Of the total power (10 W) being dissipated by the metal plate, how much of the power dissipation is going out the top and bottom of the metal plate? How much of the power dissipation is going out the sides of the metal plate?

If the electric power is suddenly lost to the room described in Problem 5, how long would it take for the room temperature to drop from 23 to 0 °C? Assume that the room (8 m × 8 m × 3 m) is relatively air tight with a heat capacity of 3.84 × 104 (W s)/oC and that the room is located at sea level.

Heatgeneration formula

\( \frac{\mathrm{d}\left({\mathrm{Heat}}_{\mathrm{Los}\backslash \mathrm{from}\kern0.17em \mathrm{Body}}\right)}{\mathrm{d}t}\propto \left[{T}_{\mathrm{Body}}-{T}_{\mathrm{Ambient}}\right] \). However, for convection cooling in general, deviations from this simple linear dependence can be observed.

HeatGeneration superpower

Show that when you bring together two identical blocks, except one has an initial temperature of T1 and the other has an initial temperature of T2, the final equilibrium temperature will be Tequilibrium = (T1 + T2)/2. [Hint show that this equilibrium temperature produces the maximum entropy change for the system.]

Heatgeneration formula inheattransfer

In heat flow analysis, the heat sink fixes the temperature at specified locations and plays a role similar to the use of ground in electrical circuits.

Thermal relaxation after a time t = 5τ is given by: \( \frac{\Delta {T}_0-\Delta T\left(t=5\tau \right)}{\Delta {T}_0}=1-\exp\;\left(\frac{5\tau }{\tau}\right)=0.993=99.3\% \). After t = 10τ, the thermal relaxation is 99.995 % complete.

Often the specific heat is subscripted as cv (specific heat at constant volume) or cp (specific heat at constant pressure). This textbook will use c = cv.

The thermal resistance (from silicon-chip junction to ambient air) for an IC package is θJA = 33 °C/W in still air. In moving air (air flow of 1 m/s) the thermal resistance reduces to 27 °C/W. Suppose that a silicon chip is placed in this IC package and that the chip dissipates a constant power of 2 W.

In Chap. 12 (Sec. 7) a failure mechanism (Hot Carrier Injection) was discussed that can be contrary to this general statement.

For the silicon chip described in Problem 5 (in still air), what is the effective heat capacity of the chip and package if the chip comes into thermal equilibrium approximately 60 s after a constant power is applied? (Assume that thermal equilibrium occurs after approximately 5 thermal time-constants.)

Heatgeneration unit

For the embedded metal plate shown, 10 W of power is being generated in the metal plate and dissipated in equilibrium through the thermal resistances shown with values: θ1 = θ2 = 5 °C/W and θ3 = θ4 = θ5 = θ6 = 40 °C/W.

Heatgeneration muscular system

For the horizontal heat flow configuration shown, where ki is the thermal conductivity and Ai is the cross-sectional area for each material with thickness S,

In general, the internal energy of a system can be changed by the flow of Heat (into or out of system) and/or by Work (done on or by system). The conservation of energy statement (dU = δHeat + δWork) is often referred to as the First Law of Thermodynamics. δHeat and δWork imply that these are not exact differentials. Thus, before we can integrate, we must know details of the processes by which the heat is changed and/or how the work is performed.

Using Eq. (18.3b) and solving for ΔT, you can see that water (due to its relatively large specific heat) is capable of absorbing significant amounts of heat with only relatively small changes in its temperature. Now you can perhaps better understand why water is an excellent cooling agent and widely used—from nuclear reactors and combustion engines to fire fighting.

McPherson, J.W. (2019). Heat Generation and Dissipation. In: Reliability Physics and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-93683-3_18

\( \underset{V}{\int}\overrightarrow{\nabla}\cdot \overrightarrow{J}\;\mathrm{d}V=\underset{A}{\int}\overrightarrow{J}\cdot \mathrm{d}\overrightarrow{A} \), where V is the volume of interest which is bounded by a surface of area A.

Recall that the volume V of interest contains all of the materials in the generation region plus all the materials between the generation region and the heat sink/ambient. A is the area of the surface that bounds the volume V of interest.

Newton’s law for cooling states that the rate of heat loss from a body is proportional to the temperature difference between the body and its surroundings:

What isheatgeneration inheattransfer

The solar constant S = 1.4 kW/m2 represents the average specific power density delivered to the earthFootnote 17 by the sun. Assuming that the distance from the earth to the sun is 1.5 × 1011 m, the radius of the sun is 6.96 × 108 m, and the emissivity of the sun is emis = 1, use the Stefan-Boltzmann equation to calculate the average temperature of the sun.

The adverse impact of temperature on device/material reliability has been emphasized often in this book. The degradation rate for most devices/materials tends to accelerate exponentially with increasing temperature. Therefore, for reliability reasons, lower temperature device operation is usually preferred. However, many devices (both electrical and mechanical) can generate significant amounts of heat as they are being operated. Once device operation begins, the rate of increase in temperature of the device/material will depend upon on the heat generation within the device, the heat capacity of the materials, and the heat dissipation from the device to the heat sink (which is often the ambient). Elevated device temperature during operation (versus the ambient temperature) creates a thermal gradient which serves to drive heat flow from the device. In thermal equilibrium the heat dissipation from the device will just match the heat generation within the device. Managing device heat dissipation may require a significant engineering effort—but the improvements in reliability can be worth the effort.

Since Tsink is assumed to be a fixed temperature, then we have used: \( \frac{\partial \left[T\left(x,t\right)-{T}_{\mathrm{Sink}}\right]}{\partial t}=\frac{\partial \left[T\left(x,t\right)\right]}{\partial t} \).

Heatgeneration in body

The actual power pulse could be more complicated than a simple rectangular shape; however, in Chap. 14 we learned how to convert complicated waveforms into rectangular equivalents.

Using the First and Second Laws of Thermodynamics, show that heat cannot spontaneously flow from a cooler object to a hotter object. [Hint bring together two identical blocks (but at different temperatures) and calculate the entropy change that would occur if heat flows from the cooler to the hotter block. Is the entropy change positive or negative?]

For the horizontal heat flow configuration shown, where ki is the thermal conductivity and Si is the thickness for each material with cross-sectional area A,

The term external energy is usually reserved for any relative motion of the macroscopic system and/or any system energy associated with external fields.

Note that the solar specific power density delivered to earth is equivalent to fourteen one-hundred watt light bulbs per square meter of the earth’s surface and represents a lot of power!

Actually, not all of the heat flow (from metal plate bottom surface area A0) is vertically downward. Part of the heat flow will be from the metal plate spreading laterally. Thus, the true heat flow across the surface area A, which bounds the volume V of interest, will be through an effective area Aeff such that Aeff > A0. However, if S0 is much less than the metal plate dimensions (length and width), then Aeff ≈ A0.

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Heatgeneration examples

On a very cold winter day (outside air temperature of −10 °C), a constant input power of 1 kW is required from an electric heater to keep the room temperature at 23 °C. Assuming that the exterior wall has a large plate glass window such that most of the heat in the room is lost through this exterior wall: (a) what is the effective thermal resistance of the exterior wall; and, (b) if the cost of the electric power is $0.1/(kW h), what is the cost to heat such a room for 1 month (30 days) under this extreme condition?

Systems with large thermal time-constants are often referred to as systems with large thermal inertia. Such systems, with large thermal inertia, have great difficulty responding quickly to short-duration power pulses. This can be good or bad depending on the details of device application.

Quench hardening is a metallurgical process whereby a piece of metal is raised to a high temperature and then suddenly quenched—rapidly lowering the temperature of the piece of metal. Suppose that a 10 kg piece of steel at 900 °C is suddenly dropped in a thermally insulated vat with 100 kg of water at 25 °C. What will be the equilibrium temperature of the steel and water? Assume that the specific heat for the steel is 0.49 kJ/(oC kg) and for the water is 4.19 kJ/(oC kg).