Page 73 - Mechanical Engineers' Handbook (Volume 4)

62 Fluid Mechanics

from p to p is from b to d, and from p to p it is from a to c. Thus, frictional duct flow
from one pressure to another results in a higher final temperature, and a lower final velocity,
in both instances. For frictional flow between given temperatures (T and T , for example),
a
b
the resulting pressures are lower than for frictionless flow (p is lower than p and p is
c
ƒ
a
lower than p ).
b
6.2 Work and Power
Power is the rate at which work is done, and is the work done per unit mass times the mass
flow rate, or the work done per unit weight times the weight flow rate.
Power represented by the work term in the energy equation is P w(VA )
w(VA )W.
Power in a jet at a velocity V is P (V /2)(VA ) (V /2g)(VA )W.
2
2
Power loss resulting from head loss is P h (VA )W.
L
Power to overcome a drag force is P FV W.
Power available in a hydroelectric power plant when water flows from a headwater
elevation z to a tailwater elevation z is P (z z )(Q ) W, where Q is the volumetric
1 2 1 2
flow rate.
6.3 Viscous Dissipation
Dissipation effects resulting from viscosity account for entropy increases in adiabatic gas
flows and the heat loss term for flows of liquids. They can be expressed in terms of the rate
at which work is done—the product of the viscous shear force on the surface of an elemental
fluid volume and the corresponding component of velocity parallel to the force. Results for
a cube of sides dx, dy, and dz give the dissipation function :
2
2
2
2
u
v
w
x y z
v w u
2
2
2
v
u
w
x y y z z x
u v 2
w
2
3 x y z
The last term is zero for an incompressible fluid. The first term in brackets is the linear
deformation, and the second term in brackets is the angular deformation and in only these
two forms of deformation is there heat generated as a result of viscous shear within the fluid.
The second law of thermodynamics precludes the recovery of this heat to increase the me-
chanical energy of the fluid.
7 CONTRACTION COEFFICIENTS FROM POTENTIAL FLOW THEORY
Useful engineering results of a conformal mapping technique were obtained by von Mises
for the contraction coefficients of two-dimensional jets for nonviscous incompressible fluids
in the absence of gravity. The ratio of the resulting cross-sectional area of the jet to the area
of the boundary opening is called the coefficient of contraction, C . For flow geometries
c
shown in Fig. 19, von Mises calculated the values of C listed in Table 4. The values agree
c
well with measurements for low-viscosity liquids. The results tabulated for two-dimensional
2
flow may be used for axisymmetric jets if C is defined by C b /b (d /d) and if d
c
jet
c
jet
and D are diameters equivalent to widths b and B, respectively. Thus, if a small round hole