Question

Each term of the Bernoulli's equation written in the form $\frac{p}{\rho} + \frac{g}{g_c}(Z) + \frac{v^2}{2g_c} = \text{Constant}$, represents the total energy per unit

• A. Mass
• B. Volume
• C. Specific weight
• D. Specific volume

Hint:

Bernoulli’s equation, in the form given, represents the total energy per unit mass, with each term (pressure, potential, kinetic) in units of \( \text{J/kg} \).

Answer 1

The Correct Option is A

Step 1: Understand Bernoulli’s equation. 
Bernoulli’s equation describes the conservation of energy for an ideal fluid (incompressible, inviscid, steady flow along a streamline). The given form is: \[ \frac{p}{\rho} + \frac{g}{g_c}(Z) + \frac{v^2}{2g_c} = \text{Constant}, \] where:
\( p \): pressure,
\( \rho \): density,
\( g \): gravitational acceleration,
\( g_c \): gravitational constant (used in some unit systems to convert units, often 1 in SI),
\( Z \): elevation (height),
\( v \): velocity.
Each term represents a form of energy per unit of a certain quantity:
\( \frac{p}{\rho} \): Pressure energy per unit,
\( \frac{g}{g_c}(Z) \): Potential energy per unit (due to elevation),
\( \frac{v^2}{2g_c} \): Kinetic energy per unit (due to velocity).
The equation states that the sum of these energies is constant along a streamline, assuming no energy losses (e.g., friction). 
Step 2: Analyze the units of each term. 
To determine what “per unit” means, examine the units of each term:
Pressure term: \( \frac{p}{\rho} \):
\( p \): Pressure (\( \text{N/m}^2 \) in SI),
\( \rho \): Density (\( \text{kg/m}^3 \)),
\( \frac{p}{\rho} \): \( \frac{\text{N/m}^2}{\text{kg/m}^3} = \frac{\text{N} \cdot \text{m}}{\text{kg}} = \frac{\text{J}}{\text{kg}} \), This is energy per unit mass (\( \text{J/kg} \)).
Potential term: \( \frac{g}{g_c}(Z) \):
\( g \): Gravitational acceleration (\( \text{m/s}^2 \)),
\( g_c \): Gravitational constant (in SI, \( g_c = 1 \), but in some systems like English units, it adjusts units),
\( Z \): Elevation (\( \text{m} \)),
Assuming \( g_c = 1 \) (SI units), \( \frac{g}{g_c}(Z) = gZ \), with units: \( (\text{m/s}^2) \cdot (\text{m}) = \text{m}^2/\text{s}^2 = \text{J/kg} \),
This is also energy per unit mass.
Kinetic term: \( \frac{v^2}{2g_c} \):
\( v \): Velocity (\( \text{m/s} \)),
\( v^2 \): \( (\text{m/s})^2 \),
\( 2g_c \): \( g_c = 1 \) in SI, so \( \frac{v^2}{2g_c} = \frac{v^2}{2} \), with units: \( (\text{m/s})^2 = \text{m}^2/\text{s}^2 = \text{J/kg} \), Again, energy per unit mass. Each term has units of energy per unit mass (\( \text{J/kg} \)), indicating that the equation represents the total energy per unit mass of the fluid. 
Step 3: Evaluate the options. 
(1) Mass: Correct, as each term (\( \text{J/kg} \)) represents energy per unit mass, so the equation gives the total energy per unit mass. Correct.
(2) Volume: Incorrect, as energy per unit volume would have units of \( \text{J/m}^3 \), but the terms are \( \text{J/kg} \). Incorrect.
(3) Specific weight: Incorrect, as specific weight (\( \rho g \), units \( \text{N/m}^3 \)) is not relevant here; the terms are per mass, not weight. Incorrect.
(4) Specific volume: Incorrect, as specific volume (\( 1/\rho \), units \( \text{m}^3/\text{kg} \)) would imply energy per unit volume if inverted, but the terms are per mass. Incorrect.
Step 4: Select the correct answer. 
Each term of Bernoulli’s equation represents the total energy per unit mass, matching option (1).