Heat conduction axial

Effect of axial heat conduction in the channel wall... 

In general, the axial heat conduction in the channel wall, for conventional size channels, can be neglected because the wall is usually very thin compared to the diameter. Shah and London (1978) found that the Nusselt number for developed laminar flow in a circular tube fell between 4.36 and 3.66, corresponding to values for constant heat flux and constant temperature boundary conditions, respectively. 

The subject of this chapter is single-phase heat transfer in micro-channels. Several aspects of the problem are considered in the frame of a continuum model, corresponding to small Knudsen number. A number of special problems of the theory of heat transfer in micro-channels, such as the effect of viscous energy dissipation, axial heat conduction, heat transfer characteristics of gaseous flows in microchannels, and electro-osmotic heat transfer in micro-channels, are also discussed in this chapter. 

The dependence of the local Nusselt number on non-dimensional axial distance is shown in Fig. 4.3a. The dependence of the average Nusselt number on the Reynolds number is presented in Fig. 4.3b. The Nusselt number increased drastically with increasing Re at very low Reynolds numbers, 10 < Re < 100, but this increase became smaller for 100 < Re < 450. Such a behavior was attributed to the effect of axial heat conduction along the tube wall. Figure 4.3c shows the dependence of the relation N /N on the Peclet number Pe, where N- is the power conducted axially in the tube wall, and N is total electrical power supplied to the tube. Comparison between the results presented in Fig. 4.3b and those presented in Fig. 4.3c allows one to conclude that the effect of thermal conduction in the solid wall leads to a decrease in the Nusselt number. This effect decreases with an increase in the... 

The numerical and experimental study of Tiselj et al. (2004) (see Fig. 4.17) was focused on the effect of axial heat conduction through silicon wafers on heat transfer in the range of Re = 3.2—84. Figure4.17 shows their calculation model of a triangular micro-channels heat sink. The results of calculations are presented in Fig. 4.18. 

The numerical experiment started at a steady-state value of 200 C for both temperature nodes with an output of 16.89% for both heaters output number 1 was then stepped to 19.00%. If both outputs had been stepped to 19%, then both nodes would have gone to 220 C. The temperature of node 5 does not go as high, and the temperature of node 55 goes too high. In the reduced order model, the time constant x represents the effect of radial heat conduction, while the time constant X2 represents the effect of axial heat conduction. SimuSolv estimates these two parameters of the dynamic model as ... 

Axial heat conduction in the polymer is neglected due to its low thermal conductivity and its relatively small magnitude, compared with axial thermal convection. 

Figure 17. Maximum temperature vs. time for rods of circular and rectangular cross section with uniform heat generation, zero initial and surface temperature, and negligible axial heat conduction. Dotted curve gives value of r0 to use in dimensionless temperature and time scale for rectangular cross sections (3)...

The axial dispersion of heat (axial heat conduction) is described by Fourier s law (5.2)... 

The boundary condition given is T = T(0) at x =0. Locally, through the tube wall, neglecting the axial heat conduction,... 

State with no entrance effects or radial velocity components body forces are neglected axial heat conduction is small compared to radial conduction Region I of Smith-Ewart kinetics (i.e., when micelles are first forming) is neglected and the initiator concentration is constant. The model may be summarized as ... 

A (ad, app) = deviation from true adiabacicity, deduced from axial heat conduction rate through the wall, as a percentage of total heat generation rate. 

Axial heat conductivity and axial diffusion in the fluid phase are neglected because of the usually large convective transport. 

Axial heat conduction in the solid phase is important, however, when reactor light-off or hot spots in the reactor have to be considered [38]. 

Simulations performed at the same conditions, but without axial heat conductivity, showed identical temperature profiles as the ones given in Fig. 10, while results at a low mass flow, typically 0.83 kg m7 sec, showed a temperature of the solid phase at the inlet of the reactor that is lower than the temperature calculated from the model with axial heat conductivity. This phenomenon was also observed by Lie et al. [35] and indicates that axial heat conduction in the solid phase can be neglected under steady-state conditions when the fluid flow is large enough. 

A study on a dilute hard-sphere gas in the transition regime using the DSMC was conducted by [44]. The simulation is for 0.02 < Kn < 2 and unity Em and Fj. They found a weak dependence of the Nusselt number on the Peclet number, which explains the weak dependence on the axial heat conduction. In the case of constant wall heat flux, a positive thermal creep, which occurs when the exit temperature is higher than the inlet temperature, tends to increase the Nusselt number while negative thermal creep tends to decrease the Nusselt number. 

We presume negligible axial heat conduction, constant wall temperature i90 and a constant temperature i9a of the fluid at the inlet of the tube. All material properties are temperature independent. 

A schematic diagram of the 4 B/D demonstration unit is shown in Fig. 12. Catalyst bed dimensions were 50 mm (2 in.) dia by 3 m (10 ft) for the DME, and 100 mm (4 in.) dia by 2.4 m (8 ft) for the ZSM-5 reactors. The linear velocities in these beds are about 10 times those in the bench-scale unit. The diameter of the ZSM-5 reactor was chosen to be 100 mm to reduce the influence of axial heat conduction along the reactor walls to a negligible value (ref. 14). Heat loss from the ZSM-5 reactor was estimated to be less than 1%. Except for size, the 4 B/D unit is very similar to the bench-scale unit. 

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. 

Equation (11.1) is essentially a solution of Eq. (11.7) and is based on a few assumptions and simplifications, e.g., no axial heat conduction, constant average heat conductivity and specific heat, constant heat source, steady-state heat transfer, one-dimensional (radial) heat flux, cylindrical geometry in the waste and in the surrounding material, e.g., salt, and no heat source in the salt. 

M. L. Michelsen, and J. Villadsen, The Graetz Problem with Axial Heat Conduction, Int. J. Heat Mass Transfer, (17) 1391-1402,1974. 

Proportional to the ratio of thermal energy convected to the fluid to thermal energy conducted axially within the fluid the inverse of Pe indicates relative importance of fluid axial heat conduction Useful in describing the thermal entrance region heat transfer results... 

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