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The image contains a question related to thermodynamics, specifically dealing with the dimensional consistency of a given equation. The equation in question is:

\[
u_1 = \sqrt{2(h_2 - h_1)}
\]

where:
- \( u_1 \) is the velocity of fluid flowing into a nozzle \((\text{m/s})\),
- \( h_1 \) and \( h_2 \) are the specific enthalpies of the fluid entering and exiting the nozzle, respectively \((\text{kJ/kg})\).

The task is to show that this equation is dimensionally consistent.

### Dimensional Consistency Check

1. **Dimensional Analysis of \( u_1 \):**
   - \( u_1 \) is a velocity, so its dimension is \([u_1] = \text{L} \, \text{T}^{-1}\) (Length per Time), which in SI units is meters per second \((\text{m/s})\).

2. **Dimensional Analysis of \( h_1 \) and \( h_2 \):**
   - \( h_1 \) and \( h_2 \) are specific enthalpies with dimensions \(\text{M} \, \text{L}^2 \, \text{T}^{-2} \text{M}^{-1}\), which simplifies to \(\text{L}^2 \, \text{T}^{-2}\) in SI units. Their units are \(\text{kJ/kg}\) or \(\text{m}^2/\text{s}^2\).

3. **Dimensional Analysis of \( 2(h_2 - h_1) \):**
   - \( h_2 - h_1 \) has the dimension of \(\text{L}^2 \, \text{T}^{-2}\) (same as \( h_1 \) and \( h_2 \)).
   - Multiplying by 2 (a dimensionless constant) doesn’t change the dimensions, so \( 2(h_2 - h_1) \) still has the dimension of \(\text{L}^2 \, \text{T}^{-2}\).

4. **Square Root of \( 2(h_2 - h_1) \):**
   - The square root of \(\text{L}^2 \, \text{T}^{-2}\) is \(\text{L} \, \text{T}^{-1}\).
   - This matches the dimension of velocity \( u_1 \).

### Conclusion
The given equation \( u_1 = \sqrt{2(h_2 - h_1)} \) is dimensionally consistent.

Would you like more details on this problem, or have any other questions?

### Related Questions:
1. What is the significance of specific enthalpy in thermodynamics?
2. How can we derive this equation using the conservation of energy principle?
3. What are common applications of this equation in engineering problems?
4. How do changes in enthalpy affect the velocity of fluid flow through a nozzle?
5. Can this equation be applied to compressible flows, and if so, how?

**Tip:** Always check units at each step to ensure that all equations used are dimensionally consistent.

Show that the equation u₁ = sqrt(2(h₂ - h₁)) is dimensionally consistent, where u₁ is the velocity of fluid flowing into a nozzle (m/s), h₁ and h₂ are the specific enthalpies of fluid entering and exiting the nozzle (kJ/kg).

Learn how to prove the dimensional consistency of the equation u₁ = sqrt(2(h₂ - h₁)) in thermodynamics. This problem involves concepts of fluid flow, specific enthalpy, and velocity analysis, suitable for undergraduate students in mechanical engineering.

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