102 Chapter 4 / Mathematical Modeling of Fluid Systems and Thermal Systemswhere steady-state liquid flow rate, m^3 sec
coefficient, m^2 sec
steady-state head, m
For laminar flow, the resistance Rlis obtained as
The laminar-flow resistance is constant and is analogous to the electrical resistance.
If the flow through the restriction is turbulent, the steady-state flow rate is given by
(4–1)
where steady-state liquid flow rate, m^3 sec
coefficient, m2.5sec
steady-state head, m
The resistance Rtfor turbulent flow is obtained from
Since from Equation (4–1) we obtain
we have
Thus,
The value of the turbulent-flow resistance Rtdepends on the flow rate and the head. The
value of Rt, however, may be considered constant if the changes in head and flow rate
are small.
By use of the turbulent-flow resistance, the relationship between QandHcan be
given by
Such linearization is valid, provided that changes in the head and flow rate from their
respective steady-state values are small.
In many practical cases, the value of the coefficient Kin Equation (4–1), which depends
on the flow coefficient and the area of restriction, is not known. Then the resistance may
be determined by plotting the head-versus-flow-rate curve based on experimental data
and measuring the slope of the curve at the operating condition. An example of such a plot
is shown in Figure 4–1(b). In the figure, point Pis the steady-state operating point. The tan-
gent line to the curve at point Pintersects the ordinate at point Thus, the slope
of this tangent line is Since the resistance Rtat the operating point Pis given by
2H the resistance Rtis the slope of the curve at the operating point.
–
Q
–
,
2H
–
Q
–
.
A0,-H
–
B.
Q=
2H
Rt
Rt=
2H
Q
dH
dQ
=
21 H
K
=
21 H 1 H
Q
=
2H
Q
dQ=
K
21 H
dH
Rt=
dH
dQ
H=
K=
Q=
Q=K 1 H
Rl=
dH
dQ
=
H
Q
H =
K =
Q =
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