at the inlet and exhaust of a turbine, the absolute pressures are 6000 kN/m^2 and 4.0 cmHg respectively. the barometric pressure is 75 cmHg. Calculate the gauge pressure for the entering steam and the vacuum gauge pressure for the exhaust steam.
Given, Find and Solution

### Given:
1. **Inlet Absolute Pressure** \( P_{\text{inlet}} \) = 6000 kN/m²
2. **Exhaust Absolute Pressure** \( P_{\text{exhaust}} \) = 4.0 cmHg
3. **Barometric Pressure** \( P_{\text{atm}} \) = 75 cmHg

### Find:
1. **Gauge Pressure for Entering Steam**
2. **Vacuum Gauge Pressure for Exhaust Steam**

### Solution:

#### 1. Gauge Pressure for Entering Steam

Gauge pressure is the difference between the absolute pressure and the atmospheric pressure:

\[
P_{\text{gauge}} = P_{\text{absolute}} - P_{\text{atm}}
\]

But first, we need to convert the atmospheric pressure from cmHg to kN/m². We can use the following conversion factor:

\[
1 \, \text{cmHg} = 1.33322 \, \text{kN/m²}
\]

Thus,

\[
P_{\text{atm}} = 75 \, \text{cmHg} \times 1.33322 \, \text{kN/m²/cmHg} = 99.9915 \, \text{kN/m²}
\]

Now, calculate the gauge pressure for the entering steam:

\[
P_{\text{gauge}} = 6000 \, \text{kN/m²} - 99.9915 \, \text{kN/m²} = 5900.0085 \, \text{kN/m²}
\]

Thus, the **gauge pressure for entering steam** is approximately **5900 kN/m²**.

#### 2. Vacuum Gauge Pressure for Exhaust Steam

For vacuum gauge pressure, we need to convert the exhaust pressure from cmHg to kN/m². Using the same conversion factor:

\[
P_{\text{exhaust}} = 4.0 \, \text{cmHg} \times 1.33322 \, \text{kN/m²/cmHg} = 5.33288 \, \text{kN/m²}
\]

The vacuum gauge pressure is the difference between the atmospheric pressure and the exhaust pressure:

\[
P_{\text{vacuum}} = P_{\text{atm}} - P_{\text{exhaust}}
\]

Substituting the values:

\[
P_{\text{vacuum}} = 99.9915 \, \text{kN/m²} - 5.33288 \, \text{kN/m²} = 94.6586 \, \text{kN/m²}
\]

Thus, the **vacuum gauge pressure for exhaust steam** is approximately **94.66 kN/m²**.

### Summary:
- **Gauge Pressure for Entering Steam:** 5900 kN/m²
- **Vacuum Gauge Pressure for Exhaust Steam:** 94.66 kN/m²

Would you like further clarification on any of the steps?

### Related Questions:
1. What are the differences between absolute and gauge pressure?
2. How do you convert atmospheric pressure from cmHg to kN/m²?
3. What is the significance of vacuum pressure in turbine operation?
4. How does barometric pressure affect turbine performance?
5. How would the calculation change if the barometric pressure were higher?

### Tip:
When converting units, always ensure the consistency of units across all parameters for accurate calculations.

At the inlet and exhaust of a turbine, the absolute pressures are 6000 kN/m^2 and 4.0 cmHg respectively. The barometric pressure is 75 cmHg. Calculate the gauge pressure for the entering steam and the vacuum gauge pressure for the exhaust steam.

Learn how to calculate the gauge pressure for entering steam and the vacuum gauge pressure for exhaust steam in a turbine. This problem covers pressure calculations, unit conversions, and is suitable for college-level engineering or physics students.

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