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Welcome back to the video course on fluid mechanics. Last lecture, we were discussing
about the pipe flow system; we have seen various aspects of lamina flow conditions at turbulent
flow conditions with respect to the pipe flow and over with respect the various losses also
with respect to pipe flow we have discussed the major losses and minor losses. In today’s
lecture, further we will discuss the various pipe losses. First, we will discuss the various
aspects of major pipe losses and then we will discuss the minor pipe process. As we discussed
earlier, in pipe flow with respect to the real fluid we have the shear stress on the
pipe wall and then shear force is there and then viscous velocity plays a major role.
One of the most important equations as far as major pipe flow is concerned is Darcy Weisbach
Equation. First, we start with the Darcy Weisbach equation the derivation of Darcy Weisbach
equation. Let us consider a pipe like this; flow is
in this direction and the diameter of the pipe is d and radius is r and it is at an
angle theta as central line theta angel theta as one here. So let us consider two sections
between 1 and 2,say, at section 1 the height of the central line the data height z1 and
that section 2 the data height z2 from the central line of the section which we consider
delta l. For such a section, let us consider the various fundamental equations are: first,
we apply the Bernoulli's equation between section 1 and 2. Here at section 1 the pressure
is given as p1 and section 2 pressure is p2 and the velocity of section 1 be v1 and velocity
section 2 be v2 and then due to the discuss effect and the shear stress effect there is
friction loss let it be represented as h is the head loss due to friction.
Let it be represented as h f. So, by applying the Bernoulli’s equation between section
1 and 2 we can write p1 by rho g plus p1 square by 2 g plus z1 equal to p2 by rho g plus v2
square by 2 g plus z2 plus h f . This is obtained from the Bernoulli’s equations. So here
h f is the loss with respect to the friction with respect to the viscosity effect between
section 1 and 2. The h f is what we want to find out from here in this equation from the
Bernoulli’s equations. So from the continuity equation we can write a1 v1 equal to a2 v2.
Now we consider the pipe flow; the diameter is same as a1 is equal to a2. So between section
1 and 2 we can see that here v1 is equal to the average velocity of section 1 and section
2 v1 is equal to v2 from the continuity equation. After using the continuity equation in this
equation number 1, we can write the head loss due to h f is equal to p1 by rho g minus p2
by rho g plus z1 minus z2 the difference in data head. This we can write as if p1 minus
p2, the pressure difference, so delta p by rho g. So h f is equal to delta p by rho g
plus delta z the diatom difference between section 1 and 2, so as in equation number
2. Now let us use the momentum theorem between
section 1 and 2. Here, we have seen that the pressure difference between section 1 and
2 is delta p. By applying the momentum equation along the flow direction, delta p into phi
r square, where r is the radius of the pipe plus rho g into phi r square in to delta l
sin 5. If forces acting are the pressure force then the weight of this fluid is obtained
as rho g into phi r square delta sin phi and then the shear force, tauw by 2 phi r into
delta l. So that should be equal to the change momentum rho is r rho q into v2 minus v1.
Since v1 is equal to v2, this is equal to 0 as the equation number 3, where q is the
discharge through the pipe and then r is the radius of the pipe. So here the forces acting
on this pipe element between section 1 and 2 are the pressure force and the weight between
section 1 and 2 and then shear force. So, the net force, the arithmetic force should
be equal to rate of change of the momentum. So rate of change of momentum is equal to
rho q is equal to v2 minus v1, where v2 is the velocity section into v1 is velocity at
section 1.
Here, we consider the flow as steady and fluid is incompressible. So with respect to all
these assumptions we can see that here since velocity v1 is equal to v2 rate of change
of momentum is 0 so that the arithmetic force is equal to 0 as in equation number 3. Now
here with respect to this figure the delta in this length. Now, z1 minus z2 that means
delta z, we can write as delta z equal to delta l sin theta.
After putting this delta z equal to delta sin theta in this equation number 3, we get
delta p into phi r square divided by rho g phi r square plus delta z is equal to toww
u into 2 phi r delta l divided by rho g phi r square, where tauw is the shear stress.
This we can simplify as delta p, this equation we can simplify as delta p by rho g plus delta
z is equal to tauw into delta l by rho g r. Now if you put in the equation number 2 by
considering this equation number 2 coming from the Bernoulli’s equations, we get h
f the head loss due to friction h f is equal to 2 tauw into delta l by rho g r.
Since delta p by rho g plus delta z is called, from the equation number 2 we get h f is equal
to 2 into tauw into delta l by rho g r. Now the wall shear stress tauw can be expressed
in functional form as, wall shear stress is functional of the density of the fluid, the
average velocity, dynamic coefficients of viscosity nu, the diameter and the reference
height.
So tauw is functional of rho v nu d and k is the average roughness height v is the average
velocity. If you do dimension analysis using Buckingham phi theorem which we discussed
earlier we can write this, we can drive an expression for this tau of u in terms of this
parameter rho v mu d k over in terms of r is number r u d and k by d. So from the Buckingham
phi theorem we get 8 tauw by rho v square is equal to as a functional of f functional
Renolds number and k by d, where k is the average reference height and d is the diametric
in the pipe and this is equal to f, where f is the friction coefficient.
We can show this 8 tauw by rho v square is the friction coefficient as per the pipe flow
is concerned. Now this is what we got from the Buckingham phi theorem. We can put it
back here with respect to tauw and then we can get an expression for the capital as due
to friction hf we can obtain.
Hence, tauw is equal to f rho v square by 8 that can be written as f by 4 into half
rho v square. We can write h f is equal to 2 into tauw delta l by rho g r. That is equal
to up to substituting for tauw hf is equal to 2 f rho v square and delta l by 8 rho g
into r.
So that is equal to 2f rho v square to delta l by 8 rho g d where d is the diameter. So
d into 2 times, r is equal to d by 2 when we write this in terms of 2 b 2 into delta
by d. So finally we get the hf as hf is equal to f l v square by 2 g d. So here delta is
replaced by the length which we consider. Now we have to derived the Darcy Weisbach
equation by considering the momentum principle and the Bernoulli equation between section
1 and 2 and then this equation is applicable for most of the fluid flow for both laminar
and trebling conditions and it is encounter of depth flow it is essential that flow should
be the circular in wash section it can be any kind of pipe flop and the Buckingham phi
equation is applicable to both laminar and turbulent conditions.
So this Darcy Weisbach equation as in the basic equation and the fundamental equation
under very much use decoration as for as the head loss calculation with respect to friction
loss between two sections in any kind of the pipe line problems or pipe flow systems.
So now the equation which is considered for Hagen-Poiseullie flow as per as laminar low
condition which we considered earlier, for Hagen-Poiseullie flow we can write v is equal
to delta p d square by 32 mu l, where delta p is the pressure difference is d is the diameter
mu is the coefficient dynameters viscosity and l is the length. So v is equal to delta
p d square by 32 mu l and head loss, we can represent as hf pc equal to p1 minus p2 by
rho g that is equal to delta p by rho g. That is equal to by using this equations here 32
mu here by d square into rho g that is equal to f l v square by 2 g d.
As far as laminar fog condition is concerned we can say that friction factor f is equal
to 64 by R r e, where r e is the known number say rho d by mu as per as the pipe flow constraints.
Finally, we get the coefficients f is equal to 64 by r e for known number considering
the Hagen-Poiseullie flow as in the case of laminar flow conditions. That way we can show
that the loss is Buckingham equations which we have derived is valid for laminar flow
as well as less treble flow conditions. Starting from this for the earlier case, consider the
Bernoulli’s equations and the momentum theorem we got the Darcy Weisbach equation. Similarly
the by consent the Hagen-Poiseullie flow also we get the same expression hf is equal to
f l v square by 2 g d.
Now, let as discuss more details about the this Darcy Weisbach equation and then the
coefficient friction, since in most of the pipe flow systems one of the most important
aspect as far as flow is considered is the coefficient of friction. So this hf is equal
to f l v square by 2 g d the head loss equation is given as Darcy Weisbach equation. Here
we can see that the head loss is given in terms of the friction factor f. You can see
that the head loss is directly proportional to the friction factor f and directly proportional
to the length of the pipe and that is proportional to the score of the velocity average velocity
and inverse proportional to the diametric d.
With respect to this Darcy Weisbach equation we can calculate the head loss, for any kind
of pipe process systems. We can calculate the head loss by using the Darcy Weisbach
equation in terms of the friction factor f in terms of the length and velocity square
and with respect to the diameter of the pipe. This Darcy Weisbach equation is considered
to be the best empirical relationship for pipe flow resistant calculation. In most of
the pipe flows which we considered, we have to calculate the head loss with respect to
the friction. So this equation the Darcy Weisbach equation is one of the most poiseuille equations
as far as head loss is considered and this equation is equivalent to the Hagen poiseuille
equation for laminar flow as we are shown.
The last light only exception is that the empirical friction factor f is introduced
here. So with respect to this here, we can see this empirical friction factor is f is
one of the important aspects as far as Darcy Weisbach equation is considered. So we have
say if f depend upon various parameter like material of the pipe and then flow conditions
and then over the pipe the various other parameters with respect to whether the material at the
pipe etc. Here by considering Darcy Weisbach equation, we have to see the friction coefficients
or friction factor as far as the considered pipe flow.
Here, with respect to the Darcy Weisbach equation hf, the major loss is the energy or head loss
h first the length units due to the friction between the moving fluid and the duct. So
here what we consider is same with respect to the pipe flow which we considered. When
we consider any kind of pipe flow like this, so here same between 2 sections so here the
head loss between if d is the diameter of the pipe and l is the length between this.
Here by considering you can see that the head loss between say section 1 1 2, 2 2 2.
We can see that here the friction factor is most of the important parameter and then we
get the energy loss, due to the friction between in mu in fluid and then deduct in terms of
as in the length unit. So friction factor f is the most important parameter in the Darcy
Weisbach equation and f is complex functional of the Renolds number and relative roughness.
So, various experiments are conducted with respect to various pipe flow systems. It was
shown that this friction factors f is depend upon the relative roughness of the pipe that
is pipe material and it whether it is smooth or rough. So f is the friction factor depends
upon the pipe materials roughness and also the Renolds number with respect to the flow
conditions. So f is functional of Renolds number and relative roughness and Renolds
number for pipe flow is rho due by mu.
So now depending upon the case whether the flow condition and also the pipe material
then whether the pipe is smooth or rough, we have to find this friction factor. For
example, now let us consider the various choices of f for smooth pipe for laminar flow that
means the Renolds number is less than 2000 is a view in the calculation we will not have
to worry about the friction factors; pipe reference is not a factor.
We can directly obtain the pipe reference the friction factor, directly calculate physical
to 64 byte Renolds number and for turbulent flow, up to this between four thousands to
the into between the range of 10 to the power of 5 to 4 four thousands Renolds number range,
blasius calculated this friction factors equal to 0.3164 divide by r e to the power 0.25.
So like this various conditions is based upon various experiment and some empirical relationship,
this friction factor has been calculated for various kinds of pipe flows for various flow,
various pipe material and reference factor. So as far as laminar flow is considered we
can directly get with respect of Renolds number friction factor f is equal to 64 by r e and
then as for turbulent flow is considered say number r is shown through experiment that
is f is equal to 0.364 by r into the Renolds number to the power 0.25.
But in other ranges also this equation is available; we will be discussing details about
this friction factor and various relationships available in literature. Now here the choice
of f for rough pipes as I mentioned, there are empirical formula and diagrams to determine
the friction factor f depending upon the roughness the Renolds number. So this is commonly used
methodology include the moody’s diagram, swamee and Jain formulae and nikuradses experiment
results and Colebrook white equations. So these are some of the commonly used methodologies
to estimate the friction factor f. We will discuss each of this methodology in details
as far as the friction factor for roughness for considered.
First let us see the moody diagram. This moody diagram has been derived by conducting large
number of experiments at various flow conditions of fluid, various Renolds number and also
fluid through various dimension diameter pipes and various smooth, rough and different kinds
of roughness gives a large number of experiments were conducted and this moody diagram has
been derived.
So this moody diagram is used for rough pipe in turbulent flow condition. We use the moody’s
diagram to obtain the re friction factor f. This moody’s diagram is obtained through
experimental data that relates the friction factor to Renolds number and then it is obtained
for fully developed pipe flow over a wide range of wall roughness.
So from the moody’s diagram depending upon the Renolds number what kind of same the turbulent
say it whether it is transmission is turbulent, deferent ranges of turbulent flow we can obtain
the friction factor with respect to the wall roughness friction factor is we can directly
obtain to do the other calculation.
The turbulent portion of moody chart we can represent by the Colebrook formula even as
1 root f is equal to minus 2 log 10 e by d divide by 3.7 plus 2 phi 51 re root f, where
f is friction factor, r is the Renolds number is the reference height d is the diameter
phi.
So this turbulent portion of moody chart experimentally shown these values and also it can be verified
that fluid values in the moody chart are very similar to the Colebrook formula even by these
equations. So here this line shows the moody’s diagram
here the x axes the Renolds number is represented and on this access the friction factor is
given and here the curves are given for various e by d ratio where e is the represent height
the relative reference e by d is the diameter e by d is equal to 0.03 like that for various
values, the roughness for various roughness are e by d issues and the Renolds number the
friction factories even here.
So there is more pipe range whenever it is almost the e is almost 0 that means smooth
pipes so here e by d is 0 that is the rearmost here and then we can consider that depending
upon the Renolds number we can obtain the friction factor with respect to various ratios
of e by d. We can obtain this pair wall this line for lamina flow and this range is for
the transition. Actually we can see that moody diagram is commonly used for to find out the
friction factor for turbulent for region is that is where we use this f factor friction
factor commonly. So it is given for various scales are given for various e by d ratio
as shown in this slide and then as a mentioned the various methodologies are available.
First one which we discussed is the moody’s diagram and then log number of other relationship
same to obtain the friction factor are available in the tracker. So few of this relationship
we will discuss here. Next one is swamee and jain formula, these
are the some of the empirical relationship derived by this swamee and jain by depositing
the large number of experiments .Here swamee and jain drawn that f is the friction factor
is f is equal to 1.325 natural log 0.27 in the e by d plus 5.74 into 1 by re to the power
0.9 over to the power minus 2 as shown this equation and this is valid friction factor
this is wide in the range of e by d ratio 10 to the power minus 6 to 0.01 and the Renolds
number range of five thousands to 3 into 10 to the power 8.
For this range this equation friction factor is valid as derived by the swamee and jain
and then head loss directly we can obtain hl is equal to 1.07 into fuel square l divide
by g in to d the power 5 and natural log e by 3.7 d plus 4.62 into new d by q to the
power 0.9 what the power minus 2 where v is the discharge are the pipe and diameter and
new is the coefficient sky metric and viscosity is the reference height, the pipe line thereon
and l is the length which is considered.
So this equation is the head loss here is the same range with respect to a factor here.
So this equation is valid at the range of e by d ratio of 10 to the power minus 6 to
0.01 and inverse number range of five thousands into three into 10 to the power 8 and then
we can observe Jain derived the equation for discharge.
So discharge q is equal to minus 0.965 g into d to the power 5 hl divided by l to the power
0.5 natural log e by 3.7 d plus 3.17 new square into l divide by g d queue into hl over to
the power 0.5. So this equation for the discharge is valid for inverse number greater than 2
thousands as given by swamee and jain and here is the reference site and new is the
kina metric viscosity and then they also got to design a pipe, the derived you can see
that while designing the pipe for the given discharge it depends upon various parameters.
So some the equations which we swamee and jain derived based upon this equation they
got an expression friction directly derived and expression for diameter for the pipe so
d is equal to 0.6 d to the power 1.25 l q square by g h l to the power 4.75 plus new
q to the power 9.4 l by g h l in to the power 0.2 to the power 0.04. Here, hl represent
to head loss this hl represent the head loss, so long as I mentioned this relationship derived
by based upon log number experiment and then using some of the available empirical relationship
already available relationship, various hagnes derived from number equations as far as the
frictions factors, head loss and also discharge over the design a pipe find out the diameter
of the pipe various relationship are derived.
So this equations shows the relationship as for as friction factor, head loss discharge
and diameter of the pipe is concerned derived by swamee and jain and then in the literature
some of the other important relationships as given by nikuradse’s through his experiments.
He also produced some charts with respect to the friction factor by relating to the
Renolds inverse number.
So nikuradse’s conducted large number of experiments for rough and smooth pipes, number
of difference, kinds of pipes and different materials are shown but for a rough pipe,
the mean height of the roughness is greater than the thickness of the laminar sub layer.
So nikuradse’s through his experiment showed that for rough pipe, mean height of roughness
is greater than the thickness of the laminar sub layer and he conducted by Nikuradse’s
conducted all these experiments by artificially referring the pipe by coating them with respect
design. So the laboratory he produced number of section the different kinds of pipes by
sand coating, artificially roughening the pipe by sand coating and then he produced
gross for f verses Renolds number for f against the friction factor against Renolds number
for range of relative roughness 1 by 30 to 1 by thousand and fourteen.
So various ranges of relative roughness Nikuradse’s produced the graph for f verses that means
friction factor f is the Renolds number stating from 1 by 30 relative reference 1 by 30 to
1 by 1000 and 14. So this graph shows different regions for various flows. So here this is
the graph derived by Nikuradse’s with respect to large number of experiments he carried
out in the laboratory. Here this is very similar to the moody’s diagram which we have seen
earlier. So here friction factor on log scale is put on this is on log scale and the y axes
and Renolds number, on log scale is new on the x axes and then relative roughness given
here like aspirant by d and r e by d in e is the roughness height.
So 1 by 31 by 60 like that for various relative roughness and Nikuradse’s produced this
graph and here this is for rough turbulence on and here is this region is for transactional
turbulence on and here this is smooth turbulence smooth by smooth by and this is the laminar
array. So after conducting large number of experiments with respect to various flow conditions,
with respect to various pipe materials, with respect to various roughnesses nikuradse’s
produced this graph. So from this graph also we can obtain then friction factor for various
Renolds number and also various relative roughness. So first we discussed moody diagram we have
seen that swamee jain formula and now third one is the Nikuradse’s plot and next one
is the Colebrook equations. Colebrook derived some of the relationship with respect to the
friction factor and Renolds number.
Here Colebrook showed that smooth pipe flow 1 by root is equal to 0.86 natural of r e
root f minus 0.8. So this is for smooth pipe flow and in transition shown that means between
the smooth to rough that transition is shown that 1 by root f is equal to minus 0.86 natural
log e by 3.7 d plus 2.51 divided by r e to power root f, where r is the Renolds number
and fuse the friction factor and d is the diametric is the height of roughness height
and then for completely turbulent zone, this can be reduced to the condition for fully
turbulent and rough zone. Here this is 1 by root f is equal to minus 0.86 natural log
e by 3.7 d this can be approximately from this equation.
So this equation is called Colebrook equations for friction factor for smooth transition
for completely rough and smooth type flow. As we have seen the moody’s diagram is very
similar to what is given by Colebrook equation. So Colebrook equation is also the most accurate
kind of equation as far as friction factors is considered and another kind of equation
is could hazen William formula for friction factor here.
This is given as f is equal to thousand fifty nine divided by c to power 1.85 d to power
0.02 re to the power 0.15 where c is the hazen William coefficient, we will discuss about
the coefficient later d is diameter r is the Renolds number, diameter in millimeter here.
This f is given by the friction factor is given like this by hazen William formula and
then another important formula using literature is barr formula.
So here the barr formula friction factor is given as 1 by root f equal to minus 4 log
time e divide by 3.71 d plus 5.1286 divided by r e to the power point 0.89. Here also
this is the relationship between the friction factors and relative roughness and the Renolds
number
Like this there are number of few more formula relationships available for friction factor
in turbulent flow conditions. Most of the equation is relates friction factors with
respect to relative roughness and the array Renolds number as we have seen Colebrook equation
are embolic equation and also as soon by Nikuradse’s. So out of the chart and equations and like
moody diagram and the Nikuradse’s charts, we can see that moody diagram is one of the
most accurate in determining the friction factor so also we can that the approximation
given by Colebrook and barr’s equation are very close to the what is given in moody’s
diagram.
In most of the design pipe design analysis we can use this moody diagram to get the friction
factor with respect to the Renolds number and the relative reference since the moody
diagram is prompt to be another most accurate relationship power chart of the variable,
the results are very similar to what we get from the Colebrook equations over the barr’s
equation. In most of the designed pipes design is considered either we can use the moody
diagram or we can use the Colebrook equation. Now we have seen this when we discussed about
the major loss as far as pipe friction is considered. One of important factor is the
roughness height which is inside in the pipe wall. So we can see that with respect to various
materials we have seen that with respect to various materials this roughness height is
changing. With respect to natural or interior surface and equivalent roughness in terms
of millimeter is given here for various materials.
Typical roughness values are given, for example, copper, lead, brass, glass, and plastic. So
phis of this material and the roughness vary from 0.001 to 0.0025 mm and then if the pipe
is riveted steel it goes from large value like 0.929 mm and steel commercial pipe this
roughness equivalent roughness varies from 0.045 to 0.09, cast iron pipe is concerned
then we can see that increment reference high varies from 0.25 to 0.8 mille meter and concrete
pipe depending upon the finishes.
So we can see that concrete or wood is considered, the roughness side or whether the pipe is
smooth or pipe is rough depends upon the finishes given to the material. Here for concrete pipe
this equivalent roughness vary from 0.323 mille meter and wood stave we considered this
varies from 0.18 to 0.9 .
So depending upon the material considered and the finishes is given and we can see that
the roughness height changes. So equivalent roughness height changes and accordingly we
can see that efficient factor all those changes as given in the various relationships various
methodology which we discussed like moody diagram is Colebrook equations like depending
upon the relative roughness the friction factors changes and according to friction factor changes
the head loss changes.
All these important aspects we have to consider in the design of pipe flow system design of
the pipes considering the considered typical system and now, we can see that one of the
most commonly used equation for the pipe design other than the Colebrook equation which is
obtain the re friction factor of ten by Colebrook other than this one of the commonly use this
equation hazen Williams equation. So hagzen William equation for velocity is given as
v is equal to k in to c in to r s to the power 0.63 s to the power 0.54. So here this equation
is valid for water at temperatures typical city water supply system range from 4 to 25
degree centigrade.
So here s is equal to h f by l h is with respect to head loss h f by l and q is then the discharge
is given as Varian velocity is v into a and r j is the if d by four that is with respect
to the a by p that means we have to d s obtain by rh is equal to d by 4 for circular pipe
and k is a unit conversion factor as far as hazen Williams formula is considered. so in
the ps system k is equal to 1.318 and si system it is 0.85 and rh is the hydraulic radius
as discussed. So hazen William’s equation is one of the formula equations as for as
pipe flow design is considered other than the Darcy Weisbach equation and Colebrook
equations which we discussed initially.
But Darcy Weisbach equation is much more accurate then the hazen William equation. So hazen
Williams method is one the popular methodology for pipe design among civil engineers because
its friction coefficient this c here this coefficients c is not a functional velocity
or duct diameter, so in literature we can see that various values for c are given.
So and hazen Williams equation is similar than d that is the Weisbach equation for calculating
the flow rate velocity or diameter depending upon the which parameter we are calculating
for which we are designing, so Darcy Weisbach equation is one of the most commonly equation
other than that hazen Williams equations is also using in the pipe design.
So, the hazen Williams friction factor which we discussed depends on the material, we can
seen in literature various values are given for c like asbestos cement c is equal to140,
cast iron 130, whether it is new old say twenty years forty years old c changes like this
concrete lined steel pipe system it would be about 140 or lined wooden forms c to 120,
or a copper pipe, copper material is concerned 130 to 140. Like this we can see this factor
c as far hazen William equation is.
So far we have discussed the major loss in a pipe systems is considered is mainly with
friction loss h f for hl, what we considered the head loss for the friction with respect
to head long friction. One of the most commonly used equation is the Darcy weisbach equation
which we discussed only things is that in Darcy Weisbach equation we have to find out
the friction factor and other commonly used equation is the hazen williams equations for
the pipe design. So for pipe design is considered we can you say the Darcy Weisbach equation
as hazen Williams equations or some of the other methodology is other equation available
in literature.
So that is as far as the major loss is considered what we discussed so far. Now you will discuss
in detail about the minor process in pipe flow, we have seen earlier that pipe flow
process are considered there are major loss due to the friction loss and minor loss, losses
due to its condition like expansion contraction or due to various is pipe fitting s like bans
than t junction harm, various connections there will be minor loss.
Now we will discuss in details about the minor loss in pipe flow as we discussed in the last
lecture the minor loss occur due to the presence of valves, elbows, joints are contracts functions
etc in the pipeline. Generally, a minor loss is especially in terms of a loss coefficient
k as a function of the velocity. So here seen this major loss already so like that in minor
loss also we represents the minor loss in time of the velocity as a functional of velocity
a multiply by a coefficient k so generally hl the head loss for minor loss is concern
hl is equal to k in to v square by 2g.
Where v is the average velocity, k is the coefficient for minor loss is the acceleration
due to gravity when k is determined experimentally for various fittings and geometric changes
of interest in piping system. As we discussed earlier depending upon a problem there would
be different kinds of joint are different kinds of contract expansion junction etc.
So depending upon the condition same for various depend upon the material which is use for
pipe construction we can have varies of this k, so either it can determine through experiment.
Sometimes depend upon the conditions manufacturing a give the values source also.
So now here we discussed various aspects of minor loss with respect to equation hl is
equal to k in to v square by 2 g. As we discussed the minor losses are in pipe line are consider
minor losses can be due to sudden expansion, sudden contraction, gradual expansion, entry
and loss exit loss and also losses the due to pipe components like loss due to valves,
bends, tee, elbow etc.
So as we have seen, for example if a connecting here the vary say time here and now if we
are connecting, putting a pipe line like this with various branches and then various joints,
you can see now in a such a system we may be distributing either for water supply or
may be for various other kinds of network.
We can seen that we may be continuing the flow like this, so such a system in the pipe
network like this we can see that here, may be connecting to another tank, so and here
this flow may be continuing so we can see here the various connections and here if this
is the tank or reservoir. So from here the flow takes place. So in such a system we can
see that there can be water enters from the tank to the pipe so there is an entry loss,
so here there is an entry loss and then here we provide that bench so there can be loss
to be bench and then here we can see that there is a junction here. There we may give
a junction, so junction loss and then we may provide an l board here over the can depend
upon the pipe line diameter changes there can be expansions, so here also we can have
expansions over here we can have contraction.
So like that and then now finally when now this pipe joins here then this tank then we
consider that again expansion, so and here we provide there is wall and then flow continue
here so this is a wall. So like this here there is a again a junction so there are number
of components there is such a flow system in there is number of components is including
the expansion, contraction etc., exit then entry loss exit loss, so we have to consider
all these losses as far as pipe flow is considered.
Now discussing in details, first let us consider the loss due to the sudden expansion. Let
us consider the figure below, here you can see that a pipe is flowing, so here the pipe
width is more diameters and then it is connected to pipe with log diameter. We can see that
will be a sudden exponential like this.
Flow is coming like this and then expanses. Here let as consider 2 sections here section
1 and section 2 and section 1 area flow section is a1 and velocity average velocity v1 and
pressure p1 and section 2 area flow section a2 and velocity is v2 and pressure is p2.
Let us consider the flow to be steady and fluid incompressible turbulent flow and velocity
is uniform with respect to the assumption.
If you assume shear force on pipe wall of short length between 1 and 2 is negligible
and mean pressure p dash of the eddying fluid in the expansion is almost equal to the pressure
p1. From Newton’s equations motion we can write p1 into a1 minus p2 into a2 plus p dash
into a2 minus a1 is equal to p1 minus p2 into a2 as in equation number 1.
So this we can simplify as this p1 a1 minus p2 a2 plus p dash a2 minus a1 equal to p1
minus p2 into a2. So here we assumed that p dash equal to p1 that is why, we got p1
minus p2 into a2 this equation and then this with respect to change of momentum. We can
write rho v2 this is equal to rho v2 into a2 v2 plus rho v1 into minus rho v1 a1 b1.
So this is the momentum. Finally we get v is the continuity equation we can obtain p1
minus p2 is equal to rho into v2 into v2 minus v1 as in equation number 1 b.
So here we consider the flow like this here one section and another is here. With respect
to this equation Newton’s equation of motion we got p1 minus p2 is equal to rho v2 into
v2 minus v1. Now the energy equation if applied between section 1 and 2 we get v1 square by
2 g plus p1 by g is equal to v2 square by 2 g plus p2 by g plus h l, where h l is the
head loss.So here is an between this section to this section if hl is the head loss we
get v1 square by 2 g plus p1 by g is equal to v2 square by 2 g plus p2 by g plus h l.
So now solving by p1 minus p2 by g from this equation and if we use this equation 1 a and
1 b and we can write v2 square minus v1 v2 by g is equal to v2 square minus v1 square
by 2 g plus hl. We obtain from equation using equation 1 a
and 1 b and equation number 2. So finally we get the head loss h l is equal to v1 minus
v2 whole square by 2 g. That can be represented as v1 square by 2 g into 1 minus a1 by a2
whole square. So here this loss coefficient here this term this already in this head loss
with represent is k into v square by 2 g. So if consider this v1 square by 2 g has a
function so we obtain here this loss coefficients k is equal to 1 minus a1 by a 2 whole square.
So here it is obvious that head loss various as this square of the velocity and it is true
for all minor losses.
So here if a2 is extremely large like pipe opens to a reservoir we can see that is a1
by a2 if a2 is argue that 0 then by obtained k is equal to 1. So like that we can find
out for various cases of exponential so with respect this k is equal to 1 minus a1 by a2
whole square. So where a1 is the rho section here at this section the smaller section and
here expansion section rho section is a2. So if a2 is extremely large, we have seen
the complete kinetic energy of flow is dissipated as here the same when the flow is going from
here this pipe flow is released to tank receiver so we can see that it is a completed expansion
so total kinetic energy of the flow is decapitated here. So it is complete expansion. So this
is the expression for the expansion. Similarly, if we consider the contraction loss due to
sudden contraction let us consider the following figure.
Here this section 1, this is the large diameter then smaller diameter, so here the velocity
is v1 at section 1 and here the velocity section 2 is v2 and then we can see that in the case
of contraction there will be a jet is formed then we can see that here as smaller section
will be there which is called in contractor. So for loss due to contraction we can write
sc is equal to v0 minus v2 r square by2 g where v0 is velocity at this in a contractor
and v is the velocity here at this junction. So we can show that sc is equal to head loss
due to contact is equal to v 0 minus v2 r square by 2 g, from continuity equation, you
can write v 0 the velocity here into cc into coefficient contraction into a2 is equal to
v2 a2, so cc is the coefficient of contraction and v0 is the velocity at section here at
in a contractor. So the head loss due to contract obtained is sc is equal to v0 minus v2 r square
by 2 g.
With respect to the vena contractor it is the section of the greatest contraction of
the jet here. We can see that the jet here, the greatest contraction and hence the head
loss can be written as hc is equal to 1 by cc minus 1 r square into v2 square by 2 g.
With respect to this appropriation here the head loss with respect to this we can write
in terms of this coefficients of contraction hc is equal to 1 by cc, where cc is the coefficients
of contraction; hc is equal to 1 by cc minus 1 whole squared into v2 square 2 g. Here you
can see that the expression is in times of the velocity v2. So Weisbach is calculated
by with respect various ratios are a2 by a1 and here is calculation of the coefficient
of contraction, so that is given the tracer here you can see here the 0.1 cc is point
a2 by a1 is 0.1, 0.6. Like that various values are ambient calculated through experiments
by Weisbach. In the next lecture, we will be discussing more about the various losses;
then we would discuss various aspects of the pipe flow design and then pipe flow system
for various conditions.