Pipe Sizing Basics Prof. Dr. Mahmoud Fouad Major & Minor Losses Under Supervision of: Prof. Dr. Mahmoud Fouad By students: Mahmoud Bakr 533 Mohammed Abdullah 511 Moaz Emad 619 Mohammed Nabil Abbas 525

Applications

How big does the pipe have to be to carry a flow of x m3/s?

Bernoulli's Equation The basic approach to all piping systems is to write the Bernoulli equation between two points, connected by a streamline, where the conditions are known. For example, between the surface of a reservoir and a pipe outlet. The total head at point 0 must match with the total head at point 1, adjusted for any increase in head due to pumps, losses due to pipe friction and so-called "minor losses" due to entries, exits, fittings, etc. Pump head developed is generally a function of the flow through the system

Bernoulli's Equation

Friction Losses in Pipes Friction losses are a complex function of the system geometry, the fluid properties and the flow rate in the system. By observation, the head loss is roughly proportional to the square of the flow rate in most engineering flows (fully developed, turbulent pipe flow). This observation leads to the Darcy-Weisbach equation for head loss due to friction

which defines the friction factor, f which defines the friction factor, f. f is insensitive to moderate changes in the flow and is constant for fully turbulent flow. Thus, it is often useful to estimate the relationship as the head being directly proportional to the square of the flow rate to simplify calculations.

Reynolds Number is the fundamental dimensionless group in viscous flow Reynolds Number is the fundamental dimensionless group in viscous flow. Velocity times Length Scale divided by Kinematic Viscosity.   Relative Roughness relates the height of a typical roughness element to the scale of the flow, represented by the pipe diameter, D.   Pipe Cross-section is important, as deviations from circular cross-section will cause secondary flows that increase the pressure drop. Non-circular pipes and ducts are generally treated by using the hydraulic diameter, in place of the diameter and treating the pipe as if it were round

For laminar flow, the head loss is proportional to velocity rather than velocity squared, thus the friction factor is inversely proportional to velocity

Turbulent flow For turbulent flow, Colebrook (1939) found an implicit correlation for the friction factor in round pipes. This correlation converges well in few iterations. Convergence can be optimized by slight under-relaxation.

The familiar Moody Diagram is a log-log plot of the Colebrook correlation on axes of friction factor and Reynolds number, combined with the f=64/Re result from laminar flow. The plot below was produced in an Excel spreadsheet

An explicit approximation

Pipe roughness pipe material pipe roughness  (mm) glass, drawn brass, copper 0.0015 commercial steel or wrought iron 0.045 Must be dimensionless! asphalted cast iron 0.12 galvanized iron 0.15 cast iron 0.26 concrete 0.18-0.6 rivet steel 0.9-9.0 corrugated metal 45 0.12 PVC

Calculating Head Loss for a Known Flow From Q and piping determine Reynolds Number, relative roughness and thus the friction factor. Substitute into the Darcy-Weisbach equation to obtain head loss for the given flow. Substitute into the Bernoulli equation to find the necessary elevation or pump head

Calculating Flow for a Known Head Obtain the allowable head loss from the Bernoulli equation, then start by guessing a friction factor. (0.02 is a good guess if you have nothing better.) Calculate the velocity from the Darcy-Weisbach equation. From this velocity and the piping characteristics, calculate Reynolds Number, relative roughness and thus friction factor. Repeat the calculation with the new friction factor until sufficient convergence is obtained. Q = VA

"Minor Losses" Although they often account for a major portion of the head loss, especially in process piping, the additional losses due to entries and exits, fittings and valves are traditionally referred to as minor losses. These losses represent additional energy dissipation in the flow, usually caused by secondary flows induced by curvature or recirculation. The minor losses are any head loss present in addition to the head loss for the same length of straight pipe. Like pipe friction, these losses are roughly proportional to the square of the flow rate. Defining K, the loss coefficient, by

. K is the sum of all of the loss coefficients in the length of pipe, each contributing to the overall head loss Although K appears to be a constant coefficient, it varies with different flow conditions Factors affecting the value of K include : the exact geometry of the component,. the flow Reynolds number , etc .

Some types of minor losses Head Loss due to Gradual Expansion (Diffuser)  diffusor angle () 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 20 40 60 80 KE

Sudden Contraction losses are reduced with a gradual contraction V2 V1 flow separation losses are reduced with a gradual contraction

Sudden Contraction Cc A2/A1 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.2 0.2 0.4 A2/A1 Cc

Entrance Losses vena contracta Losses can be reduced by accelerating the flow gradually and eliminating the vena contracta

Head Loss in Bends Kb varies from 0.6 - 0.9 High pressure Head loss is a function of the ratio of the bend radius to the pipe diameter (R/D) Velocity distribution returns to normal several pipe diameters downstream Possible separation from wall R D Low pressure Kb varies from 0.6 - 0.9

Head Loss in Valves Function of valve type and valve position The complex flow path through valves can result in high head loss (of course, one of the purposes of a valve is to create head loss when it is not fully open)

To calculate losses in piping systems with both pipe friction and minor losses use

Solution Techniques Neglect minor losses Equivalent pipe lengths Iterative Techniques Simultaneous Equations Pipe Network Software

Iterative Techniques for D and Q (given total head loss) Assume all head loss is major head loss. Calculate D or Q using Swamee-Jain equations Calculate minor losses Find new major losses by subtracting minor losses from total head loss

Solution Technique: Head Loss Can be solved directly

Solution Technique: Discharge or Pipe Diameter Iterative technique Set up simultaneous equations in Excel Use goal seek or Solver to find discharge that makes the calculated head loss equal the given head loss.

Example: Minor and Major Losses Find the maximum dependable flow between the reservoirs for a water temperature range of 4ºC to 20ºC. 25 m elevation difference in reservoir water levels Water Reentrant pipes at reservoirs Standard elbows 2500 m of 8” PVC pipe Sudden contraction Gate valve wide open 1500 m of 6” PVC pipe

Directions Assume fully turbulent (rough pipe law) find f from Moody (or from von Karman) Find total head loss Solve for Q using symbols (must include minor losses) (no iteration required) Obtain values for minor losses from notes or text

Example (Continued) What are the Reynolds number in the two pipes? Where are we on the Moody Diagram? What value of K would the valve have to produce to reduce the discharge by 50%? What is the effect of temperature? Why is the effect of temperature so small?

Example (Continued) Were the minor losses negligible? Accuracy of head loss calculations? What happens if the roughness increases by a factor of 10? If you needed to increase the flow by 30% what could you do? Suppose I changed 6” pipe, what is minimum diameter needed?