1. Exercise 6.1 The total free volume of a double stroke air compressor is 324 x \( 10^{-6} \mathrm{~m}^{3} \), the cylinder diameter is 210 mm and the stroke length of 110 mPM . It takes air in at a temperature of \( 17^{\circ} \mathrm{C} \) under a pressure of 97 kPa and delivers the afr under a pressure of 582 kPa. If the index for compression and expansion is 1 , 32 and the mechanical efficiency is \( 80 \% \), calculate: (a) The required power to drive the compressor (b) The volumetric efficiency

To solve the problem, we need to extract the known conditions and then proceed step by step to calculate the required power to drive the compressor and the volumetric efficiency. ### Known Conditions: 1. Total free volume of the compressor, \( V = 324 \times 10^{-6} \, \text{m}^3 \) 2. Cylinder diameter, \( d = 210 \, \text{mm} = 0.210 \, \text{m} \) 3. Stroke length, \( L = 110 \, \text{mm} = 0.110 \, \text{m} \) 4. Inlet temperature, \( T_1 = 17^{\circ} \text{C} = 290 \, \text{K} \) (converted to Kelvin) 5. Inlet pressure, \( P_1 = 97 \, \text{kPa} \) 6. Delivery pressure, \( P_2 = 582 \, \text{kPa} \) 7. Index for compression and expansion, \( n = 1.32 \) 8. Mechanical efficiency, \( \eta = 0.80 \) ### Step 1: Calculate the Volume of One Stroke The volume of one stroke \( V_s \) can be calculated using the formula for the volume of a cylinder: \[ V_s = \frac{\pi d^2}{4} \times L \] ### Step 2: Calculate the Number of Strokes per Minute To find the power, we need to know the number of strokes per minute. However, we need to calculate the power first, so we will use the formula for power in terms of work done per cycle and the number of cycles per minute. ### Step 3: Calculate the Work Done per Cycle The work done per cycle \( W \) can be calculated using the formula: \[ W = \frac{P_2 V}{n - 1} \left( \left( \frac{P_2}{P_1} \right)^{\frac{n - 1}{n}} - 1 \right) \] ### Step 4: Calculate the Power Required The power required \( P \) can be calculated using the formula: \[ P = \frac{W \times N}{\eta} \] where \( N \) is the number of strokes per minute. ### Step 5: Calculate the Volumetric Efficiency The volumetric efficiency \( \eta_v \) can be calculated using the formula: \[ \eta_v = \frac{V}{V_s} \] Now, let's perform the calculations step by step. #### Step 1: Calculate the Volume of One Stroke \[ V_s = \frac{\pi (0.210)^2}{4} \times 0.110 \] #### Step 3: Calculate the Work Done per Cycle \[ W = \frac{582 \times 10^3 \times 324 \times 10^{-6}}{1.32 - 1} \left( \left( \frac{582}{97} \right)^{\frac{1.32 - 1}{1.32}} - 1 \right) \] #### Step 4: Calculate the Power Required Assuming a certain number of strokes per minute \( N \), we can calculate the power. #### Step 5: Calculate the Volumetric Efficiency \[ \eta_v = \frac{V}{V_s} \] Now, let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(\left(\frac{\pi \times 0.21^{2}}{4}\right)\times 0.11\) - step1: Convert the expressions: \(\left(\frac{\pi \left(\frac{21}{100}\right)^{2}}{4}\right)\times 0.11\) - step2: Multiply the terms: \(\left(\frac{\frac{441\pi }{10000}}{4}\right)\times 0.11\) - step3: Divide the terms: \(\frac{441\pi }{40000}\times 0.11\) - step4: Convert the expressions: \(\frac{441\pi }{40000}\times \frac{11}{100}\) - step5: Multiply the fractions: \(\frac{441\pi \times 11}{40000\times 100}\) - step6: Multiply the terms: \(\frac{4851\pi }{40000\times 100}\) - step7: Multiply the terms: \(\frac{4851\pi }{4000000}\) Calculate or simplify the expression \( (582 * 10^3 * 324 * 10^{-6}) / (1.32 - 1) * ((582 / 97)^{(1.32 - 1) / 1.32} - 1) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(582\times 10^{3}\times 324\times 10^{-6}\right)}{\left(1.32-1\right)}\times \left(\left(\frac{582}{97}\right)^{\frac{\left(1.32-1\right)}{1.32}}-1\right)\) - step1: Remove the parentheses: \(\frac{582\times 10^{3}\times 324\times 10^{-6}}{1.32-1}\times \left(\left(\frac{582}{97}\right)^{\frac{1.32-1}{1.32}}-1\right)\) - step2: Divide the terms: \(\frac{582\times 10^{3}\times 324\times 10^{-6}}{1.32-1}\times \left(6^{\frac{1.32-1}{1.32}}-1\right)\) - step3: Subtract the numbers: \(\frac{582\times 10^{3}\times 324\times 10^{-6}}{1.32-1}\times \left(6^{\frac{0.32}{1.32}}-1\right)\) - step4: Divide the terms: \(\frac{582\times 10^{3}\times 324\times 10^{-6}}{1.32-1}\times \left(6^{\frac{8}{33}}-1\right)\) - step5: Multiply: \(\frac{\frac{23571}{125}}{1.32-1}\times \left(6^{\frac{8}{33}}-1\right)\) - step6: Subtract the numbers: \(\frac{\frac{23571}{125}}{0.32}\times \left(6^{\frac{8}{33}}-1\right)\) - step7: Divide the numbers: \(\frac{23571}{40}\left(6^{\frac{8}{33}}-1\right)\) - step8: Multiply: \(\frac{23571\left(6^{\frac{8}{33}}-1\right)}{40}\) - step9: Multiply: \(\frac{23571\times 6^{\frac{8}{33}}-23571}{40}\) - step10: Simplify: \(\frac{23571\sqrt[33]{6^{8}}-23571}{40}\) Calculate or simplify the expression \( (582 * 10^3 * 324 * 10^{-6}) / (1.32 - 1) * ((582 / 97)^{(1.32 - 1) / 1.32} - 1) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(582\times 10^{3}\times 324\times 10^{-6}\right)}{\left(1.32-1\right)}\times \left(\left(\frac{582}{97}\right)^{\frac{\left(1.32-1\right)}{1.32}}-1\right)\) - step1: Remove the parentheses: \(\frac{582\times 10^{3}\times 324\times 10^{-6}}{1.32-1}\times \left(\left(\frac{582}{97}\right)^{\frac{1.32-1}{1.32}}-1\right)\) - step2: Divide the terms: \(\frac{582\times 10^{3}\times 324\times 10^{-6}}{1.32-1}\times \left(6^{\frac{1.32-1}{1.32}}-1\right)\) - step3: Subtract the numbers: \(\frac{582\times 10^{3}\times 324\times 10^{-6}}{1.32-1}\times \left(6^{\frac{0.32}{1.32}}-1\right)\) - step4: Divide the terms: \(\frac{582\times 10^{3}\times 324\times 10^{-6}}{1.32-1}\times \left(6^{\frac{8}{33}}-1\right)\) - step5: Multiply: \(\frac{\frac{23571}{125}}{1.32-1}\times \left(6^{\frac{8}{33}}-1\right)\) - step6: Subtract the numbers: \(\frac{\frac{23571}{125}}{0.32}\times \left(6^{\frac{8}{33}}-1\right)\) - step7: Divide the numbers: \(\frac{23571}{40}\left(6^{\frac{8}{33}}-1\right)\) - step8: Multiply: \(\frac{23571\left(6^{\frac{8}{33}}-1\right)}{40}\) - step9: Multiply: \(\frac{23571\times 6^{\frac{8}{33}}-23571}{40}\) - step10: Simplify: \(\frac{23571\sqrt[33]{6^{8}}-23571}{40}\) ### Step 1: Calculate the Volume of One Stroke The volume of one stroke \( V_s \) is calculated as follows: \[ V_s = \frac{\pi (0.210)^2}{4} \times 0.110 \approx 0.00381 \, \text{m}^3 \] ### Step 3: Calculate the Work Done per Cycle The work done per cycle \( W \) is calculated as follows: \[ W = \frac{582 \times 10^3 \times 324 \times 10^{-6}}{1.32 - 1} \left( \left( \frac{582}{97} \right)^{\frac{1.32 - 1}{1.32}} - 1 \right) \approx 320.56 \, \text{J} \] ### Step 4: Calculate the Power Required To calculate the power required, we need to know the number of strokes per minute \( N \). However, we can express the power in terms of \( N \): \[ P = \frac{W \times N}{\eta} = \frac{320.56 \times N}{0.80} \] ### Step 5: Calculate the Volumetric Efficiency The volumetric efficiency \( \eta_v \) is calculated as follows: \[ \eta_v = \frac{V}{V_s} = \frac{324 \times 10^{-6}}{0.00381} \] Now, let's calculate the volumetric efficiency. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(324\times 10^{-6}\right)}{0.00381}\) - step1: Remove the parentheses: \(\frac{324\times 10^{-6}}{0.00381}\) - step2: Convert the expressions: \(\frac{324\times 10^{-6}}{\frac{381}{100000}}\) - step3: Evaluate the power: \(\frac{\frac{81}{250000}}{\frac{381}{100000}}\) - step4: Multiply by the reciprocal: \(\frac{81}{250000}\times \frac{100000}{381}\) - step5: Reduce the numbers: \(\frac{27}{5}\times \frac{2}{127}\) - step6: Multiply the fractions: \(\frac{27\times 2}{5\times 127}\) - step7: Multiply: \(\frac{54}{635}\) ### Final Calculations 1. **Volumetric Efficiency**: \[ \eta_v = \frac{324 \times 10^{-6}}{0.00381} \approx 0.0850 \, \text{or} \, 8.50\% \] ### Summary of Results - **Required Power to Drive the Compressor**: \[ P = \frac{320.56 \times N}{0.80} \, \text{W} \quad \text{(where \( N \) is the number of strokes per minute)} \] - **Volumetric Efficiency**: \[ \eta_v \approx 8.50\% \] If you have a specific value for \( N \) (the number of strokes per minute), you can substitute it into the power equation to find the exact power required.