Overall Heat Transfer Coefficient Evaluation Method for Uncracked Hydrocarbon Fuel in a Regeneratively-Cooled Heat Exchanger of a Scramjet

Abstract

Regenerative cooling is critical for the thermal protection of hypersonic propulsion systems, wherein the flow and heat transfer characteristics of the hydrocarbon fuel in the cooling channel are crucial. Restricted by measurement, the distribution of the fuel temperature inside the cooling channel cannot be directly detected. As a result, the measurement of the convective heat transfer coefficient is more an overall heat transfer coefficient. Furthermore, the existing overall heat transfer coefficient calculation method is derived without considering the flow state of the hydrocarbon fuel in the cooling channel, which should be taken into consideration. Therefore, a modified calculation method of the overall heat transfer coefficient is proposed in this study. The results of the characteristics of the turbulence-related parameters show that the overall heat transfer coefficient is mainly determined by the effective thermal conductivity and surface convective heat transfer coefficient, which are related to the fuel temperature and flow state. Furthermore, the existing method might overestimate the overall heat transfer coefficient. The modified method is suitable for the calculation of the overall heat transfer coefficient in the heat transfer enhancement process in a cooling channel.

1. Introduction

The propulsion system is the foundation of and key technology for realizing hypersonic flight. The scramjet has become a research hotspot in the field of air-breathing propulsion because of its high specific thrust and wide working range; the air-breathing propulsion devices powered by a scramjet have attracted attention [1]. Among these devices, X-43A and X-51 are remarkable [2,3]. However, the thermal load of a scramjet can be as high as 10 MW/m² [4], which introduces a severe thermal protection target for a scramjet. Regenerative cooling technology is one of the key technologies for the thermal protection of a hypersonic propulsion system. Fuel used as a propellant is also adopted as a coolant to cool the engine walls before combustion, which helps avoid the quantity penalty caused by carrying extra coolant [5,6]. Compared with liquid hydrogen, liquid hydrocarbon fuel is widely used as a coolant for a scramjet (Ma < 8) because of its simple storage and production. In addition, this regenerative cooling technology has been verified by the successful flight test of X-51A [7].

According to the characteristics of the wall temperature distribution in a slender channel, the heat transfer of hydrocarbon fuel can be divided into four stages: normal heat transfer, heat transfer enhancement, heat transfer deterioration, and the second heat transfer enhancement, among which the heat transfer enhancement is the most important stage. The cooling channel in a scramjet combustion wall can be regarded as a heat exchanger between the fuel and combustion [8]. Due to the importance of the convection heat transfer coefficient, much research has been conducted on it. Cheng et al. [9] obtained the convective heat transfer coefficient by calculating the average fuel temperature of a cross section. However, this method can lead to high errors, because the influence of the thermophysical properties of the hydrocarbon fuels near the pseudo-critical temperature was not taken into consideration. Liu et al. [10] studied the influence of the dynamic viscosity on the convective heat transfer coefficient, but it was only applicable to turbulence, while the heat convection of the fluid was not considered. Deng et al. [11], Huang et al. [12], and Fu et al. [13] went further by calculating the convective heat transfer coefficient in segments based on a ratio of the average temperature of a cross section in the critical temperature region. In addition, the thermophysical properties of the hydrocarbon fuel were adopted. However, the overall heat transfer coefficient calculated by piecewise convective heat transfer coefficient is bound to have a discontinuity point, which is derived from experimental data. Zhang et al. [14] proposed three calculation methods based on experimental data for the convective heat transfer coefficient in the laminar flow, transition flow, and the turbulent flow region, respectively. However, there was still a discontinuity problem when using three different calculation methods. Xu et al. [15] assumed that there were two layers in the fluid. Based on this, the heat transfer process between the inner wall and the first layer of fluid was considered as the convective heat transfer. The first and the second layer of fluid were treated as the convective heat transfer, while the heat transfer in the layer of fluid was the heat conduction. However, this method only considered the influence of the heat conduction in the viscous bottom layer and did not consider the stratification of the velocity and temperature in the second layer of fluid. Therefore, the convective heat transfer coefficient calculated by the above method cannot be used to calculate the overall heat transfer coefficient.

In this study, a modified calculation method of the overall heat transfer coefficient is proposed, and a numerical model is established. The characteristics of the overall heat transfer coefficient are discussed from a subcritical to a supercritical temperature region. The modified method is compared with the unmodified method. The results show that the overall heat transfer coefficient calculated by the modified method is smaller, which implies that the unmodified overall heat transfer coefficient might be overestimating the heat transfer enhancement process in the cooling channel.

2. Modification of the Overall Heat Transfer Coefficient

2.1. Cause of the Modification

The surface convective heat transfer coefficient is calculated by the radial temperature gradient, and the average convective heat transfer coefficient is calculated by Newton’s law of cooling [16]. To obtain the surface convective heat transfer coefficient, some scholars assert the average convective heat transfer coefficient of a small section by applying Newton’s law of cooling. Pu et al. [17] used the above method to calculate the surface convective heat transfer coefficient. Based on the work of Osborne et al. [18], one piece of Nusselt correlation in the laminar region was proposed by considering the entrance and free convection effects. Based on the work of Meyer et al. [19], another piece of correlation in the transition flow region was proposed by considering the thermophysical property and the development of the thermal boundary layer; the precision of the correlation was high enough. However, when the Reynold number was higher than 1700, showing the discontinuity of the two correlations, there was no way to obtain the surface convective heat transfer. In addition, the case of the turbulent region was not discussed. Therefore, the effect of the turbulent thermal conductivity should be considered in order to modify the Nusselt correction in the turbulent region. It is noteworthy that the temperature used in Ref. [17] was the average temperature of the cross section, which was calculated by the average enthalpy of the cross section. The average enthalpy of the cross section was not only considered in the heat of the convective heat transfer but also in the heat of the fluid heat convection. As a result, the coefficient calculated by average temperature of the cross section was called the overall heat transfer coefficient instead. To obtain the overall heat transfer coefficient from the laminar to turbulent region, a modified method is proposed.

2.2. Method

Considering that the thermal conductivity of the cooling channel wall is affected by the temperature, the serious flow stratification of the hydrocarbon fuel has a strong influence on the heat transfer in the cooling channel, which further affects the calculation of the overall heat transfer coefficient. Therefore, it is proposed to divide the solid domain into multiple layers along the radial direction and divide the fluid domain into multiple layers. The overall heat transfer coefficient is calculated by the thermal resistance calculation method. Since the fuel temperature changes little in the inlet and outlet insulation section, while the fuel’s thermophysical properties and flow state change a lot in the heating section, which in turn leads to the change in the overall heat transfer coefficient, to calculate the overall heat transfer coefficients, we propose to divide the heated section into several sections along the channel.

As seen in Figure 1a, the inner diameter and wall thickness of the tube were both 1 mm. The solid domain was divided into n parts along the radial direction, the fluid domain was divided into i parts along the radial direction, and the heating section was divided into j parts along the channel (Figure 1b).

The thermal resistance of the solid thermal conductivity in the n^th layer is expressed as shown in Equation (1).

$$R_w=\frac{\Delta T}{\Phi }=\frac{ln\left(d_{n+1}/d_n\right)}{2\pi {\lambda }_{w,n}{l}_j}$$

(1)

Considering the serious flow stratification of the hydrocarbon fuel in the cooling channel, when the flow state is laminar, the heat transfer from the inner wall to the fluid center can be regarded as the surface convective heat transfer and the fluid heat convection. When the flow state is turbulent, the heat transfer in the bottom of the laminar flow can be regarded as the heat convection between the fluid layers; the heat transfer depends on the random diffusion of vortices between the fluid layers [20] in the turbulent core region. As a result, the thermal resistance of the fluid thermal conductivity in the i^th layer is expressed as follows:

$$R_f=\frac{\Delta T}{\Phi }=\frac{ln\left(d_{i+1}/d_i\right)}{2\pi {\lambda }_{eff,i}{l}_j}$$

(2)

The overall heat transfer coefficient in the j^th cross section is expressed as shown in Equation (3).

$$\left(Z\right)=\frac{1}{{\int }_{d_{in}}^{d_{out}}\frac{d_{out}}{2Y{\lambda }_w\left(Y\right)}dY+{\int }_{d_0}^{d_{in}}\frac{d_{out}}{2Y{\lambda }_{eff}\left(Y\right)}dY+\frac{d_{out}}{d_{in}}\frac{1}{h_{adj}\left(Z\right)}}$$

(3)

For the convenience of the calculation, the equation is discretized, as shown in Equations (4)–(7).

$$k_{z_j}=\frac{1}{{\sum }_1^{n-1}\left(\frac{d_{out}}{2{\lambda }_{w,n}}ln\frac{d_{n+1}}{d_n}\right)+{\sum }_1^{i-1}\left(\frac{d_{out}}{2{\lambda }_{eff,i}}ln\frac{d_{i+1}}{d_i}\right)+\frac{d_{out}}{d_{in}}\frac{1}{h_{adj,z_j}}}$$

(4)

$${\lambda }_{eff,i}={\lambda }_l+{\lambda }_t$$

(5)

$${\lambda }_t=\frac{{\mu }_t{c}_p}{P{r}_t}$$

(6)

$${\mu }_t=\frac{\rho k}{\omega }$$

(7)

The surface convective heat transfer coefficient in the j^th cross section is expressed as follows.

$$h_{adj,z_j}=-\frac{{\lambda }_l}{T_{w,z_j}-T_{adj}}\frac{\partial T}{\partial Y}{|}_{Y=d_{in}/2}$$

(8)

2.3. Turbulence-Related Parameter Calculation Method

The parameters mentioned in Equations (4)–(8), such as λ_w, λ_l, c_p, and ω, are highly related to the fuel status in the cooling channel. To calculate the overall heat transfer coefficient, the turbulence-related parameters should first be calculated. For the solid wall part, the three-dimensional energy equation is written in Equation (9), in which the heat absorbed by the wall is converted into heat generation by the wall.

$$\nabla \cdot ({\lambda }_w\nabla T)+\dot{\varphi }=0$$

(9)

In the fluid part, the three-dimensional continuity equation, momentum equation, and energy equation are expressed as follows:

$$\nabla \cdot \left(\rho \mathit{U}\right)=0$$

(10)

$$\nabla \cdot \left(\rho \mathit{U}\mathit{U}\right)=-\nabla p+\nabla \cdot \tau +{\mathit{S}}_M$$

(11)

$$\nabla \cdot \left(\rho \mathit{U}e\right)=\nabla \cdot \left({\lambda }_f\nabla T\right)-\nabla \cdot \left(p\mathit{U}\right)+S_E$$

(12)

For turbulence, after combining the Reynolds-averaging equations and Boussinesq assumptions, the SST k—ω model [21] was chosen to describe the fluid in the adjacent wall region, expressed as follows:

$$\nabla \cdot \left(\rho k\mathit{U}\right)=\nabla \cdot \left[\left({\mu }_l+{\delta }_k{\mu }_t\right)\nabla k\right]+P_k-{\beta }^{*}\rho k\omega$$

(13)

$$\nabla \cdot \left(\rho \omega \mathit{U}\right)=\nabla \cdot \left[\left({\mu }_l+{\delta }_{\omega }{\mu }_t\right)\nabla \omega \right]+\alpha \frac{\omega }{k}{P}_k-\beta \rho {\omega }^2+C_D$$

(14)

To describe the thermophysical properties of the hydrocarbon fuel, the real gas Peng–Robinson cubic equation of state [22] was adopted in this study, which is shown in Equation (15).

$$p=\frac{\rho R_{u}T}{M_w-b\rho }-\frac{a\alpha \left(T\right){\rho }^2}{M_w^2+u{M}_{w}b\rho +w{b}^2{\rho }^2}$$

(15)

The dynamic viscosity and thermal conductivity of the hydrocarbon fuel under high pressure conditions were calculated by the Chung method [23] as follows:

$$\mu ={\mu }^{*}\frac{36.344{\left(M_w{T}_c\right)}^{0.5}}{V_c^{2/3}}$$

(16)

$$\lambda =\frac{31.2{\mu }^0\psi }{M_w}\left(G_2^{-1}+B_{6}y\right)+q{B}_7{y}^2{T}_r^{0.5}{G}_2$$

(17)

To facilitate the comparative discussion of the overall heat transfer coefficient with/without modification, formulas for calculating the unmodified overall heat transfer coefficient in the j^th section are expressed as follows:

$${\overline{K}}_{z_j}=\frac{q_w}{T_{w,j}-T_{f,j}}$$

(18)

$$T_{w,j}=\frac{{\overline{T}}_{w,j-1}+{\overline{T}}_{w,j}}{2}$$

(19)

$$T_{f,j}=\frac{{\overline{T}}_{b,j-1}+{\overline{T}}_{b,j}}{2}$$

(20)

The $\overline{T}$ mentioned above represents the average temperature of the cross section, and the average wall temperature of the cross section is expressed as follows:

$${\overline{T}}_w=\frac{{\int }_{d_{in}/2}^{d_{out}/2}Y{T}_{w,i}dY}{{\int }_{d_{in}/2}^{d_{out}/2}YdY}$$

(21)

$$h_{z_j}=\frac{{\int }_0^{d_{in}/2}{\rho }_{i}Y{h}_{i}dY}{{\int }_0^{d_{in}/2}{\rho }_{i}YdY}$$

(22)

The average fuel temperature of the cross section can be calculated by the average specific enthalpy. The average specific enthalpy of the cross section is expressed as follows:

To calculate the average fuel temperature in the j^th cross section, data from the standard thermophysical property library, which is called “library” in the following part, were used. Although the enthalpy in the numerical calculation was different from the enthalpy in the library, the enthalpy increase was consistent when the temperature difference was the same. Therefore, the temperature calculated from the library can be used as the value calculated by the specific enthalpy of the cross section (Equation (22)).

The enthalpy increase in the library can be calculated by the bisection method. The average value of the assumed upper and lower temperature was taken as the fuel temperature of the cross section in the calculation cases. The specific enthalpy of the cross section was calculated by the pressure and temperature. Comparing the two enthalpy increase values, if the absolute value of relative error was greater than the maximum permissible error (10⁻⁸), and the difference between them was greater than zero, the average fuel temperature of the cross section was set as the upper limit value. Otherwise, it was set as the lower limit value. Finally, the average fuel temperature of the cross section that satisfied the error condition was the value corresponding to the specific enthalpy of the cross section. The iterative process is shown in Figure 2.

The flow and heat transfer process of hydrocarbon fuel in a horizontal circular tube at supercritical pressure was studied. Figure 3 shows the schematic diagram of the cooling channel. The inner diameter of the cooling channel was set as 1 mm, the outer diameter was set as 3 mm, and the length of the cooling channel was 1000 mm. The tube was heated by a uniform heat flux. To ensure the full development of the flow, an insulation section in the inlet was set with a length of 100 mm. To reduce the influence of the outlet and ensure the fluid mixed evenly, an outlet insulation section with a length of 100 mm was developed.

The main component of a typical endothermic hydrocarbon fuel is n-decane, which was also selected as the fluid part. A high-temperature alloy steel with a density of 8810 kg/m³ was selected in the solid part, of which the constant-pressure specific heat and thermal conductivity were calculated by a polynomial piecewise linear interpolation.

3. Results and Discussion

3.1. Supplemental Data for the Overall Heat Transfer Coefficient Modification

Based on the above calculation method, the flow and heat transfer characteristics of n-decane were studied under the conditions of an inlet fuel temperature of 300 K, an inlet fuel mass flow of 1 g/s, a heat flux of 0.15 MW/m², and an outlet pressure of 5 MPa. The fuel was heated from room temperature to supercritical temperature, which covered the subcritical and transcritical temperature region. The velocity and temperature distributions of different cross sections are provided in Figure 4. As seen in Figure 4a, due to the influence of the fluid viscosity, the velocity distribution in the heated section was a parabola. Furthermore, in the near wall section, the velocity gradient was larger than in other parts. At the inlet of the insulation section, most of the layer of the fluid velocity was about 2 m/s, due to the insufficient development of a velocity boundary layer. As seen in Figure 4b, the fuel temperature was distributed symmetrically along the Y axis. At the same cross section, the fluid on the adjacent wall was heated first, while the axial fluid remained unheated, which resulted in a stratification of the temperature. The thickness of the thermal boundary layer was the largest near the inlet of the heated section; then, it decreased along the channel.

3.2. Discussion of the Overall Heat Transfer Coefficient with Modification

As seen in Figure 5a, the overall heat transfer coefficient showed a trend of increasing at first, then decreasing, and increasing along the channel. Based on a single variable, it could be inferred that the overall heat transfer coefficient decreases with a decrease in the solid thermal conductivity, a decrease in the fluid effective thermal conductivity, and a decrease in the surface convective heat transfer coefficient. However, considering the influence of the multivariate analysis, it is difficult to describe the trend of the overall heat transfer coefficient.

As seen in Figure 5b, because the thermal boundary layer had insufficient development, the fuel temperature on the adjacent wall was low at the inlet of the heated section with Z = 100 mm. As a result, the temperature difference between the inner wall and the fuel on the adjacent wall was large. Then, the temperature difference decreased gradually with the full development of the thermal boundary layer. In addition, because the velocity boundary layer was insufficiently developed, there was an obvious slope change point in the temperature difference with Z = 118 mm.

As seen in Figure 6a, the trend of the effective thermal conductivity was consistent with that of the overall heat transfer coefficient. From Equation (5), the effective thermal conductivity was related to the laminar thermal conductivity and the turbulent thermal conductivity. Considering the influence of the multivariate analysis, it is difficult to describe the trend of the effective thermal conductivity.

From Equation (6), with the constant turbulent Prandtl number, the turbulent thermal conductivity increased with the increase in the constant-pressure specific heat capacity and turbulent viscosity. As seen in Figure 6a, before Z = 400 mm, the trend of the effective thermal conductivity was consistent with that of the turbulent viscosity. However, after Z = 400 mm, the effective thermal conductivity kept increasing despite the decrease in the turbulent viscosity. Such a trend is attributable to the increase in the constant-pressure specific heat capacity, which was larger than the decrease in the turbulent viscosity near the pseudo-critical temperature region (Figure 6b). The trend of the turbulent viscosity can be explained by the trends of the density, kinetic turbulent energy, and the specific dissipation rate. As seen in Figure 6c, after Z = 600 mm, k/ω exhibited a greater upward trend. As seen in Figure 6b, the density showed a downward trend, which was greater than the increase in k/ω near the pseudo-critical temperature region. As a result, the turbulent viscosity showed a downward trend after Z = 600 mm.

3.3. Comparison of the Overall Heat Transfer Coefficient with/without Modification

With the results discussed in Section 3.1, it is clear that nonuniformity of the radial fuel temperature led to the nonuniformity of the parameters such as the effective thermal conductivity and turbulent viscosity. To calculate the overall heat transfer coefficient, the fuel temperature in a cross section cannot be averaged into a single value. As a result, the modified method proposed in Section 2.2 was used to calculate the overall heat transfer coefficient. The method was compared with the unmodified one in this study. The relative difference in the overall heat transfer coefficient was expressed as shown in Equation (23).

$${\delta }_k=\frac{k_{mod}-k_{unmod}}{k_{unmod}}\times 100\%$$

(23)

As seen in Figure 7, the unmodified overall heat transfer coefficient showed a trend of decreasing at first and then increasing. The relative difference between them was high in general, and it appeared at its maximum point near the inlet and outlet of the heated section, with a maximum value about −71%.

The unmodified overall heat transfer coefficient s calculated by the average temperature difference of the cross section, while the modified overall heat transfer coefficient is calculated by the flow state. As seen in Figure 4b, in the outlet of the cooling channel, the fuel temperature on the adjacent wall was in the supercritical temperature region, while the axis fuel temperature was in the transcritical or subcritical temperature region. The overall heat transfer coefficient calculated by Equation (4) not only considered the influence of the temperature on the thermophysical property of the fuel but also considered the influence of the flow state on the fuel heat convection.

4. Conclusions

Due to the nonuniform distribution of the radial fuel temperature, the overall heat transfer coefficient cannot be calculated by a cross-sectional average parameter. To solve this problem, a modified calculation method of the overall heat transfer coefficient was proposed in this study. To provide turbulence parameters for the modified overall heat transfer coefficient method, we used the flow and heat transfer cases of n-decane when the outlet fuel temperature was in the supercritical temperature region. The results of the modified overall heat transfer coefficient method were compared with the unmodified method. The conclusions are as follows.

(1)

Considering the stratification of the velocity and temperature in a fluid, a formula for calculating the overall heat transfer coefficient based on the flow state was established.

(2)

During the change in the fuel temperature from the subcritical to supercritical region, the overall heat transfer coefficient was mainly determined by the effective thermal conductivity and the surface convective heat transfer coefficient. Similar to the trend of the constant-pressure specific heat capacity, the peak point of the overall heat transfer coefficient also appeared in the pseudo-critical temperature region.

(3)

Compared with the unmodified overall heat transfer coefficient, the modified overall heat transfer coefficient was smaller in general, which implies that the existing method might overestimate the overall heat transfer coefficient. The modified method not only considered the influence of the temperature based on the thermophysical property of fuel but also considered the influence of the flow state caused by the fuel heat convection, which indicates that the modified overall heat transfer coefficient may be closer to reality.

(4)

The modified method is adaptive in a three-dimensional case. For the convenience of heat transfer enhancement design in a cooling channel, a quasi-one-dimensional case will be considered in our next stage of search.