Heat Transfer Correlations of Supercritical Fluids

*Fangbo Li, Binbin Pei and Bofeng Bai*

## **Abstract**

The drastic changes of thermophysical properties in the pseudo-critical region of supercritical fluids bring very big challenges to the traditional Dittus-Boeltertype heat transfer correlations. In this chapter, we will talk about the principles and applications of two kinds of heat transfer correlations of supercritical fluids: the empirical type and the semiempirical type. For the empirical correlations, the modification methods taking into account the variable properties and body force effects will be introduced. We will focus on the proposal of nondimensional parameters describing the buoyancy effect and flow acceleration; while for the semiempirical ones, we mainly talk about the new kind of correlation which is based on the momentum and energy conservations in the mixed convective flow.

**Keywords:** heat transfer correlation, empirical type, semiempirical type, buoyancy effect, flow acceleration

### **1. Introduction**

Establishing an accurate heat transfer correlation of supercritical fluid is the prerequisite of the design of the apparatus and devices, such as the supercritical boilers, generation IV reactors, and heat exchangers [1–4]. However, the drastic change of the thermodynamic properties in the pseudo-critical region and the complex mechanism of turbulent heat transfer lead to difficulties in predicting the heat transfer performance accurately. There are three distinguished heat transfer regimes for supercritical fluid: normal heat transfer (normal HT), improved heat transfer (improved HT), and deteriorated heat transfer (deteriorated HT). The normal HT regime is characterized by the heat transfer coefficients similar with those predicted by the Dittus-Boelter (DB) correlation at subcritical condition; the improved HT regime can be characterized by the higher heat transfer coefficients and lower wall temperatures compared with the normal HT; and the deteriorated HT regime is accompanied with lower heat transfer coefficients than the normal HT; hence, the wall temperature peak can be observed. **Figure 1** illustrates the heat transfer performances of supercritical CO2 under different mass fluxes and heat fluxes [5] of the upward flow in a circular tube with an inner diameter of 6.32 mm and heated length of 2.65 m. At relatively lower heat flux/mass flux ratio, the heat transfer will be obviously improved due to the large specific heat in the

**2. Empirical heat transfer correlations**

*Heat Transfer Correlations of Supercritical Fluids DOI: http://dx.doi.org/10.5772/intechopen.89356*

typical empirical correlations are listed in **Table 1**.

**Figure 2.**

**69**

*The approach deriving the empirical correlations.*

For the improved HT and normal HT regimes, abundant empirical correlations have been proposed, and most of them can give satisfying predictions [13–15]. Morky et al. [9] and Gupta et al. [16] proposed the correlations based on the Buckingham Π theorem, and the comparison with the experimental results confirmed their capability on the calculating the heat transfer improvement regime. However, as the increasing of the heat flux, the effect of the mixing convection increases. The fluid near the heated wall is accelerated under the effect of the buoyancy force and the flow acceleration, leading to the attenuation of heat transfer [17]. On the basis of this, researchers tried to modify the correlations by considering the nondimensional numbers reflecting the buoyancy force and the flow acceleration. Bae and Kim [10] and Komita et al. [18] analyzed the momentum balance under the mixed convection case and proposed a nondimensional parameter Bu to describe the effect of the buoyancy on the shear stress; then, they proposed a new function to describe the relationship between the Nusselt numbers and Bu based on the experimental data. This heat transfer correlation takes the form of the piecewise function divided by the value of the Bu number. Deev et al. [19] discussed the heat transfer of supercritical water in the channel. Two nondimensional criteria considering the effects of the viscous force and inertial force on heat transfer were proposed, and the weight constant treating the superposition between the forced and natural convection was introduced. Such practices were also conducted by Cheng et al. [20], Yu et al. [21], and Kuang et al. [22]. In their work, the dimensionless numbers such as the buoyancy number and the acceleration number were proposed to correct the deviation of the real heat transfer from that of the normal HT regime. **Figure 2** shows the approach establishing the empirical correlations. The dimensionless groups are firstly chosen, and the coefficients in the correlation can be obtained by the linear regression analysis. Through variable transformations the dimensionless groups are reduced, and the final form can be obtained. Some

**Figure 1.**

*Heat transfer performances of supercritical CO2 of the upward flow under different (a) mass fluxes and (b) heat fluxes in a circular tube with an inner diameter of 6.32 mm and heated length of 2.65 m. Symbols: G, mass fluxes (kg m<sup>2</sup> s 1 ); P, pressure (MPa); q, wall heat flux (kW m<sup>2</sup> ); H, enthalpy (kJ kg<sup>1</sup> ). Data in this figure come from Ref. [1].*

pseudo-critical region. The heat transfer coefficients will be higher than the ones calculated by the Dittus-Boelter correlation. As the heat flux/mass flux ratio increases, the mixing convective heat transfer occurs, and the wall temperature peak caused by the heat transfer deterioration can be observed. The experiment conducted by Kurganov and Kaptil'ny [6] offered an insight into the mechanism of heat transfer deterioration, the M-shaped velocity profile was observed, and it was considered to be closely associated with the flattened velocity gradient, reduced turbulent shear stress, and turbulent kinetic energy due to the body force (also known as the buoyancy effect). Bae et al. [7, 8] obtained the budgets of the turbulent kinetic energy in the deteriorated heat transfer by using direct numerical simulation, and the mechanism of buoyancy and flow acceleration were analyzed.

The heat transfer correlations are established on the basis of heat transfer mechanism, and a vast majority of correlations are intended for use only for normal HT and improved HT regimes. They can be categorized into two types: the empirical type and the semiempirical type. In the empirical correlations, the correction terms composed of different thermophysical properties (such as the density, thermal conductivity, specific heat, and viscosity) are introduced to the heat transfer correlation of the constant-property fluid (i.e., DB correlation) [9]. However, the performance of the empirical correlations deteriorates with the increasing of the heat flux/mass flux ratio; thus, the empirical correlations considering the nondimensional numbers which reflect the buoyancy and thermally induced flow acceleration effects have also been proposed [10]. It is shown that the predictions on the mixing convection cases can be improved by this method. In order to further improve the performances of the empirical correlations under the strong buoyancy effect, some researchers tried to establish the semiempirical correlations based on the theoretical analysis approaches [11, 12]. The existing semiempirical correlations mainly aim to derive the qualitative relationship between the heat transfer impairment and turbulent shear stress reduction; the coefficients appeared in the correlation are obtained by fitting the experimental data.

In this chapter we mainly talk about the heat transfer correlations of supercritical fluid. Both the empirical type and the semiempirical type will be introduced here. We mainly focus on the principles, applications, and comparisons of these correlations.
