Contributed by the Fluids Engineering Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Fluids Engineering Division October 27, 2000; revised manuscript received May 21, 2001. Associate Editor: L. Mondy. The necessity for fundamental investigations in forced convection through porous media by fluids with temperature-dependent viscosity cannot be overstated once we take into account the lack of information on this subject 12. Apart from the fundamental nature of the problem, which merits its study, many contemporary applications involving convection of fluids with temperature-dependent viscosity are hampered by this lack of understanding. One specific application is the new microporous enhanced cold-plate design for cooling avionics 3, specifically airborne, military, phased-array radar slats. This cold-plate operates using poly-alpha-olefin oil as coolant. Poly-alpha-olefins (PAO), a class of synthetic oils commonly used in cooling military avionics, has viscosity strongly dependent on temperature. Suitability of a particular porous material for such cold-plate design or the optimization of an existing design requires an accurate prediction of the global pressure-drop across the heated channel. The conventional procedure for predicting the pressure-drop of a fluid flowing through a porous channel using the global Hazen-Dupuit-Darcy (HDD) model (also known as the Forchheimer-extended Darcy model), fails unexpectedly, 45, when the fluid has temperature-dependent viscosity and the channel is heated/cooled. This inappropriateness of the global HDD model is because it is unable to capture the indirect viscosity effect on the global form-drag term. Alternatively, new predictive models that consider the temperature dependency of viscosity were proposed recently in 45 and were validated against numerical results. The objective of this note, by using the experimental results for poly-alpha-olefin’s (PAO) obtained with a new microporous-enhanced cold plate design for cooling avionics 6, is to verify the appropriateness of these two theoretical models as useful tools for engineering design. The verification of the models focus on the hydraulic performance of the cold-plate, in lieu of previous works 57 showing minimal temperature-dependent viscosity effect on the thermal performance of the cold-plate. Perturbation analysis presented in 5, to predict the effects of a fluid with temperature-dependent viscosity flowing through an isoflux-bounded porous medium channel is summarized as follows. The analysis considers initially the differential HDD model, for fully developed (hydrodynamic and thermal) flow situation, with constant coefficients and temperature-dependent viscosity, namely (1)C0ρK0u2+μTu−GK0=0where C0 and K0 are, respectively, the form and permeability coefficients of the porous medium, ρ and μ are the density and dynamic viscosity of the fluid, u is the local longitudinal fluid Darcy speed, and the local longitudinal pressure gradient G=dP/dx=ΔP/L, with L being the length of the porous channel. The subscript “0” on the porous medium properties K and C of Eq. (1) reminds us that these quantities are obtained through experiments under isothermal condition. In this case, the fluid viscosity is uniform throughout the channel, μT=μTin=μin. Observe that Eq. (1) incorporates the form-drag effect of the porous medium by adding the quadratic velocity term to the Darcy equation 8. The quadratic equation given by Eq. (1) when solved for u will result in a positive root, which will be a function of μT with a solution resembling u=FμT. The temperature dependency of the dynamic viscosity of the fluid can be approximated as a second-order Taylor’s series expansion enabling us to express FμT as a function of T (the local unknown temperature) and a reference (known) temperature. Assuming small change in temperature along the channel, for negligible longitudinal diffusion, zero-, first-, and second-order approximations were obtained for the transversal fluid speed u and temperature T, as well as for the global fluid-speed versus pressure-drop relation. The first- and second-order solutions (denoted with subscripts 1 and 2, respectively) for the global (cross-section averaged) fluid velocity are (2)U1=a1+a2N3(3)U2=a1+a2N3+a22N245a1−a2M15+a3N215where a1=GK02μw −1+1+4rra2=GK02μwr 1−11+4ra3=2GK0μw1+4r3/2(4)r=ρC0K02Gμw2N=q″Hke1μw dμdTTwM=q″Hke2 1μw d2μdT2Twwhere q″ is the surface heat flux, ke is the effective thermal conductivity of the medium, and H is half the distance between the plates. The viscosity μw and its derivatives are evaluated at the reference temperature Tw, the wall temperature at the exit of the channel. Observe that when the form-drag coefficient C0 is negligible then r→0. In this case, from Eq. (4), a1=a2→GK0/μw and the first-order solution Eq. (2) reduces to (5)U=ΔPL K0μTmax 1+N3a result identical to the result reported in 9, who developed a similar predictive theory for a fluid with temperature-dependent viscosity, but starting with the simpler linear Darcy flow regime, i.e., Eq. (1) replaced by u=[K0/μT]G.Numerical simulations considering convection of a fluid with temperature-dependent viscosity through a uniformly heated, parallel-plates porous channel, and including the form-drag effects, were presented recently by Narasimhan and Lage 4. In this work, the authors show the limitations of the global HDD model, viz. (6)ΔPL=μK0 U+ρC0U2in accurately predicting the pressure-drop along the channel, suggesting a modification to account for the temperature-dependent viscous effects. They also showed that the global HDD model, Eq. (6), is inappropriate because it neglects indirect effects of temperature-dependent viscosity on the form-drag term of the model, a term originally believed to be viscosity-independent. A new global model, (7)ΔPL=ζμμinK0U+ζCρC0U2that accounts for the effects of temperature dependent viscosity in both drag terms of the original HDD model, was proposed. Notice that the viscosity in Eq. (7) is evaluated at the reference inlet temperature of the channel, μin=μTin. In addition, this model retains the same form, i.e., velocity dependency, of Eq. (6). The coefficients ζμ and ζC represent, respectively, the correction for the global viscous and form-drag terms due to the local effect of temperature on viscosity, which affect directly the first-order velocity term and the fluid velocity profile (via viscosity), which in turn, affects indirectly the second-order velocity term. Obviously, for no heating (uniform viscosity), ζμ=ζC=1 and Eq. (7) becomes identical to Eq. (6). Predictive empirical relations for correcting the viscous and form drag terms, complementing this new algebraic (global) model, were obtained 4 as functions of the surface heat flux, (8)ζμ=1−Q″1+Q″0.32511+Q″18.2ζC=2+Q″0.11−ζμ−0.06with the non-dimensional heat flux Q″ given by (9)Q″=q″keK0C0μin dμdTTinObserve from Eq. (7), the N-L model is a direct modification of the original global HDD model, Eq. (6), with the introduction of two new coefficients to capture the temperature-dependent viscosity effects on both of the drag terms. Apart from physical grounds that support the introduction of two coefficients (see 4 for details), the N-L model is expected to agree better with the experimental results than the global HDD model, Eq. (6), as it allows the effects of μT to be accommodated in both of the drag terms. In contrast, the perturbation model starts with the differential HDD model and arrives at a series type solution for uy with the effects of viscosity captured in each of the terms in the series. This differential solution upon integration along y, gives the channel cross-section averaged global fluid speed U, Eq. (2) or (3). The fully developed flow situation necessary for the Nield model, although it can admit nonslug flow like profiles, does not allow changes in the velocity profile along the channel. This implies that the theory considers only the y-variation of viscosity, as the temperature difference along the channel is assumed very small. Hence, the N-L model is expected to perform better than the Nield model, especially for configurations with considerable fluid temperature variation along the channel. A microporous cold-plate with a porous insert made of a compressed aluminum-alloy porous foam sandwiched (brazed) between rectangular 102×508 mm plate sections was designed and manufactured for cooling a phased-array radar slat 310. Electric heaters generating a heat flux q″=0.59 V2, in W/m2, where V is the supply voltage in Volts, were used to heat the channel plates. The volumetric flow rate was varied from 0.5 to 5×10−5 m3/s. Pressure measurements at the inlet Pi and outlet Po gave the total PAO pressure-drop across the cold plate. Details of the experimental apparatus and procedure are found in 6. The uncertainties of the PAO flow rate Q and of the experimental pressure drop ΔP are estimated following the recommendations of 11. A conservative estimate for the uncertainty of the experimental volumetric flow rate reported in this work, UQ/Q is 5 percent. The uncertainty of the pressure-drop across the cold-plate is equal to the precision limit PΔP because both precision pressure gages were calibrated by the manufacturer using the same equipment and procedure, hence the resulting bias limit of the pressure difference is zero. This precision limit is estimated as being equal to twice the standard deviation of several measurements, or approximately 3 percent. The temperature dependence of the dynamic viscosity of PAOs can be modeled 3 as (10)μT=0.1628T−1.0868valid for 5°C⩽T⩽170°C. Within the same temperature range, the variations of density, specific heat and thermal conductivity of PAO are negligible. The effective permeability K0 and the form coefficient C0 of the porous insert can be determined by fitting the experimental no-heating (with the heaters switched off) results with a function of the type (11)ΔP0=LμinK0QAf+LρC0QAf2where L is the cold-plate length, equal to 0.102 m, Af is the flow cross-section area, equal to 5.08×10−4 m2, and μin and ρ are the PAO viscosity and density at 21°C, respectively, 5.95×10−3 kg/ms and 789.2 kg/m3. Out of the inserts tested, as reported in 6, we focus our attention on insert no.3 (initial porosity=0.88, compression ratio=3, final porosity=0.58). Using the experimental results and Eq. (11) the permeability and form-coefficient for this insert are obtained as K0=4.01×10−10 m2 and C0=33.458×103 m−1. This low permeability and high form-coefficient of the chosen porous medium make it particularly suitable for verifying the theoretical models viz. Eqs. (3) and (7), because of the negligible convective inertia and viscous diffusion effects. It is also important to verify how well the assumption of negligible longitudinal diffusion (an assumption made by both the theoretical models) is satisfied. For the present experimental tests, using the values listed previously and αPAO=8.68×10−5 m2/s, the minimum Pe´clet number, Pe=QL/Afα, equals 8,617 supporting the neglect of the longitudinal diffusion effect. Figure 1 compares the experimental pressure-drop results with that predicted by the Nield model, Eq. (3), and the Narasimhan-Lage model, Eq. (7), for a reference coolant temperature T0=21°C, and V=46.9 Vq″=1 kW/m2. To highlight the influence of form-drag effects, predictions by the linear-Darcy model, Eq. (5), is also shown. The comparison for higher heat fluxes (q″=3.9 kW/m2 and 5.8 kW/m2) is shown in Figs. 2 and 3. From Fig. 1 we see, for lower velocities Q<2×10−5 m3/s, both the perturbation theories, Eq. (3) and (5), agree well in their predictions. However, for higher velocities Q>2×10−5 m3/s, with the influence of form-drag gaining strength, the linear-Darcy model, Eq. (5), shows marked deviation, as expected. Both the predictions by Eqs. (3) and (5) yield smaller pressure-drop than the experimental result with the deviation of Eq. (3) being ∼15%. The N-L model, Eq. (7), as predicted in the earlier section, agrees extremely well with the experimental results for all heat fluxes (Figs. 123). For higher heat fluxes, the temperature distribution along the channel grows in strength making the Nield model assumption of small temperature variation along the channel invalid, leading to the systematic deviation of the second-order HDD model from the experimental results. Experiments using a low permeability, high form-coefficient porous medium sandwiched inside an isoflux-bounded parallel-plate channel were performed for verifying two recent theories for predicting the global pressure-drop/fluid-velocity relationship. The results validate the two theoretical models, subject to their respective limitations. The Nield model, Eq. (3), due to the inclusion of the form-drag effects, is definitely better than the earlier model, Eq. (5), based on the simpler Darcy equation. However, Eq. (3) is accurate only for fully developed (hydrodynamic and thermal) flow situations with very small temperature variation along the channel. Predictions from the N-L model show excellent agreement with the experimental results for all of the heat flux values tested. However, it is worth noting here that the high-heat flux correlation for ζμ and ζC, the null-global viscous drag regime as explained in 4, has not been tested because the experimental configuration is limited to moderate heat fluxes by the validity range of Eq. (10).