Temperature with internal heat source

Assume that the temperature distribution in a circular rod with internal heat source q may be represented by the following ordinary differential equation (ODE)... 

We speak of steady-state heat conduction when the temperature at every point in a thermally conductive body does not change with time. Some simple cases, which are of practical importance, have already been discussed in the introductory chapter, namely one dimensional heat flow in flat and curved walls, cf. section 1.1.2. In the following sections we will extend these considerations to geometric one-dimensional temperature distributions with internal heat sources. Thereafter we will discuss the temperature profiles and heat release of fins and we will also determine the fin efficiency first introduced in section 1.2.3. We will also investigate two- and three-dimensional temperature fields, which demand more complex mathematical methods in order to solve them, so that we are often compelled to make use of numerical methods, which will be introduced in section 2.4.6. 

Practical tests may be used to determine the internal and external temperatures of the package under normal conditions by simulating the heat source due to radioactive decay of the contents with electrical heaters. In this way, the heat source can be controlled and measured. Such tests should be performed in a uniform and steady thermal environment (i.e. fairly constant ambient tanperature, stiU air and minimum heat input from external sources such as sunlight). The package with its heat source should be held under test for sufficient time to allow the temperatures of interest to reach steady state. The test ambient temperature and internal heat source should be measured and used to adjust linearly all measured package temperatures to those corresponding to a 38 C ambient temperature. 

The associated temperature rise can be approximated by a onedimensional thermal analysis, assuming an internal heat source with a strength of q. This analysis assumes constant heat rates, which implies constant viscoelastic properties. 

The temperature at the aerosol layer in Neptune s atmosphere is about -346°F (-210°C), which is close to the temperature at the main cloud level in Uranus s atmosphere, and the effective temperatures of the atmospheres of both Uranus and Neptune were found to be close to this temperature. One would expect Neptune s visible troposphere and lower stratosphere to be about 59°F (15°C) colder than those of Uranus because of Neptune s greater distance form the Sun (30.1 a.u. vs. 19.2 a.u.) instead, the temperatures of these parts of the atmospheres of both planets are found to be about the same. Neptune s atmosphere seems to be considerably warmer than it would be if it received all or nearly all its heating from sunlight, as seems to be the case for Uranus. This is another indication that Neptune has a powerful internal heat source, unlike Uranus, which has at most a weak internal heat source (compatible with radioactivity in its interior) or none at all. Voyager 2 infrared observations confirmed this the emission to insolation ratio was found to be 2.6 from them instead of... 

Derive the differential equation for the temperature field = (r, t), that appears in a cylinder in transient, geometric one-dimensional heat conduction in the radial direction. Start with the energy balance for a hollow cylinder of internal radius r and thickness Ar and execute this to the limit Ar — 0. The material properties A and c depend on internal heat sources are not present. 

The effects of viscous dissipation on the thermal entrance problem with the uniform wall heat flux boundary condition can be found in Brinkman [27], Tyagi [6], Ou and Cheng [28], and Basu and Roy [29]. Other effects, such as inlet temperature, internal heat source, and wall heat flux variation, are reviewed by Shah and London [1] in detail. 

All of the velocity terms in the Energy Equation will disappear for a static system. If there is no variation of temperature with time and no internal heat sources, equations (5-5) or (5-6) or (5-7) will be applicable. 

Whereas the performance of a high-gain control loop is nearly independent of gain and the power regulator is merely an element of the temperature-control loop, the exceptionally wide range of conditions encountered in cryostats and calorimetry often presents problems of poor effective control, especially when the spontaneous cooling diminishes toward zero. Dependence of controlled temperature upon amplifier drifts, for the described system in the absence of internal heat sources, follows readily with defined symbols. Power regulator output emf and cryostat heat flow are [11], respectively,... 

The terrestrial planets are almost in energy balance, that is, thermal emission nearly equals absorbed solar power. On Earth only a small internal heat source exists, which manifests itself by a vertical temperature gradient in the outer layers of the crust. Early measurements, mostly from a few deep mines and bore holes, indicated a temperature increase with depth of 10-40 K km With reasonable assumptions on the thermal conductivity of rocks this corresponds to an internal heat source of approximately 2.6 x 10 W (Bullard, 1954). More recent estimates, including data from deep sea drillings, yield a slightly higher value of 4.3 0.6 x 10 W (Williams von Herzen, 1974). In contrast, solar radiation absorbed by the Earth amounts to approximately 1.2 x 10 W. The internal heat flux is, therefore, only 3.5 X lO"" of the absorbed solar radiation and, consequently, the energy balance of the Earth is approximately 1.00035. 

In Chapter 8 we associate measured quantities with the underlying physical processes. The connection between thermal equilibrium and the vertical temperature profile is investigated. Dynamical regimes are explored with emphasis on wind fields and circulations. Certain aspects of Solar System composition, internal heat sources, and the concept of global energy balance are discussed in the context of planetary evolution. 

The temperature field can be calculated on the basis of the general differential equation of thermal conduction. The general differential equation of thermal conduction describes the non-steady-state temperature field, with the absorption of the laser radiation being taken into account as an internal heat source d> ... 

In an ICES that serves both small and large buildings, the surplus internal heat from the large buildings can be used to provide source heat to smaller ones. An ICES in areas with moderate winter temperatures may use air as a heat source. Systems that use lakes or reservoirs rely on the natural collection of heat by these water sources throughout the year. 

As indicated by Fig. 23 and Fig. 24, the source function can be highly asymmetrical. For the liquid droplet corresponding to Fig. 23, one would expect the internal temperature to be higher near the back and front of the sphere because of the spikes in the source function in those regions. As a result, the evaporation rate should be enhanced at the rear stagnation point and the front of the sphere. To calculate the evaporation rate when internal heating occurs, one must solve the full problem of conduction within the sphere coupled with convective heat and mass transport in the surrounding gas. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

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