Lec-14 Thermodynamic Property Relations-II
Good morning. I welcome you to this session.
In the last class, we started the discussion on important thermodynamic property relations.
We will do that in this class, continue that. Just we started the discussion.
Today, what we will do, before continuing that, these thermodynamic property relations,
just we will recapitulate two important mathematical theorems, very simple, which we know just
as information, we have to recapitulate. These theorems are like that. Let us see that if z is expressed as a function
of two independent variables x, y. It is a very simple mathematical theorem. z is expressed
as two independent variables x and y. Then the total differential of z, dz can be written
like this; Del z divided by del x at constant y dx plus del z divided by del y at constant
x dy. Now, if this coefficient I write as M and del z divided by del y at x N dy, then
it can be written that del N divided by del y; that means, the differentiation of this
coefficient with respect to y at constant x is equal to del N divided by del x at constant
y. It is very simple, because M is nothing but
this one; del z divided by del x at y that means, other way, we can see that if we differentiate
this with respective y at constant x, what will happen? Del M divided by del y at constant
x means the differentiation of this with respective y. That means del square z at constant x,
that means this restriction will go. That means del y divided by del x. Very simple
mathematical theorem. Similarly, if we write for del N divided by
del x at y, that is differentiate this expression with x at constant y. This constant x restriction
will go. This becomes equal to del square z divided by del x del z. We know that if
z is a continuous function of x and y then these two double differentials are equal,
even if the order is different; that is del square z divided by del y del x is del square
z divided by del x del y. In any order, the second order differentiations are taken. They
are equal, provided z is a continuous function of xy. Therefore, this is one of the very
important relationships in mathematics which is used in deriving some important thermodynamic
property relations. If dz is expressed as M dx plus N dy, where
M stands for the differentiation of z with respect to x at constant y and N stands for
differentiation of this variable with respect to y at constant x. Then the differentiation
of this with respect to y at constant x has to be differentiation of this coefficient
with respect to x at constant y. Another important mathematical theorem is there. It will be
very much useful in deriving different thermodynamic property relations.
Three variables x, y, z can be expressed as a functional relationship like this; x, y,
z, the three variables are dependent on each other. It can be expressed by a functional
relationship of that so that any of these variables, for example, x can expressed as
the function of other. Similarly, y can be expressed as a function of other two and z
as a function of other two. Each one of these is related to the other two. If these three
variables are related by a functional relationship so that one can be expressed as a function
of the other two then there occurs a relationship like that, is cyclic. This is known as cyclic
relationship. If we differentiate x with respect to y at
constant z then del y divided by del z del at constant x and del z divided by del x with
these at constant y. They equal to minus 1. Let us prove this theorem is beyond the scope
of this course. So I am not going to prove it because this is not a mathematics class,
you have done it in mathematics class. But this information will be required.
So, similarly, just an immediate consequence of this in thermodynamic property relations,
first I will tell that if x, y, z is replaced by p, V, T. We know, these three, pressure,
volume and temperature are the basic measurable and sensible properties of a system. Pressure
and temperature are sense process and they are directly measurable properties. Sometimes,
we call these three are the primary properties or fundamental properties of a system. They
are related by a functional relationship for any system which is known as equation of state.
We will see afterwards that the relationship between pressure, volume and temperature for
a given mass of a system that means for a close system is known as the equation of state
of that system. Therefore, an immediate application of this
mathematical theorem tells us that del p divided by del V at constant T into del V divided
by del T at constant p into del T divided by del p at constant V is minus1 which is
again a very important relationship because this is derived using this mathematical theorem
for our thermodynamic property relation. Sometimes, this is known as cyclic relations; del p divided
by del V, del V divided by del T and del T divided by del p. So, this is one of the very
important thermodynamic relations. Now, I come to the property relations which
we started in the last class. We started with four very important extensive properties.
One is the u that is the internal energy is basically the intermolecular energy which
neglects the kinetic energy, internal energy. Another is the H that is enthalpy which equals
to u plus pV. Another is the F. F is the Helmholtz function, Helmholtz energy, or sometimes it
is known as Helmholtz function. Helmholtz function or Helmholtz energy sometimes
the concept energy comes, because the units are all energy. That way enthalpy is also
an energy unit, but this is not exactly an energy quantity. That we must know. So, far
we have learnt this thing, because the difference of enthalpy gives the energy quantity, heat
and work interactions in an open system. So, enthalpy is very similar to the energy whose
dimension is energy but is not exactly the energy.
Similar way, Helmholtz function is like this. That is why, sometimes it is known Helmholtz
energy whose definition is u minus Ts. This is a system property. All these u, T, s are
that of the system. Because earlier, we have seen that the availability function for a
close system is u plus p0V minus T0s. So they are p0, T0 at the surrounding properties,
where F is u minus Ts. Similarly, G which is defined as Gibbs energy or Gibbs function;
this is defined as H minus Ts. This H, T, s are all of system; that means, they are
all system properties. u, H, F, G is four very basic properties of
a system, where u is the internal energy which comes from the first law. H is by definition
u plus p V. F, G come from the, actually this concept of physical implications or concept
of significance of F and G. I told that this signifies the maximum amount of work that
we can extract from a closed system when it interacts with the surrounding at an isothermal
process and the temperature of system and surrounding remaining the same.
Similarly, when an open system interacts with the surrounding and isothermal process where
the temperature of these surrounding and the system remains same, then the difference of
Gibbs energy gives the maximum work done. Similarly, the difference of Helmholtz function
gives the maximum work done in closed system, as far as the physical explanation or physical
significance of Helmholtz function and Gibbs function are concerned. But at this present
moment, it will be sufficient to know, how these Helmholtz function and Gibbs function
are defined. Afterwards also, when we will start the reactive
thermodynamics or thermodynamics of reactive systems, we will study this in this course.
Then we will be able to find out the physical significance of Gibbs function in more detail.
At the present moment, if we just recognize the definition of these four functions, now
I can write the differential form or a differential expression of this. du is T ds minus p dV.
This relationship we know T ds is du plus p dV is a property relations which have been
found from the first law of thermodynamics. Similarly, dH exploiting this, the du plus
p dV plus V dp and du plus p dV is T ds. So, dH becomes T ds plus V dp. Next one, dF is
equal to, now here if I use this definition dF is du minus T ds minus s dT. So, du minus
T ds from here is minus p dV. Therefore, dF becomes minus p dV minus s dT.
Similarly for dG, by differentiating G is equal to H minus Ts equation both sides, dG
is dH minus T ds. So, dH minus T ds is V dp. So dG becomes V dp minus s dT. From the definitions
of these functions, I can write four expressions in terms of the differentials of all these
properties u, H, F, G. u, H, F, G are the point functions and they
are expressed in terms of differentials of other point functions ds, dV. In this case,
we can exploit this theorem. This means that u as a function of s V. This equation tells
that u as a function of s V. H as a function of s p. F as a function of V T. If I start
with u as a function of s V; then in that case, T is nothing but del u divided by del
s at constant V into ds. Then minus V is del u divided by del V at constant s into dV.
Similarly, if I could have expressed H as a function of s p, we could have written this
expression. Similarly F as function of V T, and G as a
function of p T; This expression shows that these point functions are expressed in terms
of the differentials of the other two point functions as independent variable. We can now just have a recapitulation that
as z is a function of x, y and if it is written dz this M dx plus N dy, where M and N are
these two coefficients. Then one can write, del M divided by del y at constant x is del
N divided by del x at constant y. So, exploiting that functional relationship
from each of this equation, now we start with the first equation that del T divided by del
V at constant s is equal to minus del p divided by del s at constant V. Now, here also, we
see that del T, del p. Here also, from the second equation del T divided by del p at
constant s is equal to del V divided by del s at constant p. From this equation, we get,
del T divided by del p at constant s is del V divided by del s at constant p.
From the third equation, in this equation both are negative. So, we can write with positive
things only; del p divided by del T at constant V is equal to del s divided by del V at constant
T. From the last equation, we can write, del
V divided by del T at constant p is minus del s divided by del p at constant T. These
four equations are very useful thermodynamic relations and they are known as Maxwell’s
relations or equations. At the same time, we can write other few important
relations. By recognizing that when I write this u as a function of s V then T is equal
to del u divided by del s at constant V. Similarly, p is minus del u divided by del V at constant
s. Then only, we can write the T ds minus p dV means T is del u divided by del s at
constant V and del u divide by del V at constant s. Similar way, T is a function of s and p.
So, this is del H divided by del s at constant p. Similarly, V is del H divided by del p
at constant s. From here also, I can write, p is minus del F divided by del V at constant
T and s is minus del F divided by del T at constant V. So, in the similar way, we can
write, V is equal to del G divided by del p at constant T. Similarly, s is equal to
minus del G divided by del T at constant p. Also
from these equations, we can construct these relationships also.
So, in deriving certain important thermodynamic relations, some of the expression will be
required. So, we can write all those equations from these four basics. If we can express
these four equations in this form, these three, four basic functions then we derive these
equations. These four sets of equations are known as Maxwell’s equations.
This is the basic definition from which we wrote this, because this is del u divided
by del s at constant V. This is del u divided by del V at constant S so that del T divided
by del V at constant s is del p divided by del s at constant V. Okay please third one, this one Yes place where is the minus p is equal to
this thing. Oh! say yes, yes, yes, one this will be a
minus. Very good, there will be a minus, very good,
and then it is all right? Now, we will derive few other important equations
in thermodynamics which are known as T minus ds equations. These are very important equations, we must
know. So, there is nothing much of thermodynamics in concept which we discussed earlier; the
first law and second law and their corollaries. These are more towards primary label algebra
rather than thermodynamic concepts. But we have to do, because we have to know the important
property relations in thermodynamic. The other most important property relations
in thermodynamics are known as T minus ds equations. If I express now, first T minus
ds equation, now let us express s as a function of T and V for a single face substance, s
as a function of T and V then we can write ds is del s divided by del T at constant volume
into dT plus del s divided by del V at constant T into dV. If I multiply both the sides by
T then I get T into del s divided by del T at constant V into dT plus del s divided by
del V at constant T into dV. I can write the first term as cvdT, T into del s divided by
del T as cv. V how? Yes, the question comes how definitely.
So we cannot appreciate this. Now cv, what is cv? cv is some delta Q divided
by delta T at constant volume, when delta T tends to 0 per unit mass. Because it is
a heat capacity or if it be becomes per unit mass delta Q divided by m, then it will be
this specific heat. Let us consider heat capacity, because we
are considering about the whole system, not for unit mass. All are extensive properties.
Now we know that this expression at del Q is T ds. Now T ds plus p dV now at constant,
du is T ds minus p dV and we know that delta Q is du plus p dV. Now, when we heat it reversibly
with only work, because in defining specific heat there will not be any work except p dV.
But at constant volume p dV work will be 0. So, del Q is du, du is Tds.
So, del Q divided by del T will be du divided by by del T is equal to T into ds divided
by del T and that is at constant volume. It becomes T; this can be expressed at del s
divided by del T at constant volume. So, T into del s divided del T at constant volume
gives the value of specific a heat capacity at constant volume, because small amount of
heat del Q is a change in the du internal energy. So, del Q divided by del T is du divided
by delta T, delta T tends to 0 at constant volume. This expression at constant volume
when delta T tends to 0 becomes del s divided by del T. Here, delta T tends to 0. So, that
is T into del s divided by del T. T is always there, So T into del s divided by del T. Clear
or any doubt? Clear? du divided by delta T, limit delta T tends to 0 at constant volume
is T into ds divided by delta T, limit delta T tends to 0 at constant volume. This becomes
T into del s divided by del T plus. Now, what is del s divided by del V at constant T? Now, del s divided by del V at T is del V
divided by del T at V; del s divided by del V at T is del V divided by del T at V, that
is from Maxwell’s equation. Sorry one T will be there, you are correct,
plus T into del p divided by del T at V. Any problem? You ask me. It is mathematics only
that is nothing dV. Very good that is nothing to be exited and may be some silly mistake
or you can just detect me. This is the expression known as first T minus ds equation. cv dT
in terms of T and V plus T into del p divided by del T at constant V into dV. Now, next
we will derive the second T minus ds equation. This is first T minus ds equation. Now, second T minus ds equation is T minus
ds is expressed in terms of T and p; that means, we start with s as a function of T
and p, temperature and pressure. So, we can write ds change in del s divided by del T
at constant p into dT plus del s divided by del p at constant T i0 dp.
So, if we multiply with Tds is T into del s divided by del T at constant p into dT plus
T into del s divided by del p at constant T into dp. Again, I can write this as cpdT.
Let me again check, how I can write this cpdT. We know the definition of cp is delta Q divided
by delta T at constant pressure when delta T tends to 0. At constant pressure, delta
Q is nothing but delta H; rather, I should write dH at constant pressure. That we have
already know, because H by 1, the definition of H and del Q. We know that at constant pressure,
heat addition is the change in the enthalpy, because delta Q is du plus p dV and d into
u plus pV. Here, we have to remember that whenever we define this specific heat is a
property of a system. How to define? It is very important.
Here is the concept. These are very simple. I am telling you through out your life you
will see. Mathematics is just a tool, just to follow it a little bit of merit will do
but what is tough is the physic that is the concept, the concept is clear everything is
clear. What is specific heat? Sometimes, by the basic
definition, we are not clear. When we define the specific heat of a substance, we consider
a closed system comprising that substance. Then add heat in such a way that there is
no work transfer except a reversible work, because of the system boundary displacement.
We add heat reversibly to infinite small temperature difference slowly, and allowing for no work
transfer between the system and the surrounding, except the reversible displacement work. What
is reversible displacement? P dV work. So, when we heat is at a constant volume p
dV, work automatically becomes 0. So, absolutely there is no work transfer. So, always we write
the first law as del Q is du plus p dV. Because whenever I write the first law, del Q is du
plus pdV. Probably in the examination, in the class test, there are so many expressions
when we are told under what conditions they are right.
So, del Q is du plus dW is the general statement of first law, but del Q is du plus p dV. Whenever
I write these as first law that means always we are assuming that closed system as work
interactions, only through reversible displacement work that means pdV work.
When we define the specific heat at constant pressure, we allow the p dV work to maintain
the pressure constant not by anything else. We cannot have any other work transfer to
maintain the constant pressure. That is not the definition of the specific heat at constant
pressure. Now, the pressure will be held constant by allowing only the reversible displacement
work. So, p dV and when the pressure remains constant, this becomes this, this becomes
dH. So, this is again and again recapitulation. If we recapitulate earlier things then our
concept will be more or less clear. Now, if we define this, with this if we write
the expressions cp then it will be simply coming as del H divided by del p. So, dH again
we can write dH is Tds. Now, a property relation V dp, because at constant pressure dp is 0,
so that dH, I can write ds. So, del Q at constant p is nothing but T ds at constant pressure.
By this definition, del Q divided by del T at constant pressure T. We can write in terms
of the differentiation; del s divided by del T at constant pressure.
No Book will derive this when they discuss the Tds equations. I am telling because they
know that we should know these things from the very basic definition so that it may be
little confusing that how they are substituting this, but I am doing it. Now be clear that
T into del s divided by del T at constant p is cp. Again, the similar way this expression should
be here. These expressions, del s divided by del p at constant T is minus del V divided
by del T at constant p. So, I can write minus T del V divided by del
T at constant p into dp and this is known as the second Tminusds equation. Now, from
these two T minus ds equations, I will derive a very important relationship. Now, let us
write the first T minus ds equation and second T minus ds equation side by side. This is the first T minus ds equation T ds
is cv dT plus T into del p divided by del T at constant V into dV. I am writing the first T minus ds equation
and second T minus ds equation. Let me write the second equation first. T ds is cp dT minus
T into del V divided by del T at constant p into dp, whereas first Tds set equation
is this. Instead of that it will be cv dT. It will be plus T and this will be del p divided
by del T at constant V into dV. These are the two T minus ds equations.
Now, if we deduct one from other then we get cp minus cv into dT minus T del V divided
by del T at constant p into dp minus T del p divided by del T at constant V into dV is
equal to 0, or we can write dT is equal to T; T divided by cp minus cv, these things
if we take this side, del V divided by del T at constant p into dp plus T divided by
cp minus cv del p divided by del T at constant V into dV.
So now I am writing dT is equal to T this side or these two terms I am taking this side,
T divided by cp minus cv del V divided by del T at constant p into dp plus T divided
by cp minus, I am not taking this as common, T divided by cp minus cv. It can be written
in that fashion, but I am writing in this fashion, del p divided by del T at constant
V into dV. This term, we can write this way, dT is equal
to del T divided by del p by expressing T as a function of p and V. That means we can
write del T divided by del p at constant volume dp plus del T divided by del V at constant
pressure dV which means that this term is equals this, this term must equal this. I
can write this term that T divided by cp minus cv into del V divided by del T at constant
p must be equal to del T divided by del p at constant V.
Similarly, T divided by cp minus cv into del p divided by del T at constant V is nothing
but del T divided by del V at constant p. This coefficient can be written as del T divided
by del p at constant V, as if T is expressed in terms of p and V. This coefficient can
be written as del T divided by del V at p. Now, we get cp minus cv is T into del V divided
by del T at constant p into del p divided by del T at constant V and the same thing
we get from here. cp minus cv is T into del V divided by del
T at constant p and del p divided by del T at constant V. So, from both the equations,
we arise equation is correct. We arrive the same expression for cp minus cv. Now we go
back well, any problem, again I am telling. T is this del T divided by del p at constant
V; del T divided by del V at constant p because this expresses T as a function of p and V.
I can equate this with this one and I can equate this with del T divided by del V at
constant p. So, if I just use this equation del p divided
by del V, I want to substitute this. So, del p divided by del T is minus del p divided
by del T at constant V. This side is minus del V divided by del T at constant p and del
p divided by del V at constant T. I can write therefore cp minus cv.
Can you see all these three equations? You can see this. You can also see this I
think. If I use this cyclic relationship to replace
del p divided by del T at constant V by these two terms then I get cp minus cv. There is
a minus; minus T. This becomes square. del V divided by del T at constant p square into
del p divided by del V at constant T. So, this is one of the very important relationships.
This is cp minus cv; that is, difference in heat capacity is minus T del V divided by
del T whole square at constant p into del p divided by del V at constant T.
Now, we see this term is always positive. Square of any real term is always positive.
Now, del p divided by del V for all the substances are negative
which means that cp minus cv is always greater than 0, which means cp is greater than cv.
Now this is a general thermodynamic relation. I can derive or small relationships can be
derived in terms of the characteristic parameter of a system, if I know the equation of state.
We know the equation of state is the equation between p, V and T.
For example, in case of an ideal gas, the equation of state tells that p into V is m
into R into T. For an ideal gas of mass m, the equation of state states that in an ideal
gas, the pressure of a close system into its volume equals to m into R is some constant
known as characteristic gas constant into T. Therefore, in this equation, if we use
this as equation of state and find out these parameters del V divided by del T at constant
p then it becomes a very simple exercise; del p divided by del V at constant T, we will
arrive cp minus cv is equal to mR. We have to only find out del V divided by
del T at constant p is equal to mR by p. Similarly, del p divided by del V at constant T is minus
mRT by V square, because we take del p divided by del V at constant T. If we use this equation,
then it can be shown that cp minus cv is m into R. this is a special case, if we have
this equation between p V T. That means if we know the equation of state for different
systems, I can find out cp minus cv in the simple expression, valid for that particular
system. For an ideal gas, for particular system then ideal gas cp minus cv is mR, but this
is the basic thermodynamic relation. Now, another interesting fact we see that
cp is greater than cv always, but there is a point where cp exactly becomes equal to
cv when this becomes 0. When will this become 0? del p divided by
del V never becomes 0. For example, when del V divided by del T at
constant pressure becomes 0, cp=cv and this is the case fulfilled by water at 40C because
water reaches its minimum volume for a given mass that is minimum specific volume so del
V divided by del T is 0. So these are few an important relationships. Now, after that we define beta known as volume
expansivity that is defined 1 by V into del V divided
by del T at constant pressure with change in volume with respect to temperature is known
as the volume expansivity. Another expression is known as KT that is
isothermal compressibility. Now, we know, already we have read in fluid mechanics that
when we compress a system that means the apply pressure to change its volume, the relationship
between the pressure change and the change in volume depends upon the way which this
change is brought. It depends upon the process constant; whether the temperature is kept
constant or not, whether the temperature is not kept constant, some other properties kept
constant or not, that depends upon the constant of the process, because when the pressure
changes, the volume also changes and other properties may also change.
So, the relationship between change in pressure and change in volume which defines the modulus
of rigidity or the compressibility depends upon the process constant. So, two such process
constants are very important. One is isothermal compressibility.
If we change the pressure at constant temperature then this quantity, change of pressure with
change of volume with respect to volume at constant temperature divided by the initial
volume 1 by V. This is known as isothermal compressibility; that is, change in pressure
divided by the change in volume per original volume at constant temperature.
Similar is the case that if we make it isentropic. Yes please any question? No.
Isentropic compressibility, sorry, I am extremely sorry del V divided by del p.
Compressibility, with a minus sign, very good you are very thorough.
del V, I am extremely sorry otherwise this reverse is this the modulus of rigidity very
good, minus 1 by V into del V divided by del p, because these are the scalar quantities
with positive sign and the reciprocal of this are defined as modulus of rigidity. These
definitions are very important. Now, I will derive another one important relationship;
that is, the ratio of cp by cv. If I write the two T minus ds equations, that
one is Tds is cp dT minus T into del V divided by del T at constant p into dp and the first
Tds equation Tds is cv dT plus T into del p divided by del T at constant V into dV.
I can exploit these two equations to derive the ratio between cp and cv. If we consider
isentropic process, ds is 0. From here, we get cp is equal to T into del V divided by
del T at constant p and this dp will be del p divided by del T at constant s, because
I am using the constant entropy Tds that this becomes 0. So, dp divided by dT at constant
s will be defined as del p divided by del T at constant s. I can write, from this equation
at constant entropy, ds is equal to 0 at constant s.
Similarly, I can write from the second equation which is basically the first T minus ds equation
that cv is equal to T into del p divided by del T at constant V into del V divided by
del T at constant s, with a minus sign; cp will be minus.
So cp by cv will be T into del V divided by del T at constant p into del p divided by
del T at constant s divided by this; that means, del T divided by del V at constant
s. T will be cancelled, I am sorry, into del T divided by del p at constant V.
That is del V divided by del T at constant p into del p divided by del T at constant
s into del T divided by del V at constant s into del T divided by del p at constant
V. So, this can be written as, these two together
as, minus del V divided by del T at constant p and then del T divided by del p at constant
V into del p divided by del V at constant s.
Now first of all we have to find out del p divided by del V at constant s; del p del
Vs where from you get del p del Vs? well del p del Vs Just a minute del p del Vs, we find out del
p del V, no, we have to find out del p del Vs is there, del T del pV, oh sorry I am extremely
sorry. del T del pV has to be replaced from the equation del T from the cyclic equations.
I am sorry, this is not be used, let me call the cyclic equations, cyclic equations, sorry
del T del p del T del pV that means del T del pV del T del pV. I use this cyclic equation, del p divided
by del V at constant T into del V divided by del T at constant p; that means, del T
divided by del V at constant p is del V divided by del T at constant p. This will be cancelled.
So, we see that del T divided by del p at constant V is this is cancelled. del V divided
by del p will go there; that means, minus1 by del V divided by del p. This becomes del p divided by del V at constant
s divided by del p divided by del V at constant T. So this minus sign will go because this
del V divided by del T, del V divided by del T at p and del V divided by del T at constant
p, this will be minus1 by this. So, this comes here. So, I am replacing del T divided by
del p at constant V from this cyclic equation. So, this becomes equal to… as I have defined
earlier that the isentropic compressibility by isothermal compressibility. Since cp is
greater than cv, this will be greater than 1 which proves that isentropic compressibility
is greater than the isothermal compressibility. Now, I will start a very important aspect
of thermodynamics is Joule-Kelvin expansion. What is Joule Kelvin expansion? So far, we
have discussed about the property relations. These are nothing but the mathematical relations,
some algebraic things without any thermodynamic concept much of any thermodynamic concept.
Now, what is Joule minus Kelvin expansion which is very important or Joule minus Kelvin
effect? Let us consider that a fluid is while passing
through a tube is being restricted to flow by a valve. What is this physically? Let this
tube and these valves are all insulated. We know the different types of valves are
inserted in a pipe, the basic functions of the valve is to control the flow that is known
to everybody even who does not have any much qualifications are not gone through engineering
studies they know that. If there is a flow of fluid, so if we have to regulate the flow
in the down stream direction, we have a valve which gives a restricted passage. If the valve
is completely closed then the passage is completely closed and the flow cannot come here. So,
this becomes a static condition; the flow comes here, strikes here and exerts a pressure.
There will be no flow here. As we gradually open, the valve allows the
flow to take place that means to restricted passage, the flow starts taking place. Now,
in such a condition of a partial opening or partial closer of the valve, when the fluid
comes through this and flow through the valve some at steady state, some flow is maintain.
Then we know that pressure across this valve is dropped; that means there is a certain
drop or decrease of pressure across the valve. If one keeps a pressure gauge here, let this
section is p1 and also keeps the temperature t1 and keep a temperature t2. This is the
state two and keeps a pressure gauge p2. So, if we place pressure and temperature measuring
instruments at off stream, further off stream from the valve. So, that is where this stream
is undisturbed. Again, this downstream location will be at somewhere when again there will
be no disturbance from the valve. At a certain distance from here where again
the steady state is there, same flow is flowing through this, but there is a decrease in the
pressure p1. p2 is always less than p1. This process is known as Throttling process.
This is a colloquial term in engineering term used in soft floor. That fluid is throttle
means the flow is throttle so that there is a large difference of pressure. The fluid
flows through this special difference. So, flow rate is automatically reduced to take
care of the frictional pressure down. So, this pressure is dropped because of the friction
in the form. Entire thing is adiabatic that is insulted.
Now, try to understand the physics of this process. Why the pressure is being reduced?
Because of the fact that there is the frictional dissipation, pressured energy is being converted
into intermolecular energy. So, basically, this is an irreversible process.
But what will be T2? Will it be less than T1 or will it be greater than T1? Now the
answer to this question is, T2 may be less than T1, may be greater than T1 depending
upon the situations which will be discussed. That is because there are two countering effects
taking place in changing the temperature from initial to the final state. When there is
a change in the pressure, because of the friction, some of the pressure energy, the part of the
mechanical energy is being dissipated into intermolecular energy or heat. Therefore,
the fluid temperature or the system temperature should increase. But at the same time, when
the pressure is decreased depending upon its equation of state, its temperature will also
decrease. Because of a decrease in pressure, the temperature will decrease. So, temperature
will decrease because of a decrease in pressure. Following its equation of state, we do not
know, what is extend or decrease in temperature that depends upon the typical equation of
state. At the same time, because of some fraction
of the pressure energy is a part of mechanical energy being converted in an intermolecular
energy temperature should increase. So, these two countering effects decide that which one
will dominate over the other, so that there will be either an increase or a decrease in
temperature. Under usual conditions, our common practice is that whenever we throttle a fluid
so that its pressure is being reduced to allow it to flow through a restricted passage. Usually
temperature also drops, but it is not always the case.
Now, let us analyze this case by taking a control volume like this. Now tell me, what
is this process actually? This is an irreversible process. We can write, always the first law
of thermodynamics or the conservation of energy for any process, reversible and irreversible
process. If I write this in steady flow energy equations, at one and two, flow is going out.
For this control volume, what can we write? Now, steady flow energy equations we can write
for unit mass basics. We can simply write h1 is h2. Why? Because v1 is v2. Because the
flow rate is same, the cross sectional area is same.
So, if we recall the complete energy equations, h1 is h2 is the outcome of the energy equations.
The applications of this steady flow energy equations for this case because the change
in kinetic energy is 0, change in potential energy is 0. So, simply h1 is h2. There is
no other energy interaction; neither heat nor work. Therefore, this throttling process
is an isenthalpic process, where the entropy of the process remains same. Enthalpy of the process remains the same,
but one very important thing is that this process is an irreversible process and not
a reversible process. So, the initial and final enthalpy remains the same. This process
ensures the initial and final enthalpy to remain same and it is an irreversible process. Now, if I draw this process in Tp curve but
immediately we must ask Sir, how can you draw a process in a thermodynamic coordinate plane
if the process is an irreversible one? So, I cannot draw this process. Therefore the
question does not come but what I can draw in TP plane that if I go on varying these
inlet condition p1, t1 at different values. No sorry, if I i am not going sorry, I am
extremely sorry. If I keep this p1 t1 fixed, but if I go on varying the position of the
valve if I throttle to different pressures and temperatures, that means starting from
full with close position, if I go on slowly opening the valve that means at different
valve settings, creating different restricted passage areas which means that if I throttle
to different pressure and temperature, you will see that throttling pressure and temperature
will be different. For keeping the same initial condition, if
I throttle at different valve settings so that I get different pressures and temperatures,
I can tell that since the basic energy equation for this process is h1 is h2, all the values
of enthalpy corresponding to different pressures and temperatures will correspond to the enthalpy
at 1 which is kept fixed. That means I can generate the different state points with varying
p and T, but with the same enthalpy by doing so. We can draw a curve which is nothing but
the locus of same enthalpy and the curve will look like that. If I do so, the curve will
look like that. This is an experimental fact. What is this
curve? Let this point is p1T1 and this is h2 is equal to h1 constant. By doing these
experiments, I can plot different points on pT diagram and this can be joined by a dotted
line, because this does not show a thermodynamic process; rather this shows a curve of same
enthalpy; locus of equal enthalpy point. That means I can generate number of pairs of p
and T points whose enthalpy equals to the enthalpy h1 which is the enthalpy corresponding
to the initial state p1 and T1. If we generate this way, a number of pairs
of points with p and T having the same enthalpy and we can draw the locus of a constant enthalpy
in Tp plane. This will increase first, have a maximum and then it will decrease. So, this
way, we can generate a family of such curves if we now change this initial condition p1
and T1. If we change now initial condition p1, T1
that means we are going to a different enthalpy. Again, at that same initial condition p1 and
t1 set to a different enthalpy, we can throttle to different pressures and temperature, and
generate again differentiates of p and T values corresponding to these initial enthalpy. This
way, we can generate a family of curves which represent the constant enthalpy in Tp plane.
This we will discuss in the next class because time is up for today’s class
Joule-Kelvin expansion, we will again discuss in the next class.