# Forced Convection Heat Transfer for Stratospheric Airship Involved Flight State
## 3 Results and Discussions
### Formula Fitting
#### 3.2.2 State Modification  
The problem of changes of attack angle of a stratospheric airship is an important subject on the research of the Stratospheric Airship Platform. [PERSON] [38] pointed out that the airship reaches its stable attack angle in a short time after released, and that the attack angle is usually large. Then, the airship ascends at its stable attack angle. So, it is necessary to investigate the forced convection heat transfer of the airship at a certain attack angle. The definition of \(K\) is introduced in this part and can be expressed in the form of  
\[K(Re,e,\alpha)=Nu(Re,e,\alpha)/Nu(Re,e) \tag{7}\]  
Figure 7 shows the relation between \(K\) and \(\alpha\) (\(Re=1\times 10^{7}\)). Clearly, \(K\) increases with the increase of \(\alpha\) regardless of the value of \(e\), the larger the attack angle, the greater the difference. \(K\) increases with the increase of the \(e\) for a given \(\alpha\), especially at higher \(\alpha\). Figure 8 shows the relation between \(K\) and \(Re\) (\(e=2\)). Clearly, \(K\) remains nearly the same as the increase of \(Re\) regardless of the value of \(\alpha\). For example, \(K\) reduces from 1.052 to 1.033 with \(Re\) changes from \(1\times 10^{7}\) to \(2.0\times 10^{8}\) when \(\alpha=20^{\circ}\). This result means that the effect of \(Re\) on \(K\) can be ignored. \(K\) increases with the increase of \(\alpha\) for a given \(Re\). For instance, the value of \(K\) at \(\alpha=20^{\circ}\) and \(\alpha=5^{\circ}\) is 1.053 and 1.004, respectively, with a difference of 4.8% when \(Re=1\times 10^{7}\). Figure 8: Correction coefficient \(K\) vs. \(Re\) (\(e=2\)). Figure 7: Correction coefficient \(K\) vs. \(\alpha\) (\(Re=1.0\times 10^{7}\)). The correction factor \(K\) can be simplified to \(K\left(e,\alpha\right)\) based on the above analysis. The mean valueof \(K\) corresponding to different \(Re\) is taken as the final value of \(K\). Based on the data obtained from the above simulation results, a correlation of the correction factor \(K\) with a determination coefficient (\(r^{2}\)) of 0.9995 and a RMSE of 0.0005286 is proposed via MATLAB R2018a. The formula can be written as  
\[K=10^{-4}\Big{(}2.17e^{2}+1.274\alpha^{2}-7.653e-1.01\alpha+9991\Big{)} \tag{8}\]  
where \(e\in\)[2; 6] and \(\alpha\in\)[0\({}^{\circ}\), 20\({}^{\circ}\)]. According to the above analysis, the forced convection heat transfer for stratospheric airship involved flight state can be calculated by the following empirical correlation  
\[Nu=0.0161 KRe^{0.8543}e^{-0.0454} \tag{9}\]  
The correctness of Formula (9) is likewise validated, and the results are listed in Table 5.  
As mentioned above, for the external forced convection heat transfer of an ellipsoidal airship flying with a certain angle of attack, there is no specific correlation available yet. Most of the researchers conducted their researches without considering the effect of flight state. Figure 9 compares the values of the average Nusselt number obtained from Formula (9) with those taken from the literatures [11; 30; 39]. The geometry models used in these literatures are the same as or close to the one used in present study. They are all ellipse. It can be noted that the results of the present study show a consistent performance trend with literatures. Generally, the present results with \(\alpha=20^{\circ}\) are higher than those of existing correlations, especially at a higher Reynolds number, where the difference reaches a maximum of about 30 percent compared with Dai. In the thermal design period, the effect of flow state cannot be ignored. Figure 9: Comparison of values of the Nusselt number from the present study with \(\alpha=20^{\circ}\), with those from the literatures.