Assignment in Hydromechanics (VVR090) sample problems to be handed in B

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.