1. Mathematical Solution of Non-linear equations : Newton Raphson method
Hassam
Mathematical Solution of Non-linear equations :
Newton Raphson method

2. NEWTON-RAPHSON METHOD
Where 𝑥 𝑖 = known approximation for unknowns while 𝑥 𝑖+1 = the next approximation
Here, J(x) = the Jacobian matrix defined as :

3. Example

4. Creating mass flow equations
From last slide:
K = flow conductance , 𝜌= density, p = pressure, n= constant based on flow (0.5 for turbulent)

5. Derivative on each nodes

6. Newton- Raphson iteration
𝐽 −1 𝑥 = 1 𝐽 𝑥 𝑎𝑑𝑗 𝐽(𝑥)
Initial guess
𝑒.𝑔 𝑖𝑓 𝐴= 𝑎 𝑏 𝑐 𝑑
𝐴 −1 = 1 𝐴 𝑎𝑑𝑗 𝐴
𝐴 =𝑎𝑑−𝑐𝑏
𝑎𝑑𝑗 𝐴 = 𝑑 −𝑏 −𝑐 𝑎

7. Pressure Change
Nodes
Branches

8. Mass flow

9. Mathematical Solution of Non-linear equations : HarDy Cross Method
Saqlain
Mathematical Solution of Non-linear equations :
HarDy Cross Method

10. Hardy Cross method

11. Pressure Variation

12. 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.

13. Example ∆𝑝 1 = ∆𝑝 2 =0 (for closed loops)
Since it is a closed network so
We don’t need external nodes or
Pseudo-loops.
Fully turbulent flow : q = 2

14. First Iteration

15. Continued Iteration

16. Thanks