In a centrifugal compressor with inlet guide vanes, air leaving the guide vanes has a velocity of 91.5 m/s at 70 degree the tangential direction. Determine the inlet relative Mach number assuming frictionless flow through the guide vanes and impeller total head isentropic efficiency. The other operating conditions are - (i) Impeller diameter at inlet = 457 mm (ii) Impeller diameter at exit = 762 mm (iii) Radial component of velocity at exit = 53.4 m/s (iv) Slip factor = 0.9 (v) Impeller speed = 11000 r.p.m. (vi) Static pressure at impeller exit = 223 kPa (abs) Po1=1.013 bar. (vii) Take To1 = 288 K

To determine the inlet relative Mach number in the given centrifugal compressor, we will follow these steps:

1. **Calculate the tangential and axial components of the absolute inlet velocity:**

   Given:
   - Absolute velocity at inlet, \( V_1 = 91.5 \, \text{m/s} \)
   - Angle of velocity at inlet, \( \alpha_1 = 70^\circ \)

   The tangential and axial components of the absolute inlet velocity can be calculated as:
   \[
   V_{1t} = V_1 \cos(\alpha_1) = 91.5 \cos(70^\circ)
   \]
   \[
   V_{1r} = V_1 \sin(\alpha_1) = 91.5 \sin(70^\circ)
   \]

   Calculating these:
   \[
   V_{1t} = 91.5 \cos(70^\circ) \approx 31.29 \, \text{m/s}
   \]
   \[
   V_{1r} = 91.5 \sin(70^\circ) \approx 85.97 \, \text{m/s}
   \]

2. **Calculate the impeller inlet speed (blade speed) using the diameter and speed:**

   Given:
   - Impeller diameter at inlet, \( D_1 = 457 \, \text{mm} = 0.457 \, \text{m} \)
   - Impeller speed, \( N = 11000 \, \text{rpm} \)

   The impeller inlet speed (blade speed) is:
   \[
   U_1 = \pi D_1 \frac{N}{60}
   \]

   Calculating this:
   \[
   U_1 = \pi \times 0.457 \times \frac{11000}{60} \approx 263.58 \, \text{m/s}
   \]

3. **Determine the relative velocity at the inlet:**

   The relative velocity at the inlet, \( W_1 \), is given by the vector difference between the absolute velocity and the blade speed. Using the components calculated above:
   \[
   W_1 = \sqrt{(V_{1r})^2 + (U_1 - V_{1t})^2}
   \]
   \[
   W_1 = \sqrt{(85.97)^2 + (263.58 - 31.29)^2}
   \]
   \[
   W_1 = \sqrt{7382.58 + 54093.14} \approx \sqrt{61475.72} \approx 247.94 \, \text{m/s}
   \]

4. **Determine the speed of sound at the inlet:**

   Given:
   - Total temperature at inlet, \( T_{01} = 288 \, \text{K} \)
   - Specific gas constant for air, \( R = 287 \, \text{J/(kg \cdot K)} \)
   - Specific heat ratio for air, \( \gamma = 1.4 \)

   The speed of sound, \( a_1 \), is:
   \[
   a_1 = \sqrt{\gamma R T_{01}}
   \]
   \[
   a_1 = \sqrt{1.4 \times 287 \times 288} \approx 340.3 \, \text{m/s}
   \]

5. **Calculate the inlet relative Mach number:**

   The relative Mach number at the inlet, \( M_{r1} \), is:
   \[
   M_{r1} = \frac{W_1}{a_1}
   \]
   \[
   M_{r1} = \frac{247.94}{340.3} \approx 0.73
   \]

Hence, the inlet relative Mach number \( M_{r1} \) is approximately \( 0.73 \).

Would you like more details or have any questions on this calculation?

### Additional Related Questions
1. How does the slip factor affect the overall performance of a centrifugal compressor?
2. What is the significance of the radial component of velocity at the exit?
3. How is the impeller's total head isentropic efficiency determined?
4. What role does the static pressure at the impeller exit play in the compressor's performance?
5. How do changes in impeller diameter affect the compressor's operation?
6. What is the impact of the impeller speed on the compressor efficiency?
7. Can you explain the importance of the total temperature at the inlet?
8. How does the specific heat ratio (γ) influence the speed of sound in air?

### Tip
When analyzing compressor performance, always consider the effects of both absolute and relative velocities on the flow dynamics within the compressor stages.

Learn how to calculate the inlet relative Mach number in a centrifugal compressor using given operational parameters such as inlet velocity, impeller speed, and temperature. This detailed solution covers vector addition, speed of sound calculation, and the concept of relative Mach number in fluid mechanics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation