The velocity of blood in the aorta is determined by a number of factors. The work done by the heart during ventricular contraction is

W = DP V,

but not all of that energy goes into moving the blood. Some of it is stored as potential energy in the increased blood pressure, some is stored as elastic energy in the walls of the aorta, and some is lost to dissipation:

W = K + U _{blood pressure} + U _{aortal walls} + E _{dissapation},

where K is the kinetic energy of the blood. We will attempt to compute the maximum velocity which the blood in the aorta achieves. Note that we will make a number of approximations. The numerical ones will correspond, in essence, to a choice of patient. We will discuss the qualitative ones at the end of this section.

The pressure difference DP is the difference between the maximum and residual ventricular pressures (the ventricle never empties completely). Assuming a maximum normal (systolic) pressure of 120 mmHg and a residual pressure of 9 mmHg, the pressure difference in the ventricle is 111 mmHg, or 1.5 x 10^{ 5} dynes / cm ^{2.} The stroke volume V (the amount of blood expelled into the aorta during ventricular contraction) is about 80 cm^{ 3}. This means that the heart does about 1.18 x 10^{ 7} ergs of work during a ventricular contraction. Only about 70% of that work is done before the blood velocity reaches its maximum. So the amount of energy available to move blood at its maximum velocity is 8.29 x 10 ^{6} ergs.

The potential energy of the blood increases as the blood pressure increases from its diastolic to its systolic levels. Assuming a normal diastolic pressure of 80 mmHg, the pressure difference in the blood is 40 mmHg, or 5.3 x 10^{ 4 }dynes / cm ^{2} . Using 70 % of the stroke volume above, this means that 2.99 x 10^{ 6} ergs are stored as potential energy in the increased blood pressure. In addition, potential energy is stored in the arterial walls as they expand. If we assume Hookean behavior with a "spring constant" of k = 1.25 x 10^{ 6} dynes / cm and a variation in radius of .2 cm in the aorta, 2.5 x 10^{ 4} ergs are stored in its walls as elastic potential energy. This energy is released as the aorta walls contract and the blood flows into the rest of the system. Note that it is negligible compared to the work done by the heart and the blood pressure potential energy.

In order to compute the energy lost to dissipation, we will use Poiseuille's Equation to calculate the pressure loss along the aorta, and then compute the energy loss using the stroke volume. The flow rate (assuming 72 beats per minute) is 96 cm^{ 3} / s. The average radius of the aorta is 1.25 cm, and its length is approximately 30 cm. From these assumptions, we find the pressure drop along the aorta to be 120.2 dynes / cm^{ 2}. This means that we lose 9.6 x 10^{ 3} ergs to dissipation. Clearly we can afford to ignore dissipation in the aorta for the purpose of computing aortal velocity.

To compute the velocity, we simply conserve energy (ignoring elastic energy and dissapation from our above computations):

K = W - U _{blood pressure},

or

(1/2) r (.7 V) v^{ 2} = 8.29 x 10^{ 6} - 2.99 x 10^{ 6} ergs,

where r (.7 V) is the mass of the blood moved during the heartbeat, giving

v = 4.25 x 10^{ 2} cm / s!

It should be noted at this juncture that this is an order of magnitude computation only; reconsider the caveats used throughout this chapter. It is consistent in magnitude with the average velocities computed during the problems in section C: during pulsatile flow, the velocity rises sharply and drops again to zero within the first quarter of the heartbeat, in a normal patient. But even so, it is a surprisingly large order of magnitude. Might it be large enough for turbulent flow to develop?

Fluid flow in a pipe crosses the threshold from laminar to turbulent flow when a dimensionless parameter called "Reynold's Number" ("Re") reaches about 2000. It is defined as

Re = r l v / h,

where v is the velocity of the fluid and l is the characteristic "length" of the pipe: its diameter. Re is essentially the ratio of the inertial forces (tending to keep the fluid flowing) to the viscous forces (tending to slow the motion due to contact with adjacent layers) experienced by a layer of fluid. Its value indicates the relative unimportance of viscosity (ie., low Re corresponds to very viscous situations). Re can also can be expressed as the ratio of viscous diffusion time (the time required for effects to diffuse through fluid layers) to eddy advection time (the time required for eddies or vortices to form). Using the value for aortal velocity computed above, we see that Re reaches a value of over 28,000 in the aorta!

This value is misleading, however. While Re reaches values which might indicate the presence of turbulence in the aorta, it is not clear how long Re remains that large. If it is not that high for an adequate time to form macroscopic eddies, we would not expect to detect the turbulence. The eddy advection time is

t_{ a} = Sqrt ( h Dt V / K),

where Dt is the time duration of a heartbeat, K is kinetic energy of the blood and V is the stroke volume. Using the results of the computations above, we find t_{ a} = 0.7 milliseconds. The eddy length scale, which tells us the size of an eddy which can form in that time, is

l_{ a} = (h^{ 3} Dt V / r ^{2 }K)^{ 1/4},

which here comes out to 5.2 x 10^{ - 3} cm. These two numbers are characteristic values for our patient, indicating that small eddies are forming and dissipating in very short lengths of time. What then will happen during the much longer time during which blood flows?

In normal patients, the velocity reaches its peak and falls in approximately 0.2 s. If, as seems likely from the dependence of t_{ a} and l_{ a} on energy, the length scales as the square root of the time, then the expected aortal eddy size is .09 cm. The magnitude of this number indicates that normal flow in the aorta is laminar, but on the verge of turbulence. Of course, now it is time for an experiment to determine just how the length scales with the advection time. While such results are highly suspect due to the order of magnitude nature of the calculation, they are in fact qualitatively correct.

During these computations, we have made several qualitative assumptions and idealizations which we must now examine. Our first assumption concerned the energy output of the heart:

W = DP V.

This equation is only valid if the ventricular contraction is instantaneous, since it assumes that the volume is constant. That is, we assume that a constant volume of blood undergoes a change in pressure, and then is placed instantly into the aorta. While this is obviously a simplification, it is necessary in order to avoid more complex mathematical treatment. Our heuristic approach to compensating for this simplification was to say that only 70 % of the work done by the heart is done by the time the blood reaches its maximum velocity.

Our next major idealization was treating the walls of the aorta as Hookean "springs". While the aortal walls have very high resilience, their stress versus strain curves are not linear throughout the range of radii which the aorta assumes during pulsatile flow. Since the normal radial variation is on the order of plus or minus eight percent, however, we feel justified in saying that they do not deviate too far from Hookean behavior.

Finally, in using Poiseuille's Equation and Reynold's Number, we have committed the nearly unforgiveable sin of treating pulsatile flow in flexible blood vessels with equations designed for constant flow through straight, rigid pipes! In the case of Poiseuille's Equation, we should not feel so bad: the results, even though they are only reasonable to order of magnitude, are of negligible order. If the dissipative pressure drop in the aorta was significantly higher, we should have much less faith in our work. Similarly, Reynold's Number is only useful in situations where the velocity is essentially constant over time. This is why we had to examine the eddy time and length scales; since the velocity changes during pulsatile flow, we must look deeper into the conditions required for macroscopic turbulence. One final caveat: the eddy advection time as presented here is really only valid for small eddies. This is due to the assumption that viscous dissipation is proportional to the velocity gradient. Our scaling of the expected eddy length with the square root of the time of non-zero blood flow is an assumption with no theoretical basis. It is used here to illustrate the use of such computations in deciding what to expect when designing an experiment.

Using the "Pulsatile Flow" Mathematica notebook, we can attempt to improve upon our computations in this section. We will input a series of blood pressure values representing one cardiac cycle. It will also be necessary to choose a model for the rate of volume flow out of the heart. The notebook will then split our volume into small "parcels" of varying size as a function of time, and compute the velocity as the quotient of the power and the force applied to the blood already in the aorta. This effectively obviates the necessity to assume that the ventricular contraction is instantaneous.

In evaluating the results of the model, it is important to remember that our assumptions are useful primarily for the educational experience; this is not professionally publishable work! So when interpreting the expected eddy size computed using the graphical output of the notebook, you should only compare it to the results in the previous section. That is, values much larger or smaller than 0.09 cm may indicate that, compared to the "standard" calculation above, the qualitative behavior is different for your patient. For values near 0.09 cm, however, it is best to evaluate the results as "not substantially differing from the standard". Do not place too much faith in these numbers until you compare the results with actual experimental data. After all, construction of a theoretical model is pointless unless it is calibrated with empirical data. Only then is it allowable (possibly!) to extrapolate the model's results.

The next chapter is about electricity.

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©1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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