ESSENTIAL MATHS METHODS CAS 3&4 PDF

Unit 3 begins with graphs of different functions, how to sketch them and find their equations. You will explore how to transform the functions graphically and algebraically. The different functions we study include: polynomials parabolas, cubics and so on logarithmic, exponential and trigonometry functions sine, cos and tan. In Calculus we look at the rates of change in these functions. You will use algebraic skills and graphs to analyse how rates of change affect each function. Calculus then continues from Unit 3 into Unit 4, where you study integration — how to find the area under the curve of a graphed function.

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Specific Gravity of ethyl alcohol is: Temperature is a measure of the internal energy level in a fluid, usually expressed in units of degrees Centigrade C or degrees Fahrenheit F. Other fluid properties such as viscosity and density can also be affected by temperature changes. Thus a cooling of the product in the discharge line could have a significant effect on the pumping of a fluid. The temperature of a fluid can also have a significant affect on the selection of any elastomeric materials used.

A temperature conversion table is given in section The connection between the velocity and the capacity of a fluid similar to water in different tube sizes is shown in table Under some conditions the fluid will appear to flow as layers in a smooth and regular manner.

This can be illustrated by opening a water tap slowly until the flow is smooth and steady. This type of flow is called laminar flow. If the water tap is opened wider, allowing the velocity of flow to increase, a point will be reached whereby the Alfa Laval Pump Handbook 13 Terminology and Theory stream of water is no longer smooth and regular, but appears to be moving in a chaotic manner. This type of flow is called turbulent flow. The type of flow is indicated by the Reynolds number.

In these instances, a certain velocity may be required to prevent solids from settling in the pipework, which could result in blockages and changes in system pressure as the actual internal diameter of the pipe is effectively decreased, which could impact on pump performance.

The fluid moves through the pipe in concentric layers with the maximum velocity in the centre of the pipe, decreasing to zero at the pipe wall. The velocity profile is parabolic, the gradient of which depends upon the viscosity of the fluid for a set flow-rate.

Turbulent Flow This is sometimes known as unsteady flow with considerable mixing taking place across the pipe cross section. The velocity profile is more flattened than in laminar flow but remains fairly constant across the section as shown in fig. This is a ratio of inertia forces to viscous forces, and as such, a useful value for determining whether flow will be laminar or turbulent.

For a given set of flow conditions, the Reynolds number will not vary when using different units. It is important to use the same set of units, such as above, when calculating Reynolds numbers. Re less than Laminar Flow Viscous force dominates - high system losses Transitional Flow Critically balanced forces Turbulent Flow Inertia force dominates - low system losses Re in range to Re greater than Where transitional flow occurs, frictional loss calculations should be carried out for both laminar and turbulent conditions, and the highest resulting loss used in subsequent system calculations.

The vapour pressure of a fluid is the pressure at a given temperature at which a fluid will change to a vapour and is expressed as absolute pressure bar a or psia - see 2. Temperature 0 C 32 F o o Vapour pressure bar 0.

In general terms Pvp: Is dependent upon the type of fluid. Increases at higher temperature. Is of great importance to pump inlet conditions. Should be determined from relevant tables.

The Pvp for water at various temperatures is shown in section Special attention should be given regarding any abrasive solids with respect to pump type and construction, operating speed and shaft seals. Size of solids is also important, as when pumping large particles the pump inlet should be large enough for solids to enter the pump without bridging the pump inlet.

Also the pump should be sized so the cavity created in the pump chamber by the pump elements is of sufficient size to allow satisfactory pump operation. This is usually a known value dependent on the actual process. A 1 Fig. This derived unit is called the Pascal Pa. Conversion factors between units of pressure are given in section Different Types of Pressure For calculations involving fluid pressures, the measurements must be relative to some reference pressure.

Normally the reference is that of the atmosphere and the resulting measured pressure is called gauge pressure.

Pressure measured relative to a perfect vacuum is called absolute pressure. Atmospheric Pressure The actual magnitude of the atmospheric pressure varies with location and with climatic conditions. The range of normal variation of atmospheric pressure near the earths surface is approximately 0. At sea level the standard atmospheric pressure is 1. It is a measure of the force per unit area exerted by a fluid, commonly indicated in units of barg bar gauge or psig psi gauge.

Absolute Pressure Is the total pressure exerted by a fluid. It equals atmospheric pressure plus gauge pressure, indicated in units of bar a bar absolute or psia psi absolute. This is a measure of the difference between the measured pressure and atmospheric pressure expressed in units of mercury Hg or units of psia.

Inlet Suction Pressure This is the pressure at which the fluid is entering the pump. The reading should be taken whilst the pump is running and as close to the pump inlet as possible. This is expressed in units of absolute bar a psia or gauge bar g psig depending upon the inlet conditions. Outlet Discharge Pressure This is the pressure at which the fluid leaves the pump.

Again this reading should be taken whilst the pump is running and as close to the pump outlet as possible. The reading is expressed in units of gauge bar psig. Differential Pressure This is the difference between the inlet and outlet pressures.

For inlet pressures above atmospheric pressure the differential pressure is obtained by subtracting the inlet pressure from the outlet pressure. For inlet pressures below atmospheric pressure the differential pressure is obtained by adding the inlet pressure to the outlet pressure.

It is therefore the total pressure reading and is the pressure against which the pump will have to operate. Power requirements are to be calculated on the basis of differential pressure. The same vertical height will give the same pressure regardless of the pipe configuration in between.

Water Slurry Solvent 35 m ft. Static Head The static head is a difference in fluid levels. Static Suction Head This is the difference in height between the fluid level and the centre line of the pump inlet on the inlet side of the pump.

Static Discharge Head This is the difference in height between the fluid level and the centre line of the pump inlet on the discharge side of the pump.

Friction Head This is the pressure drop on both inlet and discharge sides of the pump due to frictional losses in fluid flow. Dynamic Head This is the energy required to set the fluid in motion and to overcome any resistance to that motion.

Total Suction Head The total suction head is the static suction head less the dynamic head. Where the static head is negative, or where the dynamic head is greater than the static head, this implies the fluid level will be below the centre line of the pump inlet ie suction lift.

Total Discharge Head The total discharge head is the sum of the static discharge and dynamic heads. Total Head Total head is the total pressure difference between the total discharge head and the total suction head of the pump.

The head is often a known value. It can be calculated by means of different formulas if the installation conditions are specified. Pressure Drop Manufacturers of processing equipment, heat exchangers, static mixers etc, usually have data available for pressure drop. These losses are affected by fluid velocity, viscosity, tube diameter, internal surface finish of tube and tube length.

The different losses and consequently the total pressure drop in the process are, if necessary, determined in practice by converting the losses into equivalent straight length of tube which can then be used in subsequent system calculations. For calculations on water like viscosity fluids, the pressure drop can be determined referring to the Pressure Drop Curve see For higher viscosity fluids, a viscosity correction factor is applied to the tube fittings by multiplying the resultant equivalent tube length by the figures shown below - see Example 2.

Table 2.

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Specific Gravity of ethyl alcohol is: Temperature is a measure of the internal energy level in a fluid, usually expressed in units of degrees Centigrade C or degrees Fahrenheit F. Other fluid properties such as viscosity and density can also be affected by temperature changes. Thus a cooling of the product in the discharge line could have a significant effect on the pumping of a fluid. The temperature of a fluid can also have a significant affect on the selection of any elastomeric materials used.

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