Maximum CPU temperature determination using newtons' Law of cooling! Please help and give detailed information!!

This is for my maths assignment someone please help!!! I'm not experienced in this so any assistance will be appreciated and rewarded. Thanks in advance.


Home Comforts Ltd is a company which makes control systems for home automation. Each system is based on a standard CPU processor chip.

The system works fine when they test it on a laboratory bench. The air temperature in the laboratory is about 20 °C. When they turn the system on, they find that the processor warms up and that after a while it reaches a steady temperature of about 35 °C. Note that the system includes an aluminium heatsink, which is attached to the processor and which helps the heat disperse into the air, and that 35 °C is within the operating range of the processor.

But when the system is delivered, it is installed in plastic case. The air inside the case heats up, so that the processor then overheats. The processor gets damaged and stops working.

The company design department is considering various ways of solving the problem. One suggestion is to fit a small fan to blow air at room temperature into the case (and to make holes in the case to let the hot air out). The sales team complain that this will make the system noisy, but the CPU includes a temperature sensor, so the designers are thinking of only using the fan when the processor temperature exceeds a particular value (the value to be decided by simulating the system).

Your task is to model the system to better understand what is happening and what can be done about it. You need to model (a) the laboratory setup, (b) the existing setup in the plastic case, and (c) the proposed setup with a fan. You will need some additional information: some of this you can find by research (including in data sheets for components), other information (such as the size of the case) will be supplied to your group during the assignment. To explain your recommendation, you need to produce a report and give a presentation to the management of the company.


The processor is an Intel Core i3-4130T. The electrical power it consumes is 35 watts.
The aluminium heatsink has a mass of 23 grams.
The dimensions of the plastic case are 408 × 355 × 197 mm

1. Create a mathematical model of the system, including first-order differential equations. An essential aspect of mathematical modelling is to simplify the problem: for this assignment creating a “compartmental model” and using Newton’s Law of Cooling will lead to suitable differential equations.


1) The mathematical model of the system, including differential equations using appropriate mathematical notation.

2) Graph(s) showing the existing behaviour.

3) Graphs showing the predicted behaviour for relevant values of the coefficients in the equations.

4) An explanation of the model and of the results.

5) A recommendation for solving the problem.
4 answers Last reply
More about maximum cpu temperature determination newtons law cooling give detailed information
  1. Hate mathematics...LOL :D
  2. Ah, do your own homework?
  3. Well if i wouldnt be any easier will it....someone please not good at this as I have tried to understand it but i cant gett any answers or examples through which i can solve this problem..My research has been done but i cant find any please help
  4. hello,
    i am not a cpu expert. For eleminating heat ,[cpu heating ] if possible try
    1.use a set of heat sinks which can evaporate heat into atmosphere.For that ,
    A)calculate heat evaporation from specified heat sink .Calculate multiples if needed.
    2.Thermally connect heat sinks. i.e. heat sink set must be in one tempereture.
    3. set heat sink firmally to plastic case.
    4.use nets for ventillation in plastic case to atmospere [avoiding small creatures and rats etc.
    if this is possible with plastic case dimensions ,fans may [far possibility ...]avoided.

    # used to denote directly proportional to [in physics]

    temperature outside [atmospheric] # tempereture inside plastic case. # tempereture at heatsink set .[or one heat sink as u use...]
    1.temperture inside plastic case willbe slightly more than outside temperture.
    2.temperture of heat sink will be more than temperture inside plastic case .

    use NEWTON'S LAW OF COOLING for heat transfer to temperature determination inside plastic case..

    @ means - almost directly proportional [strong relation ]

    HEAT IN HEATSINK willbe # current inside CPU. # temperature at heatsink. @ wattage of processor.
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