Brit. J. Anaesth. (1965), 37, 967 PHYSICS APPLIED TO ANAESTHESIA III: THE PROPERTIES OF LIQUIDS, GASES AND VAPOURS BY D. W. HILL Research Department of Anaesthetics, Royal College of Surgeons of England, London Density The pressure exerted by a column of liquid or gas depends on its density. Density is defined as mass per unit volume. The density of water is 1 g per ml at 4 C, and that of mercury 13.6 g per ml at 20 G. Density is temperature-dependent and, strictly speaking, a value for a density must always include a statement of the ambient temperature. Although static columns of liquid are much used in manometers for measuring pressure, the majority of applications of liquids and vapours in surgery and anaesthesia are concerned with the movement of these substances. In Part II of this series, the work cost of moving a given volume of gas was calculated. In practice, we are mainly concerned with the movement of gases and liquids through pipes or tubes, and it is necessary to know which properties are important. Viscosity It can be shown that for a given pressure drop developed across a given pipe, the volume of liquid or gas flowing per unit time depends on its viscosity. In a viscous fluid, the velocity of adjacent layers of the fluid will differ. The viscosity of a liquid is defined as the resistance which it exhibits to the flow of one layer over another. The layers in contact with the walls are assumed to be at rest, and the linear velocity of the fluid along the length of the tube increases towards the centre. Consider two layers of area A situated at distances x and (x+dx) from one wall. Let the linear velocities of the planes be V and (V+dV). Due to the internal friction between the layers, there exists a tangential viscous force between them. The value of this force is given by 17 A This is Newton's law for viscous flow (for streamline as opposed to turbulent flow). The quantity T is termed the coefficient of viscosity of the fluid. d V For moderate pressure gradients, the individual molecules of the fluid progress down the pipe in orderly fashion. This is known as laminar or streamline flow. Under these conditions, the volume of fluid passing per second Q is given by the wellknown Poiseuille's formula * Srjl where P is the pressure gradient across the pipe, r is the radius of the pipe, and / is the length. It is seen that for a given gas, pressure gradient, and length of the pipe, the volume of gas that can be passed depends on the fourth power of the radius. Thus doubling the radius will increase the flow of gas by a factor of 2*= 16. It is for this reason that the bore of endotracheal tubes and connectors should be kept as large as possible. For sick patients breathing spontaneously with a small respiratory effort, the length and bore of any breathing tubes may be of importance. A pneumotachograph head is basically a resistance to gas flow, so designed that the pressure drop produced across it is linearly related to the volume flow rate over a stated range. The well-knowa design due to Professor Fleisch of Lausanne consists of a large number of small-bore pipes in parallel. The flow in each is small enough to be streamline. From Poiseuille's equation it is seen that the volume of gas passing in unit time is directly proportional to the pressure drop, providing the flow pattern remains streamline. Increasing the applied pressure above a certain value for the tube will cause the orderly motion of the molecules to break down. At higher values of applied pressure the flow becomes turbulent, the molecules swirling around in eddies. This is a less efficient way of transporting the gas or liquid. In between the fully streamline and fully turbulent conditions, there is a transitional region. The velocity V e at which fully turbulent flow sets in is known as the critical velocity.
968 BRITISH JOURNAL OF ANAESTHESIA It can be shown that v c = k -l where k is a constant known as Reynold's Number, r is the radius of the tube, and p is the density of the fluid. At low fluid velocities, when the motion is streamline, the volume flow is proportional to the pressure gradient. When the velocity is increased above the critical value, the volume flow increases less rapidly with pressure. It soon becomes independent of the viscosity of the fluid, and dependent mainly on its density. When fully turbulent, the volume flow is nearly proportional to the square root of the pressure. The pressure is now used to overcome the turbulent motion and in communicating kinetic energy to the fluid. Examples of turbulent flow are to be found in both the respiratory and circulatory systems. The sound heard during quiet breathing in a healthy person, using a stethoscope, constitutes the normal respiratory murnr.ir. The inspiratory murmur is a soft blowing sound of a frequency averaging 350 c.p.s. (Graves and Graves, 1964). It is related to turbulence as air flows from the small air passages into the alveoli. It is heard only because it takes place so near to the stethoscope. In the case of the circulation, turbulent flow is usually seen only at the root of the aorta, and possibly at the heart valves where it may be responsible for the sound of heart murmurs. Elsewhere the flow of blood is streamline, except in the smaller vessels, the arterioles and the capillaries. Here the bloodflowis faster than would be expected on the basis of Poiseuille's equation (Nightingale, 1959). The red cells tend to congregate along the axis of the tube, leaving a layer of plasma along the walls. The plasma has a lower viscosity than does whole blood, and this allows the cells to pass more rapidly than would uniform whole blood. The anomalous viscosity of blood is well discussed by Bayliss (1962). Turbulent flow conditions may arise from the rapid flow of blood through dilated vessels, if the force and speed of the blood passing is sufficient. It may be possible to hear sounds usually referred to as "bruits"; the bruit generally pulses in time with the heartbeat. Vascular bruits arising from the local turbulence may also be produced by such things as sudden changes in the smoothness of the vessel wall or diameter of the vessel. Turbulence occurring when a vessel branches is discussed by Krovetz (1965). Units of viscosity. The units of viscosity can be readily determined by substituting the appropriate dimensions for the other terms in Poiseuille's equation. Thus Q = volume per unit time (L 3 /T), P=pressure, force per unit area, (M.L.T.~ 2 /L 2 ), so that since on substituting the appropriate dimensions we have ML L 4 T _ M. T 2 L 2 0 L = TL The dimensional formula for viscosity is thus M.L^T- 1. This can be rewritten as MLT- 2.T.L.~ 2 or dynes per second per square centimetre. The unit of viscosity on the c.g.s. system is the poise where one poise equals 1 dyne per second per square centimetre. This is further divided into centipoises where 100 centipoises equal 1 poise. A mean value for the viscosity of whole blood in vivo is 2.70 centipoises. That of water is 1 centipose at 20 C. ' The onset of turbulence as expressed in the concept of critical velocity depends on the kinematic viscosity which is given by viscosity _ r\ density p for the gas or liquid. The dimensional formula for kinematic viscosity is L 2 T -1. The unit on the c.g.s. system is the Stokes where 1 Stokes is equal to 1 square centimetre per second. The kinematic viscosity for oxygen is approximately twice that for nitrous oxide, so that for a given tube and pressure drop one would expect that the flow for nitrous oxide would become turbulent at a considerably lower flow rate than would be the case for oxygen. The flow of fluids through orifices. In a pipe or tube, the length is large compared with the radius but in the case of an orifice, the length is less than that of the radius. Theflowof gas through an orifice is always partly turbulent and the rate of flow is dependent on the density of the gas rather than the viscosity. The smaller the density, the less is the pressure drop required across the orifice to maintain a given gas flow. For this reason, a helium-oxygen mixture is sometimes ad-
PHYSICS APPLIED TO ANAESTHESIA ffl ministered to patients suffering from respiratory obstruction, since this mixture can be respired with less effort than air. Measuring orifice C0 2 ABsorber Critical orifice To pump Sample Inlet tlg. I Critical orifice carbon dioxide analyzer. The critical orifice. An interesting case arises in that of the critical orifice. If the pressure on one side of a thin, sharpwalled, orifice is less than 0.58 times the absolute pressure on the upstream side of the orifice, the gas passing through the orifice attains the velocity of sound in the gas. The velocity of sound in a gas is given by v = / cm/sec, V p where y is the ratio of the specific heats at constant pressure and constant volume for the gas, P is the pressure at the orifice in dynes per sq.cm and p is the density of the gas in g/ml. If the cross-sectional area of the orifice is A cm2, then the volume flow rate V of gas through the orifice is given by V P A simple carbon dioxide analyzer based on this principle is shown in figure 1 (Mead and Collier, 1959). The critical orifice ensures a constant outflow of gas from the carbon dioxide absorber. The pressure across the inlet, non-critical, orifice depends on the rate at which gas enters the arrangement which in turn depends on the rate of absorption of carbon dioxide. The manometer can thus be calibrated in terms of the carbon dioxide concentration in the gas stream. Surface tension The attractive forces existing between molecules give rise, in liquids, to a phenomenon known as surface tension. In the bulk of the liquid, a mole- 969 cule is surrounded by others on all sides, and the attractive forces between molecules will tend to cancel. However, a molecule at the surface will receive a net inward force from those below it. The fact that a liquid surface contracts spontaneously shows that there is free energy associated with it, and that work must be done to extend the surface. The surface of a liquid acts at all times as though it has a thin membrane stretched over it, and for any specified volume of matter, a sphere has a smaller surface area than any other geometrical figure. Hence droplets of water and soap bubbles assume a spherical shape as they fall through the air. The work done in extending by 1 cm a surface which is pulling with a tension of S dynes per cm is 5 ergs per sq.cm. An erg is a dyne cm, and the c.g.s. unit of surface tension is dynes per cm. For example, the surface tension of water at 20 C is 73 dynes per cm...... It can be shown that the pressure existing inside a bubble is given by 2S\r dynes per sq.cm, where 5 is the surface tension of the liquid coating of the bubble, and r is the bubble radius in cm. The pressure inside the bubble is raised by the action of the compressive tension of the film. This force can be high when the bubble radius is small, resulting in a stable bubble. Such droplets may cause trouble when it is desired to vaporize a liquid such as halothane in order to produce accurate calibration concentrations of halothane vapour For example, the saturation vapour pressure of halothane cooled to -30 C is 16.8 mm Hg. Assuming a barometric pressure of 760 mm Hg, this will correspond to a halothane concentration of 2.2 per cent v/v. However, a rather higher value is produced due to the formation of droplets unless the issuing vapour is passed through a condensing coil immersed in the coolant bath, to trap out any small droplets of liquid halothane. Without this precaution, the small droplets are swept along with the vapour stream, and evaporate to produce an enhanced concentration. Surface tension plays an important part in the contraction of the lungs. The alveoli can be considered as a large number of bubbles, each communicating with the lung volume via a duct. If the surface tension of the alveolar lining was constant, then the pressure in the smaller alveoli would be higher than that in the larger alveoli. On this basis, one would have an unstable situation,
970 BRITISH JOURNAL OF ANAESTHESIA with the smaller alveoli tending to discharge into the larger ones. Brown, Johnson and Clements (1959), and Clements and associates (1961) have shown that the surface tension of the lung is not constant, but decreases markedly on compression resulting from deflation. With changes of surface area of less than 50 per cent the lung surface tension approaches a limiting value of 10-15 dynes per cm. On re-expansion of the surface, an upper limiting tension of 40-50 dynes per cm is approached. Considerable hysteresis occurs, and indicates a major alteration of the surface material when compressed beyond 50 per cent. The diminution in surface tension with reduction of area is thought to be due to the action of a lipoprotein which is known as the "anti-atelectatic substance" or "surfactant". Diffusion of gases. Gas molecules will move, by a process known as diffusion, from a place where there is a higher partial pressure of the gas concerned, to one where there is a lower partial pressure. This is a molecular process, and is not to be confused with processes such as convection in which a movement occurs of the gas in bulk. Thus gas molecules can diffuse through pores of a porous membrane to make the partial pressures equal on either side. The oxygenation of the venous blood in the lung capillaries arises from the diffusion of oxygen into the blood from the alveolar air. Similarly, carbon dioxide diffuses from the venous blood into the alveoli. Another example occurs with diffusion respiration. This was reported in the dog by Draper and Whitehead (1944), who found that under certain conditions, blood will be normally oxygenated from the lungs in the absence of respiratory movements and rhythmical pressure changes in the lungs. Draper and Whitehead (1949) explain the phenomenon as follows. The marked affinity of oxygen for haemoglobin causes the amount of oxygen removed from the alveoli during apnoea to exceed the amount of carbon dioxide that simultaneously leaves the blood. The net result is to reduce the pressure in the alveoli to below atmospheric, thus giving rise to a continuous inward flow of the deadspace gas and the external atmosphere. Essential conditions for diffusion respiration are a high percentage of oxygen in the lungs and in the deadspace, a free airway and an adequate circulation. The technique of diffusion respiration has been applied to man by Enghoff, Holmdahl and Risholm (1951), Payne (1962) and others. The quantitative treatment of diffusion is based upon Pick's law of diffusion, which is analogous to Ohm's law of conduction of electricity. Fields law states that the rate of diffusion is proportional to the concentration gradient, that is to the change of concentration per unit length in the direction of diffusion. An important case arises in the diffusion of oxygen and carbon dioxide across the alveolar membrane. Carbon dioxide diffuses rapidly, widi the result that equilibration is normally reached between the alveolar air and the capillary blood. Oxygen may not always achieve equilibrium, and this fact is reflected in the diffusion capacity for oxygen. Fields law may be stated in the form dq = - dt dx where dq is the volume of oxygen diffusing across the alveolar membrane in a time interval dt, k is the diffusion constant for oxygen and the alveolar membrane, 5 is the area of membrane through which the oxygen diffuses, and dc\dx is the concentration gradient across the membrane of thickness dx. For a thin membrane, dc ci cz dx Thus._ -ksfa-c&t By means of the Bunsen solubility coefficient for oxygen, the concentrations c\ and ci can be related to the gas tensions on either side of die membrane, i.e. and B where a is the Bunsen solubility coefficient, B is the barometric pressures and/>i and/>2 are the tensions. Pick's law equation becomes dq -ksa dt fi~ (pl ^ or
PHYSICS APPLIED TO ANAESTHESIA 971 where D is called the pulmonary diffusing capacity. In practice, the diffusing capacity for oxygen can be denned as DOz = where V02 is the volume of oxygen diffusing per minute, PAO, is the oxygen tension in the alveoli, and PCo, is the effective oxygen tension in the capillary blood. The units of pulmonary diffusing capacity for oxygen are ml of 02/per minute/mm Hg pressure difference across the lung. A typical value would be 21 ml Oz/min/mm Hg for D02 in a normal adult subject at rest. A clear account of diffusion processes in the lungs is given by Comroe and his colleagues (1962). Graham's law of diffusion states that the rates of diffusion of gases through certain membranes is inversely proportional to the square root of their molecular weight. The molecular weight of oxygen is 32, and that of carbon dioxide 44. The square roots of the molecular weights are in the ratio of 1.2:1, so that oxygen will diffuse faster by a factor of 20 per cent through a dry porous membrane than will carbon dioxide. The nature of the membrane, however, exerts an important influence on the rates of diffusion. For example, carbon dioxide diffuses rather more rapidly than oxygen through a rubber-lined Douglas bag. Mills (1952) reports losses in the range 0.22 per cent v/v to 0.45 per cent v/v of carbon dioxide per day from Douglas bags due to diffusion. As mentioned in the calculation on pulmonary diffusing capacity, the effective tension of gases on either side of the moist alveolar membrane depends on the solubility of the gases involved. The diffusion rate of carbon dioxide across the alveolar membrane is high, due to the marked solubility of carbon dioxide in water (23 times that of oxygen). Thus sufficient excretion of carbon dioxide occurs, even though the difference is small between the mixed venous carbon dioxide tension of 46.5 mm Hg, and the 40 mm Hg partial pressure of alveolar air. Oxygen diffuses in the reverse direction under the influence of the pressure gradient between the mixed venous oxygen tension of 40 mm Hg and the partial pressure of oxygen in the alveolar air of 110 mm Hg, causing the arterial blood to have a normal tension of 95 mm Hg of oxygen. Nitrogen does not diffuse across the alveolar membrane since the tensions in venous blood, arterial blood and the partial pressure in alveolar air are all the same, i.e. 573 mm Hg. REFERENCES Bayliss, L. E. (1962). The rheology of blood; in The Handbook of Physiology, Vol. 2, section 2 (Circulation), p. 137. Washington: American Physiological Society. Brown, E. S., Johnson, R. P., and Clements, J. A. (1959). Pulmonary surface tension. J 1. appl. Physiol: 14, 717. Clements, J. A., Hustead, R. F., Johnson, R. P., and Gribetz, I. (1961). Pulmonary surface tension and alveolar stability. J. appl. Physiol., 16, 444. Comroe, J. H., Forster, R. E., Dubois, A. B., Briscoe, W. A., and Carlsen, Elizabeth (1962). The Lung, 2nd ed., p. 111. Chicago: Year Book. Draper, W. B., and Whitehead, R. W. (1944). Diffusion respiration in the dog anesthetized with Pentothal sodium. Anesthesiology, 5, 262. (1949). The phenomenon of diffusion respiration. Curr. Res. Anesth. Analg., 28, 307. Enghoff, H., Holmdahl, M. H:Son., and Risholm, L. (1951). Diffusion respiration in man. Nature, 168, 830. Graves, G., and Graves, Valerie (1964). Medical Sound Recording. London: Focal. Krovetz, L. J. (1965). The effect of vessel branching on haemodynamic stability. Phys. in Med. Biol., 10,417. Mead, J., and Collier, C. (1959). Relation of volume history of lungs to respiratory mechanism in anesthetized dogs. J. appl. Physiol., 14, 669. Mills, J. N. (1952). The use of an infra-red analyser in testing the properties of Douglas bags. J. Physiol. (Lond.), 116, 22P. Nightingale, A. (1959). Physics and Electronics in Physical Medicine. London: Bell. Payne, J. P. (1962). Apnoeic oxygenation in anesthetized man. Acta anaesth. scand., 6, 129.