# Step-by-Step Solution: Dimensional Analysis for Natural Ventilation ## **1. Identify Variables and Their Dimensions** Let's list the physical quantities involved and their dimensions in the MLT (Mass-Length-Time) system: | Quantity | Symbol | Units | Dimensions | |----------------|--------|---------------|----------------------| | Building height| \( h \) | m | \( [L] \) | | Building width | \( b \) | m | \( [L] \) | | Wind velocity | \( V \) | m/s | \( [L][T]^{-1} \) | | Air viscosity | \( \mu \) | kg/(m·s) | \( [M][L]^{-1}[T]^{-1} \) | | Pressure diff. | \( \Delta P \) | N/m² or Pa | \( [M][L]^{-1}[T]^{-2} \) | ## **2. Express the Problem** We seek a relationship for the pressure difference (\( \Delta P \)) as a function of the other variables: \[ \Delta P = f(h, b, V, \mu) \] ## **3. Count Variables and Fundamental Dimensions** - Variables: \( \Delta P, h, b, V, \mu \) → **5 variables** - Fundamental Dimensions: **3** (M, L, T) Number of dimensionless groups (\( \pi \) terms) = \( 5 - 3 = 2 \) ## **4. Choose Repeating Variables** Pick 3 variables that cover all base dimensions. Good choices: - \( V \) (covers L, T) - \( b \) (covers L) - \( \mu \) (covers M, L, T) ## **5. Form Dimensionless Groups (\( \pi \) Terms)** ### **First Group (\( \pi_1 \)): Involving \( \Delta P \)** Let: \[ \pi_1 = \Delta P \cdot V^a \cdot b^c \cdot \mu^d \] Write dimensions: - \( [\Delta P] = [M][L]^{-1}[T]^{-2} \) - \( [V] = [L][T]^{-1} \) - \( [b] = [L] \) - \( [\mu] = [M][L]^{-1}[T]^{-1} \) So, \[ [\pi_1] = [M][L]^{-1}[T]^{-2} \cdot [L]^a [T]^{-a} \cdot [L]^c \cdot [M]^d [L]^{-d} [T]^{-d} \] Combine exponents: - \( M: 1 + d \) - \( L: -1 + a + c - d \) - \( T: -2 - a - d \) Set exponents to zero (since dimensionless): 1. \( 1 + d = \implies d = -1 \) 2. \( -1 + a + c - d = \implies a + c = -1 \) (since \( d = -1 \)) 3. \( -2 - a - d = \implies -2 - a + 1 = \implies a = -1 \) Now, \( a + c = -1 \implies -1 + c = -1 \implies c = \) So, \[ \pi_1 = \Delta P \cdot V^{-1} \cdot b^{} \cdot \mu^{-1} = \frac{\Delta P}{\mu V} \] ### **Second Group (\( \pi_2 \)): Involving \( h \)** Let: \[ \pi_2 = h \cdot V^a \cdot b^c \cdot \mu^d \] Dimensions: - \( [h] = [L] \) - \( [V] = [L][T]^{-1} \) - \( [b] = [L] \) - \( [\mu] = [M][L]^{-1}[T]^{-1} \) So, \[ [\pi_2] = [L] \cdot [L]^a [T]^{-a} \cdot [L]^c \cdot [M]^d [L]^{-d} [T]^{-d} \] \[ = [M]^d [L]^{1 + a + c - d} [T]^{-a - d} \] Set exponents to zero: - \( M: d = \implies d = \) - \( L: 1 + a + c = \) - \( T: -a = \implies a = \) So, \( 1 + + c = \implies c = -1 \) \[ \pi_2 = h \cdot V^ \cdot b^{-1} \cdot \mu^ = \frac{h}{b} \] ## **6. Summary of Dimensionless Parameters** The two main dimensionless groups describing the system are: 1. **Pressure group:** \[ \boxed{\frac{\Delta P}{\mu V}} \] 2. **Geometric aspect ratio:** \[ \boxed{\frac{h}{b}} \] Alternatively, you could also use the Reynolds number (\( Re = \frac{\rho V b}{\mu} \)) if density is relevant, but based on your variables, the above groups are most appropriate. ## **7. Final Answer** ### **Dimensionless Parameters:** \[ \boxed{ \frac{\Delta P}{\mu V}, \quad \frac{h}{b} } \] **These dimensionless groups describe how building height, width, wind velocity, and air viscosity influence the pressure changes across the building for natural ventilation.** --- **Note:** If you include density (\( \rho \)), you could form a Reynolds number, but with the given variables, \(\frac{\Delta P}{\mu V}\) and \(\frac{h}{b}\) are the relevant dimensionless parameters.