Assignment in Hydromechanics (VVR090) (sample problems to be handed in) 1. The discharge conditions in a proposed, fairly broad-crested weir with a width B=1.8 m will be investigated in a physical model (see figure below). The discharge in the prototype should be in the interval 5-30 m3/s and the weir head H in the model should not be smaller than 20 mm (otherwise the discharge will be affected by surface tension). The maximum possible discharge in the laboratory is 90 l/s. Determine the interval for possible geometric scales λ in the physical model, so that the abovementioned two conditions are not violated. Assume that the discharge from the proposed weir approximately follows the equation for a sharp-crested, rectangular weir: Q = 0.407 2 g BH 3/ 2 . 2. A fluid flows out of a container through a small, circular tube (see figure on next page). The average velocity vm into the atmosphere is assumed to depend on the distance h to the liquid surface, the tube diameter d, the acceleration due to gravity g, the density of the liquid ρ, and the dynamic viscosity μ. a) Show by using Buckingham’s Π-theorem that: h⎞ ⎛ vm = 2 gh ⋅ function ⎜ Re, ⎟ d⎠ ⎝ hv Re = m ν b) Measurements for a specific water flow from the container show that: vm=2.9 m/s for h=0.7 m when the opening is small (d/h<<1). What will the value of the function be in this case and what is the main reason why this value is less than one? 3. A settling tank is 10 m long, 3 m wide, and 1.5 m deep (see figure below). The flow rate Q, containing small particles that should settle, is introduced equally distributed across the upstream end of the tank and moves with constant velocity towards the downstream end of the tank, where the flow is discharged. What is the maximum possible Q if spherical particles with a diameter of 0.5 mm and a density of 1100 kg/m3 should reach bottom before the water is discharged from the tank? 4. Water is discharged from a lake in a long and 30 m wide channel with a rectangular cross section and a bottom slope of 1:400 (see figure on next page). The flow rate in the channel is 160 m3/s. Calculate the water level H in the lake if: a) the Manning coefficient M=1/n = 80 (smooth concrete) b) the Manning coefficient M=1/n = 40 (blasted channel in rock) Consideration should be given to the inlet head loss (acceleration loss coefficient ka=0.25). The hydraulic radius can be approximated with the water depth. 5. A channel connects two reservoirs, where the water level of the upper reservoir is located 1.0 m above the bottom of the inlet section. At the outlet to the lower reservoir is the water depth 2.0 m. The upper part of the channel is long and the lower part is short with L2=200 m. The channel cross section has got a rectangular shape with B=2 m. Further data on the channel is given in the figure below. What will the flow rate Q be and where will a hydraulic jump form? What are the water depths immediately upstream and downstream the jump? 6. The outlet from a settling tank consists of four sheet metal flumes on the same level and with horizontal bottoms and top edges. The flumes discharge into a concrete collection flume (BC) having a free outflow at C and a horizontal bottom, which is 20 cm below the bottom of the sheet metal flumes (see attached figure). The sizes and locations of the flumes are shown in the figure on the following page. Calculate the necessary height of the metal sheet flumes (the same depth for all four of them), if the flow rate through the settling tank is 0.12 m3/s and the top edge of the flumes should be 0.05 m above the highest water level in the flumes. 7. Water is pumped to a channel from a reservoir with the water level at +3 m (see figure below). The pump pipeline is L=50 m long with a diameter of D=0.35 m and a friction coefficient of f=0.02. The channel has a rectangular cross section with a width B=0.7 m and a horizontal bottom at the level +2.7 m. There is a weir a short distance downstream in the channel. The water level is assumed to be horizontal in the channel up to the weir. a) What will the flow rate Q from the pump be, if the weir is a sharp-crested one with the height P=1 m above the bottom? b) What will the flow rate Q from the pump be, if the weir is a sharp-crested 90o triangular (Thomson) one with the apex at P=1 m above the bottom? For simplicity, assume that the weir is fully contracted. 8. A 1500-m long (L) pipeline with a diameter of D=0.2 m and a constant friction coefficient f=0.020 connects two reservoirs (see figure below). A Venturi meter is connected to the pipeline. What will the reading Δx be on the manometer? The only head losses that arise are due to friction in the pipeline.