In this screencast we will develop a transfer function model for a process starting from
a material and or energy balance, in this case we'll focus on an energy balance. So
first we have the problem statement provided, and what we're going to do is as we read this
we're going to develop our diagram of this process. So we have a stirred well mixed tank,
which has volume V. The liquid enters and exits at the same volumetric flow rate F.
We are given the fact that heat capacity and density are independent of temperature and
pressure, and we are also given the fact that the tank is well insulated, meaning the fact
that there is no heat being added or removed from the system. What the question is asking
us to do is to develop a transfer function that shows how changes in the inlet temperature,
which we'll show here as T(in), impact changes in the outlet temperature, T(out). So in order
to do this analysis we have to develop appropriate balance. If we were to do a mass balance what
we would be showing here is that accumulation equals in minus out where in is F and out
is F, so therefore we're showing the fact that the accumulation is zero and that our
mass balance is at steady state, which for our analysis of this process to show how a
change in input temperature impacts output temperature really doesn't help us very much,
but for our energy balance we have that accumulation equals in minus out plus the energy from reaction
plus energy exchanged with the surroundings. Problem statement does not state anything
about a reaction occurring so we'll assume the fact that there is no energy from a reaction
because there is no reaction, and it is also stated that the tank is perfectly insulated
which means the fact that there is no energy change with the surroundings. Therefore our
accumulation can be represented as the inlet enthalpy minus the outlet enthalpy. So recall
that enthalpy equals MCp times T minus T(ref). So therefore for our system here we can state
that the inlet enthalpy is going to be the inlet mass multiplied by the heat capacity
of the inlet, multiplied by T(in) minus T(ref). We can do a similar relationship for the outlet.
If you recall in the problem statement we were given a volumetric flow rate F, but in
order to do this analysis we need to know the inlet flow rate, M(in). So fortunately
the volumetric flow rate and the mass flow rate can be related via the density, which
we said was independent of temperature, so we have the fact that this is going to be
F(in) rho(in), and we can do the same thing for the second term. What we'll now do is
take advantage of a few assumptions, which are the fact that F(in) equals F(out), and
that the density and heat capacities are independent of temperatures, so in other words we don't
need the subscripts rho(in) and rho(out). So we can simplify this as F rho Cp times
T(in) minus T(ref), minus F rho Cp times T(out) minus T(ref). What we'll notice here is the fact that
the T(ref) terms are going to cancel out identically, and that leaves us with our right hand side
as F rho Cp multiplied by T(in) minus T(out). Now for the accumulation here what we want
to find is the amount of energy in the tank at a given time, and that will simply be the
mass that we have multiplied by the energy per mass, which is the enthalpy. The mass
that we have in this system is going to be the volume V multiplied by the density, and
then the energy will be the enthalpy, which will just be Cp times T minus T(ref), and
what we're trying to evaluate here is the change in energy in the tanks, this is the
accumulation term, so therefore this will become a derivative, or d this whole thing,
dt equals, we can make a few modifications to the left hand side, T(ref) is a constant,
so therefore dT(ref) dt will be zero, so that will go away, and we can also take out V rho
and Cp, for those are all constants, and that will leave us with rho V Cp dT dt equals F
rho Cp times T(in) minus T(out), but there is one more thing that we have to take into
account here is that when we did our energy balance inside the tank the temperature here
represents the temperature inside the tank, and we know the fact that for our system is
well mixed the temperature of the fluid inside the tank will be the same as our outlet temperature,
so therefore we'll appropriately add the subscript dT(out) here in order to get a transfer function
model we need to take the Laplace transform of this and rearrange, but before we take
the Laplace transform we want to put this into deviation variables. The reason why is
the fact that it will make subsequent calculations a bit easier because of the fact that in the
real differentiation theorem most supplementary terms will be zero. What we can note here
is the fact that all variables are first order. So what this means is the fact that all of
our variables, which we'll call as independent variable x, can be replaced by their deviation
form x prime without other modifications. If we had a t and squared term, or a multiplication
of T(in) times T(out) for example, our two variables here, we would have needed to linearize
this which would have made it unable for us to evaluate this system so simply. So when
we put this into deviation variables we have rho V Cp dT(out) prime dt equals F rho Cp multiplied
by T(in) prime minus T(out) prime. So when we take the Laplace transform of this we end
up with rho V Cp multiplied by the Laplace variable, which I'll indicate with a slightly
different looking letter T(out) prime, multiplied by s equals F rho Cp multiplied by the Laplace
of the inlet temperature, and the Laplace of the outlet temperature. Now that we have
taken the Laplace transform of all of our variables we can now begin to begin to develop
a transfer function model. To do this what we want to do is put our output variables
on one side, in this case the s, and we want to put our input variable, or variables, in
this case we have one, T(in) prime, on the other side. So when we do this, before we
go any further we are going to make one quick simplification, and we could have done this
earlier, and that is we're going to get rid of all of the densities and heat capacities
because it is present in all the terms. So what we can now do is factor out the T(out)
prime. We are very close to having our transfer function model. In order to get our transfer
function model we will divide both sides by the V s plus F on the left hand side, and
the T(in) prime thats on the right hand side. So when we do this we end up with the left
hand side becoming T(out) prime over T(in) prime equaling F divided by Vs plus F, and
this is a transfer function model, but an important point to note here is the fact that
this is a non-standard transfer function model. The reason why it is non-standard is because
in the denominator the constant is not one, it is in fact the value F, so if we want to
standardize this we want to multiply both the numerator and the denominator by one over
F. So when we multiply both the top and bottom by one over F we'll end up with our transfer
function model in standard form, so therefore we could evaluate this system for if the change
in the inlet temperature wasn't instant for example, instantaneous, increase of five degrees
Celsius, or a ramp of one degree Celsius per minute, or a rectangular pulse, using this
transfer function model determine how the outlet temperature reacts to these changes.