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Mathematical Solution of Non-linear equations : Newton Raphson method
Hassam Mathematical Solution of Non-linear equations : Newton Raphson method
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NEWTON-RAPHSON METHOD
Where π₯ π = known approximation for unknowns while π₯ π+1 = the next approximation Here, J(x) = the Jacobian matrix defined as :
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Example
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Creating mass flow equations
From last slide: K = flow conductance , π= density, p = pressure, n= constant based on flow (0.5 for turbulent)
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Derivative on each nodes
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Newton- Raphson iteration
π½ β1 π₯ = 1 π½ π₯ πππ π½(π₯) Initial guess π.π ππ π΄= π π π π π΄ β1 = 1 π΄ πππ π΄ π΄ =ππβππ πππ π΄ = π βπ βπ π
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Pressure Change Nodes Branches
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Mass flow
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Mathematical Solution of Non-linear equations : HarDy Cross Method
Saqlain Mathematical Solution of Non-linear equations : HarDy Cross Method
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Hardy Cross method
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Pressure Variation
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Comparison with Newton Raphson
Hardy cross The parameter value in the next step is computed The correction term is evaluated for all the loops simultaneously Convergence is not always guaranteed but depends on the problem parameters e.g. initial values etc. The change in parameter value in the next step is computed The correction term is evaluated for each loop independently and is considered same for each loop.
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Example βπ 1 = βπ 2 =0 (for closed loops)
Since it is a closed network so We donβt need external nodes or Pseudo-loops. Fully turbulent flow : q = 2
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First Iteration
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Continued Iteration
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Thanks
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