## 1. Introduction

The range of use of heat exchangers is being expanded to extensive applications in various fields. In particular, supercritical fluids and high‐speed air, that is,, compressible fluids, are suitable as working fluids.

Supercritical fluid is a phase of substances, in addition to the solid, liquid, and gas phases. In particular, in the vicinity of the critical point, many physical properties behave in an unusual way. For example, the density, viscosity, and thermal conductivity drastically change at the critical point, the specific heat and thermal expansion ratios diverge at the critical point, and the sound velocity is zero at the critical point. The physical properties of a supercritical fluid must be evaluated by the appropriate equation of state and equation of the transport properties.

On the other hand, a compressible flow can be assumed as an ideal gas, but additional dynamic energy, that is, the Mach‐number effect, must be considered. Therefore, three types of pressures (static, total, and dynamic), four types of temperatures (static, total, dynamic, and recovery), the difference between laminar and turbulent boundary layers, etc., should be distinguished and treated.

## 2. Equations of state and transport properties of supercritical fluid

**Figure 1** shows the *P*–*T* diagram of a pure substance (water in this case), which is also called a phase diagram. The sublimation curve divides the solid and gas phases, the melting curve divides the solid and liquid phases, and the vaporization curve divides the liquid and gas phases. Two phases coexist on these three curves. When the pressure and/or temperature change across these three coexistence curves of solid‐gas, solid‐liquid, and liquid‐gas, the density discontinuously changes. These three coexistence curves meet at the triple point, which is the unique point where solid, liquid, and gas coexist in equilibrium.

The vaporization curve ends at the critical point. On the vaporization curve, liquid is called the saturation liquid, and gas is called the saturation gas (vapor). When approaching the critical point along the vaporization curve, the density of the saturation liquid decreases, and the density of the saturation gas (vapor) increases. Finally, they meet at the critical point. Fluid overtaking the critical point in temperature and pressure is called the “supercritical fluid.”

The phase is thermodynamically determined by the Gibbs free energy

Where

That is, the phase is determined by the balance between the diffusivity caused by the thermal mobility of the molecules and the condensability by intermolecular forces. The diffusivity caused by thermal mobility increases with the temperature. The condensability by intermolecular forces increases with the density. In general, the following relationships hold:

Solid | Diffusivity << condensability |

Liquid | Diffusivity < condensability |

Supercritical fluid | Diffusivity ≈ condensability |

Gas (vapor) | Diffusivity > condensability |

Gas (ideal gas) | Diffusivity >> condensability |

In **Figure 1**, the first‐order differentials of the Gibbs free energy

are discontinuous across the three coexistence curves, but the first‐order differentials of the Gibbs free energy are continuous at the critical point. In addition, the second‐order differentials of the Gibbs free energy are discontinuous at the critical point.

Here,

**Figures 2** and **3** show the isobaric and isothermal changes of the density, viscosity, and kinematic viscosity by using the data from references [2, 3]. Both the density (derived from the equation of state) and the viscosity (derived from the equation of the transport properties) drastically change at the critical point, and the derivatives with respect to temperature and pressure diverge at the critical point. The kinematic viscosity (combined with the density and viscosity) has an extremum value at the critical point. The equations of state and the transport properties should consider these types of tricky features in the vicinity of the critical point for transcritical‐ and supercritical‐fluid flows.

The most important substances in practical applications are carbon dioxide and water, although all substances have a supercritical‐fluid phase. Recently, accurate correlations for the equations of state and the transport properties containing the critical point have been proposed.

For carbon dioxide, Span and Wagner proposed the equation of state from the triple point to 1100 K at pressures up to 800 MPa [2]. Their equation of state is briefly introduced here. They expressed the fundamental equation in the form of the Helmholtz energy *A*:

with two independent variables—the density

Where

Pressure

Entropy

Internal energy

Isochoric specific heat

Enthalpy

Isobaric specific heat

Saturated specific heat

then,

Speed of sound

etc.

Here,

For carbon dioxide, Vesovic et al. proposed transport properties in the temperature range of 200–1500 K for the viscosity

For water, Wagner and Pruß proposed the equation of state for the temperature range of 251.2–1273 K and pressures up to 1000 MPa [4]. Huber et al. proposed the transport properties from the melting temperature to 1173 K at 1000 MPa [5, 6].

## 3. Heat transfers between supercritical fluid flow and solid

As mentioned in Section 2, the kinematic viscosity of a supercritical fluid is less than those of a liquid and gas; therefore, the Reynolds number, Re, of a supercritical fluid flow is higher than those of a liquid and gas flow with the same velocity, and a turbulent flow is easily formed. For heat transfer in a turbulent flow, Dittus and Boelter proposed a correlation of the Nusselt number using the **Figure 4**.

Here, the superscript

(23) |

Here,

Ito et al. proposed an airfoil heat exchanger, which is applied between a compressible airflow and a liquid or a supercritical fluid flow [9]. It has an outer airfoil shape suitable for high‐speed airflow and contains several tubes for a high‐pressure liquid or a supercritical fluid flow. The researchers installed a cascade of airfoil heat exchangers into a subsonic wind tunnel at a temperature of

These correlations are very simple and similar to the Dittus‐Boelter correlation in Eq. (20) but have sufficient accuracy. Ito et al. used accurate equations of state and the transport properties, as mentioned in Section 2. They said in reference [9] that ordinary correlations (of course, containing the Dittus‐Boelter correlation) for liquid and gas can be used when sufficiently accurate equations of state and the transport properties are used. However, the physical properties at a temperature and pressure in the vicinity of the critical point continuously change throughout the tube because of the heat input and/or pressure loss; therefore, changes in these physical properties throughout the tube should be sufficiently considered. For example, the present author recommends the numerical integration of local heat transfer correlations using local accurate physical properties for the entire tube.

## 4. Thermofluid dynamics of compressible flow on solid wall

### 4.1. Meanings of temperature and pressure of compressible flow

A stationary fluid pressure of ^{3}/kg], and constant temperature

The “pressure” (often called “static pressure”)

in addition to

Here, *P* decreases, and *P* increases, and *P* remains constant,

Next, we consider thermal energy. A stationary fluid at an isochoric specific heat of

Here, the internal energy is an index of the thermal energy level contained in a stationary fluid. In the case of a constant

The “temperature” (often called “static temperature”)

In cases where a fluid is assumed as an ideal gas,

where

A stationary fluid at an isochoric specific heat of

In the case of a constant

where

Here, *h* decreases, *T* increases, and

### 4.2. Isentropic change and sound speed of ideal gas

The specific heat ratio

This equation is substituted into Eqs. (40) and (44). Then,

The change in the entropy

When isentropic change

We totally differentiate Eq. (41), obtaining the following:

We substitute the final equation of Eq. (59) and Eq. (45) into the rightmost part of Eq. (55):

We integrate Eqs. (57) and (62):

The sound speed

Eqs. (63) and (41) are substituted into Eq. (65), yielding the following:

### 4.3. Relationships of static and total values in isentropic compressible flow

The one‐dimensional energy equation of an isentropic flow at an arbitrary cross section is derived by using Eq. (46) as:

When the enthalpy and velocity are

This relationship is true even if cross section 1 corresponds to the stagnant cross section 0 (

Eqs. (44) and (50) are substituted into Eq. (69), yielding the following:

Eq. (66) is substituted into Eq. (70):

We multiply by

At the stagnant cross section 0, the static temperature

### 4.4. Relationships of local Mach number, pressure and temperature of flows on adiabatic walls

**Figure 5** shows the pressure distribution on a plane and an airfoil. On both the plane and the airfoil, boundary layers are formed. The pressure

The pressure distribution on a solid wall is usually expressed by pressure coefficient

but is sometimes expressed by another pressure coefficient

The two expressions are related as follows:

On a plane,

**Figure 6** shows the temperature distribution on an adiabatic plane and an airfoil. In flows on an adiabatic wall, the total temperature

Here, the static temperature

On an adiabatic plane,

### 4.5. Recovery temperature definition in boundary layer in compressible flow on adiabatic, heating and cooling walls

Eckert surveyed and organized the heat transfer in a boundary layer in a compressible flow on a wall [10]. In a boundary layer on an adiabatic plane, the adiabatic‐wall temperature reaches **Figure 7** shows a schematic of the total temperature **Figure 7**.

In cases where a thermal boundary layer is completely inside a momentum boundary layer, that is, Pr ≥ 1 the heat generated by the braking effect uses the rise of the static temperature

Here,

where

Here, Eckert's theory is extended to the recovery temperature

(86) |

where

## 5. Mach‐number distribution on solid walls with various shapes

As described in Section 4 4, the local Mach number

the distribution of the local pressure coefficient **Figure 8**), the distribution of the local pressure coefficient

Ito et al. obtained distributions of _{2}I_{8b})10 airfoils, as shown in the right frame of **Figure 9**, from **Figure 9** [10].

## 6. Air‐temperature distribution in boundary layers on solid walls

Nishiyama described in his book [11] that a developing boundary layer transforms from a laminar boundary layer to a turbulent boundary layer at **Figure 9**, a developing boundary layer may transform at **Figure 10**). This

## 7. Heat transfer through practical heat exchanger with complex shape

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

Second, they assumed *n* parts of the air‐contacted outer surface of the airfoil heat exchanger, as well as one heat‐transfer coefficient

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

Using these procedures, Ito et al. obtained an air Nusselt number correlation _{2}I_{8b})10 airfoils, as shown in **Figure 8** [12].

They also obtained a supercritical‐fluid Nusselt number correlation

Moreover, the heat‐transfer rate

where

Here, **Figure 8**.

Here,

Here,

Here,

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

The actual heat‐exchange rate is estimated as

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.

## 8. Conclusion

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.