Flow across a surface elementFlow across a surface element
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How many flux lines pass through aunit area?
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How many flux lines pass through aunit area?
How many flux lines pass through aunit area?
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In this case,qnormal = qactual cos θ
The number of flux lines passing through a unit area depends on the geometrical relationship between the direction of the flux vector and the orientation of the area.
This, in turn, is abbreviated.qn = Area . qactual cos θbecomes:
qn = A . qactual
(Note that vectors are shownin bold, "areas" have direction,and this vector product is calleda "scalar", "dot" or "inner" product)
Application of these concepts tostreamflow discharge.
Application of these concepts tostreamflow discharge.
Consider a stream in plan (2D) view.Consider a stream in plan (2D) view.
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What is the total stream discharge Q?What is the total stream discharge Q?
First, . . . we need to ask how is stream discharge Q measured?
First, . . . we need to ask how is stream discharge Q measured?
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Standard (USGS) proceduresemploy rotating cups.
(Lance Ramsbey - USGS)
Standard (USGS) proceduresemploy rotating cups.
(Lance Ramsbey - USGS)
What is the total stream discharge Q?What is the total stream discharge Q?
A rotating cup (pin-wheel) flow-meterA rotating cup (pin-wheel) flow-meter
(Number of turns/minute is proportional to flow velocity)
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This is a scalar flow measurement.
This is a scalar flow measurement.
A practical application: Suppose the hydrologist has limitedaccess to the stream. Only this profile, because of depth, etc..
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But, ... the hydrologist wants to know the total discharge Q at C-D;and, for estimating erosion effects, the mean velocity of theflow at C-D.
Suppose we know the cross-sectional area A of the stream, and the average velocity of the stream V, only along the profile A-B.
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Determine Flux:Q (stream discharge) = V A cos θ A = Cross-sectional area
A Question (?):Assume that V, the average velocity of the stream, is 1 ft/s.What would be the magnitude of that component of V normal (i.e. perpendicular) to the line A-B?
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A Question (?):Assume that along A-B, the cross-sectional area of the stream is 500 ft2, and the average velocity of the stream is 1 ft/s.What is the total discharge Q across the line A-B?
A practical application: Suppose the hydrologist has limitedaccess to the stream.
Determine Total Flux:Q (stream discharge) = V A cos θ A = Cross-sectional area
Application of the concept of residence timeto a reservoir.
Application of the concept of residence timeto a reservoir.
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A Question (?):Assuming a steady-state total volume of200,000,000 m3, and an average inflow ofstreamflow, groundwater flow and precipitationof 20,000,000 m3 / year, what is the averageresidence time of a drop of water in thereservoir?
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Another Question (?):What is the partitioning of discharge from the reservoir
a) over the spillway,b) consumptive use,c) evaporation,d) groundwater ”seepage"?
Another Question (?):What is the partitioning of discharge from the reservoir
a) over the spillway,b) consumptive use,c) evaporation,d) groundwater ”seepage"?
(This is a question for later.)(This is a question for later.)
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We next use conservation of mass to quantify the water cycle.
We next use conservation of mass to quantify the water cycle.
