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.

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.

TABLE 14.1

Physical Quantities

The fundamental quantities in physics are mass, length and time.

Mass (m) is defined as the amount of matter in a body. The unit of mass is the kilogram (kg), for which the standard is a block of platinum held in a Physics Reference Laboratory.

Length (l) is defined as the distance between two points. The SI unit is the metre (m), which is defined as the distance occupied by a specified number of wavelengths of light.

Time (t) is measured in seconds. The reference standard for time is based on the frequency of resonation of the caesium atom.

From these basic definitions, several units of measurement may be derived:

Volume has units of m³.

Density is defined as mass per unit volume:

Velocity is defined as the distance travelled per unit time:

Acceleration is defined as the rate of change of velocity:

Force is that which is required to give a mass acceleration:

The SI unit of force is the newton (N). One newton is the force required to give a mass of 1 kg an acceleration of 1 m s^–1:

1 N = 1 kg m s^–2

Weight is the force of the earth’s attraction for a body. When a body falls freely under the influence of gravity, it accelerates at a rate of 9.81 m s^–2 (g):

Momentum is defined as mass multiplied by velocity:

momentum = m × v

Work is undertaken when a force moves an object:

work = force × distance

= F × l N m (or joules, J)

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.

Power (P) is the rate of doing work. The SI unit of power is the watt, which is equal to 1 J s^− 1:

power = work per unit time

= joules per second = watt (W)

Pressure is defined as force per unit area:

= pascal (Pa)

As 1 Pa is a rather small unit, it is more common in medical practice to use the kilopascal (kPa): 1 kPa ≈ 7.5 mmHg.

FLUIDS

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.

Behaviour of Gases

The Gas Laws

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.

Boyle’s law states that, at constant temperature, the volume (V) of a given mass of gas varies inversely with its absolute pressure (P):

PV = k₁

Charles’ law states that, at constant pressure, the volume of a given mass of gas varies directly with its absolute temperature (T):

V = k₂T

The third gas law (sometimes known as Gay-Lussac’s law) states that, at constant volume, the absolute pressure of a given mass of gas varies directly with its absolute temperature:

P = k₃T

Combining these three gas laws:

PV = kT

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.

GAS REGULATORS

Pressure Relief Valves

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:

FIGURE 14.2 A pressure relief valve.

force (F) = gas pressure (P) × disc area (A)

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.

Pressure-Reducing Valves (Pressure Regulators)

Pressure regulators have two important functions in anaesthetic machines:

They reduce high pressures of compressed gases to manageable levels (acting as pressure-reducing valves).

They minimize fluctuations in the pressure within an anaesthetic machine, which would necessitate frequent manipulations of flowmeter controls.

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.

FIGURE 14.3 A simple pressure-reducing valve.

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):

p.A = P.a

By tensing the spring, a force F is produced which offsets the closing effect of the valve. Thus, p may be increased by increasing the force in the spring.

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.

Pressure Demand Regulators

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.

FIGURE 14.4 The Entonox two-stage pressure demand regulator.

Flow of Fluids

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.

Hence:

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

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.

In a tube in which laminar flow occurs, the relationship between flow and pressure is given by the Hagen–Poiseuille formula:

where is the flow, ∆P is the pressure gradient along the tube, r is the radius of the tube, η is the viscosity of fluid and l is the length of the tube.

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.

Turbulent Flow: Flow of Fluids Through Orifices

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.

FIGURE 14.5 (A) Diagrammatic illustration of laminar flow. (B) Velocity gradient.

FIGURE 14.6 Diagrammatic illustration of turbulent flow.

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: