Axial heat conduction in a moving semi-infinite fluid

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Abstract

This paper considers the steady state conduction of heat from a wall to a fluid moving at a uniform velocity. The wall is heated by a step change in temperature. Although this appears to be a classical heat conduction problem, its application to various convective heat transfer problems is new. The mathematical procedure leads to the computation of the temperature field and the heat transfer coefficient. In the presence of a step change in the wall temperature, it is shown that the Stanton number is a function of the Peclet number alone. The acquired analytical results show that, near the thermal entrance location, heat conduction dominates and the local heat flux becomes independent of velocity. This phenomenon applies to classical convection problems in various-shaped ducts.

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 yˆ=0. 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 xˆ<0 and suddenly changes to Tw when xˆ>0. 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 = kT/∂yy=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), becomesSt=qwk(Tw-Ti)αU=-12ωθ(x,y)yy=0Following standard substitutions, using Eqs. (20a), (20b), the Stanton number isSt=e-Pex/22πK0(Pex/2)-K1(Pex/2)whenx<0andSt=ePex/22πK0(Pex/2)+K1(Pex/

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 xˆzˆ-plane with W = 1 m is dQ×1=h(Tw-Ti)×1×dxˆ. Then, the heat flux per unit length in z-direction is obtainable by integration of dQ over xˆ,Q1,2=xˆ1xˆ2h(Tw-Ti)dxˆIn dimensionless space, when ζ = Pex/2, Eq. (24a) becomesQ1,2k(Tw-Ti)=2αkUζ1ζ2hdζ=2ζ1ζ2StdζThis 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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