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.
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More about maximum cpu temperature determination newtons law cooling give detailed information
  1. for cpu cooling purposes water cooling methods are their.

    1. Take processor heat generation at your specific degree Celsius
    This heat to be given out to surroundings.

    go --------

    read that.following is part of it. That page also contains various other details.

    A heat sink transfers thermal energy from a higher temperature device to a lower temperature fluid medium. The fluid medium is frequently air, but can also be water, refrigerants or oil. If the fluid medium is water, the heat sink is frequently called a cold plate. In thermodynamics a heat sink is a heat reservoir that can absorb an arbitrary amount of heat without significantly changing temperature. Practical heat sinks for electronic devices must have a temperature higher than the surroundings to transfer heat by convection, radiation, and conduction.
    To understand the principle of a heat sink, consider Fourier's law of heat conduction. Joseph Fourier was a French mathematician who made important contributions to the analytical treatment of heat conduction.[2] Fourier's law of heat conduction, simplified to a one-dimensional form in the x-direction, shows that when there is a temperature gradient in a body, heat will be transferred from the higher temperature region to the lower temperature region. The rate at which heat is transferred by conduction, q_k, is proportional to the product of the temperature gradient and the cross-sectional area through which heat is transferred.
    q_k = -k A \frac{dT}{dx}

    Figure 2: Sketch of a heat sink in a duct used to calculate the governing equations from conservation of energy and Newton’s law of cooling.
    Consider a heat sink in a duct, where air flows through the duct, as shown in Figure 2. It is assumed that the heat sink base is higher in temperature than the air. Applying the conservation of energy, for steady-state conditions, and Newton’s law of cooling to the temperature nodes shown in Figure 2 gives the following set of equations.
    \dot{Q} = \dot{m}c_{p,in}(T_{air,out} - T_{air,in}) (1)
    \dot{Q} = \frac{T_{hs} - T_{air,av}}{R_{hs}} (2)
    T_{air,av} = \frac{T_{air,in} + T_{air,out}}{2} (3)
    Using the mean air temperature is an assumption that is valid for relatively short heat sinks. When compact heat exchangers are calculated, the logarithmic mean air temperature is used. \dot{m} is the air mass flow rate in kg/s.
    The above equations show that
    When the air flow through the heat sink decreases, this results in an increase in the average air temperature. This in turn increases the heat sink base temperature. And additionally, the thermal resistance of the heat sink will also increase. The net result is a higher heat sink base temperature.
    The increase in heat sink thermal resistance with decrease in flow rate will be shown in later in this article.
    The inlet air temperature relates strongly with the heat sink base temperature. For example, if there is recirculation of air in a product, the inlet air temperature is not the ambient air temperature. The inlet air temperature of the heat sink is therefore higher, which also results in a higher heat sink base temperature.
    If there is no air flow around the heat sink, energy cannot be transferred.
    A heat sink is not a device with the "magical ability to absorb heat like a sponge and send it off to a parallel universe".[3]
    Natural convection requires free flow of air over the heat sink. If fins are not aligned vertically, or if fins are too close together to allow sufficient air flow between them, the efficiency of the heat sink will decline.


    After step 1, modelling first order differential's not a big problem

    3. Allow proper ventilation in plastic case.

    if u need further assistance email :
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