3.25 Pressure form of Bernoulli Equation


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1 CEE 3310 Control Volume Analysis, Oct 3, Review The Energy Equation Q Ẇshaft = d dt CV ) (û + v2 2 + gz ρ d + (û + v2 CS 2 + gz + ) ρ( v n) da ρ where Q is the heat energy transfer rate, Ẇ shaft is the shaft power (work rate), û is the internal energy per unit mass, v 2 /2 is the kinetic energy per unit mass, and gz is the potential energy per unit mass. 1D Steady State Head Form of the Energy Equation ( ) ( ) 2g + z = out 2g + z h f + h p h s in where h f is the friction head loss, h p is work added by pumps and h s is work removed by shafts ressure form of Bernoulli Equation If we multiply our head form of the Bernoulli equation by the specific weight we arrive at the pressure form of the Bernoulli Equation: + ρ v2 2 + γz = t where we call the first term the static pressure, the second the dynamic pressure, the third the hydrostatic pressure, and the righthandside the total pressure. Hence the Bernoulli Equation says that in inviscid flows the total pressure along a streamline is constant. If we remain at a constant elevation the above equation reduces to + ρ v2 2 = s where we refer to s as the stagnation pressure. Thus by definition the stagnation pressure is the pressure along horizontal streamlines when the velocity is zero.
2 Stagnation oint and ressure Consider the flow around a circular cylinder: + ρ v2 2 + γz = t We see that the stagnation pressure is simply the conversion of all kinetic energy to potential energy and hence there is a subsequent pressure rise. The elevation head simply accounts for any change in the potential energy due to vertical changes in elevation itotstatic Tube The static and stagnation pressures can be measured simultaneously using a itotstatic tube. Consider the following geometry: Now, we see the streamlines around the tube, either at the tip or away from the tip (but not around the curved front end), are horizontal. If the tube is not very long it is very reasonable to assume friction is negligible for this analysis. Now the velocity at the tip of the itotstatic tube is zero hence the pressure at this point (and hence along the entire horizontal leg of the itot tube as this portion of the device is known) is the stagnation pressure. The holes perpendicular to the flow are similar to piezometers  they simply
3 CEE 3310 Control Volume Analysis, Oct 3, measure the static pressure in the fluid flow. If we write the equation for the itot tube we have 1 + ρ v2 1 2 = 2 = γh Now, at the static tube we have the freestream pressure, 2o but at this point, which is along the streamline from point 1 to point 2, the velocity is the same as it is for point 1 (assuming the itotstatic tube is small and does not affect the flow) and there is no elevation change so the Bernoulli Equation gives us: 1 = 2o = γh Substituting this expression into the equation for the itot tube we arrive at the itot formula or in terms of heads V 1 = ρ V = 2g(H h) Example Flow accelerating out of a reservoir 2gh V 2 = 1 ( ) 2 A2 A and if A 1 A 2 1 ( A2 A 1 ) 2 1 V 2 2gh, again! Let s look at this a bit further by asking the question what speed will a parcel of fluid dropped a distance h be traveling at? v = t g dt = gt! h = t v dt = t gt dt = 1 2 gt2
4 86 Therefore t = 2h g and v = gt = 2gt aha! Thus we see that in an inviscid flow, which by definition has no frictional energy losses, we simply convert potential energy to kinetic energy and hence the same result, v = 2gh keeps showing up. This was first noted by Torricelli Irrotational Flow and Bernoulli Consider 2g + z = h v2 2g + 0 = h 0 = v2 2g h 2 γ γ + v2 2g h 2 = h 0 h 3 γ γ + v2 2g h 3 = h 0 hγ γ + 0 h = 0 h 0 Notice that in the unsheared regions (uniform flow) h 0 = a constant across streamlines while where shear exists (e.g., shear is nonzero), h 0 varies across the streamlines. More strictly speaking we actually want to know if the flow is rotational. Our test is if we stick a small neutrally buoyant + shaped probe in the flow and see if it will rotate. In uniform flow it will not, in a linear shear, like the shear profile shown here, it will. Hence we say that h 0 is constant in irrotational (non rotational) flows. This allows us to connect Bernoulli points that are not on the same streamline in flows that are irrotational, further expanding the power of the Bernoulli equation but also the opportunities to misuse it!
5 CEE 3310 Control Volume Analysis, Oct 3, Energy and the Hydraulic Grade Line As we have seen we can write the head form of the Energy equation as + z = H = Energy Grade Line (EGL) 2g In the case of Bernoulli flows the energy grade line is simply a constant since by assumption energy is conserved (there is no mechanism to gain/lose energy). For other flows it will drop due to frictional losses or work done on the surroundings (e.g., a turbine) or increase due to work input (e.g., a pump). Note that this is the head that would be measured by a itot tube. We can also write γ + z = Hydraluic Grade Line (HGL) and we see that the HGL is due to static pressure the height a column of fluid would rise due to pressure at a given elevation or in other words the head measured by a static pressure tap or the piezometric head. Example Venturi Flow Meter Consider 2g h Q = A 2 V 2 = A 2 ( A2 1 A 1 ) 2 1 2
6 Frictional Effects If we have abrupt losses, say at a contraction, a simple way of accounting for this is through a discharge coefficient. We can write a modified form of Torricelli s formula for incompressible flow V = Q A = C d 2gh where C d is the discharge coefficient and is 1 for frictionless (inviscid) flow and can range down to about 0.6 for flows strongly effected by friction. Note we can handle nonuniform (violation of 1D assumption) flow effects with a C d as well Vena Contracta Effect For a flow to get around a sharp corner there would need to be an infinite pressure gradient, which of course does not happen. Hence if the boundary changes directions too rapidly at an exit, the flow separates from the exit and forms what is known as a vena contracta Clearly A j /A 1. For a round sharpcornered exit the coefficient is C c = A j /A = 0.61 and typical values of the coefficient fall in the range 0.5 C c 1.0.
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