Solved problems 4 th exercise

Transcription

1 Soled roblem th exercie Soled roblem.. On a circular conduit there are different diameter: diameter D = m change into D = m. The elocity in the entrance rofile wa meaured: = m -. Calculate the dicharge and mean elocity at the outlet rofile (ee fig. ). Determine alo tye of flow in both conduit rofile (whether the flow i laminar or turbulent) temerature of water T = C Figure S o l u t i o n Dicharge Q and conequently elocity can be calculated from the continuity equation. D Q S 9,5m Q Q 9,5, m S D To determine tye of flow in conduit, the Reynold number Re D will be ued. For laminar flow: Re 0 For turbulent flow Re 0 Kinematic icoity of water of C:,0 6 m. (ee Tab. ) For the conduit: D. Re 88,.0 6,.0 D,. Re 5.0 6, turbulent flow turbulent flow K HYAE exercie

2 Soled roblem.. A horizontal ieline i attached to the wall of reeroir (ee fig. ). The ieline ha different rofile. The water leel in the uer reeroir i in the height H =.5 m aboe the ieline axi. From the lower end of the ieline water flow out to the oen ace. Diameter and length of ieline reache are: D = 0. m, L = m, D = 0. m, L = m, D = 0. m, L = m. Calculate dicharge in the ieline and draw the coure of energy line (EL) and reure line (PL). Reole the roblem: a) Neglecting loe (i.e. conider the liquid to be ideal) b) Conidering loe for water of temerature 0 C. Steel ieline conider to be after uage (lightly ruted). S o l u t i o n a) Neglecting loe Figure Bernoulli equation (BE) and continuity equation will be ued to ole the roblem. Uing BE to calculate dicharge, it will be the mot conenient to tate the datum (reference) leel at the axi of the horizontal ie, and to write then BE for the uer water leel (rofile 0 reure on the leel i known - a ), and for the centre of outlet rofile (it oition, referred to the datum leel, i h = 0). The uer reeroir can be conidered to be large (it dimenion are not mentioned becaue they hae no imortance) and the effect of aroach elocity u n can be therefore neglected (i.e. n = 0). Then: BE 0 : a n H g g a 0 g g Arranging thi equation, mean elocity in outlet rofile can be calculated gh 9,8,5 5,m. Ma dicharge i calculated from continuity equation 0, Q S 5, 0,06m. Now it i oible to ue continuity equation again, to calculate elocitie in other rofile and to calculate thu alo correonding elocity head.,56, 0,09m g 9,6 Q 0,06.,56m S 0, K HYAE exercie

3 7,8,0m g 9,6 Q 0,06. 7,8m S 0, A diameter in the ingle reache of the ieline are contant, alo elocitie are contant in thee reache. The energy line and reure line will therefore be arallel and, becaue of conidering the liquid to be ideal, they will be horizontal. Then the BE can be written for rofile - the firt one ituated at the water leel of the uer reeroir, the econd one ituated e.g. in the middle of length of the firt reach of ie (tatic reure there i till unknown): BE 0 : a H g 0 0 g g after it rearrangement: g g a H g,5 0,09,06m (oerreure head) Similarly, for the econd reach of the ieline: g g a H g,5,0,6m (underreure head) In the outlet rofile there i the atmoheric reure which, in thi cae, will be alo in the whole length of the lat ie reach. Both elocitie and reure are contant in ingle reache of ieline, energy and reure line are therefore arallel, a mentioned. In rofile, where the ie diameter change, the reure line change it oition uddenly. Energy line and reure line are een on fig.. Figure K HYAE exercie

4 b) Conidering loe A in a), Bernoulli equation and continuity equation will be ued to ole the roblem. To calculate dicharge, the mot adantage rocedure again i to write Bernoulli equation for rofile of water leel in reeroir (rofile 0) and for outlet rofile (rofile ). The datum leel can be conidered at the axi of the horizontal ie. The uer reeroir can be conidered to be large and the effect of aroach elocity can be neglected (i.e. n = 0). Conidering the Corioli number =,0, it can be then written: a H g n g a 0 g g Z Loe Z are calculated a a um of friction loe Z t and local loe Z m. Friction lo i exreed by Darcy-Weibach equation Z L t D, local lo can be g exreed a, where i coefficient of local lo and i mean elocity of flow g in the rofile of ie fitting. The coefficient of friction lo can be determined uing D Moody diagram, in deendency on Reynold number Re (... kinematic icoity of flow of liquid) and in deendency on relatie roughne (... hydraulic D roughne of a ie). Coefficient of local loe: inlet to ieline: inlet = 0,5 (related to elocity behind the inlet) D 0, contraction of ie from D to D : 0, 7 contr. 0, 5 D 0, (related to diameter D ) D 0, enlargement of ie from D to D :, enl arg. 0, 9 D 0, Coefficient of friction lo: (related to diameter D ) A neither dicharge nor elocitie of flow in ingle reache are known, it i not oible to tate alue of Reynold number in adance. That i why in the firt te of calculation coefficient of friction loe i will be determined for ingle reache i only according relatie roughne, uoing rough turbulent quadratic zone of Di friction loe. Mean hydraulic roughne for gien ie (teel ie after uage - lightly ruted) i 0,5 mm. From Moody diagram alue of coefficient of friction loe in ingle reache are: K HYAE exercie

5 0,5 reach : 0,0008 0, 0, D 0 0,5 reach : 0,0050 0, 00, D 00 0,5 reach : 0,007 0, 09. D 0 Bernoulli equation for rofile of water leel in reeroir and for rofile at the end of the third reach can be written a L L L inlet contr. enl arg g g D g D g D H. Velocitie i in ingle reache can be exreed, following continuity equation, from Di dicharge Q and from cro ection area, Si : Q Q L Q L Q L inlet contr. enl arg S g S g D S g D S g D H. From Bernoulli equation arranged in thi form it i oible to exre dicharge Q: Q S g S g inlet L D S g H contr. L D S g enl arg. L D Q 0,050 m. Now it i neceary to erify alue of coefficient of friction loe in the firt te of calculation determined under the reumtion of rough turbulent quadratic zone 6 (kinematic icoity of water with temerature of 0 C,0 m ): D Q D 0,050 0, 5 Re,0 S 0, 6,0 D Q D 0,0500,0 5 Re,00 S 0,0 6,0 D Q D 0,0500, 5 Re,80 S 0, 6,0 0,05 0,0 0,09 K HYAE 5 exercie

6 A alue of coefficient of friction loe for the firt and econd ie reach differ from thoe one calculated reiouly, it i neceary to recalculate dicharge Q with thee changed alue uing the ame rocedure of calculation: Q 0,09 m. A the recalculated alue of dicharge differ only little comared with alue obtained from the firt te of calculation (arox. about 0, %), it i not neceary to make further correction of coefficient of friction loe and the dicharge Q 0,09 m. can be conidered to be the final reult. Energy line (EL) in thi cae come out from water leel in reeroir (energy horizon) and in direction of flow decreae becaue of loe. Preure line (PL) i in lower oition, it ertical ditance from EL i gien by alue of elocity head. A there are contant elocitie of flow in ingle reache, the energy line and reure line are arallel (ee cheme - fig. ). Figure K HYAE 6 exercie

7 Soled roblem.. Water flow in the ieline (ee fig. 5). Calculate maximum elocity u max in the ie axi and dicharge Q. Determine whether the flow i laminar or turbulent (T = o C). The mercury differential manometer ( Hg = 600 kgm - ) how the difference between leel in Pitot tube H m = 0,0 m. Diameter of the ie i D = 0,5 m. Velocity coefficient of Pitot tube i =,0. Mean elocity i conidered to be = 0,8 u max. S o l u t i o n Figure 5 A the flow i teady, dicharge Q i contant. Diameter of the ie i alo contant, therefore elocity i contant, too. Pitot tube ere to meaure oint elocitie. The rincial lie in a change of tye of mechanical energy between two rofile (Bernoulli equation) change of elocity head caue a change of reure head. The difference between elocity head in rofile and (in rofile the elocity head i gien by elocity u max, in rofile the elocity head i zero, becaue in manometer there i no elocity) will caue a difference between reure head. Conequently, it caue a change in mercury leel in manometer. To determine the oint elocity, combination of equation of reure balance (at conenient urface area in manometer) will be ued in combination with Bernoulli equation for rofile and. Determination of reure difference (balance of tatic reure at urface area): g H g H Hg m w m H m g Hg w w g H m Hg w () Figure 6 K HYAE 7 exercie

8 Determination of oint elocity (from Bernoulli equation for rofile and, datum leel at ie axi):. umax umax Z, where the lo Z.,. g. g. g.g after equation arrangement, including introduction of a new coefficient come to u max.. g (). g, it If lo i neglected, ς = 0 φ =, combination of equation () and () gie u max..g.h m Hg w.. 9, 8. 0, 0., 6, m. Uing the relation between oint and mean elocity = 0,8 u max, mean elocity will be determined and, conequently, continuity equation will be ued to calculate dicharge Q. 0,8. u,868m. max. D Q. S,868. 0,0m Flow regime will be determined uing the Reynold number:. D,868.0,5 Re 59 6,.0 The alue of Reynold number Re > 0 flow in the ieline i turbulent.. K HYAE 8 exercie