MENG443HEATING VENTILATION AND AIR

CONDITIONING

ASSOC.PROF.DR.HASAN HACISEVKI

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Formulas to Calculate Water Flow Rates for Different Pipe Sizes

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Limitations on the Hazen Williams Formula for Water Flow Rate Calculations

The Hazen Williams formula is used for water flow rate calculations. Either S.I. units or U.S. units can be used.

The Hazen Williams formula is an empirical equation that can be used for turbulent flow of water at typical ambient temperatures.

Most practical applications of water transport in pipes are in the turbulent flow regime.

Reynolds Number and Laminar & Turbulent Flow

Pipe flow may be either laminar flow or turbulent flow. Laminar flow is characterized by low flow velocity and high viscosity. Turbulent flow is characterized by high flow velocity and low viscosity. For Reynolds number < 2300, flow is laminar. For Reynolds number > 4000, flow is turbulent.

The typical criterion for whether pipe flow is laminar or turbulent is the value of the Reynolds Number. The Reynolds number for pipe flow is defined as Re = ρDV/μ, where D is the pipe diameter, V is the average flow velocity in the pipe, ρ is the density of the flowing fluid and μ is the dynamic viscosity of the flowing fluid. Re is a dimensionless number.

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FLOW REGIME

For Re < 2300, it is laminar flow.

For Re > 4000, it is turbulent flow.

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ENTRANCE LENGTH

The Reynolds number isused to calculate theentrance length needed toreach fully developed flowfor turbulent flow or forlaminar flow in a pipe. Atthe end of the entrancelength the pipe flow entersthe fully developed flowregion, where the velocityprofile remains constant.

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Equations for analyzing pipe flow, such as theDarcy Weisbach equation for frictional headloss, often apply only to the fully developedflow portion of the pipe flow. If the total pipelength is large compared to the entrancelength, then the effect of the entrance lengthcan usually be neglected and the total pipelength can be used in calculations. If the totalpipe length is relatively short in comparisonwith the entrance length, however, then theentrance region may need to be analyzedseparately.

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ENTRANCE LENGTH CALCULATION

For turbulent flow the entrance length, Le, can be estimated from the equation:

Le/D = 4.4Re1/6.

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For Laminar Flow, the entrance length, Le, can be estimated from the equation:

Le/D = 0.06 Re.

The Friction Factor & Frictional Head Loss

Frictional head loss (or pressure drop) inpipe flow is related to the friction factorand flow velocity by the Darcy Weisbachequation. Reynolds number is needed tofind friction factor value. Fully developedturbulent flow is needed in order to usethe friction factor equation for pipe flow.

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The Darcy Weisbach Equation, which willbe discussed in this article, is commonlyused for a variety of calculations involvingfrictional head loss, pipe diameter, flowrate or velocity, and several otherparameters. The friction factor, which isused in the Darcy Weisbach equation,depends upon the Reynolds number andthe pipe roughness.

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The Darcy Weisbach Equation for Frictional Head Loss

hL = f (L/D)(V2/2g), where

hL = frictional head loss, [ft-lb/lb]

L = pipe length, [ft]

D = pipe diameter, [ft]

V = average flow velocity of fluid (= Q/A), [ft/sec]

g = acceleration due to gravity = 32.2 [ft/sec2]

f = friction factor, a dimensionless empirical factor that is a function of Reynolds Number (Re = DVρ/μ) and/or ε/D, where

ε = an empirical pipe roughness, [ft]10

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The friction factor (also sometimes called the Moodyfriction factor) can be determined for known values ofReynolds number and ε/D from empirically derivedcharts and/or equations. A commonly used chart is theMoody friction factor chart, shown in the diagram on theleft. Clicking on the chart will give you a larger scalediagram, so you can see it better.

The Friction Factor - Charts and Equations

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This chart helps to illustrate how the frictionfactor, f, depends upon Reynolds number andpipe roughness/pipe diameter (ε/D). Thestraight line at the upper left on the diagramrepresents laminar flow, in which f isindependent of ε/D and depends only on Re.The portion of the chart with horizontal lines iscalled the completely turbulent region, in whichf depends only on ε/D. For the rest of the graph,the transition region, f depends upon both Reand ε/D. The dark solid line represents "smoothpipe turbulent flow", in which f depends only onRe.

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There are equations available for friction factor for each of the four regions of the chart identified above as follows.

For laminar flow (Re < 2100):

f = 64/Re

For the completely turbulent region:

f = [1.14 + 2 log10(D/ε)]-2

For smooth pipe turbulent flow:

f = 0.316/Re1/4

For the transition region:

f = {-2 log10[(ε/D)/3.7 + 2.51/Re(f1/2)]}-2

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MOODY CHART

The Hazen Williams formula

The Hazen Williams formula is used for waterflow rate calculations. Either S.I. units or U.S.units can be used.

There are several different forms of the HazenWilliams Formula in use for water flow ratecalculations. It can be written in terms ofwater velocity or water flow rate, in terms ofpressure drop or head loss, and for severaldifferent sets of units.

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U.S. units: V = 1.318 C R0.633 S0.54, where:

V = water flow velocity in ft/sec

C = Hazen Williams coefficient, dependent on thepipe material and pipe age

R = Hydraulic radius, ft (R = cross-sectionalarea/wetted perimeter)

S = slope of energy grade line = head loss/pipelength = hL/L, which is dimensionless

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S.I. units: V = 0.85 C R0.633 S0.54, where:

V is in m/s and R is in meters

The Hazen Williams Formula is used primarily forpressure flow in pipes, for which the hydraulic radius isone fourth of the pipe diameter (R = D/4). Using thisrelationship and Q = V(πD2/4), for flow in a circularpipe, the Hazen Williams formula can be rewritten asshown in the next section.

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For flow of water under pressure in a circular pipe, the Hazen Williams formula shown above can be rewritten into the following convenient form:

in U.S. units:

Q = 193.7 C D2.63 S0.54, where:

Q = water flow rate in gal/min (gpm)

D = pipe diameter in ft

C and S are the same as above

Water Flow Rates for Pipe Sizes over a Range of Diameters with the Hazen

Williams Formula

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in S.I. units:

Q = 0.278 C D2.63 S0.54, where

Q is in m3/s and D is in meters.

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The Hazen Williams formula can also be expressed in terms of the pressure difference (ΔP) instead of head loss (hL) across the pipe length, L, using

ΔP = ρghL:

In S.I. units, a convenient form of the equation is:

Q = (3.763 x 10-6) C D2.63 (ΔP/L)0.54, where

Q is water flow rate in [m3/hr],

D is pipe diameter in [mm]

L is pipe length in [m]

ΔP is the pressure difference across pipe length, L, in [kN/m2]

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In U.S. units:

Q = 0.442 C D2.63 (ΔP/L)0.54, where

Q is water flow rate in , [gpm]

D is pipe diameter in , [inches]

L is pipe length in , [ft]

ΔP is the pressure difference across pipe length, L, in , [psi]

This is a form of the Hazen Williams formula that is convenient to use for estimating water flow rates for pipe sizes and lengths in U.S. units,

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Values for the Hazen Williams Coefficient

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Limitations on the Hazen Williams Formula for Water Flow

Rate Calculations

The Hazen Williams formula is an empiricalequation that can be used for turbulent flowof water at typical ambient temperatures. Theturbulent flow requirement is not verylimiting. Most practical applications of watertransport in pipes are in the turbulent flowregime.

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Strictly speaking, the Hazen Williamsformula applies to water at 60oF, but itworks quite well for a reasonable rangeof water temperatures above or below60oF. For fluids with viscosity differentfrom water, or for water temperaturesfar above or below 60oF, the DarcyWeisbach Equation works better thanthe Hazen Williams Formula.

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PRESSURE DROP CALCULATION

Calculate pressure drop for known flow rate

Calculate flow rate for known pressure drop

Friction factor calculation

Friction and minor losses calculation

Darcy equation based

Laminar and turbulent flow

Circular and rectangle pipe

For incompressible flow only

Calculation includes flow velocity

Pipe flow calculators

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PIPE DIAMETER CALCULATION

Calculate pipe diameter for known flow rate and velocity

Calculate flow velocity for known pipe diameter and flow rate

For liquids and ideal gases

Convert from volumetric to mass flow rate

Calculate volumetric flow rate of ideal gas at different conditions of pressure and temperature

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CONTROL VALVE SIZING CALCULATION

Select liquid control valve size

Calculate flow coefficient Cv or Kv of liquid control valve for known flow capacity and pressure difference

Calculate maximum flow capacity of control valve for known flow coefficient Cv or Kv and pressure difference

Calculate pressure difference for know flow coefficient Cv or Kv and flow capacity

Convert from volumetric to mass flow rate

Input either diameter or flow velocity

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Resistance coefficient K and equivalent length L/D calculation for valves and fittings

Calculator can be used for resistance coefficient (orresistance factor) K and equivalent length L/Dcalculation for valves and fittings. Resistancecoefficient K and equivalent length LD(head loss) andsquare velocity of fluid flowing through valves andfittings like elbow, bend, reducer, tee, pipe entranceand exit, as follows:

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COLEBROOK EQUATIION

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Where:

f: friction factor [-]

ε: roughness [m]

Dh: hydraulic diameter - for pipes is equal to the internal diameter [m]

Re: Reynolds number [-]

For laminar flow (R < 2300 inpipes), f can be deduced analytically.The answer is,

For turbulent flow (R > 4000 in pipes), f is determined from experimental curve fits. One such fit is provided by Colebrook,

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USEFUL FORMULAS

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STEADY FLOW ENERGY EQUATION WITH PUMP AND TURBINE APPLICATIONS

TYPES OF FLUID PROBLEMS

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Calculate the Reynolds number

Determine if the flow is Laminar or Turbulent

Calculate the friction factor for either Laminar flow or Turbulent flow

Calculate the fluid head resistance to overcome the flow through the pipe work

Determine the ‘K’ factors for the fittings within the pipe work layout

Calculate the fluid head resistance to overcome the flow through the fittings

Determine which lengths & components within the pipe work layout are significant in establishing the maximum fluid head to be considered (branch lines may be important).

Calculating the fluid head

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