Module b: Flow in Pipes Darcy-Weisbac Robert Pitt University o Alabama and Sirley Clark Penn State-Harrisburg Darcy-Weisbac can be written or low (substitute V Q/A, were A (π/4)d in te above equation): 8 π D 5 g Q Darcy-Weisbac can be rewritten to solve or velocity: V g D Based upon teory. Used to estimate energy loss due to riction in pipe. V D g Were ead loss (eet) lengt o pipe (eet) riction actor D internal diameter o pipe (eet) V /g velocity ead (eet) Friction actor or : Based upon te ynolds number, N R or, and a dimensionless parameter called te relative rougness, e/d or ε/d (ε absolute rougness; d diameter). For laminar low: 64 N R 64 For turbulent low, riction actor must be read o a Moody diagram (or o a relative rougness vs. riction actor diagram or completely turbulent low). 1
V D g To use Darcy-Weisbac to calculate ead loss, need : A polymeric coagulant, undiluted, as an absolute viscosity o 0.48 kg/(m-sec) [0.01 lb-sec/t] and a speciic gravity o 1.15. Tis luid is to be pumped at te rate o.78 /min (1 gallon/min) troug 15.5 m (50 t) o ½-inc diameter scedule 40 pipe (ID 0.6 in 15.8 mm 0.0158 m). Wat is te ead loss due to riction? Example: Equivalent sand rougness diagram [(.78 /min)(1 m /1000 )(1 min/60 sec)] [(π / 4)(0.0158 m) ] V 0.1m / sec or V 0.m / sec V I te low is laminar, te riction actor can be calculated. Oterwise, it must be looked up o te cart. Need to determine velocity in order to calculate ynolds number (determine i low is laminar or turbulent). By Continuity Equation: Q VA or V Q/A Substituting: Moody Diagram
µ ρ Friction Slope Friction Slope: 6.7 m 1.75 175 % 15.5 m (5.7)(15.5 m)(0.1 m/sec) (0.0158 m)[()(9.806 m/sec )] 6.7m Substituting into Darcy-Weisbac equation: (0.1 m/sec)(0.0158 m)(1.15)(1000 kg/m ) (0.48 kg/m sec) 1.15 Deinition o Speciic Gravity Fluid Density/Water Density Density o Water ρho 1000 kg/m Substituting: Calculate te ynolds number to see i low is laminar or turbulent. ε/d 0.000 From te relative rougness diagram (assume similar to aspalted cast iron, read ε/d. A 4-inc class I ductile iron pipe (ID 0.6 m 4.79 in.) 90 m long wit a neat cement lining (aspalted cast iron) carries a low o water at 1.5 m/sec (5.9 t/sec). Wat is te riction loss in te pipe? Example: Darcy-Weisbac Equation 64 64 1.15 5.7 HAVE AMINAR FOW!! Tereore, te riction actor or Darcy-Weisbac is calculated as ollows:
Calculate ynolds number. Assuming tat te luid is water: ν 1.00 x 10-6 m /sec at 0 o C. Find velocity o low. Q 1.5 m / sec V A π ( 0.6m ) 4 V 4.81 m / sec Find te ynolds number..0x 10 ( 4.81m /sec)( 0.6m ) 6 ν ( 1.00x 10 m /sec) 6 Moody Diagram ading rom te Moody Diagram: (ε/d 0.000 &.0 x 10 6 ) 0.014 Substituting into Darcy-Weisbac: (0.014)(90 m)(4.81m /sec) (0.6 m)()(9.806 m /sec.6 m ) 4 Example: Determine te lowrate in a 500-m section o a 1-m diameter commercial steel pipe wen tere is a -m drop in te energy grade line over te section. Given: 500 m D 1 m Commercial steel pipe m Want to use Darcy-Weisbac equation wit low rate. 8 Q 5 π D g
1/ 1/ 4 π π 4 V.79m / sec.147m / sec Q VA V (1.0m ) ν (.79 m / sec)(1.0 m ) 1.19 x10 6 m / sec.40 x10 6 Calculate te ynolds number. (Note tat no temperature is given or te luid. Assume te luid is water and te temperature is 15oC). At 15oC, ν 1.19 x 10-6 m/sec Substituting: Must ceck assumption o ully turbulent low (i.e., was te selected rom te igure acceptable?). Using te continuity equation: (m)π (9.81m / sec )(1m) 5 Q 8(0.0105)(500m) Q.147m / sec Substituting : 8 π gd 5 8 Q π D 5 g π gd 5 Q 8 Q Solve Darcy-Weisbac or low rate: 0.0105 As a irst assumption about te low in te pipe, will assume ully turbulent low. Using Moody diagram or relative rougness in turbulent low: Need to ind te riction actor. Darcy-Weisbac Equation 5
1/ Are we done? (m)π (9.81m / sec )(1m) 5 Q 8(0.0115)(500m) Q.05m / sec π gd 5 Q 8 Substituting : 1/ Substituting back into Darcy-Weisbac or low: ε/d (0.000046 m)/ (1.0 m) 0.000046 In order to use te ull Moody diagram, need te relative rougness. For commercial steel, ε 0.000046 m. Calculate: Darcy-Weisbac Equation (.61m / sec)(1.0m) ν 1.19 x10 6 m / sec.x106 Calculate te ynolds number associated wit tis velocity: Q.05m / sec A π (1.0m) 4 V.61m / sec V To ceck tis low rate, repeat te process. By continuity: Using te ull Moody diagram, or.40 x 106 and ε/d 0.000046: 0.0115 Moody Diagram: Draw imaginary line interpolating at ε/d 0.000046 6
Need to solve or V and A to get Q, te low. Example: A 14-inc scedule 80 pipe (commercial steel) as an inside diameter o 1.5 in (17.5 mm). How muc low can tis pipe carry i te allowable ead loss is.5 m in a lengt o 00 m? Using te ull Moody diagram, or.0 x 106 and ε/d 0.000046: 0.0115 Moody Diagram: Draw imaginary line interpolating at ε/d 0.000046 ε/d 0.0046 mm/ 17.5 mm 0.000014 In order to use te ull Moody diagram, need te relative rougness. For commercial steel, ε 0.000046 m. Calculate: Darcy-Weisbac Equation Are we done? Yes. Friction actors approximately te same between last two iterations. Q.05 m/sec Friction actors approximately te same between last two iterations. Can use value rom previous iteration as Q. 7
er to Moody Diagram 8 For tis ε/d, ranges rom 0.0087 to 0.04, depending on te low (expressed as te ynolds number). Assume 0.01 (in range near low end). I 0.01, ten te ynolds number is approximately 4 x 10 6. Substituting tis into Darcy-Weisbac: [ V V V g D ] [ ] [(.5 m)()(9.806 m /sec )(0.175 m)] [(00 m)(0.01)].0m /sec Are we done? Now need to ceck te riction actor assumption: Calculate ynolds number or V.0 m/sec. ν.0m / sec(0.175m ) 6 1.00x 10 m /sec 6 1.04x 10 er to Moody Diagram From te Moody diagram, te or tis ynolds number is 0.01.
[ ] [ g D ] Calculate ε/d. D 46 cm 0.46 m ε 0.001 t (0.048 m/t) 0.00005 mm Tereore ε/d 0.000 m/0.46 m 0.00066 Example: Determine te ead loss in a 46-cm concrete pipe wit an average velocity o 1.0 m/sec and a lengt o 0 m. Darcy-Weisbac Equation V [(.5 m)()(9.806 m / sec )(0.175 m)] [(00 m)(0.01)] V.01m / sec V calculate using Darcy-Weisbac: ν Yes. ν (1 m / sec)(0.46 m) 1.00 x10 6 m / sec 4.58 x105 Calculate ynolds number: Are we done? π Q VA (.01m / sec ) (0.175 m ) 4 Q 0.8 m / sec COSE ENOUGH, So pipe will carry: (.01 m / sec)(0.175 m) 1.00x10 6 m / sec 0.95 x106 Calculate te ynolds number or V.01 m/sec. 9
er to Moody Diagram 10 From te Moody diagram, te or tis ynolds number is 0.019. Substituting into Darcy-Weisbac: D 0.06 V (0.019) m g 0 0.46 m m (1 m (9.806 / sec) m / sec ) Are we done? Yes.