Principles of Food and Bioprocess Engineering (FS 231)
Fluid Flow
Stress: It is the force per unit area and its units are N/m2 or Pascal (Pa). Normal stress (pressure)
is the stress applied perpendicular to a surface. Shear stress is the stress applied parallel to a
surface. A fluid does not deform when a normal stress is applied. However, it deforms (or flows)
when a shear stress is applied. Higher the shear stress, higher the flow velocity.
Viscosity: It is the fluid property that denotes the resistance between internal layers of a fluid to
movement. Its units are Pa s (N s/m2) or poise (dyne s/cm2). The viscosity of water is 0.01 poise
= 1 centipoise = 10 -3 Pa s (at ~ 20 /C). It varies by ~ 3% for every 1 /C change in temperature.
The viscosity of water at different temperatures is given in Table A.4.1 (pages 814-815 of
textbook) and the viscosity of different liquid foods are given in Table A.2.4 (page 803 of
textbook).
Newtonian Fluids: For Newtonian fluids, the shear stress (F) is proportional to the shear rate ( ).
Shear stress is measured in Pascals and shear rate in s-1. The ratio of shear stress to shear rate is
the viscosity of the Newtonian fluid.
Non-Newtonian fluids: These are fluids that don’t obey Newton’s law of viscosity. i.e., the shear
stress is NOT directly proportional to the shear rate.
The following equation (Herschel-Bulkley model) is used to describe Newtonian and nonNewtonian fluids (time-independent fluids):
In the above equation, K is the consistency coefficient (in Pa sn), n is the flow behavior index,
and F0 is the yield stress (minimum stress required to initiate flow of the liquid).
Flow curve for Newtonian and non-Newtonian fluids
Newtonian
: K=:, n=1, F0 = 0 (water, milk, fruit juices)
Dilatant
: K=K, n>1, F0 = 0 (homogenized peanut butter)
Pseudoplastic
: K=K, n<1, F0 = 0 (mayonnaise, mustard, soups)
Bingham
: K=:, n=1, F0 > 0 (Toothpaste, tomato paste)
Plastic
: K=K, n<1, F0 > 0 (Fish paste, raisin paste)
Note: If the shear rate (and hence viscosity) of a fluid changes with time under the application of
a constant shear stress, the fluid is called a time-dependent fluid.
Viscometer: A viscometer is a device used to measure the viscosity of a fluid. In a tube
viscometer, fluid is pumped through a tube and pressure drop is measured for different
volumetric flow rates. The following equation is then used to determine the viscosity of the fluid:
If gravitational force provides the pressure in a (capillary) tube viscometer, )P = DgL
In a rotational viscometer, fluid is placed between two coaxial cylinders and one of the
cylinders is made to rotate. The torque (T) required to rotate the cylinder is measured for
different rotational speeds (N). The following equation is then used to calculate the viscosity:
Rs: Radius of spindle
Rc: Radius of cup
Apparent viscosity: It is the ratio of the shear stress to shear rate for a non-Newtonian fluid.
Thus, apparent viscosity (:app) is the viscosity of a non-Newtonian fluid, with the assumption that
the non-Newtonian fluid behaves as a Newtonian fluid. Apparent viscosity MUST be expressed
along with the value of shear rate.
Effect of temperature on viscosity: It is described by the following equation:
Note that the temperature MUST be expressed in Kelvin.
BA (in Pa s) and Ea (in J/kg mol) are Arrhenius parameters and Rg = 8,314 J/kg mol K is the universal
gas constant. For water, there is about a 3% change in viscosity for every 1 /C change in temperature.
Mass flow rate and volumetric flow rate:
Mass flow rate ( ) = DA
Its units are kg/s
Volumetric flow rate ( ) = A
Its units are m3/s
A = Cross-sectional area of tube = BR2 for a tube of circular cross-section (units: m2)
D = Density of fluid
(units: kg/m3)
= Average velocity of fluid
(units: m/s)
Reynolds number (for a Newtonian fluid)
D = Density of fluid, kg/m3
= Average velocity of fluid, m/s
D = Diameter of tube, m
: = Viscosity of fluid, Pa s
Laminar, Transition, and Turbulent flow: At low flow rates, fluid flow in a pipe is streamline or
laminar (NRe < 2,100). When NRe > 4,000, the flow is completely erratic and is called turbulent
flow. For Reynolds numbers between 2,100 and 4,000, the flow is said to be in transition.
Poiseuille flow (Pressure driven flow in a cylindrical pipe): Parabolic velocity profile
(Valid for Laminar flow of Newtonian fluids in straight round conduits)
Flow measuring devices: The flow rate of a fluid in a pipe can be measured using a pitot tube,
orifice meter, venturi meter or variable area meter (a rotameter is a commonly used variable area
meter). The first 3 devices make use of a U-tube manometer to measure pressure difference.
Friction factor: The shear stresses on the fluid near the wall results in friction and this results in a
loss of pressure as fluid is being pumped through a pipe. This loss is described in terms of a
dimensionless quantity called friction factor (f). For flow in a pipe, the loss in energy due to
friction in a straight pipe, fittings, contraction, and expansion are determined respectively by:
The friction factor for laminar flow is given by f = 16/NRe and for turbulent flow, it is determined
from the Moody diagram (pg. 98 of textbook) based on the Reynolds number and relative
roughness (,/D) of the pipe. For fittings such as elbows and valves in pipes, the loss in energy
due to friction can be determined using the above equation with an appropriate Cff value (page
115 of textbook). The pressure loss due to contraction and expansion in the pipeline system are
given by equations 2.90 - 2.92 (page 114 of textbook).
Bernoulli’s equation:
Where ) PE is the change in potential energy and ) KE is the change in kinetic energy
The above equation can also be written in the following form:
Ep = Energy supplied by pump
Ef = Total frictional losses
" = 0.5 for laminar flow
" = 1.0 for turbulent flow
All the terms in Bernoulli’s equation are energy per unit mass and hence have the units J/kg
which is also the same as m2/s2.
Power required by a pump to overcome these energies =
J/s or Watts or kg m2/s3
Types of pumps: In a centrifugal pump, product enters the center of an impeller and due to
centrifugal force, moves to the periphery. At this point, the liquid experiences maximum pressure
and is forced out into the pipeline. For a centrifugal pump, the volumetric flow rate is directly
proportional to the pump speed; the total head varies as the square of the pump speed; and the
power required varies as the cube of the pump speed. In a positive displacement pump (rotary,
reciprocating, axial flow pumps), direct force is applied to a confined liquid to make it move.
Pump selection: Some of the factors involved in pump selection are given below.
1. Flow rate of fluid
2. Net positive suction head required (NPSHR) -- depends on impeller design. It is required to
maintain stable operation of pump including avoiding cavitation.
3. Net positive suction head available (NPSHA) -- depends on absolute pressure, vapor pressure
of liquid, static head of liquid above center line of pump, friction loss in the suction system.
4. Properties of fluid (such as density, viscosity).
5. Characteristic pump curves (Graph of head, power consumption, and efficiency versus
volumetric flow rate).