In
fluid dynamics, the entrance length is the distance a flow travels after entering a
pipe
Pipe(s), PIPE(S) or piping may refer to:
Objects
* Pipe (fluid conveyance), a hollow cylinder following certain dimension rules
** Piping, the use of pipes in industry
* Smoking pipe
** Tobacco pipe
* Half-pipe and quarter pipe, semi-circular ...
before the flow becomes fully developed.
[''ES162_08_Notes02a_Flow_In_Pipes_Changtamu.Pdf''. 1st ed. Cambridge: J. R. Rice, 2017. Print.] Entrance length refers to the length of the entry region, the area following the pipe entrance where effects originating from the interior wall of the pipe propagate into the flow as an expanding
boundary layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...
. When the boundary layer expands to fill the entire pipe, the developing flow becomes a fully developed flow, where flow characteristics no longer change with increased distance along the pipe. Many different entrance lengths exist to describe a variety of flow conditions. Hydrodynamic entrance length describes the formation of a
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
profile caused by
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
forces propagating from the pipe wall. Thermal entrance length describes the formation of a temperature profile.
Awareness of entrance length may be necessary for the effective placement of instrumentation, such as fluid
flow meters
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
.
Hydrodynamic entrance length
The hydrodynamic entrance region refers to the area of a pipe where fluid entering a pipe develops a velocity profile due to
viscous forces
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
propagating from the interior wall of a pipe.
This region is characterized by a non-uniform flow.
The fluid enters a pipe at a uniform velocity, then fluid particles in the layer in contact with the surface of the pipe come to a complete stop due to the
no-slip condition
In fluid dynamics, the no-slip condition for viscous fluids assumes that at a solid boundary, the fluid will have zero velocity relative to the boundary.
The fluid velocity at all fluid–solid boundaries is equal to that of the solid boundary. C ...
. Due to viscous forces within the fluid, the layer in contact with the pipe surface resists the motion of adjacent layers and slows adjacent layers of fluid down gradually, forming a velocity profile. For the
conservation of mass
In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass can ...
to hold true, the velocity of layers of the fluid in the center of the pipe increases to compensate for the reduced velocities of the layers of fluid near the pipe surface. This develops a velocity gradient across the cross-section of the pipe.
Boundary layer
The layer in which the shearing viscous forces are significant, is called the
boundary layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...
.
This boundary layer is a hypothetical concept. It divides the flow in pipe into two regions:
# Boundary layer region: The region in which viscous effects and the velocity changes are significant.
# The irrotational (core) flow region: The region in which viscous effects and velocity changes are negligible, also known as the inviscid core.
When the fluid just enters the pipe, the thickness of the boundary layer gradually increases from zero moving in the direction of fluid flow and eventually reaches the pipe center and fills the entire pipe. This region from the entrance of the pipe to the point where the boundary layer covers the entire pipe is termed as the hydrodynamic entrance region and the length of the pipe in this region is termed as the hydrodynamic entry length. In this region, the velocity profile develops and thus the flow is called the hydrodynamically developing flow. After this region, the velocity profile is fully developed and continues unchanged. This region is called the hydrodynamically fully developed region. But this is not the fully developed fluid flow until the normalized temperature profile also becomes constant.
In case of
laminar flow, the velocity profile in the fully developed region is parabolic but in the case of
turbulent flow
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between t ...
it gets a little flatter due to vigorous mixing in radial direction and eddy motion.
The velocity profile remains unchanged in the fully developed region.
''Hydrodynamic Fully Developed velocity profile Laminar Flow :''
(where
is in the flow direction).
Shear stress
In the hydrodynamic entrance region, the wall
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
(τw ) is highest at the pipe inlet, where the boundary layer thickness is the smallest. Shear stress decreases along the flow direction.
That is why the
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
drop is highest in the entrance region of a pipe, which increases the average
friction factor for the whole pipe. This increase in the friction factor is negligible for long pipes.
In a fully developed region, the pressure gradient and the shear stress in flow are in balance.
Calculating hydrodynamic entrance length
The length of the hydrodynamic entry region along the pipe is called the hydrodynamic entry length. It is a function of
Reynolds number of the flow. In case of laminar flow, this length is given by:
Where,
is the Reynolds number and
is the diameter of the pipe.
But in the case of turbulent flow,
Thus, the entry length in turbulent flow is much shorter as compared to laminar one. In most practical engineering applications, this entrance effect becomes insignificant beyond a pipe length of 10 times the diameter and hence it is approximated to be:
Other authors give much longer entrance length, e.g.
* Nikuradse recommends 40 D
* Lien et al. recommend 150 D for high Reynolds number turbulent flow.
Entry length for pipes with non-circular cross-sections
In the case of a non-circular cross-section of a pipe, the same formula can be used to find the entry length with a little modification. A new parameter “
hydraulic diameter The hydraulic diameter, , is a commonly used term when handling flow in non-circular tubes and channels. Using this term, one can calculate many things in the same way as for a round tube. When the cross-section is uniform along the tube or channel ...
” relates the flow in non-circular pipe to that of circular pipe flow. This is valid as long as the cross-sectional area shape is not too exaggerated. Hydraulic Diameter is defined as:
Where,
is the area of cross-section and
is the Perimeter of the wet part of the pipe
Average velocity of fully developed flow
By doing a force balance on a small volume element in the fully developed flow region in the pipe (Laminar Flow), we get velocity as function of radius only i.e. it does not depend upon the axial distance from the entry point.
The velocity as the function of radius comes out to be:
Where
By definition of Average velocity,
where
is Area of cross section
Thus,
For fully developed flow, the maximum velocity will be at r=0.
Thus,
Thermal entrance length
The thermal entrance length is the distance for incoming flow in a pipe to form a temperature profile with a stable shape. The shape of the fully developed temperature profile is determined by temperature and heat flux conditions along the inside wall of the pipe, as well as fluid properties.
Overview
Fully developed heat flow in a pipe can be considered in the following situation. If the wall of the pipe is constantly heated or cooled so that the
heat flux from the wall to the fluid via
convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the conve ...
is a fixed value, then the bulk temperature of the fluid increases steadily at a fixed rate along the flow direction.
An example can be a pipe entirely covered by an electrical heating pad with the flow being introduced after a uniform heat flux from the pad is achieved. At some distance away from the entrance of the fluid, fully developed heat flow is achieved when the
heat transfer coefficient of the fluid becomes constant and the temperature profile has the same shape along the flow. This distance is defined as the thermal entrance length, which is important for engineers to design efficient heat transfer processes.
Laminar flow
For laminar flow, the thermal entrance length is a function of pipe diameter and the dimensionless
Reynolds number and
Prandtl number.
The Prandtl number modifies the hydrodynamic entrance length to determine thermal entrance length. The Prandtl number is the
dimensionless number
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
for the ratio of momentum diffusivity to
thermal diffusivity
In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It measures the rate of transfer of heat of a material from the hot end to the cold end. It has the SI ...
.
The thermal entrance length for a fluid with a Prandtl number greater than one will be longer than the hydrodynamic entrance length, and shorter if the Prandtl number is less than one. For example, molten sodium has a low Prandtl number of 0.004, so the thermal entrance length will be significantly shorter than the hydraulic entrance length.
For turbulent flows, thermal entrance length may be approximated solely based on pipe diameter.
where:
Heat transfer
The development of the temperature profile in the flow is driven by heat transfer determined conditions on the inside surface of the pipe and the fluid.
Heat transfer may be a result of a constant heat flux or constant surface temperature. Constant heat flux may be caused by
joule heating from a heat source, like heat tape, wrapped around the pipe. Constant temperature conditions may be produced by a phase transition, such as
condensation of saturated
steam on a pipe surface.
Newtons law of cooling describes convection, the main form of heat transport between the fluid and the pipe:
where:
Constant surface heat flux result in
becoming a constant as the flow develops and constant surface temperature results in
approaching zero.
Thermally fully developed flow
Unlike hydrodynamic developed flow, a constant profile shape is used to define thermally fully developed flow because temperature continually approaches ambient temperature.
Dimensionless analysis of change in profile shape defines when a flow is thermally fully developed.
Requirement for thermally fully developed flow:
Thermally developed flow results in reduced heat transfer compared to developing flow because the difference between the surface temperature of the pipe and the mean temperature of the flow is greater than the temperature difference between surface temperature of the pipe and the temperature of the fluid near the pipe boundary.
Concentration entrance length
The
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', '' number concentration'', ...
entrance length describes the length needed for the concentration profile in a flow to be fully developed. The concentration entrance length can be determined by relating it to the hydrodynamic entrance length with the
Schmidt number
Schmidt number (Sc) is a dimensionless number defined as the ratio of momentum diffusivity ( kinematic viscosity) and mass diffusivity, and it is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convec ...
or by experimental techniques. The Schmidt number describes the ratio of momentum diffusivity to mass diffusivity.
where:
Applications
Understanding the entrance length is important for the design and analysis of flow systems. The entrance region will have different velocity, temperature, and other profiles than exist in the fully developed region of the pipe.
Flow meters
Many types of flow instrumentation, such as
flow meters
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
, require a fully developed flow to function properly.
Common flow meters, including vortex flow meters and differential-pressure flow meters, require hydrodynamically fully developed flow. Hydraulically fully developed flow is commonly achieved by having long, straight sections of pipe before the flow meter. Alternatively,
flow conditioners and straightening devices may be used to produce the desired flow.
Wind tunnels
Wind tunnels use an inviscid flow of air to test the aerodynamics of an object. Flow straighteners, which consist of many parallel ducts which limit turbulence, are used to produce inviscid flow. Entrance length must be considered in the design of wind tunnels, because the object being tested must be located in the irrotational flow region, between the flow straighteners and the entrance length.
Exit length
Similar to the development of flow at the entrance of the pipe, the flow velocity profile changes before the exit of a pipe. The exit length is much shorter than the entrance length, and is not significant at moderate to high Reynolds numbers.
Hydraulic exit length for laminar flows may be approximated as:
See also
*
Fluid dynamics
*
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
*
Laminar flow
*
Thermal entrance length
*
Turbulent flow
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between t ...
*
Viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
References
{{reflist
Fluid dynamics