Lecture 11 Introduction to Settling Velocity

Transcription

1 Lecture 11 Introduction to Settlin Velocity Settlin elocity i one o thoe thin that eem to hae deeloped a whole academic indutry around it people hae worried or a ery lon time how to calculate ettlin elocity, and how to do it accurately. Thi i not entirely illy; all ranulometry deice that meaure ettlin elocity a a proxy or rain ize (read ettlin tube) require a precie knowlede o the ettlin elocity o phere, or example. Settlin elocity will become a primary input or bedload tranport tudie, a well. Gien how important ettlin elocity i to ediment tranport, it not urpriin that many, many people hae taken a crack at olin thi problem or once and or all. It eem o deceptiely imple conider ediment to be pherical, and balance the weiht o the phere pullin it down aaint the riction o water ruhin pat the phere holdin it up. How hard can thi be? Well, the irt part i not hard. The weiht o a phere i jut the ma time the acceleration o raity, and the ma i jut the olume time the denity. SO, π = ( ρ )

2 So our only problem i what hold the phere back. The irt ormulation o thi wa by Stoke in The orce he elt held back the phere wa the icou reitance o the luid on the urace o the phere. Thi i related to the urace area o the phere, the icoity o the luid, and the elocity o the luid. = πµ To et ettlin elocity, jut balance the two orce, and ole or! π ( ρ ) = π µ 1 18 ( ρ ) µ Ok, thi i ine o lon a icou orce are the only one lowin the phere. HOWEVER, there another orce to concern ourele with the impact o water trikin the phere. Picture an extreme example. You are a particle. You are prayed with a irehoe. Are you bein puhed backward by icoity, or by impact o water? You ueed it riot police don t ue irehoe becaue they pray really icou water. So, we need a ormula that handle the impact o all thoe little particle o water on our phere. Conider a phere bein upported on a ountain o water:

3 I eery particle o water trike the phere dead on, and ully dichare it momentum into the phere, then the impact i related to the ma o water per unit time that trike the phere and the elocity. In math: i π = ρ 4 Relatin thi to the weiht o the phere ie a elocity: ( ρ ) ρ which i, enu trictu, the impact law. Ok, there two problem with thi. One, we can t combine thee two ormulae yet. Two, we aumed that the particle o water all releaed all their momentum to the phere, and we know or a act that they don t the particle alon the perimeter, or example, barely raze the phere, o why are they iin any momentum at all? The olution to the irt problem i eay. Rubey irt thouht thi one up the orce balancin the weiht o the phere i the combination o the impact and the riction! Hey, who knew? Thu: and π π ( ρ ) = πµ + ρ 4 ( ρ ) ρ + µ µ ρ Ok, or our econd problem. We need ome way o talkin about how water impart momentum to the phere (and wouldn t it be nice i we could eneralize thi to thin other than phere?). One olution i to create a eneral orce o dra, and make it account or both impact and icou dra. Such a ormulation would look a lot like the impact

4 ormula you d need the projected area, the luid denity, and the elocity: 1 = C Aρ the only addition i thi dra coeicient, that uppoed to talk about how important icoity i, and to handle how water trike the object (note that C i unitle). So, all we need i ome way to talk about how water trike an object, and how important icoity i in all thi. or now, take it on aith people did a lot o experiment on thi and came up with a raph that deine how dra coeicient chane with chane in relatie elocity. Here it i: Notice that we hae a new Reynold number here the elocity i the ettlin elocity and the lenth cale i particle diameter and not low

5 depth. Thi Reynold number doe baically the ame thin a beore it tell u when low around an object i laminar (and thereore riction dominated), and when low i turbulent (and thereore impact dominated). Here a chart that explain omethin about how low behae around a phere or dierent particle Reynold number: or phere, the plot o C. R i pretty interetin or ery low R, C behae a a unction o R, wherea or relatie lare R, C become contant at about 0.5. Thi i neat C how the dominance o riction at low R, and the dominance o impact at hih R. It alo how omethin called the dra crii that happen to phere at ery hih R. Here, the particle boundary Reynold number become turbulent, and the dra on the phere uddenly drop. Ok! So inally, then, we hae a eneral ormula or ettlin elocity that only ha thi one naty in it. 4 1 C ρ ρ Thi i commonly reerred to a impact law, althouh it a more eneral orm than true impact law.