7. Heat transfer through practical heat exchanger with complex shape

experimental results of Slocal or ηlocal have been reported for single airfoils and cascades of

Ito et al. obtained distributions of Mlocal around an airfoil in a cascade of NACA65-(12A2I8b)10 airfoils, as shown in the right frame of Figure 9, from Slocal, which is shown in the left frame of

Nishiyama described in his book [11] that a developing boundary layer transforms from a laminar boundary layer to a turbulent boundary layer at Rex≅104 in regions with adverse

Figure 10. Recovery-temperature distribution assumed according to the pressure coefficient and local Mach number

airfoils. The distributions of Mlocal are calculated using Eqs. (75), (76), and (78).

Figure 9. Local Mach-number distributions assumed from pressure-coefficient distribution.

142 Heat Exchangers– Advanced Features and Applications

6. Air-temperature distribution in boundary layers on solid walls

Figure 9 [10].

distributions in Figure 9.

Ito et al. evaluated the rate of heat transfer from a hot compressible airflow to a cold supercritical-fluid flow through an airfoil heat exchanger, as shown in Figure 8 [10]. Heat is transferred from the hot compressible airflow to the outer surface of the airfoil heat exchanger and is conducted from the outer surface to the five inner surfaces in the airfoil heat exchanger. Then, heat is transferred from the five inner surfaces of the airfoil heat exchanger to the cold supercritical-fluid flow inside the five tubes.

First, Ito et al. conducted wind-tunnel experiments. They installed n thermocouples into the airfoil heat exchanger and experimentally measured the temperature at n points inside the exchanger. Simultaneously, the air temperature and the air Mach number at the inlet, the supercritical-fluid temperature and pressure at the inlet, and the supercritical-fluid temperature at the outlet were experimentally measured.

Second, they assumed n heat-transfer coefficients hair, <sup>1</sup> to hair,<sup>n</sup> for the n parts of the aircontacted outer surface of the airfoil heat exchanger, as well as one heat-transfer coefficient hsfc for the supercritical-fluid-contacted five inner surfaces of the airfoil heat exchanger.

Third, they performed an inverse heat-conduction analysis. The boundary conditions were set according to the experimental results for the distribution of the recovery temperature using the methods described in Sections 4.6, as well as the inlet supercritical-fluid temperature and pressure. Using these boundary conditions, heat-conduction calculations for the airfoil heat exchanger were conducted, and the temperatures at the n points in the airfoil heat exchanger and the outlet supercritical temperature were numerically obtained.

Finally, the n þ 1 numerically obtained temperatures were compared with n þ 1 experimentally obtained temperatures. If the temperatures were equal, the assumed hair,<sup>1</sup> to hair,<sup>n</sup> and hscf were true. Otherwise, the assumed hair,<sup>1</sup> to hair,<sup>n</sup> and hscf were corrected, and the inverse heatconduction analysis was repeated.

Using these procedures, Ito et al. obtained an air Nusselt number correlation Nuair for a cascade of NACA65-(12A2I8b)10 airfoils, as shown in Figure 8 [12].

$$\mathrm{Nu\_{air}} = 4.9410^{-3} \mathrm{Re\_{air\&i}} \mathrm{M}\_{\mathrm{in}}^{1.44} \tag{91}$$

They also obtained a supercritical-fluid Nusselt number correlation Nuscf for the tube flow given by Eq. (33).

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

$$Q\_{\text{centre}} = \psi \kappa A\_{\text{sf}} \Delta T\_{\text{lm, entire}},\tag{92}$$

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 resistance.

$$\psi = \frac{0.1236[0.02093|\xi| + 1]}{\phi\_{\rm scf} - \exp[-0.5 \text{min}\{1, \varepsilon\_{\rm SA}\}]} + 1\tag{93}$$

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

$$\phi\_{\text{scf}} = \frac{T\_{\text{scf,out}} - T\_{\text{scf,in}}}{T\_{\text{air,in}} - T\_{\text{scf,in}}} \tag{94}$$

$$\phi\_{\rm air} = \frac{T\_{\rm air,in} - T\_{\rm air,out}}{T\_{\rm air,in} - T\_{\rm scf,in}} \tag{95}$$

Here, φscf and φair are positive for an air-cooled system and negative for an air-heated system. εSA is the ratio of the heat-capacity rates.

$$
\varepsilon\_{\rm SA} = \frac{m\_{\rm scf} \mathbb{C}\_{\rm P,sf}}{m\_{\rm air} \mathbb{C}\_{\rm P,air}} \tag{96}
$$

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 without thermal resistance.

$$\kappa = \frac{1}{\frac{1}{h\_{\text{ref}}} + \frac{1}{h\_{\text{air}}} \frac{A\_{\text{ref}}}{A\_{\text{air}}}} \tag{97}$$

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:

$$
\Delta T\_{\text{lm, entrie}} = \Phi[T\_{\text{air,in}} - T\_{\text{scf,in}}] \tag{98}
$$

Φ is the ratio of the logarithmic mean temperature difference to the temperature difference between the inlet air temperature and the supercritical-fluid temperature.

$$\begin{aligned} \boldsymbol{\Phi} &= \mathbf{1} & \text{for} & \boldsymbol{\varepsilon}\_{\rm SA} = \mathbf{1} \\ \boldsymbol{\Phi} &= \frac{|\boldsymbol{\phi}\_{\rm scf}| - |\boldsymbol{\phi}\_{\rm air}|}{\ln\left[\frac{\boldsymbol{\phi}\_{\rm air}}{\boldsymbol{\phi}\_{\rm scf}}\right]} & \text{for} & \boldsymbol{\varepsilon}\_{\rm SA} \neq \mathbf{1} \end{aligned} \tag{99}$$

The actual heat-exchange rate is estimated as Qentire½numberofairfoils�.

For example, Ito et al. performed cycle calculations for an intercooled and recuperated jet engine employing several pairs of airfoil heat exchangers whose heat-transfer performance is evaluated by Eqs. (91)–(99) [13].

These examples can be used for a cascade of airfoil heat exchangers; therefore, the air Nusselt number correlation in Eq. (91) or thermal resistance in Eq. (93) might be further modified in the near future according to the progress of research, as knowledge in this field is still developing.
