular (ABC) is the answer for “k” where the coordination of gravity centre is:

t = 13.5 (min)

α – cut = 0.54

According to Appendix II or III, we can see that the answer for “k” is approximately 0.055 per min. Thus we have:

 

In the result, the temperature of pot water will be about 8.5 degrees C after one hour.

Now, you can do an experimental test at your home to prove this method.

Perhaps, we say that the solution of this case is very easy because Mr. X can measure the temperature of pot water after next 5 minute then he can solve above function to get “k”. Yes, it is true but we should consider that there are some adjacent areas into the space or even into the earth with different temperatures where we have not access to measure the temperatures during the period of the time.

 

Note:  “All spreadsheets and calculation notes are available. The people, who are interested in having my spreadsheets of this method as a template for further practice, do not hesitate to ask me by sending an email to: soleimani_gh@hotmail.com or call me on my cellphone: +989109250225. Please be informed these spreadsheets are not free of charge.”

Appendix I

 

Appendix II

Appendix III