CIEG 305 DERIVATION OF BERNOULLI

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CIEG 305 DERIVATION OF BERNOULLIS EQN FROM
LINEAR MOMENTUM ALONG A STREAMLINE
Gory Details
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PdP
1
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Take a narrow stream tube at some angle (shown
on previous slide Require cross-sections to be
small to that we take P and V essentially
constant Neglect friction and assume
incompressible.
Now conserve mass using RTT for control volume
defined by stream tube
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OR
REFERENCING EVERYTHING TO STATION 1
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FROM MASS CONSERVATION AND ASSUMPTION THAT VALUES
ARE MORE OR LESS EQUAL
Product rule
This is the differential form of mass
conservation along streamline
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LINEAR MOMENTUM EQUATION IN STREAMWISE DIRECTION
The last part comes from ugly chain rule, product
rule and substitution of mass conservation
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WHAT ARE THE FORCES??
GRAVITY
PRESSURE
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SUBSTITUTE INTO THE LINEAR MOMENTUM EQUATION
By mass continuity
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Divide by ?A and rearrange
THIS IS BERNOULLIS EQUATION FOR UNSTEADY
FRICTIONLESS FLOW ALONG A STREAMLINE
INTEGRATE ALONG STREAMLINE
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If we consider only steady state and
incompressible flow
THIS IS BERNOULLIS EQUATION FOR STEADY,
FRICTIONLESS, INCOMPRESSIBLE FLOW ALONG A
STREAMLINE