07 Analisa Dimensional dan Similaritas Part2

Summary

The script discusses the method of dimensional analysis using the variable repetition method, primarily in fluid mechanics. The process involves identifying all relevant variables, formulating their dimensions, and determining dimensionless parameters (P). The script goes through the steps of creating these parameters systematically by selecting repeating variables, raising them to powers, and ensuring independence across variables. The example of analyzing pressure drop in a pipe is used to explain the method, with further examples on drag analysis of a plate in a fluid flow. The key focus is on simplifying complex phenomena into dimensionless terms for easier analysis.

Takeaways

• 😀 The method of dimensional analysis is used to form dimensionless parameters, crucial for fluid mechanics analysis.
• 😀 The key method discussed for determining dimensionless parameters is the 'repeating variable method', which helps systematically form dimensionless parameters.
• 😀 Step 1 involves listing all the variables involved in the analysis, considering factors such as fluid properties and geometry.
• 😀 Step 2 requires expressing each variable in terms of its fundamental dimensions (e.g., force, length, time, mass).
• 😀 Step 3 uses the Pi theorem to determine the number of dimensionless groups, where the number of such groups is the number of involved variables minus the number of fundamental dimensions.
• 😀 Step 4 establishes the repeating variables, ensuring that they are dimensionally independent.
• 😀 Step 5 involves forming dimensionless parameters (Pi groups) by multiplying the repeating variables raised to appropriate powers with non-repeating variables.
• 😀 Step 6 requires repeating Step 5 for all non-repeating variables to form all necessary Pi groups.
• 😀 Step 7 ensures that each formed Pi group is indeed dimensionless, checking the dimensional consistency of each group.
• 😀 Step 8 involves simplifying the dimensionless parameters if needed to make the analysis more straightforward, possibly identifying key dimensionless numbers like Reynolds number.
• 😀 The final example demonstrates using dimensional analysis to derive dimensionless parameters for a pipe flow scenario, with Reynolds number being a key result for fluid flow analysis.

Q & A

What is the purpose of determining dimensionless parameters (P) in fluid mechanics?

The purpose of determining dimensionless parameters is to analyze fluid behavior systematically. These parameters allow us to simplify complex physical phenomena and make comparisons across different cases by removing units, making the analysis more efficient and generalized.

Why is the method of repeating variables (repetition of variables method) preferred for determining P?

The repetition of variables method is preferred because it helps to systematically form the dimensionless parameters (P) in a way that ensures their dimensional independence. This approach allows for easier analysis of fluid dynamics.

What is the first step in applying the repetition of variables method?

The first step is to list all the variables involved in the analysis. This includes variables related to geometry, fluid properties, and external influences. The correct identification of variables requires experience and knowledge of the phenomenon being analyzed.

What does the second step of the repetition of variables method involve?

In the second step, each variable is expressed in terms of its basic dimensions, such as force (F), length (L), time (T), or mass (M). This allows the variables to be categorized and analyzed based on their dimensions.

What is the significance of determining the number of dimensionless parameters (k - r)?

Determining the number of dimensionless parameters (k - r) is essential because it tells us how many independent dimensionless groups will be formed. Here, 'k' represents the number of variables, and 'r' is the number of fundamental dimensions involved.

What does the step of determining the number of repeating variables (step 4) depend on?

The number of repeating variables is determined by the number of fundamental dimensions involved in the system. If three fundamental dimensions (M, L, T) are involved, then three repeating variables must be chosen.

Why must the repeating variables be independent of each other?

The repeating variables must be independent in terms of their dimensions because one repeating variable should not be formed by a combination of the others. This ensures the validity of the dimensionless parameters formed.

What is the role of the fifth step in the repetition of variables method?

In the fifth step, the dimensionless parameters (P) are formed by multiplying the non-repeating variables with the products of the repeating variables, each raised to a power. This combination ensures the parameters are dimensionless.

How is the consistency of the dimensionless parameters checked?

The consistency of each dimensionless parameter is checked by ensuring that its dimensional analysis results in zero powers for all fundamental dimensions (M, L, T). This ensures that the parameter is indeed dimensionless.

In the pipe flow example, which variables are identified as influencing pressure drop per unit length?

In the pipe flow example, the variables influencing pressure drop per unit length are the pipe diameter (D), fluid density (ρ), fluid viscosity (μ), and flow velocity (V).

What are the results of the dimensional analysis of the pressure drop in the pipe flow example?

The dimensional analysis reveals that the pressure drop per unit length can be expressed as a function of the Reynolds number (a dimensionless quantity) and the ratio of diameter to velocity squared.

How does the method of dimensional analysis apply to the drag on a plate in fluid flow?

For drag on a plate, dimensional analysis is used to express the drag force as a function of the plate's width, height, fluid viscosity, density, and velocity. The analysis identifies the necessary dimensionless parameters, such as the Reynolds number and aspect ratio, for experimental validation.