## 133 Heat fluxes

The heat, Q, flowing into the fluid through the cellular medium per unit width is related to the heat transfer coefficient by:

where L is the length of the foam layer in Figure 13.1. Here ATlm is the logarithmic mean temperature. It is related to the temperature of the heat source T1 as well as fluid temperature at the inlet, T0, and that at the outlet, Te by:

Usually, T1 and T0 are specified by the application. Accordingly, Te must be assessed in order to determine Q. For preliminary estimates, the approximation

may be used. Explicit determination requires either experimental measurements or application of the following expressions governing the fluid flows. The temperature in the fluid along the x-direction varies as

where I is a transfer length governed by the properties of the cellular metal, the fluid and the substrate. In the absence of a thermal resistance at the attachments, this length is:

PaCpbVf

where cp is the specific heat of the fluid and q = 1 — 0.22(p/ps). The exit temperature may thus be determined by introducing I from equation (13.12) into (13.11) and setting x = L, whereupon Tf = Te.

Expected trends in the heat flux, Q, dissipated by cellular metals (in W/m2) can be anticipated by using the above formulae. Typically, this is done using non-dimensional parameters, as plotted in Figure 13.2 with air as the cooling fluid. The parameters are defined in Table 13.1. The principal feature is the substantial increase in heat dissipation that can be realized upon either decreasing the cell edge diameter, d, or increasing the relative density, p/ps. Eventually, a limit is reached, governed by the heat capacity of the cooling fluid.

Prif2

1000 800 650 400 200

1000 800 650 400 200 Present materials

Figure 13.2 The heat flux Q = Q/ks[Tj — T0 ] into the fluid, plotted as a function of the relative density, p, and the dimensionless cell-edge diameter, d = d/L

### Present materials

Figure 13.2 The heat flux Q = Q/ks[Tj — T0 ] into the fluid, plotted as a function of the relative density, p, and the dimensionless cell-edge diameter, d = d/L

 Heat flux Q = -- Q/ks[T1 - T0] Prandtl number Pr = = Va/^a Reynolds number lie II Cell wall thickness d = d/L Foam thickness b = b/L Nusselt number Nu = Biks/ka Thermal conduction K f — y/ka/ks Power dissipation P = ApvfbL 2/Pavl

aa = thermal diffusivity of cooling fluid.

aa = thermal diffusivity of cooling fluid.

0 0