To determine The power input to the compressor, in horse power. Explanation Here, the nitrogen flows at steady state through the compressor. Hence, the inlet and exit mass flow rates are equal. m ˙ 1 = m ˙ 2 (I) Write the formula for inlet mass flow rate in terms of ideal gas equation. m ˙ 1 = A 1 V 1 p 1 ( R ¯ / M ) T 1 (II) Here, the cross sectional area is A , the velocity is V , the pressure is p , universal gas constant is R ¯ , the molar mass of the nitrogen is M , the temperature is T and the suffix 1 indicates the inlet state. Write the formula for area of the duct whose cross section is circle. A 1 = π d 2 4 (III) Here, the diameter of the duct is d . Write the energy rate balance equation. 0 = Q ˙ cv − W ˙ cv + m ˙ [ ( h 1 − h 2 ) + ( V 1 2 − V 2 2 ) 2 + g ( z 1 − z 2 ) ] (IV) Here, the rate of heat transfer is Q ˙ cv , rate of work transfer is W ˙ cv , rate of mass flow is m ˙ , the enthalpy is h , velocity is V , gravitational acceleration is g , the elevation is z and the suffixes cv, 1 and 2 indicates control volume, initial and final state of the system respectively. Neglect the heat transfer and potential energy effect i.e. Q ˙ cv = 0 and g ( z 1 − z 2 ) = 0 . Then the Equation (IV) becomes as follows to obtain the work done. W ˙ cv = m ˙ [ ( h 1 − h 2 ) + ( V 1 2 − V 2 2 ) 2 ] (V) Refer Table A-23, “Ideal Gas Properties of Selected Gases”. It gives the enthalpy values in molar basis. When, the molar mass ( M ) of nitrogen is introduced, the enthalpy h 1 , h 2 will become as follows. h 1 = h 1 ¯ M h 2 = h 2 ¯ M Here, the h 1 ¯ , h 2 ¯ are molar enthalpies of nitrogen. Rewrite the Equation (V) in terms of molar enthalpies. Substitute h 1 ¯ M for h 1 and h 2 ¯ M for h 2 in Equation (V). W ˙ cv = m ˙ [ ( h 1 ¯ M − h 2 ¯ M ) + ( V 1 2 − V 2 2 ) 2 ] = m ˙ [ ( h 1 ¯ − h 2 ¯ M ) + ( V 1 2 − V 2 2 ) 2 ] (VI) At inlet: Refer Table A-23E, “Ideal Gas Properties of Selected Gases”. The enthalpy ( h 1 ¯ ) of nitrogen corresponding to the temperature of 50 ° F ( 510 ° R ) is 3541.75 Btu/lbmol (using linear interpolation). At exit: Refer Table A-23E, “Ideal Gas Properties of Selected Gases”. The enthalpy ( h 2 ¯ ) of nitrogen corresponding to the temperature of 198 ° F ( 658 ° R ) is 4571.86 Btu/lbmol (using linear interpolation). The molar mass of nitrogen is 28.01 lb/lb ⋅ mol . Conclusion: Substitute 6 in . for d in Equation (III)

Fundamentals of Engineering Thermodynamics

8th Edition

ISBN: 9781118832318

Author: MORAN

Publisher: WILEY

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Chapter 4.12, Problem 54P

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