132 Heat transfer coefficient

The cellular metal is envisaged as a system that transfers heat from a hot surface into a fluid. Thermal performance is characterized by an effective heat transfer coefficient, Hc, which is related to the heat flux, per unit area, q, from the hot surface in the standard manner (e.g. Holman, 1989):

where AT is a representative temperature drop, roughly equal to the temperature difference between the hot surface and the incoming fluid. A more precise definition is given later. The goal is to develop a cellular system with large Hc that also has acceptable pressure drops and occupies a small volume (compact).

The determination of the heat transfer coefficient, Hc, can be approached in several self-consistent ways. The one presented in this chapter regards the cellular metal as a geometric variant on a staggered bank of cylinders (BOC) oriented normal to the fluid flow. The modified BOC solution has proportionality coefficients that reflect the geometric differences between the foam and the cylinder. This approach has been validated experimentally and the unknown coefficients calibrated. Only the results are given here.

The heat transfer coefficient, Hc, for the cellular metal is (Lu et al., 1998):

2b d

Here p = p/ps, kef is an effective thermal conductivity related to the actual thermal conductivity of the constituent metal, ks, by:

and b is the thickness of the medium (Figure 13.1). The coefficient of 0.28 has been determined by experimental calibration, using infrared imaging of the cellular medium (Bastawros and Evans, 1997; Bastawros et al., in press). The heat transfer that occurs from the metal ligaments into the fluid can be expressed through a non-dimensional quantity referred to as the Biot number:

dks where h is the local heat transfer coefficient. The Biot number is governed by the dynamics of fluid flow in the cellular medium. The established solutions for a staggered bank of cylinders (e.g. Holman, 1989) are:

where Re, the Reynolds number, is

Va with vf the free stream velocity of the fluid, va its kinematic viscosity, ka its thermal conductivity and Pr is the Prandtl number (of order unity). For the cellular metal, Bi will differ from equation (13.5) by a proportionality coefficient (analogous to that for the thermal conductivity) resulting in an effective value:

where the coefficient 1.2 has been determined by experimental calibration (Bastawros et al., in press).

This set of equations provides a complete characterization of the heat transfer coefficient. The trends are found upon introducing the properties of the foam (d, p/ps and ks), its thickness b, and the fluid properties (va, ka and Pr), as well as its velocity vf. The caveat is that the proportionality constants in equations (13.3) and (13.7) have been calibrated for only one category of open cell foam: the DUOCEL range of materials. Open-cell foams having different morphology are expected to have different coefficients. Moreover, if p/ps and d are vastly different from the values used in the calibration, new domains of fluid dynamics may arise, resulting again in deviations from the predictions.

The substrate attached to the cellular medium also contributes to the heat transfer. In the absence of a significant thermal constriction, this contribution may be added to Hc. Additional interface effects can reduce Hc, but we shall ignore these.

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