# Chapter (6) Energy Equation and Its Applications

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1 Chapter (6) Energy Equation and Its Applications

2 Bernoulli Equation Bernoulli equation is one of the most useful equations in fluid mechanics and hydraulics. And it s a statement of the principle of conservation of energy along a stream line. Bernoulli Equation can be written as following: P ρg + v 2g + z = H = constant All these terms have a unit of length (m) = pressure energy per unit weight = pressure head We know that P = ρgh h = P ρg (m). = kinetic energy per unit weight = velocity head We know that K. E = 1 2 mv divided by weight 1 2 mv mg = v 2g (m). z = potential energy per unit weight = (potential elevation head) We know that P. E = mgz divided by weight mgz mg = z (m). H = total energy per unit weight = total head (m). By using principle of conservation of energy, we can apply Bernoulli equation between two points (1 and 2) on the streamline: Total head at (1) = Total head at (2) 2g + z = 2g + z But!!, this equation no energy losses ( e.g. from friction) or energy gains (e.g. from a pump) along a stream line, so the final form for Bernoulli equation is: 2g + z + h = 2g + z + h + h Page (116) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

3 h = q = Energy supplied by pump per unit weight (m) h = w = work done by turbine per unit weight (m) h = Total friction losses per unit weight (m) Representation of Energy Changes in a Fluid System (HGL and EGL) It is often convenient to plot mechanical energy graphically using heights. Hydraulic Grade Line or (HGL): HGL = P ρg + z It is the line that joins all the points to which water would rise if piezometric tubes were inserted. Energy Grade Line or (Total Energy Line) EGL: EGL = total head = P ρg + v 2g + z It is the line that joins all the points that represent the total head (i.e. the EGL is always above HGL by a velocity head ). Important Notes (Guidelines) for drawing HGL and EGL: EGL and HGL are falls in the direction of flow due to friction losses (h L ). EGL and HGL are vertically downward due to minor losses such as: [ entrance, exit, elbow, valve, increasing or decreasing diameter] if exist. EGL and HGL are vertically upward due to the head supplied by pump (if exist) and tend vertically downward due to head consumed by turbine (if exist). If there exist a reservoir, the EGL and HGL are coinciding with the surface of fluid in the reservoir because the velocity is zero and the pressure is atmospheric pressure (zero gauge pressure). But if the water exits from a nozzle with a specific velocity the HGL will coincide with the water because the pressure is zero, but the EGL will be above HGL by a velocity head. Another notes will be described in the problems that we will solve. Page (117) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

4 Problems 1. Water flows through the horizontal branching pipe shown below. Calculate: a) The water speed at section (2). b) The pressure at section (3). c) Flow rate at section (4). Assume no losses and branch in horizontal plane A = m = 35 kpa a) (v =?? ) v = Q = 0.28 = 3.11 m/s A 0.09 By applying Bernoulli equation between sections (1) and (2): 2g + z = 2g + z z = z = 0.0 (system is in horizontal plane) = v v = 8.9 m/s. Page (118) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

5 b) ( =?? ) By applying Bernoulli equation between sections (1) and (3): 2g + z = 2g + z z = z = 0.0 (system is in horizontal plane) = = Pa = kpa. c) (Q =?? ) Q = A v = = m /s Q = A v = = m /s Q = Q (continuity equation) Q = Q + Q + Q 0.28 = Q Q = m /s. Page (119) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

6 2. For the shown figure below, if D 1 > D 2. Draw the HGL and EGL. Page (120) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

7 3. For the shown figure below, if D 1 > D 2. Draw the HGL and EGL. Page (121) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

8 4. For the shown figure below, if D 3 >D 1 > D 2. Draw the HGL and EGL. Page (122) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

9 5. For the shown figure below. Draw the HGL and EGL. Page (123) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

10 6. For the system shown below. a) What is the direction of flow? b) What kind of machine is at A? c) Do you think both pipes, AB and CA, are the same diameter? d) Sketch the EGL for the system. Page (124) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

11 Applications of Bernoulli Equation There are several applications of Bernoulli equation and will be discussed in the problems of this chapter. Important notes before solving problems: Three equations should be in mind: Bernoulli equation. Continuity Equation. Up down method (Ch.2) When the fluid is flowing from section (1) to section (2), some losses of energy will occur between sections (1) and (2), so we include a coefficient of discharge (C d ) to get the actual discharge: Q = C Q Q = the calculated discharge = Av This will be clarified in problems later. Pump is a machine that consuming electricity power to produce a head of water. Turbine is a machine that consuming energy from water to produce power elevtricity. If we want to calculate the power: Power = ρqgh (Watt) 746 Watt = 1 horse power Power consumed by pump = ρqgh Power generated by turbine = ρqgh If we need the power supplied by any object (machine, reservoir, jet, etc ) just calculate the total head and apply in power equation above. We can also calculate the efficiency of pump or turbine or any system: η = Power Power For pump: Power = ρqgh Power = electricity power For turbine: Power = electricity power Power = ρqgh Page (125) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

12 Problems 1. What is the head that must be supplied by the pump to the water to pump 0.4 m 3 /s from the lower to the upper reservoir? Hint: [The head loss in the pipes is given in the figure as a function of water velocity]. 2 1 Firstly we calculate the velocities of each pipe, assume the pipe in left side is pipe (1) and the pipe in right side is pipe (2). Q = 0.4 m /s v = Q = 0.4 A π = m/s v = Q 0.4 = A π = m/s Now we calculate the head loss for each pipe (from given formulas): h, = 22.5 V = g = m h, = 45 V = 45 2g = m h, = = m Now apply Bernoulli equation between points (1) and (2): 2g + z + h = 2g + z + h + h h = 0.0 (because there is no turbine between 1 and 2) Page (126) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

13 2g + z + h = 2g + z + h = = 0.0 (1 and 2 are exposed to atmosphere) v = v = 0.0 (surface of each reservoir is constant) h = h = m. 2. A pipe of a diameter 6 mm connects tank A and open container B as shown in figure below. The liquid having a specific weight of 9780 N/m 3 and a viscosity of kg/m.s. The pressure P A =34.5 kpa and the head loss from pipe CD is h = 25 a) Which direction will the water flow and what is the flow rate? b) Is the flow in pipe CD laminar or turbulent? 1 2 a) We calculate the head loss for pipe CD: h = 25 v 2g = 25 v = 1.274v Assume the fluid is moving from container B to tank A. Now apply Bernoulli equation between points (1) and (2): 2g + z + h = 2g + z + h + h h = h = 0.0 (because there is no pump and turbine between 1 and 2) Page (127) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

14 2g + z = 2g + z + h = 0.0 (1 is exposed to atmosphere) = 34.5 kpa = Pa v = v = 0.0 (surface of each reservoir is constant) z (from datum) = sin45 = 4.44m z (from datum) = 1m = v2 v 2 = v = = 0.262m/s The negative sign means, the fluid is moving opposite to the assumed direction (i.e. the flow is from tank A to container B). Q = A v = π = m /s. b) R = ρ V D μ ρ = 9780 = kg/m R = = 1959 < 2000 Laminar Page (128) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

15 3. When the pump shown in figure below draws 0.06 m 3 /s of water from the reservoir, the total friction head loss is 5 m. The flow discharges through a nozzle to the atmosphere. Estimate the pump power in kw delivered to the water. We apply Bernoulli Equation between points (1) and (2) Q = 0.06 m /s v = Q A = 0.06 π = 30.6 m/s v = v v = 0.0 (fluid is static in reservoir) = = 0.0 (exposed to atm. ) Assume the datum at point (1), so: z = 0.0, z = 2m 2g + z + h = 2g + z + h + h h = 0.0 (because there is no turbine between 1 and 2) 2g + z + h = 2g + z + h Apply now: h = h = 54.72m The pump power = ρqgh = = Watt The pump power = 32.2 kw. 1 2 Page (129) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

16 4. For the water system shown, the velocity at section (1) is v 1 =0.6m/s. Calculate the mercury manometer reading, h. Notes: Losses are neglected. Section 2 is opened to atmosphere. ρ = kg/m, ρ = 1000 kg/m Apply Bernoulli equation between points (1) and (2): 2g + z + h = h = h = h = 0.0 2g + z + h + h 2g + z = 2g + z Assume datum is passing through centerline of section (1) z = 0.0, z = 3m v = 0.6m/s (apply continuity equation to find v ) A v = A v v = A v A = π π = 5.4 m/s = 0.0 (opened to atmosphere) To calculate, use up down method: h = 0.0 = h 5886 Now apply in Bernoulli equation: h h = m = Page (130) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

17 5. Kerosine flows through the pump shown in figure below at m 3 /s. Head losses between 1 and 2 are 2.4 m, and the pump delivers 8 hp to the flow. What should the mercury manometer reading h be? ρ = kg/m, ρ = 804 kg/m 2 1 y Apply Bernoulli equation between points (1) and (2): 2g + z + h = 2g + z + h + h h = 0.0 h = 2.4 m Power = 8 hp = = 5968 Watt Power = ρqgh 5968 = h h = m Assume datum is passing through centerline of section (1) z = 0.0, z = 1.5m v = Q = A π = 14.7m/s v = Q = A π = 3.68m/s Page (131) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

18 Apply in Bernoulli equation: = = Eq. (1) Now apply up down method between 1 and 2: + ρ gy + ρ gh ρ g(h + y + 1.5) = = h Substitute from Eq. (1): h = h = 1.23 m. 6. Oil flows through a pipe as shown in figure below. If the coefficient of discharge (C d ) for the orifice in the pipe is Calculate the discharge of oil in the pipe. y Firstly we apply Bernoulli equation between points (1) and (2): 2g + z = 2g + z z = z = 0.0 (by assuming the datum passing through points 1 and2) = v v ρg 2g Now, by applying continuity equation between (1) and (2): A v = A v Page (132) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

19 A = π = m A = π = m v = v v = 0.16 v = v 0.16v = 0.84v ρg 2g ρg 2g Now, by applying up down method between (1) and (2): y y = = Pa = 0.84v v ρg 2g = m/s Q = A v = = m /s Q = Q C = = 0.053m /s. Page (133) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

20 7. A weir of cross section shown below is used to measure the flow rate. Calculate this value of flow rate (Q). We divide this cross section for rectangle with dimensions of (0.2 x 0.4) and triangle with dimensions of (0.4 x 0.1) and then we sum the two values of Q. For rectangle: The flow rate for the shown element is: Q = A v A = 0.4 h v = 2gh Q = 0.4 2gh h But the value of h is caries from 0 to 0.2m, so the value of v also varies and thereby Q is varies with h, so, to find the total discharge for this rectangle we integrate from 0 to 0.2m to get total amount of discharge:.. Q = 0.4 2gh h = 0.4 2g h. h Q = 1.77 h. (from 0 to 0.2) 1.5 Page (134) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

21 Q = = m /s For triangle: Here the width of the triangle b(h) varies with depth h so we find b(h) as a function of h from trigonometry: b(h) 0.3 h = 0.4 b(h) = 1.2 4h 0.1 Q = A v Q = (1.2 4h) h 2gh Q = 2g (1.2h. 4h. ) h Now the integration will be from (h=0.2 to h=0.3). Q = 2g (1.2h. 4h. ) h. Q = h h. (from h = 0.2 to h = 0.3) 2.5 Q = Q = 4.3 ( ) = m /s The total value of Q is : Q + Q = = 0.147m /s. Page (135) Dr.Khalil Al-astal Eng. Ahmed Al-Agha Eng. Ruba Awad

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