Axial Conduction in the Fluid

For q = const, q is the heat flux on the wall), we introduce new variables 

Transferring Eq. (4.15) to divergent form and integrating this equation through the micro-channel cross-section we obtain  

Assuming that the exit of the micro-channel is connected to an adiabatic section the boundary conditions are  

Taking into account conditions (4.17) we obtain from (4.16)  

The effect of axial conduction on heat transfer in the fluid in the micro-channel can be characterized by a dimensionless parameter 

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It shows that close to the micro-channel inlet, heat losses to the cooling inlet due to axial conduction in the fluid are dominant. In the second case parameter M is ... 

The existence of heat transfer due to axial conduction in the fluid leads to increasing difference between wall and fluid temperatures and decreasing value of the Nus-... 

In Fig. 3, the variation of local Nusselt number along the constant wall temperature tube is presented as a function of Peclet number, representing axial conduction in the fluid. For Pe = 50, which represents a case with negligible axial conduction, the solution of the classical Graetz problem, Nu = 3.66, is reached [44], while for Pe = 1, Nu = 4.03 [45] is obtained as the fully developed values of Nu. The temperature gradient at the wall decreases at low Pe values, thus the local and fully developed Nu values increase with decreasing Pe. 

By including the effect of axial conduction in the fluid, the fully developed average Nusselt number becomes a function of the Peclet number. An approximate relation by means of which Nux can be calculated as a function of Peclet number is the following ... 

Tunc and Bayazitoglu [3] have calculated for the T case the fully developed Nusselt numbers for microtubes through which a rarefied gas flows by taking into account the viscous dissipation but neglecting axial conduction in the fluid and the flow work. They defined the Brinkman number (Eq. 20) with ATref = Te — Tip and used in the slip boundary conditions (Eqs. 8 and 19) (x = = a., = a, = 1. The values of the Nusselt... 

Axial conduction in the fluid leads to an increased temperature difference between the wall and the fluid. Therefore, Nu decreases in the entrance region. Axial... 

For the T3 boundary condition the average Nusselt number for fuUy developed laminar flow with negligible external volume forces (fent = 0), axial heat conduction (Pe —> oo), viscous dissipation (Br = 0), flow work (p = 0) and thermal energy sources (Sg = 0) within the fluid is a function of the dimensionless wall thermal resistance / the values of Nut3 as a function of / w are quoted in Tab. 2. When Rw tends to zero the Nusselt number tends to the value taken under T boundary conditions. On the contrary, for Rv, oo the Nusselt number tends to the value of the H boundary condition (48/11). In Tab. 3 the values of Nut3 are tabulated as a function of the dimensionless wall thermal resistance / w and the Peclet number by taking into account the axial conduction in the fluid. 

Aoto et al. (2014) presented a summary of recent SFR developments. SabharwaU et al. (2012) investigated the effects of axial conduction in the fluid of a liquid metal reactor and concluded that the effect is small for the fluids of interest and natural-circulation... 

Mala GM, Li D, Werner C (1997b) Flow characteristics of water through a micro-channel between two parallel plates with electro kinetic effects. Int J Heat Fluid Flow 18 491 96 Male van P, Croon de MHJM, Tiggelaar RM, Derg van den A, Schouten JC (2004) Heat and mass transfer in a square micro-channel with asymmetric heating. Int J Heat Mass Transfer 47 87-99 Maranzana G, Perry I, Maillet D (2004) Mini- and micro-channels influence of axial conduction in the walls. Int J Heat Mass Transfer 47 3993 004 Maynes D, Webb BW (2003) Full developed electro-osmotic heat transfer in microchannels. Int J Heat Mass Transfer 46 1359-1369... 

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

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. 

In the following sections, the main deviations from classical theory for the calculation of heat transfer in microchaimels are discussed. This discussion includes axial heat conduction in the fluid, conjugate heat transfer, surface roughness, viscous dissipation, thermophysical property variations, electric double layers, entrance region and measurement accuracy. Whenever possible, the reader is referred to design criteria and Nu correlations when the different aspects have to be taken into account. 

The contribution of the axial heat conduction in the chaimd walls to the total heat transfer depends on the ratio of the conductivities of the wall and the fluid, on the ratio of the wall thickness and channel diameter and on the Pedet number. As the wall thickness in macroscale applications is of small size compared with the chaimel diameter, axial conduction in the channel walls is neglected [41]. 

The IR technique also yielded temperature distributions (Fig. 2.17) in the symmetry plane at Re = 30 and g = 19 x lO W/m. The wall temperature decreases by axial conduction through the solid walls in the last part of the micro-channel (x/L > 0.75) since this part is not heated. Neither the wall nor the fluid bulk temperature distribution can be approximated as linear. 

Heat transfer in micro-channels occurs under superposition of hydrodynamic and thermal effects, determining the main characteristics of this process. Experimental study of the heat transfer in micro-channels is problematic because of their small size, which makes a direct diagnostics of temperature field in the fluid and the wall difficult. Certain information on mechanisms of this phenomenon can be obtained by analysis of the experimental data, in particular, by comparison of measurements with predictions that are based on several models of heat transfer in circular, rectangular and trapezoidal micro-channels. This approach makes it possible to estimate the applicability of the conventional theory, and the correctness of several hypotheses related to the mechanism of heat transfer. It is possible to reveal the effects of the Reynolds number, axial conduction, energy dissipation, heat losses to the environment, etc., on the heat transfer. 

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. 

In this lecture, a variety of results for convective heat transfer in microtubes and microchannels in the slip flow regime under different conditions have been presented. Both constant wall temperature and constant wall heat flux cases have been analyzed in microtubes, including the effects of rarefaction, axial conduction, and viscous dissipation. In rough microchannels the abovementioned effects have also been investigated for the constant wall temperature boundary condition. Then, temperature-variable dynamic viscosity and thermal conductivity of the fluid were considered, and the results were compared with constant property results for microchannels, with the effects of rarefaction and viscous dissipation. 

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