High blood pressure (HBP), also called hypertension, has been described as “a serious condition that can lead to coronary heart disease (also called coronary artery disease), heart failure, stroke, kidney failure and other health problems” (See: What Is High Blood Pressure? US Department of Health & Human Services, www.nhlbi.nih.gov/..hbp/). We shall re-examine the adequacy of this statement in subsequent articles. But what is Blood Pressure?
Blood pressure (BP) is a measure of the force per unit area exerted by the flowing blood against the walls of the arteries as the heart beats. Based on the description in the cited reference above, it is often measured in two ways: systonic BP (taken when the heart beats) and diatonic BP (taken when the heart is at rest or in-between beats). Thus BP numbers are often written as BP110/75 where the first number (i.e. 110) refers to systonic BP while the second number (i.e. 75) refers to diatonic BP. The unit of measurement is millimeter mercury written as mmHg. Whether the mmHg is written or nor, it is implied in all BP records. The same applies to BP110/75 which can be written as just 110/75. From the table provided in the cited reference, it is reasonable to state that high pressure starts when these values are greater than BP120/80.
A visit to the clinic, discussion with medical experts, study of write ups, listening to enlightenment programme on HBP would reveal some or all of the following as the causes of HBP. These are age, too much salt, high cholesterol, lack of exercise, obesity, excessive drinking of alcohol, coffee intake and smoking” (For further details, see: High Blood Pressure, www.medicinenet.com/../article.htm). I once suffered from HBP and my various doctors placed me on medication coupled with regular check-ups.
Today, I reflect on the whole process only to discover that the main source of HBP has been largely ignored by medical experts. This is the motivation for this write up. What is this main cause of HBP? It is the quantity of blood (Q).
Let us go into the Mathematics:
In all pumping operations, the following simplified equation suffices (For further details, see: J. F. Douglas, J. M. Gasiorek & J.A. Swaffield - Fluid Mechanics, 2nd Ed, ELBS, pp 586 – 635; R. K. Sharma & T. K. Sharma – A Textbook Of Water Power Engineering, S. Chand, India, 2003, pp 412 - 430):
P = ρgHQ /η (i)
Where:
P = Effective power available from the pump (This power is constant and does not change), ρ = the density of fluid (also a constant for a given fluid at a particular temperature), g = acceleration due to gravity (also a constant at a particular location and time), η = efficiency of the pump measured as a percentage (also a constant), Q = discharge from pump or flow per unit time, and H = the head to which the flow (Q) can be successfully lifted taking into consideration distance, height, the size or diameter of conduit or pipe, and friction factors associated with the lining of the inner walls of the conduit.
The hydraulic head H, also called the total energy head, has been defined mathematically using Daniel Bernoulli’s equation as (For further details, see: J. F. Douglas, J. M. Gasiorek & J.A. Swaffield - Fluid Mechanics, ibid; Mark J. Hammer & Mark J. Hammer (Jr.) – Water and Wastewater Technology, 4th Ed, Prentice-Hall, India, 2004, pp 93 – 122)
H = Z + Pr/w + V^2/(2g) ..(2)
Where:
Z = Elevation head or effective difference in elevation between two points / locations connected by the conduit or pipe, Pr = Pressure head at a particular point (Please note that Pr has been used instead of P to distinguish pressure from pump power), w = ρg (see equation 1), and V = velocity head at a specific point, and ^ = exponent or power function. There are limits to Bernoulli’s equation. These limits include the possibility of (See: Bernoulli’s Principle, Wikipedia, www.enwikipedia.org/wiki/Bernoulli's_p...):
(i) cavitation at too low pressure regimes (e.g. near zero or negative values),and
(ii) variation in density at very high speeds of flow common with gases and sounds.
When two points of interest are located on the same elevation, the elevation head Z = 0. What drives the flow is then governed by equation (2a) below:
Hi = Pr/w + V^2/(2g) (2a)
Thus when one’s BP is checked using Sphygmomanometer, the cuff strapped over the arm close to the arm pit is virtually situated at the same level with the heart, thus reducing elevation head Z = 0. Consequently the BP measurement taken reduces to the sum of pressure and velocity heads given by equation (2a) only (For further details on blood pressure instruments, Sphygmomanometer and cuff, see: David Werner et al - Where There Is No Doctor, Revised Ed, Macmillan, London, 1993, pp 38 - 39). Thus Hi is the same thing as H in BP measurements.
When the pump operates, it transfers flow from one elevation to another and from one location to another through a conduit or pipeline connecting the pump to the point(s) of its discharge. The natural corollary is the heart working in conjunction with the arteries and blood as well as all the various organs of the body.
The heart is a natural pump with total power output P while the quantity of blood pumped per unit time is the discharge (Q). The arteries constitute the conduit through which the flow Q passes to meet the requirements of the various organs of the body with total head H. This H is called the BP (blood pressure) when normal and HBP (high blood pressure) when higher than normal. It is normal to assume that the efficiency factor η = 1 in a natural process.
Thus we can re-write equation 1 as:
P = ρgHQ (a)
Since P is constant in all pumps, we can only play with the right hand side of equation (1a) above. In the right hand side, both ρ and g are equally constants. Thus, we can only play with both H and Q. Let H increase by dH and Q by dQ. It follows from above equation that:
P = ρg (H + dH)(Q + dQ) ..(2)
P = ρgHQ + ρgHdQ + ρgQdH + ρgdHdQ (2a)
Subtracting equation (1a) from equation (2a), then dividing through by ρg, and after removing very negligible values / terms such as dHdQ (as usual in Mathematics), we have:
HdQ + QdH = 0 (3)
Rearranging equation 3, we have:
dH/H = - dQ/Q (4a)
dH = - (dQ/Q)H (4b)
dQ = -(dH/H)Q (4c)
dH/dQ = - H/Q (4d)
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