The Three Reservoir Problem - Maple Help
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The Three Reservoir Problem


Three reservoirs at different elevations are connected through a piping network at a single point, with an outflow from the common junction.  This application will calculate the flow rates, flow directions, and head at the common junction.


The Bernoulli equation (ignoring the losses associated with pipe fittings) for Reservoir 1 is


 z1=f1 L1D1V122 g+H


where z1 is the reservoir elevation, f1 is the friction factor, L1 is the pipe length, D1 is the pipe diameter, V1 is the liquid velocity, and H is the head at the common junction.  However, since flow can be either into or out of a reservoir (that is, a positive or negative velocity), we rewrite the Bernoulli equation thus


z1=f L1D1V1 V12 g+H


Similar equations can be defined for the other reservoirs.  These equations, along with the continuity equation,




and a correlation for the friction factor, can be used to find the flow rates, flow directions, and the head at the junction.




Liquid viscosity and density





Reservoir elevations

z__190 m:z__285 m:z__360 m:


Lengths, diameters, and cross-sectional areas of the pipes connecting the reservoirs to the junction

L__12000 m:L__21500 m:L__33000 m:

D__10.3 m:D__20.25 m:D__30.25 m:

A__1π D__12/4: A__2π D__22/4: A__3π D__32/4:

Roughness of the pipe



Outflow from the junction

Qoutflow0.03m3 s1:


Gravitational constant

g9.81m s2:

Bernoulli Equations for the Three Reservoir System

Bernoulli analysis of the system

sys z__1&equals;fric__1L__1 Q__1 Q__1&sol;2 gD__1 A__12&plus;H&comma; z__2&equals;fric__2L__2 Q__2 Q__2&sol;2 gD__2 A__22&plus;H&comma; z__3&equals;fric__3 L__3 Q__3 Q__3&sol;2 gD__3 A__32&plus;H&comma; Q__1&plus;Q__2&plus;Q__3&equals;Qoutflow&comma; Rey__1 &equals; 4 absQ__1 &rho;&sol;π D__1 &mu;&comma; Rey__2 &equals; 4 absQ__2 &rho;&sol;π D__2 &mu;&comma; Rey__3 &equals; 4 absQ__3 &rho;&sol;π D__3 &mu;&comma; fric__1 &equals; piecewiseRey__1<2500&comma; 64&sol;Rey__1&comma; 1&sol;1.8 log10e&sol;3.7 D__11.11&plus;6.9&sol;Rey__12&comma; fric__2 &equals; piecewiseRey__2<2500&comma; 64&sol;Rey__2&comma; 1&sol;1.8 log10e&sol;3.7 D__21.11&plus;6.9&sol;Rey__22&comma; fric__3 &equals; piecewiseRey__3<2500&comma; 64&sol;Rey__3&comma; 1&sol;1.8 log10e&sol;3.7 D__31.11&plus;6.9&sol;Rey__32&colon;

estimatesH&equals;1 m&comma;Q__1=0.1 m3 s1&comma;Q__2=0.1 m3 s1&comma;Q__3=0.1 m3 s1&comma;Rey__1=1&comma;Rey__2&equals;1&comma;Rey__3&equals;1&comma;fric__1=1&comma;fric__2&equals;1&comma;fric__3&equals;1&colon;

Calculation of Flow Rates, Flow Directions, and Junction Head

Positive flow rates indicate flow out of a reservoir, while negative flow rates indicate flow into a reservoir