Constitutive relations heat flux

The well-known dual-phase-lag heat conduction model introduces time delays to account for the responses among the heat flux vector, the temperature gradient and the energy transport. The dual-phase-lag heat conduction model has been used to interpret the non-Fourier heat conduction phenomena. The onedimensional dual-phase-lag constitutive equation relating heat flux to temperature gradient is expressed as (Xu, 2011 Zhou et al., 2009)... 

Examples of constitutive information are Newtai s law of viscosity, which relates shear stress to shear rate, and Former s law of heat conduction, which relates heat flux to temperature gradient... 

The relation between the emf of the thermoelectric pile and the heat flux from the calorimeter cell will be first established. Let us suppose (Fig. 8) that the process under investigation takes place in a calorimeter vessel (A), which is completely surrounded by n identical thermoelectric junctions, each separated from one another by equal intervals. The thermocouples are attached to the external surface of the calorimeter cell (A), which constitutes the internal boundary (Eint) of the pile and to the inside wall of the heat sink (B), constituting the external boundary (Eext) of the thermoelectric pile. The heat sink (B) is maintained at a constant temperature (6e). 

As a first approximation, the stresses for the solid, ice and gel water can be formulated with the help of a linearized Hookean type law, where the depression of the gel water below the macroscopic freezing point of water must be considered. This can be done by including the micro-ice-lens model of Set-zer [1] in the constitutive relations for the aforementioned stress tensor. The gas phase can be described as an ideal gas. Concerning the constitutive assumptions for the liquid stresses, the heat flux and the interactions, the reader is referred to de Boer et al. [4], There a ternary model for the numerical simulation of freezing and thawing processes is discussed. 

Olir discussion on diffusion will be restricted primarily to binary systems containing only species A and B. We now wish to determine how the molar diffusive flux of a species (i.e., Ja) is related to its concentration gradient. As an aid in the discussion of the transport law that is ordinarily used to describe diffusion, lesll similar laws ftom other trans K)it processes. For example, in conductive heat transfer the constitutive equation relating the heat flux q and the temperature gradient is Fourier s law ... 

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. 

For diffusion in isothermal multicomponent systems the generalized driving force was written as a linear function of the relative velocities (m/ — My). In the general case, we must allow for coupling between the processes of heat and mass transfer and write constitutive relations for and q in terms of the (m — My) and V(l/r). With this allowance, the complete expression for the conductive heat flux is... 

Chapter 1 serves to remind readers of the basic continuity relations for mass, momentum, and energy. Mass transfer fluxes and reference velocity frames are discussed here. Chapter 2 introduces the Maxwell-Stefan relations and, in many ways, is the cornerstone of the theoretical developments in this book. Chapter 2 includes (in Section 2.4) an introductory treatment of diffusion in electrolyte systems. The reader is referred to a dedicated text (e.g., Newman, 1991) for further reading. Chapter 3 introduces the familiar Fick s law for binary mixtures and generalizes it for multicomponent systems. The short section on transformations between fluxes in Section 1.2.1 is needed only to accompany the material in Section 3.2.2. Chapter 2 (The Maxwell-Stefan relations) and Chapter 3 (Fick s laws) can be presented in reverse order if this suits the tastes of the instructor. The material on irreversible thermodynamics in Section 2.3 could be omitted from a short introductory course or postponed until it is required for the treatment of diffusion in electrolyte systems (Section 2.4) and for the development of constitutive relations for simultaneous heat and mass transfer (Section 11.2). The section on irreversible thermodynamics in Chapter 3 should be studied in conjunction with the application of multicomponent diffusion theory in Section 5.6. 

Since we have just verified that both the viscous stresses and the heat conduction terms vanish for equilibrium flows, the constitutive stress tensor and heat flux relations required to close the governing equations are determined. That is, substituting (2.233) and (2.234) into the conservation equations (2.202), (2.207) and (2.213), we obtain the Euler equations for isentropic flow ... 

A statement of the constitutive relation analogous to those for mass, heat, and momentum is that the flux due to migration in an electric field is proportional to the force acting on the particle multiplied by the particle concentration. The molar flux in stationary coordinates is then... 

Note 3.4 (On the constitutive relation and the Second Law of Thermodynamics). The Second Law of Thermodynamics essentially gives a relationship between the heat flux q that is externally supplied and the induced temperature field. Some scientists have stated that constitutive relations that depend on fields other than the temperature can also be derived by the Second Law however, as shown above, the Second Law does not consider fields other than the heat flux and temperature. All the constitutive relations can be derived from the First Law of Thermodynamics, internal energy and thermodynamic potentials induced by Legendre transformations (see Sect. 3.4). ... 

Using the subsequent first-order perturbation solution for ft, the stress tensor and heat-flux vector in the gas can be evaluated explicitly in terms of F, A and B, and thus the transport coefficients, which are the proportionality factors in the phenomenological constitutive equations for the gas, may be determined. It turns out that F is related only to the bulk viscosity of the gas, A to the thermal conductivity and B to the viscosity. Specifically the bulk viscosity is given by... 

The most common types of models in chemical engineering are those related to the transport of mass, heat, and momentum. In addition to the balance equation, a constitutive equation that relates the flux of interest to the dependent variable (e.g. mass flux to concentration) is needed. These relations (in simple ID form) for the microscopic level and for flow at the porous media level are given in Table 3.1. It should be noted that all these relations have the general form... 

The mass transfer flux law is analogous to the laws for heat and momentum transport. The constitutive equation for Ja, the diffusional flux of A resulting from a concentration difference, is related to the concentration gradient by Pick s first law ... 

A system may contain energy when it possesses the ability to store it, as already stated. This ability takes the form of two constitutive properties, one for each subvariety of energy inductance for the inductive subvariety, and capacitance for the capacitive one. These names are borrowed from electrodynamics and generalized to all energy varieties. So, there is a translational mechanical inductance (inertial mass), a rotational mechanical inductance (inertia), a hydrodynamical capacitance (compressibility integrated over the volume), a thermal capacitance (which depends on the specific heat), and so on. In electrodynamics, inductance and capacitance feature components called inductor (or self-inductance) and capacitor, respectively. Electric inductance relates current to the quantity of induction (induction flux) and electric capacitance relates potential to charge. 

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