Which of the following are dimensionless physical quantities?
Thermal conductivity
Temperature
Amount of substance
Density
Dynamic viscosity
Reynolds No. (
Dryness fraction
Specific enthalpy
Ratio of specific heat capacities
Mach No. ( Ma)
Frequency
Specific heat transfer
Entropy (S)
Strain
The Correct Answer and Explanation is :
Dimensionless Physical Quantities from the Given List:
The following quantities are dimensionless:
- Reynolds Number (Re)
- Dryness Fraction (x)
- Ratio of Specific Heat Capacities (γ = Cp/Cv)
- Mach Number (Ma)
- Strain
Explanation:
Dimensionless physical quantities are those that do not have any physical units and are represented as pure numbers. They are often formed by the ratio of similar physical quantities, canceling out the units.
- Reynolds Number (Re)
- Defined as ( Re = \frac{\rho vL}{\mu} ), where ( \rho ) is density, ( v ) is velocity, ( L ) is characteristic length, and ( \mu ) is dynamic viscosity.
- Since it is a ratio of forces (inertial to viscous), all units cancel out, making it dimensionless.
- Dryness Fraction (x)
- Defined as ( x = \frac{\text{Mass of vapor}}{\text{Total mass}} ).
- Since it is a ratio of masses, it has no dimensions.
- Ratio of Specific Heat Capacities (γ = Cp/Cv)
- Specific heats (( C_p ) and ( C_v )) have the same unit (J/kg·K).
- Their ratio is dimensionless.
- Mach Number (Ma)
- Defined as ( Ma = \frac{v}{c} ), where ( v ) is the speed of the object and ( c ) is the speed of sound.
- Since it is a ratio of two velocities, it has no dimensions.
- Strain
- Defined as ( \text{Strain} = \frac{\Delta L}{L} ), where ( \Delta L ) is change in length and ( L ) is original length.
- Since it is a ratio of lengths, it has no dimensions.
Physical Quantities that are NOT Dimensionless:
- Thermal Conductivity (W/m·K) → Has dimensions
- Temperature (K) → Has dimensions
- Amount of Substance (mol) → Has dimensions
- Density (kg/m³) → Has dimensions
- Dynamic Viscosity (Pa·s or kg/m·s) → Has dimensions
- Specific Enthalpy (J/kg) → Has dimensions
- Frequency (Hz or s⁻¹) → Has dimensions
- Specific Heat Transfer (J/kg·K) → Has dimensions
- Entropy (S) (J/K) → Has dimensions
Conclusion:
Only Reynolds Number, Dryness Fraction, Ratio of Specific Heats, Mach Number, and Strain are dimensionless quantities in the given list. These are typically formed as ratios of similar properties where the units cancel out, making them independent of measurement systems.