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In this screencast we are going through an example of an energy balance on an open system
at steady state. In this example we are looking at a Ventricular assistance device (VAD) which
is used on patients who have weakened hearts in order to reduced the work the heart performs
on pumping blood. The VAD is hooked up as seen in the diagram to assist either ventricle
or both at the same time by circulating blood from the ventricle to the aorta. This reduces
the amount of work that the ventricle and atrium has to do. This allows a heart that
is in a weakened condition to heal. The problem specifically looks at 2 different devices.
We are going to imagine a patient is in need of one of them and has the choice between
the two. We need to figure out which one results in less work that the ventricle and atrium
has to perform. VAD 1 uses the centrifugal motor design while VAD 2 is pulsatile and
uses a positive displacement plump. VAD 1 requires 0.0004 hp and is 90% efficient in
converting energy to shaft work. We are going to assume the rest is loss to heat to its
surroundings and does not influence the fluid. VAD 2 requires 0.001 hp and converts this
all to shaft work. However, it can only handle 25% of the flow rate of blood that VAD 1 can.
Other information that we need to be able to solve this includes the flow rate. In this
case blood flow rate is 4 L/min and the density of blood is 1.056 g/mL. The average fluid
velocity in the pulmonary vein is 30 cm/s and the internal pressure is 10 mm Hg. The
same in the aorta is 30 cm/s for the fluid velocity and 100 mmHg for the pressure. At
this point for any material and energy balance we should start with drawing a diagram. In
this diagram we have an inlet flow for the blood with an volumetric flow of 4 L/min,
a density, and we are told the velocity and pressure. We know what it is entering the
aorta as well as you can see on the outlet side. The blood is split between atrium and
the ventricle and the VAD. Both perform work on the blood. The first place to start is
to pick what area we want to do a material and energy balance on. It makes sense doing
an overall balance. The material balance basically in this case we are just looking at one component,
the blood. We have mass in equals mass out. Pretty straight forward. Mass in is just going
to be the density times the volumetric flow rate. In this case we are going to us 1.056
kg/L. Multiply that by out 4 L/min. This gives us a mass flow rate of 4.22 kg/min. If we
write a general energy balance we know that energy cannot be destroyed or create and we
are going to write it as a change in enthalpy plus the change in kinetic energy plus the
change in potential energy is going to be equal to the amount of heat transferred to
the fluid plus the amount of work done on the fluid. We are given the fluid velocity
at the inlet and the outlet and it is the same so we can neglect any change in kinetic
energy. We are also assuming that there is no change in height of the fluid from start
to finish so we are going to neglect the potential energy. We are also doing to assume that no
heat is actually transferred to or from the fluid in this case. It may not be the safest
assumption but we are going to do it here. Our energy balance breaks down to the change
in enthalpy is equal to the shaft work. We know enthalpy is also written as the internal
energy plus the fluid work. If we assume that the temperature of the fluid does not change
from start to finish then we can probably neglect any change in the internal energy
so we are left with a change in enthalpy equaling a change in the pressure volume term. Our
overall energy balance thus becomes V delta P plus P delta V is equal to the shaft work
performed on the fluid. If we also assume that the fluid is incompressible we are going
to get rid of that term so we are left with V delta P is equal to the shaft work. We could
calculate the specific volume that is just the volume of the fluid divided by the mass
or the inverse of the density. This comes out to be 0.947 L/kg. We know our delta P
of our system if we look back at our diagram we start out at 10 mmHg and we are going to
100 mmHg. P final minus P initial is 90 mmHg. This must be equal to the work done on the
fluid. If we look back at out energy balance here the shaft work is provided as a rate
where as the left side we need to multiply by our mass flow. To do that on the left side
here we know that we have 4.22 kg/min. At this point it is just a exercise in unit conversion.
I will go ahead and write that out. When you do this I have kept everything in SI units
we get that our fluid work term is equal to 48 newton meters per minute which is also
equivalent to 0.8 J/s or 0.8 Watts. We know that our shaft work needs to be 0.8 Watts
to accomplish the change in blood pressure to go from the pulmonary vein to the aorta.
Now the question is which device is going to reduce the amount of shaft work performed
by the atrium and ventricle as we are leaving the patient. If we look at VAD 1 we are told
that is can convert 90% of the energy to shaft work. If we look at this we know the conversion
of hp. This tells us that VAD 1 will provided 0.27 W of shaft work to the fluid. Thus the
atrium and ventricle would have to provide the other part. Now VAD 2 we are told requires
0.001 hp and converts it all to shaft work. So if we do the same conversion we did before
we get that VAD 2 can perform 0.75 W of shaft work on the blood. However we are also told
that VAD 2 can only handle 25% of the blood that VAD 1 can. Even though we are told that
VAD 2 only handles a fourth of the amount of blood that VAD 1 handles it doesn't actually
matter. We look at our overall energy balance. We know exactly how much energy is required
to add to the blood to take it from point A which is the pulmonary vein to point B,
The aorta. We are told that that's the shaft work from the VAD plus whatever shaft work
is done by the atrium and ventricle. Because these are rates of energy transfer it does
matter the actual amount of fluid that that is done on. We are just looking at the energy
balance. VAD 2 has more energy than the shaft work that the atrium and have to do. Ends
up being only 0.05 Watts. When we compare this to VAD 1 it is pretty clear that VAD
2 wins. Of course there might be other considerations in terms of how much you actually need to
reduce the work or what a safe amount of reduction might be in of course cost. Hopefully this
gives you a good example of an open energy balance at steady state.