# Numerical Heat Transfer Simulation of Oil Shale Large-Size Downhole Heater
## 3 Results
### Comprehensive Performance  
Because of the incompressibility of gas, it adopts a plunger flow pattern constrained by a helical baffle, as observed in Figures 8 and 9. The gas flow rate in the shell is uniform, and there is no flow dead zone. The pressure distribution in the shell decreases linearly along the Z-axis, yet noticeable localized pressure anomalies occur at the front of the center tube and at the exit. This is primarily attributed to variations in fluid flow diameter. Upon entering the interior of the heater, the space becomes significantly larger, increasing several times compared to the space in the gas pipeline, resulting in fluctuations in the gas flow rate. At the exit, the gas flow is precisely the opposite of that at the entrance. When combined with the temperature distribution shown in Figures 4 and 5, the pressure abnormal area does not adversely impact the heater's heat transfer performance. However, it is more likely to compromise the support of the center tube to the helical baffle, and if lost cannot provide a helical flow constraint to the injected gas.  
The heater shell process gas follows the following equations during flow:  
The Prandtl number expression is defined as follows:  
\[P_{r}=\frac{C_{p}\cdot\mu}{\lambda} \tag{5}\]  
Figure 8: Pressure distribution in the heater shell.  
Figure 9: Flow rate distribution of gas in the heater.  
The Reynolds number (Re) is defined as follows:  
\[R_{e}=\frac{d_{e}u_{s}\rho}{\mu} \tag{6}\]  
The Nusselt number (\(N_{u}\)) is defined as follows:  
\[N_{u}=CR_{e}{}^{m}P_{r}{}^{\frac{1}{3}} \tag{7}\]  
where \(C_{p}\) is isobaric specific heat capacity (\(\frac{\mathrm{I}}{\mathrm{g}}\cdot\mathrm{K}\)), \(\mu\) is gas viscosity (Pa\(\cdot\)s), \(\lambda\) is thermal conductivity (\(\frac{\mathrm{W}}{\mathrm{m}}\cdot\mathrm{K}\)), \(d_{e}\) is hydraulic diameter (m), \(u_{s}\) is the velocity of the shell side (m/s), and \(\rho\) is density (kg/m\({}^{3}\)), \(C=0.59\), \(m=0.25\).  
According to Equations (5)-(7), as the gas is gradually heated within the heater shell, the temperature rises, and the Prandtl number of the gas decreases, resulting in an increase in gas viscosity. As a consequence, the Nusselt number of the heater decreases. Furthermore, as the gas undergoes heating, its viscosity steadily increases along the gas flow path. Concurrently, the local differential pressure resistance and friction resistance of the heater in the well increase, resulting in a further reduction in the flow rate of the injected gas. Owing to these various impacts, the duration of gas heating in the rear end of the shell is prolonged. Subsequently, the heat transfer efficiency of the heater's rear section decreases, resulting in the formation of high-temperature hot spots on the surface of the heater rod. This, in turn, affects the long-term stable operation of the heater.  
The heat transfer coefficient \(h/\Delta p^{\frac{1}{3}}\) serves as a key indicator for evaluating heat transfer performance. The primary objective of a downhole heater is to achieve efficient heat transfer. The significance of the average heat transfer coefficient at the heater rod wall surpasses that of the heater pressure drop. Heat transfer is most efficient at 100 m\({}^{3}\)/h and is significantly influenced by heating power at 200 m\({}^{3}\)/h, as depicted in Figure 10. The former outperforms the latter by approximately 6%. Moreover, at 160 m\({}^{3}\)/h, the heat transmission performance is at its most stable. It can be deduced that the heat transfer structure significantly influences heat transfer performance, and so do the power of the heating rod and the gas injection condition. Therefore, by adjusting the injected gas flow rate and heating power across a wide range, not only can the heating requirements of different methods be met, but efficient heat transfer cracking can also be achieved.  
Figure 10: Comprehensive heat transfer performance at different working conditions.