## 2 The Quick Preliminary Design Model
### Propulsion Power  
The propulsion power, \(\dot{W}_{\text{prop}}\), is calculated as follows,  
\[\dot{W}_{\text{prop}}=\frac{R\cdot v}{\eta} \tag{1}\]  
where \(R\) represents the drag of the vehicle; \(v\) its speed; and \(\eta\) stands for the efficiency of the propulsion chain, that is, the combined efficiency of the electric motor, shaft, and propeller.  
\(R\) is calculated by Equation (2) [18]:  
\[R=\frac{1}{2}\cdot\rho\cdot S\cdot v^{2}\cdot C_{d} \tag{2}\]  
where \(\rho\) is the density of sea water, \(S\) represents the wetted surface of the hull, and \(C_{d}\) stands for the drag coefficient of the vehicle.  
The calculation of an accurate value of \(C_{d}\) is a complex problem that depends on many factors as the shape and size of the hull, control surfaces, domes, discontinuities in the hull, or the interactions between the hull and the appendages, among others. There are research works that seek to address this matter by using of computational fluid dynamics (CFD) techniques, supplemented with empirical experiments involving towing tanks [16; 18; 19]. In this work, a simplified approach has been used where the drag coefficient of the bare hull is increased by means of a margin coefficient that covers the effect of the appendages and other irregularities of the hull, \(C_{dMrg}\). The value of \(C_{dMrg}\) was set at 0.60 following the analysis of the drag generated by a torpedo-shaped AUV carried out by the Underwater Systems Lab National Oceanography Centre of Southampton, England [20]. That work observed an increase of 57% in the value of the drag coefficient for a torpedo-shaped AUV when comparing the bare-hull values with the final vehicle. As a security margin, this percentage was rounded up to 60% for this work.  
\[C_{d}=(1+C_{dMrg})\cdot(C_{vBH}+C_{a}) \tag{3}\]  
where \(C_{dMrg}\) is the margin coefficient applied to the calculation of \(C_{d}\), \(C_{vBH}\) is the viscous coefficient of the bare hull, and \(C_{a}\) represents the allowance corresponding to the roughness of the surface of the hull.  
To calculate \(C_{vBH}\), the correlation proposed by the Massachusetts Institute of Technology [21] is used:  
\[C_{vBH}=C_{F}\cdot\left[1+1.5\cdot\left(\frac{\dot{\rho}}{L}\right)^{\frac{3}{ 2}}+7\cdot\left(\frac{\dot{\rho}}{L}\right)^{3}+0.002\cdot(C_{P}-0.6)\,\right] \tag{4}\]  
where \(C_{F}\) is the friction coefficient of the AUV; \(\dot{\rho}\) stands for the diameter of the AUV; \(L\) is the total length of the AUV; \(C_{P}\) is the prismatic coefficient of the AUV, calculated by means of Equation (5); and \(V_{AUV}\) represents the volume enclosed by the hull of the AUV.  
\[C_{P}=\frac{V_{AUV}}{\pi\cdot\left(\frac{\dot{\rho}}{2}\right)^{2}\cdot L} \tag{5}\]  
The calculation of \(C_{a}\) is made in accordance with the ITTC recommended procedure 7.5-02-03-01.4 for performance prediction [22],  
\[C_{a}=0.044\cdot\left(\left(\frac{k_{s}}{L}\right)^{\frac{1}{3}}-10\cdot \text{Re}^{\frac{-1}{3}})\right)+0.000125 \tag{6}\]where \(k_{s}\) is the roughness of the surface and Re stands for the Reynolds number of the AUV considering \(L\) as the characteristic dimension. For this work, as no data on the roughness are available, the value recommended by ITTC in the cited procedure is used: 150 \(\upmu\)m.  
Finally, to estimate \(C_{F}\), the correlation line ITTC1957, proposed by the International Towing Tank Conference, was used [23].  
\[C_{F}=\frac{0.075}{\left(\log_{10}\text{Re}-2\right)^{2}} \tag{7}\]  
Note that wave resistance is not computed as the vehicle is considered to travel far away from the water surface.