## 2 Materials and Methods
### Scale model--Selection of variables  
The purpose of this scale model study was to evaluate natural ventilation characteristics (wind direction) and alternative ridge designs. The baseline for the scale model was a CBP barn located in Kentucky. The airflow rate, air patterns and system performance under natural ventilation are affected by fluid properties including viscous forces, inertia forces and buoyancy forces. These properties were represented by Reynolds and by building geometry factors. Assuming that the same phenomenon governs performance in the scale model, key variables affecting ventilation characteristics were compiled (Table 1) and reduced to basic dimensions of length (L), mass (M), temperature (\(\theta\)) and time (T). Variables selections were determined based on four major assumptions: incompressible flow, negligible generation of gases and moisture from the building environment and significant internal heat generation from the compost pack. According to the Buckingham \(\pi\) theorem, a group of ten (14 variables minus 4 dimensions) independent \(\pi\)-terms was available. The relationships among \(\pi\)-terms for fluid flow and heat transfer are expressed in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline
**Variable** & **Description** & **Units (S.I)** & **Symbols** & **Dimensional Symbol** \\ \hline
1 & Building length & m & \(l\) & L \\
2 & Building width & m & \(w\) & L \\
3 & Building height & m & \(h\) & L \\
4 & Gravitational acceleration & m s\({}^{-2}\) & g & L T\({}^{-2}\) \\
5 & Coefficient of heat transfer & kg s\({}^{-3}\) K\({}^{-1}\) & \(h_{c}\) & M T\({}^{-3}\) @\({}^{-1}\) \\
6 & Heat & kg m\({}^{2}\) s\({}^{-2}\) & \(Q\) & M L\({}^{2}\) T\({}^{-2}\) \\
7 & Wind velocity & m s\({}^{-1}\) & \(v_{w}\) & L T\({}^{-1}\) \\
8 & Temperature of air & K & \(T_{a}\) & \(\theta\) \\
9 & Density of air & kg m\({}^{-3}\) & \(\rho\) & M L\({}^{-3}\) \\
10 & Specific heat of air & m\({}^{2}\) s\({}^{-2}\) K\({}^{-1}\) & \(c_{p}\) & L\({}^{2}\) T\({}^{-2}\) @\({}^{-1}\) \\
11 & Heat transfer coefficient & kg s\({}^{-3}\) K\({}^{-1}\) & \(U\) & M T\({}^{-3}\) @\({}^{-1}\) \\
12 & Dynamic viscosity of air & kg m\({}^{-1}\) s\({}^{-1}\) & \(\mu\) & M L\({}^{-1}\) T\({}^{-1}\) \\
13 & Thermal conductivity of air & kg m s\({}^{-3}\) K\({}^{-1}\) & \(k\) & M L T\({}^{-3}\) @\({}^{-1}\) \\
14 & Coef. of thermal expansion & K\({}^{-1}\) & \(\beta_{\rm e}\) & \(\theta^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Variables affecting natural ventilation characteristics. \begin{table}
\begin{tabular}{c c c c} \hline
**Term n\({}^{\circ}\)** & \(\pi\)**-term** & **Term n\({}^{\circ}\)** & \(\pi\)**-term** \\ \hline
1 & \(\pi_{1}=w\cdot l^{-1}\) & 6 & \(\pi_{6}=c_{p}v_{w}{}^{-2}\cdot T_{a}\) \\
2 & \(\pi_{2}=h\cdot l^{-1}\) & 7 & \(\pi_{7}=U\cdot v_{w}{}^{-3}\cdot T_{a}\cdot p^{-1}\) \\
3 & \(\pi_{3}=g\cdot l^{-1}\cdot\rho^{-1}\) & 8 & \(\pi_{8}=\mu\cdot l^{-1}\cdot v_{w}{}^{-1}\cdot p^{-1}\) \\
4 & \(\pi_{4}=v_{w}{}^{-3}\cdot T_{a}\cdot\rho^{-1}\cdot h_{c}\) & 9 & \(\pi_{9}=k\cdot l^{-1}\cdot v_{w}{}^{-3}\cdot T_{a}\cdot\rho^{-1}\) \\
5 & \(\pi_{5}=Q\cdot l^{-3}\cdot v_{w}{}^{-2}\cdot\rho^{-1}\) & 10 & \(\pi_{10}=\beta_{c}T_{a}\) \\ \hline \end{tabular}
\end{table} 
Table 2: List of \(\pi\)-terms. Design conditions 1 through 5 (Table 2) require a scale ratio 1:\(n\) (\(n\) = 16, the geometric length scale) multiplied by the prototype geometric dimensions. Design conditions 6 through 10 (Table 2) establish the ratio of prototype wind velocity to model wind velocity. Using the same fluid in both the scale model and prototype systems requires the model wind velocity to be \(n\) times the prototype wind velocity. The tests were performed with the airflow in the scale model geometrically and dynamically similar to a full-scale CBP barn. To ensure the similarity of the characteristics of the airflow in the scale model and real scale CBP barn, the flow regime, expressed by the Reynolds number, must be equal in both cases: \([\text{Re}]_{\text{Model}}\) = \([\text{Re}]_{\text{Full}}\).