CEE 3310 Control Volume Analysis, Oct. 7, 2015 81 3.21 Review 1-D Steady State Head Form of the Energy Equation ( ) ( ) 2g + z = 2g + z h f + h p h s out where h f is the friction head loss (which combines the net change in internal energy and heat flux terms), h p is work added by pumps and h s is work removed by shafts. 2-D Flow can approxiate as 1-D and correct for error with kinetic constant, α. For laminar pipe flow (with a parabolic velocity profile) α = 2.0. in 3.22 Bernoulli Equation Bernoulli wrote down a verbal form of his famous equation in 1738 and Euler completed the analytic derivation in 1755. The differential form of the Bernoulli Equation is known as the Euler Equation. Consider our 1-D head form of the energy equation and let s apply it along a streamline of a flow. If the flow is steady then the integral over the control volume vanishes. Further, since by definition there is no flow normal to the streamline we only have flux terms at the starting and ending points along the streamline (really we are talking about a volume and hence a streamtube just a cylindrical volume element defined by a family of streamlines - a virtual pipe!). Clearly there is no shaft work along the streamline. If we assume the flow is frictionless (i.e., inviscid or ν = 0) h f =0 and we have ( ) ( ) 2g + z = 2g + z out This is the Bernoulli equation. Clearly anywhere along a streamline, as long as no work is done between analysis points and the assumption of frictionless flow is good, we can write 2g + z = h 0 in
CEE 3310 Control Volume Analysis, Oct. 7, 2015 82 where the constant h 0 is referred to as the Bernoulli constant and varies across streamlines. The Bernoulli Equation can be derived by considering Newton s second law F = m a along a streamline (conservation of linear momentum). This leads to the steady form of the Bernoulli equation. If we add conservation of mass we can derive the unsteady form. Bernoulli Equation Assumptions Flow along single streamline different streamlines, different h 0. Steady flow (can be generalized to unsteady flow). Incompressible flow. Inviscid or frictionless flow, very restrictive! No w s between analysis points on streamline. No q between points on streamline.
CEE 3310 Control Volume Analysis, Oct. 7, 2015 83 3.22.1 Illustrations of Valid and Invalid Regions for the Application of the Bernoulli Equation 3.23 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 right-hand-side the total pressure. Hence the Bernoulli Equation says that in inviscid flows the total pressure along a streamline is constant.
CEE 3310 Control Volume Analysis, Oct. 7, 2015 84 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. 3.23.1 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.
CEE 3310 Control Volume Analysis, Oct. 7, 2015 85 3.24 itot-static Tube The static and stagnation pressures can be measured simultaneously using a itot-static 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 itot-static 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 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 free-stream 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 itot-static 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 = 2 2 1 ρ V = 2g(H h)
CEE 3310 Control Volume Analysis, Oct. 7, 2015 86 3.24.1 Example Flow accelerating out of a reservoir 2gh V 2 = 1 ( ) 2 A2 A 1 1 2 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? Therefore v = t 0 g dt = gt! h = t = 2h g and t 0 v dt = t v = gt = 2gt aha! 0 gt dt = 1 2 gt2 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. 3.25 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)
CEE 3310 Control Volume Analysis, Oct. 7, 2015 87 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. 3.25.1 Example Venturi Flow Meter Consider 2g h Q = A 2 V 2 = A 2 ( A2 1 A 1 ) 2 1 2
CEE 3310 Control Volume Analysis, Oct. 7, 2015 88 3.26 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 non-uniform (violation of 1-D assumption) flow effects with a C d as well. 3.27 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 sharp-cornered 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.