Biomedical Engineering Reference
In-Depth Information
(4.1)
mm
=
out
in
For practical purposes where it is easy to measure volumetric flow, mass is
written as the product of density and volumetric flow rate as
(
)
(
)
ρ
QQ
=
ρ
(4.2)
out
in
Further, the volumetric flow rate is written as a product of the average velocity
and cross-sectional area of the tubing, which is of particularly useful to deal with
the evaluation of force. Momentum is the product of mass and velocity, which in-
dicates the tendency of an object in motion to keep moving or not to slow down.
For a simple case of one inlet and one outlet
mVA VA
=
ρ
=
ρ
out
out
out
in
in
in
For incompressible fluids, conservation of mass simplifies to
(4.3)
QQ
=
in
out
If there is one inlet but three outlets, (4.1) can be extended to
mm
=
+
m
+
m
(4.4)
in
1,
out
2,
out
3,
out
Equation (4.4) is useful when there are multiple branches from one manifold,
for example, the airway in the lung dividing into different lobes.
4.2.2 Inertial and Viscous Forces
When a fluid flows, there are several forces acting on it including pressure, gravi-
tational, and surface tension. Similar to solids, there is an inertial force as fluids
have a mass, given by Newton's second law. A force applied to a body causes an
acceleration (
a
) of that body of a magnitude proportional to the force, in the direc-
tion of the force, and inversely proportional to the body's mass. This is termed the
inertial force, which tends to keep the fluid flowing. In the mathematical form, this
can be written as
(
)
dQ V
dt
ρ
dV
Fma Qam
dt
(
)
==
ρ
=
=
In the
English engineering system
(lb
m
, lb
f
, ft, and s), gravitational constant
g
c
is used to convert lb
m
to slugs (the mass units in the British gravitational system),
and
g
c
is 32.2 (lb
m
ft)/(lb s
2
). In the SI system,
g
c
is 1. Hence,
g
c
is not included in
subsequent discussion in this chapter.
Another force depends on the stickiness of the fluids. The velocity of blood in
contact with the stationary pipe wall is not the same as in the center of the pipe.
