PGE 421K Spring 2010 HWSolution3

# PGE 421K Spring - PGE 421K Spring 2010 HOMEWORK 3 100 points Due M 22 Feb 20 pts 1 Knowing the terms in the mass and energy balances can help you

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PGE 421K, Spring 2010 HOMEWORK 3 100 points Due M 22 Feb 1. Knowing the terms in the mass and energy balances can help you figure out appropriate (i.e. physically reasonable) forms of correlations, design equations, etc. The chart above appears in a Dowell Engineers Handbook from the 1950s and shows the volumetric flow rate that must be supplied to operate fluid jets through orifices of various sizes, as a function of the pressure drop across the jet. a. Start with the general form of the energy balance and derive the relationship between Q and Δ P from the energy balance. State the assumptions you make in the derivation. Assume steady state, isentropic, no shaft work, no change in potential energy. Mass balance: gG ±² = G ´ µ¶ − G ´ ·¸¹ = 0 G ´ µ¶ = G ´ ·¸¹ Steady state energy balance across the jet: 0 = º ´ + » ´ ¼ + G ´ µ¶ ½¾ ¿ µ¶ + À µ¶ Á 2 + ÂÃ µ¶ Ä − G ´ ·¸¹ ½¾ ¿ ·¸¹ + À ·¸¹ Á 2 + ÂÃ ·¸¹ Ä 20 pts

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Combine energy and mass balances: 0 = g G ±² ³´ µ ±² − ´ µ ¶·¸ + ¹ ±² º − ¹ ¶·¸ º 2 » Assume incompressible fluid. Then: ´ µ ±² − ´ µ ¶·¸ = ¼ ½ ¾¿ ±² − ¿ ¶·¸ À and g G ±² = ¹ ±² Á ±² ¼ ½ = ¹ ¶·¸ Á ¶·¸ ¼ ½ where A in , A out are cross sectional areas of pipe and orifice. Hence: ¹ ¶·¸ = ¹ ±² Á ±² Á ¶·¸ Note: Á ¶·¸ < Á ±² so ¹ ¶·¸ > ¹ ±² . Substituting into the energy balance, we obtain: ¼ ½ ∆¿ = 1 2 ¾¹ ¶·¸ º − ¹ ±² º À = 1 2 ³¹ ¶·¸ º − ¹ ¶·¸ º Â Á ¶·¸ Á ±² Ã º » Solving for ¹ ¶·¸ : ¹ ¶·¸ = Ä ½ ∆¿ 1 − Å Á ¶·¸ Á ±² Æ º Flow through orifice is Q=u out A out so we have: Ç = Á ¶·¸ Ä ½ ∆¿ 1 − Å Á ¶·¸ Á ±² Æ º orifice
b. Use the result of (a) to explain the shape of the curves in the chart. That is, what is the connection between the mathematical form of (a) and family of curves in the chart.

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## This note was uploaded on 02/13/2012 for the course PGE 421K taught by Professor Bryant during the Spring '11 term at University of Texas at Austin.

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PGE 421K Spring - PGE 421K Spring 2010 HOMEWORK 3 100 points Due M 22 Feb 20 pts 1 Knowing the terms in the mass and energy balances can help you

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