In this screencast we will look at how to convert mathematical models in the Laplace
domain into a block diagram. To illustrate this we will use the example shown on the
screen, a case of two tanks presented in series where we have two separate inlet streams,
one at temperature T(1) entering the first tank, and one at temperature T(3) in the second
tank. In this problem we are assuming that the flow rates are constant and that there
is no accumulation. We have the inlet temperature T(1), which can vary, so therefore a variation
T(1) can cause a variation in T(2). Also entering this tank is recycle from our outlet of the
second tank, T(4), therefore T(2) can vary. We have some of the flow from the first tank
entering the second tank, and we have our inlet, which is at temperature T(3), which
also can vary coming into our second tank, so therefore we have four variables here,
the respected temperatures, and when you conduct the energy balances you end up with the relationships
shown here. These would be functions of the volume of the tank, flow rates, heat capacities,
densities, etc., but to show the point of the block diagram we're going to ignore developing
the mathematical model. The question now becomes is how do we develop a block diagram from this? So
the first step that we would probably want to do is look at our variables, and identify
them as either input variables, output variables, our intermediate variables. So if we look at
this T(1) and T(3) would both qualify as input variables because they are things that can
vary that are not modified by the process. T(4) would be our output variable, and T(2)
is something which is directly affected by input variables, but is not the desired output,
so therefore T(2) represents an intermediate variable. So lets start with a few basic ideas
in terms of developing a block diagram. An important point to remember is the fact that
a block diagram is a mathematical representation of the process. The way that the process connects
to one another has to be shown in this block diagram. There is a connection between how
the system physically is ordered with how the system is mathematically ordered. So because
of that generally what you try to do is have output variables on the right, and input variables
on the perimeter, you would want the perimeter consistent with its location in the process,
so for example its the first thing that you see is the change in a particular input variable
going from left to right, then that should probably be located at the left hand side.
If its a variable which comes in part way through the process, it will probably come in somewhere
in the middle of the diagram, and it will come from either the top or the bottom. So
to show this lets look at our equations we have here. The top equation comes from the
first tank balance, and the bottom equation comes from the second tank balance going from
left to right. So there are a number of ways you can approach this, whether you start with
the first equation, or the second equation. Here we'll start with the second equation, because
we know that T(4) is one of our output variables, so therefore we put on the right hand side
of our diagram and we'll indicate that this is T(4). By looking at our diagram we see
that it is a sum of two separate transfer functions, so what that means is, is in order
to get to T(4) we need to put in a summing point. Since both are positive we'll have
a plus for both. T(2) is and intermediate variable, so therefore we would want to put
that somewhere in the middle of our process, so therefore we'll keep that in this linear
flow. This will be the transfer function K(3) over tau(2) s plus one, and what will come
in is T(2). T(3) is an input variable somewhere in the middle of the process, so therefore
it should exist in the perimeter. Thats a good starting place, try to put it somewhere
in the top. So here we'll have T(3), and then this transfer function will be K(4) over tau(2)
s plus one, so if we look at this in terms of block diagram algebra we have T(2) prime
times K(3) over tau(2) s plus one, plus K(4) over tau(2) s plus one and the sum equals
T(4) prime. That is exactly what we have here, so that has taken care of the second equation,
so now lets look at the first equation. It is for T(2), an intermediate variable that
we have already defined. So similar to the T(4) relationship we have an a sum of two
terms, which requires again a summing point, which is the sum of T(1), which is one of
our input variables, so since it is the first variable we see, generally we would put that
on the far left hand side. We have a block here to show that a transfer function, so that
will be K(1) over tau(1) s plus one, and then we have a K(2) over tau(1) s plus one times
T(4). The important point here is T(4) is already in the diagram, so we do not want to draw
something along this line, this would be incorrect because T(4) is already in the process, so
this is incorrect. Why? Because T(4) is already located here, so you have to keep everything
self consistent with one another in order to mathematically appropriately represent
the process. So what we should do instead is since this stream represents T(4) we use
a branch point, remembering that a branch point everything represents the same, so therefore
this is also T(4), so we have the branch point and then we form a block here, which would
represent the relationship between T(2) and T(4), and we have it reach the summing point,
so therefore, looking at this summing point, we have that T(1) prime multiplied by K(1)
over tau(1) s plus one, plus K(2) over tau(1) s plus one multiplied by T(4) prime, gets
you to T(2) prime, and that is consistent with our diagram here. So if we look at the
second part of the question, we go back to what we had is that the question asks not
only to turn the equations into the block diagram, but also to use the diagram to find
the relationship between T(4) and T(1). So an important point to note here is when we're
trying to find this relationship is when we're evaluating T(4) prime over T(1) prime and
you're looking at how one output variable is affected by one input variable you assume
all other inputs are zero, so therefore we have a second input here of T(3), so when
we do this analysis we will ignore it. So to do this analysis what we'll see is that
we have T(1) here, so we'll actually have the same relationship we developed in the
first equation for T(2), so we'll get the fact that we know that this term here is representative
of our first differential equation. So in other words this is T(1) prime multiplied
by K(1) over tau(1) s plus one, plus T(4) prime over K(2) over tau(1) s plus one. To
continue going through the diagram, it passes through a block, so therefore this K(3) over
tau(2) s plus one is multiplied by both terms. The summation is no longer necessary because
since T(3) prime is zero that means that this value here is zero, so therefore when we send
it through this block that will just equal T(4). So what we'll have is that K(1) over
tau(1) s plus one, multiplied by K(3) over tau(2) s plus one, multiplied by T(1) prime, plus
K(2) over tau(1) s plus one, multiplied by K(3) over tau(2) s plus one, multiplied by
T(4) prime, and that will equal T(4) prime. So we'll gather like terms together. We end
up with T(4) prime over T(1) prime equals the, this term here, the term in front of
the T(1) prime, divided by, you factor out the T(4) prime here, that is just a one, minus
this whole term in front of the second T(4) prime term. We could simplify this a little
bit by getting a least common denominator for the denominator. The denominators in the
numerator and denominator are going to cancel, that leaves us with a transfer function in
general form, T(4) prime over T(1) prime equals K(1), K(3), over tau(1) s plus one times tau(2)
s plus one, minus K(2) K(3). This analysis to find this transfer function contain the
identical answer just using the equations, but for systems, particularly as the models
block diagrams get more complicated, the block diagram method will usually be easier to find
relationships between input and output variables, versus trying to sift through multiple equations.
This screencast showed a methodology by which you can develop a block diagram from a number
of equations that are used to describe a mathematical model for a process.