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Figure PH. 12 Expiratory flow curve curve is shown in Figure PH.12. Here the flow rate is not constant but it varies with time and can be written as a function Q(t). To find the expired volume, the patient's total expiratory time is first split into small intervals of length 5t. The flow rate can be considered approximately constant over each time interval, 5t. The incremental volume, 5V, which flows during each 5t is given by:

The expired volume is then the sum of these individual increments in volume (Z represents 'the sum of): Expired vol nine = v Q(t) x fir where Q(t) is the volume flow rate at any instant in time, and is, thus, the height of the rectangular strips in Figure PH.12.

Note that the flow rate is not constant over each increment in time, particularly when it is changing rapidly during the rise and fall of the curve. However, in the mathematical operation of integration, the width of each rectangular strip is allowed to become infinitely small. In this case flow rate becomes effectively constant for this instant in time. The sum of instantaneous flow rates is then written:

infinitely narrow strips of Q(t). The numbers t1 and t2 at the top and bottom of the integral sign are the times for starting the summing process, and finishing it.

### Evaluation of an Integral

If a graph can be described by a mathematical function, then the integral of this function can be found and used to determine the area under the curve. Once the integral of a function has been found it is relatively simple to calculate the integral (area under any section of the curve) by substituting values into the integral.

Differentiation

The symbol ^' is called the integral sign, and is a stylised ' S', indicating that one is performing a sum of

Consider the problem of determining gas flow rate by using a spirometer that measures expired volume. The recording from the spirometer is a graph of total expired volume as a function of time. If the average flow is required, a simple ratio of total expired volume over expiratory time is taken:

However, to determine the flow rate at each instant in time, the following approach is taken. The expiratory time is again split up into small increments, each of duration 5t, during which the change in expired volume is 8V. Then at any given time, the average flow over this small interval, dt is given by:

which also represents the average gradient of the graph over 8t.

The mathematical operation of differentiation then allows the time interval, 8t, to become infinitely short, so that the above flow becomes an instantaneous flow. This can be written as:

The letter 'd' is used to show that the intervals in V and t have become infinitely small. This mathematical process is called differentiation' and can be applied to any function. Differentiation of a function produces another function called the 'derivative'. This can be written as:

The derivative defines the gradient of the function at any point on its curve. Differentiation can be thought of as the inverse of integration. For example, if the function sin(t) is differentiated, the result is cos(t). Integration of cos(t) then gives sin(t), the original function. This holds true for any function.

### Mechanics

Mechanics is the science and study of forces and their effects on the motion of bodies and the stresses and strains within those bodies. At the heart of mechanical principles are Newton's three laws of motion:

• A body remains at rest or in a state of uniform motion in a straight line unless acted on by an external force.

• If a force acts on a body, it produces an acceleration equal to the force divided by the mass of the object.

• If two bodies, A and B, interact with each other, then any force that A imparts on B will be matched by an equal and opposite force which B imparts on A.

Mechanical principles can be applied to any system where forces act. In static systems, such as the skeletal structure of a person holding up a heavy weight, mechanics can be used to find forces and stresses involved. Alternatively, the forces may produce motion, and mechanics can inform as to the magnitude and nature of the motion.

Mass, Force and Acceleration

Mass is rather vague in its definition. A common view is that it the amount of 'matter' that a body possesses. However, mass must be distinguished from volume, but mass gives a body inertia and, therefore, makes it difficult to start it moving, or if it is already moving, to stop it. There is some confusion of the terms 'mass' and 'weight', and they are often used interchangeably. The weight of an object is the force on that object when it is placed in a gravitational field. While this force will be proportional to the mass for a given gravitational field strength, the force would change if the object were taken to the Moon (where the gravitational field is different from the Earth). Thus, the mass is a fundamental property of an object, but its weight depends on the gravitational field in which it is weighed.

In a frictionless system (in Outer Space), a force will cause a mass to accelerate. Mass is a scalar quantity, but force is a vector since it is applied in a particular direction. The direction of the applied force is the same as the direction of the resulting acceleration. Acceleration is also a vector quantity.

### Work and Energy

Work and energy both have the same units (joules), and both have common and rather imprecise meanings, as well as rigorous definitions in physics. Useful mechanical work can be expended in doing two things: either accelerating a body or overcoming a resistive force. In isometric muscle activity, neither of these two things is happening.

In Figure PH.13, a constant force f is applied to an object of mass m, and the object is moved from point A to point B. The work done, W (measured in watts), is the product of the applied force and the distance moved:

Figure PH.13 Movement of a mass from points A to B (separated by a distance 'x')

by applying a constant force 'f'

If at point B the object has gathered speed, then the work (or energy) expended will have given the mass kinetic energy. If, however, the force was applied to overcome a resistance, perhaps because the object was rubbing on the floor, then the work done will have caused some heat to be generated due to friction.

The principle of conservation of energy states that energy may be transformed from one type to another, but cannot be destroyed in the process. The work (energy) put into the system above is transformed into either kinetic energy or heat energy, or some combination of the two.

Potential energy (PE) is the energy that a body possesses because of its position in a gravitational field. The work done, and the potential energy (PE) gained, in raising a heavy object of mass m through a height h, is equal to the product of the gravitational force and the height:

where g is the gravitational force per unit mass. From Figure PH.14 it can be appreciated that this is simply the product of the force and distance seen in the work equation above. The energy could be recovered later by attaching the mass to a pulley system, and using its fall to drive a motor.

Figure PH. 14 Movement of a mass 'm' through a height 'h'. The constant gravitational force on it equals m x g

Kinetic Energy.

A body has kinetic energy (KE) when it is in motion. It is proportional to the mass of the object and to the square of the its velocity, v:

KE = Yimv2

The squared velocity results from the fact that energy is being measured in terms of the velocity and not the distance required to accelerate the body to that velocity.

### Heat Energy.

Heat is energy in the form of the microscopic motion of molecules making up the structure of a body. Heat energy can be extracted by placing a hot body in contact with a cooler one. Heat energy will then always flow from the hot body into the cooler one.

### Chemical Energy

When chemical bonds form, energy may be given out in the process; this would be termed an exothermic reaction. On the other hand, energy may be required to form the bonds (an endothermic reaction). The energy given out or taken in during a chemical reaction may be in the form of heat and light (such as during the burning of methane gas), mechanical work (during muscular contraction) or electrical potential.

If energy is put in to form chemical bonds, it will be given out when these bonds are broken. The sun provides energy to synthesize glucose in plants, and this energy is recovered later by oxidation within the body.

### Power and Efficiency

Power is the rate at which energy is transferred. Efficiency is the proportion of useful energy produced compared with the energy that is wasted. If a motor car has a very small engine, the force that it can apply to the wheels is also small and the car will accelerate poorly. A high-powered engine will consume fuel at a faster rate, but the car will achieve the desired speed much sooner.

Heat

The nature of heat and the meaning of temperature eluded experimenters until comparatively recently. It was not until the early 19th century that heat and other forms of energy were regarded as equivalent. Before that time heat was thought of as an invisible fluid, called caloric, which flowed from hot bodies to cold ones. It was hypothesized that heat was simply another form of energy by several independent workers, and each was either derided or ignored at the time.

### Heat Capacity

The heat capacity of a body is the amount of heat energy needed to raise the temperature of the body by 1°C. For a given amount of heat, Q, and an associated temperature rise, T. The heat capacity, C, is thus:

Heat capacity is a property of a particular object. The specific heat capacity of a material allows prediction of the heat capacity of any object made from that material. Specific heat capacity, c, is the amount of heat energy required to raise the temperature of 1 kg of a material by 1°C. For a mass, m, of material

Using this equation, it is possible to calculate the amount of heat needed to raise the temperature of an object if the mass and the material from which the object is made are known.

### Latent Heat

In some circumstances, it is possible to supply heat energy to a material without raising its temperature, for example, when solids are melted or when liquids boil to form vapour. In the case of melting, the molecular kinetic energy changes from being the vibrational energy of molecules in the solid matrix to energy due to the free rotation and linear motion of the molecules. In addition, the chemical bonds that joined the molecules in the solid are broken. The heat supplied goes towards breaking these bonds, but the average molecular kinetic energy, and hence the temperature, does not change. The heat supplied is known as the latent heat of fusion. Similarly, when a liquid vaporizes to form a gas, the energy supplied (the latent heat of vaporization) breaks the bonds that held the molecules together in the liquid, allowing free motion in the gas. The latent energy is surprisingly large compared with the energy required to change the temperature. For example, it takes approximately the same heat energy to change ice at O°C to water at the same temperature as it does to heat the same water from 0 to 100°C. It would then take more than five times the heat energy to boil that water into steam at 100°C.

### Boiling Point

The terms 'gas' and 'vapour' are used almost interchangeably although, strictly, they should not. If a vapour exists in a container at a certain pressure, and then the pressure is increased sufficiently (at a constant temperature), some of the vapour will return to the liquid phase. A gas is simply a vapour that is above its critical temperature, the temperature at which no amount of increased pressure will cause it to liquefy. While oxygen, for example, is a true gas at room temperature, water is a vapour, and does not become a gas until it reaches 400°C. A saturated vapour is one that exists in equilibrium with its liquid form, so that at a given pressure as many molecules are evaporating from the liquid as are returning to the liquid through condensation. If the pressure of a saturated vapour is raised, then this results in further condensation until a new equilibrium is reached with fewer molecules in the vapour phase. At the same time, the pressure reduces due to fewer molecules being in the vapour phase until it is back to the original pressure. Similarly, if the pressure is reduced, then evaporation occurs to bring the liquid and vapours back to equilibrium and the pressure is raised in the process. The pressure at equilibrium is known as the saturated vapour pressure, and varies with the temperature of the liquid/vapour. The relationship between temperature and saturation vapour pressure for water is shown in Figure PH.15.

Boiling is a visible form of evaporation, occurring when there is a sudden change away from the saturation equilibrium. This can occur when a large amount of heat is supplied to the liquid, or when the pressure above the liquid suddenly drops. Boiling can be thought of as the formation of small regions (bubbles) where the local internal pressure just falls below the saturation vapour pressure due to local temperature fluctuations. It follows from Figure PH.15 that the boiling temperature of a liquid varies with pressure. By artificially increasing the pressure above a liquid, the boiling point is increased above the 'normal' boiling point at atmospheric pressure. The water vapour under these conditions is known as 'superheated' and is used practically in pressure cookers to reduce the cooking time and in steam sterilization equipment to speed the process.

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Figure PH.15 Variation of the saturation vapour pressure for water with temperature

### Cooling by Evaporation

In hot climates, the most important way in which excess heat is lost from the body is by evaporation of perspiration from the surface of the skin. The film of water on the surface of the skin contains molecules in rapid motion. Some of the more energetic molecules may be ejected from the liquid to form vapour, leaving the less energetic ones behind. In this way, the average kinetic energy of the water will be reduced (its temperature will have fallen) and it is able to absorb heat from the body. In very humid weather the air already contains a good deal of water vapour, and water will tend to condense on, as well as evaporate from, the skin. It is much more difficult to dissipate body heat.

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### Mechanisms of Heat Energy Transfer

It is often said that either conduction, convection or radiation transfers heat. However, convection is not really a mechanism for heat transfer, but rather for material transport. The heat is not transferred from the fluid being convected, but the fluid is moved and the heat energy it contains is taken with it. There are analogous mechanisms in the human body that involve the active transport of heat carrying material, principally blood flow. Blood is used to transport many substances around the body, but is also used to redistribute heat away from the organs and muscles which generate heat, either for dissipation or for the maintenance a more even temperature throughout the body. Conduction occurs when two materials are in physical contact and are at different temperatures. Heat flows from the hotter body to the cooler one. More accurately, heat will flow through a body when a temperature difference exists across it. Thus, when blood is brought close to the surface of the skin, and the skin surface is at a lower temperature than the blood, then heat will flow through the cells of the skin by conduction. The skin surface is kept cool by evaporation of water from the surface, and fresh, warm blood is continuously being pumped up to the skin, so heat continues to flow and to be lost from the body. When an object is at temperature above absolute zero (- 273.15°C), it will lose heat by radiation. Since radiation is the loss of energy by emission of photons (small packets of energy), there needs to be no physical contact between the object and its surroundings in order for energy to be lost by this mechanism. However, any object will also gain heat by absorbing the energy radiated by surrounding objects. The efficiency with which objects emit or absorb radiation is called the emmissivity which depends on the surface properties of the object.

If the object is at a higher temperature than the surroundings, then it loses net energy, while if it is at a lower temperature, it will receive net energy.

Gases

Pressure, Volume and Temperature Molecular View of Gas Pressure

Within a gas, the molecules are in constant, rapid motion. The molecules collide with each other and also with the walls of any container in which the gas is retained (Figure PH.16). It is these constant and countless collisions that result in a force against the container which cause the gas pressure. In each collision, the impact is perfectly elastic, and as long as the gas and the container are at the same temperature, there is no net exchange of energy between them. Thus, the gas pressure will not change over time in these circumstances.

For a container of volume V containing n moles of gas, the relationship between pressure (P), volume and temperature (T) is given by the Universal Gas Law:

Figure PH. 16 A gas molecule inside a container constantly colliding with the walls of the vessel

This law does not depend on the actual gas in the container, it applies to any gas. The constant, R, is called the universal gas constant. There are, however, some restrictions on the applicability of the Universal Gas Law. It is derived with the assumption that the molecules remain in the gaseous phase. At low temperatures or high pressure, this relationship breaks down as the gas starts to condense into a liquid. Furthermore, it does not take into account the forces between molecules (van der Waal's forces), nor the fact that molecules are of a finite size. Small correction factors may be introduced to account for these, but for most purposes they can be neglected.

### Absolute and Relative Pressure

All pressure gauges measure differences in pressure. For example, a mercury barometer measures the difference between atmospheric pressure and the (almost) zero pressure above a column of mercury. For many of the pressure gauges used in anaesthesia, the difference is that between the pressure to be measured and atmospheric pressure. The mercury barometer is said to measure absolute pressure (since it is the difference from zero), while the latter gauges measure relative pressure (since the reading is relative to atmospheric pressure). Thus, when the pressure measured is atmospheric pressure, the reading on a relative pressure gauge is always zero, but the reading on an absolute pressure gauge will be about 1 atm depending on the weather.

### The Gas Laws

Before molecular theory could account for the relationship between pressure, volume and temperature, some of the relationships between these variables were discovered experimentally. Boyle determined what happened to the pressure and volume of a gas at constant temperature. From the universal gas law, it is easy to see that i'V = cmiMLiin (lioyle'j Liw)

Similarly, Charles and Gay-Lussac showed that at constant volume, a given mass of gas obeys the relationship: — = constant (Charles's law}

As long as the temperature is measured on the absolute temperature scale, these two laws are special cases of the universal gas law.

Dalton's Law of Partial Pressures

Gases are rarely composed from a single type of molecule, but they contain mixtures of gases. Dalton's Law (Figure PH.17) states that the contribution which each constituent gas makes to the overall pressure is in proportion to the number of molecules of that constituent. Putting this another way, if each gas existed separately in containers all of the same size and shape, then the sum of the pressures in the individual containers would equal the pressure if all the gases were put together in just one of containers. The pressure that would be exerted by one of the constituent gases is known as the partial pressure of that gas.

Figure PH.17 The sum of the individual pressures of the separate gases in the three bottles would equal the pressure in one bottle if all three gases were put into that bottle

Henry's Law and Graham's Law

The atmosphere is made up of several gases mixed in different proportions; the constituents and their relative contributions are shown in Figure PH.18. The atmosphere also contains varying amounts of water vapour according to the relative humidity and temperature.

The alveolar concentration of CO2 is much greater than atmospheric and alveolar gas is saturated with water vapour at body temperature. The air is in effect diluted by water vapour, and as a result the partial pressures of alveolar gases are lower than might be expected. Gases move from the atmosphere into the blood by diffusion across membranes, primarily the alveolar capillary membrane, and a net movement of gases occurs when there is a difference in partial pressure across the membrane. If there is a gas on one side of the membrane and a liquid on the other, then the gas will diffuse across the membrane and dissolve in the liquid until the partial pressure in the liquid is the same as the gas pressure. This is an illustration of Henry's Law which states that the partial pressure exerted by a dissolved gas is proportional to the concentration of gas molecules in solution. If a gas and liquid (containing dissolved gas) are at equilibrium, and the gas pressure is suddenly reduced, then gas will diffuse out of the liquid, reducing the pressure in the liquid until they are once more at equilibrium. Similarly, more gas will be dissolved in the liquid if the gas pressure is increased. The pressure in a liquid can be taken as the gas pressure above a liquid at equilibrium. The rate at which gas molecules diffuse is governed by Graham's Law, which states that the larger the molecules the slower their diffusion rate. The diffusion rate is in fact inversely proportional to the square root of the molecular weight. Graham's Law comes into play as gases diffuse across the alveolar membrane and are dissolved in the blood. As the membrane is very thin, complete transport of gases occurs in < 0.5 s.

 GASES IN THE ATMOSPHERE AND INSIDE THE ALVEOLI Gas Relative pressure in atmospheric air Relative pressure in alveolar gas Nitrogen 78% 75% Oxygen 21% 13% Argon 1% 1% Carbon dioxide 0.03% 5% Water vapour Variable 6%

Figure PH.18 Hydrodynamics

Figure PH.18 Hydrodynamics

Gases, Liquids and Fluid Behaviour

Although gases and liquids differ considerably in their physical properties, they display similar behaviour under flow conditions, and can both be described as being "fluid'. The following points are similarities in behaviour between gases and liquids:

• Liquids and gases both fill the shape of their container, and are subject to constraints imposed by gravity. However, because of their lower density, gases are less affected by gravity than liquids

• The flow behaviour of gases and liquids is largely determined by density and viscosity, although gases have much lower density and viscosity than liquids

• Flow is produced in both gases and liquids, by the application of a pressure gradient

The similarity between gases and liquids in flow behaviour has led to the development of "fluid mechanics', which is the study of fluids in motion and applies equally to both gases and liquids.

### Viscosity

Viscosity may be thought of as the "stickiness' of a fluid. It takes much more effort to pull a spoon quickly though treacle than to pull it through water, because treacle is much more viscous than water. Similarly, when different fluids flow through a tube the more viscous the fluid, the slower the flow through the tube. Gases are far less viscous than liquids and viscous effects only become apparent at much higher flow velocities in gases compared with liquids.

### Shear Stress and Shear Rate

A viscous force, or drag, is felt on any object when it moves through a viscous fluid, or when the fluid moves past a stationary object. Figure PH.19 shows a thin, flat plate with a fluid flowing past it. Away from the plate, the fluid flows faster, but closer in towards the plate, the fluid slowed down by the presence of the plate until at the surface the fluid is not moving. This happens at any surface because of adhesion between the fluid and the solid surface; it is known as the 'no-slip' condition. Near to the surface, the flow pattern is deformed from one of uniform flow velocity, to one in which layers of fluid parallel to the direction of flow 'slip' against each other giving rise to a drag effect or 'shearing' action. This shearing action at the surface gives a drag force per unit area of the plate, which is called the 'shear stress'.

Figure PH.19 Shear stress

The lengths of the arrows represent the velocity of the fluid, which diminishes to zero next to the plate. The velocity of flow, thus, varies between these fluid layers, i.e. a velocity gradient perpendicular to the direction of flow, or 'shear rate, is produced.

### Coefficient of Viscosity

The viscosity of a fluid can be defined by considering laminar flow (see below) in which two parallel layers of fluid are slipping against each other. This produces two effects, a shear stress between the layers, and a velocity gradient at right angles to the direction of flow (shear rate). The coefficient of viscosity (or viscosity) is defined by:

The units of viscosity are 'poises' after Poiseuille, who discovered the laws governing the flow of fluids through tubes. Water has a viscosity of 0.0101 poises at 20°C, while air has a viscosity of 0.00017 poises at 0°C.

Viscosity varies with temperature. Liquids generally become less viscous with increasing temperature, while gases become more viscous as temperature rises.

Newtonian fluids are fluids in which viscosity, n, is constant, regardless of the velocity gradients produced during flow. Many fluids including water, are Newtonian. Some fluids, however, do not behave in this way, such as the shear thinning fluids whose viscosity falls as the shear rate between layers increases, and the rheotropic fluids, which become more viscous the longer the shearing persists. Blood is a shear-thinning fluid.

### Measurement of Viscosity

Viscometers are used to obtain a measurement for the coefficient of viscosity. The simplest form of viscometer allows fluid to flow under the influence of gravity down a fine-bore calibrated tube. The rate of fall of the fluid meniscus is detected by photocells from which viscosity can be calculated.

A more complicated device uses the viscous drag created by spinning a small drum containing a sample of fluid. A pointer is mounted on a float suspended in the sample and is displaced by the torque due to the viscous drag. This records the viscosity measurement on a scale.

### Viscosity and the Damping of Fluid Flow

The viscous shearing action in a flowing fluid dissipates energy as heat and is analogous to frictional effects between two solid surfaces rubbing against each other. This dissipative effect dampens the motion of fluid in a system, and thus viscous effects form a major component of 'damping' in any hydrodynamic system. As with mechanical or electrical systems damping is an important factor in determining the behaviour of system.

Viscous effects can also affect the pattern of flow, since fluid flow can occur with two different basic patterns:

Laminar flow where flow is smooth, with one layer of fluid sliding evenly over the one next to it. All fluid velocity vectors at any cross-section tend to be parallel to the overall direction of flow.

Turbulent flow turbulent flow contains eddies of swirling fluid, which disrupt the flow and give greater drag. Flow velocity vectors may occur in all directions including perpendicular to the overall direction of flow.

### Flow Through Tubes

When a pressure difference is applied between the ends of a tube, fluid will flow from the high-pressure end to the low-pressure end. In the case of a liquid, gravity may also play a part in driving fluid downwards.

Using a simple manometer the pressure difference across the ends of a tube can be visualized. This will depend on flow velocity, tube length and diameter, viscosity, and density of the fluid. An analogy may be made with an electrical circuit: fluid flow (electrical current) occurs along the tube (conductor) because of a driving pressure (voltage), and energy is dissipated by the viscous drag between the fluid and the tube (electrical resistance).

### Hagen-Poiseuille Law

Hagen (in 1839) and Poiseuille (in 1840) discovered the laws governing laminar flow through a tube. Consider a pressure, P, applied across the ends of a tube of length, l, and diameter, d. Then the flow rate, Q, produced is proportional to:

• Fourth power of the tube diameter (d4)

• Reciprocal of fluid viscosity (1/n) This is often combined as:

and attributed to Poiseuille, a surgeon, who verified this relationship experimentally.

### Kinematic Viscosity

As noted above, the viscosity of a fluid influences its flow pattern by creating a damping effect. However, the inertial properties of the fluid (dependent on fluid density) also affect the flow pattern. Thus, the relative effects of inertial and viscous forces can determine the nature of fluid flow in any given situation. This is taken into account by using the kinematic viscosity (p), which is defined as the ratio of the viscosity to the density (p). If the kinematic viscosity is high, rapid irregular flow patterns in a fluid will be well-damped, but if it is low then disturbances such as swirling eddies may persist for a long time.

Reynolds' Number

The Reynolds' number (Re) is used to determine whether the flow will be laminar or turbulent in any given situation. It includes the kinematic viscosity, p, and the ratio of the inertial forces to the viscous damping forces in the fluid and is given by:

where v = mean flow velocity for flow through a

] = a characteristic length of the system, such si [he diameter of a tube.

At low Re, the viscous forces dampen minor irregularities in the flow, resulting in a laminar pattern. A high Re means that the inertial forces dominate, and any eddies in the flow will be easily created and persist for a long time, creating turbulence. For flow though a tube, an Re < 2000 tends to give laminar flow, while between 2000 and 4000 the flow may be a mixture of laminar and turbulent depending on the smoothness of the fluid entering the tube. At an Re > 4000, flow will certainly be turbulent.

Velocity Profiles for Laminar and Turbulent Flow in a Tube

In considering the velocities across a tube (the velocity profile), shapes are different for laminar and turbulent flow (Figure PH.20).

Figure PH.20 Velocity profiles in laminar and turbulent flow

Figure PH.20a shows the profile for laminar flow. This is much more pointed than the flat central portion of turbulent flow. The arrows are flow velocity vectors and are all parallel to the axis of the tube. There is a gradual decrease in flow velocity as the walls of the tube are approached.

In turbulent flow (Figure PH.20b) the tube contains swirling eddies and the velocity varies continuously in time and, therefore, the velocity profile (broken line) is averaged in time. For the same net flow rate as in the laminar case, the flow velocity at the centre of the tube is flatter across the centre of the tube and has a lower peak value. However, the velocity gradient at the walls is steeper because of an increased viscous drag associated with the turbulence.

When flow is slow it remains laminar and both viscous drag and pressure drop along the tube, increase in proportion with flow velocity. As flow velocity increases, there is an increased tendency for small eddies to disrupt the flow until at higher velocities the flow becomes turbulent. When flow is turbulent there is an abrupt change in the viscous forces, as reflected by an increased pressure drop along the tube. The slope of a graph plotting pressure drop against flow velocity becomes steeper at the laminar-turbulent transition. This transition is illustrated in Figure PH.21 Around this point, Re exceeds the threshold of about 2000.

A mixture of laminar and turbulent gas flow patterns is found in the airways of the lung during normal breathing. Turbulent flow occurs in the trachea and main bronchi at peak flow rates during quiet breathing,

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Figure PH.21 Transition between laminar and turbulent flow

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Figure PH.21 Transition between laminar and turbulent flow while flow in the small airways remains laminar under virtually all conditions. Pressure and Velocity

In many situations flow occurs through tubes which have a varying cross-sectional area (Figure PH.22).

The fluid is assumed to be incompressible, an assumption which is clearly valid for liquids, and which under normal circumstances remains valid for gases. The volume flow rate is the product of the area of the tube and the average flow velocity, and since no fluid leaves or enters the tube, the volume flow rate must be the same at point 1 as it is at point 2. This statement can be written as:

Figure PH.22 Flow through a tube of varying diameter

As the fluid moves from point 1 to 2, the velocity increases from v1 to v2.

The fluid has kinetic energy because it is moving, and since it is moving faster at point 2 than at point 1, its kinetic energy at point 2 is higher. For a gain in kinetic energy to occur, some work must have been done on the fluid between points 1 and 2. It is the pressure at point 1 that is driving the fluid forward and which gives it increased kinetic energy. It follows that an increased velocity at point 2 must be balanced by a reduced pressure. The relationship between pressure (P), and velocity (v) at any point in a fluid is given by:

This is Bernoulli's Equation for incompressible flow, assuming no change in potential energy due to gravity (i.e. flow does not occur uphill or downhill). This is a good approximation for gases in which gravitational effects are usually negligible or liquid flow in horizontal tubes.

Bernoulli's Equation shows that as the velocity of a fluid increases, the pressure falls, or alternatively, if the pressure of a gas flow falls, it gains velocity. This is illustrated by the example of gas escaping from a cylinder at high pressure through a nozzle to the atmosphere. The gas in the cylinder acquires a high speed as it exits through the nozzle to atmospheric pressure. The energy initially contained in the gas due to it being compressed has been converted to kinetic energy as the pressure falls to atmospheric pressure.

Bernoulli's principle is applied in the Venturi flow meter to estimate flow velocity from the drop in pressure at a tube constriction. However, as the pressure drop is not directly proportional to the flow velocity, the device needs careful calibration.

### Electricity

Electricity has been recognized for several centuries, but the importance that it holds today really began with the work of Michael Faraday towards the mid-19th century. Early work was concerned with electrical charge, and was restricted to the study of 'static' electricity. It was noticed that some materials (such as glass and silk) when rubbed together tend to cling to each other. If two pieces of glass are rubbed on silk in the same way, then they repel each other. The materials are said to be 'charged' by the rubbing, and it was discovered that there are two types of charge, termed 'positive' and 'negative'. Materials that have the same charge repel each other, while oppositely charged materials attract. Furthermore, charge could be transferred from one object to another by bringing them into contact.

The invention of the electrical cell allowed Faraday and other workers to study moving charges, and it led directly to the invention of the electrical generator, electric motors and, ultimately, to modern electronics and communications systems.

### Electrical Charge

It is known now that a negative electrical charge results from an excess of electrons, and a positive charge from a deficit of electrons. When the number of electrons is matched to the number of protons in a material, that material is said to be 'neutral'. The measure of the number of electrons in surplus or deficit is the coulomb (C). When electrons are crowded together in an region of surplus, the repulsive forces between electrons impels them to move towards regions where the surplus is not so great; this driving 'force' in electrical circuits is called the electrical potential difference, electromotive force or voltage.

### Electrical Current

The flow of electrons is known as an electrical current (measured in amperes, symbol A). The purpose of electrical cells or generators is to cause an imbalance between the numbers of electrons and protons in a material and, thus, give a driving voltage. Although in electrical circuits it is the electrons that flow from the negatively charged cathode to the positively charged anode, the current is said to flow from the anode to the cathode. This is an historical anomaly, which came about before subatomic particles were classified. In Figure PH.23, if the current I flows for t seconds, then the charge (Q) which passes any point in the electrical circuit is: