## 2 The Quick Preliminary Design Model
### Fuel Cell Stack Sizing and Consumables Calculation  
As explained in Section 2.6, the nominal power, delivered by the fuel cell and batteries, is calculated according to the method proposed by [PERSON] et al. [26]. The power demand for each phase of the operational profile is calculated by adding the propulsion power, \(\dot{W}_{prop}\), to the power demand of the equipment onboard, \(\dot{W}_{EQ}\).  
\[\dot{W}=\dot{W}_{prop}+\dot{W}_{EQ} \tag{8}\]  
where \(\dot{W}_{EQ}\) is equal to the sum of the power of each device multiplied by its utilization factor, \(K_{U}\).  
\[\dot{W}_{EQ}=\sum_{i}\dot{W}_{EQ_{i}}\times K_{U_{i}} \tag{9}\]  
To size the fuel cell, DMFC-AUV\(\backslash\)QpDM makes use of a database of DMFC stacks. This database has been created by using the software developed by the research group PICOHIMA from the Universidad Politecnica de Madrid Aero-Marine DMFC Designer(r) [33]. This tool allows designing DMFC stacks based on a polarization curve and a series of parameters and restrictions with special emphasis on saving weight, volume, and/or fuel, depending on the design criteria. In this case, the tool has been configured to design stacks with the highest efficiency. This leads to larger and heavier stack designs, but in the AUVs studied herein the impact of the loaded fuel, oxygen, and products on the weight and volume of the vehicle is greater than that of the installed fuel cells. Thus, to reduce the global weight and volume it is advisable to use high efficiency stacks. After calculating the power to be delivered by the fuel cells, DMFC-AUV\(\backslash\)QpDM searches the database for a whole number of stacks that can meet the demand and selects the highest efficiency solution. The solution can be comprised of one or more stacks based on the concept of the Multi-Stack Fuel Cell [34]. In the last case, the stacks are identical in order to avoid incompatibilities, and the tool allows defining the maximum number of stacks that can be used, _maxNStacks_. For this work, the maximum number of stacks was limited to two units connected as shown in Figure 6. The polarization curve used is that published by [PERSON] et al. of a DMFC stack using pure oxygen, a working temperature equal to 353.15 K (80 \({}^{\circ}\)C), a methanol concentration equal to 1 M, and a working pressure equal to 0.1 MPa [35].  
Aero-Marine DMFC Designer(r) calculates the methanol and oxygen consumption rates of the stack at the indicated working point, \(\dot{m}_{MOH}\) and \(\dot{m}_{O_{2}}\), respectively. Once the stack has been selected and using the value of the endurance and the density of methanol and compressed oxygen, \(\rho_{CH_{3}OH}\) and \(\rho_{O_{2}}\), respectively, the mass and volume of the embarked methanol and oxygen are calculated, \(m_{CH_{3}OH}\) and \(m_{O_{2}}\), respectively. It must be noted that the endurance is used as an input rather than an output. This may appear confusing, but, in the end, it is equivalent to the definition of the capacity of methanol of the AUV and allows a more intuitive configuration of DMFC-AUV\(\backslash\)QpDM.  
The mass and volume of CO\({}_{2}\) and water produced, \(m_{CO_{2}}\), \(V_{CO_{2}}\), \(m_{H_{2}O}\), and \(V_{H_{2}O}\), respectively, are calculated in accordance with the reactions that occur in a DMFC, i.e., 1 mol of CO\({}_{2}\) and 2 mol of H\({}_{2}\)O for each mole of methanol [14].