Basic Fluid Mechanics Chapter 5: Application of Bernoulli Equation 4/16/2018 C5: Application of Bernoulli Equation 1 5.1 Introduction In this chapter we will show that the equation of motion of a particle can be integrated (under certain circumstances) and related to the potential and kinetic energies of the particle. Initially we will consider incompressible flow, but later with the use of the first law of thermodynamics a similar form of the equations of motion for compressible flow can be developed. The following are different forms of the Bernoulli equation. Simple Form of the Bernoulli Equation Assumptions: 1. Steady 2. Incompressible 3. Inviscid 4. Along a streamline 5. Inertial reference frame 4/16/2018 C5: Application of Bernoulli Equation 2 1
5.1 Introduction Bernoulli Equation with Conservative Forces Assumptions: 1. Steady 2. Incompressible 3. Inviscid 4. Along a streamline 5. Inertial reference frame 6. Conservative Body Force U Bernoulli Equation with Conservative Forces Assumptions: 1. Steady 2. Incompressible 3. Inviscid 4. Along a streamline 5. Inertial reference frame 6. Conservative Gravitational Force 4/16/2018 C5: Application of Bernoulli Equation 3 5.1 Introduction Unsteady Bernoulli Equation Assumptions: 1. Unsteady 2. Incompressible 3. Inviscid 4. Along a streamline 5. Inertial reference frame Extended Bernoulli Equation Assumptions: 1. Steady 2. Incompressible 3. Viscous (i.e., resistance) W f 4. Along a streamline 5. Shaft work, W s 6. Inertial reference frame 4/16/2018 C5: Application of Bernoulli Equation 4 2
5.1 Introduction Problem 5.1: Determine the pressure difference between two stations 1 and 2 as a function of the flow rate Q and density. V 1 V 2 A 1 Solution: 1) Assume steady state 2) Assume uniform flow 3) Assume inviscid fluid 4) Assume = constant 5) Apply the conservation of mass to determine V 2. 6) Apply the simple Bernoulli equation A 2 Recall Conservation of Mass: 4/16/2018 C5: Application of Bernoulli Equation 5 5.1 Introduction For an Incompressible fluid, the Volume flow rate (Q) = constant and the conservation of mass reduces to continuity eq: Apply Bernoulli Eq; Rewriting Bernoulli Eq (5.2b); From continuity eq; Eq (EX5.1) Substituting for V 1 into the pressure difference equation (EX5.1); 4/16/2018 C5: Application of Bernoulli Equation 6 3
5.1 Introduction ====The End==== 4/16/2018 C5: Application of Bernoulli Equation 7 5.2 Extended Bernoulli s Equation Consider the presence of non-conservative body forces (f nc ) as well as the presence of shear and normal stresses arising from friction (f f ). Including these forces in Bernoulli's Equation (5.4) leads to; A Assumptions: 1. Steady 2. Incompressible 3. Along a streamline 4. Inertial reference frame B (5.8) 4/16/2018 C5: Application of Bernoulli Equation 8 4
5.2 Extended Bernoulli s Equation A B Term "A represents the useful work done on the fluid by f nc (e.g., moving of pump vane or turbine blade) where W s is the shaft work performed or energy expended per unit mass. W s = Where W s is considered "+" for a pump and "-" for a turbine blade, since they add and remove energy from the flow respectively. Term B represents the work done by the fluid in overcoming viscous resistance (not useful work) which is always a loss in energy. W f = 4/16/2018 C5: Application of Bernoulli Equation 9 5.2 Extended Bernoulli s Equation Therefore, Where each term is written as an energy/unit mass. In hydraulic engineering, the terms are rearranged and expressed as energy (or work)/unit weight g g (5.10) (5.11) Where each term is written with units of length, enabling the use of the term head. 4/16/2018 C5: Application of Bernoulli Equation 10 5
5.2 Extended Bernoulli s Equation M pump work H f friction loss Y elevation head static pressure head velocity head h t total head h t2 = h t1 + M h f (5.12) The total head after the flow process is equal to the total head before the process plus any mechanical work performed minus losses due to friction. 4/16/2018 C5: Application of Bernoulli Equation 11 5.2 Extended Bernoulli s Equation Aero engineers multiply Eq (5.10) by to obtain Bernoulli Equations in terms of work/unit volume. In this form, all terms have the dimensions of a pressure; so is the dynamic pressure total pressure (p t); (analogous to total head) Total pressure is also known as the stagnation pressure since it is the pressure which would result if a fluid stream was brought to rest isentropically. Each term in Eq (5.10) represents energy/unit mass. When multiplied by the mass flow rate (mass/time) one obtains the power () in watts. energy/unit mass * mass/time = energy/time = 4/16/2018 C5: Application of Bernoulli Equation 12 6
5.2 Extended Bernoulli s Equation Similarly, each term in Eq. 5.12 represents energy/unit weight. When multiplied by the weight flow rate, gq (weight/time), where Q is the volume flow rate, one also obtains power (). The function of a pump is to increase the total head of the flow, h t. The associated power () required for this increase Recall, the fluid power is considered positive for a pump and negative for a turbine. The actual power, act, necessary to run a pump is always larger than the ideal fluid power due to friction, internal leakage, and a variety of other losses. The ratio / act is called the pump "efficiency." Likewise, the ratio act / is turbine efficiency. 4/16/2018 C5: Application of Bernoulli Equation 13 5.3 Engineering Application of Bernoulli s Equation Often it is possible to assume that the flow is "one-dimensional," i.e., no velocity variation normal to the axis of the conduit. This uniform velocity is taken to be equal to the average velocity over the flow cross section. This would be true of flow in pipe circuits. 4/16/2018 C5: Application of Bernoulli Equation 14 7
Problem 5.2: A constant diameter pipe (d) of length l is attached to a reservoir whose water level is y 1. For a length of pipe equal to one diameter the frictional loss is f, the friction coefficient, is constant. Determine the distributions of static pressure and total head, as well as the outlet velocity for the system shown. You may neglect frictional losses at the pipe entrance. The following info is available: f = 0.02 l /d = 1000 y 1 y 2 = 100 m 4/16/2018 C5: Application of Bernoulli Equation 15 Total Head Hydraulic gradeline y 1 y 1 <0 h f >0 Solution: y Pipe elevation y 2 y 3 Distance along the pipe Solution: 1. Apply the extended Bernoulli equation between points 1 & 2. 2. Note that no pump work is performed. 3. Assume that the reservoir is large so that the velocity at 1 due to the discharge is very small. 4. Assume that the pressures at 1 & 2 are equal to atmospheric. 5. Solve for the discharge velocity at 2. 4/16/2018 C5: Application of Bernoulli Equation 16 8
6. Write the extended Bernoulli equation between station 1 and some arbitrary station along the pipe, x. 7. Note that since the pipe diameter is constant, the pipe velocity is constant. 8. Solve for the total head as a function of x position. Apply Bernoulli's Eq (5.12 and 11), (work/unit weight) between points 1 and 2 written in terms of total head h t2 = h t1 + M h f (5.12) g g (5.11) but in the absence of a pump or turbine, M = 0 and if we assume the free surface at 1 is large; V 1 = 0 and p 2 = p 1, so 4/16/2018 C5: Application of Bernoulli Equation 17 /. /.. / 4/16/2018 C5: Application of Bernoulli Equation 18 9
One can now determine the total pressure distribution along the pipe between station 1 and some arbitrary station "x". h tx h t1 h fx h tx h t1 This equation shows that the total head decreases linearly with distance along the pipe. Substituting for the total head g h t1 4/16/2018 C5: Application of Bernoulli Equation 19 From continuity and the fact that the pipe diameter is constant, V x = V 2 and let y 1 = 0 and p 1 = p atm = 0 gage g therefore the static pressure is a linear function of pipe length x. The group (p/g + y) is sometimes called the piezometric head, and the graph of the piezometric head along the pipe is called the hydraulic grade line. Note: If the elevation of the pipe at any section exceeds that of the hydraulic grade line, the static pressure will fall below the atmospheric pressure. 4/16/2018 C5: Application of Bernoulli Equation 20 10
Note: If the elevation of the pipe at any section exceeds that of the hydraulic grade line, the static pressure will fall below the atmospheric pressure. Total Head Hydraulic gradeline y 1 y 1 <0 h f >0 y 3 Solution: y Pipe elevation y 2 Distance along the pipe ====The End==== 4/16/2018 C5: Application of Bernoulli Equation 21 Problem 5.5: Find the head across the pump and power necessary to transport water between the reservoirs shown. Given are the frictional losses between designated points, the respective elevations and volume flow rate. h f1 2 = h f3 4 = 5 m; y 1 = 20 m; y 5 =40 m; Q p = 0.5 m 3 /s; A 4 = 0.04 m 2 1 4 y1 y 4 y 5 4/16/2018 C5: Application of Bernoulli Equation 22 11
Solution Strategy: 1 4 y1 y 4 y 5 1. Apply extended Bernoulli eq between points 1 & 5; solve for M. 2. Assume that the areas at stations 1 & 5 are large, therefore their respective velocities can be assumed small and neglected, use this to simplify the above expression. 3. Write the extended Bernouli equation between points 4 and 5. 4. Use the hydrostatic eq to write an expression for pressure at (4). 5. Solve for the losses between 4 and 5. 4/16/2018 C5: Application of Bernoulli Equation 23 Solution: To determine the head across the pump, it is only necessary to write the extended Bernoulli equation between points 5 and 1. h t5 = h t1 + M h f1 2 h f3 4 h f4 5 (5.5A) Recall the pressure acting on the two liquid surfaces (points 1 & 5) are equal, and the surface areas at these locations are large, which allows us to assume the velocities are small and thus can be neglected. g Since V 5 = V 1 0 and g g and g (5.5B) 4/16/2018 C5: Application of Bernoulli Equation 24 12
Solve for the total pressure difference (energy/unit volume),( ) in Eq. (5.5.B) recalling that p 1 = p 5 g or solving for h t in terms of an energy/weight viewpoint therefore, Eq 5.5.A becomes M = y 5 y 1 h f1 2 h f3 4 h f4 5 (5.5C.1) Except for h f4 5, all frictional losses are given. To estimate h f4 5, one assumes that as the jet discharges at point 4, the jet spreads and it's velocity goes to zero. 4/16/2018 C5: Application of Bernoulli Equation 25 Writing Bernoulli's Equation between points (4) and (5). h t5 = h t4 + M h f4 5 since the pump is located between points 2 and 3, M in the above equation is zero. Also recall V 5 = 0 g h f4 5 g g + h f4 5 (5.5D) But if the fluid in the reservoir surrounding the jet is approximately stationary, then the pressure is distributed hydrostatically and one can use the hydrostatic pressure equation. g or g g 4/16/2018 C5: Application of Bernoulli Equation 26 13
If we work in terms of atmospheric pressure (gage pressure), p 5 = 0 g (5.5E) Substituting Eq. (5.5.E) into Eq. (5.5.D) h f4 5 Now substituting h f4 5 into Eq. (5.5.C1) M = y 5 y 1 h f1 2 h f3 4 + (5.5F) Thus using the given conditions / M = 40m 20m + 5m + 5m +. / M = 30m +. /.. / = 38.0 m (5.5B2) 4/16/2018 C5: Application of Bernoulli Equation 27 The ideal pump power () required is; = Force/area volume/time = Total Pressure volume flow rate or = g(h t ) Q = p t Q = 1000 9.81 m 0.5 186.4 Nm/s = 80.4 W ====The End==== 4/16/2018 C5: Application of Bernoulli Equation 28 14