Axial Conduction in the Wall

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. 

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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... 

The dimensionless quantity on the left-hand side of Eq. (7.3) labeled as M by the above-cited authors, allows for the comparison of heat transfer by axial conduction in the wall to the convective heat transfer in the flow. 

Maranzana, G., Perry, I., Maillet, D. Mini- and microchannels influence of axial conduction in the walls, Int. J. Heat Mass Transfer 47, (2004) 3993-4004. 

G. Maranzana, I. Perry, D. Maillet, Mini-and microaxial conduction in the walls. International Journal of Heat and Mass Transfer, 2004,47, 3993-4004. 

WALL TEMPERATURE RESPONSE FOR HEAT TRANSFER WITH THE AMBIENT. The following partial differential equation derived from the first law of thermodynamics describes the transient in-wall temperature due to a transfer of heat between the wall and the ambient and the pressurizing gas and also axial conduction in the wall itself ... 

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 this section, the results will be presented in tabular and graphical forms, for Nusselt number for both constant wall temperature and constant wall heat flux cases with variable Kn, Br, Pe values to investigate the effects of rarefaction, viscous dissipation, and axial conduction in the slip-flow regime for microtubes. Table 1 presents the effect of rarefaction on laminar flow fully developed Nu values for constant wall temperature (Nut) and constant wall heat flux (Nuq) cases, where viscous dissipation and axial... 

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. 

It is very well possible that the axial heat conduction in the wall has an influence on the temperature in the region close to the wall. A detailed description of the temperature in the bed as a function of the radial and axial positions can be obtained by extending the model to two dimensions. However, it is also possible to get more insights into the behaviour by adjusting the one-dimensional model. Instead of accounting for the heat capacity of the wall in the accumulation term, an extra energy balance for the wall is included in the model, in which heat is exchanged with the packed bed ... 

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... 

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]. 

At low Re and when conjugate effects have to be considered, the temperature distribution along the microchannel is not linear. Under constant heat flux boundary conditions, Nu decreases with decreasing ratio of outer to inner channel diameter, approaching the constant temperature solution. A decrease in Nu is also seen with increasing wall conductivity. For constant temperature boundary conditions, Nu will increase approaching the constant heat flux solution with axial heat conduction in the wall. The values for local Nusselt number for the conjugated problem lie between the values for the two boundary conditions constant heat flux and constant temperature. 

In microchannels, conjugate heat transfer leads to a complex three-dimensional heat flow pattern and Poiseuille flow may no longer be accurate [43]. Numerical simulations show that axial conduction in the channel wall does lower the Nusselt number but it is still in the range of conventional values [38]. The work of Gamrat et al. [44], in contrast, could not explain the lower Nusselt number by the axial conduction in the channel walls by numerical simulations. 

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. 

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. 

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 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 reactor configuration we propose, shown in Figs. 1 and 2, allows rapid heat transfer along the axial direetion of the reactor by conduction through the wall made of high conductivity metal such as copper or aluminum. The catalyst can be packed into the honeycomb cells or wash coated on the walls of the cells. 

With decreasing cell size, the temperature difference between the wall of the cell and the eatalyst partiele in the cell would decrease to zero. For sufficiently small cell dimensions, we may assume the two temperatures are the same. In this case, the heat conduction through the wall becomes dominant and affects the axial temperature profile. As the external heat exchange is absent and the outside of the reactor is normally insulated, the temperature profile is flat along the direction transverse to the reactant flow, and the conditions in all channels are identical to each other. The energy balance is... 

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