Heat channel cross-section

A number of authors have considered channel cross-sections other than rectangular [102-104]. Figure 2.17 shows some examples of cross-sections for which friction factors and Nusselt numbers were computed. In general, an analytical solution of the Navier-Stokes and the enthalpy equations in such channel geometries would be involved owing to the implementation of the wall boundary condition. For this reason, usually numerical methods are employed to study laminar flow and heat transfer in channels with arbitrary cross-sectional geometry. 

In addihon to those more or less regular channel cross-sections, Richardson et al. [104] studied heat transfer in some more exohc channels as displayed on the right side of Figure 2.17. They numerically computed Nusselt numbers and expressed them as a dimensionless enhopy generation rate, defined as... 

Kolb et al. [52] applied small externally heated sandwich-type reactors for catalyst screening for propane steam reforming. Two plates of 2 mm thickness were attached to each other and bonded by laser welding. The reactors where 41 mm long and 10 mm wide carrying 14 channels each, which were 25 mm long, 500 pm wide and 250 pm deep on each plate, thus resulting in a total channel cross-section of 500 pm x 500 pm when mounted. 

In this equation L = spiral length D = effective diameter, characteristic of the channel cross section AT = temperature difference between melt and channel wall Ap = pressure drop p = density of the solid polymer X = heat conductivity of the solid polymer 77 = viscosity of the melt AH = enthalpy difference between melt and solid. 

When the flow is through the annulus of a double-pipe heat exchanger, Eqs. (13.15) and (13.19) can be used to estimate the frictional pressure drop, provided that the inside diameter, D of the tube or pipe is replaced by the hydraulic diameter, D , which is defined as 4 times the channel cross-sectional area divided by the wetted perimeter. For an annulus, Du = >2 - ),. 

The electric potential

electrokinetic flow can be split into two terms, i.e., (p= (p + ij/, where 4>, the electric potential externally applied along the axis of a channel, produces Joule heating in Eq. 1, and ij/, the double-layer potential internally induced over the channel cross-section, causes liquid electroosmosis via Eqs. 4 and 5. These two electric potentials satisfy, respectively, the Laplace equation and the Poisson equation... 

The heating rate of a flowing fluid is usually reported as the Nusselt number instead of giving the solution to the temperature field across the channel cross section. The Nusselt number is a dimensionless quantity characterizing the efficiency of heat transfer and is defined by... 

Here, is the mean velocity, Ac is the area of the channel cross section, and Abottom is the area of the heated substrate wall. 

Let us consider the heated microchannel subjected to the HI boundary condition. By integrating Eq. 2 over the channel cross-sectional area f2, one obtains the following result ... 

In this section, we formulate a ID model with interphase mass and heat transfer coefficients. These lumped models [103] describe the axial variation of concentration and temperature (which are averaged over the channel cross section). The diffusion processes in the transverse directions (represented by differential terms) are replaced by a transfer term, associated with a given driving force. The use of ID models is widespread throughout the literature on monoUth reactor modeling. Chen et al. [3] reviewed some specific appUcations including simulation of simultaneous heat transfer in monofith catalysts... 

Here q, denotes the heat transfer coeflflcient in W/m K, 5 the circumference of the channel cross section, th the mass flow rate, and Cp the heat capacity of the fluid, respectively. The wall temperature and fluid temperature in Jz are given by Tw(z) and 7fl(z), respectively (Figure 4a). Of course, the above expression is valid only for the heating of an incompressible fluid but nonetheless is useful for the arguments that we wish to make below. Introducing the continuity equation m = p u a (where a is the cross-sectional area of the duct) and the dimensionless Stanton number (St) (25,26) into Eq. (11), we obtain... 

Coreless furnaces derive their name from the fact that the coil encircles the metal charge but, in contrast to the channel inductor described later, the cod does not encircle a magnetic core. Figure 8 shows a cross section of a typical medium sized furnace. The cod provides support for the refractory that contains the metal being heated and, therefore, it must be designed to accept the mechanical loads as well as the conducted thermal power from the load. 

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