8. Conclusion

They also obtained a supercritical-fluid Nusselt number correlation Nuscf for the tube flow

where ψ is a correction factor for the airfoil heat exchanger, and ψ is the ratio of the actual heattransfer rate to the heat-transfer rate of the ideal counter-flow heat exchanger without thermal

Here, ξ is an incidence of air at the inlet. The incidence is a flow-direction angle from the airfoil camber (center) line at its leading edge, corresponding to an angle of attack of α ¼ 9:47� for the

> <sup>φ</sup>scf <sup>¼</sup> <sup>T</sup>scf,out−Tscf,in Tair,in−Tscf,in

> <sup>φ</sup>air <sup>¼</sup> <sup>T</sup>air,in−Tair,out Tair,in−Tscf,in

Here, φscf and φair are positive for an air-cooled system and negative for an air-heated system.

<sup>ε</sup>SA <sup>¼</sup> <sup>m</sup>scfCP, scf mairCP, air

Here, mscf and mair are the mass flow rates of a supercritical-fluid and air, respectively, for an airfoil heat exchanger, and CP, scf and CP,air are the specific heats of a supercritical-fluid and air, respectively. κ is the overall heat-transfer coefficient for an ideal counter-flow heat exchanger

> <sup>κ</sup> <sup>¼</sup> <sup>1</sup> 1 <sup>h</sup>scf <sup>þ</sup> <sup>1</sup> hair Ascf Aair

Here, Ascf and Aair are areas of supercritical-fluid-contact and air-contact surfaces, respectively, for an airfoil heat exchanger. ΔTlm, entire is the logarithmic mean temperature difference:

Φ is the ratio of the logarithmic mean temperature difference to the temperature difference

between the inlet air temperature and the supercritical-fluid temperature.

ΔTlm, entire ¼ Φ½Tair,in−Tscf,in� (98)

<sup>ψ</sup> <sup>¼</sup> <sup>0</sup>:1236½0:02093jξj þ <sup>1</sup>�

cascade in Figure 8. φscf and φair indicate the temperature effectiveness, as follows:

Qentire ¼ ψκAscfΔTlm, entire, (92)

<sup>φ</sup>scf−exp½−0:5minf1, <sup>ε</sup>SAg� <sup>þ</sup> <sup>1</sup> (93)

(94)

(95)

(96)

(97)

Moreover, the heat-transfer rate Qentire of an airfoil heat exchanger is estimated as follows:

given by Eq. (33).

144 Heat Exchangers– Advanced Features and Applications

resistance.

εSA is the ratio of the heat-capacity rates.

without thermal resistance.

The Nusselt number between supercritical fluid flows and solid walls can be estimated by appropriate conventional correlations using the Reynolds and Prandtl numbers if sufficiently accurate physical properties are used for each local point through the region of supercritical fluid flows. Thus, a numerical integration of local heat flow rate is required when you calculate the entire heat flow rate in a heat exchanger between supercritical fluid flows and solid walls.

The recovery temperature should be considered for the estimation of heat transfer between compressible flows and solid walls. For compressible flows on adiabatic airfoil surfaces, the local recovery temperature varies by each point on the airfoil surface, owing to the accelerating and decelerating effects of the main flow outside of the boundary layer on the airfoil surface. In addition, for compressible flows on cooling and heating airfoil surfaces, the local total temperature on airfoil surfaces in the boundary layer also varies at each point because of cooling and heating effects. The accelerating and decelerating effects can be estimated from the local Mach number distribution on the airfoil shape. The cooling and heating effects can also be estimated when a numerical integration of elapsed variation of the local total temperature along the boundary layer from the leading edge if the detailed solid temperature distribution on the airfoil surface is known. To obtain the detailed solid temperature distribution on the airfoil surface, detailed experimental measurements or an accurate CFD analysis may be required.

To estimate conjugate heat transfer through a practical heat exchanger with a complex shape, one solution is a combination of experimental results in wind tunnel tests and an inverse heat conduction method. The other solution is CFD analysis validated by experimental results in wind tunnel tests. Empirical correlations are very limited for conjugate heat transfer through a practical heat exchanger with complex shape because knowledge in this field is still developing.
