# Fluid Flow in Pipes The Darcy-Weisbach Equation and the Fluid-Flow Calc v1.0 Tool for Engineers

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1 Fluid Flow in Pipes The Darcy-Weisbach Equation and the Fluid-Flow Calc v1.0 Tool for Engineers by Lawrence H. Smith, Sr., P.E. Copyright 2011 Lawrence H. Smith, Sr., P.E.

2 Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 2 of 36

3 A. Introduction This course is intended for mechanical and civil engineers who want to learn more about Sizing Piping Systems with the Darcy-Weisbach equation. Design issues covered include, understanding the equations, purpose and history, sizing residential, commercial and industrial piping systems. This course qualifies for four (4) hours Professional Engineering CEU credits. Upon completion of this course, you will have a thorough understanding of the design aspects related to Sizing Piping Systems with the Darcy- Weisbach equation and others relating to its application. B. General Because of the great variety of fluids handled in modern industrial processes (heating and air conditioning, piping systems, etc.) a single equation used for any flow in piping systems would obviously have an advantage over other equations. Such an equation is the Darcy-Weisbach equation for liquids only in this course. The equation can be derived rationally by means of dimensional analysis; however, one variable in the equation, the friction factor, must be determined experimentally. Today with the invention of the calculator, computers and software and spreadsheets, etc. this is not a problem. This equation has a wide application in the field of fluid mechanics. The Fluid Flow Calc tool was developed using the Darcy-Weisbach and other equations and utilizes the following variables: = Density of the fluid, lb/ cu ft = Viscosity of the fluid, cp Q = Quantity of flow, gpm, gph, cfs and B V = Velocity of flow, mean velocity of the flow in the pipe, fps D = Pipe inside diameter, in and ft L = Fitting total K coeff s = Fitting total L/D values = Length of pipe, ft P = Pressure drop, feet of fluid and psi in fluid R = Roughness factor,, ft The input looks at the data and tracts it to determine proper input. Five types of calculations can be performed as follows: 1. Sizing pipe for Pressure Drop, requires,, Q, D, L and R 2. Sizing pipe for Maximum Velocity for Flow, requires,, V, D, L and R Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 3 of 36

4 3. Sizing pipe for Maximum Velocity for Pipe Diameter, requires,, Q, V, L and R 4. Sizing pipe for Maximum Pressure Drop, requires,, Q, L, P and R 5. Sizing pipe for Maximum Flow, requires,, D, L and R The input tracks and displays one of the following messages: 1. Too Much Data 2. Default Used 3. Need to Check Your Input If the data input is entered correctly, one of the following messages will be displayed. 1. Sizing pipe for Pressure Drop 2. Sizing pipe for Maximum Velocity for Flow 3. Sizing pipe for Maximum Velocity for Pipe Diameter 4. Sizing pipe for Maximum Pressure Drop 5. Sizing pipe for Maximum Flow There are three buttons on the screen, one for Default Density data, one for Viscosity, and one for the Roughness factor. Density is lb/cu ft. Viscosity is 1.1 at 60 o F and Roughness factor Note: The Microsoft Excel Spreadsheet has three (3) Macros for the three Default Buttons and the following security screen will appear. The Enable Macros button should be pressed to operate correctly if the Security Level is Set above (Low). C. Units The following units will be used during this course: gpm to cfs = gpm / gph to gpm = gph / 60 cfs to gpm = cfs x gpm to B/h = gpm x Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 4 of 36

5 gph to B/h = gph x 42 D. Physical Properties of Fluids The solution to any fluid problem requires an understanding and knowledge of the physical properties of the fluid under consideration. Therefore, an accurate value for the properties affecting the fluid is essential to problem solving. Numerous publications have tables and charts for this purpose. The following is the fluid physical properties that we will be using throughout this course: 1. Viscosity, cp 2. Weight density, lb/cu ft Viscosity: Viscosity expresses the readiness with which a fluid flows when it is acted upon by an external force. The coefficient of absolute viscosity or, simply, the absolute viscosity of a fluid, is a measure of its resistance of a fluid which is being deformed by either shear stress or tensile stress. In everyday terms (and for fluids only), viscosity is thickness or internal friction. Thus, water is thin, having a lower viscosity, while honey is thick, having higher viscosity. Put simply, the less viscous the fluid is, the greater its ease of movement. Although most fluids are predictable in their viscosity, in some, the viscosity depends upon the previous working of the fluid. Wood pulp slurries and ketchup are examples of fluids possessing such thixotropic properties of viscosity. Note: Thixotropic is the property of certain gels or fluids that are thick (viscous) under normal conditions, but flow (become thin, less viscous) over time when shaken, agitated, or otherwise stressed. Considerable confusion exists concerning the units used to express viscosity; therefore, proper units must be employed whenever substituting values of viscosity into formulas. In the CGS or cgs (centimeter-gram-second) or metric system (proposal made in 1832 by the German mathematician Carl Friedrich Gauss), the unit of absolute viscosity is the poise (it is named after Jean Louis Marie Poiseuille) and is often used with the (metric prefix centi-) which is equal to 100 centipoises. The centipoise is properly abbreviated cp, but the alternative abbreviations cps, cp and cps are also commonly seen. The poise has the dimensions of dyne seconds per square centimeter or of grams per centimeter second. It is believed that less confusion concerning units will prevail if the centipoises is used exclusively as the unit of viscosity. For this reason and since most handbooks and tables follow the same procedure, all viscosity data in this course is expressed in centipoises. Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 5 of 36

6 The English units most commonly employed are slug per foot second or pound force second per square foot ; however, pound mass per feet second or poundal second per square foot may also be encountered. The viscosity of water at a temperature of 68 o F is: = 1 centipoises = 0.01 poise = 0.01 gram per cm second = 0.01 dyne second per sq cm e = pound mass per foot second poundal second per square foot ' e = slug per foot second pound force second per square ft See Table 3, in Appendix A for some viscosity of liquid s. Kinematic viscosity is the ratio of the absolute viscosity to the mass density. In the metric system the unit of kinematic viscosity is the stoke. The stoke has dimensions of square centimeters per second and is equivalent to 100 centistokes centistoke s = = ' S Specific gravity: the specific gravity S, in the foregoing formula is based upon water at a temperature of 39.2 o F (4 o C), whereas specific gravity used throughout this course is based upon water at 60 o F. In the English system, kinematic viscosity has dimensions of square feet per second. Weight density, specific volume, and specific gravity: The weight density or specific weight of a substance is its weight per unit volume. In the English system of units, this is expressed in pounds per cubic foot and the symbol designation used in the course is (Rho). In the metric system, the unit is grams per cubic centimeter and the symbol designation used is centimeter and the symbol designation used is (Rho prime). The specific volume V, being the reciprocal of the weight density, is expressed in the English system as the number of cubic feet of space occupied by one pound of the substance, this is: Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 6 of 36

7 V 1 Computation in the metric system is not commonly referred to in terms of specific volume; however, the number of cubic centimeters per gram of a substance can readily be expressed as the reciprocal of the weight density, that is: V 1 ' The variations in weight density as well as other properties of water with changes in temperature are shown in Table 1, in Appendix A. The weight densities of other common liquids are shown in Table 2, in Appendix A. Unless very high temperatures are considered, the effect of pressure on the weight of liquids is of no practical importance in flow problems. Specific gravity is a relative measure of weight density. Since pressure has an insignificant effect upon the weight density of liquids, temperature is the only condition that must be considered in designating the basis for specific gravity. The specific gravity of a liquid is its weight density at 60 o F (unless otherwise specified) to that of water at standard temperature of 60 o F. S p2 p1 Where p1 = any liquid at 60 o F unless otherwise specified p2 = water at 60 o F The specific gravity is measured with the hydrometer and there are three scales used commonly in this country. The API scale which is used for oil. The Baume (Be) scales, one for liquids heavier than water and one for liquids lighter than water. The relationship between the scales and specific gravity are: For oils: Degrees API = (141.5 / G) G = / ( Degrees API) Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 7 of 36

8 For liquids lighter than water: Specific Gravities at 60 o F = 140 / (160 Deg. Be) 60 o F For liquids heavier than water: Specific Gravities at 60 o F = 145 / (145 Deg. Be) 60 o F E. Darcy-Weisbach Equation The Darcy-Weisbach equation is named after Henry Darcy of France and further refined into the form used today by Julius Weisbach of Saxony in Historically this equation arose as a variant on the Prony equation (Gaspard Clair Francois Marie Riche de Prony, ). Initially, data on the variation of f with velocity was lacking, so the Darcy-Weisbach equation was out performed at first by the empirical Prony equation in many cases. Although in the later years it was eschewed in many special case situations in favor of a variety of empirical equations valid only for certain flow regimes. Notably the Hazen-Williams equation named after (Allen Hazen and Gardner Stewart Williams) or the Manning equation (Gauckler-Manning formula) named after (Philippe Gauckler in 1867 and Robert Manning in 1890) most of which were significantly easier to use in calculation. The name of the equation through time is also curious and may be tracked in hydraulic and fluid mechanics textbooks. Early texts generally do not name the equation. Starting in the mid 20th century some authors including at least one German named it "Darcy's Equation", an obvious confusion point with "Darcy's Law". Rouse in 1946 appears to be the first to call it "Darcy-Weisbach", but that naming did not become universal until the late 1980's. It is a good enough name, but as pointed out previously, it leaves out many important contributions. From a practical standpoint, the Darcy-Weisbach equation has only become popular since the advent of the electronic calculator, computer and software like the spreadsheet. It requires a lot of number crunching compared to empirical relationships, such as the Hazen-Williams equation, which are valid over narrow ranges. However, because of its general accuracy and complete range of application, the Darcy-Weisbach equation should be considered the standard and the others should be left for the historians. A recent interesting discussion on the topic was presented by Liou (1998), Christensen (2000), Locher (2000) and Swamee (2000). Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 8 of 36

9 The Darcy-Weisbach equation is considered the best empirical relation for pipe-flow resistance. In terms of head units it is: Pipe Friction 2 L v h f f D 2 g where h f = friction resistance in feet of fluid f = friction factor, dimensionless L = length of pipe, (in feet) D = internal diameter of pipe, (in feet) v = average velocity in feet per second, (in fps) g = acceleration due to gravity in feet per second per second, feet/second/second The Darcy-Weisbach equation can be written in terms of pressure loss: L v2 h p f D 2g where Velocity = fluid density at mean temperature, (in lb/cu ft) The following are equations used to calculate mean velocity of flow in several units: v = A q q = d Q = d where B v = d A = cross section area of pipe in square feet B = rate of flow in barrels (42 gallon) per hour (B) Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 9 of 36

10 Q = rate of flow in gallons per minute (gpm) q = rate of flow in cubic feet per second (cfs) v = mean velocity of flow, in feet per second (fps) V = specific volume of fluid, in cubic feet per pound d = inside diameter of pipe, (in inches) Relative Pipe Roughness Factor /d = /d where = absolute roughness of pipe surfaces d = inside diameter of pipe, (in inches) The relative pipe roughness is the ratio of the pipe surface roughness, to its diameter, d, or /d. See Table 4, in Appendix A for some common pipe relative roughness factors. Reynolds Number dv Re where d = inside diameter of pipe, (in inches) v = average velocity in feet per second, (in fps) = fluid density at mean temperature, (in lb/cu ft) = viscosity, (in cp) Friction Factor The Darcy-Weisbach formula can be rationally derived by dimensional analysis, with the exception of the friction factor f, which must be determined experimentally. The friction factor for laminar flow condition (Re < 2000) is a function of Reynolds number only: whereas, for turbulent flow (Re > 4000), is also a function of the character of pipe walls. A region known as the "critical zone" occurs between Reynolds number of approximately 2000 and In this region, the flow may be either laminar or turbulent depending upon several factors; these include changes in section or direction of flow and obstructions, such as valves in the upstream piping. The friction factor in the region is Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 10 of 36

11 indeterminate and has lower limits on laminar flow and upper limits based on turbulent flow conditions. At Reynolds numbers above approximately 4000, flow conditions become more stable and a definite friction factor can be established. This is important because it enables the spreadsheet to determine the flow characteristics of any fluid flowing in a pipe providing the viscosity and weight density at flowing conditions are known. If the flow is laminar (Re < 2000); the friction factor may be determined from the following: f 64 Re When the flow is turbulent (Re > 4000), the friction factor depends not only upon the Reynolds number but also upon the relative roughness, /D the roughness of the pipe wall, as compared to the diameter of the pipe D. For very smooth pipes such as drawn brass tubing, the friction factor decreases more rapidly with increasing Reynolds number than for pipe with comparatively rough walls. Since the character of the internal surface of commercial pipe is practically independent of the diameter, the roughness of the walls has a greater effect on the friction factor in the small sizes. Consequently, pipe of small diameters will approach the very rough conditions and, in general, will have higher friction factors than large pipes of the same material The most useful and widely accepted data of friction factors for use with the Darcy- Weisbach equation was presented by L. F. Moody. Professor Moody improved upon the well established Piggott and Kemler friction factor diagram by incorporating more recent investigations and developments of many outstanding scientists. See the Moody Friction Factors Diagram in Appendix A. For turbulent flow smooth pipe with the Blasius equation f for Re up to Re for 105 < Re < 3 X 106 at least Re = For Fully Rough Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 11 of 36

12 1 / f = log( D / ) Transition between this case and smooth-wall friction factor is represented by Colebrook s natural function: 1 / f = log (D/ ) 2 log Re / D f E. Associated Items Which Have Pressure Losses Valves and Fitting The pressure drop through valves and fittings are usually handled by finding the length of straight pipe of the same diameter that would have the same pressure drop as the appropriate fitting at the same flow rate. This length is referred to as the equivalent length and is added to the actual pipe length in using the Darcy-Weisbach equation. So-called minor losses occurring at the entrance to pipe, change in pipe diameter or other changes in shape are usually expressed by the following: 2 v h f K 2g therefore L K f for turbulent flow D where h f = frictional resistance in feet of fluid v = average velocity in feet/second in a pipe of corresponding diameter g = feet/second/second K = resistance coefficient for valve or fitting L/D = is the equivalent length in pipe diameters of straight pipe which will cause the same pressure drop as the valve or fitting under the same flow conditions Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 12 of 36

13 The K is a factor that varies for each case. Values of K are available in fluid mechanics texts as well as in handbooks as monographs that allow determination of equivalent length of pipe and fittings. The K resistance coefficient for a given line of valves or fittings, tend to vary with the size as does the friction factor f for straight pipe, and that the equivalent length L/D tend toward a constant for the various sizes of a given line of valves or fittings. L/D Values for Fitting and Valves The L/D values are available in fluid mechanics texts as well as in handbooks in Table form based on a particular fitting type. In the flow range of complete turbulence as defined by the friction factor chart, the K coefficient for a given size and the L/D is constant and that K varies in the same manner as the friction factor. However, since the tendency is in this direction, it is believed to provide more accurate solutions than would the assumption that K is constant for all Reynolds numbers. Therefore, within this course we will be using the fittings and valves L/D values. See Table 9 in Appendix A for typical valves and fittings L/D values and Table 8 in Appendix A for Typical Pipe Entrance and Exits K Coefficients. When we have a fitting or valve with a K coefficient the L/D would be: K L / D for turbulent flow f where f = friction factor, dimensionless K = resistance coefficient for valve or fitting L/D = is the equivalent length in pipe diameters of straight pipe which will cause the same pressure drop as the valve or fitting under the same flow conditions L D s Re 1000 L D t for laminar flow Re < 1000 where f = friction factor, dimensionless K = resistance coefficient for valve or fitting Re = Reynolds number Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 13 of 36

14 (L/D) s = (L/D) t = subscript (s) refers to the equivalent length in pipe diameters of straight pipe under laminar flow conditions where Reynolds number is less than subscript (t) refers to the equivalent length in pipe diameters determined from tests in the turbulent flow range. In some branches of the valve industry, particularly in connection with the control valves, express the capacity of a valve and the terms of flow coefficient Cv. The Cv of a valve is defined as the flow of water at 60 o F, in gallons per minute at a pressure drop of one pound per square inch d Cv = L f D 29.9d 2 K Strainer Pressure Drop Strainer pressure drop should be obtained from the manufacturer data sheets to determine proper value to use. The pressure drop depends on the type of strainer, size, flow rate, service, filtration (screen loss factor for the mesh used) and viscosity of the fluid. Increase in Friction Loss Due to Aging of Pipe The deterioration of pipes with age depends upon the chemical properties of the liquid flowing and the characteristics of the material from which the pipe is made. In general, the flow carrying capacity of a pipe line decreases with age due to roughening of the interior surface caused by corrosive products, tubercules and the like or an actual reduction in area caused by chemical deposits. The effect corresponds to a variation in friction factor due to increasing relative roughness. A wide variation in waters over the country makes it impossible for any precise estimation of this aging effect. No reputable authority will go on record to endorse friction factors for other than new pipe. This fact, however, does not eliminate the deterioration of friction factor and some means of estimation is required. Whenever records are available on the aging effects of local or similar waters it is recommended they be studied and applied as a correction to the computation of friction loss for new pipe from the Darcy-Weisbach equation or any other. This is a sound and logical approach for a specific problem. In many instances, either the economics of the project does not warrant the expense of this detailed investigation or there are no available records on local or similar waters. For Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 14 of 36

15 those occasions, Table 7 in Appendix A may be used with caution and discretion. It is based upon the best available data. It will be obvious there is no sudden increase in aging effect between 10 inch and 12-inch pipe as indicated from Table - 7. The values shown are composites of many tests grouped by the experimenter. A reasonable amount of interpretation and logic must be used in selecting and applying a multiplier for each specific problem. It must also be borne in mind that some test data on aging of pipe may vary up to fifty percent from the averages as shown in Table - 7. Therefore, based upon the above-mentioned aging information it is recommended that 15 percent be added to pipe friction loss. Recommended Water Maximum Velocities The velocities recommended for water piping depend on two conditions: 1. The service for which the pipe is to be used. 2. The effects of erosion. Table - 5 in Appendix A, lists recommended velocity ranges for different services. The design of the water piping system is limited by the maximum permissible flow velocity. The maximum values listed in Table 6 is based on established permissible sound levels of moving water and entrained air, and on the effects of erosion. Erosion in water piping systems is the impingement on the inside surface of the tube or pipe of rapidly moving water containing air bubbles, sand or other solid matter. In some cases, this may mean complete deterioration of the tube or pipe walls, particularly on the bottom surface and at the elbows. Since erosion is a function of time, water velocity, and suspended materials in the water, the selection of design water velocity is a matter of judgment. The maximum water velocities presented in Table 6 is based on many years of experience and they ensure the attainment of optimum equipment life under normal conditions. Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 15 of 36

16 Sample Problem - 1 Fluid Flow in Pipe The Darcy-Weisbach Equation The Kentucky Water Processing Plant is going to build a new two tank atmospheric system for storing water at 70 of in a Tank designated as A and then have 0.56 cfs of water pumped to the second floor to another Tank designated as B. The piping system will have Schedule S-40 pipe with a roughness factor of ,ft and the pipe is 100 feet long and is 3 inches with a inside diameter of inches. The system has a square-edged entry from Tank A, 3-90 deg elbows, 2 gate valves, and 1 check valve in the piping, and free discharge at Tank B. See Figure 1 below for the geometry of the piping system. Calculate the pump head in feet. Figure 1 The engineer for the project, used a density of lb/cu ft, 1.0 cp and L/D of 135 for the check valve, L/D of 13 for each of the gate valves, L/D of 20 for the 3-90 deg elbows which totaled 221 L/D and a K of 0.5 for the square-edged entry. He used the Fluid Flow Calc v1.0 tool to calculate the velocity, total equivalent length, pressure drop, relative roughness, Reynolds number and the friction factor. Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 16 of 36

17 Input to the Fluid Flow Calc tool Density of flow = lb/cu ft Viscosity = 1.0 cp Quantity of flow = cfs Pipe Diameter = inches Fitting Total K Coeff s = 0.50 Fitting Total L/D values = 221 Length of Pipe = 100 ft Roughness Factor = , ft Output from the Fluid Flow Calc tool Velocity = fps Total equivalent length = ft Pressure drop = feet of fluid Relative roughness = /D Reynolds number = 258,324 Re Friction factor = f Total Static head hs = ( ) = 65 Total Head Loss Th = hs + h f Th = Th = Feet of fluid Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 17 of 36

18 Sample Problem - 2 Fluid Flow in Pipe The Darcy-Weisbach Equation The Kentucky Water Processing Plant is going to pump 600 Barrels/hr of oil through a straight horizontal steel pipeline 450 feet long to an open tank. The tank is 20 feet above the pump discharge with a free discharge into the tank. The velocity shall be 7 fps maximum and with a check valve, 1 gate valve, and 3-90 deg elbows in the piping system. See Figure 2 below for the geometry of the piping system. Fluid properties are as follows: Density = 42 lb/cu ft Viscosity = 30 cp Figure 2 Determine the pipe size, pump total head with the fittings: The engineer for the project, used a schedule 40 steel pipe with, ft of , L/D of 135 for the check valve, L/D of 13 for the gate valves and L/D of 20 for the 3-90 deg elbows, which totals 208 L/D. He used the Fluid Flow Calc v.1.0 tool to calculate the maximum pipe size for the 7 fps mean velocity, Pressure drop, Relative roughness, Reynolds number and Friction factor for the above conditions. Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 18 of 36

19 Input to the Fluid Flow Calc tool Density = 42 lb/cu ft Viscosity = 30 cp Quantity of flow = 600 Barrels/hr Velocity of flow = 7.00 fps Fitting Total L/D values = 208 Length of Pipe = 450 Roughness Factor = , ft Output from the Fluid Flow Calc tool Pipe inside diameter = inches Total equivalent length = ft Pressure drop = feet of fluid Relative roughness = /D Reynolds number = 6,010 Re Friction factor = f Fluid Flow in Pipe The Darcy-Weisbach Equation The system requires a pipe larger than inches. Therefore, the engineer selected a 5-inch pipe to meet the velocity requirements. Input to the Fluid Flow Calc tool New Conditions inches in the Pipe inside diameter Output from the Fluid Flow Calc tool Velocity of flow = 6.73 fps Total equivalent length = ft Pressure drop = feet of fluid Relative roughness = /D Reynolds number = 5,896 Re Friction factor = f Total Static head hs = 20 Total Head Loss Th = hs + h f Th = Th = Feet of fluid Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 19 of 36

20 Sample Problem - 3 Fluid Flow in Pipe The Darcy-Weisbach Equation The Kentucky Water Processing Plant is going to install a new chiller to supply two cooling coils. The chiller will have a flow of 175 gpm with a pressure drop of 20 feet of fluid. Coil - A will have a flow of 100 gpm with 12 feet of fluid pressure drop and Coil B will have a flow of 75 gpm with 12 feet of fluid pressure drop. Each coil will have a control valve to control the flow with a pressure drop at the flows of 3 feet of fluid each. The piping will be schedule S-40 steel and will be designed with a maximum pressure drop of 4 feet of fluid per 100 feet. See Figure 3 below for the piping geometry and nodes. Node 0 1 from the Chiller will have 2 gate valves, 6 90 deg elbows, 1 check valve and 1 tee with flow through run and 300 feet of pipe. Node 1 2 for Coil A will have 2 gate valves, 1 control valve and 1 balancing valve, 2 - tee with flow through branch, 8 90 deg elbows and 250 feet of pipe. Node 1 2 for Coil B will have 2 gate valves, 1 control valve and 1 balancing valve, 2 - tee with flow through run, 6 90 deg elbows and 310 feet of pipe. Node 2 3 to the pump will have 2 gate valves, 1 strainer, 1 - tee with flow through run, 6 90 deg elbows and 600 feet of pipe. Determine the piping sizes and pump total head. Figure - 3 The engineer for the project, used a water density of lb/cu ft, viscosity of 1.00 cp,, ft of for the pipe, L/D of 13 for the gate valves, L/D of 18 for the balancing valves, 135 for the check valve, L/D of 20 for the 90 deg elbows, L/D of 20 for the tees with flow through run, L/D of 60 for the tees with flow through branch and 0.5 feet through the strainer. He used the Fluid Flow Calc v.1.0 tool to calculate the maximum pressure drop size for the 4 feet of fluid pressure drop for the above four-pipe section. Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 20 of 36

21 The project engineer then input the data into the Fluid Flow Calc tool for each of the Nodes to determine the pipe sizes. He entered this information into a spreadsheet as shown below. Node 0 1 Input to the Fluid Flow Calc tool Density = lb/cu ft Viscosity = 1.00 cp Quantity of flow = 175 gpm Length of Pipe = 100 ft feet of fluid = 4.00 Roughness Factor = , ft Node 0 1 Output from the Fluid Flow Calc tool Pipe inside diameter = inches Pressure drop = 4.00 feet of fluid The project engineer then input the flow quantities with each of the pipe sizes along with the fittings, L/D valves and length of pipe for each node section into the Fluid Flow Calc tool. After getting the results, he entered this data into his spreadsheet as shown below. As can be seen below, Coil A has the maximum pressure drop along with Nodes 0 1 and 2 3. The pump total head is feet of fluid. Node 0 1 Input to the Fluid Flow Calc tool Density = lb/cu ft Viscosity = 1.0 cp Quantity of flow = 175 gpm Pipe Diameter = inches Fitting Total L/D values = 301 Length of Pipe = 300 ft Roughness Factor = , ft Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 21 of 36

22 Node 0 1 Output from the Fluid Flow Calc tool Velocity = 4.41 fps Total equivalent length = ft Pressure drop = 6.98 feet of fluid Relative roughness = /D Reynolds number = 137,040 Re Friction factor = f Fluid Flow in Pipe The Darcy-Weisbach Equation Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 22 of 36

23 Sample Problem 4 The Kentucky Water Processing Plant needed a new city water distribution system. The pipeline will be 1,500 feet in length to connect the plant to the city water main. The pipeline has 2 Gate valves, 4-90 deg elbows, 1 tee with flow through branch. The total water flow will be 400 gallons per minute (gpm) for the normal operation and the Fire Department required an additional capacity of 1,500 gallons per minute. The maximum total pressure drop from the main to the plant shall not be over 22 (psi) with a maximum flow of 3,900 gpm from a flow test. The engineer for the project selected asphalt coated cast iron pipe with a roughness factor of Water density of lb/cu.ft. and cp of For the fittings, he used an L/D of 13 for the two gate valves, L/D of 20 for the 4-90 degree elbows and L/D of 60 for the tee, which totals 166 L/D. He applied a safety factor of 15% to the fluid flow. He used the Fluid Flow Calc v1.0 tool to calculate the pipe size required for the project. Input to the Fluid Flow Calc tool Density = lb/cu ft Viscosity = 1.00 cp Quantity of flow = 2,185 gpm Fitting Total L/D values = 166 Length of Pipe = 1,500 ft Pressure drop = 22 psi Roughness factor = , ft Output from the Fluid Flow Calc tool Total equivalent length = 1,633.3 (ft) Pipe inside diameter = inch Relative roughness = /D Reynolds number = 715,912 Re Friction factor = f Pressure drop = psi of fluid With the above information the engineer selected a 10-inch diameter pipe with an inside diameter of 10 inches. Again, he entered the new pipe size into the Fluid Flow Calc tool to determine the maximum flow with the limitations of 22 psi. Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 23 of 36

24 Input to the Fluid Flow Calc tool inches in the Pipe inside diameter Column Output from the Fluid Flow Calc tool Quantity of flow = 2,414 gpm Total equivalent length = 1,638.3 (ft) Relative roughness = /D Reynolds number = 761,916 Re Friction factor = f Pressure drop = psi of fluid The new flow of 2,414 gpm would allow for a plant growth capacity of only 229 gpm. Therefore, the engineer looked at a 12-inch pipe to allow for plant growth and he used the Fluid Flow tool to determine the maximum pressure drop through this pipe size and maximum flow of 3,900 gpm. Input to the Fluid Flow Calc tool Quantity of flow = 3,900 gpm Pipe inside diameter = 12.0 inches Output from the Fluid Flow Calc tool Total equivalent length = 1,666.0 (ft) Relative roughness = /D Reynolds number = 1,025,778 Re Friction factor = f Pressure drop = psi of fluid The engineer reported that a 12-inch pipe would be of adequate size to meet the current flow requirements and future growth of the plant to 1,715 gpm maximum. The project manager decided to accept the recommendations and install a 12-inch pipe line. Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 24 of 36

25 Sample Problem - 5 The Kentucky Water Processing Plant has new a sump pump with a capacity of 400 gpm with a total dynamic head of 95 feet that needs to be installed in a 12-foot deep sump and pump the discharge to a gravity sewer located 1,200 feet from the top of the sump at elevation 0 feet. The discharge pipe will have 1 check valve, 1 gate valve, degree long radius elbows and a sharp edged exit discharged into the sewer. The pipe material will be schedule 40 PVC. The velocity shall not be under 2 fps minimum and the maximum shall not be over 8 fps. See Figure 4 below for the geometry of the piping system. Fluid properties are as follows: Density = lb/cu ft Viscosity = 1.00 cp Determine the pipe size for the above velocity restrictions and be within the total dynamic head of the pump. Figure - 4 The engineer for the project used a roughness factor of 0.000,005, ft for the pipe, 135 L/D for the check valve, L/D of 13 for the gate valves and L/D of 20 for the degree elbows, which totals 348 L/D. He used a K of 1.0 for the sharp edged exit to the sewer. He applied a safety factor of 15% to the feet of fluid pressure drop. The total dynamic head less the 30 feet of static head witch equals feet of fluid pressure drop maximum (95 30) * 0.85 He used the Fluid Flow Calc v1.0 tool to calculate the required pipe size for the project velocity restrictions. Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 25 of 36

26 Input to the Fluid Flow Calc tool Density = lb/cu ft Viscosity = 1.00 cp Quantity of flow = 400 gpm Fitting total K coeff s = 1.00 Fitting Total L/D values = 348 Length of Pipe = 1,200 ft Pressure drop = feet of fluid Roughness factor = 0.000,005, ft Output from the Fluid Flow Calc tool Velocity = 8.03 fps Pipe inside diameter = inches Total equivalent length = 1,356.1 ft Pressure drop = feet of fluid Relative roughness = /D Reynolds number = 279,977 Re Friction factor = f Fluid Flow in Pipe The Darcy-Weisbach Equation With the above information the engineer selected a 5-inch diameter pipe with an inside diameter of inches. Again, he entered the new pipe size into the Fluid Flow Calc tool to determine the pressure drop. Input to the Fluid Flow Calc tool New Conditions inches in the Pipe inside diameter Output from the Fluid Flow Calc tool Velocity of flow = 6.49 fps Total equivalent length = 1,373.2 ft Pressure drop = feet of fluid Relative roughness = /D Reynolds number = 251,734 Re Friction factor = f Total Static head hs = 30 Total Pump Head Th = hs + h f Th = (32.40 *.15%) Th = Feet of fluid < 95 feet Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 26 of 36

27 Sample Problem 6 Fluid Flow in Pipe The Darcy-Weisbach Equation The Kentucky Water Processing Plant has a new 8-inch S.A.E 70 Lube Oil line that will supply 600 barrels per hour through 200 feet of Schedule 40 pipe 8-inch pipe, in which an 8-inch conventional globe valve is installed. Fluid properties are as follows: Density = 56.2 lb/cu ft Viscosity = 470 cp Determine the pressure loss through the pipe and valve. The engineer for the project used a pipe inside diameter of in, roughness factor of , ft and L/D of 340 for the globe valve and used the Fluid Flow Calc v1.0 tool to calculate the total equivalent length of pipe, relative roughness, Reynolds number, friction factor and pressure drop. Input to the Fluid Flow Calc tool Density = 56.2 lb/cu ft Viscosity = 470 cp Quantity of flow = 600 (B) Pipe Diameter = (in) Fitting Total L/D values = 340 Length of Pipe = 200 ft Roughness Factor = , ft Output from the Fluid Flow Calc tool Total equivalent length = (ft) Relative roughness = /D Reynolds number = 322 Re Friction factor = f Pressure drop = 9.69 feet of fluid Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 27 of 36

28 Sample Problem 7 Fluid Flow in Pipe The Darcy-Weisbach Equation The Kentucky Water Processing Plant has a system that needs to pump 2.5 cfs of water at 50 o F from a reservoir with an elevation of 50 feet and needs to pump the water to a reservoir with an elevation of 150 feet. The suction pipe to the pump is 8 inches diameter with a length of 1,000 feet and the discharge pipe is 6 inches diameter and is 2,000 feet long. The suction pipe has 1 - gate valve, sharp edged entrance and the discharge pipe has 1 gate valve, 1 check valve, decrees long radius elbows and sharp edged exit. The pipe material shall be clean cast iron pipe. See Figure 4 below for the geometry of the piping system. Fluid properties are as follows: Density = lb/cu ft Viscosity = 1.3 cp What is the maximum feet of fluid loss for the two pipes with and without the static head of the system? Figure - 4 The engineer for the project selected a roughness factor of for the pipe. For the fittings, he used an L/D of 13 for the two gate valves, L/D of 20 for the degree elbows, L/D of 135 for the check valve and used a K of 0.50 for the sharp edged entrance and a K of 1.00 for the sharp edged exit. He used the Fluid Flow Calc v1.0 tool to calculate the pressure drop for the two pipes. The suction pipe total K is 0.50, L/D is 13 and pipe inside diameter is 8 inches. Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 28 of 36

29 Input to the Fluid Flow Calc tool Density = lb/cu ft Viscosity = 1.30 cp Quantity of flow = 2.5 cfs Pipe Diameter = 8.00 (in) Fitting total K Coeff s = 0.50 Fitting Total L/D values = 13 Length of Pipe = 1,000 ft Roughness Factor = , ft Output from the Fluid Flow Calc tool Total equivalent length = 1,024.1 (ft) Relative roughness = /D Reynolds number = 340,640 Re Friction factor = f Pressure drop = feet of fluid Fluid Flow in Pipe The Darcy-Weisbach Equation The discharge pipe total K of 1.0, L/D of 388 and pipe inside diameter of 6 inches. Input to the Fluid Flow Calc tool Density = lb/cu ft Viscosity = 1.30 cp Quantity of flow = 2.5 cfs Pipe Diameter = 6.00 (in) Fitting total K Coeff s = 1.00 Fitting Total L/D values = 388 Length of Pipe = 2,000 ft Roughness Factor = , ft Output from the Fluid Flow Calc tool Total equivalent length = 2,215.8 (ft) Relative roughness = /D Reynolds number = 454,187 Re Friction factor = f Pressure drop = feet of fluid The total feet of fluid loss for the two pipes is = = ft The total feet of fluid loss with the static head of the system = (150-50) or ft. Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 29 of 36

30 Conclusion The objective for this course has been to develop a better understanding of the use of Darcy- Weisbach equation for flow of fluids in piping systems and to give attendees a new tool for salving problems and answering questions. With the use of calculators and computers along with software the Darcy-Weisbach equation should be considered the standard predictor of flow in pipes and now the Fluid Flow Calc v1.0 Tool makes it easier to use the Darcy-Weisbach equation along with the appropriate other equations in this course than ever before. A final word of caution: Never rely entirely on this or any software package to give you the final answers to engineering questions. Use it to optimize and explore nut than perform your own computations to verify and cross-check results. As a professional engineer our profession exists to protect the safety, well-being and other interests of the general public. So your integrity is being viewed when performing computations and designed. Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 30 of 36

31 Temperature of Water Appendix A Note: Weight per gallon is based on 7.48 gallons per cubic feet Specific gravity of water at 60 of = 1.00 Liquid Physical Properties of Water Specific Weight Volume Density Cu lb/cu ft ft/pound Table 1 Physical Properties of Water Weight Pound per Gallon Weight Density and Specific Gravity of Liquids Temp Weight Deg. Density F lb/cu ft Specific Gravity Mile to Olive Oil SAE 10 Lube SAE 30 Lube SAE 70 Lube Table 2 Weight Density and Specific Gravity of Liquids Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 31 of 36

32 Viscosity of Liquids Liquid Centipoise cp Temp Deg. F Water Gasoline Kerosene Bunker C Fuel Fuel 5 (Max) or 6 (Min) SAE 10 Lube SAE 30 Lube SAE 70 Lube Table 3 Viscosity of Liquids Relative Roughness of Pipe Materials Material e, ft Glass, new commercial pipe surfaces, drawn typing 0.000,005 (brass, copper, lead) Asphalted cast iron Cast iron Commercial steel or wrought iron Concrete Drawn Tubing 0.000,005 Galvanized iron Riveter steel Schedule 40 PVC 0.000,005 Wood stave Table 4 Relative Roughness of Pipe Materials Recommended Water Velocity Service Velocity Range (fps) Pump discharge 8 12 Pump suction 4 7 Drain line 4 7 Header 4 15 Riser 3 10 General service 5 10 City water 3 7 Table 5 Recommended Water Velocity Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 32 of 36

33 Maximum Water Velocity To Minimize Erosion Normal Operation hr/year Water Velocity (fps) Table 6 Maximum Water Velocity To Minimize Erosion Increase In Friction Loss Due To Aging Of Pipe Age of Pipe in Years Small Pipes 4-in to 10-in Large Pipes 12-in to 60-in New Note: Multiplies for use with new pipe loss. Use this table with extreme care. Table 7 Increase in Friction Loss Due To Aging of Pipe Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 33 of 36

34 Resistance Due to Pipe Entrance and Exit Type Picture K Inward Projecting Pipe Entrance 0.78 Sharp Edged Entrance 0.50 Slightly Rounded Entrance 0.23 Well Rounded Entrance 0.04 Projecting Pipe Exit 1.00 Sharp Edged Exit 1.00 Rounded Exit 1.00 Table 8 Typical Pipe Entrance and Exits K Coefficients Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 34 of 36

35 Equivalent Length in Pipe Diameters (L/D) of Valves and Fittings Type Description L/D Angle valves with no obstruction in flat, bevel, or plug type seat 145 with wing or pin guided disc 200 Globe valves with no obstruction in flat, bevel, or plug type seat 340 with wing or pin guided disc 450 Gate valves Wedge, Disc, Double Disc, or Plug Disc Fully open 13 Three-quarters open 35 One-half open 160 One-quarter open 900 Check valves Conventional Swing 135 Clearway Swing 50 Globe Lift or Stop; Stem Perpendicular to Run or Y-Pattern Same as Globe Angle Lift or Stop Same as Globe In-Line Ball 150 Foot Valves with With poppet lift-type disc 420 Strainer With leather-hinged disc 75 Butterfly Valve 8-inch and larger 40 Cocks Straight-Through 44 Three-Way 140 Fittings Elbows 90 Degree Standard Elbow Degree Standard Elbow Degree Long Radius Elbow Degree Street Elbow Degree Street Elbow 26 Square Corner Elbow 57 Fitting Standard With flow through run 20 Tee With flow through branch 60 Fitting Bands Close Pattern Return 50 Table 9 Typical Valves and Fitting L/D Values Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 35 of 36

36 Moody Friction Factors Diagram Copyright 2011 Lawrence H. Smith, Sr., P.E. Page 36 of 36

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