design_method.md:

- Corrected cabin slenderness ratio equation
- Fixed some formatting

docs/documentation/sizing/fuselage_design/design_method.md

@@ -27,63 +27,64 @@ In which
 #### Calculate cabin width
 The cabin width $w_{\text{cabin}}$ (in inch) can then be calculated:
 $$
-    w_{\text{cabin}} = w_{\text{aisle}} + w_{seat\_bench\_left} + w_{seat\_bench\_right} + 2 \cdot w_{seat\_space}
+    w_{\text{cabin}} = w_{\text{aisle}} + w_{\text{bench,left}} + w_{\text{bench,right}} + 2 \cdot w_{\text{seat space}}
 $$
 
 In which
 
-- $w_{aisle}$ - passenger aisle width
-- $w_{seat\_space}$ - lowest class seat space
+- $w_{\text{aisle}}$ - passenger aisle width
+- $w_{\text{seat space}}$ - lowest class seat space
 
-In case of a **wide-body aircraft configuration** there is an additional row in the middle of the aircraft as well as an additional passenger aisle. The width of the seat bench $w_{seat\_bench\_center}$ can be calculated using an equation similar to that in the previous section.
-$
-    w_{seat\_bench\_center} = n_{seats} \cdot w_{seat} + 2 \cdot w_{armrest\_outer} + (n_{seats} - 1) \cdot w_{armrest\_inner}
-$
+In case of a **wide-body aircraft configuration** there is an additional row in the middle of the aircraft as well as an additional passenger aisle. The width of the seat bench $w_{\text{bench,center}}$ can be calculated using an equation similar to that in the previous section.
+$$
+    w_{\text{bench,center}} = n_{\text{seats}} \cdot w_{\text{seat}} + 2 \cdot w_{\text{armrest,outer}} + (n_{\text{seats}} - 1) \cdot w_{\text{armrest,inner}}
+$$
 
 In which
-- $w_{seat}$ - seat width (from lowest class seat parameters of right side)
-- $w_{armrest\_outer}$ - width of outer armrest (from lowest class seat parameters of right side)
-- $w_{armrest\_inner}$ - width of inner armrest (from lowest class seat parameters of right side)
+
+- $w_{\text{seat}}$ - seat width (from lowest class seat parameters of right side)
+- $w_{\text{armrest,outer}}$ - width of outer armrest (from lowest class seat parameters of right side)
+- $w_{\text{armrest,inner}}$ - width of inner armrest (from lowest class seat parameters of right side)
 
 The equation for the cabin width estimation must be adapted accordingly:
-$
-    w_{cabin} = w_{aisle} + w_{seat\_bench\_left} + w_{seat\_bench\_right} + 2 \cdot w_{seat\_space} + w_{aisle} + w_{seat\_bench\_center}
-$
+$$
+    w_{\text{cabin}} = w_{\text{aisle}} + w_{\text{bench,left}} + w_{\text{bench,right}} + 2 \cdot w_{\text{seat space}} + w_{\text{aisle}} + w_{\text{bench,center}}
+$$
 
 ### Cabin slenderness ratio <sup>[1]</sup>
-The cabin slenderness ratio describes the ratio of cabin width to cabin length $\frac{w_{cabin}}{l_{cabin}}$.
-Whilst the cabin width is already known, the cabin length can be determined using the following equation:
-$
-    l_{cabin} = \frac{n_{PAX\_per\_class}}{ab} \cdot \left[ sp + \frac{a_{service}}{w_{seat}} + \frac{a_{bulk}}{\frac{w_{aisle}}{ab} + w_{seat}} + x \cdot w_{exit} \cdot \left( \frac{ab}{n_{PAX\_per\_class}} + \frac{sp}{d_{exits}} \right)  \right]
-$
+The cabin slenderness ratio describes the ratio of cabin width to cabin length and can be determined using the following equation:
+$$
+    \frac{w_{\text{cabin}}}{l_{\text{cabin}}} = \frac{n_{\text{PAX per class}}}{ab} \cdot \left[ sp + \frac{a_{\text{service}}}{w_{\text{seat}}} + \frac{a_{\text{bulk}}}{\frac{w_{\text{aisle}}}{ab} + w_{\text{seat}}} + x \cdot w_{\text{exit}} \cdot \left( \frac{ab}{n_{\text{PAX per class}}} + \frac{sp}{d_{\text{exits}}} \right)  \right]
+$$
 
 In which
+
 - $x$ - factor (1 for single-aisle, 2 for wide-body)
-- $n_{PAX\_per\_class}$ - number of PAX per class
+- $n_{\text{PAX per class}}$ - number of PAX per class
 - $ab$ - seat abreast
 - $sp$ - seat pitch
-- $a_{service}$ - service area per PAX
-- $a_{bulk}$ - bulk area per PAX
-- $w_{exit}$ - exit width
-- $d_{exits}$ - maximum distance between two exits
+- $a_{\text{service}}$ - service area per PAX
+- $a_{\text{bulk}}$ - bulk area per PAX
+- $w_{\text{exit}}$ - exit width
+- $d_{\text{exits}}$ - maximum distance between two exits
 
 ### Cabin length
 Knowing the cabin width and the cabin slenderness ratio, the cabin length (in inch) can be calculated:
-$
-    l_{cabin} = \frac{w_{cabin}}{\frac{w_{cabin}}{l_{cabin}}}
-$
+$$
+    l_{\text{cabin}} = \frac{w_{\text{cabin}}}{\frac{w_{\text{cabin}}}{l_{\text{cabin}}}}
+$$
 
 ### Cabin wall thickness
 The cabin wall thickness can be estimated using the following calculation:
-$
-    t_{wall} = 0.02 \cdot w_{cabin} + 2.5"
-$
+$$
+    t_{\text{wall}} = 0.02 \cdot w_{\text{cabin}} + 2.5"
+$$
 
 ### Cabin floor thickness
 With the use of the cabin wall thickness, the cabin floor thickness can be calculated:
-$
-    t_{floor} = 1.5 \cdot t_{wall}
-$
+$$
+    t_{\text{floor}} = 1.5 \cdot t_{\text{wall}}
+$$
 
 ## Determine fuselage geometry {#fuselage-geometry}
 With the calculated cabin the fuselage dimensions can be estimated.
@@ -92,85 +93,92 @@ With the calculated cabin the fuselage dimensions can be estimated.
 The fuselage length can be determined via regression formulas using the cabin length (in meter).
 
 For single-aisle aircraft:
-$
-    l_{fuselage} = \frac{l_{cabin}}{0.23482756 \cdot \log l_{cabin} - 0.05106017}
-$
+$$
+    l_{\text{fuselage}} = \frac{l_{\text{cabin}}}{0.23482756 \cdot \log l_{\text{cabin}} - 0.05106017}
+$$
 
 For wide-body aircraft:
-$
-    l_{fuselage} = \frac{l_{cabin}}{0.1735 \cdot \log l_{cabin} - 0.0966}
-$
+$$
+    l_{\text{fuselage}} = \frac{l_{\text{cabin}}}{0.1735 \cdot \log l_{\text{cabin}} - 0.0966}
+$$
 
 ### Fuselage diameters
 The fuselage does not necessarily have a circular cross-section. It is more common to design elliptical cross-sections. Because of that, there are several values that must be determined:
+
 - Fuselage diameter in y-direction
 - Fuselage diameter in negative z-direction
 - Fuselage diameter in positive z-direction
 
 #### Fuselage diameter in y-direction
-The fuselage diameter in y-direction $d_{fuselage\_y}$ can be calculated in the following way:
-$
-    d_{fuselage\_y} = w_{cabin} + 2 \cdot t_{wall}
-$
+The fuselage diameter in y-direction $d_{\text{fuselage,y}}$ can be calculated in the following way:
+$$
+    d_{\text{fuselage,y}} = w_{\text{cabin}} + 2 \cdot t_{\text{wall}}
+$$
 
 #### Fuselage diameter in negative z-direction
-The fuselage diameter in negative z-direction $d_{fuselage\_z\_neg}$ is determined by the cargo accommodation. It can be calculated in the following way.
+The fuselage diameter in negative z-direction $d_{\text{fuselage,z,neg}}$ is determined by the cargo accommodation. It can be calculated in the following way.
+
 At first, the distance to the cargo bottom is calculated:
-$
-    d_{to\_cargo\_bottom} = h_{max} + t_{floor} + d_{container\_to\_ceil} + o_{cabin\_floor}
-$
+$$
+    d_{\text{to cargo bottom}} = h_{\text{ULD,max}} + t_{\text{floor}} + d_{\text{container to ceil}} + o_{\text{cabin floor}}
+$$
 
 In which
-- $h_{max}$ - maximum height of unit load device
-- $d_{container\_to\_ceil}$ - distance from the container to the ceiling
-- $o_{cabin\_floor}$ - offset cabin floor
+
+- $h_{\text{ULD,max}}$ - maximum height of unit load device
+- $t_{\text{floor}}$ - floor thickness
+- $d_{\text{container to ceil}}$ - distance from the container to the ceiling
+- $o_{\text{cabin floor}}$ - offset cabin floor
 
 Afterwards, the distance to the lower compartment edge is estimated:
-$
-    d_{to\_lower\_compartment\_edge} = d_{container\_to\_wall} + 0.5 \cdot w_{max\_at\_base}
-$
+$$
+    d_{\text{to lower compartment edge}} = d_{\text{container to wall}} + 0.5 \cdot w_{\text{base,max}}
+$$
 In which
-- $d_{container\_to\_wall}$ - distance from container to wall
-- $w_{max\_at\_base}$ - maximum width at container base
+- $d_{\text{container to wall}}$ - distance from container to wall
+- $w_{\text{base,max}}$ - maximum width at container base
 
 Based on the Pythagorean theorem, the inner fuselage diameter (that equals the hypotenuse) can be calculated:
-$
-    d_{inner\_fuselage\_z\_neg} = \sqrt{(d_{to\_cargo\_bottom})^2 + (d_{to\_lower\_compartment\_edge})^2}
-$
+$$
+    d_{\text{fuselage,z,neg,inner}} = \sqrt{(d_{\text{to cargo bottom}})^2 + (d_{\text{to lower compartment edge}})^2}
+$$
 
 Adding the wall thickness results in the fuselage diameter in negative z-direction:
-$
-    d_{fuselage\_z\_neg} = d_{inner\_fuselage\_z\_neg} + t_{wall}
-$
+$$
+    d_{\text{fuselage,z,neg}} = d_{\text{fuselage,z,neg,inner}} + t_{\text{wall}}
+$$
 
 #### Fuselage diameter in positive z-direction
-The fuselage diameter in positive z-direction $d_{fuselage\_z\_pos}$ is determined by the passenger accommodation. It can be calculated in the following way.
+The fuselage diameter in positive z-direction $d_{\text{fuselage,z,pos}}$ is determined by the passenger accommodation. It can be calculated in the following way.
+
 Firstly, the inner fuselage height (equals outer cabin height) can be determined:
-$
-    d_{inner\_fuselage\_z\_pos} = h_{aisle\_standing} - o_{cabin\_floor} + h_{system\_bay}
-$
+$$
+    d_{\text{fuselage,z,pos,inner}} = h_{\text{aisle,standing}} - o_{\text{cabin floor}} + h_{\text{system bay}}
+$$
 
 In which
-- $h_{aisle\_standing}$ - passenger aisle standing height
-- $o_{cabin\_floor}$ - cabin floor offset
-- $h_{system\_bay}$ - system bay height above cabin
+
+- $h_{\text{aisle,standing}}$ - passenger aisle standing height
+- $o_{\text{cabin floor}}$ - cabin floor offset
+- $h_{\text{system bay}}$ - system bay height above cabin
 
 Adding the wall thickness leads to the fuselage diameter in positive z-direction.
-$
-    d_{fuselage\_z\_pos} = d_{inner\_fuselage\_z\_pos} + t_{wall}
-$
+$$
+    d_{\text{fuselage,z,pos}} = d_{\text{fuselage,z,pos,inner}} + t_{\text{wall}}
+$$
 
 ### Fuselage height
 The total height of the fuselage can be determined by summing up the fuselage diameters in positive and negative z-direction:
-$
-    h_{fuselage} = d_{fuselage\_z\_pos} + d_{fuselage\_z\_neg}
-$
+$$
+    h_{\text{fuselage}} = d_{\text{fuselage,z,pos}} + d_{\text{fuselage,z,neg}}
+$$
 
 !!! note 
     If the `force_circle_cross_section` mode is selected, fuselage height and width are set to the maximum of both.
 
 ## Mass estimation {#mass-estimation}
 The following masses are estimated:
+
 - Fuselage structure
 - Operator items
 - Furnishing
@@ -183,8 +191,10 @@ Please refer to _Synthesis of Subsonic Airplane Design_ by E. Torenbeek<sup>[3]<
 
 ## Generate fuselage shape {#generate-shape}
 The fuselage shape is generated using the calculated data and the reference ellipses (see the [getting started](getting_started.md) page for more information). The final geometry is written to the `fuselage_design_ellipses.json` file.
+
 The aircraft is divided into three sections: A cockpit section, followed by a constant section, and the tail section. 
 The steps of the shape generation are basically the same for all aircraft sections:
+
 1. Calculate the section length as a percentage of the fuselage length<sup>*</sup>.
 2. Proportionally adjust the given reference geometry to match the actual geometry using scaling factors. Therefore, separate scaling factors are calculated for
     - the x-direction (lengthwise),