Basic Physics for the Anaesthetist
K nowledge of some physics is required in order to understand the function of many items of apparatus for anaesthesia delivery and physiological monitoring. This chapter emphasizes the more elementary aspects of physical principles and it is hoped that the reader may be stimulated to study some of the books written specifically for anaesthetists and which examine this topic in greater detail (see ‘Further reading’). Sophisticated measurement techniques may be required for more complex types of anaesthesia, in the intensive care unit and during anaesthesia for severely ill patients, and an understanding of the principles involved in performing such measurements is required in the later stages of the anaesthetist’s training.
This chapter concentrates on the more common applications, including pressure and flow in gases and liquids, electricity and electrical safety. However, it is necessary first to consider some basic definitions.
It is now customary in medical practice to employ the International System (Système Internationale; SI) of units. Common exceptions to the use of the SI system include measurement of arterial pressure and, to a lesser extent, gas pressure. The mercury column is used commonly to calibrate electronic arterial pressure measuring devices and so ‘mmHg’ is retained. Pressures in gas cylinders are also referred to frequently in terms of the ‘normal’ atmospheric pressure of 760 mmHg; this is equal to 1.013 bar (or approximately 1 bar). Low pressures are expressed usually in the SI unit of kilopascals (kPa) whilst higher pressures are referred to in bar (100 kPa = 1 bar). The basic and derived units of the SI system are shown in Table 14.1.
Energy is the capacity for undertaking work. Thus it has the same units as those of work. Energy can exist in several forms, such as mechanical (kinetic energy [KE] or potential energy [PE]), thermal or electrical and all have the same units.
Substances may exist in solid, liquid or gaseous form. These forms or phases differ from each other according to the random movement of their constituent atoms or molecules. In solids, molecules oscillate about a fixed point, whereas in liquids the molecules possess higher velocities and therefore higher kinetic energy; they move more freely and thus do not bear a constant relationship in space to other molecules. The molecules of gases possess even higher kinetic energy and move freely to an even greater extent.
Both gases and liquids are termed fluids. Liquids are incompressible and at constant temperature occupy a fixed volume, conforming to the shape of a container; gases have no fixed volume but expand to occupy the total space of a container. Nevertheless the techniques for analysing the behaviour of liquids and gases (or fluids in general) in terms of their hydraulic and thermodynamic properties are very similar.
In the process of vaporization, random loss of liquid molecules with higher kinetic (thermal) energies from the liquid occurs while vapour molecules randomly lose thermal (kinetic) energy and return to the liquid state. Heating a liquid increases the kinetic energy of its molecules, permitting a higher proportion to escape from the surface into the vapour phase. The acquisition by these molecules of higher kinetic energy requires an energy source and this usually comes from the thermal energy of the liquid itself, which leads to a reduction in its thermal energy as vaporization occurs and hence the liquid cools.
Collision of randomly moving molecules in the gaseous phase with the walls of a container is responsible for the pressure exerted by a gas. The difference between a gas and a vapour will be discussed later.
There are three gas laws which determine the behaviour of gases and which are important to anaesthetists. These are derived from the kinetic theory of gases; they depend on the assumption that the substances concerned are perfect gases (rather than vapours), and they assume a fixed mass of gas.
where suffixes 1 and 2 represent two conditions different in P, V and T of the gas. Note that where a change of conditions occurs slowly enough for T1 = T2, conditions are said to be isothermal, and the combined gas law could be thought of as another form of Boyle’s law.
The behaviour of a mixture of gases in a container is described by Dalton’s law of partial pressures. This states that, in a mixture of gases, the pressure exerted by each gas is the same as that which it would exert if it alone occupied the container. Dalton’s law can be used to compare volumetric fractions (concentrations) to calculate partial pressures, which are an important concept in anaesthesia. Thus, in a cylinder of compressed air at a pressure of 100 bar, the pressure exerted by nitrogen is equal to 79 bar, as the fractional concentration of nitrogen is 0.79.
These data are useful in calculating, for example, the quantity of gas produced from liquid nitrous oxide. The molecular weight of nitrous oxide is 44. Thus, 44 g of N2O occupy a volume of 22.4 L at standard temperature and pressure (STP). If a full cylinder of N2O contains 3.0 kg of liquid, then vaporization of all the liquid would yield:
If the temperature of a vapour is low enough, then sufficient application of pressure to it will result in its liquefication. If the vapour has a higher temperature, implying greater molecular kinetic energy, no amount of compression liquefies it. The critical temperature of such a substance is the temperature above which that substance cannot be liquefied by compression alone. A substance in such a state is considered a gas, whereas a substance below its critical temperature can be considered a vapour.
The critical temperature of oxygen is −118 °C, that of nitrogen is −147 °C, and that of air is −141 °C. Thus, at room temperature, cylinders of these substances contain gases. In contrast, the critical temperature of carbon dioxide is 31 °C and that of nitrous oxide is 36.4 °C. The critical pressures are 73.8 and 72.5 bar respectively; at higher pressures, cylinders of these substances at UK room temperature contain a mixture of gas and liquid.
A ‘full’ cylinder of oxygen on an anaesthetic machine contains compressed gaseous oxygen at a pressure of 137 bar (2000 lb in–2) gauge pressure. If the cylinder of oxygen empties and the temperature remains constant, the volume of gas contained is related linearly to its pressure (by Boyle’s law). In practice, linearity is not followed because temperature falls as a result of adiabatic expansion of the compressed gas; the term adiabatic implies a change in the state of a gas without exchange of heat energy with its surroundings.
By contrast, a nitrous oxide cylinder contains liquid nitrous oxide in equilibrium with its vapour. The pressure in the cylinder remains relatively constant at the saturated vapour pressure for that temperature as the cylinder empties to the point at which liquid has totally vaporized. Subsequently, there is a linear decline in pressure proportional to the volume of gas remaining within the cylinder.
Filling Ratio: The degree of filling of a nitrous oxide cylinder is expressed as the mass of nitrous oxide in the cylinder divided by the mass of water that the cylinder could hold. Normally, a cylinder of nitrous oxide is filled to a ratio of 0.67. This should not be confused with the volume of liquid nitrous oxide in a cylinder. A ‘full’ cylinder of nitrous oxide at room temperature is filled to the point at which approximately 90% of the interior of the cylinder is occupied by liquid, the remaining 10% being occupied by nitrous oxide vapour. Incomplete filling of a cylinder is necessary because thermally induced expansion of the liquid in a totally full cylinder may cause cylinder rupture. Because vapour pressure increases with temperature, it is necessary to have a lower filling ratio in tropical climates than in temperate climates.
Entonox: Entonox is the trade name for a compressed gas mixture containing 50% oxygen and 50% nitrous oxide. The mixture is compressed into cylinders containing gas at a pressure of 137 bar (2000 lb in–2) gauge pressure (see below). The nitrous oxide does not liquefy because the two gases in this mixture ‘dissolve’ in each other at high pressure. In other words, the presence of oxygen reduces the critical temperature of nitrous oxide. The critical temperature of the mixture is −7 °C, which is called the ‘pseudocritical temperature’. Cooling of a cylinder of Entonox to a temperature below −7 °C results in separation of liquid nitrous oxide. Use of such a cylinder results in oxygen-rich gas being released initially, followed by a hypoxic nitrous oxide-rich gas. Consequently, it is recommended that when an Entonox cylinder may have been exposed to low temperatures, it should be stored horizontally for a period of not less than 24 h at a temperature of 5 °C or above. In addition, the cylinder should be inverted several times before use.
Although the use of SI units of measurement is generally accepted in medicine, a variety of ways of expressing pressure is still used, reflecting custom and practice. Arterial pressure is still referred to universally in terms of mmHg because a column of mercury is still used occasionally to measure arterial pressure and also to calibrate electronic devices.
Measurement of central venous pressure is sometimes referred to in cm H2O because it can be measured using a manometer filled with saline, but it is more commonly described in mmHg when measured using an electronic transducer system. Note that, although we colloquially speak of ‘cm H2O’ or ‘mmHg’, the actual expression for pressure measured by a column of fluid is P = ρ.g.H, where ρ is fluid density, g is acceleration due to gravity and H is the height of the column. Because mercury is 13.6 times more dense than water, a mercury manometer can measure a given pressure with a much shorter length of column of fluid. For example atmospheric pressure (PB) exerts a pressure sufficient to support a column of mercury of height 760 mm (Fig. 14.1).
In considering pressure, it is necessary to indicate whether or not atmospheric pressure is taken into account. Thus, a diver working 10 m below the surface of the sea may be described as compressed to a depth of 1 atmosphere or working at a pressure of 2 atmospheres absolute (2 ata).
In order to avoid confusion when discussing compressed cylinders of gases, the term gauge pressure is used. Gauge pressure describes the pressure of the contents above ambient pressure. Thus, a full cylinder of oxygen has a gauge pressure of 137 bar, but the contents are at a pressure of 138 bar absolute.
The Heidbrink valve is a common component of many anaesthesia breathing systems. In the Magill breathing system, the anaesthetist may vary the force in the spring(s), thereby controlling the pressure within the breathing system (Fig. 14.2). At equilibrium, the force exerted by the spring is equal to the force exerted by gas within the system:
Modern anaesthesia systems contain a variety of pressure relief valves, in each of which the force is fixed so as to provide a gas escape mechanism when pressure reaches a preset level. Thus, an anaesthetic machine may contain a pressure relief valve operating at 35 kPa, situated on the back bar of the machine between the vaporizers and the breathing system to protect the flowmeters and vaporizers from excessive pressures. Modern ventilators contain a pressure relief valve set at 7 kPa to protect the patient from barotrauma. A much lower pressure is set in relief valves which form part of anaesthetic scavenging systems and these may operate at pressures of 0.2–0.3 kPa to protect the patient from negative pressure applied to the lungs.
Modern anaesthetic machines are designed to operate with an inlet gas supply at a pressure of 3–4 bar (usually 4 bar in the UK). Hospital pipeline supplies also operate at a pressure of 4 bar and therefore pressure regulators are not required between a hospital pipeline supply and an anaesthetic machine. In contrast, the contents of cylinders of all medical gases (i.e. oxygen, nitrous oxide, air and Entonox) are at much higher pressures. Thus, cylinders of these gases require a pressure-reducing valve between the cylinder and the flowmeter.
The principle on which the simplest type of pressure-reducing valve operates is shown in Figure 14.3. High-pressure gas enters through the valve and forces the flexible diaphragm upwards, tending to close the valve and prevent further ingress of gas from the high-pressure source.
If there is no tension in the spring, the relationship between the reduced pressure (p) and the high pressure (P) is very approximately equal to the ratio of the areas of the valve seating (a) and the diaphragm (A):
Without the spring, the simple pressure regulator has the disadvantage that reduced pressure decreases proportionally with the decrease in cylinder pressure. The addition of a force from the spring considerably reduces but does not eliminate this problem, and in order to overcome it, newer pressure regulators contain an extra closing spring. During high flows, the input to the valve may not be able to keep pace with the output. This can cause the regulated pressure to fall. A two-stage regulator can be employed in order to overcome this. Simple one-stage regulators are often designed for use with a specific gas. A universal regulator, in which the body is used for all gases but has different seatings and springs fitted for each specific gas, is now available.
These are regulators in which gas flow occurs when an inspiratory effort is applied to the outlet port. The Entonox valve is a two-stage regulator and its mode of action is demonstrated in Figure 14.4. The first stage is identical to the reducing valve described above. The second-stage valve contains a diaphragm. Movement of this diaphragm tilts a rod, which controls the flow of gas from the first-stage valve. The second stage is adjusted so that gas flows only when pressure is below atmospheric.
Viscosity (η) is the constant of proportionality relating the stress (τ) between layers of flowing fluid (or between the fluid and the vessel wall), and the velocity gradient across the vessel, dv/dr.
In this context, velocity gradient is equal to the difference between velocities of different fluid layers divided by the distance between layers (Fig. 14.5B). The units of the coefficient of viscosity are Pascal seconds (Pa s).
Fluids for which η is constant are referred to as Newtonian fluids. However, some biological fluids are non-Newtonian, an example of which is blood; viscosity changes with the rate of flow of blood (as a result of change in distribution of cells) and, in stored blood, with time (blood thickens on storage).
Viscosity of liquids diminishes with increase in temperature, whereas viscosity of a gas increases with increase in temperature. An increase in temperature is due to an increase of kinetic energy of fluid molecules. This can be thought of as causing a freeing up of intermolecular bonds in liquids, and an increase in intermolecular collisions in gas.
Laminar flow through a tube is illustrated in Figure 14.5A. In this situation, there is a smooth, orderly flow of fluid such that molecules travel with the greatest velocity in the axial stream, whilst the velocity of those in contact with the wall of the tube may be virtually zero. The linear velocity of axial flow is twice the average linear velocity of flow.
The Hagen–Poiseuille formula applies only to Newtonian fluids and to laminar flow. In non-Newtonian fluids such as blood, increase in velocity of flow may alter viscosity because of variation in the dispersion of cells within plasma.
In turbulent flow, fluid no longer moves in orderly planes but swirls and eddies around in a haphazard manner as illustrated in Figure 14.6. Essentially, turbulent flow is less efficient in the transport of fluids because energy is wasted in the eddies, in friction and in noise (bruits). Although viscosity is the important physical variable in relation to the behaviour of fluids in laminar flow, turbulent flow is more markedly affected by changes in fluid density.
It may be seen from Figure 14.7 that the relationship between pressure and flow is linear within certain limits. However, as velocity increases, a point is reached (the critical point or critical velocity) at which the characteristics of flow change from laminar to turbulent. The critical point is dependent upon several factors, which were investigated by the physicist Osborne Reynolds. These factors are related by the formula used for calculation of Reynolds’ number:
Studies with cylindrical tubes have shown that if Reynolds’ number exceeds 2000, flow is likely to be turbulent, whereas a Reynolds’ number of less than 2000 is usually associated with laminar flow. However, localized areas of turbulent flow can occur at lower Reynolds’ numbers (i.e. at lower velocities) when there are changes in fluid direction, such as at bends, or changes in cross sectional area of the tube.
A tube can be thought of as having a length many times its diameter. In an orifice by contrast, the diameter of the fluid pathway exceeds the length. The flow rate of a fluid through an orifice is much more likely to be turbulent and is described by the factors discussed above.
In upper respiratory tract obstruction of any severity, flow is inevitably turbulent downstream of the obstruction; thus for the same respiratory effort (driving pressure), a lower tidal volume is achieved than when flow is laminar. The extent of turbulent flow may be reduced by reducing gas density; clinically, it is common practice to administer oxygen-enriched helium rather than oxygen alone (the density of oxygen is 1360 kg m− 3 and that of helium is 160 kg m− 3). This reduces the likelihood of turbulent flow and reduces the respiratory effort required by the patient.
In anaesthetic breathing systems, a sudden change in diameter of tubing or irregularity of the wall may be responsible for a change from laminar to turbulent flow. Thus, tracheal and other breathing tubes should possess smooth internal surfaces, gradual bends and no constrictions.
Resistance to breathing is much greater when a tracheal tube of small diameter is used (Fig. 14.8). Tubes should be of as large a diameter and as short as possible.
In a variable orifice flowmeter, gas flow at low flow rates is predominantly laminar. Flow depends on viscosity. At higher flow rates, because the flowmeter behaves as an orifice, turbulent flow dominates and density is more important than viscosity.
A venturi is a tube with a section of smaller diameter than either the upstream or the downstream parts of the tube. The principles governing the behaviour of fluid flow through a venturi were formulated by Bernoulli in 1778, some 60 years earlier than Venturi himself. In any continuum, the energy of the fluid may be decribed by the Bernoulli equation, which suggests that the sum of energies possessed by the fluid is constant, i.e.:
assuming that the predominant fluid flow is horizontal such that gravitational potential energy can be ignored.
In a venturi, in order that the fluid flow be continuous, its velocity must increase through its narrowed throat (v2 > v1). This is associated with an increase in kinetic energy and Bernoulli’s equation shows that where this occurs, there is an associated reduction in potential energy and therefore in pressure. Beyond the constriction, velocity decreases back to the initial value and the pressure rises again. The principle is illustrated in Figure 14.9