Films heat conduction across

Fig. 20.1-1. Steady heat conduction across a thin film. Heat conduction across a thin film is like diffusion across a membrane (see Section 2.2). The resulting temperature profile is linear, and the flux is constant and inversely proportional to the film thickness /.

The generalized form for steady-state heat conduction across a thin film (where we allow the insulating material a thickness 8) is given by... 

The gap resistance was further rationalized in subsequent work. It was shown that the associated heat transfer mechanism relies primarily on heat conduction across the stagnant gas film, trapped in the gap between tbe monolith and the inner reactor tube wall. In fact, the gap resistance was inversely proportional to the gap size evaluated under the reaction conditions (so differential thermal expansion of the monolith and tube materials should be considered), and directly proportional to the gas-phase conductivity, as evidenced by heat transfer measurements with N2-He mixtures of different compositions. Estimates for h in excess of 700 W/(m K) were obtained when using pure He. 

To calculate heat fluxes or temperature profiles, we make energy balances and then combine these with Fourier s law. The ways in which this is done are best seen in terms of two examples heat conduction across a thin film and into a semi-infinite slab. The choice of these two examples is not casual. As for diffusion, they bracket most of the other problems, and so provide limits for conduction. 

Steady Heat Conduction Across a Thin Film... 

The limits of heat conduction across a thin film and into a thick slab are the two most important cases of a rich variety of examples. This variety largely consists of solutions of Eq. 20.1-16 for different geometries and boundary conditions. The geometries include slabs, spheres, and cylinders, as well as more exotic shapes like cones. The boundary conditions are diverse. For example, they include boundary temperatures that vary periodically because this is important for diurnal temperature variations of the earth. They include boundary conditions in which the heat flux at the surface is related to the temperature of the surroundings, Tsurrl for example. 

Equations (18) and (19) are valid too, in hydrod5mamic regions for calculating surface temperature, if we assume that viscous heating concentrates on the middle layer of lubricating films and temperature varies linearly across the film [20]. TheFourier law of heat conduction gives rise to the following expressions ... 

Next, consider the gradients of temperature. If the reaction is exothermic, the center of the particle tends to be hotter, and conversely for an endothermic reaction. Two sets of gradients are thus indicated in Figure 8.9. Heat transfer through the particle is primarily by conduction, and between exterior particle surface (Ts) and bulk gas (Tg) by combined convection-conduction across a thermal boundary layer, shown for convenience in Figure 8.9 to coincide with the gas film for mass transfer. (The quantities T0, ATp, A7y, and AT, are used in Section 8.5.5.)... 

One objection to a Forster and Zuber assumption has been given by Zwick (Zl). Forster and Zuber state that the principal mechanism for heat transfer to a growing bubble is conduction across the film resistance. Zwick points out that heat can also flow by mass transport and that this convection should be included in the equations. 

Hint The process of heat exchange across an interface can be treated in the same way as the exchange of a chemical at the interface. To do so, we must express the molecular thermal heat conductivity by a molecular diffusivity of heat in water and air, Z)thw and Z)tha, respectively. At 20°C, we have (see Appendix B) flthw = 1.43 xl(T3 cm2 s-1, Dlh a = 0.216 cm2 s 1. Use the film model for air-water exchange with the typical film thicknesses of Eq. 20-18a. 

There are upper and lower limits of applicability of the equation above. The lower limit results because natural-convection heat transfer governs at low temperature differences between the surface and the fluid. The upper limit results because a transition to film boiling occurs at high temperature differences. In film boiling, a layer of vapor blankets the heat-transfer surface and no liquid reaches the surface. Heat transfer occurs as a result of conduction across the vapor film as well as by radiation. Film-boiling heat-transfer coefficients are much less than those for nucleate boiling. For further discussion of boiling heat transfer, see Refs. 5 and 6. 

Heal flux meters use a very sensitive device known as a thermopile lo measure the temperature difference across a thin, heat conducting film made of kapton (k = 0.345 W/m K). If the thermopile can delect temperature differences of 0.1 °C or more and the film thickness is 2 mm, what is the minimum heal flux this meter can detect Ansv/en 17.3 W/rrf... 

Suppose that on a vertical wall whose temperature is constant and equal to Ts, stagnant dry saturated vapor is condensing. Let us consider the steady-state problem under the assumption that we have laminar waveless flow in the condensate film. According to [200], we make the following assumptions the film motion is determined by gravity and viscosity forces the heat transfer is only across the film due to heat conduction there is no dynamic interaction between the liquid and vapor phases the temperature on the outer surface of the condensate film is constant and equal to the saturation temperature Tg the physical parameters of the condensate are independent of temperature and the vapor density is small compared with the condensate density the surface tension on the free surface of the film does not affect the flow. 

What is the thermal conductivity of silicon nanowires, n-alkane single molecules, carbon nanotubes, or thin films How does the conductivity depend on the nanowiie dimension, nanotube chirality, molecular length and temperature, or the film thickness and disorder More profoundly, what are the mechanisms of heat transfer at the nanoscale, in constrictions, at low tanperatures Recent experiments and theoretical studies have dononstrated that the thermal conductivity of nanolevel systems significantly differ from their macroscale analogs [1]. In macroscopic-continuum objects, heat flows diffusively, obeying the Fourier s law (1808) of heat conduction, J = -KVT, J is the current, K is the thermal conductivity and VT is the temperature gradient across the structure. It is however obvious that at small scales, when the phonon mean free path is of the order of the device dimension, distinct transport mechanisms dominate the dynamics. In this context, one would like to understand the violation of the Fourier s... 

Rigorous modelling must take the selected geometry into consideration and this usually requires CFD. It must, in addition to heat transfer across the wall and film mass and heat transfer between the gases and catalysts, also include axial heat conduction in the metals due to steep temperature profiles. In addition transient behaviour and interaction between the steep temperature profiles must be understood for a proper design, especially when a reasonable catalyst deactivation is... 

The basis of the method is to divide the film into 3D cells with a node at the centre of each cell. The flow across the cell boundaries is found. A heat balance is then formed considering internal dissipation and the heat flux convected and conducted across the boundaries. 

Heat transfer between contacting particles is assumed to occur by a static process controlled by conduction across stagnant gas fillets at the point of contact, and a dynamic process involving a series mechanism of solid conduction, film convection and turbulent mixing,as in Fig. (6). The static and dynamic processes occur in parallel, thus... 

Throughout this book various transport properties and transfer coefficients have been used. These include effective diffusivity and thermial conductivity for mass and heat transport in catalyst pellets, film transfer coefficients for mass and heat transfer across the pellet-bulk fluid interface, transport properties for the degree of dispersion of mass and heat in the reactor, and heat transfer coefficients for heat exchange between the cooling medium and the reactor. In this chapter these transport properties and transfer coefficients are treated in detail, including experimental methods for obtaining these properties. 

How does the flux vary with physical properties for the thick slab as compared with the thin fibril Doubling the temperature difference doubles the heat flux in both cases. Doubling the thermal conductivity increases the flux by /2 for the thick slab and by 2 for the thin film. Doubling the heat capacity increases the flux by >/2 for the thick slab, but has no effect for the steady-state conduction across a thin film... 

If you double the thermal conductivity, how much will the heat flux across a thin film change ... 

Heat conduction also occurs, but free convection does not. Only gas 1 and gas 2 are in the film, and the thermal conductivity is constant. Also assume that the thermal conductivity at the boundaries is much greater than in the bulk. Find the heat transfer coefficient across this thin film in three steps (a) Find the concentration profiles in the film, (b) Find the temperature profile corrected for mass transfer, (c) Find the heat flux at the boundary z = 0. 

For turbulent flow of a fluid past a solid, it has long been known that, in the immediate neighborhood of the surface, there exists a relatively quiet zone of fluid, commonly called the Him. As one approaches the wall from the body of the flowing fluid, the flow tends to become less turbulent and develops into laminar flow immediately adjacent to the wall. The film consists of that portion of the flow which is essentially in laminar motion (the laminar sublayer) and through which heat is transferred by molecular conduction. The resistance of the laminar layer to heat flow will vaiy according to its thickness and can range from 95 percent of the total resistance for some fluids to about I percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core each offer a resistance to beat transfer which is a function of the turbulence and the thermal properties of the flowing fluid. The relative temperature difference across each of the layers is dependent upon their resistance to heat flow. 

The overall heat transfer coefficient, U, is a measure of the conductivity of all the materials between the hot and cold streams. For steady state heat transfer through the convective film on the outside of the exchanger pipe, across the pipe wall and through the convective film on the inside of the convective pipe, the overall heat transfer coefficient may be stated as ... 

Heat is transferred from one fluid stream to a second fluid across a heat transfer surface. If the film coefficients for the two fluids are, respectively, 1.0 and 1.5 kW/m2 K, the metal is 6 mm thick (thermal conductivity 20 W/m K) and the scale coefficient is equivalent to 850 W/m2 K. what is the overall heat transfer coefficient ... 

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