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CVE 372 Hydromechanics 1/20
Dr. Bertuğ AkıntuğDepartment of Civil EngineeringMiddle East Technical University
Northern Cyprus Campus
CVE 372CVE 372HYDROMECHANICSHYDROMECHANICS
FLOW IN CLOSED CONDUITSFLOW IN CLOSED CONDUITSII
CVE 372 Hydromechanics 2/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
Overview
2.1 General Characteristics of Flow in Closed Conduits
2.1.1 Definition of Laminar and Turbulent Flows
2.1.2 Entrance Region and Fully Developed Flow
2.1.3 Head Losses in Pipes
CVE 372 Hydromechanics 3/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
The governing equations for mass, momentum and energy transfer are developed in CVE 371 Fluid Mechanics.
Now, we will apply them to viscous, incompressible fluids in pipes or ducts
Typical pipe system components
CVE 372 Hydromechanics 4/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
We assume that pipe is fully filled with the fluid.Main driving force is likely to be a pressure gradientalong the pipe.If the pipe is not full, it is not possible to maintain this pressure difference, p1-p2.
(a) Pipe flow, (b) Open-channel flow
CVE 372 Hydromechanics 5/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
2.1.1. Definition of Laminar and Turbulent FlowOsborne Reynold (1842-1912), a British scientists and mathematician was the first to distinguish the difference.In pipe flow:
(a) Experiment to illustrate type of flow. (b) Typical dye streaks.
Large Re
Small Re
Intermediate Re
µρ DVa=Re
Re ≤ 2100 Laminar2100 < Re < 4000 TransitionRe ≥ 4000 Turbulent
CVE 372 Hydromechanics 6/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
2.1.1. Definition of Laminar and Turbulent Flow
Time dependence of fluid velocity at a point
CVE 372 Hydromechanics 7/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
The time-averaged, ū, and fluctuating, u', description of a parameter for tubular flow.
General Characteristics of Flow in Closed Conduits
2.1.1. Definition of Laminar and Turbulent Flow
∫=T
dtuT
u0
1
u(t): instantaneous velocity in the x-directionu’(t): fluctuating part of u(t)
)()( tuutu ′+=
CVE 372 Hydromechanics 8/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
2.1.1. Definition of Laminar and Turbulent Flow
Example 1:Water at a temperature of 10°C flows through a pipe of diameter D=1.85 cm. a) Determine the minimum time taken to fill a 355 cm3 glass with
water if the flow in the pipe is to be laminar.b) Determine the maximum time taken to fill the same glass if the
flow is to be turbulent.Repeat the calculation if the water temperature is 60°C.
Solved in the class room
CVE 372 Hydromechanics 9/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
Example 2:Water is flowing through capillary tubes A and B into tube C. IfQA=2x10-3 lt/s in tube A, what is the largest QB allowable in tube B for laminar flow in tube C? If the water has a temperature of 40°C with the calculated QB, what kind of flow exists in tubes A and B?(DA= 5 mm, DB= 4 mm, and DC= 6 mm)
Solved in the class room
A
B
C
CVE 372 Hydromechanics 10/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
2.1.2. Entrance Region and Fully Developed Flow
Entrance length Re06.0=Dle
Entrance length
Laminar flow
Turbulent flow6/1(Re)4.4=Dle
accelerated decelerated
u(r), du/dr, and τ are fixed
CVE 372 Hydromechanics 11/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
2.1.2. Entrance Region and Fully Developed FlowPressure and Shear Stress
If viscosity was zero, pressure would not vary along x direction.The physical properties of the shear stress are quite different for laminar flow then for turbulent flow.
Pressure distribution along a horizontal pipe
( )dydu
dydu
tµµτ
µτ
+=
=
turbulent
laminar
µ = molecular viscosityµt = turbulent viscosity
CVE 372 Hydromechanics 12/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
2.1.3. Head Losses in PipesWe need to determine the head loss. For convenience, we will consider two types of energy losses; minor and major loss
Total head loss, hL
hf : Friction (viscous, major) losshm :Local (minor) loss
Note that major and minor do not necessarily reflect the magnitude of the energy losses
mfL hhh +=
CVE 372 Hydromechanics 13/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
2.1.3. Head Losses in PipesDarcy – Weisbach Equation for friction loss
ifgV
DLfh
2
2
f =
22
2
5f 216 KQ
gQ
DLfh ==
π
( ) 42
2
222
2
2
22 16
4/ DQ
D
QAQV
ππ===
f: Darcy-Weisbach friction factorL: pipe length (m)D: Pipe diameter (m)V: average velocity (m/s)Q: Discharge (flow rate) (m3/s)
where
528DgfLK
π=
CVE 372 Hydromechanics 14/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
2.1.3. Head Losses in PipesHazen – Williams Equation for friction loss
or85.1
165.185.1f8.6 VDL
Ch =
C: Hazen-Williams coefficient of roughness (unitless)L: pipe length (m)D: Pipe diameter (m)V: average velocity (m/s)Q: Discharge (flow rate) (m3/s)
85.187.485.1f
6.10 QDL
Ch =
CVE 372 Hydromechanics 15/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
2.1.3. Head Losses in PipesMajor losses are due to energy loss in long straight sections of pipe which can be calculated by use of the friction factor.However, most pipe systems are not straight and have valves, bends, etc. which result in additional losses. These energy losses are collectively called minor losses.Typically, minor losses are evaluated from
where KL is the loss coefficient which depends on both the geometry of the system and the Reynolds number. For large Re, it simply becomes a function of the geometry.
gVKh m 2
2
m =
CVE 372 Hydromechanics 16/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
2.1.3. Head Losses in PipesLocal (minor) LossEmpirical equation (except for sudden enlargement)
gVKh m 2
2
m =
K: constantV: average velocity (m/s)
Entrance flow conditions and loss coefficient. (a) Reentrant, Km = 0.8, (b) Sharp-edged, Km = 0.5, (c) Slightly rounded, Km = 0.2, (d) Well-rounded, Km = 0.04
CVE 372 Hydromechanics 17/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
General Characteristics of Flow in Closed Conduits
2.1.3. Head Losses in PipesLocal (minor) Loss
Loss coefficient for a sudden contraction.
CVE 372 Hydromechanics 18/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
2
2
1 )1(AAKL −=
General Characteristics of Flow in Closed Conduits
2.1.3. Head Losses in PipesLocal (minor) Loss
CVE 372 Hydromechanics 19/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS
Character of the flow in a 90° bend and the associated loss coefficient
2.1.3. Head Losses in PipesLocal (minor) Loss
General Characteristics of Flow in Closed Conduits
CVE 372 Hydromechanics 20/20
2. FLOW IN CLOSED CONDUITS2. FLOW IN CLOSED CONDUITS