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Applications

Understanding the entrance length is important for 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.

Many types of flow instrumentation, such as flow meters, require a fully developed velocity profile to function properly.

Entrance length

From Wikipedia, the free encyclopedia

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In fluid dynamics, the entrance length is the distance a flow travels after entering a pipe before the flow becomes fully developed. 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. When the boundary layer expands to fill the entire pipe, the developing flow transitions to fully developed flow, where flow characteristics no longer change with increased distance along the pipe. Many different entrance lengths exist for a variety of flow conditions. Hydrodynamic entrance length describes the formation of a velocity profile caused by viscous fources 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 meters.

Contents

  null hide 

• 1Hydrodynamic Entrance Length
  • 1.1Boundary layer
  • 1.2Shear stress
  • 1.3Entry Length
  • 1.4Entry Length For Pipes with Non-circular Cross-Section
  • 1.5Average velocity
• 2Thermal Entrance Length
  • 2.1Laminar Flow
  • 2.2Turbulent Flow
  • 2.3Heat Transfer
  • 2.4Thermally Fully Developed Flow
• 3Exit Length
• 4See also
• 5References

Hydrodynamic Entrance Length[edit | edit source]

The hydrodynamic entrance region refers to the area of a pipe where fluid entering a pipe develops a velocity profile due to viscous forces 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. Due to viscosity of the fluid, the layer in contact with the pipe surface, resists the motion of adjacent layers and slows them down gradually. For the conservation of mass to hold true the velocity of middle layers of the fluid in the pipe increases (since the layers of fluid near the pipe surface have reduced velocities). This develops a velocity gradient across the cross section of the pipe.

Boundary layer[edit | edit source]

The layers in which the shearing viscous forces are significant, is called boundary layer. This boundary layer is a hypothetical concept. It divides the flow in pipe into two regions:

1. Boundary layer region: The region in which viscous effects and the velocity changes are significant.
2. The irrotational (core) flow region: The region in which viscous effects and velocity changes are negligible.

When the fluid just enters the pipe, the thickness of the boundary layer gradually increases from zero as we move in direction of fluid flow and eventually it reaches the pipe centre and fills the entire pipe. This region from the entrance of pipe to the point where the boundary layer covers the entire pipe is termed as the hydrodynamic entrance region and 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 termed the (hydrodynamically) fully developed region. But this is not the fully developed fluid flow until the normalised 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 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  :  

(where  is the streamwise or flow direction). The developing velocity profile of a fluid entering a pipe.

Shear stress [edit | edit source]

In the hydrodynamic entrance region, the wall shear stress (τw ) is highest at the pipe inlet where the boundary layer thickness is the smallest and it decreases along the flow direction. That is why the pressure drop is the highest in entrance region of a pipe and hence it always increases the average friction factor for the whole pipe. This increase in the friction factor is negligible for long pipes.

In fully developed region the pressure gradient and the shear stress in flow are in balance. Variation of Shear Stress with distance from the entry point.

Entry Length[edit | edit source]

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 Reynold's 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-Section[edit | edit source]

In 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” relates the flow in non-circular pipe to that of circular pipe flow. This is valid until 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[edit | edit source]

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,

In a fully developed flow, the maximum velocity will be at r=0.

Thus,

Thermal Entrance Length[edit | edit source]

The thermal entrance length describes the distance for incoming flow in a pipe to form a temperature profile of 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.

Laminar Flow[edit | edit source]

For laminar flow, the thermal entrance length is a function of pipe diameter and the dimensionless Reynolds number and Prandtl number.

where:

The Prandtl number modifies the hydrodynamic entrance length to determine thermal entrance length. The Prandlt number is the dimensionless number for the ratio of momentum diffusivity to thermal diffusivity. 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.

Turbulent Flow[edit | edit source]

For turbulent flows, thermal entrance length may be approximated solely based on pipe diameter.

where:

Heat Transfer[edit | edit source]

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 thermal tape, wrapped around the pipe.

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[edit | edit source]

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.

Exit Length[edit | edit source]

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[edit | edit source]

• Fluid Dynamics
• Laminar Flow
• Turbulent Flow
• Viscosity
• Heat Transfer

References[edit | edit source]

1. ^ Jump up to:^a b c ES162_08_Notes02a_Flow_In_Pipes_Changtamu.Pdf. 1st ed. Cambridge: J. R. Rice, 2017. Print.
2. Jump up^ 
3. ^ Jump up to:^a b 
4. ^ Jump up to:^{a b c d e} 
5. Jump up^ Nikuradse, J., Gesetzmäßigkeiten der turbulenten Strömung in glatten Rohren, Forschung auf dem Gebiet des Ingenieurwesens, 3, 1932, 1–36, (Translated in NASA TT F-10, 359,1966.
6. Jump up^ 
7. ^ Jump up to:^{a b c d e f} 
8. Jump up^ 1924-1978., Perry, Robert H.,; W., Green, Don (2008-01-01). Perry's chemical engineers' handbook. McGraw-Hill. ISBN 0071593136. OCLC 72470708.

Old Draft

In fluid dynamics, the entrance length is the distance a flow travels after entering a pipe before the flow no longer changes with increased length. The entrance length is characterized by an expanding boundary layer of material which is affected by the the boundary of the pipe. When the boundary layer expands to fill the entire pipe, the developing flow transitions to fully developed flow, where properties are no longer dependent on distance along the pipe. Entrance lengths exist for a variety of flow properties. Hydrodynamic entrance length describes the formation of a velocity profile caused by viscous forces of 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 meters.

Hydrodynamic Entrance Length

When a fluid is entering a pipe at a uniform velocity, the fluid particles in the layer in contact with the surface of the pipe come to a complete stop due to the no-slip condition. Due to viscosity of the fluid, this layer in contact with the pipe surface, resists the motion of adjacent layers and slows them down gradually. For the conservation of mass to hold true the velocity of middle layers of the fluid in the pipe increases (since the layers of fluid near the pipe surface have reduced velocities). This develops a velocity gradient across the cross section of the pipe.

Boundary layer

The layers in which the shearing viscous forces are significant, is called boundary layer. This boundary layer is a hypothetical concept. It divides the flow in pipe into two regions:

1. Boundary layer region: The region in which viscous effects and the velocity changes are significant.
2. The irrotational (core) flow region: The region in which viscous effects and velocity changes are negligible.

When the fluid just enters the pipe, the thickness of the boundary layer gradually increases from zero as we move in direction of fluid flow and eventually it reaches the pipe centre and fills the entire pipe. This region from the entrance of pipe to the point where the boundary layer covers the entire pipe is termed as the hydrodynamic entrance region and 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 termed the (hydrodynamically) fully developed region. But this is not the fully developed fluid flow until the normalised 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 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  : ${\frac {\partial u(r,x)}{\partial x}}=0\quad \Rightarrow u=u(r)$
(where $x$ is the streamwise or flow direction).

We can see in this image the developing velocity profile of a fluid entering a pipe.

Shear stress $(\tau _{w})$

In the hydrodynamic entrance region, the wall shear stress (τw ) is highest at the pipe inlet where the boundary layer thickness is the smallest and it decreases along the flow direction. That is why the pressure drop is the highest in entrance region of a pipe and hence it always increases the average friction factor for the whole pipe. This increase in the friction factor is negligible for long pipes.
In fully developed region the pressure gradient and the shear stress in flow are in balance.

In this image we can see that the Shear Stress is maximum right at the entry point and becomes uniform as the fluid flow develops.

Entry Length

The length of the hydrodynamic entry region along the pipe is called hydrodynamic entry length. It is a function of Reynolds Number of the flow. In case of laminar flow, this length is given by:

$L_{h,laminar}=0.06R_{e}D$

Where, $R_{e}$ is the Reynold's number and $D$ is the diameter of the pipe.
But in the case of turbulent flow,

$L_{h,turbulent}=1.359D(R_{e})^{1/4}$

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 :

$L_{h,turbulent}\approx 10D$
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-Section

In 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” relates the flow in non-circular pipe to that of circular pipe flow. This is valid until the cross sectional area shape is not too exaggerated. Hydraulic Diameter is defined as:

$D_{h}={\frac {4A}{P}}$

Where, $A$ is the area of cross-section and $P$ is the Perimeter of the wet part of the pipe

Average velocity

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:

$u(r)=-{\frac {R^{2}}{4\mu }}{\frac {dP}{dx}}(1-{\frac {r^{2}}{R^{2}}})$
Where

${\frac {dP}{dx}}=constant$

By definition of Average velocity,

$V_{avg}={\frac {\int u\mathrm {d} A}{A_{c}}}$ where $A_{c}$ is Area of cross section

Thus,

$V_{avg}={\frac {2}{R^{2}}}\int _{0}^{R}u(r)r\mathrm {d} r$

$=-{\frac {2}{R^{2}}}\int _{0}^{R}{\frac {R^{2}}{4\mu }}{\frac {dP}{dx}}(1-{\frac {r^{2}}{R^{2}}})r\mathrm {d} r$

$=-{\frac {R^{2}}{8\mu }}{\frac {dP}{dx}}$

In a fully developed flow, the maximum velocity will be at r=0.
Thus,

$U_{max}=2V_{avg}$

Thermal Entrance Length

The thermal entrance length describes the distance for incoming flow in a pipe to form a temperature profile of 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.

For laminar flow, the thermal entrance length is a function of pipe diameter and the dimensionless Reynolds number and Prandtl number.

$({\frac {x_{fd,t}}{D}})_{lam}\approx 0.05Re_{D}Pr$

where:

$x_{fd,t}={\text{Thermal Entrance Length}}$

${\text{D}}={\text{pipe inside diameter}}$

${\text{Re}}={\text{Reynolds Number}}$

${\text{Pr}}={\text{Prandtl number}}$

The Prandtl number modifies the the hydrodynamic entrance length to determine thermal entrance length. The Prandlt number is the dimensionless number for the ratio of momentum diffusivity to thermal diffusivity. The thermal entrance length for a fluid with a Prandlt number greater than one will be longer than the hydrodynamic entrance length, and shorter if the Prandle number is less than one.

For turbulent flows, thermal entrance length may be approximated solely based on pipe diameter.

$({\frac {x_{fd,t}}{D}})_{turb}\approx 10$

where:

$x_{fd,t}={\text{Thermal Entrance Length}}$

${\text{D}}={\text{pipe inside diameter}}$

References

Category:Fluid dynamics 2/1/17 Draft your Article

I am improving on an existing article. The content of the article focuses exclusively on the hydrodynamic entrance length. I plan on changing the structure of the article, and adding sections about thermal entrance regions, and concentration entrance regions. There are a lot of conditions, such as pipe shape that affect the entrance length and are used specifically in that purpose. I am also planning on focusing how fluid mechanics numbers including Reynolds number, Prandtl number, Nusselt number, and Schmidt number relate to the the entrance region. So far, the best source I have found for describing non hydrodynamic entrance lengths is Fundamentals of heat and mass transfer by Bergman. I plan on restructuring the sections under the different varieties of entrance lengths. Most of the current content will evaluated, cited, revised, and incorporated into a section on hydrodynamic entrance length. Under the section on hydrodynamic entrance length will be subtopics such as Boundary Layer, Laminar Flow, Turbulent Flow, Shear Stress, Entry Length, Conditions effecting entry length, and importance in design. Similar subtopics will exist for the other variations of entrance length. The first paragraph will be heavily changed to represent the more general idea of entrance length and include examples and information demonstrating entrance length's relevance to fluid mechanics. The goals of improving readability and adding citations will also be focused on.

Draft Changes to contents and 1st Paragraph

The entrance region refers to a portion of flow through a pipe between the entrance and the point where fluid properties are fully developed. This region is characterized by a non-uniform flow. When a fluid is entering a pipe at a uniform velocity, the fluid particles in the layer in contact with the surface of the pipe come to a complete stop due to the no-slip condition. Due to viscosity of the fluid, this layer in contact with the pipe surface, resists the motion of adjacent layers and slows them down gradually. For the conservation of mass to hold true the velocity of middle layers of the fluid in the pipe increases (since the layers of fluid near the pipe surface have reduced velocities). This develops a velocity gradient across the cross section of the pipe. Other properties including, temperature and concentration, also have entrance regions before transitioning to a fully developed profile. Awareness of entrance length may be necessary for the effective placement of instrumentation, such as fluid meters.

Contents

null hide 

• 1.Hydrodynamic Entry Region
  • 1.1 Boundary layer
  • 1.2 Shear stress 
  • 1.3 Hydrodynamic Entrance Length
• 2. Thermal Entry Region
• 3. Concentration Entry Region
• 5. References

1/23/17 Choose topic/Find your sources

In your sandbox, write a few sentences about what you plan to contribute to the selected article.

• Think back to when you did an article critique. What can you add? Post some of your ideas to the article's talk page. -I'm planning on adding several sections that focus on better describing the characteristics and mathematics of entrance length. Possible topics include: Boundary Layer, Hydrodynamic Entry Region, Laminar Flow, Turbulent Flow, Effect on Heat Transfer, Relation to pipe and fluid conditions, Derivation, and additional descriptions about each of the regions located around the entry region. I will also add sections focusing on the importance of entrance length and the topic's context in engineering and fluid mechanics. I will discuss the importance of entrance length for calculation and design. One example I found was for accounting for entrance length when positioning flow instrumentation. I will also explore the circumstances when entrance length is relevant to calculations. Finally, entrance length will be given context in it's relationship to other fluid mechanics topics such as Reynolds Number, Prandtl Number, Nusselt Number, Turbulence, head loss, velocity profiles, and others.
• Compile a list of relevant, reliable books, journal articles, or other sources. Post that bibliography to the talk page of the article you'll be working on, and in your sandbox. Make sure to check in on the Talk page to see if anyone has advice on your bibliography. Potential Sources
• 1. 1924-, Stewart, Warren E.,; 1925-, Lightfoot, Edwin N., (2002-01-01). Transport phenomena. J. Wiley. ISBN 0471410772. OCLC 46456316.
• 2. L., Bergman, T.; P., Incropera, Frank (2011-01-01). Fundamentals of heat and mass transfer. Wiley. ISBN 9780470501979. OCLC 713621645.
• 3. 1961-, Lienhard, John H., (2011-01-01). A heat transfer textbook. Dover Publications. ISBN 9780486479316. OCLC 853622802.
• 4. M., Cimbala, John (2006-01-01). Fluid mechanics : fundamentals and applications. McGraw-HillHigher Education. ISBN 0072472367. OCLC 56481360.
• 5. 1939-, Okiishi, T. H. (Theodore Hisao),; W., Huebsch, Wade; 1959-, Rothmayer, Alric P.,. Fundamentals of fluid mechanics. John Wiley & Sons, Inc. ISBN 1118399714. OCLC 781279071.
• 6. M., Cohen, Ira (2008-01-01). Fluid mechanics. Academic Press. ISBN 9780123737359. OCLC 647911370.
• 7. Taher., Schobeiri, Mohammed (2010-01-01). Fluid Mechanics for Engineers A Graduate Textbook. Springer Berlin. ISBN 3642115934. OCLC 873659245.
• 8. Marghitu, Dan (2001). Mechanical Engineer's Handbook. Elsevier

Peer Review

Hello Krbuw,

Good start on your article! I like that the language is encyclopedia and objective. Overall, the spelling, grammar, and sentence structures look sound, and I like that the lead section has concise sentences (for example, the definition of entrance length). I would expand on your subheading "Thermal Entrance Length" (cover any content gaps) and define the variables in that equation. Also, I would find more sources for the content under "Average velocity," "Entry Length for Pipes with Noncircular Crossections," "Shear Stress," and "Boundary layers."

MissAndrea (talk) 04:09, 14 February 2017 (UTC)

Peer Review Response

Thanks for the review MissAndrea. You can see the article on the Wikipedia main page. I'm filling in the content gaps and improving the citations for existing content. I'm trying my best on the lead page to keep the definition of entrance length general. Most sources seem to focus on hydrodynamic entrance length. Much of the content for the hydrodynamic entrance length came from an author from many years ago and is based off of one textbook. I'll cite the claims that are made and add additional sources. The graphics are from another Wikipedian, but the info they are based on appears to come from Fluid mechanics : fundamentals and application, I'm currently researching and writing about applications. It is difficult because entrance length/entry region is rarely mentioned by name. Looking into wind tunnel design and heat transfer applications. Krbuw (talk) 02:45, 23 February 2017 (UTC)