A water distribution system consisting of two centrifugal pumps in two parallel

Question

A water distribution system consisting of two centrifugal pumps in two parallel flow channels is to be designed. The total flow rate V is the sum of the flow rates Vl and V2 in the two paths. Therefore,

V = Vl + V2

Also, the characteristics of the two pumps are given in terms of the pressure difference P as

P = P1 - A (Vl)^2 P = P2 - B (V2)^2

where P1 and P2 are the maximum pressures generated (for no flow conditions) and A, B are constants. The energy balance, considering elevation change H and the friction losses, gives

P = H + C (V)^2

Take the base, or design, values of P1, P2, H, A, B, and C as 470, 700, 135, 10, 20 and 5, respectively, in SI units. Develop script-m or function-m files to achieve the following.

1. For these base values, compute the flow rates and the pressure difference P, using the Newton-Raphson and the modified Gauss-Seidel (successive substitution) methods. Elimination may be used to reduce the number of equations to 2 for Newton's method and to 1 (root solving) for the other method.

2. Using either of the above two methods, calculate the pressure P and the total flow rate V if the design variables P1, P2, A, B and C are varied by up to ± 20% from their base values. In each case, vary only one variable at a time, keeping the others constant at the design, or base, values. Plot the results.

Explanation / Answer

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