Physical Chemistry of Foods

local flow velocity is decreasing, but less than proportionally, which implies
that the local velocity gradientCis increasing. Flow of a liquid means
dissipation of kinetic energy, and theenergy dissipation rate(in J?m
^3 ?s
^1 ),
also known as thepower density(in W?m
^3 ), is simply given by

power density¼ZC^2 ð 5 : 3 Þ

where C is thelocal velocity gradient. This implies that small eddies
dissipate more kinetic energy per unit volume than do large ones, and below
a certain size all energy is dissipated into heat. This also means that in
turbulent flow more energy is dissipated than in laminar flow, which means,
in turn, that it costs more energy to produce flow. It is thusas ifthe viscosity
of the liquid were larger in turbulent than in laminar flow.
Turbulent flow thus generates rapid local velocity fluctuations.
According to the law of conservation of energy (Section 2.1), the sum of
the kinetic energy of the flowing liquid and its potential energy must remain
the same. The potential energy per unit volume is simply given by the
pressurep(unit Pa¼N=m^2 ¼J=m^3 ) and the kinetic energy byð 1 = 2 Þmv^2
(wheremis mass). This leads to theBernoulli equation

pþð 1 = 2 Þrv^2 ¼constant ð 5 : 4 Þ*

where r is mass density (kg=m^3 ). Consequently, a high liquid velocity
implies a low pressure, and turbulent flow thus generates rapid local
pressure fluctuations. These causeinertial forcesto act on any particle
present near an eddy. For low-viscosity liquids like water, the stresses
caused by these inertial forces tend to be much higher than the frictional
forces caused by laminar flow. (These aspects are further discussed in
Section 11.3.3.)
What are the conditions for flow to become turbulent? This depends
on the preponderance of inertial stresses—proportional to rv^2 —over
frictional or viscous stresses. The latter are equal toZCin laminar flow;
Cis proportional tov/L, whereLis a characteristic length perpendicular to
the direction of flow. The ratio is proportional to the dimensionless
Reynolds number, given by

Re:

Lvvr

Z

ð 5 : 5 Þ

Herevvis the average flow velocity, i.e., the volume flow rate (flux) divided by
the area of the cross section of the flow channel. The characteristic length is,
for instance, a pipe diameter. If now Re is larger than a critical value Recr,
turbulence will set in. Table 5.1 gives the Reynolds number equation for