International Journal of Heat and Mass Transfer
Axial heat conduction in a moving semi-infinite fluid
Introduction
Conduction of heat in the presence of a moving wall is widely available in the literature [1], [2]. The emphasis of this paper is to determine the effect of axial conduction of heat, in the flow direction parallel to a wall. Also, the fluid can be viewed as a stationary semi-infinite domain with a moving wall temperature at . A simple model, as shown in Fig. 1, is selected to show this effect and it describes a fluid flowing at a uniform velocity over an infinite plate whose temperature is Ti when and suddenly changes to Tw when . This mathematical model provides useful information as to the behavior of the heat transfer near the thermal entrance location within the flow in ducts.
Knowledge of heat flux near the location where heating begins has application to convective heat transfer in ducts when the axial conduction effect is significant. This phenomenon has been studied in the past for free flow through ducts using various numerical schemes [3], [4], [5], [6], [7] and a complete extended Graetz solution is in [8]. The contribution of axial thermal conduction in porous passages, with walls at uniform temperature, are numerically determined and reported in [9] for parallel plate channel and in [10] for circular ducts. Also, it has applications in micro-devices [11] and cooling of electronic systems, especially when the ducts are filled with fluid saturated porous materials [12], [13]. Accordingly, the flow in channels filled with fluid saturated hyperporous metallic foam [14] is of interest for cooling of electronic devices. Experimental data in [15] show that the Peclet number in channels filled with fluid saturated porous aluminum foam can become relatively small and, under this condition, the contribution of axial conduction becomes significant.
Lahjomri and Oubarra present a procedure in [8] for determination of heat transfer in the presence of axial conduction to free flow through parallel plate channels and circular ducts using a modified Graetz solution method. Also, a study of the effect of axial thermal conduction to the heat transfer phenomena in these ducts, filled with fluid saturated porous materials, is in Minkowycz and Haji-Sheikh [16].
The case of slug flow appears in fluid flow through porous passages when the permeability is small. The studies reported here show that, for other cases, this slug flow assumption provides an asymptotic value for heat transfer near the location where the wall temperature changes. The following mathematical formulations show that both slug flow and no flow conditions approach the same values in the vicinity of thermal entrances location. This implies that near the thermal entrance location the velocity effect becomes negligibly small.
Section snippets
Mathematical formulation
For study of heat transfer at very small x values, a non-series solution is desirable. Alternatively, two limiting solutions are sought: one assumes no flow condition as a lower limit and the other assumes a slug flow as the upper limit. The actual solution is expected to fall between these two solutions. The subsequent formulation considers flow in a semi-infinite region, shown in Fig. 1, whose solution provides these limiting cases. The steady state energy equation assuming constant
Determination of local wall heat flux
Using the heat flux definition, qw = −k∂T/∂y∣y=0, it is observed that Eqs. (20a), (20b) are related to the wall heat flux. These quantities are used to determine expressions for the wall heat flux and then to get the local heat transfer coefficient h = qw/(Tw − Ti); then, the local Stanton number, St = h/(ρcpU), becomesFollowing standard substitutions, using Eqs. (20a), (20b), the Stanton number isand
Total wall heat flux
The total heat flux is needed for the design of cooling devices when the wall is heated discretely. The heat flux leaving a differential element W × dx in the -plane with W = 1 m is . Then, the heat flux per unit length in z-direction is obtainable by integration of dQ over ,In dimensionless space, when ζ = Pex/2, Eq. (24a) becomesThis definition of total heat flow in conjunction with Eqs. (23a), (23b) yields the
Results and discussion
An examination of Eqs. (19a), (19b) shows that there is a significant amount of energy transport by conduction to the fluid before arrival to the heated region. This causes the temperature of the fluid to rise before passing through the x = 0 plane. The temperature when x = 0 is obtainable from Eq. (19a) or Eq. (19b). The limiting value of the second term within the square brackets in Eq. (19a), as x → 0, is π/2. Therefore, when x = 0, the integration of Eq. (19a) or Eq. (19b) yields the function θ(0, y
Conclusion
It is shown that the heat transfer rate at sufficiently small values of x is independent of the magnitude of the velocity, represented as the Peclet number. Indeed, even the no flow and slug flow conditions provide reasonable limiting values near the thermal entrance location. Additionally, this limiting concept is valuable for determination of wall heat flux at the entrance regions of parallel plate ducts, circular pipes, and those with two-dimensional cross sections such as rectangular and
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