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e layers of fluid near the pipe surface this develops a velocity gradient across the cross section of the pipe 5 boundary layer edit the layer in which the shearing viscous forces are significant is called the boundary layer 6 this boundary layer is a hypothetical concept it divides the flow in pipe into two regions 6 boundary layer region the region in which viscous effects and the velocity changes are significant 6 the irrotational core flow region the region in which viscous effects and velocity changes are negligible also known as the inviscid core 2 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 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 6 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 laminar flow u r x x 0 u u r displaystyle frac partial u r x partial x 0 quad rightarrow u u r 6 where x displaystyle x is in the flow direction the developing velocity profile of a fluid entering a pipe 7 shear stress edit in the hydrodynamic entrance region the wall shear stress τ w displaystyle tau _ w is highest at the pipe inlet where the boundary layer thickness is the smallest shear stress decreases along the flow direction 6 that is why the pressure 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 6 in a fully developed region the pressure gradient and the shear stress in flow are in balance 6 variation of shear stress with distance from the entry point 7 calculating hydrodynamic entrance length edit 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 l h l a m i n a r 0 0575 r e d d displaystyle l_ h laminar 0 0575re_ d d 2 where r e displaystyle r_ e is the reynolds number and d displaystyle d is the diameter of the pipe but in the case of turbulent flow l h t u r b u l e n t 1 359 d r e d 1 4 displaystyle l_ h turbulent 1 359d re_ d 1 4 8 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 t u r b u l e n t 10 d displaystyle l_ h turbulent approx 10d 6 other authors give much longer entrance length e g nikuradse recommends 40 d displaystyle 40d 9 and lien et al recommend 150 d displaystyle 150d for high reynolds flows 10 entry length for pipes with non circular cross sections edit 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 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 d h 4 a p displaystyle d_ h frac 4a p 6 where a displaystyle a is the area of cross section and p displaystyle p is the perimeter of the wet part of the pipe average velocity of fully developed flow edit 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 6 the velocity as the function of radius comes out to be u r r 2 4 μ d p d x 1 r 2 r 2 displaystyle u r frac r 2 4 mu frac dp dx left 1 frac r 2 r 2 right 6 where d p d x textstyle frac mathrm d p mathrm d x is constant by definition of average velocity is given by v a v g u d a a c displaystyle v_ avg frac int u mathrm d a a_ c where a c displaystyle a_ c is cross sectional area thus v a v g 2 r 2 0 r u r r d r 2 r 2 0 r r 2 4 μ d p d x 1 r 2 r 2 r d r r 2 8 μ d p d x displaystyle begin aligned 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 end aligned 6 for fully developed flow the maximum velocity will be at r 0 displaystyle r 0 thus u m a x 2 v a v g displaystyle u_ max 2v_ avg 6 thermal entrance length edit 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 2 overview edit 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 is a fixed value then the bulk temperature of the fluid steadily increases or decreases respectively 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 11 this distance is defined as the thermal entrance length which is important for engineers to design efficient heat transfer processes laminar flow edit for laminar flow the thermal entrance length is a function of pipe diameter and the dimensionless reynolds number and prandtl number 2 x f d t d l a m i n a r 0 05 r e d p r displaystyle left frac x_ fd t d right _ laminar approx 0 05re_ d pr 2 where r e d displaystyle re_ d is the reynolds number based on the pipe diameter and p r displaystyle pr is the prandtl number the prandtl number modifies the hydrodynamic entrance length to determine thermal entrance length the prandtl number is the dimensionless number for the ratio of momentum diffusivity to thermal diffusivity 5 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 12 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 2 x f d t d t u r b u l e n t 10 displaystyle left frac x_ fd t d right _ turbulent approx 10 2 where x f d t displaystyle x_ fd t is the thermal entrance length and d displaystyle d is the pipe inner diameter heat transfer edit 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 2 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 13 constant temperature conditions may be produced by a phase transition such as condensation of saturated steam on a pipe surface 14 newtons law of cooling describes convection the main form of heat transport between the fluid and the pipe q s h t s t m displaystyle q _ s h t_ s t_ m 2 where q s displaystyle q _ s is the heat flux into the fluid h displaystyle h is the convection coefficient t s displaystyle t_ s is the surface temperature and t m displaystyle t_ m is the mean stream temperature constant surface heat flux result in t s t m displaystyle t_ s t_ m becoming a constant as the flow develops and constant surface temperature results in t s t m displaystyle t_ s t_ m approaching zero 2 thermally fully developed flow edit unlike hydrodynamic developed flow a constant profile shape is used to define thermally fully developed flow because temperature continually approaches ambient temperature 2 dimensionless analysis of change in profile shape defines when a flow is thermally fully developed requirement for thermally fully developed flow x t s t t s t m f d t 0 displaystyle frac partial partial x biggl frac t_ s t t_ s t_ m biggr _ fd t 0 2 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 2 concentration entrance length edit the 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 or by experimental techniques 15 the schmidt number describes the ratio of momentum diffusivity to mass diffusivity 2 x f d c 0 05 d r e d s c displaystyle x_ fd c approx 0 05dre_ d sc 2 where x f d c displaystyle x_ fd c is the concentration entrance length d displaystyle d is the pipe inner diameter r e d displaystyle re_ d is the reynolds number based on the pipe diameter and s c displaystyle sc is the schmidt number applications edit 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 edit illustration of a venturi flow meter an example of a differential pressure flow meter 16 many types of flow instrumentation such as flow meters require a fully developed flow to function properly 3 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 17 wind tunnels edit 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 18 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 19 exit length edit similar to the development of flow at the entrance of the 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 20 hydraulic exit length for laminar flows may be approximated as 20 x d l a m 1 2 low r e 0 r e 100 displaystyle left frac x d right _ lam approx begin cases frac 1 2 text low re 0 re 100 end cases where x displaystyle x is the exit length d displaystyle d is the pipe inner diameter and r e displaystyle re is the reynolds number see also edit fluid dynamics heat transfer laminar flow thermal entrance length turbulent flow viscosity references edit 1 2 3 es162_08_notes02a_flow_in_pipes_changtamu pdf 1st ed cambridge j r rice 2017 print 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 bergman t l incropera frank p 2011 01 01 fundamentals of heat and mass transfer wiley isbn 9780470501979 oclc 713621645 1 2 marghitu dan 2001 mechanical engineer s handbook elsevier pp section 402 1 page 6 via knovel stewart warren e lightfoot edwin n 2007 01 01 transport phenomena j wiley isbn 978 0471410775 oclc 77079936 1 2 white frank m 2006 01 01 viscous fluid flow mcgraw hill higher education isbn 978 0072402315 oclc 693819619 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cimbala yungas a çengel john m 2006 fluid mechanics fundamentals and applications 1st ed boston mcgraw hill higher education pp 321 322 323 324 325 326 327 328 329 isbn 978 0072472363 1 2 cimbala john m 2006 01 01 fluid mechanics fundamentals and applications mcgraw hill higher education isbn 978 0072472363 oclc 834846067 çengel yunus a 2018 fluid mechanics fundamentals and applications john m cimbala fourth ed new york ny isbn 978 1 259 69653 4 oclc 958355753 cite book cs1 maint location missing publisher link 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 lien k et al 2004 the entrance length for fully developed turbulent channel flow pdf 15th australasian fluid mechanics conference mills a f 1999 heat transfer 2nd ed prentice hall upper saddle river new jersey richardson j f backhurst j r harker j h 1995 01 01 fluid flow heat transfer and mass transfer butterworth heinemann isbn 9780750644440 oclc 505103987 g1408 electric heat cable for farm and home use university of missouri extension extension missouri edu archived from the original on 2017 03 12 retrieved 2017 03 11 ashrae 2016 01 01 hvac systems and equipment 2016 si edition heating ventilating and air conditioning systems and equipment amer society of heating isbn 9781939200273 oclc 947795042 j electrochem soc 160 2013 e5 e11 doi 10 1149 2 015302jes formanavt 2009 02 07 русский устройство трубы вентури retrieved 2017 03 11 flowmeter piping requirements flow control network flow control network 2010 09 26 retrieved 2017 03 11 farell c youssef s 1996 experiments on turbulence management using screens and honeycombs asme j fluids eng talay t a january 1975 chapt3 history nasa gov retrieved 2017 03 11 1 2 perry robert h don w green 2008 01 01 perry s chemical engineers handbook mcgraw hill isbn 978 0071593137 oclc 72470708 v t e fluid mechanics fluid statics hydraulics archimedes principle fluid dynamics computational fluid dynamics aerodynamics navier stokes equations boundary layer entrance length dimensionless numbers alfvén mach archimedes atwood bagnold bejan biot bodenstein bond brinkman capillary cauchy chandrasekhar damköhler darcy dean deborah dukhin eckert ekman eötvös euler froude galilei graetz grashof...