# System characteristic curve

The system characteristic curve (see Characteristic curve) represents the relationship between the system head (H_{sys}) and the flow rate (Q). It is often parabola-shaped and does not generally pass through the origin of the H/Q coordinate system. The curve becomes progressively steeper as throttling increases. *See Fig. 1 System characteristic curve and Fig. 4 Operating point*

The intersection of the pump-specific H/Q curve with the system-specific curve Hsys/Q determines the operating point. *See Fig. 1 Operating point *

The shape and position of the system characteristic curve result from the equation used to determine the system head (Hsys):

p Static pressure

v Flow velocity

z Geodetic height

H_{L} Head loss (see Pressure loss, Pressure head)

ρ Density of fluid handled

g Acceleration due to gravity

### Location-specific subscripts

e Defined inlet cross-section (suction tank)

*See Head Fig. 2*

a Defined outlet cross-section

e,s Relate to the system's suction side, i.e. to the portion between the cross-sections e and s

*See Head Fig. 2*

d,a Relate to the system's discharge side, between the cross-sections d and a

The expression (v_{a}^{2} – v_{e}^{2})/(2 ∙ g) is a negligible quantity if the system's cross-sections in e and a are of adequate size or of approximately the same size.

In practice, this expression is seldom of any significance. The expressions (p_{a} – p_{e})/(ρ ∙ g) and (z_{a} – z_{e}) are independent of the pump's flow rate (Q).

Therefore, the relationship between the system head (H_{sys}) and the flow rate (Q) is evidenced mainly in the head losses (H_{L}) which can be calculated by means of the following equation:

ζ Loss coefficient (Head loss)

v Flow velocity in a characteristic cross-section

(of cross-sectional area A)

As the flow velocity (v) is the quotient of the flow rate (Q) and the cross-sectional area (A), and assuming a constant loss coefficient (ζ) and sufficiently high Reynolds numbers (see Model laws), we have: H_{L} ~ Q^{2}.

The reason for the system curve's parabola shape becomes clear. For the vertex of the system characteristic curve at Q = 0, we have:

From the above equation it follows that the system characteristic curve shifts vertically in the H/Q_{sys} coordinate system if the system's tank pressures (pa, pe) and the geodetic head H_{geo} = z_{a} – z_{e} vary. H_{sys,0} is often referred to as Hstat in the scholarly literature.

Thus for instance, we have the following equations for a cooling water system (see Cooling water pump) comprising a pipe drawing water out of a river, a cooling water pump and a discharge line leading into a cooling tower basin: *See Fig. 1 System characteristic curve*

H_{A} = z_{a} - z_{e} + H_{v.e,s} + H_{v.d,a}

H_{A,0} = z_{a} - z_{e}

p_{a} = p_{e} = p_{b} (see Atmospheric pressure)

v_{a} = 0 (negligible flow velocities at a)

v_{e} = 0 (negligible inlet velocities in the intake structure from

the river)

H_{L} Head loss (pressure losses at inlet and outlet, pressure

losses through valves or elbows,

pressure losses caused by pipe friction, passage

through the condenser

and by abrupt changes of cross-section etc.)

z_{a} – z_{e} Difference in geodetic head of the water level in the

cooling water basin and in the river bed.

As the water level in the river (z_{e}) fluctuates, the system characteristic curves will shift accordingly.*See Fig. 1 System characteristic curve*