Pressure variation with direction. Pressure variation with location How can we calculate the total force on a submerged surface?

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1 Definitions and Alications CVEN Statics: no relative motion between adjacent fluid layers. Shear stress is ero Only ressure can be acting Gravity force acts on the fluid ( body force) Alications: Pressure variation within a reservoir orces on submerged surfaces Tensile stress on ie walls (submarine) Buoyant forces Motivation? Ustream face of Hoover Dam What are the ressure forces behind the Hoover Dam? Tall: 0 m (76 ft) Crest thickness:.7 m (50 ft) Base thickness: 0 m (660 ft - two football fields) WHY??? Ustream face of Hoover Dam in 95 What do you think? What do we need to know? Pressure variation with direction Pressure variation with location How can we calculate the total force on a submerged surface? Lake Mead, the lake behind Hoover Dam, is the world's largest artificial body of water by volume (5 km ). Is the ressure at the base of Hoover Dam affected by the volume of water in Lake Mead?

2 Pressure Variation with Direction (Pascal s law) Equation of Motion y Surface forces = ma Body forces s s y m a a 0 y y Pressure ield In the absence of shearing forces (no relative motion between fluid articles) what causes ressure variation within a fluid? y - s s sin s sin y y - y 0 s Indeendent of direction! y y Which has the highest ressure? Small element of fluid in ressure gradient with arbitrary acceleration. Pressure is at center of element i i HG kk j Pressure ield I K J H I K y orces acting on surfaces of element y y y y y H I K y HG I K J Mass m y Aly Newton s Second Law ma y m y Same in! Obtain a general vector eression relating ressure gradient to acceleration and write the comonent equations. Mass of element of fluid y ya Substitute into Newton s nd Law a a kˆ Tet version of eq. a ay a comonent equations y d g At rest (indeendent of and y) d Simlify the eression for the force acting on the element y y I Same in y! HG J I HG J y K y K y y y y This begs for vector notation! HG y I K J i j k y y y y i j k y orces acting on element e e of fluid due to ressure gradient Pressure Variation When the Secific Weight is Constant What are the two things that could make secific weight ()varyinafluid? a = g Comressible fluid - changing density d d Changing gravity is constant constant t Pieometric head constant

3 Eamle: Pressure at the bottom of a Tank of Water? Units and Scales of Pressure Measurement h = h Does the ressure at the bottom of the tank increase if the diameter of the tank increases? NO!!!! h atmoshere 0.5 kpa 4.7 si 0.4 m H mm Pa = si Local barometer reading Gage ressure Absolute ressure Standard atmosheric ressure Local atmosheric ressure Suction vacuum (gage ressure) Absolute ero (comlete vacuum) Mercury Barometer A few imortant constants! What is the local atmosheric ressure (in kpa) when R is 750 mm? S.6 P = vaor ressure R S S R water R Proerties of water Density: 000 kg/m Viscosity: 0 - N s/m Ns/m Secific weight: 9800 N/m Proerties of fth the atmoshere Atmosheric ressure 0. kpa Height of a column of water that can be suorted by atmosheric ressure 0. m N / m 0.75m 00, Pa So-erfect Design Oil from Dee water? Integrate / from 0 to

4 Deewater record Perdido 8000/9600ft Suction Anchors Pressure Measurement Standard Manometers Barometers Weight or ressure Manometers Standard Differential i Pressure Transducers What is the ressure at A in terms of h? = h Pressure in water distribution ib i systems commonly varies between 5 and 00 si (75 to 700 kpa). How high would the h water rise in a manometer connected to a ie containing water at 500 kpa? h = / h = 500, Pa/9800 N/m h = 5 m Not very ractical! A ieometer tube container Manometers for High Pressures Differential Manometers ind the gage ressure in the center of the shere. The shere contains fluid with and the manometer contains fluid with. What do you know? P = 0 h? Use statics to find other ressures. P + h -h =P or small h use fluid with high density. U-tube manometer h Mercury! Water orifice h h ind the dro in ressure between oint and oint. h Mercury + h w -h -h w = - =(h -h ) w +h - = h ( - w )

5 Procedure to kee track of ressures Start at a known oint or at one end of the system and write the ressure there using an aroriate symbol Add to this the change in ressure to the net meniscus (lus if the net meniscus is lower, and minus if higher) Continue until the other end of the gage gg is reached and equate the eression to the ressure at that oint + = Pressure Transducers Ecitation: 0 Vdc regulated Outut: 00 millivolts l Accuracy: ±% S ull Scale Proof Pressure: 40 kpa (0 si) for 7 kpa model No Mercury! Can be monitored easily by comuter Myriad of alications Volume of liquid in a tank low rates Process monitoring and control Summary for Statics Statics eamle Pressure is indeendent of direction Pressure increases with deth constant density Pressure scales units datum Pressure measurement d dd =h What is the air ressure in the cave air ocket?