Similarly for isentropic pumps or turbines using

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Similarly for isentropic pumps or turbines using CPIGs with negligible changes in kinetic and potential energies, Eq. (9.39) reduces to w cv = integraldisplay 2 1 vdP, (9.52) = k k 1 ( P 2 v 2 P 1 v 1 ) , (9.53) = k k 1 R ( T 2 T 1 ) , (9.54) = kRT 1 k 1 parenleftbigg T 2 T 1 1 parenrightbigg , (9.55) = kRT 1 k 1 parenleftBigg parenleftbigg P 2 P 1 parenrightbigg k - 1 k 1 parenrightBigg . (9.56) An isothermal pump or compressor using a CPIG has w cv = integraldisplay 2 1 vdP, (9.57) = integraldisplay 2 1 RT P dP, (9.58) = RT 1 ln P 2 P 1 , (9.59) = P 1 v 1 ln P 2 P 1 . (9.60) CC BY-NC-ND. 2011, J. M. Powers.
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274 CHAPTER 9. SECOND LAW ANALYSIS FOR A CONTROL VOLUME 9.2.3 Torricelli’s formula Let us consider a special case of the Bernoulli equation, known as Torricelli’s formula, devel- oped by Evangelista Torricelli, 2 the inventor of the barometer, and for whom the pressure unit torr is named (1 torr = 133 . 322 Pa = 1 / 760 atm. ) Torricelli is sketched in Fig. 9.6. Figure 9.6: Evangelista Torricelli (1608-1647), Italian physicist and mathematician; image from Torricelli . Consider the scenario of Fig. 9.7. Here a fluid is in an open container. The container has a small hole near its bottom. The fluid at the top of the container, z = z 1 , is at P 1 = P atm . The leaking fluid exhausts at the same pressure P 2 = P atm . The fluid leaks at velocity v 2 at a hole located at z = z 2 . The fluid at the top of the container barely moves; so, it has negligible velocity, v 1 0. The fluid exists in a constant gravitational field with gravitational acceleration g , as sketched. Assume the fluid is incompressible and all of the restrictions of Bernoulli’s law are present. Let us apply Eq. (9.44): P 1 ρ + 1 2 v 2 1 bracehtipupleftbracehtipdownrightbracehtipdownleftbracehtipupright 0 + gz 1 = P 2 ρ + 1 2 v 2 2 + gz 2 . (9.61) Setting P 1 = P 2 = P atm and ignoring v 1 gives gz 1 = 1 2 v 2 2 + gz 2 , (9.62) v 2 = radicalbig 2 g ( z 1 z 2 ) , (9.63) Torricelli’s formula. 2 E. Torricelli, 1643, De Motu Gravium Naturaliter Accelerato , Firenze. CC BY-NC-ND. 2011, J. M. Powers.
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9.2. BERNOULLI’S PRINCIPLE 275 P 1 = P atm 1 P 2 = P atm 2 z 1 z 2 v 2 v 1 ~ 0 g Figure 9.7: Fluid container with hole. Notice rearranging Torricelli’s formula gives 1 2 v 2 2 bracehtipupleftbracehtipdownrightbracehtipdownleftbracehtipupright kinetic energy = g ( z 1 z 2 ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright potential energy . (9.64) It represents a balance of kinetic and potential energy of the fluid, and thus is concerned only with mechanical energy. Example 9.3 Let us design a liquid water fountain by cutting a hole in a high pressure water pipe. See Fig. 9.8. We desire the final height of the water jet to be 30 m . The jet rises against a gravitational field with g = 9 . 81 m/s 2 . The atmospheric pressure is 100 kPa . Water has density ρ = 997 kg/m 3 . Find the necessary pipe gauge pressure P 1 and jet exit velocity v 2 .
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