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

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

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."

Figure 1. Phase chart on P-T diagram (for water).


The phase is thermodynamically determined by the Gibbs free energy G:

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

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

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."

2. Equations of state and transport properties of supercritical fluid

is the unique point where solid, liquid, and gas coexist in equilibrium.

distinguished and treated.

126 Heat Exchangers– Advanced Features and Applications

Figure 1. Phase chart on P-T diagram (for water).

$$\mathbf{G} = \mathbf{H} \mathbf{-} T \mathbf{S} = \mathbf{U} \mathbf{-} T \mathbf{S} + \mathbf{P} \mathbf{V} \tag{1}$$

Where H is the enthalpy, S is the entropy, U is the internal energy, and V is the specific volume [1].

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:

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

$$dG = -SdT + VdP\tag{2}$$

$$\left(\frac{dG}{dT}\right)\_{\text{P}} = -\mathbb{S} \tag{3}$$

$$\left(\frac{dG}{dP}\right)\_{\rm T} = V = \frac{1}{\rho} \tag{4}$$

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.

$$\left(\frac{d^2G}{dT^2}\right)\_\text{P} = -\frac{1}{T}\left(\frac{dH}{dT}\right)\_\text{P} = -\frac{1}{T}\text{C}\_\text{P} \tag{5}$$

$$
\left(\frac{d^2G}{dP^2}\right)\_\mathrm{T} = \left(\frac{dV}{dP}\right)\_\mathrm{T} = -VK\_\mathrm{T} \tag{6}
$$

Here, ρ is the density, C<sup>P</sup> is the isobaric specific heat, and K<sup>T</sup> is the isothermal compressibility. At the critical point, the density drastically changes, the specific heat and thermal expansion ratio diverge, and the sound velocity is zero.

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

Figure 2. Isobaric changes of the density, viscosity, and kinematic viscosity near the critical point where Tcritical = 304.1282 K and Pcritical = 7.3773MPa (for carbon dioxide).

Figure 3. Isothermal changes of the density, viscosity, and kinematic viscosity near the critical point where Tcritical = 304.1282K and Pcritical = 7.3773MPa (for carbon dioxide).

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:

$$A = \mathcal{U}\text{--}TS = H\text{-}RT\text{-}TS \tag{7}$$

with two independent variables—the density ρ and temperature T. The dimensionless Helmholtz energy φ ¼ A=RT is divided into a part obeying the ideal gas behavior φ � and a part that deviates from the ideal gas behavior φ<sup>r</sup> [2]:

Heat Transfer of Supercritical Fluid Flows and Compressible Flows http://dx.doi.org/10.5772/65931 129

$$
\phi(\delta,\tau) = \phi^\circ(\delta,\tau) + \phi^\mathbf{r}(\delta,\tau),
\tag{8}
$$

Where δ ¼ ρ=ρ<sup>c</sup> is the reduced density; τ ¼ Tc=T is the inverse reduced temperature; and ρ<sup>c</sup> and T<sup>c</sup> are the density and temperature, respectively, at the critical point. Then, all of the other thermodynamic properties can be obtained by the combined derivatives of Eq. (7) using the Maxwell relations [1].

Pressure

$$P(T,\rho) = -\left(\frac{\partial A}{\partial V}\right)\_T \text{ then} \frac{P(\delta,\tau)}{\rho RT} = 1 + \delta \phi\_8^\tau \tag{9}$$

Entropy

$$S(T,\rho) = -\left(\frac{\partial A}{\partial T}\right)\_V \text{ then} \frac{S(\delta,\tau)}{R} = \tau[\phi\_\tau^\circ + \phi\_\tau^\circ] \neg \phi^\circ - \phi^\circ \tag{10}$$

Internal energy

$$\mathcal{U}I(T,\rho) = A - T\left(\frac{\partial A}{\partial T}\right)\_V \text{ then} \frac{\mathcal{U}(\delta,\tau)}{RT} = \tau[\phi\_\tau^\circ + \phi\_\tau^\circ] \tag{11}$$

Isochoric specific heat

$$\text{Cov}(T,\rho) = \left(\frac{\partial \mathcal{U}}{\partial T}\right)\_V \text{ then} \\ \frac{\mathbb{C}\_V(\boldsymbol{\delta}, \boldsymbol{\tau})}{R} = -\boldsymbol{\tau}^2 [\boldsymbol{\phi}\_{\boldsymbol{\tau}\boldsymbol{\tau}}^\circ + \boldsymbol{\phi}\_{\boldsymbol{\tau}\boldsymbol{\tau}}^\circ] \tag{12}$$

Enthalpy

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

Figure 3. Isothermal changes of the density, viscosity, and kinematic viscosity near the critical point where Tcritical =

Figure 2. Isobaric changes of the density, viscosity, and kinematic viscosity near the critical point where Tcritical = 304.1282

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

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

with two independent variables—the density ρ and temperature T. The dimensionless Helm-

A ¼ U−TS ¼ H−RT−TS (7)

�

and a part that

state and the transport properties containing the critical point have been proposed.

expressed the fundamental equation in the form of the Helmholtz energy A:

holtz energy φ ¼ A=RT is divided into a part obeying the ideal gas behavior φ

transcritical- and supercritical-fluid flows.

304.1282K and Pcritical = 7.3773MPa (for carbon dioxide).

K and Pcritical = 7.3773MPa (for carbon dioxide).

128 Heat Exchangers– Advanced Features and Applications

deviates from the ideal gas behavior φ<sup>r</sup> [2]:

$$H(T,\rho) = A - T\left(\frac{\partial A}{\partial T}\right)\_V - V\left(\frac{\partial A}{\partial V}\right)\_T \text{ then } \frac{H(\delta,\tau)}{RT} = 1 + \tau[\phi\_\tau^\circ + \phi\_\tau^\mathbf{f}] + \delta\phi\_\delta^\mathbf{f} \tag{13}$$

Isobaric specific heat

$$\mathbb{C}\_{\rm P}(T,\rho) = \left(\frac{\partial H}{\partial T}\right)\_{\rm P} \text{then} \frac{\mathbb{C}\_{\rm P}(\delta,\tau)}{R} = -\tau^2 [\phi\_{\rm \tau \tau}^\circ + \phi\_{\rm \tau \tau}^\circ] + \frac{\left[1 + \delta \phi\_8^\rm f - \delta \tau \phi\_{\rm \delta \tau}^\circ\right]^2}{1 + 2\delta \phi\_8^\rm f + \delta^2 \phi\_{8\delta}^\rm f} \tag{14}$$

Saturated specific heat

$$\mathbf{C}\_{\sigma}(T) = \left(\frac{\partial H}{\partial T}\right)\_{\mathrm{P}} + T \left(\frac{\partial P}{\partial T}\right)\_{\mathrm{V}} \left(\frac{\partial P\_{\mathrm{sat}}}{\partial T}\right) / \left(\frac{\partial P}{\partial V}\right)\_{\mathrm{T}\_{\mathrm{sat}}} \tag{15}$$

then,

$$\frac{\mathbf{C}\_{\rm{o}}(\delta,\tau)}{R} = -\tau^{2}[\boldsymbol{\phi}\_{\rm{rr}}^{\circ} + \boldsymbol{\phi}\_{\rm{rr}}^{\mathrm{r}}] + \frac{1 + \delta\phi\_{\rm{b}}^{\mathrm{r}} - \delta\tau\phi\_{\rm{b}\tau}^{\mathrm{r}}}{1 + 2\delta\phi\_{\rm{b}}^{\mathrm{r}} + \delta^{2}\phi\_{\rm{b}\boldsymbol{\delta}}^{\mathrm{r}}} \left[ \{1 + \delta\phi\_{\rm{b}}^{\mathrm{r}} - \delta\tau\phi\_{\rm{b}\boldsymbol{\tau}}^{\mathrm{r}} \} - \frac{\rho\_{\rm{c}}}{R\delta} \frac{dP\_{\rm{sat}}}{dT} \right] \tag{16}$$

Speed of sound

$$w(T,\rho) = \sqrt{\left(\frac{\partial P}{\partial \rho}\right)\_{\text{S}}} \text{ then} \frac{w^2(\delta,\tau)}{RT} = 1 + 2\delta\phi\_{\text{\\$}}^{\text{r}} + \delta^2\phi\_{\text{\\$}\delta}^{\text{r}} - \frac{\left[1 + \delta\phi\_{\text{\\$}}^{\text{r}} - \delta\tau\phi\_{\text{\\$}\tau}^{\text{t}}\right]^2}{\tau^2[\phi\_{\text{\\$}\tau}^{\text{\\$}} + \phi\_{\text{\\$}\tau}^{\text{t}}]} \tag{17}$$

etc.

Here,

$$\phi\_{\eth} = \left(\frac{\partial \phi}{\partial \delta}\right)\_{\tau}, \phi\_{\eth \delta} = \left(\frac{\partial^2 \phi}{\partial \delta^2}\right)\_{\tau}, \phi\_{\tau} = \left(\frac{\partial \phi}{\partial \tau}\right)\_{\delta}, \phi\_{\tau \tau} = \left(\frac{\partial^2 \phi}{\partial \tau^2}\right)\_{\delta} \text{ and } \phi\_{\eth \tau} = \left(\frac{\partial^2 \phi}{\partial \delta \partial \tau}\right)\_{\delta}.$$

For carbon dioxide, Vesovic et al. proposed transport properties in the temperature range of 200– 1500 K for the viscosity μ and in the temperature range of 200–1000 K for the thermal conductivity k [3]. Their equations of the transport properties μ and k are briefly introduced. Their fundamental equation combines three independent parts: a part obeying the ideal gas behavior μ � ðTÞ and k � ðTÞ, a part with excess properties because of the elevated density Δμðρ, TÞ and Δkðρ, TÞ, and a part with an enhancement in the vicinity of the critical point ΔcμðTÞ and ΔckðTÞ:

$$
\mu(\rho, T) = \mu^\circ(T) + \Delta\mu(\rho, T) + \Delta\_c \mu(T) \tag{18}
$$

$$k(\rho, T) = k^\circ(T) + \Delta k(\rho, T) + \Delta\_c k(T) \tag{19}$$

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 Re and Prandtl number, Pr, for a liquid flow in a circular automobile radiator [7] as shown in Figure 4.

$$\text{Nu}\_{\text{local},\text{turb}} = 0.023 \text{Re}\_{\text{local}}^{0.8} \text{Pr}\_{\text{local}}^{\text{n}} \tag{20}$$

$$\text{Re}\_{\text{local}} = \frac{\mu\_{\text{local}}D}{\nu\_{\text{local}}} = \frac{\rho\_{\text{local}}\mu\_{\text{local}}D}{\mu\_{\text{local}}} = \frac{4m}{\pi D \mu\_{\text{local}}} \tag{21}$$

$$\text{Pr}\_{\text{local}} = \frac{\nu\_{\text{local}}}{\kappa\_{\text{local}}} = \frac{\mu\_{\text{local}} \mathbf{C}\_{\text{P,local}}}{k\_{\text{local}}} \tag{22}$$

Here, the superscript n ¼ 0:3 for Twall < Tfluid or n ¼ 0:4 for Twall > Tfluid, ulocal is the average velocity across the cross section, D is the diameter of the tube, μ is the viscosity, m is the mass

Figure 4. Heat transfer between a supercritical fluid flow and a circular solid tube wall.

wðT, ρÞ ¼

130 Heat Exchangers– Advanced Features and Applications

τ

<sup>φ</sup><sup>δ</sup> <sup>¼</sup> <sup>∂</sup><sup>φ</sup> ∂δ � �

�

radiator [7] as shown in Figure 4.

etc. Here,

μ � ðTÞ and k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂P ∂ρ � �

, <sup>φ</sup>δ<sup>δ</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup><sup>φ</sup>

the melting temperature to 1173 K at 1000 MPa [5, 6].

s

S

∂δ<sup>2</sup> � �

τ

μðρ, TÞ ¼ μ

kðρ, TÞ ¼ k

3. Heat transfers between supercritical fluid flow and solid

Relocal <sup>¼</sup> <sup>u</sup>local<sup>D</sup>

νlocal

Prlocal <sup>¼</sup> <sup>ν</sup>local

�

�

, <sup>φ</sup><sup>τ</sup> <sup>¼</sup> <sup>∂</sup><sup>φ</sup> ∂τ � �

then <sup>w</sup><sup>2</sup>ðδ, <sup>τ</sup><sup>Þ</sup>

RT <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>δφ<sup>r</sup>

δ

For carbon dioxide, Vesovic et al. proposed transport properties in the temperature range of 200– 1500 K for the viscosity μ and in the temperature range of 200–1000 K for the thermal conductivity k [3]. Their equations of the transport properties μ and k are briefly introduced. Their fundamental equation combines three independent parts: a part obeying the ideal gas behavior

Δkðρ, TÞ, and a part with an enhancement in the vicinity of the critical point ΔcμðTÞ and ΔckðTÞ:

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

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 Re and Prandtl number, Pr, for a liquid flow in a circular automobile

Nulocal,turb <sup>¼</sup> <sup>0</sup>:023Re<sup>0</sup>:<sup>8</sup>

κlocal

Here, the superscript n ¼ 0:3 for Twall < Tfluid or n ¼ 0:4 for Twall > Tfluid, ulocal is the average velocity across the cross section, D is the diameter of the tube, μ is the viscosity, m is the mass

<sup>¼</sup> <sup>ρ</sup>localulocal<sup>D</sup> μlocal

localPr<sup>n</sup>

<sup>¼</sup> <sup>μ</sup>localCP,local klocal

<sup>¼</sup> <sup>4</sup><sup>m</sup> πDμlocal

<sup>δ</sup> <sup>þ</sup> <sup>δ</sup><sup>2</sup> φr δδ−

, <sup>φ</sup>τ<sup>τ</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup><sup>φ</sup>

ðTÞ, a part with excess properties because of the elevated density Δμðρ, TÞ and

∂τ<sup>2</sup> � �

δ

ðTÞ þ Δμðρ, TÞ þ ΔcμðTÞ (18)

ðTÞ þ Δkðρ, TÞ þ ΔckðTÞ (19)

<sup>½</sup><sup>1</sup> <sup>þ</sup> δφ<sup>r</sup>

τ<sup>2</sup>½φ�

δ−δτφ<sup>r</sup> <sup>δ</sup><sup>τ</sup>� 2

<sup>τ</sup><sup>τ</sup>� (17)

<sup>τ</sup><sup>τ</sup> <sup>þ</sup> <sup>φ</sup><sup>r</sup>

and <sup>φ</sup>δ<sup>τ</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup><sup>φ</sup>

local (20)

(21)

(22)

∂δ∂τ � � :

> flow rate, κ is the thermal diffusivity, and k is the heat conductivity. For liquid and gas flows, the fluid properties and flow conditions can be regarded as constant throughout the entire region in most practical cases because the fluid properties are insensitive to temperature and pressure changes in the tube. Therefore, the inlet values of the physical properties and flow conditions can be used, and Re and Pr can be regarded as constant throughout the entire tube. On the other hand, the fluid properties of a supercritical fluid are very sensitive to temperature and pressure changes in the tube. Thus, in the tube, the density gradually changes because of the heat input and/or pressure loss, the local average velocity changes, and even Re and Pr change. Unfortunately, the Dittus-Boelter correlation with the inlet values of the physical properties and flow conditions cannot be directly used for heat transfer in a supercritical turbulent flow. Liao and Zhao measured the rate of the heat transfer between a supercritical carbon dioxide flow and a circular solid tube wall for Twall < Tfluid [8]. Their tube was set in the horizontal direction. They proposed a correlation of area-averaged Nusselt numbers as functions of the Reynolds and Prandtl numbers defined at the temperatures of the mean bulk and the wall.

$$\mathrm{Nu\_{ave, turb}} = 0.128 \mathrm{Re\_{wall}^{0.8}} \mathrm{Pr\_{wall}^{0.3}} \left[ \frac{\mathrm{Gr}}{\mathrm{Re\_{bulk}^2}} \right]^{0.205} \left[ \frac{\rho\_{\mathrm{bulk}}}{\rho\_{\mathrm{wall}}} \right]^{0.437} \left[ \frac{\mathrm{C\_{P,bulk}}}{\mathrm{C\_{P,wall}}} \right]^{0.411} \text{ forCO}\_2 \tag{23}$$

$$\text{Gr} = \frac{[\rho\_{\text{wall}} - \rho\_{\text{bulk}}]\rho\_{\text{bulk}}gD^3}{\mu\_{\text{bulk}}^2} \tag{24}$$

$$\text{Re}\_{\text{bulk}} = \frac{4m}{\pi D \mu (T\_{\text{bulk}}, P\_{\text{bulk}})} \tag{25}$$

$$
\rho\_{\text{bulk}} = \rho(T\_{\text{bulk}}, P\_{\text{bulk}}) \tag{26}
$$

$$
\mu\_{\text{bulk}} = \mu(T\_{\text{bulk}}, P\_{\text{bulk}}) \tag{27}
$$

$$\text{Re}\_{\text{wall}} = \frac{4m}{\pi D \mu (T\_{\text{wall}}, P\_{\text{wall}})} \tag{28}$$

$$\text{Pr}\_{\text{wall}} = \frac{\nu(T\_{\text{wall}}, P\_{\text{wall}})}{\kappa(T\_{\text{wall}}, P\_{\text{wall}})} = \frac{\mu(T\_{\text{wall}}, P\_{\text{wall}}) \text{C}\_{\text{P}}(T\_{\text{wall}}, P\_{\text{wall}})}{k(T\_{\text{wall}}, P\_{\text{wall}})} \tag{29}$$

$$
\rho\_{\text{wall}} = \rho(T\_{\text{wall}}, P\_{\text{wall}}) \tag{30}
$$

$$\mathcal{C}\_{\text{P,wall}} = \mathcal{C}\_{\text{P}}(T\_{\text{wall}}, P\_{\text{wall}}) \tag{31}$$

$$\mathcal{C}\_{\text{P,bulk}} = \mathcal{C}\_{\text{P}}(T\_{\text{bulk}}, P\_{\text{bulk}}) \tag{32}$$

Here, Tbulk ¼ ½Tin þ Tout�=2 and Twall are constant. This correlation is applicable in the range of 7.4 MPa <Pbulk < 12.0 MPa, 20�C < Tbulk< 110�C, 2�C < Tbulk−Twall < 30�C, 0.02 kg/min < m\_ < 0.2 kg/min, 1025 < Gr=Re<sup>2</sup> bulk < 1022 for the horizontal long tubes of 0.50 mm <d < 2.16 mm.Re<sup>0</sup>:<sup>8</sup> wall and Pr<sup>0</sup>:<sup>3</sup> wall were originally derived from the Dittus-Boelter correlation. Gr=Re<sup>2</sup> bulk is the effect of buoyancy in the radial direction of a horizontal tube. The density of fluid at a temperature and pressure in the vicinity of the critical point is very sensitive to changes in temperature; thus, the effect of the buoyancy derived from the temperature difference between the bulk and wall cannot be ignored. This effect is enhanced as the diameter of the tube increases.

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 Tair and measured the heat-transfer coefficient of a liquid or a supercritical fluid flow at a temperature of Tscf < Tair in a vertical tube. They derived correlations for supercritical carbon dioxide and compressed water at a pressure of Pscf≤30 MPa as follows:

$$\mathrm{Nu\_{ave, turb}} = \begin{cases} 0.0230 \mathrm{Re}^{0.808} \mathrm{Pr}^{0.300} & \text{forH}\_2 \mathrm{O} \\ 0.0231 \mathrm{Re}^{0.823} \mathrm{Pr}^{0.300} & \text{forCO}\_2 \end{cases} \tag{33}$$

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 P [Pa], specific volume of V [m3 /kg], and constant temperature T stores a mechanical energy of epre [J/kg]. Here,

$$
\sigma\_{\text{pre}} = PV.\tag{34}
$$

The "pressure" (often called "static pressure") P is the potential of the mechanical energy level contained in a stationary fluid. A motional fluid has an additional dynamic energy edyn [J/kg]:

$$
\varepsilon\_{\rm dyn} = \frac{1}{2} \mu^2. \tag{35}
$$

.in addition to epre; therefore,

ρwall ¼ ρðTwall, PwallÞ (30)

CP,wall ¼ CPðTwall, PwallÞ (31)

CP,bulk ¼ CPðTbulk, PbulkÞ (32)

wall

(33)

/kg], and constant temperature T

bulk is the effect of

bulk < 1022 for the horizontal long tubes of 0.50 mm <d < 2.16 mm.Re<sup>0</sup>:<sup>8</sup>

Here, Tbulk ¼ ½Tin þ Tout�=2 and Twall are constant. This correlation is applicable in the range of 7.4 MPa <Pbulk < 12.0 MPa, 20�C < Tbulk< 110�C, 2�C < Tbulk−Twall < 30�C, 0.02 kg/min < m\_ < 0.2

buoyancy in the radial direction of a horizontal tube. The density of fluid at a temperature and pressure in the vicinity of the critical point is very sensitive to changes in temperature; thus, the effect of the buoyancy derived from the temperature difference between the bulk and wall

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 Tair and measured the heat-transfer coefficient of a liquid or a supercritical fluid flow at a temperature of Tscf < Tair in a vertical tube. They derived correlations for supercritical carbon dioxide and compressed water at a pressure of Pscf≤30 MPa as follows:

Nuave,turb <sup>¼</sup> <sup>0</sup>:0230Re<sup>0</sup>:808Pr0:<sup>300</sup> forH2O

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 correla-

0:0231Re<sup>0</sup>:823Pr0:<sup>300</sup> forCO2

wall were originally derived from the Dittus-Boelter correlation. Gr=Re<sup>2</sup>

cannot be ignored. This effect is enhanced as the diameter of the tube increases.

tions using local accurate physical properties for the entire tube.

4.1. Meanings of temperature and pressure of compressible flow

A stationary fluid pressure of P [Pa], specific volume of V [m3

stores a mechanical energy of epre [J/kg]. Here,

4. Thermofluid dynamics of compressible flow on solid wall

kg/min, 1025 < Gr=Re<sup>2</sup>

132 Heat Exchangers– Advanced Features and Applications

and Pr<sup>0</sup>:<sup>3</sup>

$$
\varepsilon\_{\rm pre} + \varepsilon\_{\rm dyn} = PV + \frac{1}{2}\mu^2 = V[P + P\_{\rm dyn}] = VP\_{\rm tot} = \varepsilon\_{\rm mech}.\tag{36}
$$

$$P\_{\rm dyn} = \frac{1}{2V}u^2 = \frac{1}{2}\rho u^2. \tag{37}$$

$$P\_{\text{tot}} = P + P\_{\text{dyn}}.\tag{38}$$

.Here, Pdyn [Pa] is called "dynamic pressure" and is an index of the dynamic mechanical energy level contained in a motional fluid. Further, emech [J/kg] is called "total mechanical energy." Moreover, Ptot [Pa] is called the "total pressure" and is an index of the total mechanical energy level contained in a motional fluid. Some processes are reversible between mechanical energies of epre and edyn in cases where epre and edyn transform in the equilibrium processes. For example, using a nozzle, Pdyn increases, P decreases, and Ptot is constant in an acceleration section, and Pdyn decreases, P increases, and Ptot is constant in a deceleration section. Some process are irreversible between epre and edyn in cases where epre and edyn transform in nonequilibrium processes. For example, because of friction, P remains constant, Pdyn decreases, and Ptot also decreases.

Next, we consider thermal energy. A stationary fluid at an isochoric specific heat of C<sup>V</sup> [J/(kg K)] stores a relative internal energy of e [J/kg] from e<sup>0</sup> at the standard temperature T0:

$$
\mathfrak{e} \cdot \mathfrak{e}\_0 = \int\_{T\_0}^{T} \mathbb{C}\_V dT.\tag{39}
$$

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

$$
\epsilon \text{--} \mathbf{e}\_0 = \mathbf{C} \mathbf{v} [T - T\_0]. \tag{40}
$$

The "temperature" (often called "static temperature") T is an index of the energy level contained in a stationary fluid.

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

$$PV = RT \Leftrightarrow P = \rho RT,\tag{41}$$

where R is the gas constant, and ρ is the density, which is equal to 1=V. Then,

$$R = \frac{P\upsilon}{T} = \frac{P\_0 \upsilon\_0}{T\_0} \tag{42}$$

.A stationary fluid at an isochoric specific heat of C<sup>V</sup> [J/(kg K)] stores a relative enthalpy of h [J/kg] from h<sup>0</sup> at the standard temperature of T0, a pressure of P0, and a specific volume of V0. Videlicet, enthalpy is a combination of internal energy and mechanical energy. Here,

$$h \cdot h\_0 = [\varepsilon + PV] \cdot [\varepsilon\_0 + P\_0 V\_0] = \int\_{T\_0}^{T} \mathbb{C}\_V dT + \int\_{T\_0}^{T} R dT. \tag{43}$$

In the case of a constant CV,

$$h \cdot h\_0 = \mathbb{C}\_{\mathcal{V}}[T - T\_0] + R[T - T\_0] = \mathbb{C}\_{\mathcal{P}}[T - T\_0],\tag{44}$$

$$\mathbf{C}\_{\rm P} = \mathbf{C}\_{\rm V} + \mathbf{R},\tag{45}$$

where C<sup>P</sup> is the isobaric specific heat. A motional fluid has an additional dynamic energy edyn [J/kg], as shown in Eq. (35). If a motional fluid suddenly stops, dynamic energy can be converted into enthalpy. Then, the following equation applies:

$$hh - h\_0 + \mathfrak{e}\_{\rm dyn} = \mathbb{C}\_{\rm P}[T - T\_0] + \frac{1}{2}u^2 = \mathbb{C}\_{\rm P}[T + T\_{\rm dyn} - T\_0] = \mathbb{C}\_{\rm P}[T\_{\rm tot} - T\_0].\tag{46}$$

$$T\_{\rm dyn} = \frac{1}{2\mathbb{C}\_P} \mu^2. \tag{47}$$

$$T\_{\text{tot}} = T + T\_{\text{dyn}}.\tag{48}$$

Here, Tdyn [K] is called the "dynamic temperature" and is an index of the dynamic energy level contained in a motional fluid. Moreover, Ttot [K] is called the "total temperature" and is an index of the total energy level contained in a motional fluid. Some processes are reversible between mechanical energies of h and edyn in cases where edyn transforms into PV in equilibrium processes. For example, using a nozzle, in an acceleration section, Tdyn increases, h decreases, Ttot is constant, and vice versa. Some processes are irreversible in cases where edyn transforms into CVT in nonequilibrium processes. For example, because of friction, Tdyn decreases, T increases, and Ttot remains constant; however, T cannot be converted into Tdyn again.

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

The specific heat ratio γ is defined as:

$$\gamma = \frac{\mathbf{C\_P}}{\mathbf{C\_V}} \tag{49}$$

.From Eqs. (45) and (49),

PV ¼ RT⇔P ¼ ρRT, (41)

(42)

where R is the gas constant, and ρ is the density, which is equal to 1=V. Then,

<sup>h</sup>−h<sup>0</sup> ¼ ½<sup>e</sup> <sup>þ</sup> PV�−½e<sup>0</sup> <sup>þ</sup> <sup>P</sup>0V0� ¼ ∫

converted into enthalpy. Then, the following equation applies:

<sup>h</sup>−h<sup>0</sup> <sup>þ</sup> <sup>e</sup>dyn <sup>¼</sup> <sup>C</sup>P½T−T0� þ <sup>1</sup>

Here,

again.

In the case of a constant CV,

134 Heat Exchangers– Advanced Features and Applications

<sup>R</sup> <sup>¼</sup> Pv

<sup>T</sup> <sup>¼</sup> <sup>P</sup>0v<sup>0</sup> T0

T

<sup>C</sup>VdT <sup>þ</sup> ∫

h−h<sup>0</sup> ¼ CV½T−T0� þ R½T−T0� ¼ CP½T−T0�, (44)

T

T0

C<sup>P</sup> ¼ C<sup>V</sup> þ R, (45)

<sup>u</sup><sup>2</sup> <sup>¼</sup> <sup>C</sup>P½<sup>T</sup> <sup>þ</sup> <sup>T</sup>dyn−T0� ¼ <sup>C</sup>P½Ttot−T0�: (46)

Ttot ¼ T þ Tdyn: (48)

: (47)

RdT: (43)

T0

.A stationary fluid at an isochoric specific heat of C<sup>V</sup> [J/(kg K)] stores a relative enthalpy of h [J/kg] from h<sup>0</sup> at the standard temperature of T0, a pressure of P0, and a specific volume of V0. Videlicet, enthalpy is a combination of internal energy and mechanical energy.

where C<sup>P</sup> is the isobaric specific heat. A motional fluid has an additional dynamic energy edyn [J/kg], as shown in Eq. (35). If a motional fluid suddenly stops, dynamic energy can be

2

<sup>T</sup>dyn <sup>¼</sup> <sup>1</sup> 2CP u2

Here, Tdyn [K] is called the "dynamic temperature" and is an index of the dynamic energy level contained in a motional fluid. Moreover, Ttot [K] is called the "total temperature" and is an index of the total energy level contained in a motional fluid. Some processes are reversible between mechanical energies of h and edyn in cases where edyn transforms into PV in equilibrium processes. For example, using a nozzle, in an acceleration section, Tdyn increases, h decreases, Ttot is constant, and vice versa. Some processes are irreversible in cases where edyn transforms into CVT in nonequilibrium processes. For example, because of friction, Tdyn decreases, T increases, and Ttot remains constant; however, T cannot be converted into Tdyn

$$\mathbb{C}\_{\text{V}} = \frac{R}{\gamma - 1}, \; \mathbb{C}\_{\text{P}} = \frac{\gamma R}{\gamma - 1} \tag{50}$$

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

$$e \multimap e\_0 = \frac{R}{\gamma - 1} [T - T\_0], \quad h - h\_0 = \frac{\gamma R}{\gamma - 1} [T - T\_0] \tag{51}$$

$$de = \frac{R}{\gamma - 1} dT, dh = \frac{\gamma R}{\gamma - 1} dT \tag{52}$$

The change in the entropy ds is defined as:

$$T\,\mathrm{ds} = de + p dV,\\
\mathrm{Tds} = dh \mathrm{-V} \mathrm{d}P \tag{53}$$

$$ds = \frac{d\varepsilon + pdV}{T}, ds = \frac{dh - VdP}{T} \tag{54}$$

When isentropic change ds ¼ 0,

$$0 = ds = \mathcal{C}\_V \frac{dT}{T} + R\frac{dV}{V},\\ 0 = ds = \mathcal{C}\_P \frac{dT}{T} \text{--} R\frac{dP}{P} \tag{55}$$

$$0 = \frac{R}{\gamma - 1} \frac{dT}{T} - R \frac{d\rho}{\rho}, \\ 0 = \frac{\gamma R}{\gamma - 1} \frac{dT}{T} - R \frac{dP}{P} \tag{56}$$

$$\frac{1}{\gamma - 1} \frac{dT}{T} = \frac{d\rho}{\rho}, \frac{\gamma}{\gamma - 1} \frac{dT}{T} = \frac{dP}{P} \tag{57}$$

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

$$dP = RTd\rho + \rho TdR + \rho RdT\tag{58}$$

$$\frac{dP}{P} = \frac{d\rho}{\rho} + \frac{dR}{R} + \frac{dT}{T} = \frac{d\rho}{\rho} + \frac{dT}{T} \Leftrightarrow \frac{dT}{T} = \frac{dP}{P} - \frac{d\rho}{\rho} \tag{59}$$

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

$$00 = ds = \mathbf{C}\_{\text{P}} \frac{dT}{T} \text{-} \mathbf{R} \frac{dP}{P} = \mathbf{C}\_{\text{P}} \frac{dP}{P} \text{-} \mathbf{C}\_{\text{P}} \frac{d\rho}{\rho} \text{-} \mathbf{R} \frac{dP}{P} = \mathbf{C}\_{\text{V}} \frac{dP}{P} \text{-} \mathbf{C}\_{\text{P}} \frac{d\rho}{\rho} \tag{60}$$

$$0 = \frac{R}{\gamma - 1} \frac{dP}{P} - \frac{\gamma R}{\gamma - 1} \frac{d\rho}{\rho} \tag{61}$$

$$
\gamma \frac{d\rho}{\rho} = \frac{dP}{P} \tag{62}
$$

$$\frac{dP}{d\rho} = \gamma \frac{P}{\rho} \tag{63}$$

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

$$\frac{T}{\rho^{\gamma - 1}} = \text{const}, \quad \frac{T}{P^{\gamma}} = \text{const}, \quad \frac{P}{\rho^{\gamma}} = \text{const} \tag{64}$$

The sound speed a is defined as:

$$a^2 = \left[\frac{dP}{d\rho}\right]\_S \tag{65}$$

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

$$a^2 = \gamma \frac{P}{\rho} = \gamma RT\tag{66}$$

#### 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:

$$h + \frac{1}{2}u^2 = \text{const} \tag{67}$$

When the enthalpy and velocity are h<sup>1</sup> and u<sup>1</sup> at an arbitrary cross section 1,

$$
\hbar\_1 + \frac{1}{2}\mu\_1^2 = \hbar + \frac{1}{2}\mu^2\tag{68}
$$

This relationship is true even if cross section 1 corresponds to the stagnant cross section 0 (h<sup>0</sup> and u<sup>0</sup> ¼ 0); therefore,

$$h\_0 = h + \frac{1}{2}u^2\tag{69}$$

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

Heat Transfer of Supercritical Fluid Flows and Compressible Flows http://dx.doi.org/10.5772/65931 137

$$\frac{\gamma RT\_0}{\gamma - 1} = \frac{\gamma RT}{\gamma - 1} + \frac{1}{2}\mu^2 \tag{70}$$

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

0 ¼ ds ¼ C<sup>P</sup>

136 Heat Exchangers– Advanced Features and Applications

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

The sound speed a is defined as:

derived by using Eq. (46) as:

and u<sup>0</sup> ¼ 0); therefore,

dT <sup>T</sup> <sup>−</sup><sup>R</sup> dP

T

ργ<sup>−</sup><sup>1</sup> <sup>¼</sup> const, <sup>T</sup>

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

<sup>P</sup> <sup>¼</sup> <sup>C</sup><sup>P</sup>

<sup>0</sup> <sup>¼</sup> <sup>R</sup> γ−1 dP <sup>P</sup> <sup>−</sup> <sup>γ</sup><sup>R</sup> γ−1 dρ

> γ dρ <sup>ρ</sup> <sup>¼</sup> dP

dP <sup>d</sup><sup>ρ</sup> <sup>¼</sup> <sup>γ</sup>

P γ−1 γ

<sup>a</sup><sup>2</sup> <sup>¼</sup> <sup>γ</sup> P

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

h þ 1 2

When the enthalpy and velocity are h<sup>1</sup> and u<sup>1</sup> at an arbitrary cross section 1,

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

h<sup>1</sup> þ 1 2 u2 <sup>1</sup> ¼ h þ

<sup>a</sup><sup>2</sup> <sup>¼</sup> dP dρ 

The one-dimensional energy equation of an isentropic flow at an arbitrary cross section is

This relationship is true even if cross section 1 corresponds to the stagnant cross section 0 (h<sup>0</sup>

1 2

h<sup>0</sup> ¼ h þ

1 2

P

<sup>¼</sup> const, <sup>P</sup>

S

dP <sup>P</sup> <sup>−</sup>C<sup>P</sup> dρ <sup>ρ</sup> <sup>−</sup><sup>R</sup> dP

<sup>P</sup> <sup>¼</sup> <sup>C</sup><sup>V</sup>

dP <sup>P</sup> <sup>−</sup>C<sup>P</sup> dρ

<sup>ρ</sup> (61)

<sup>P</sup> (62)

<sup>ρ</sup> (63)

<sup>ρ</sup> <sup>¼</sup> <sup>γ</sup>RT (66)

<sup>u</sup><sup>2</sup> <sup>¼</sup> const (67)

u<sup>2</sup> (68)

u<sup>2</sup> (69)

ργ <sup>¼</sup> const (64)

<sup>ρ</sup> (60)

(65)

$$\frac{a\_0^2}{\gamma - 1} = \frac{a^2}{\gamma - 1} + \frac{1}{2}u^2 \tag{71}$$

We multiply by <sup>γ</sup>−<sup>1</sup> <sup>a</sup><sup>2</sup> and substitute Eq. (66), obtaining the following:

$$\frac{a\_0^2}{a^2} = \frac{RT\_0}{RT} = \frac{T\_0}{T} = 1 + \frac{\gamma - 1}{2} \frac{u^2}{a^2} \tag{72}$$

At the stagnant cross section 0, the static temperature T<sup>0</sup> is equal to the total temperature Ttot; therefore,

$$\frac{T\_{\text{tot}}}{T} = 1 + \frac{\gamma - 1}{2} \mathbf{M}^2, \ T = \frac{T\_{\text{tot}}}{1 + \frac{\gamma - 1}{2} \mathbf{M}^2},\tag{73}$$

where M is the local Mach number. From Eqs. (64) and (73),

$$\frac{P\_{\text{tot}}}{P} = \frac{T\_{\text{tot}}^{\frac{\gamma}{\gamma - 1}}}{T^{\frac{\gamma}{\gamma - 1}}} = \left[\frac{T\_{\text{tot}}}{T}\right]^{\frac{\gamma}{\gamma - 1}} = \left[1 + \frac{\gamma - 1}{2} \mathbf{M}^2\right]^{\frac{\gamma}{\gamma - 1}},\\P = \frac{P\_{\text{tot}}}{\left[1 + \frac{\gamma - 1}{2} \mathbf{M}^2\right]^{\frac{\gamma}{\gamma - 1}}}\tag{74}$$

#### 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 Plocal,bound in a boundary layer is almost equal to the pressure Plocal,main in a main flow outside of the boundary layer; therefore, the pressure in a boundary layer can be expressed by using relationships of the isentropic main flow. That is, Plocal,bound ¼ Plocal,main. Afterwards, both the pressures are expressed as Plocal.

The pressure distribution on a solid wall is usually expressed by pressure coefficient Slocal, which is defined as:

$$S\_{\rm local} = \frac{P\_{\rm tot,in} - P\_{\rm local}}{\frac{1}{2}\rho u\_{\rm in}^2}, P\_{\rm local} = P\_{\rm tot,in} - \frac{1}{2}\rho u\_{\rm in}^2 S\_{\rm local} \tag{75}$$

but is sometimes expressed by another pressure coefficient ηlocal, which is defined as:

$$
\eta\_{\rm local} = \frac{P\_{\rm in} - P\_{\rm local}}{\frac{1}{2}\rho u\_{\rm in}^2},
\\
P\_{\rm local} = P\_{\rm in} - \frac{1}{2}\rho u\_{\rm in}^2 \eta\_{\rm local}.\tag{76}
$$

The two expressions are related as follows:

Figure 5. Pressure distributions of flows on a plane and an airfoil.

$$S\_{\rm local} = \frac{P\_{\rm tot,in} - P\_{\rm local}}{\frac{1}{2}\rho u\_{\rm in}^2} = \frac{P\_{\rm tot,in} - P\_{\rm in} + P\_{\rm in} - P\_{\rm local}}{\frac{1}{2}\rho u\_{\rm in}^2} = \frac{\frac{1}{2}\rho u\_{\rm in}^2 + P\_{\rm in} - P\_{\rm local}}{\frac{1}{2}\rho u\_{\rm in}^2} = 1 + \eta\_{\rm local}.\tag{77}$$

On a plane, Slocal is unity everywhere; thus, Plocal is constant everywhere. On the other hand, on an airfoil, Slocal varies with the location; thus, Plocal varies.

Figure 6 shows the temperature distribution on an adiabatic plane and an airfoil. In flows on an adiabatic wall, the total temperature Ttot,local remains constant at the inlet total temperature Ttot,in. For incompressible flows, that is, with the Mach number Mlocal ¼ 0, the static temperature Tlocal is always the same as Ttot,local. Then, Tlocal remains constant everywhere on both an adiabatic plane and an airfoil. On the other hand, for compressible flows, the Mach number Mlocal varies according to the following equation, which is derived from Eq. (74):

$$\mathbf{M}\_{\text{local}} = \sqrt{\frac{2}{\gamma - 1} \left\{ \left[ \frac{P\_{\text{tot,local}}}{P\_{\text{local}}} \right]^{\frac{\gamma - 1}{\gamma}} - 1 \right\}} \tag{78}$$

Here, the static temperature Tlocal varies with respect to Mlocal for compressible flows.

$$T\_{\text{local}} = \frac{T\_{\text{tot,local}}}{1 + \frac{\gamma - 1}{2} \mathbf{M}\_{\text{local}}^2} \tag{79}$$

On an adiabatic plane, Mlocal is constant. Thus, Tlocal remains constant anywhere on an adiabatic plane, even in cases of compressible flows. On the other hand, on an adiabatic airfoil, Mlocal varies with the location; therefore, Tlocal varies in cases of compressible flows.

Figure 6. Temperature distributions of flows on an adiabatic plane and an airfoil.

<sup>S</sup>local <sup>¼</sup> <sup>P</sup>tot,in−Plocal 1 <sup>2</sup> ρu<sup>2</sup> in

138 Heat Exchangers– Advanced Features and Applications

Figure 5. Pressure distributions of flows on a plane and an airfoil.

<sup>¼</sup> <sup>P</sup>tot,in−Pin <sup>þ</sup> <sup>P</sup>in−Plocal 1 <sup>2</sup> ρu<sup>2</sup> in

Mlocal varies according to the following equation, which is derived from Eq. (74):

2 γ−1

Here, the static temperature Tlocal varies with respect to Mlocal for compressible flows.

Mlocal varies with the location; therefore, Tlocal varies in cases of compressible flows.

<sup>T</sup>local <sup>¼</sup> <sup>T</sup>tot,local <sup>1</sup> <sup>þ</sup> <sup>γ</sup>−<sup>1</sup> <sup>2</sup> <sup>M</sup><sup>2</sup> local

On an adiabatic plane, Mlocal is constant. Thus, Tlocal remains constant anywhere on an adiabatic plane, even in cases of compressible flows. On the other hand, on an adiabatic airfoil,

Mlocal ¼

on an airfoil, Slocal varies with the location; thus, Plocal varies.

On a plane, Slocal is unity everywhere; thus, Plocal is constant everywhere. On the other hand,

Figure 6 shows the temperature distribution on an adiabatic plane and an airfoil. In flows on an adiabatic wall, the total temperature Ttot,local remains constant at the inlet total temperature Ttot,in. For incompressible flows, that is, with the Mach number Mlocal ¼ 0, the static temperature Tlocal is always the same as Ttot,local. Then, Tlocal remains constant everywhere on both an adiabatic plane and an airfoil. On the other hand, for compressible flows, the Mach number

¼ 1 <sup>2</sup> ρu<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ptot,local Plocal � �<sup>γ</sup>−<sup>1</sup>

γ −1

vu ( ) ut (78)

in þ Pin−Plocal 1 <sup>2</sup> ρu<sup>2</sup> in

¼ 1 þ ηlocal: (77)

(79)

#### 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 Tr. This is called the "recovery temperature." Figure 7 shows a schematic of the total temperature Ttot and static temperature T profiles, as well as the recovery temperature T<sup>r</sup> in the vicinity of an adiabatic solid surface with a boundary layer in a compressible flow. As described in Section 4.1, a compressible flow has a measurable dynamic energy; then, the static temperature T in a boundary layer increases because of the braking effect, which converts a dynamic energy to a thermal energy. At the same time, heat generated by the braking effect conducts to the outside of the boundary layer. Therefore, the static temperature T and total temperature Ttot in the boundary layer approach the recovery temperature T<sup>r</sup> on the wall, as shown in the middle in 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 T. The recovery temperature T<sup>r</sup> on an adiabatic wall is equal to the total temperature Ttot,main of the main flow outside of the boundary layer. On the other hand, in cases where a thermal boundary layer protrudes from the edge of a momentum boundary layer, that is, Pr < 1, only part of the heat generated by the braking effect uses the rise of the static temperature T; then, the recovery temperature T<sup>r</sup> on an adiabatic wall has an intermediate value between the total temperature Ttot,main and the static temperature Tmain of the main flow outside of the boundary layer. Eckert proposed an equation for the local recovery temperature Tr,local on an adiabatic wall.

Figure 7. Total-, static-, and recovery-temperature profiles in the vicinity of cooling, adiabatic, and heating solid surfaces with a boundary layer in a compressible flow.

$$T\_{\rm r,local} = T\_{\rm main} + [T\_{\rm tot,main} - T\_{\rm main}]r\_{\rm local} = T\_{\rm main} + T\_{\rm dyn,main}r\_{\rm local} \tag{80}$$

$$r\_{\text{local}} = \begin{cases} \min(1, \text{Pr}^{1/2}) & \text{for a laminar boundary layer} \\ \min(1, \text{Pr}^{1/3}) & \text{for a turbulent boundary layer} \end{cases} \tag{81}$$

Here, rlocal is the "temperature recovery factor," which is the ratio of the recovery temperature to the dynamic temperature of the main flow. Eckert mentioned that heat flux qlocal in a boundary layer in a compressible flow should be defined as:

$$\eta\_{\text{local}} = h\_{\text{local}}[T\_{\text{r,local}} - T\_{\text{solid,local}}] \tag{82}$$

,where hlocal is the local heat transfer coefficient between a compressible flow and a solid wall. In the case where qlocal ¼ 0, the local wall temperature Tsolid,local equals the recovery temperature Tr,local and is called the "adiabatic wall temperature."

Here, Eckert's theory is extended to the recovery temperature T<sup>r</sup> on a heating and cooling wall. In Eq. (80), the first term expressed by the static temperature Tmain represents the internal energy that a local boundary layer originally has, and the second term expressed by the dynamic temperature Tdyn,mainrlocal represents the net dynamic energy that is used to increase the temperature in a local boundary layer. When a local boundary layer is heated or cooled, the first term is affected, but the second term remains constant. The first term should be replaced by the appropriate form suitable for the heating or cooling of a boundary layer. Heating or cooling affects only a thermal boundary layer; therefore, the local total temperature Ttot,bound, <sup>x</sup> at the location x in the flow direction is defined as follows:

$$T\_{\text{tot},\text{bound},\text{x}} = T\_{\text{tot},\text{in}} + \int\_0^\mathbf{x} \frac{q\_\mathbf{x}}{\rho\_\mathbf{x} C\_\text{P} \delta\_\mathbf{x} u\_{\text{ave},\text{x}}} d\mathbf{x} \tag{83}$$

$$T\_{\text{bound},\text{x}} = T\_{\text{tot},\text{bound},\text{x}} - T\_{\text{dyn},\text{main},\text{x}} \tag{84}$$

$$
\rho\_\mathbf{x} = \frac{P\_\mathbf{x}}{RT\_{\text{bound},\mathbf{x}}} \tag{85}
$$

$$
\delta\_{\mathbf{x}} = \begin{cases}
 \left. \frac{\mathbf{v}}{\left[ \mu\_{\min, \mathbf{v}} \text{Pr} \right]} \right|\_{\mathbf{x}}^{0.5} & \text{for a laminar boundary layer} \\
 0.37 \left[ \frac{\mathbf{v}}{\mu\_{\min, \mathbf{v}} \text{Pr}} \right]^{0.2} \text{x}^{0.8} & \text{for a turbulent boundary layer}
\end{cases} \tag{86}
$$

$$
\mu\_{\mathbf{x}} = \begin{cases}
 0.5u\_{\min, \mathbf{x}} \text{ for a laminar boundary layer} \\
 0.8u\_{\min, \mathbf{x}} \text{ for a turbulent boundary layer}
\end{cases} \tag{87}
$$

where Tbound, <sup>x</sup> and ρ<sup>x</sup> are the static temperature and density, respectively, in a heated or cooled boundary layer, and δ<sup>x</sup> and u<sup>x</sup> are thermal boundary layer thickness and average velocity, respectively. Here, evaluations of δ<sup>x</sup> and u<sup>x</sup> are used for a plane, but more appropriate expression for a particular target flow field can be used. Finally, Eq. (80) is replaced by the following equation for the local recovery temperature of a heated or cooled boundary layer.

$$T\_{\rm r,local} = T\_{\rm bound,x} + T\_{\rm dyn,main,x} \\ r\_{\rm local} = T\_{\rm bot,bound,x} - T\_{\rm dyn,main,x} [1 - r\_{\rm local}] \tag{88}$$

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

Tr,local ¼ Tmain þ ½Ttot,main−Tmain�rlocal ¼ Tmain þ Tdyn,mainrlocal (80)

Here, rlocal is the "temperature recovery factor," which is the ratio of the recovery temperature to the dynamic temperature of the main flow. Eckert mentioned that heat flux qlocal in a

Figure 7. Total-, static-, and recovery-temperature profiles in the vicinity of cooling, adiabatic, and heating solid surfaces

,where hlocal is the local heat transfer coefficient between a compressible flow and a solid wall. In the case where qlocal ¼ 0, the local wall temperature Tsolid,local equals the recovery tempera-

Here, Eckert's theory is extended to the recovery temperature T<sup>r</sup> on a heating and cooling wall. In Eq. (80), the first term expressed by the static temperature Tmain represents the internal energy that a local boundary layer originally has, and the second term expressed by the dynamic temperature Tdyn,mainrlocal represents the net dynamic energy that is used to increase the temperature in a local boundary layer. When a local boundary layer is heated or cooled, the first term is affected, but the second term remains constant. The first term should be replaced by the appropriate form suitable for the heating or cooling of a boundary layer. Heating or cooling affects only a thermal boundary layer; therefore, the local total temperature Ttot,bound, <sup>x</sup>

x

qx ρxCPδxuave, <sup>x</sup>

Tbound, <sup>x</sup> ¼ Ttot,bound,x−Tdyn,main,<sup>x</sup> (84)

dx (83)

(85)

0

RTbound,<sup>x</sup>

<sup>ρ</sup><sup>x</sup> <sup>¼</sup> <sup>P</sup><sup>x</sup>

Þ for a laminar boundary layer

Þ for a turbulent boundary layer

qlocal ¼ hlocal½Tr,local−Tsolid,local� (82)

(81)

<sup>r</sup>local <sup>¼</sup> minð1, Pr<sup>1</sup>=<sup>2</sup>

boundary layer in a compressible flow should be defined as:

ture Tr,local and is called the "adiabatic wall temperature."

at the location x in the flow direction is defined as follows:

<sup>T</sup>tot,bound,<sup>x</sup> <sup>¼</sup> <sup>T</sup>tot,in <sup>þ</sup> ∫

with a boundary layer in a compressible flow.

140 Heat Exchangers– Advanced Features and Applications

minð1, Pr<sup>1</sup>=<sup>3</sup>

As described in Section 4 4, the local Mach number Mlocal is constant on a plane but varies with the location on a single airfoil or an airfoil in a cascade. For a single airfoil, when the inlet Mach number Min, the Reynolds number Reairfoil with a representative length of the airfoil chord LC, and the angle of attack α are fixed, as

$$\mathbf{M}\_{\rm in} = \frac{\mu\_{\rm in}}{a\_{\rm in}} \tag{89}$$

$$\text{Re}\_{\text{airfoil}} = \frac{\mu\_{\text{in}} L\_{\text{C}}}{\nu\_{\text{in}}},\tag{90}$$

the distribution of the local pressure coefficient Slocal or ηlocal is uniquely obtained. In cases of a cascade of airfoils, when the stagger angle β and the solidity σ are fixed (see Figure 8), the distribution of the local pressure coefficient Slocal or ηlocal is obtained. Fortunately, many

Figure 8. Flow field through a cascade of airfoils, where θ is the turning angle.

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

experimental results of Slocal or ηlocal have been reported for single airfoils and cascades of airfoils. The distributions of Mlocal are calculated using Eqs. (75), (76), and (78).

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 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 Rex≅104 in regions with adverse

Figure 10. Recovery-temperature distribution assumed according to the pressure coefficient and local Mach number distributions in Figure 9.

pressure gradients, but a developing boundary layer transforms at Rex≅108 in regions with favorable pressure gradients. This means that a developing boundary layer transforms across the minimum pressure point, that is, the maximum of the pressure coefficient Slocal or <sup>η</sup>local on the airfoil surface in cases of Reairfoil≅106 . According to the left graph of Figure 9, a developing boundary layer may transform at x=LC≅0:025 on the lower concave surface and at x=LC≅0:6 on the upper convex surface. Then, the local recovery temperature Tr,local is assumed by using Eqs. (81) and (88) (see Figure 10). This Tr,local can be used for the evaluation of the local heat flux qlocal using Eq. (82) if an adequate heat-transfer coefficient hlocal is employed.
