Concepts of Thermodynamics
Prof. Suman Chakraborty Department of Mechanical Engineering
Indian Institute of Technology, Kharagpur
Lecture – 19
Control Volume Conservation Reynolds Transport Theorem So, far we have discussed about first law
of thermodynamics in the context of situations where we are considering control mass system. That means a system of fixed mass and fixed
identity. Now, we will try to consider situations where
we will apply the first law of thermodynamics for not a control mass system but for a flow
process. So, when there is a fluid flow taking place
the situation may no more be addressed by a control mass system and the reason is as
follows. So, if you have a pipe like this and say some
fluid is flowing so this fluid can no more be considered as a control mass system, because
within the pipe not the same fluid is flowing at all the times. So, if you have by same fluid I mean same
fluid molecules. So, if you have a specified region in the
pipe, then in this specified region what is happening is some fluid is entering, some
fluid is leaving. At a given instant of time maybe the mass
within this, may not change, because if it is a steady flow at the same rate the mass
enters that is the same rate at which the mass leaves, but the identity of the mass
has changed. So, remember the definition of a control mass
system is something of fixed mass and fixed identity. So, the molecules here, the total mass maybe
fixed at a given instant of time may or may not be. It is fixed, if it is a steady flow, but different
molecules are occupying this at different instants of time and therefore it is not a
control mass system anymore. So, this is better represented by something
called as control volume. So, what is this control volume? So, this control volume is a specified region
in space across which there is flow of, there is transport of mass, there is transport of
momentum, there maybe transport of energy. So, there is some kind of transport across
this region in space. So, what is identified here is not these particles
or molecules, but a region across which the fluid flow is taking place. So, physically it is as if you are sitting
with a camera and focusing your camera on this specified region. So, some mass is entering, some mass is leaving,
some momentum is entering, some momentum is leaving, some energy is entering, some energy
is leaving. So, this is alright so, we have a shift of
paradigm from control mass approach to control volume approach. In short sometimes for control volume, we
call it C.V. So, from control mass to control volume approach,
when we make this shift of paradigm, then the question is that are our equations representing
conservation of mass, conservation of momentum, conservation of energy. Are these equations remaining the same or
are these equations getting altered? The principles remain the of course the same,
but their mathematical expressions tend to get altered. Why their mathematical expressions tend to
get altered? The reason is straight forward, our classical
ways of expressing the conservation laws are all based on control mass approach like, Newton’s
second law of motion. It is a traditionally expressed using a control
mass approach. Now, when you have to change the approach
from control mass to control volume, you have to make some adjustments in your mathematical
formulation and we will learn what kind of adjustments you will require in your mathematical
formulation, which is realized through a theorem called as Reynolds transport theorem. So, Reynolds transport theorem is a theorem
that commits a control volume based conservation principle or conservation law to the same
conservation law applied to a with reference to a control mass system. So, to understand this theorem, let us say
that this is the control mass system at time t. So, within this there are some molecules or
particles whatever you call it, macroscopically particles. Now, after some time let us say that this
occupies this configuration, just arbitrary. This is control mass system at time t plus
delta t. If delta t is very small, then these two configurations
are almost overlapping. So, this two different configurations are
drawn just for clarity in distinguishing their positions, but imagine that when delta t is
very small, they are almost overlapping with each other. So, when they are almost overlapping with
each other, their common zone can easily be identified and this common intersection region
is the identified region in space across which we are studying the transport so this is our
control volume. So, see so nicely the control volume picture
is emerging out of a control mass system picture. So, we have a control mass system at time
t, you have the same control mass system over a time change delta t so that the system has
only infinitely changed its configuration. So, the common intersection between these
two is maybe your region of interest, which you call as control volume. So, then, let capital N be an extensive property. So, extensive property you have already understood
the definition. Extensive property of the system means a property
of the system which depends on the total extent or total mass. For example; total mass, total momentum, total
energy like that and small n is capital N per unit mass. Now, let us divide this entire domain which
is there into three parts- the 1st part is this one, which we call as I , the 2nd part
is this one, which we call as II and the 3rd part is this one, which we call as III. So, when the system, when the control mass
system is sort of sweeping in this direction, then this I region is like the region across
which the fluid is entering the control volume and III is the region across which fluid is
leaving the control volume. So, this is like inflow and this is like out
flow. So, now we are interested in calculating dN
dt of the system, that is the rate of change of N, capital N with respect to time for the
system. So, when we do that, remember that we are
interested for the expression for rate of change, when expressed in terms of the control
mass system ok. So, what is our objective? Our objective is to express this in terms
of a corresponding rate of change of N, but with respect to the control volume. So, system based analysis to control based
analysis and vice versa. So, by definition this is as good as limit
as delta t tends to zero. N at t plus delta minus N at t divided by
delta t. Now, when you say limit as delta tends to
0 N at t plus delta t minus N at t by delta t this N comprises N at I and N at II, because
at time t the region occupied is I plus II. So, this is limit as delta t tends to 0 and
when you write the sorry, this is N t plus delta means II and III right. So, at t plus delta t it is II plus III and
at t, it is I plus II. So, N II t plus delta t plus N III plus delta
t II, plus III these two regions make it at time t plus delta t minus . So, 2 plus 3 is
a delta t, t plus delta t, I plus II at t. So, now you can isolate this two terms with
a commonality. What is the commonality? Both of these represent the region II, which
is the control volume. So, dN dt of the system, plus when you write
plus you have this term and this term divided by delta t. So, what is this term divided by delta t? This term divided by delta t is the rate of
transport of the physical quantity across this three and that essentially means outflow,
if it is sweeping in this direction and this similarly means inflow, so this plus rate
of outflow minus inflow of property. So, finally, what is this? This is the rate of change of N with respect
to time as seen from the control volume. See very interestingly we write it as a partial
derivative with respect to time, because we are freezing up position and then finding
rate of change with respect to time. So, rate of change with respect to time for
a given position, whatever position we are fixing up. So, it means that you have dN dt of the system,
this is the control mass of the system base rate of change, this is the dN dt with respect
to control volume, this is the control volume based rate of change of N and this is rate
of outflow minus inflow of property. Now, what is this del N del t with respect
to control volume? So, you can take a small volume dV, so then
this is rate of change of the property within the control volume. So, within dV the property is, within dV the
mass is rho into dV, small n is property per unit mass. Similarly we have to write an expression for
the rate of outflow minus inflow, which is the final task remaining before we are able
to complete the mathematical description of this entire term, because it is till now qualitative
lead, we have to write a quantitative expression for that. So, imagine that there is a control volume. So, let me erase this figure so, that I can
draw a fresh figure, where we will have only the control volume. So, you have a control volume like this, where
this is an outflow boundary that means fluid is flowing out. See this, the essence of transport phenomenon
is like when fluid is flowing in or out it is carrying with it some transport quantity,
like it is carrying with it mass, momentum, energy. So, that is why, that is how fluid flow is
connected with say, for example, heat transfer and work transfer in thermodynamics. So, fluid flow is carrying away some energy
and that energy transfer may lead to some heat transfer, work done, mass transfer etc
and that is how the fluid flow and energy transfer is related. So, here let us say that the velocity vector
is u. This velocity is essentially velocity of the
fluid relative to the control volume, because that is what which essentially matters for
the transport of mass, momentum, energy whatever. Let eta be the unit normal vector perpendicular
to this surface, we have taken a small area dA, so that you know you can have a fixed
eta otherwise eta will change across the surface. The normal direction will change and this
is a very important vector, because the direction of this vector essentially describes the orientation
of this area ok. So, what we can write is that the rate of
outflow. What is that? Rate of outflow of property, integral. So, what is the rate of flow here? So, this is the rate of flow and rate of flow
of property is this times the property n, which is property per unit mass. So, this is integrated over the surface of
the control volume which is called as control surface. Now so far it is fine for describing the outflow,
but what about inflow. So, for inflow
you have a surface like this with fluid entering the surface. The difference between inflow and outflow
boundary is, for the outflow boundary, the outward normal is the unit normal outward
to the surface is oriented in a similar direction of that as the flow. Whereas, in this case the outward normal is
not in the similar direction. Therefore, this dot product will give positive,
but this dot product will give negative algebraic expression. So, because this negative term is here, then
a negative value of this will automatically imply that it is inflow. Because if you take this algebraically outflow
minus inflow, we do not care about whether it is this expression is they are separately
for outflow and inflow. If it comes out to be plus it will be automatically
outflow, if it comes out to be minus it will be automatically inflow. So, we can write eventually. Now before concluding this lecture I will
make a change of the symbols because some of the symbols in fluid mechanics conflict
with some of the symbols in thermodynamics. For example, u in thermodynamics we have earlier
preserved for internal energy right. So, it is better not to use u for velocity
here. We will better use V for velocity instead
of u. So, we will switch from u to V, but when you
switch from u to V there was another term which contained V as a symbol and this is
volume. So, what we will do is we will just put a
different symbol to indicate volume here. So, the change in symbol is accounted for
here and it becomes dN dt of the system is equal to partial derivative with respect to
time rho n dV, this V is for volume and then rho V relative dot eta dA into n. So, what we have successfully done here, is
we have converted an expression of the rate of change of N with respect to a control mass
system to a rate of change with respect to a control volume based system. So, this is what is the essence of the Reynold’s
Transport Theorem. In the next class what we will do is, we will
try to use this Reynolds Transport Theorem for conservation of mass and conservation
of energy, which are essentially the hallmarks of applying the 1st law of thermodynamics
for a flow process. Thank you.
