Fourier s law of heat transfer

From the extensive experimental and model development work performed at CSM (during a period of over 15 years), it has been demonstrated that a heat transfer controlled model is able to most accurately predict dissociation times (comparing to laboratory experiments) without any adjustable parameters. The current model (CSMPlug see Appendix B for details and examples) is based on Fourier s Law of heat transfer in cylindrical coordinates for the water, ice, and hydrate layers, and is able to predict data for single- and two-sided depressurization, as well as for thermal stimulation using electrical heating (Davies et al 2006). A heat transfer limited process is controlled by the rate of heat supplied to the system. Therefore, a measurable intermediate (cf. activated state) is not expected for heat transfer controlled dissociation (Gupta et al., 2006). 

Familiar examples of the relation between generalized fluxes and forces are Fick s first law of diffusion, Fourier s law of heat transfer, Ohm s law of electricity conduction, and Newton s law of momentum transfer in a viscous flow. 

On the other hand, heat conductivity means the transfer of energy caused by temperature gradients. Fourier s law of heat transfer is expressed as... 

The heat flow, Q, can be calculated using Fourier s law of heat transfer in cylindrical coordinates ... 

In Section 15.2.1 we noted that Pick derived his model for mass transfer pardy by analogy to Fourier s law of heat transfer and that one reason Pick s model was rapidly accepted was this close analogy to Fourier s law. Shordy after Pick s developments, Osborne Reynolds (yes, the Reynolds number is named after him) stated that heat or mass transfer in a moving fluid should be the result of both normal diffusion processes and eddies caused by the fluid motion. At the time, he had not yet discovered the difference between laminar motion (only normal diffusion operates) and turbulent motion (both molecular and eddy diffusion occur). We now know that Reynolds was correct only for turbulent flow. Since eddies depend on fluid velocity, the easiest functional form is to assume that eddy diffusion is linearly dependent on velocity. Then the equation for mass transfer becomes... 

Analogous to Newton s law of momentum transport and Fourier s law of heat transfer by conduction. Pick s first law for mass transfer by steady-state equimolar diffusion, is... 

If you look closely at this equation and Fourier s law of heat transfer, they look very similar. This similarity is not casual because Pick s observations, based on mass transfer, gave similar results to those observed for heat transfer. This similarity among heat, mass, and momentum transfer are very relevant for chemical and bioprocess engineering and makes it possible to group them into the so-called discipline of transport phenomena. 

The formulae for the thermal conductance of elements of type 2 and 4 given In Figure 6 were derived from Fourier s law of heat transfer,... 

Techniques based on Fourier s law of heat transfer, Eq. (IX are not adept at handling molten polymers nor the effect of pressure. Molten polymers often degrade during the time required to reach equilibrium. Molten polymers cannot be used if the heat source is the gas from a boiling liquid. While pressure can be applied when the sample is contained between two flat dates, most ctHnmercial units are not equipped to handle significant levels of pressure. 

Fourier s Law of Heat Conduction. The heat-transfer rate,, per unit area,, in units of W/m (Btu/(ft -h)) transferred by conduction is directly proportional to the normal temperature gradient ... 

The Tube Wall Tubular heat exchangers are built using a number of circular (or noncircular) tubes thus, the heat-transfer rate across tubular walls, following Fourier s law of heat conduction, becomes... 

Thermal conductivity is a physical property of the solid through which the heat is being transferred. It is a measure of the material s ability to conduct heat. Insulators have a low thermal conductivity and conductors have a high thermal conductivity. The rate of heat transfer has magnitude and direction. This is represented mathematically by the negative sign that appears in Fourier s law of heat conduction. 

At the simplest level, as Griskey (1) notes, Pick s law of diffusion for mass transfer and Fourier s law of heat conduction characterize mass and heat transfer, respectively, as vectors, i.e., they have magnitude and direction in the three coordinates, x, y, and z. Momentum or flow, however, is a tensor which is defined by nine components rather than three. Hence, its more complex characterization at the simplest level, in accordance with Newton s law, is... 

Thermal conductivity is the intensive property of a material that indicates its ability to conduct heat. For one-dimensional heat flow in the x-direction the steady state heat transfer can be described by Fourier s law of heat conduction ... 

At this point we retrace our development slightly to introduce a different conceptual viewpoint for Fourier s law. The heat-transfer rate may be considered as a flow, and the combination of thermal conductivity, thickness of material, and area as a resistance to this flow. The temperature is the potential, or driving, function for the heat flow, and the Fourier equation may be written... 

B Understand Ihe basic mechanisms of heat transfer, v/hich are conduction, convection, and radiation, and Fourier s law of heat conduction, Newton s law of cooling, and the Stefaa-Boltzmann law of radiation,... 

When there is sufficient information about energy interactions at a surface, it may be possible to determine Ihe rate of heat transfer and thus the heat flux q (heat transfer rate per unit surface area, W/m ) on that surface, and this information can be used as one of the boundary conditions. The heat flux in the positive x-direction anywhere in the medium, including the boundaries, can be expressed by Fourier s law of heat conduction as... 

Consider a long cylindrical layer (such as a circular pipe) of inne.r radius r outer radius rz, length L, and average thermal conductivity k (Fig. 3-24). The two surfaces of the cylindrical layer arc maintained at constant temperatures T, and Tz- There is no heat generation in the layer and the thermal conductivity is constant. For one-dimensional heat conduction through the cylindrical layer, we have T r). Then Fourier s law of heat conduction for heat transfer through the cylindrical layer can be expressed as... 

The rate of heat transfer from the fin can he determined again from Fourier s law of heat conduction ... 

You will recall that heat Is transferred by conduction, convection, and radiation. Mass, however, is transferred by conduction (called diffusion) and convection only, and tliere is no such thing as mass radiation (unless there is something Scotty knows that we don t when he beams people to anywhere in space at the speed of light) (Fig. 14-5). The rate of heat conduction in a direction X is proportional to the tempetatuve gradient dT/d.x in that direction and is expressed by Fourier s law of heat conduction as... 

It is valid for each continuum independent of the individual material properties and is therefore one of the fundamental equations in fluid mechanics and subsequently also in heat and mass transfer. The movement of a particular substance can only be described by introducing a so-called constitutive equation which links the stress tensor with the movement of a substance. Generally speaking, constitutive equations relate stresses, heat fluxes and diffusion velocities to macroscopic variables such as density, velocity and temperature. These equations also depend on the properties of the substances under consideration. For example, Fourier s law of heat conduction is invoked to relate the heat flux to the temperature gradient using the thermal conductivity. An understanding of the strain tensor is useful for the derivation of the consitutive law for the shear stress. This strain tensor is introduced in the next section. 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the steady state distribution of heat or concentration across the slab or the material in which the experiment is performed. This steady state process involves solving second order ordinary differential equations subject to boundary conditions at two ends. Whenever the problem requires the specification of boundary conditions at two points, it is often called a two point boundary value problem. Both linear and nonlinear boundary value problems will be discussed in this chapter. We will present analytical solutions for linear boundary value problems and numerical solutions for nonlinear boundary value problems. 

Fourier s law of heat conduction states that heat transfer by molecular interactions at any point in a solid or fluid is proportional in magnitude and coincident with the direction of the negative gradient of the temperature field [48] ... 

The Landauer expression for heat transfer (Equation 12.13) assumes the absence of inelastic scattering processes in the system, and the two opposite phonon flows of different temperatures are out of equilibrium with each other. This leads to an anomalous transport of heat, where (classically) the energy flux is proportional to the temperature difference, Tl - T, rather than to the temperature gradient VT, as asserted by the Fourier s law of heat conductivity. 

Foams, in addition to being useful as cushioning, can be used to provide thermal insulation for products. A frozen product, for example, might be packaged with ice (or dry ice or gel packs) to provide cooling, and encased in a foam container to help reduce the conduction of heat from the surroundings into the container. Often the temperature inside and outside the container can be regarded as relatively constant, and the heat transfer process can considered essentially one-dimensional. In such cases, Fourier s law of heat conduction reduces to its one-dimensional steady-state form ... 

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