Heat flow/transport

Conduction is the process by which heat flows by molecular transportation along or through a material or from one material to another, the material receiving the heat being in contact with that from which it receives it. Conduction takes place in solids, liquids and gases and from one to another. The rate at which conduction occurs varies considerably according to the substance and its state. 

Introduction.—Statistical physics deals with the relation between the macroscopic laws that describe the internal state of a system and the dynamics of the interactions of its microscopic constituents. The derivation of the nonequilibrium macroscopic laws, such as those of hydrodynamics, from the microscopic laws has not been developed as generally as in the equilibrium case (the derivation of thermodynamic relations by equilibrium statistical mechanics). The microscopic analysis of nonequilibrium phenomena, however, has achieved a considerable degree of success for the particular case of dilute gases. In this case, the kinetic theory, or transport theory, allows one to relate the transport of matter or of energy, for example (as in diffusion, or heat flow, respectively), to the mechanics of the molecules that make up the system. 

Where large samples of reactant are used and/or where C02 withdrawal is not rapid or complete, the rates of calcite decomposition can be controlled by the rate of heat transfer [748] or C02 removal [749], Draper [748] has shown that the shapes of a—time curves can be altered by varying the reactant geometry and supply of heat to the reactant mass. Under the conditions used, heat flow, rather than product escape, was identified as rate-limiting. Using large ( 100 g) samples, Hills [749] concluded that the reaction rate was controlled by both the diffusion of heat to the interface and C02 from it. The proposed models were consistent with independently measured values of the transport parameters [750—752] whether these results are transfenable to small samples is questionable. 

For the lower heat transfer surfaces in Fig. 2.60 to contribute to the energy transport, the solid should be an effective conductor of heat through its thickness. In other words, conjugate heat transfer effects should not create a more significant resistance to heat flow than that of the fluid in the channel. Since the heat transfer coefficient is generally a maximum at CHF, this leads to... 

The variable gap method is a steady-state method, with the merit that transport of heat by radiation can be separated from the total heat flow ... 

The effective diffusivity depends on the statistical distribution of the pore transport coefficients W j. The derivation shows that the semi-empirical volume-averaging method can only be regarded as an approximation to a more complex dynamic behavior which depends non-locally on the history of the system. Under certain circumstances the long-time (t —> oo) diffusivity will not depend on t (for further details, see [191]). In such a case, the usual Pick diffusion scenario applies. The derivation presented above can, with minor revisions, be applied to the problem of flow in porous media. When considering the heat conduction problem, however, some new aspects have to be taken into accoimt, as heat is transported not only inside the pore space, but also inside the solid phase. 

Monte Carlo heat flow simulation, nonequilibrium molecular dynamics, 73-74, 77-81 multiparticle collision dynamics hydrodynamic equations, 105-107 macroscopic laws and transport coefficients, 102-104 single-particle friction and diffusion, 114-118... 

Terminal velocity, linear thermodynamics intermediate regimes and maximum flux, 25-27 regression theorem, 18-20 Test particle density, multiparticle collision dynamics, macroscopic laws and transport coefficients, 100-104 Thermodynamic variables heat flow, 58-60... 

The fundamental concept of heat transport controlled moisture uptake [17] is shown in Eq. (22), where the rate of heat gained at the solid/vapor surface (W AH) is balanced exactly by the heat flow away from the surface (Q). The term All is the heat generated by unit mass of water condensed on the surface. The two most probable sources of heat generation are the heat of water condensation and the heat of dissolution. A comparison of the heat of water condensation (0.58 cal/mg water) with the heat of dissolution for a number of salts indicates that the heat of dissolution can be neglected with little error for many materials. 

The total heat flow (Q) at the film surface is equal to the mass flow rate (W k) times the heat of condensation (AH). That is, the heat generated by condensation at the surface must be equal to the heat transported away by conduction and radiation for steady state to be achieved. In mathematical terms this results in the equation. 

By Dalton s law, Equation (2.9), the mixture pressure, p, is Y i= Pi- The ternl Y I PiVjhj is sometimes considered to be a heat flow rate due to the transport of enthalpy by the species. (This is not the same as q" arising from VT which is called the Dufour effect and is generally negligible in combustion.) With the exception of the enthalpy diffusion term, all the sums can be represented in mixture properties since ph = Ya i Pihi However, it is convenient to express the enthalpies in terms of the heat of formation and specific heat terms, and then to separate these two parts. 

The transport process abont which most of us have an intnitive nnderstanding is heat transfer so we will begin there. In order for heat to flow (from hot to cold), there must be a driving force, namely, a temperature gradient. The heat flow per unit area (Q/A) in one direction, say the y direction, is the heat flux, qy. The temperature difference per unit length for an infinitesimally small unit is the temperature gradient, dT/dy. According to Eq. (4.1), there is then a proportionality constant that relates these two quantifies, which we call the thermal conductivity, k. Do not confuse this quantity with... 

The thermoelectric effect is due to the gradient in electrochemical potential caused by a temperature gradient in a conducting material. The Seebeck coefficient a is the constant of proportionality between the voltage and the temperature gradient which causes it when there is no current flow, and is defined as (A F/A7) as AT- 0 where A Fis the thermo-emf caused by the temperature gradient AT it is related to the entropy transported per charge carrier (a = — S /e). The Peltier coefficient n is the proportionality constant between the heat flux transported by electrons and the current density a and n are related as a = Tr/T. 

At the phenomenological level, there are enough further relations between the 14 variables to reduce the number to 5 and make the problem determinate. These further relations are the thermodynamic ones and Stokes and Newton s laws of viscosity and heat flow. These lead from the transport equations to the Navier-Stokes equations. It is noted that these are irreversible. 

Irreversible thermodynamics thus accomplishes two things. Firstly, the entropy production rate EE t allows the appropriate thermodynamic forces X, to be deduced if we start with well defined fluxes (eg., T-VijifT) for the isobaric transport of species i, or (IZT)- VT for heat flow). Secondly, through the Onsager relations, the number of transport coefficients can be reduced in a system of n fluxes to l/2-( - 1 )-n. Finally, it should be pointed out that reacting solids are (due to the... 

The quantity, h, in Equation 5 is not likely to be greatly different from its value in a plane adiabatic combustion wave. Taking x as the coordinate normal to such wave, h becomes the integral of the excess enthalpy per unit volume along the x-axis, so that the differential quotient, dh/dx, represents the excess enthalpy per unit volume in any layer, dx. Assuming the layer to be fixed with respect to a reference point on the x-axis, the mass flow passes through the layer in the direction from the unbumed, w, to the burned, 6, side at a velocity, S, transporting enthalpy at the rate Sdh/dx. Because the wave is in the steady state, heat flows by conduction at the same rate in the opposite direction, so that... 

Component 1, which is unconstrained, is diffusing along a long bar while the temperature everywhere is maintained constant. Find an expression for the heat flow that would be expected to accompany this mass diffusion. What role does the heat of transport play in this phenomenon ... 

An extremely fine localization of primordial 3He injection, on a 10-m scale, has also been observed. It has been suggested that lower-than-expected conductive heat flow at oceanic ridges could be due to significant heat transport by hydrothermal circulation (e.g., Talwani, Windisch Langseth, 1971), in which recently emplaced hot rock drives convection of local sea water. On the basis of temperature-salinity relationships, Weiss et al. (1977) made the first identification of hydrothermal circulation in the open ocean, observing several plumes (temperature differential <0.2°C)... 

To explain the imbalance, O Nions and Oxburgh (1983) and Oxburgh and O Nions (1987) proposed that a barrier, which is suggested to exist between the upper and the lower mantle from seismic observation, has trapped helium in the lower mantle and retarded the heat transport from the lower mantle to the upper mantle. O Nions et al. (1983) suggested, from a semiquantitative discussion, that delayed heat transfer from the lower mantle to the upper mantle with a time constant of about 2Ga would enhance the present heat flow by a factor of two. McKenzie and Richter (1981) made numerical calculation on a two-layered mantle convection and showed that heat transfer from the lower mantle to the upper mantle is considerably retarded to give rise to an enhancement of the present surface heat flow up to a factor of two. If the thermal barrier not only retards the heat transfer and hence enhances the present surface heat flow but also essentially prevents the 4He flux from the lower to the upper mantle, this would qualitatively explain the imbalance. If this indeed were the case, we would expect a large amount of 4He accumulation in the lower mantle. However, it is difficult to conclude such a large accumulation of 4He in the lower mantle from the currently available scarce noble gas data derived from mantle-derived materials. 

Radiative heat transport through olivine has been discussed extensively (e.g., Fukao et al., 1968 Shankland, 1970 Schatz and Simmons, 1972 Scharmeli, 1979 Shankland et al., 1979). The radiative thermal conductivity, Kt of forsteritic olivine increases with rising temperature and would contribute to heat flow in the Upper Mantle (Shankland et al., 1979). However, values of Kt for olivine are considered to be rather low to satisfactorily explain the dissipation of the Earth s internal heat by radiation and lattice conduction alone. Note, however, that Fe2 CF transitions in almandine, pyroxenes (M2 site) and, perhaps, silicate perovskites absorb strongly in the wavelength range 1,250 to... 

A very important feature of heat pipe (HP) is the ability to transport a large amounts of energy over the length of heat pipe with a small temperature drop by means of liquid evaporation at the heat pipe evaporator (heat source) and vapour condensation at the condenser (heat sink) and liquid movement in the opposite direction inside a wick by capillary force. Essential is a possibility to change the direction of a heat flow along the heat pipe in time and to use heat pipes for cooling and heating alternatively. 

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