# Design Equations for Laminar and Turbulent Flow in Pipes

Dimensions of standard steel pipe

Since the purpose with a pipe is the transport of fluids like water, oil and many other products, the most import pipe property is the capacity, or in reality, the inside diameter of the pipe.

The nominal diameter of a pipe is therefore related to the inside diameter.

Appendix A.5-1

If we take a look at : NPS ( nominal pipe size, nominal pipe thread straight) 2” pipe schedule 40 : inside diameter is 2.067″. And wall thickness is 0.154 in. NPS 2” and schedule 80 pipe has 1.939“ inside diameter and wall thickness is 0.218”

Both inside diameters are close to 2″. The outside diameters for both schedules are 2.375″.

Since the outside diameter of a single nominal pipe size is kept constant, the inside diameter of a pipe will depend on the “schedule”, or the thickness, of the pipe. (Advantage…..same size fittings are used)

The schedule and the actual thickness of a pipe will vary with size of pipe.

It is common to identify pipes in inches by using NPS or “Nominal Pipe Size”.

The metric equivalent is called DN or “diametre nominel”. The metric designations conform to International Standards Organization (ISO) usage and apply to all plumbing, natural gas, heating oil, and miscellaneous piping used in buildings.

Tubes

The nominal dimensions of tubes are based on the outside diameter. A.5-2

The inside diameter of a tube will depend on the thickness of the tube. The thickness is often specified as a gauge.(BWG number)

The tolerances are higher with tubes compared to pipes. Tubes are often more expensive to produce than pipes.

For all pipe sizes the outside diameter (O.D.) remains relatively constant. The variations in wall thickness affects only the inside diameter (I.D.).

Steady state laminar flow in a pipe for Newtonian fluid: Hagen-Poiseuille equation; Pressure loss due to friction

vave=ΔPf (D2) /32 ΔL μ

ΔPf= 32μvΔL/(D2) eq 1

Ff= ΔPf/ρ Mechanical energy loss due to friction and part of ∑F.

Fanning friction factor, f: drag force per wetted surface unit area (ts at surface) divided by the product of density times velocity head or ½ρv2

ts =ΔPfπR2 (force)/2πRΔL( wetted surface area)

f= ts /½ρv2= (ΔPfπR2 /2πRΔL)/½ρv2

ΔPf=4fρ*(ΔL/D)*v2/2 eq 2

Ff= ΔPf/ρ=4f*(ΔL/D)*v2/2 (Design equation: mechanical energy loss due to friction in straight pipe laminar flow)

ΔPf/ρg=Ff/g=hL (Head loss)( In the analysis of piping systems, pressure losses are commonly expressed in terms of the equivalent fluid column height, called the head loss hL) equate eq 1 and eq 2

Then 32μvΔL/(D2)=4fρ*(ΔL/D)*v2/2

f=16μ/Dvρ=16/NRe for laminar flow where NRe<2100

Fanning friction factor, ( Used in Geankoplis) named after the American engineer John Fanning,(1837–1911)] In some other texts( Like Çengels) the Darcy–Weisbach friction factor, named after the Frenchman Henry Darcy (1803–1858) and the German Julius Weisbach (1806–1871) is used.

Only difference is

fdarcy=4*ffanning

Also keep in mind that

Ff= ΔPf/ρ=4ffanning*(ΔL/D)*v2/2

Ff= ΔPf/ρ=fdarcy*(ΔL/D)*v2/2

Turbulent flow in pipe

Unlike laminar flow, the expressions for the velocity profile in a turbulent flow are based on both analysis and measurements, and thus they are semi-empirical in nature with constants determined from experimental data.

the velocity profile is parabolic in laminar flow but is much fuller in turbulent flow, with a sharp drop near the pipe wall.

Turbulent flow along a wall can be considered to consist of four regions;

The very thin layer next to the wall where viscous effects are dominant is the viscous (or laminar or linear or wall) sublayer.( The velocity profile in this layer is very nearly linear, and the flow is streamlined.)

# Next to the viscous sublayer is the buffer layer, in which turbulent effects are becoming significant, but the flow is still dominated by viscous effects.

Above the buffer layer is the overlap (or transition) layer, also called the inertial sublayer, in which the turbulent effects are much more significant, but still not dominant.

Above that is the outer (or turbulent) layer in the remaining part of the flow in which turbulent effects dominate over molecular diffusion (viscous) effects

Flow characteristics are quite different in different regions, and thus it is difficult to come up with an analytic relation for the velocity profile for the entire flow as we did for laminar flow.

The best approach in the turbulent case turns out to be to identify the key variables and functional forms using dimensional analysis, and then to use experimental data to determine the numerical values of any constants

Despite the small thickness of the viscous sublayer (usually much less than 1 percent of the pipe diameter), the characteristics of the flow in this layer are very important since they set the stage for flow in the rest of the pipe.

Any irregularity or roughness on the surface disturbs this layer and affects the flow.

Therefore, unlike laminar flow, the friction factor in turbulent flow is a strong function of surface roughness

It should be kept in mind that roughness is a relative concept, and it has significance when its height є is comparable to the thickness of the laminar sublayer (which is a function of the Reynolds number).

All materials appear “rough” under a microscope with sufficient magnification

In fluid mechanics, a surface is characterized as being rough when the hills of roughness protrude out of the laminar sublayer.

A surface is said to be smooth when the sublayer submerges the roughness elements.

Glass and plastic surfaces are generally considered to be hydrodynamically smooth

The friction factor in fully developed turbulent pipe flow depends on the Reynolds number and the relative roughness є /D, which is the ratio of the mean height of roughness of the pipe to the pipe diameter. (Prandtl 1933)

f=f(NRe, є/D)

In 1939, C. F. Colebrook combined the available data for transition and turbulent flow in smooth as well as rough pipes into the following implicit relation known as the Colebrook equation:

This equation was plotted in the early 1940s into the now famous Moody chart, named after L. F. Moody; It presents the friction factors for pipe flow as a function of the Reynolds number and є/D over a wide range.

It is probably one of the most widely accepted and used charts in engineering.

Although it is developed for circular pipes, it can also be used for noncircular pipes by replacing the diameter by the hydraulic diameter

Commercially available pipes differ from those used in the experiments in that the roughness of pipes in the market is not uniform, and it is difficult to give a precise description of it.

these values are for new pipes, and the relative roughness of pipes may increase with use as a result of corrosion, scale buildup, and precipitation. (may increase by a factor of 5 to 10) Actual operating conditions must be considered in the design of piping systems.

Also, the Moody chart and its equivalent Colebrook equation involve several uncertainties (the roughness size, experimental error, curve fitting of data, etc.), and thus the results obtained should not be treated as “exact.” It is usually considered to be accurate to % 5 percent over the entire range in the figure.

The Colebrook equation is implicit in f, and thus the determination of the friction factor requires some iteration unless an equation solver such as EES is used.

An approximate explicit relation for f was given by S. E. Haaland in 1983 as

The results obtained from this relation are within 2 percent of those obtained from the Colebrook equation. ( get a value from Haaland and use in Colebrook as starting value for the iteration)

For laminar flow, the friction factor decreases with increasing Reynolds number, and it is independent of surface roughness.

The friction factor is a minimum for a smooth pipe (but still not zero because of the no-slip condition) and increases with roughness

The transition region from the laminar to turbulent regime (2100 The flow in this region may be laminar or turbulent, depending on flow disturbances, or it may alternate between laminar and turbulent, and thus the friction factor may also alternate between the values for laminar and turbulent flow. The data in this range are the least reliable.

At small relative roughnesses, the friction factor increases in the transition region and approaches the value for smooth pipes.

At very large Reynolds numbers the friction factor curves are nearly horizontal, and thus the friction factors are independent of the Reynolds number

The flow in that region is called fully rough flow, or completely (or fully) turbulent flow. This is because the thickness of the laminar sublayer decreases with increasing Reynolds number, and it becomes so thin that the surface roughness protrudes into the flow.

The Colebrook equation in the completely turbulent zone (Re → ∞) reduces to the von Karman equation expressed as which is explicit in f

Types of Fluid Flow Problems

In the design and analysis of piping systems that

involve the use of the Moody chart (or the Colebrook

equation), we usually encounter three types of problems(the fluid and the roughness of the pipe are assumed to be specified in all cases)

ΔPf=4ffanning* ρ(ΔL/D)*v2/2

1- First type: straightforward

and can be solved directly

by using the Moody chart.

2- second type

ΔPf=4ffanning* ρ(ΔL/D)*v2/2

1-Assume NRe is very large so f=f(ϵ/D) only

2-Find fassumed for a given ϵ/D from Moody’s chart

3- calculate v from design equation

4- calculate NRe and find fcalculated from Moody’s Chart

5- check if fassumed = fcalculated if not

6- use fcalculated as fassumed and start from 3rd step and continue until they are equal

3-third type

ΔPf=4ffanning* ρ(ΔL/D)*v2/2

third type: ( D is unknown ) Reynolds number and the relative roughness cannot be calculated.

1-Assume D

2-calculate v ( from vol flow rate) , NRe and ϵ/D then from Moody’s Chart find f

3-use design equation to calculate Dcalculated.

4-Check if Dassumed is equal to Dcalculated .

5- If not use Dcalculated as Dassumed for second trial and start from 1

# Pressure drop and friction in flow of gases

Methods described for turbulent and laminar flow of incompressible fluids can be used with gases with average density

( arithmatic mean of density) if density change ( or pressure change do not exceed 10%)

Effect of heat transfer on friction factor

Friction factor f in fig 2.10-3 is given for isothermal flow( ie. No heat transfer)

if fluid is heated or cooled, μ is affected significantly

Method of Sieder and Tate can be used for correction

1-calculate mean bulk temp ta=(tin+tout)/2

2-Calculate NRe using μa at ta and get f from fig 2.10-3

3-determine μw at tw ( tube wall temp)

4-calculate ψ for appropriate case

Ψ=( µa/µw)0.17 heating NRe>2100

Ψ=(µa/µw)0.11 cooling NRe>2100

Ψ=(µa/µw)0.38 heating NRe<2100

Ψ=(µa/µw)0.23 cooling NRe< 2100

5- fcorrected=f/ Ψ

if heating Ψ >1 then f decreases

if cooling Ψ < 1 then f increases

Friction losses in expansion, contraction and pipe fittings

In pipe line, in addition to friction losses in straight pipe segments which calculated using Fanning ( or Darcys) friction factor and design equation, if the velocity of the fluid is changed in direction or magnitude, additional friction losses occur.

Sudden enlargement losses

If the cossection of the pipe enlarges suddenly, it results in additional energy losses due to eddies formed by the jet expanding in the enlarged section.

Sudden contraction losses

Losses in fittings and valve

hf=Kfv12/2α

Kf: loss factor for fittings and valves

(Table 2.2.10-1 Kf and Le/D for turbulent flow)

( Table 2.10-2 Kf for laminar and turbulent flow)

There is considerable uncertainty in these values since the loss coefficients, in general, vary with the pipe diameter, the surface roughness, the Reynolds number, and the

details of the design.

The loss coefficients of two seemingly identical valves by two different manufacturers, for example, can differ by a factor of 2 or more.

Therefore, the particular manufacturer’s data should be consulted in the final design of piping systems rather than relying on the representative values in handbook

Equivalent lenght method

Effect of Flushing on Flow Rate from a Shower

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