The objective of this screencast is to develop a mathematical model for a chemical process.
For the case that we have here the information is provided on the screen. We have a cylindrical
tank with cross sectional area A, where a fluid with constant density and constant heat
capacity is entering and exiting a tank at the volumetric flow rates of Qi and Q respectively.
As the question notes there is no temperature change or chemical reaction occurring in the
tank. So the question asks is develop a model for determining the height of the tank. For
our case here we have no temperature change, which means the fact that an energy balance for
this problem is unnecessary. So our focus now is going to be on a mass balance. So for
the mass balance what we'll start with the general mass balance which is accumulation
equals in minus out, plus generation, minus reaction. As it notes in the problem statement,
there is no generation and no reaction, so therefore there is no chemical reaction occurring,
so therefore these terms go away. So an important point to note here is the fact that what we
have here is a mass balance. Therefore we want to be able to determine the amount of
material that is coming in, out, and accumulating, but the slight problem we have right now is
the fact that we need, for our in and out terms, we need a mass flow rate, but we have
a volumetric flow rate. Fortunately the mass flow rate and the volumetric flow rate
can easily be related by the density of the fluid, such that a mass flow rate equals the
volumetric flow rate, multiplied by rho, the mass, in this case, the mass density. Do a quick units look on this,
we see the fact that the mass flow rate is mass per time, the volumetric flow rate is
volume per time, and rho is mass per volume, so therefore the volumes cancel out, and we
have mass per time on both sides of the equation, which is what we want. Revisiting our mass
balance, which is now simplified as accumulation equals in minus out, we can now find the in
term and the out term, so the in term, our inlet flow rate is coming in at Qi, and we
multiply by rho, and what is now coming out is the volumetric flow rate out, Q, multiplied
by rho, where our accumulation term is d mass dt. We now have to find is the mass over where
things are accumulating. In this case the mass that we're looking at is the mass of
fluid inside the tank, so what we now need to do is develop a relationship to be able
to find what the mass of fluid in the tank is. We know the fact that inside the tank
at any given time there will be a fluid of volume V. Similar to our volumetric flow rates
we have to multiply this by rho in order to get into mass terms. The question explicitly
asks to develop a model that will be able to give us the idea of the height as a function
of time, not the volume, but we know the fact that volume is a function of height. So here
we are going to further break down the volume term, and break it down into the height multiplied
by the cross sectional area, which is how we can find a volume, and the rho term still
stays, so therefore the mass of the fluid in the tank is h times A, or the volume, times
rho. Now that we have solved for the mass part in the accumulation equation we can just
substitute that in, and we have d multiplied by h A rho dt, equals qi rho minus q rho.
We can simplify this a bit more by looking at whats inside of our derivative. If we look
at our derivative we have h, which will vary as a function of time, because the height
of the tank can change depending on what the flow rates in and out are. However, A, which
is the cross sectional area of the cylindrical tank, and rho, the density of the fluid, are two things that
are not going to change as a function of time, therefore these are both constants, and what
that allows us to do is it allows us to remove them from the derivative. Removing them from
the derivative leaves us with A rho dh dt equals qi rho minus q rho. The rhos are all
the same, so therefore they can all cancel out, and that leaves us with our final answer,
our final mathematical model to represent the height here, which is A times dh dt equals
qi minus q. As a quick units check we can take a look at what we have here. A is a cross
sectional area, h is a length per time, so thats where the derivative is, and the right
hand side both terms are volumetric flow rates, which are volume per time. For area times
length is a volume we are dimensionally consistent and this model does make dimensional sense.