Parallel-plate thermal conductivity

The problem of axial conduction in the wall was considered by Petukhov (1967). The parameter used to characterize the effect of axial conduction is P = (l - dyd k2/k ). The numerical calculations performed for q = const, and neglecting the wall thermal resistance in radial direction, showed that axial thermal conduction in the wall does not affect the Nusselt number Nuco. Davis and Gill (1970) considered the problem of axial conduction in the wall with reference to laminar flow between parallel plates with finite conductivity. It was found that the Peclet number, the ratio of thickness of the plates to their length are important dimensionless groups that determine the process of heat transfer. 

One particular characteristic of conduction heat transfer in micro-channel heat sinks is the strong three-dimensional character of the phenomenon. The smaller the hydraulic diameter, the more important the coupling between wall and bulk fluid temperatures, because the heat transfer coefficient becomes high. Even though the thermal wall boundary conditions at the inlet and outlet of the solid wall are adiabatic, for small Reynolds numbers the heat flux can become strongly non-uniform most of the flux is transferred to the fluid at the entrance of the micro-channel. Maranzana et al. (2004) analyzed this type of problem and proposed the model of channel flow heat transfer between parallel plates. The geometry shown in Fig. 4.15 corresponds to a flow between parallel plates, the uniform heat flux is imposed on the upper face of block 1 the lower face of block 0 and the side faces of both blocks... 

A variety of studies can be found in the literature for the solution of the convection heat transfer problem in micro-channels. Some of the analytical methods are very powerful, computationally very fast, and provide highly accurate results. Usually, their application is shown only for those channels and thermal boundary conditions for which solutions already exist, such as circular tube and parallel plates for constant heat flux or constant temperature thermal boundary conditions. The majority of experimental investigations are carried out under other thermal boundary conditions (e.g., experiments in rectangular and trapezoidal channels were conducted with heating only the bottom and/or the top of the channel). These experiments should be compared to solutions obtained for a given channel geometry at the same thermal boundary conditions. Results obtained in devices that are built up from a number of parallel micro-channels should account for heat flux and temperature distribution not only due to heat conduction in the streamwise direction but also conduction across the experimental set-up, and new computational models should be elaborated to compare the measurements with theory. 

In the parallel-plate method, the heat flux downward is measured hy a fluxmeter under which the thermal bond material and the molten salt of interest are located. Because the thermal transfer is only conductive if the thermal contacts are perfect, the balance of heat flow through the... 

The initial conditions are at t = 0, T = To, andp = 0. The parameter n characterizes the dimensions of the volume for a parallel plate reactor n = 0 for a cylindrical reactor n = 1 and for a spherical reactor n = 2. In these equations, x is a space coordinate A. is the coefficient of thermal conductivity r is the characteristic size of the reactor k is the heat transfer coefficient and To is the initial temperature of the initial medium. 

Couette flow is the model for flow between parallel plates. The plates are separated by a distance L, and filled with a fluid with density p, viscosity i and thermal conductivity k, Fig. 8.1. The upper plate moves at a constant velocity U and causes the fluid particles to move in the direction parallel to the plates. The upper and lower plates are kept at uniform temperatures Ti and T0 respectively. 

Problem Two black infinite parallel plates separated by a transparent medium of thickness b and thermal conductivity k. Plate 2 is at temperature T2, and a known amount of energy Q(/A is added per unit area to plate 1 and removed at plate 2. What is the temperature Ti of plate 1 ... 

Figure 4.2 shows a Couette flow of a fluid of constant density p, viscosity p, and thermal conductivity k between parallel plates. The bottom plate is at rest, while the top plate is moving at a constant velocity Vj. The upper and lower plates are at uniform temperatures Tx and r2, respectively. The equation of motion for fully developed flow in the x-direction is... 

The treatment of thermal conductivity parallels quite closely the treatment given for viscosity. Let us suppose that we have two parallel plates (Fig. VIII.3) at fixed temperatures Ti and T2, with a fluid between them. 

Fig. VIII.3. Thermal conduction between parallel plates. ...

Martin and Lang, by the parallel plate method, found for water at 7°-60 and 1 atm. kx 103=1 394(1+0 00230. The thermal conductivity of water was measured to 270° by Schmidt and Sellschopp it has a maximum at 130°. Mohanty connected k for water with the degree of association. 

Consider steady heat transfer between two large parallel plates at constant temperatures of Ti = 300 K and Tz - 200 K that are t = 1 cm apart, as shown in Fig. 1-41. Assuming the surfaces to be black (emissivity e = 1), determine the rate of heat transfer between the plates per unit surface area assuming the gap between the plates is (a) filled with atmospheric air, (b) evacuated, (c) (illed. vvith urethane insulation, and (d) filled with superinsulation that has an apparent thermal conductivity of 0 00002 W/m K... 

In the case of the thermal-conductivity, there are three main techniques those based on Equation (1) and those based on a transient application of it. Prior to about 1975, two forms of steady-state technique dominated the field parallel-plate devices, in which the temperature difference between two parallel disks either side of a fluid is measured when heat is generated in one plate, and concentric cylinder devices that apply the same technique in an obviously different geometry. In both cases, early work ignored the effects of convection. In more recent work, exemplified by the careful work in Amsterdam with parallel plates, and in Paris with concentric cylinders, the effects of convection have been investigated. Indeed, the parallel-plate cells employed in Amsterdam by van den Berg and his co-workers have the unique feature that, because the temperature difference imposed can be very small and the horizontal fluid layer very thin, it is possible to approach the critical point in a fluid or fluid mixture very closely (mK). 

The traditional way to measure thermal conductivity is with steady-state instruments, in which a measured heat flux is compared to a temperature difference between surfaces. Most often the geometry is coaxial cylinders, a thin wire inside a cylinder, or parallel plates. In such instruments, eliminating convection currents is crucial many old data taken with steady-state instruments are unreliable because of convection. Multiple experiments at different heat fluxes are often performed to verify the absence of convection. With good design and operation, such instruments may achieve accuracy in the 1% to 3% range. 

Commercial instruments exist for cylindrical probes for transient measurements and for steady-state parallel-plate measurements. These generally require calibration with reference fluids of known thermal conductivity. 

In this work, heat and fluid flow in some common micro geometries is analyzed analytically. At first, forced convection is examined for three different geometries microtube, microchannel between two parallel plates and microannulus between two concentric cylinders. Constant wall heat flux boundary condition is assumed. Then mixed convection in a vertical parallel-plate microchannel with symmetric wall heat fluxes is investigated. Steady and laminar internal flow of a Newtonian is analyzed. Steady, laminar flow having constant properties (i.e. the thermal conductivity and the thermal diffusivity of the fluid are considered to be independent of temperature) is considered. The axial heat conduction in the fluid and in the wall is assumed to be negligible. In this study, the usual continuum approach is coupled with the two main characteristics of the microscale phenomena, the velocity slip and the temperature jump. 

Although the heat flow and fluid flow in packed beds are quite complex, the heat transfer characteristics can be described by a simple concept of effective thermal conductivity Ke that is based on the assumption that on a macroscopic scale the bed can be described by a continuum. Effective thermal conductivity is a continuum property that depends on temperature, bed material, and structure. It is usually determined by evaluating the steady-state heat flux between two parallel plates separated by a packed bed. The effective thermal conductivity applies very accurately to steady-state heat transfer and to unsteady-state heat transfer if (t d2p) > 1.94 x 107 s/m2 [27] in other cases, for unsteady state heat transfer the thermal... 

Let us first recall briefly the classical Benard-Rayleigh problem of thermal convection in an isotropic liquid. When a horizontal layer of isotropic liquid bounded between two plane parallel plates spaced d apart is heated from below, a steady convective flow is observed when the temperature difference between the plates exceeds a critical value A 7. The flow has a stationary cellular character with a spatial periodicity of about 2d. The mechanism for the onset of convection may be looked upon as follows. A fluctuation T in temperature creates warmer and cooler regions, and due lO buoyancy effects the former tends to move upwards and the latter downwards. When AT < AT, the fluctuation dies out in time because of viscous effects and heat loss due to conductivity. At the threshold the energy loss is balanced exactly and beyond it instability develops. Assuming a one-dimensional model in which T and the velocity (normal to the layer) vary as exp (i j,y) with x ji/rf, the threshold is given by the dimensionless Rayleigh number... 

Most of the conventional techniques of thermal conductivity measurements are based on the steady-state solution of Equation (5.1), i.e. establishing a stationary temperature difference across a layer of liquid or gas confined between two cylinders or parallel plates (Kestin and Wakeham, 1987). In recent years, the transient hot-wire technique for the measurement of the thermal conductivity at high temperatures and high pressures has also widely been employed (Assael et al, 1981, 1988a,b, 1989, 1991, 1992, 1998 Nagasaka and Nagashima, 1981 Nagasaka et al., 1984, 1989 Mardolcar et al., 1985 Palavra et al, 1987 Roder and Perkins, 1989 Perkins et al, 1991, 1992 and Roder et al, 2000). 

In Table 5.1 all available experimental thermal conductivity data sources at high temperatures (above 200 °C) and high pressures are presented. As one can see from this table, all data were derived by the parallel-plate and the coaxial-cylinder techniques, except only two datasets for LiBr by Bleazard et al. (1994) and DiGuilio and Teja (1992) which were obtained by the transient hot-wire technique. We further note that almost all investigators quote an uncertainty of better than 2%. In this section a brief analyses of these methods is presented. The theoretical bases of the methods, and the working equations employed is presented, together with a brief description of the experimental apparatus and the measurements procedure of each technique. For a more thorough discussion of the various techniques employed, the reader is referred to relevant literature (Kestin and Wakeham, 1987 Wakeham et al., 1991 Assael et al, 1991, and Wakeham and Assael, in press). 

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