fluid mechanics - Dimensional analysis

What is the primary difference between a ‘dimension’ and a ‘unit’ in physical measurement?

a dimension is a measure of a physical quantity and a unit is a way to assign a numerical value to the dimension

how many primary dimensions are there

7

what primary dimension does L represent

Length

Which primary dimension uses the symbol m and is measured in kilograms

mass

the primary dimension ‘amount of matter’ is represented by the symbol ‘N’. what is its primary si unit

mole

what is the primary unit used to measure mass

pound-mass (lbm)

In the context of temperature what are the primary si and english units respectively

kelvin (K) and rankine (R)

Dimension ‘amount of light’ uses the symbol ‘C’ what is the associated si unit

candela (cd)

apart from ‘t’ what other symbol is sometimes used to represent the primary dimension of time

T

Express the base dimensions of acceleration (a) using the symbols for Length (L) and Time (t).

Lt^(-2) or L/t²

What are the base dimensions of Force (F), calculated as mass×acceleration?

mLt^(-2) or mL/t²

What are the base dimensions of density (ρ)?

mL^(-3) or m/L³

Calculate the base dimensions of Pressure (P), defined as force/area

mL^(-1)t^(-2) or m/t²L

State the base dimensions of Power, derived from the formula Force⋅Velocity.

mL²t^(-3) or mL²/t³

Explain why Torque (or moment) and Energy share the same base dimensions.

both are calculated as force . distance, resulting in the base dimensions

mL²t^(-2)

Given the heat equation Q=mcΔT, what are the base dimensions of specific heat (c)?

L²t^(-2)T^(-1) (where T or theta is temperature)

What are the base dimensions of dynamic viscosity (μ)?

mL^(-1)t(-1) or m/Lt

The note in the source states that two specific physical quantities are dimensionally identical. what are they?

stress and pressure

Name the three necessary conditions for complete similarity between a model and a prototype

geometric similarity, kinematic similarity and dynamic similarity

Define geometric similarity

the model must be the same shape as the prototype but may be scaled by a constant factor

define kinematic similarity

the velocity at any point in the model flow must be proportional to the velocity at the corresponding point in the prototype flow

define dynamic similarity

all forces in the model flow scale by a constant factor to the corresponding forces in the prototype flow

What is the most popular and simplest method for generating non-dimensional parameters ($\Pi$s)?

The Method of Repeating Variables (Buckingham Pi Theorem)

in step one of the method of repeating variables, what is the value of ‘n’?

the total number of parameters in the problem

in step three of the method of repeating variables, how do you calculate the expected number of dimensionless groups (k)?

k=n−j, where j is the reduction (number of primary dimensions).

what is the requirement for the selection of repeating parameters in step four of the buckingham pi method

All primary dimensions involved in the problem (usually M, L, and t) must be included within the selected parameters.

In the analysis of pressure loss in a pipe, why is the 'Roughness ratio' (η) considered unique among the parameters?

it is already dimensionless

What are the primary dimensions of Pressure loss (Δp)?

ML^(-1)t^(-2) (the same as pressure)

what is the buckingham pi theorem

it states that a relationship between n variables with j primary dimensions can be expressed as a relationship between n-j dimensionless groups.

The Reynolds number (Re) represents the ratio of which two types of forces?

inertial forces to viscous forces

State the formula for the Reynolds number using density (ρ), average velocity (Vavg​), diameter (D), and viscosity (μ).

Re= ρV^(avg)​D​/μ

under what conditions is flow in a pipe generally considered to be laminar

Re < 2300

Under what condition is flow in a pipe generally considered to be turbulent

Re > 4000

How is the volumetric flow rate (Q˙​) calculated using area (A) and average velocity (Vavg​)?

Q˙​=A×Vavg​

In the heat exchanger example, calculate the area (A) for a single pipe of diameter D=6×10−3 m.

A= [π(6×10−3)² ]/4​ = 0.00002826 m²

in the heat exchanger comparison, why was the flow for design a classified as laminar

Because its Reynolds number (Re=142) was less than 2300

For 'Design B' with 9 parallel tubes, the calculated Reynolds number was _____, indicating laminar flow.

15.7

in step 2 of the method of repeating variables, what must be listed for each of the n parameters?

their primary dimensions

what are the base dimensions for shear stress? (note it is identical to pressure)

mL^(-1)t^(-2) or m/t²L

in fluid mechanics, what is the base dimension of shear rate

1/time or t^(-1)

what is step 6 in the method of repeating variables

write the final functional relationship and check the algebra

When calculating Re for a square building of 5 m×5 m in 7 m/s air flow, what dimension represents 'D' or 'L'?

the side length of the building (5m)

What happens in Step 5 of the Method of Repeating Variables?

the k dimensionless groups are constructed and manipulated as necessary