Lecture 21: Pressure and Fluid Dynamics

Abstract

Pressure is the force (in N) being exerted, divided by the area (in m²) on which the force acts. Note that pressure is NOT a force (nor is it a length, though one unit is “mm Hg”).

1 Definition and Units for Pressure

Pressure is the force (in N) being exerted, divided by the area (in m²) on which the force acts. Note that pressure is NOT a force (nor is it a length, though one unit is “mm Hg”).

$$ P=F/A\kern1em \mathrm{or}\kern1em \mathrm{Pressure}=\mathrm{Force}\div \mathrm{Area} $$

The unit is N/m² = Pascal (Pa).

(When performing CPR, why use the heel of the hand rather than the whole palm to press down on the sternum?)

• One pascal is a small unit, so pressure is usually expressed in kilopascals (kPa).

Other Units

• Millimetres of mercury (mm Hg) – for BP!

• Centimetres of water (cm H₂O)

• Standard atmospheres (atm.) – for hyperbaric!

• Bars (hence baroreceptors and barometers)

• Pounds per square inch (psi) – for car tyres!

Conversion Factors

• 1 kPa = 1000 Pa = 7.5 mm Hg = 10.23 cm H₂O

• 1 mm Hg = 0.133 kPa = 1.36 cm H₂O

$$ \left(120/80\ \mathrm{mm}\ Hg\equiv 16/10.6\ kPa\right) $$

2 Atmospheric Pressure and Pressure Shown on the Gauge

The air around us (79% N₂, 21% O₂, 0.5% H₂O, 0.04% CO₂) exerts a pressure due to the weight of the atmosphere above us, called atmospheric pressure.

It was first measured by Torricelli using a mercury barometer (atm. P. was measured by how high – in mm – a column of mercury could be supported by the pressure of the atmosphere).

The standard atmospheric pressure on 1 m² of area at sea level is due to the weight of air in the column rising to the top of the atmosphere. The column contains about 10,100 kg of air (which has a weight of about 101,000 N), but the amount varies from day to day.

• 1 atm = 101,000 N/m² = 1.01 × 10⁵ Pa

• 101 kPa = 1010 hPa = 1010 millibars 760 mm Hg = 1030 cm H₂O = 14.7 psi

Atmospheric pressure decreases with altitude above sea level. At 3000 m, atm. P = 70 kPa. Our middle ear contains air that is separated from the atmosphere by our eardrum but may be vented by the Eustachian tube. The middle ear is sensitive to rapid changes in air pressure.

Atmospheric pressure is used as a relative zero. Pressures greater than atm. press. are “positive”. Those less than atm. Press. are “negative” (suction).

The readings on pressure gauges show pressure values above or below atm. press. (and are called “gauge pressure”).

That is, a systole pressure of 120 mm Hg means 120 above atmospheric and is actually 120 + 760 = 880 mm Hg. And car tyre pressures are 192 kPa (28 psi) greater than atm. press.

3 Suction

Suction involves creating a negative pressure so that atmospheric pressure will push the fluid in the direction of the negative pressure (e.g. drinking with a straw, filling a syringe, wound drainage using a Bellovac or RediVac or PortaVac bottle, taking blood using a Vacuette).

4 Boyle’s Law

“The pressure in a fixed amount of gas will increase as its volume decreases (pressure is inversely proportional to volume).”

$$ P\propto 1/V $$

Examples are expanding the lungs to breathe, a syringe and other “bicycle pump” style devices.

5 Hydrostatic Pressure

Pressure in liquids increases with depth because the weight of liquid above increases with depth (about 1 atm. per 10 m).

Pressure due to the weight of liquid or the “head” of liquid above a point is called hydrostatic pressure (HP), stated in units “cm H₂O”, e.g. pressure at the cannula of an intravenous (IV) saline infusion with a head of liquid of 40 cm

$$ {\displaystyle \begin{array}{c}P=9.8\times \mathrm{density}\times \mathrm{head}\\ {}=9.8\times 1000\kern0.5em \mathrm{kg}/{\mathrm{m}}^3\times 0.4\kern0.5em \mathrm{m}\\ {}=3920\kern0.5em \mathrm{Pa}=3.9\kern0.5em kPa\kern0.5em \left(=29\kern0.5em \mathrm{m}\mathrm{m}\kern0.5em Hg\right)\end{array}} $$

This pressure must be greater than blood pressure for IV liquid to flow in.

While standing, there is a pressure difference between feet and head due to gravity acting on the “head of liquid”. Hence, when standing, blood pressure in the feet will be increased by the head of the blood contained in the vessels (blood pressures in the body are stated as if the body is supine).

Note that:

• 1. At the surface of the liquid (where depth is zero), pressure is atmospheric.

• 2. Pressure difference, due to the head of liquid, depends only on the depth in the liquid, not on the amount of the liquid nor on the shape of the vessel.

• 3. The pressure at any point in a liquid that is at rest acts equally in all directions (otherwise, the liquid would not be at rest).

• 4. PASCAL’S PRINCIPLE: Because liquids are incompressible, “pressure applied to an enclosed liquid at rest is transmitted undiminished to every portion of the liquid and to the walls of the containing vessel”.

(For example, relief of pressure sores; Queckenstedt’s test of cerebrospinal fluid (CSF) pressure; squeezing an IV bag that cannot be hung up; increased eyeball pressure in glaucoma; distribution of pressure in the knee joint, thanks to enclosed synovial fluid; foetus enclosed by amniotic fluid)

Soon after the heart’s ventricles fill with blood, they contract, and pressure on the blood in the left ventricle (LV) rises rapidly from about 0 mm Hg to 80 mm Hg. The atrioventricular (AV) valves (mitral and tricuspid) close immediately, but the semilunar aortic (and pulmonary) valves do not open until LVP = 80 mm Hg (and RVP = 8 mm Hg). That is, tension builds up in the cardiac muscle until the pressure exerted by the (incompressible) blood builds up enough to push open the semilunar valves (which are, until then, held shut by the pressure of the blood on the aorta side of the valve).

Pressure considerations in flowing fluids are slightly different than in static fluids.

6 Gases Dissolved in Water

Soluble gases like oxygen (O₂) and carbon dioxide (CO₂) dissolve in body fluids if the gas is in contact with the fluid.

The concentration of dissolved gas depends on the solubility of the gas and on its partial pressure.

• Henry’s Law

$$ {\mathrm{Conc}}^{\mathrm{n}}\ \mathrm{of}\ \mathrm{dissolved}\ \mathrm{gas}={P}_{\mathrm{gas}}\times \mathrm{solubility}\ \mathrm{coeff} $$

This means that the higher is the partial pressure of gas adjacent to a liquid surface (e.g. in the lungs), the greater the concentration of gas that will dissolve (→ oxygen therapy!).

Concentrations of blood gases are usually stated as “partial pressures” rather than as “dissolved concentrations”.

This means that if the partial pressure of O₂ in the alveolar air is 100 mm Hg (13 kPa) after inhalation, the arterial blood that leaves the lungs will contain dissolved O₂ with a concentration of ~100 mm Hg.

Partial pressure is another way of expressing solution concentration (for gases in a solution).

7 Pressure Gradient

If the pressure between two places is different, pressure gradient = pressure difference ÷ the distance between two places (unit = kPa/m).

Fluids (= liquids or gases) flow from places of high pressure to places of lower pressure. They flow down the pressure gradient. (Why do we press down hard with the hand during CPR?)

So blood flows from the LV to the arteries. Thus, air moves into and out of the lungs.

The rate of fluid flow (unit ml/s) is proportional to the pressure gradient (ΔP ÷ l).

Molecules dissolved in a gas or liquid undergo net diffusion along their concentration gradient: e.g. if 100 mm Hg O₂ is dissolved in alveolar fluid and 40 mm Hg O₂ is dissolved in the venous blood of the alveolar capillary, diffusion occurs from the alveolar fluid to the venous blood.

8 Factors That Affect Volume Flow Rate

Volume flow rate (V) is the number of litres flowing past per minute (rather than the speed of flow). It is the same thing as cardiac output:

$$ CO\ \left(\mathrm{ml}/\min \right)= HR\ \left(\mathrm{bpm}\right)\times SV\ \left(\mathrm{ml}\right). $$

• 1. Pressure gradient is the difference in pressure in the fluid at either end of the tube divided by the length of the tube (units kPa/m or mm Hg/m).

• 2. Resistance to flow through a tube depends on the length of the tube (l), the radius of the tube (R) and the internal friction between fluid particles (the fluid’s viscosity, η “eta”).

Poiseuille’s law relates the pressure gradient (ΔP/l) and resistance to flow to the volume flow rate V.

$$ \left( CO=\right)\kern0.5em V=\frac{\Delta P\times {R}^4}{l\times 2.54\times \eta}\kern1em \left(\mathrm{in}\ {\mathrm{m}}^3/\mathrm{s}\right) $$

“The volume flow rate, V, equals the pressure drop ΔP, divided by the resistance to flow (2.54 η l) ÷ R⁴.”

$$ V\ \upalpha\ 1/l,\kern1em V\ \upalpha\ {R}^4,\kern1em V\ \upalpha\ 1/\eta \kern1em V\ \upalpha\ \Delta P $$

• Pressure Gradient in the Body (V α ΔP/l)

  • If the pressure gradient doubles, other things being equal, so does the flow rate (V α ΔP/l).

  • While lying horizontally, the pressure gradient is due to the pumping action of the heart.

(16 kPa (120 mm Hg) in the aorta to 4 kPa (30 mm Hg) at the start of the capillaries

2 kPa (15 mm Hg) at the end of the capillaries about 530 Pa (4 mm Hg) or less at the right atrium)

• Radius of Tube (V α R⁴⁾

  • A wider tube produces less friction; in fact, if the radius is doubled (vasodilation), the flow rate increases 16 times! (V ∝ R⁴).

Conversely, if the radius is halved, the flow rate decreases to 1/16! (Coronary arteries narrowed by plaque lead to angina and myocardial infarction (MI).)

(vasoconstriction and vasodilation are potent mechanism to alter blood flow)

• Blood viscosity (η) (V α 1/η)

It is ≈ 0.004 Pa s for blood (0.001 Pa s for water) but changes with its speed. Blood vessels are not rigid tubes. Flow is pulsatile.

The higher the viscosity, the lower the flow rate (V α 1/η).

Friction increases with the percentage of cells in the blood (99% of cells are red blood cells (RBC)).

A “normal” man has about 42% of his blood volume taken up by cells (haematocrit of 42).

• Haematocrit of 15 = anaemia.

• Haematocrit of 70 = polycythemia.

(dehydration or hypothermia will cause blood friction to increase)

• Length of Blood Vessels (V α 1/l)

There is greater resistance in the peripheral circulation than in the pulmonary circulation because of the greater length of the blood vessels.

9 Capillaries

The purpose of the circulatory system is to deliver blood to the capillaries.

The primary function of the capillaries is the exchange of gases, nutrients and waste products between body cells and the blood.

• Average length = 1 mm

• Average diameter = 8–10 μm (RBC slip-through in a single file)

O₂, CO₂, amino acids, glucose, lipids, wastes, drugs, electrolytes and hormones pass between the capillary blood and interstitial fluid by diffusion along their concentration gradient (from where they are in high concentration towards where they are in lower concentration).

10 Fluid Movement Through Capillaries

“Equation of continuity”:

$$ \mathrm{Volume}\ \mathrm{flow}\ \mathrm{rate}\kern0.5em (V)=\mathrm{Cross}\hbox{-} \mathrm{section}\kern0.5em (CS)\kern0.5em \mathrm{area}\times \mathrm{speed}. $$

In the aorta, the cross-section area = 4 cm², and the speed of flow = 30 cm/s.

∴ V = 30 × 4 = 120 ml/s.

The volume flow rate must be the same in the capillaries, but the CS area = 4500 cm², so in the capillaries:

$$ \mathrm{Speed}=V\div CS\ \mathrm{area}=120\div 4500=0.03\ \mathrm{cm}/\mathrm{s}!\kern0.5em \left(=0.3\ \mathrm{mm}/\mathrm{s}\right). $$

This slow speed of blood flow allows exchange through the capillary wall to occur.

The capillary “bed” (extensive branching networks of capillaries) provide a large surface area for diffusion and filtration; ∴ the exchange of materials is rapid.

Movement Across the Capillary Wall

Blood enters the capillary bed from the terminal arteriole at ~35 mm Hg. Precapillary sphincters open to allow blood into the capillary.

More sphincters open as more blood flow is required (if closed, blood bypasses the bed via the “thoroughfare channel”).

Blood exits the bed into the post-capillary venule at ~17 mm Hg.

In addition, bulk fluid flow (H₂O) through capillary walls occurs due to pressure differences between the inside and outside of the wall.

• At the arteriole end:

  • HP (35 mm Hg) − OP (26 − 1) = 10 mm Hg

  • (hence, fluid moves OUT of the capillary).

• At venule end:

  • HP (17 mm Hg) − OP (26 − 1) = −8 mm Hg

  • (hence, fluid moves INTO the capillary).

• Excess fluid returns to the blood via lymph vessels.

11 Type of Flow

Laminar (smooth, streamlined) flow is usual in long smooth vessels.

Turbulent flow is when fluid continually mixes in swirls and eddies. Turbulence adds to the resistance to blood flow by increasing friction. It will occur if blood velocity is high (e.g. 30 cm/s in the aorta) and when blood passes through a narrowing or constriction in a vessel (a stenosis).

Turbulent flow is “noisy”.

• The opening and closing of the heart valves produce turbulent flow (heart valve sounds).

• Auscultatory blood pressure measurements “listen” for turbulent blood flow. The cuff causes turbulent flow in the squashed brachial artery (Korotkoff sounds).

12 Pressure Lecture Revision: Homework Exercise 14

• 1. Define pressure and explain how it is different from force.

• 2. Convert a blood pressure measurement of 130 mm Hg/80 mm Hg to units of kPa.

• 3. Use P = F/A to determine the pressure exerted during CPR for the two cases below. In each case, the resuscitator can apply a force of 200 N.

  • (a) If the whole hand (area = 140 cm² = 0.014 m²) is used

  • (b) If only the “heel” of the hand is used (area = 40 cm²)

• 4. What is meant by positive pressure?

• 5. Write out Pascal’s principle. Apply it to bed sore prevention or glaucoma or Queckenstedt’s test or the knee or to a foetus surrounded by amniotic fluid.

• 6. Use Henry’s law to describe why breathing air enriched to 30% O₂ is often a beneficial therapy.

• 7. Describe inhalation and exhalation using Boyle’s law and pressure gradient.

• 8. In about four sentences, describe and explain the movement of fluid into and out of capillaries.