Question

According to Buckingham's Pi theorem, if there are 'n' variables in the problem and these variables contain 'm' dimensions, the equation relating all the variables will have the following dimensionless groups

• A. (n - m)
• B. (n + m)
• C. (n x m)
• D. (n m)

Hint:

Dimensional Analysis: Buckingham's $\pi$-Theorem states that number of dimensionless groups = $\mathbf{(n - m)}$ (Total variables minus fundamental dimensions).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Dimensional analysis and Buckingham's $\pi$-theorem: if a physical phenomenon involves $n$ independent physical variables expressed in $m$ fundamental dimensions ($M, L, T, heta$), the relationship can be expressed as a function of exactly $(n - m)$ independent dimensionless $\pi$-groups.
Key Formula or Approach:
\[ \mathbf{Number \text{ } of \text{ } Dimensionless \; \pi\text{-Terms}} = \mathbf{n - m} \quad [n = \text{Total Variables}, \; m = \text{Fundamental Dimensions}] \]

Step 2: Detailed Explanation:
In dimensional analysis, fluid mechanics, and heat transfer modeling (Buckingham's $\pi$-Theorem, Edgar Buckingham 1914):
- Buckingham's Pi Theorem states:
1. If a dimensionally homogeneous equation involves $n$ physical variables (dependent and independent variables, such as velocity, pressure drop, viscosity, diameter, density):
\[ f(q_1, q_2, q_3, \dots, q_n) = 0 \]
2. And these $n$ variables contain $m$ fundamental primary dimensions (such as Mass $M$, Length $L$, Time $T$, Temperature $\theta$):
3. Then the equation can be reformulated and simplified into an equivalent relationship involving exactly (n - m) independent dimensionless groups ($\pi$-terms) (A):
\[ \phi(\pi_1, \pi_2, \dots, \pi_{n-m}) = 0 \]
- (This is the foundational mathematical theorem used to derive the Reynolds number $Re$, Nusselt number $Nu$, and Prandtl number $Pr$ in dairy thermal engineering).

Step 3: Final Answer:
Therefore, the equation will have (n - m) dimensionless groups, corresponding to option (A).