Fluid Flow
Nature of Fluids
• Definition: fluid is a substance that does not permanently
resist distortion.
 Fluids (liquids and gases) are a form of matter that cannot
achieve equilibrium under an applied shear stress but deform
continuously, or flow, as long as the shear stress is applied.
 Attempt to change the shape of a mass of fluid results in
layers of fluid sliding over one another.
 During change in shape, shear stress exist.
 Shear stress magnitudes depend on the viscosity of the fluid
and the rate of sliding.
 Shear stress disappear when the fluid reaches its final shape.
 A fluid in equilibrium is free from shear stress.
• Fluids have definite densities at a given temperature
and pressure.
 Density is measured in pounds(lb)/f𝑡 3 or kg/𝑚3 .
 The density of incompressible fluids is little affected
by moderate changes in temp. and pressure.
 The density of compressible fluids is sensitive to
changes in temp. and pressure.
SOME PROPERTIES OF FLUIDS
• Viscosity: Viscosity is a property that characterizes the flow behavior of
a fluid, reflecting the resistance to the development of velocity
gradients within the fluid.
• Its quantitative significance may be explained by reference to the
Figure 1 . A fluid is contained between
two parallel planes each of area A 𝑚2
and distance h m apart.
 The upper plane is subjected to a shear
force of F N and acquires a velocity of
u m 𝑠𝑒𝑐 −1 relative to the lower plane.
Fig. 1
 The shear stress, t, is F/A (N 𝑚−2
 The velocity gradient or rate of shear is
given by u/h or, more generally, by the differential coefficient du/dy,
where y is a distance measured in a direction perpendicular to the
direction of shear.
Newtonian fluids
• For gases, simple liquids, true solutions, and dilute disperse
systems, the rate of shear is proportional to the shear stress.
•
𝐹
𝐴
=𝑡 = 𝜇
𝑑𝑢
𝑑𝑦
• The proportionality constant 𝜇 is the dynamic viscosity of the
fluid: the higher its value, the lower the rates of shear induced
by a given stress.
• The dimensions of dynamic viscosity are M 𝐿−1 𝑇 −1 .
• For the SI system of units, viscosity is expressed in N ⋅s 𝑚−2 .
For the centimeter-gram-second (CGS) system, the unit of
viscosity is the poise (P). One N ⋅s 𝑚−2 is equivalent to10 P.
• The viscosity of water at room temperature is about 0.01 P or
1 centipoise(cP). Pure glycerin at this temperature has a value
of about 14 P. Air has a viscosity of 180 X 10−6 P.
• Kinematic Viscosity:
 It is defined as the ratio of viscosity (𝜇) and the
density (ρ) of the liquid.
 Kinematic viscosity = 𝜇/ ρ
 Unit of Kinematic viscosity is stokes and centistokes.
 CGS unit of Kinematic viscosity is cm2/ sec.
 Kinematic viscosity is used by most official books like,
BP, USP, and National formularies.
• For gases at low density the viscosity increases with
increasing temperature.
• For liquids the viscosity usually decreases with
increasing temperature.
• When the flow of a fluid parallel to some boundary is considered, it
is assumed that no slip occurs between the boundary and the fluid,
so the fluid molecules adjacent to the surface are at rest (u = 0).
• As shown in Figure 2, the velocity
gradient du/dy decreases from a
maximum at the boundary (y = 0) to
zero at some distance from the boundary
(y=y′) when the velocity becomes equal to
the undisturbed velocity of the fluid (u = u′)
The shear stress must, therefore, increase
from zero at this point to a maximum at the
Fig. 2
boundary.
• A shear stress, opposing the motion of the fluid and sometimes
called fluid friction, is therefore developed at the boundary.
• The region limited by the dimension y′ is called the boundary layer.
Non-Newtonian Flow
• Emulsions, suspensions and semisolids have complex rheological
behavior and thus do not obey Newton ’s law of flow and thus they
are called non Newtonian liquids.
• They are further classified as
A)Plastic flow
B)Pseudo-plastic flow
C)Dilatant flow
A)Plastic flow
• The substance initially behaves like an elastic body and fails to flow when
less amount of stress is applied. Further increase in the stress leads to a
nonlinear increase in the shear rate which then turns to linearity.
• Extrapolations of the linear plot gives ‘x’ intersect which is called yield
value. This curve does not pass through the origin. As the curve above
yield value tends to be straight, the plastic flow is similar to the Newtonian
flow above yield value.
B) Pseudo-plastic Flow
• Here the relationship between shear stress and the shear rate
is not linear and the curve starts from origin. Thus the viscosity
of these liquids can not be expressed by a single value.
• Normally, pseudo plastic flow is exhibited by polymer
dispersions like:
® Tragacanth water
® Sodium alginate in water
® Methyl cellulose in water
® Sodium carboxy methyl cellulose in
water
Dilatant Flow
• In this type of liquids resistance to flow (viscosity) increases
with increase in shear rate. When shear stress is applied their
volume increases and hence they are called Dilatant. This
property is also known as shear thickening.
• Dilatant flow is observed in suspensions containing
more than 50% v/v of solids.
Graph representing the dilatant flow
11
Thixotropy
• Thixotropy is defined as the isothermal slow reversible conversion of gel to
sol.
• Thixotropic substances on applying shear stress convert to sol(fluid) and on
standing they slowly turn to gel (semisolid).
Thixotropic substances are now a day’s more used in suspensions to give stable
suspensions. As Thixotropic substances on storage turn to gel and thus that their
viscosity increases infinitely which do not allow the dispersed particles to settle
down giving a stable suspension. When shear stress is applied they turn to sol and
thus are easy to pour and measure for dosing. So Thixotropic substances solve
both the problems, stability and pourability.
12
Negative Thixotropy And Rheopexy:
• Negative Thixotropy is a time dependent increase
in the viscosity at constant shear.
Suspensions containing 1 to 10% of dispersed
solids generally show negative Thixotropy.
• Rheopexy is the phenomenon where sol forms a
gel more rapidly when gently shaken than when
allowed to form the gel by keeping the material
at rest.
• In negative Thixotropy, the equilibrium form is sol
while in Rheopexy, the equilibrium state is gel.
13
Compressibility
• Deformation is not only a shear-induced phenomenon.
• If the stress is applied normally and uniformly over all boundaries,
then fluids, like solids, decrease in volume.
• The decrease in volume yields a proportionate increase in density.
• Liquids can be regarded as incompressible, and changes of density
with pressure can be ignored.
• This is not possible in the study of gases if significant changes in
pressure occur.
Surface Tension
• Surface tension is a property confined to a free surface.
• not applicable to gases.
• is derived from unbalanced intermolecular forces near the surface
of a liquid.
• may be expressed as the work necessary to increase the surface by
unit area.
FLUIDS AT REST—HYDROSTATICS
• The study of fluids at rest is based on two principles:
1. Pressure intensity at a point, expressed as force per
unit area, is the same in all directions.
2. Pressure is the same at all points in a given
horizontal line in a continuous fluid.
 The pressure, P, varies with depth, z, in a manner
expressed by the hydrostatic equation:
dP = −ρg dz
where ρ is the density of the fluid and g is the
gravitational constant. integration between the limits
P1 and P2, z1 and z2, gives
P1 − P2 = − ρg(z1 − z2)
THE MEASUREMENT OF PRESSURE INTENSITY
• Application of the hydrostatic equation to the column of liquid shown in
Figure Gives:
PA − P1 = −ρgh and P1 = PA+ ρgh
• The U-tube manometer may be used for the measurement of higher
pressures with both liquids and gases. The density of the immiscible
liquid in the U-tube, ρ1 is greater than the density of the fluid in the
container, ρ2. The gauge pressure is given by
P = h1ρ1g − h2ρ2g
FLUIDS IN MOTION
• If the flow at any position does not vary with time, it is steady
and the streamlines retain their shape.
• In the regions on the upstream side of the cylinder, the
velocity of the fluid is increasing.
• On the downstream side, the reverse occurs.
• The maximum velocity occurs in the fluid adjacent to regions
B and D. At points A and C, the fluid is at rest.
• As the velocity increases, the pressure decreases. The
pressure field around the cylinder is therefore the reverse of
the velocity field.
BERNOULLI’S THEOREM
• At any point in flowing fluid, the total mechanical energy can be
expressed in terms of the following components: potential energy,
pressure energy, and kinetic energy.
• The potential energy of a body is its capacity to do work by reason
of its position relative to some center of attraction.
• For a unit mass of fluid at a height z above some reference level,
Potential energy = zg where g is the acceleration due to gravity.
• The pressure energy or flow energy is an energy form peculiar to
the flow of fluids. For an incompressible fluid, for a unit mass of
fluid:
Pressure energy =
𝑝
𝜌
• The kinetic energy is a form of energy possessed by a body by
reason of its movement. If the mass of the body is m and its
velocity is u, the kinetic energy is 1/2 m𝑢2 , and for a unit mass of
𝑢2
fluid, Kinetic energy =
2
• The total mechanical energy of a unit mass of fluid is,
therefore,
𝑢2
2
+
𝑃
𝜌
+𝑍𝑔
• The mechanical energy at two points, A and B, will be the
same if no energy is lost or gained by the system. Therefore,
we can write
𝑢2 𝑎
2
+
𝑃𝑎
𝜌
+ 𝑍𝑎 g =
𝑢2 𝑏
2
+
𝑃𝑏
𝜌
+ 𝑍𝑏 g
• If the energy lost during flow between A and B is E, then the
equation becomes:
𝑢2 𝑎
2
+
𝑃𝑎
𝜌
+ 𝑍𝑎 g =
𝑢2 𝑏
2
+
𝑃𝑏
𝜌
+ 𝑍𝑏 g +𝐸
• This is a form of Bernoulli’s theorem, restricted in application
to the flow of incompressible fluids.
• The dimensions are 𝐿2 𝑇 −2 . In practice, each term is divided
by g(𝐿 𝑇 −2 ) to give the dimension of length. The terms are
then referred to as velocity head, pressure head, potential
head, and friction head, the sum giving the total head of the
fluid:
𝑢2 𝑎
2𝑔
+
𝑃𝑎
𝜌𝑔
+ 𝑍𝑎 =
𝑢2 𝑏
2𝑔
+
𝑃𝑏
𝜌𝑔
𝐸
+ 𝑍𝑏 +
𝑔
• A second modification may be made to the equation if
mechanical energy is added to the system at some point by
means of a pump. If the work done, in absolute units, on a
unit mass of fluid is W, then
𝑊
𝑔
+
𝑢2 𝑎
2𝑔
+
𝑃𝑎
𝜌𝑔
+ 𝑍𝑎 =
𝑢2 𝑏
2𝑔
+
𝑃𝑏
𝜌𝑔
+ 𝑍𝑏 +
𝐸
𝑔
• When pump is introduced to the system
and taking into consideration the
a
energy loss due to friction inside
the pump then the net work
Tank A
to the fluid is: 𝒘𝒑 − 𝒉𝒇𝒑
b
Tank B
• In practice the pump efficiency (𝛈) is used which is defined by the
equation:
𝛈=
𝒘𝒑 −𝒉𝒇𝒑
𝒘𝒑
• The mechanical energy delivered to the fluid is, then, 𝛈Wp
• Bernoulli equation can be modified as:
𝜼𝑾𝒑 +
𝒖𝟐 𝒂
𝟐𝒈
+
𝑷𝒂
𝝆
+ 𝒁𝒂 =
𝒖𝟐 𝒃
𝟐𝒈
+
𝑷𝒃
𝝆
+ 𝒁𝒃 +𝒉𝒇𝒑
.
• The sum of heads, ΔH (for a unit mass), is the total head
against which the pump must work. Therefore,
𝑊
𝑔
= ∆𝐻
• The power required to transfer mass(m) in time(t)is:
∆𝐻𝑔𝑚
𝑃𝑜𝑤𝑒𝑟 =
𝑡
• Since the volume flowing in unit time, Q , is m/ρ t, then
𝑃𝑤𝑒𝑟 = 𝑄∆𝐻𝑔𝜌
LAMINAR AND TURBULENT FLOWS
• Flow of fluids can be laminar (and may be depicted by
streamlines) or turbulent.
• The distinction between Laminar
and turbulent was first
Demonstrated by Osborne
Reynolds(1883).
• The apparatus, shown in the Figure
consisted of a straight glass tube through
which the fluid was allowed to flow.
• The nature of flow was examined by introducing a dye into
the axis of the tube.
• Reynolds found that:
 At low flow rate the dye formed a coherent thread that grew very little
in thickness with distance down the tube. The behavior of the color
band showed clearly that the water was flowing in parallel straight
lines and that the flow was called “Laminar”.
 When flow rate was increased, a velocity called the critical velocity was
reached at which the thread of color disappeared and the color
diffused uniformly throughout the entire cross section of the stream of
water. This behavior of colored water showed that the water no longer
flowed in laminar motion but moved erratically in the form of cross
currents and eddies. This type of motion was called ”Turbulent”.
 In the Laminar region, flow was ordered, always moving parallel to the
walls of the tube.
 Generalizing, in laminar flow the instantaneous velocity at a point is
always the same as the mean velocity in both magnitude and direction.
 In turbulent flow, order is lost and irregular motions are imposed upon
the main steady motion of the fluid.
Reynolds Number and transition from Laminar to Turbulent
• Reynolds found that the critical velocity at which laminar flow
changes into turbulent depends on four quantities:
 The diameter of the tube
 The viscosity of the fluid
 The density of the fluid
 The average velocity of the fluid
• These four quantities can be combined into one group:
𝑁𝑅𝑒 =
𝐷úρ
μ
=
𝐷ú
υ
 Laminar flow is always encountered at NRe < 2100
 Under ordinary conditions of flow the flow is turbulent at
NRe > 4000
 2100 < NRe < 4000 a transition or dip region
PUMPS
• Basically pumps can be divided into two main categories:
 positive displacement pumps, which may be reciprocating or
rotary. Positive displacement pumps displace a fixed volume
of fluid with each stroke or revolution.
 Impeller pumps, on the other hand, impart high kinetic
energy to the fluid that is subsequently converted to pressure
energy.
 Equipment for pumping gases and liquids is essentially similar.
Machines delivering gases are commonly called compressors
or blowers.
 Compressors discharge at relatively high pressures and
blowers at relatively low pressures.
• The lower density and viscosity of gases lead to the use of
higher operating speeds and, to minimize leakage, smaller
clearance between moving parts.
Type of Pumps
Pump Classification
Classified by operating principle
Pumps
Dynamic
Centrifugal
Others (e.g.
Impulse, Buoyancy)
Special effect
Internal
gear
Positive
Displacement
Rotary
External
gear
Reciprocating
Lobe
Slide
vane
27
Type of Pumps
Positive Displacement Pumps
• For each pump revolution
• Fixed amount of liquid taken from one end
• Positively discharged at other end
• If pipe blocked
• Pressure rises
• Can damage pump
• Used for pumping fluids other than
water
28
Type of Pumps
Positive Displacement Pumps
• Reciprocating pump
• Displacement by reciprocation of piston
plunger
• Used only for viscous fluids and oil wells
• Rotary pump
• Displacement by rotary action of gear, cam
or vanes
• Several sub-types
• Used for special services in industry
29
Type of Pumps
Dynamic pumps
• Mode of operation
• Rotating impeller converts kinetic energy
into pressure or velocity to pump the fluid
• Two types
• Centrifugal pumps: pumping water in
industry – 75% of pumps installed
• Special effect pumps: specialized conditions
30
Type of Pumps
Centrifugal Pumps
How do they work?
•
Liquid forced into
impeller
•
Vanes pass kinetic
energy to liquid: liquid
rotates and leaves
impeller
•
Volute casing converts
kinetic energy into
pressure energy
Type of Pumps
Centrifugal Pumps
Rotating and stationary components
32
Type of Pumps
Centrifugal Pumps
Impeller
•
Main rotating part that provides centrifugal
acceleration to the fluid
•
Number of impellers = number of pump stages
•
Impeller classification: direction of flow, suction type
and shape/mechanical construction
Shaft
•
Transfers torque from motor to impeller during pump
start up and operation
33
Type of Pumps
Centrifugal Pumps
Casings
• Functions
• Enclose impeller as “pressure vessel”
• Support and bearing for shaft and impeller
• Volute case
• Impellers inside casings
• Balances hydraulic pressure on pump shaft
• Circular casing
• Vanes surrounds impeller
• Used for multi-stage pumps
34
Heat Transfer
Nature of heat flow
• heat flows from the object at the higher temperature to
that at lower temperature.
• The flow is always in the direction of the temperature
decrease.
• The mechanisms by which the heat may flow are three:
conduction, convection, and radiation:
1. Conduction :
 if a temperature gradient exists in a continuous substance,
heat can flow unaccompanied by any observable motion of
matter.
 In metallic solids, thermal conduction results from the
motion of un bounded electrons, and there is close
correspondence between thermal conductivity and
electrical conductivity.
 In gases conduction occurs by the random motion of
molecules.
2. convection :
The motion of fluids transfers heat between the fluids
by conviction.
 In natural convection the movements is caused by
buoyancy forces induced by variations in the density of
the fluid caused by differences in temperature.
 In forced convection the movements is created by an
external energy source; such as a pump.
3. Radiation :
 All bodies with a temperature about the absolute zero
radiate heat in the form of electromagnetic waves.
 Radiation may be transmitted, reflected or absorbed by
matter.
Heat transfer between fluids
• The transfer of heat from one fluid to another across a solid
boundary is of great importance.
• The system, which frequently varies in nature from one
process to another, can be divided into constituent parts and
each part characterized in its resistance to the transfer of
heat.
total temp diff.
• Rate at which heat is transferred α
total thermal resistance
• Example: Hot liquid passing through a metal pipe:
 Heat transfer from liquid to the pipe
 Conduction through the wall
 Heat loss to the surrounding by natural convection
Heat transfer through a wall
• Heat transfer by conduction through walls follows the basic
relation given by Fourier’s equation where the rate of heat
flow, Q, is proportional to the temperature gradient, dT/dx,
and to the area normal to the heat flow, A:
Q=-KA
𝒅𝑻
𝒅𝒙
Q: the rate of heat flow.
T: temperature
A: area
X: distance
K: the thermal conductivity.
 As the distance, x, increases, the temperature, T, decreases.
Hence, measuring in the x direction, dT/dx is algebraically
negative.
• Steady non directional heat transfer
through a plane wall of thickness x
and area A is represented in the figure.
 If the thermal conductivity does not
change with temperature, the
temperature gradient is linear and equal
to (T1 - T2)/x, where T1 is the
temperature of the hot face and T2 is
the temperature of the cool face.
 Fourier’s Equation then becomes:
𝑄
𝑑𝑇
= -K
𝐴
𝑑𝑥
𝑇2
𝑑𝑇=
𝑇1
•
•
𝑄 𝑋2
𝑑𝑥
𝐾𝐴 𝑋1
𝑄
(T1-T2)= (x2-x1)
𝐾𝐴
𝑄
(𝑇1−𝑇2)
=K
𝐴
𝑥2−𝑥1
-
• The equation can be written as:
𝛥𝑇
𝑅
Q= A , R: is the thermal resistance of the solid
between points 1&2.
𝛥𝑥
=
𝑘𝐴
R
• An increase in thermal resistance will reduce the heat
flow promoted by a given temperature difference.
• Fig.b demonstrates a composite wall. The rate of heat
transfer is the same for both materials, if steady- state
heat transfer exists.
𝐾1𝐴(𝑇1−𝑇2)
Q=
𝑋1
𝑇1−𝑇3
Q=A
𝑥1 𝑥2
+
𝑘1 𝑘2
=
𝐾2𝐴(𝑇2−𝑇3)
Or
𝑥2
Heat transfer in pipes and tubes
• Fourier’s law:
𝑑𝑇
𝑑𝑇
, Ar= 2πrL, Qr = -2πKrL
𝑑𝑟
𝑑𝑟
2𝜋𝐾𝐿 𝑇𝑖
2𝜋𝐾𝐿
𝑑𝑡,
Lnr0-lnri=
(Ti-T0)
𝑇0
𝑄
𝑄
Qr = -KAr
𝑟0 𝑑𝑟
𝑟𝑖 𝑟
Q=
=
2𝜋𝐾𝐿(𝑇𝑖−𝑇0)
𝑟0
ln ( 𝑟𝑖 )
𝑟0
the thermal resistance: Rth=
ln( 𝑟𝑖 )
2𝜋𝐾𝐿
• The thermal- resistance concept may be used for multiple- layer cylindrical
walls, for three layer system:
Q=
2𝜋𝐿(𝑇1−𝑇4)
𝑟2
ln 𝑟1
𝐾𝐴
𝑟3
ln 𝑟2
+ 𝐾𝐵
𝑟4
ln 𝑟4
+ 𝐾𝑐
• spherical systems are treated as one dimensional when temp. is a function
of radius. The heat flow is:
• q=
4𝜋𝐾(𝑇𝑖−𝑇0)
1 1
−
𝑟𝑖 𝑟0
Example(1)
An exterior wall of a house may be approximated by 4-in layer of common brick
(k=0.7 w/m.°c) followed by a 1.5-in layer of gypsum plaster (k= 0.48 w/m.°c). what
thickness of loosely packed rock wool insulation (K= 0.065 w/m.°c) should be added to
reduce the heat loss (or gain) through the wall by 80% ?
Solution:
-overall heat loss will be given by:
𝜟𝑻
Q= 𝜮𝑹𝑻𝒉
-heat loss with rock – wool will be 20%
𝑸𝒘𝒊𝒕𝒉 𝒊𝒏𝒔𝒖𝒍𝒂𝒕𝒊𝒐𝒏
𝑸 𝒘𝒊𝒕𝒉𝒐𝒖𝒕 𝒊𝒏𝒔𝒖𝒍𝒂𝒕𝒊𝒐𝒏
𝒎
𝟒 𝒊𝒏∗𝟎.𝟎𝟐𝟓𝟒𝒊𝒏
Rb =
𝟎.𝟕𝐰/𝐦.°𝒄
𝟏.𝟓∗𝟎.𝟎𝟐𝟓𝟒
=
𝟎.𝟒𝟖
= 0.2 =
𝜮 𝑹 𝒕𝒉 𝒘𝒊𝒕𝒉𝒐𝒖𝒕 𝒊𝒏𝒔𝒖𝒍𝒂𝒕𝒊𝒐𝒏
𝜮 𝑹 𝒕𝒉 𝒘𝒊𝒕𝒉 𝒊𝒏𝒖𝒍𝒂𝒕𝒊𝒐𝒏
= 0.145 𝒎𝟐 .°c / w
Rp =
0.079 𝒎𝟐 .°c / w
R = Rb+Rp = 0.145+0.079 = 0.224 𝒎𝟐 .°c / w
Rwith insulation = 0.224/0.2 = 1.122 𝒎𝟐 .°c / w
1.122 = 0.224 + Rrw
𝜟𝒙
𝜟𝒙
Rrw = 0.898 = 𝑲 = 𝟎.𝟎𝟔𝟓
Δxrw = 0.0584 m = 2.3 in
Example (2)
A thick – walled tube of stainless steel (18%cr, 8% Ni, K=
19 w/m.°c) with 2 cm ID and 4 cm OD is covered with a 3
cm layer asbestos insulation ( K= 0.2 w/m.°c) . if the
inside wall temp. of the pipe is maintained at 600 °c and
the outside of the insulation at 100 °c , calculate the heat
loss per foot of length?
𝒒
𝒍
=
=
𝟐 𝝅 (𝑻𝟏−𝑻𝟐)
𝒓𝟐
𝒓𝟏
𝑲𝒔
𝒍𝒏
+
𝒓𝟑
𝒓𝟐
𝑲𝒂
𝒍𝒏
𝟐 𝝅 (𝟔𝟎𝟎−𝟏𝟎𝟎)
𝟏
𝒍𝒏
𝟐
𝟏𝟗
𝟓
𝒍𝒏
+ 𝟎.𝟐𝟑
= 680 w/m
Convection heat transfer:
• Convection heat transfer can be calculated
from the following equation:
q conv. = h A(Tw-T∞)
Tw [=] the temperature of the wall (plate)
T∞ [=] the temperature of the fluid.
h [=] the convection heat transfer coefficient
q conv. =
𝑻𝒘−𝑻∞
𝟏/𝒉𝑨
, where,
1/hA is the convection resistance.
Heat exchange between a fluid and a solid boundary
𝑲𝑨 (𝑻𝟏𝒘−𝑻𝟐𝒘)
• q = h1A(T1-T1w) =
𝜟𝒙
= h2A(T2w-T2)
• over all transfer process may be
represented by the resistance net
work as:
q=
𝑻𝟏−𝑻𝟐
𝟏
𝜟𝒙
𝟏
+
+
𝒉𝟏𝑨 𝑲𝑨 𝒉𝟐𝑨
• the overall heat transfer by combined conduction and convention is
frequently expressed in terms of an overall heat transfer coefficient
U, defined by the relation :
q = UxAxΔT overall.
• For hollow cylinder exposed to convection environment on its inner
and outer surfaces the resistance analogy is:
q=
𝑻𝑨−𝑻𝑩
𝒓𝟎
𝒍𝒏 𝒓𝒊
𝟏
+
𝒉𝒊𝑨𝒊 𝟐𝝅𝑲𝒍
𝟏
+𝒉𝟎𝑨𝟎
Heat transfer by radiation
• Continuous change of energy between bodies by the emission and absorption
of radiation.
• Bodies should have different temperatures to have heat exchange between
them.
• If two adjacent surfaces are at different temperatures, the hotter surface
radiates more energy than it receives and its temperature falls. The cooler
surface receives more energy than it emits and its temperature rises.
• Ultimately thermal equilibrium is reached. Interchange of energy continues,
and gains and losses are equal.
• Of the radiation which falls on a body, a fraction, a, is absorbed, a fraction, r,
is reflected, and a fraction, t, is transmitted. These fractions are called the
absorptivity, the reflectivity, and the transmissivity, respectively.
• Most industrial solids are opaque, so the transmissivity is zero and
a+r= 1
• Reflectivity and, therefore, absorptivity depend greatly on the nature of the
surface.
• Black body = a body which absorbs all and reflects none of the incident
radiation.
Exchange of radiation
• The exchange of radiation is based on two laws:
1. First law :
 Kirchhoff’s law: the ratio of the emissive power to the absorptivity is the same for all
bodies in thermal equilibrium.
 The emissive power of a body E, the radiant energy emitted from unit area in unit
time(j/𝑚2 .s)
 Body (1) with area A1 and emissivity E1, emits energy at area E1A1. If the radiation fall
on unit area of another body is Eb, the rate of absorption is Eb a1A1 (a1= absorptivity)
then:
 At thermal equilibrium: Eba1A1 = E1A1
 For another body in the same environment: Eba2A2= E2A2. therefor,
𝑬𝟏
𝑬𝟐
Eh= 𝒂𝟏 = 𝒂𝟐
 a for blackbody = 1
 the emissivity of a surface is defined as the ratio of the emissive power, E, of the
surface to the emissive power of blackbody at the same temp.
𝑬
e = 𝑬𝒃
 Emissivity is numerically equal to the absorptivity.
 Emissive power varies with wave length, so the ratio should be at particular wave
length.
 Gray bodies: the emissive power is constant fraction of the blackbody radiation; i.e.,
the emissivity is constant .
2.
Second law:
• Boltzmann law: the rate of energy emission from a
blackbody is proportional to the fourth power of the
absolute temp. T.
E = δ 𝑻𝟒 , where,
E: the total emissive power.
δ : Boltzmann constant (5.676 × 10−8 j/𝑚2 . S. 𝐾 4 )
• the heat emitted in unit time, Q from a blackbody of area
A is:
Q = δ A 𝑻𝟒
And for a body which is not black:
Q= δ e A 𝑻𝟒 (e: emissivity)
• Net energy gained or lost can be estimated by these
laws.
• Gray body in blackbody surroundings is the simplest case.
 None of the energy emitted by the body is reflected back in
case of a body radiating to atmosphere.
 If the absolute temp. of a body is T1, the rate of heat loss is
δ e 𝑇1 4 . Surrounding at temp.
T2 will emit radiation proportional to δ 𝑇2 4
Q will be = δ a A 𝑇2 4
 Since absorptivity and emissivity are equal:
 Net heat transfer rate = δ e A(𝑇1 4 - δ 𝑇2 4 )
 If part of the energy emitted by the surface is emitted back,
equations of various configurations take the general form:
Q = F1F2δA(𝑇𝐴 4 -𝑇𝐵 4 ). Where,
F1,F2 = factors determined by configuration and emissivity
of surfaces at TA and TB
Example:
A mercury in glass thermometer having E= 0.9
hangs in a metal building and indicates a temp. of 20 °c.
The walls of the building are poorly insulated and have
temperature of 5°c. The value of h for thermometer = 8.3
w /𝑚2 . °c. Calculate the true air temp. ?
Solution:
h A(𝑇∞ -𝑇𝑡 ) = δ A E(𝑇𝑡 4 -𝑇𝑠 4 )
8.3 (𝑇∞ -293) = (5.676 × 10−8 )(0.9) (2934 − 2784 )
T∞ = 301.6 °K = 28.6 °c
Mass Transfer
Mass transfer
• It is conceptually and mathematically analogous to heat
transfer.
• Some materials can be mechanically separated:
 Solids may be separated from liquids by arrest in a bed
permeable to the fluid (filtration process).
 Difference in density of two phases permits separation
(sedimentation and centrifugation).
 Many processes operate by a change in the composition of
phase due to the diffusion of one component in another
(diffusional or mass transfer process).
 Examples are: Distillation, dissolution, drying and
crystallization.
• Diffusion is the result of difference in the concentration of the
diffusing substance.
• Components move from a region of high concentration to
region of low concentration under the influence of
concentration gradient.
• In mass transfer operations, two immiscible phases are normally
present, one or both of which are fluid.
• In general, these phases are in relative motion and the rate at
which a component is transferred from one phase to the other is
greatly influenced by the bulk movement of the fluids.
 In most drying processes, for example, water vapor diffuses from a
saturated layer in contact with the drying surface into a turbulent
airstream.
 In the laminar layer, movement of water vapor molecules across
streamlines can only occur by molecular diffusion.
 In the turbulent region the movement of relatively large units of
gas, called eddies, from one mregion to another causes mixing of
the gas components. This mixing is called eddy diffusion.
 Eddy diffusion is a more rapid process, and although molecular
diffusion is still present, its contribution to the overall movement of
material is small.
 In still air, eddy diffusion is virtually absent and evaporation occurs
only by molecular diffusion.
Molecular diffusion in gases
•
•
•
Transfer of material in stagnant fluid or cross the streamlines of a fluid in laminar
flow occurs by molecular diffusion.
In the “Fig.” if the partition is removed some
molecules of A will move toward the region of B.
concurrently molecules of B diffuse towards regions
occupied formerly by A.
Variation in concentration of A is expressed as
𝒅𝑪𝑨
𝒅𝒙
•
•
(CA: conc. Of A)
𝒅𝑪𝑩
And variation in conc. Of B is: 𝒅𝒙 (CB: conc. Of B).
The rate of diffusion of A depends on the concentration gradients and on the
average velocity with which the molecules of A move in the x direction.
Ficks’ law expressed this relationship:
𝒅𝑪𝑨
NA = -DAB 𝒅𝒙 . 𝑊ℎ𝑒𝑟𝑒,
D: the diffusivity of A and B
NA: the diffusion rate of A
 D is dependent on temp. and pressure of the gases.
 NA is expressed as moles per 𝑚2 per sec. (moles/ 𝑚2 .sec)
 Units of D are 𝑚2 . 𝑠 −1
Equimolecular counter diffusion:
• If no bulk flow occurs in the element of length dx, shown in the Figure, the
rates of diffusion of the two gases, A and B, must be equal and opposite:
NA = -NB (if no bulk flour occurs in the dx)
• Partial pressure of A changes by dPA over dx and B by dPB
𝒅𝑷𝑨
=
𝒅𝒙
𝒅𝑷𝑩
(no bulk flow).
𝒅𝒙
• For ideal gases: PA V= nA RT. Where nA is the number of moles of gas A in a
volume V. Since the molar concentration, CA, is equal to nA/V,
PA= CART
𝑫𝑨𝑩 𝒅𝑷𝑨
𝑹𝑻 𝒅𝒙
𝑫𝑩𝑨 𝒅𝑷𝑩
NB = 𝑹𝑻 𝒅𝒙
DAB= DBA = D
𝒙𝟐
NA RT 𝒙𝟏 𝒅𝒙 =
NA = •
𝑷𝑨𝟐
-D 𝑷𝑨𝟏 𝒅𝑷𝑨
NA RT (x2-x1) =-D (PA2-PA1)
NA =−
𝑫 (𝑷𝑨𝟐−𝑷𝑨𝟏)
𝑹𝑻
𝒙𝟐−𝒙𝟏
DIFFUSION THROUGH A STATIONARY, NONDIFFUSING GAS
• An important practical case arises when a gas A diffuses through a gas B,
there being no overall transport of gas B.
• It arises, for example, when a vapor formed at a drying surface diffuses
into a surrounding gas.
 At the liquid surface, the partial pressure of A is dictated by the
temperature. For water, it would be 12.8 mmHg at 298 K. Some distance
away the partial pressure is lower and the concentration gradient causes
diffusion of A away from the surface.
 Similarly, a concentration gradient for B must exist, the concentration
being lowest at the surface. Diffusion of this component takes place
toward the surface. There is, however, no overall transport of B, so
diffusional movement must be balanced by bulk flow away from the
surface.
 The total flow of A is, therefore, the diffusional flow of A plus the transfer
of A associated with this bulk movement.
Molecular diffusion in liquids
• Equations are similar to those of gases
• The rate of diffusion of material A in a liquid is given by the
equation:
𝒅𝑪𝑨
𝒅𝒙
NA = -D
• Ficks’ law for steady state, equimolal counter diffusion is
then:
𝑪𝑨𝟐−𝑪𝑨𝟏
.
𝒙𝟐−𝒙𝟏
NA = -D
Where CA2 and CA1 are the molar
concentrations at points x2 and x1, respectively.
• Diffusivity in liquids varies with concentration from point to
point
• The total molar concentration varies from point to point.
• Diffusivities in liquids are very much less than diffusivities
in gases, commonly by a factor of 104 .
Mass transfer in turbulent and laminar flows:
𝑑𝐶𝐴
NA= -DAB
𝑑𝑥
• In the laminar flow region the equation
is applicable.
• In turbulent flow region the boundary layer is then considered to
consist of three flow regimes.
• In the most distant from the surface region, flow is turbulent and
mass transfer is the result of interchange of large portions of the
fluid.
• As the surface is approached, transition from turbulent to laminar
flow occurs. In this region mass transfer from eddy diffusion and
molecular diffusion.
• In a fluid layers at the surface,
laminar flow conditions persist.
This laminar sub layer offers the main
resistance to mass transfer.
• As flow becomes more turbulent the
thickness of laminar sub layer and its
resistance to mass transfer decrease.
• In general, the gas content of the laminar layer is so small that
PA and PAg are virtually equal.
• If molecular diffusion were solely responsible for diffusion,
PAg would be reached at some fictitious distance, x′, from the
surface, over which the concentration gradient (PAi - PAg)/x′
exists.
• The molar rate of mass transfer would then be
𝑫 (𝑷𝑨𝒊−𝑷𝑨𝒈)
𝑹𝑻
𝑿′
NA =
• However, x′ is not known and this equation may be written
𝑲𝒈
NA =
(PAi -PAg)
𝑹𝑻
• Kg: the mass transfer coefficient (m 𝑠 −1 )
𝑃𝐴
𝑅𝑇
• Since CA = , so
NA = Kg(CAi – CAg). where CAi and CAg are the gas
concentrations at either side of the film. Similar equations
describe the diffusion of B in the opposite direction.
• Diffusion across a liquid film is:
NA= k1(CAi – CAl). Where CAi is the concentration of
component A at the interface and CAl is its concentration
in the bulk of the phase.
• Mass transfer coefficient depends on the diffusivity of
the transferred material and the thickness of the
effective film.
• The thickness is determined by Re of the moving fluid
which is dependent on (µ, ρ ,V)
• Dimensional analysis gives the relation:
𝑲𝒅
μ
𝒒
= 𝑹𝒆 ( ). Where, Re: Reynolds number, K: mass
𝑫
𝝆𝑫
transfer coefficient ,D: diffusivity, d: a dimension
geometry of the system
𝐾𝑑
𝐷
μ
𝜌𝐷
= Nusselt group in heat transfer
= Schmidt number (sc)
INTERFACIAL MASS TRANSFER
• So far, only diffusion in the boundary layers of a single phase has been
discussed.
• In practice, however, two phases are normally present, and mass transfer
across the interface must occur.
• On a macroscopic scale the interface can be regarded as a discrete
boundary.
• On the molecular scale, however, the change from one phase to another
takes place over several molecular diameters.
• Due to the movement of molecules, this region is in a state of violent
change, the entire surface layer changing many times a second.
• Transfer of molecules at the actual interface is, therefore, virtually
instantaneous, and the two phases are at this point in equilibrium.
• Since the interface offers no resistance, mass transfer between phases can
be regarded as the transfer of a component from one bulk phase to
another through two films in contact, each characterized by a mass
transfer coefficient.
• This is the two-film theory and is the simplest of the theories of interfacial
mass transfer.
• For the transfer of a component from a gas to a liquid, the theory is
described in then Figure.
• Across the gas film, the concentration, expressed as partial pressure, falls
from a bulk concentration PAg to an interfacial concentration PAi.
• In the liquid the concentration falls from an interfacial value CAi to a bulk
value CAl.
• At the interface equilibrium conditions
exist.
• The break in the curve is due to the
different affinity of component A for
the two phases and the different units
expressing concentration.
• The bulk phases are not, of course, at
equilibrium, and it is the degree of
displacement from equilibrium conditions that provides the
driving force for mass transfer.
• Transfer of a component from one mixed phase to another, as described
previously, occurs in several processes. Liquid-liquid extraction, leaching,
gas adsorption, and distillation are examples.
• In other processes, such as drying, crystallization, and dissolution, one
phase may consist of only one component.
• Concentration gradients are set up in one phase only with the
concentration at the interface given by the relevant equilibrium
conditions.
• In drying, for example, a layer of air in equilibrium, (i.e., saturated with the
liquid) is postulated at the liquid surface, and mass transfer to a turbulent
𝑲𝒈
airstream is described by the equation NA =
(PAi -PAg).
𝑹𝑻
• The interfacial partial pressure is the vapor pressure of the liquid at the
temperature of the surface.
• Similarly, dissolution is described by equation NA= k1(CAi – CAl) , the
interfacial concentration being the saturation concentration.
• The rate of solution is determined by the difference between this
concentration, the concentration in the bulk solution, and the mass
transfer coefficient.