Flow Table Test

This is a laboratory test, which gives an indication of the quality of concrete with respect to consistency, cohesiveness and the proneness to segregation. This test is as per IS: 5512 – 1983. In this test, a standard mass of concrete is subjected to jolting. The spread or the flow of the concrete is measured and this flow is related to workability.

Flow Table Apparatus

The flow table top is constructed from a flat metal of minimum thickness 1.5 mm. The top is in plan 700 mm x 700 mm. The centre of the table is marked with a cross, the lines which run parallel to and out to the edges of the plate and with a central circle 200 mm in diameter. The front of the flow table top is provided with a lifting handle and the total mass of the flow table top is about 16 ± 1 kg. The flow table top is hinged to a base frame using externally mounted hinges in such a way that no aggregate can become trapped easily between the hinges or hinged surfaces.

Fig.1 Flow Table Apparatus

The front of the base frame shall extend a minimum 120 mm beyond the flow table top in order to provide a top board. An upper stop similar to that is provided on each side of the table so that the lower front edge of the table can only be lifted 40 ± 1mm. The lower front edge of the flow table top is provided with two hard rigid stops which transfer the load to the base frame. The base frame is so constructed that this load is then transferred directly to the surface on which the flow table is placed so that there is minimal tendency for the flow table top to bounce when allowed to fall.

Accessory Apparatus

Mould

The mould is made of metal readily not attacked by cement paste or liable to rust and of minimum thickness 1.5 mm. The interior of the mould is smooth and free from projections, such as protruding rivets and is free from dents. The mould shall be in the form of a hollow frustum of a cone having the internal dimensions as shown in Fig. 2. The base and the top is open and parallel to each other and at right angles to the axis of the cone. The mould is provided with two metal foot pieces at the bottom and two handles above them.

Fig. 2 Mould for Flow Test

Tamping Bar

The tamping bar is made of a suitable hardwood.

The table top is cleaned of all gritty material and is wetted. The mould is kept on the centre of the table, firmly held and is filled in two layers. Each layer is rodded 25 times with a tamping rod 1.6 cm in diameter and 61 cm long rounded at the lower tamping end. After the top layer is rodded evenly, the excess of concrete which has overflowed the mould is removed. The mould is lifted vertically upward and the concrete stands on its own without support. The table is then raised and dropped 12.5 mm 15 times in about 15 seconds. The diameter of the spread concrete is measured in about 6 directions to the nearest 5 mm and the average spread is noted. The flow of concrete is the percentage increase in the average diameter of the spread concrete over the base diameter of the mould.

The value could range anything from 0 to 150 per cent. A close look at the pattern of spread of concrete can also give a good indication of the characteristics of concrete such as tendency for segregation. As well as getting an accurate measurement of the workability of the concrete, the flow test gives an indication of the cohesion. A mix that is prone to segregation will produce a noncircular pool of concrete. Cement paste may be seen separating from the aggregate. If the mix is prone to bleeding, a ring of clear water may form after a few minutes.

Procedure

1. The table is made level and properly supported. Before commencing the test, the table-top and inner surface of the mould is wiped with a damp cloth.
2. The 700 mm square flow table is hinged to a rigid base, provided with a stop that allows the far end to be raised by 40 mm.
3. A cone, similar to that used for slump testing but truncated, is filled with concrete in two layers.
4. Each layer is tamped 10 times with a special wooden bar and the concrete of the upper layer finished off level with the top of the cone. Any excess is cleaned off the outside of the cone.
5. The cone is then raised allowing the concrete to flow out and spread out a little on the flow table.
6. The table top is then raised until it meets the stop and allowed to drop freely 15 times.
7. This causes the concrete to spread further, in a roughly circular shape.
8. The diameter of the concrete spread shall then be measured in two directions, parallel to the table edges.
9. The flow diameter is the average of the maximum diameter of the pool of concrete and the diameter at right angles.

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Vee-Bee Consistometer Test

This is a good laboratory test to measure indirectly the workability of concrete as per IS: 1199– 1959. The test equipment, which was developed by Swedish Engineer V. Bahrner, is shown in Fig.1. It consists of a vibrating table, a cylindrical pan, a slump cone and a glass or plastic disk attached to a free-moving rod, which serves as a reference endpoint. The cone is placed in the pan. After it is filled with concrete and any excessive concrete is struck off, the cone is removed. Then, the disk is brought into a position on top of the concrete cone and the vibrating table is set in motion and simultaneously a stop watch started. The vibration is continued till such a time as the conical shape of the concrete disappears and the concrete assumes a cylindrical shape. This can be judged by observing the glass disc from the top for disappearance of transparency. Immediately when the concrete fully assumes a cylindrical shape, the stop watch is switched off. The time required for the shape of concrete to change from slump cone shape to cylindrical shape in seconds is known as Vee Bee Degree. This method is very suitable for very dry concrete whose slump value cannot be measured by slump test but the vibration is too vigorous for concrete with a slump greater than about 50 mm.

The Vee bee test is a good laboratory test, particularly for very dry mixes. This is in contrast to the compacting factor test where error may be introduced by the tendency of some dry mixes to stick in the hoppers. The Vee bee test also has the additional advantage that the treatment of concrete during the test is comparatively closely related to the method of placing in practice. Moreover, the cohesiveness of concrete can be easily distinguished by Vee bee test through the observation of distribution of the coarse aggregate after vibration. Table 1 shows the relationship between workability and Vee Bee values.

Fig.1 Vee bee Test Setup

Procedure

1. Mix the dry ingredients of the concrete thoroughly till a uniform colour is obtained and then add the required quantity of water.
2. Pour the concrete into the slump cone with the help of the funnel fitted to the stand.
3. Remove the slump mould and rotate the stand so that transparent disc touches the top of the concrete.
4. Start the vibrator on which cylindrical container is placed.
5. Due to vibrating action, the concrete starts remoulding and occupying the cylindrical container. Continue vibrating the cylinder till concrete surface becomes horizontal.
6. The time required for complete remoulding in seconds is the required measure of the workability and it is expressed as number of Vee-bee seconds.

Table 1 Relationship between Workability and Vee Bee Test Results

Workability Description	Vee-bee Time in Seconds
Extremely dry	32 – 18
Very stiff	18 – 10
Stiff	10 – 5
Stiff plastic	5 – 3
Plastic	3 – 0
Flowing	-

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Kelly Ball Test

This is a simple field test consisting of the measurement of the indentation made by 15 cm diameter metal hemisphere weighing 13.6 kg connected to a handle with a ruler, when freely placed on fresh concrete. ASTM C360 covers the Kelly ball penetration test. The hammer is fixed on a box container through a pin. When taking measurements, the box is placed on the top of the concrete to be tested with the surface of the hammer touching the concrete. When the pin is removed, the hammer will sink into the fresh concrete by its own weight. The depth of the hammer penetration can be read from the ruler and is used as an index of workability. A concrete with higher consistency leads to a deeper ball penetration. The penetration test is usually very quick and can be done on site, right in the formwork, provided it is wide enough. The ratio of slump value to penetration depth is from 1.3 to 2.0.

The test has been devised by Kelly and hence known as Kelly Ball Test. This has not been covered by Indian Standards Specification. The advantages of this test is that it can be performed on the concrete placed in site and it is claimed that this test can be performed faster with a greater precision than slump test. The disadvantages are that it requires a large sample of concrete and it cannot be used when the concrete is placed in thin section. The minimum depth of concrete must be at least 20 cm and the minimum distance from the centre of the ball to nearest edge of the concrete is 23 cm. The surface of the concrete is struck off level, avoiding excess working, the ball is lowered gradually on the surface of the concrete. The depth of penetration is read immediately on the stem to the nearest 6 mm. The test can be performed in about 15 seconds and it gives much more consistent results than Slump Test.

Fig. 1 Kelly Ball Apparatus

Test Procedure

• The concrete which is presumed to be tested should be poured into a container such as a buggy or actually in the form which should be up to a depth of 200mm (20cm). Then once the concrete is poured the top surface should be levelled.
• On the surface of the concrete the Kelly ball apparatus should be placed. The handles of the hemisphere (ball) should be placed in such a way that the frame touches the surface of the concrete. Away from the containers end the minimum lateral dimension of frame should be around 230 mm (23 cm).
• Then once it is done, the handle should be released slowly and the ball should be allowed to penetrate through the concrete by its own weight.
• Once the ball (hemisphere) is relieved, the penetration depth of the ball will be signified on the scale.
• Then the reading of the graduated scale should be noted down. (in which the penetration is showed).

The same procedure should be repeated at different portions for at least three times in the container and the then the average values of these readings should be taken down.

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Compaction Factor Test

The compacting factor test is designed primarily for use in the laboratory but it can also be used in the field. It is more precise and sensitive than the slump test and is particularly useful for concrete mixes of very low workability as are normally used when concrete is to be compacted by vibration. Such dry concrete is insensitive to slump test. The compacting factor test has been developed at the Road Research Laboratory U.K. and it is claimed that it is one of the most efficient tests for measuring the workability of concrete. This test works on the principle of determining the degree of compaction achieved by a standard amount of work done by allowing the concrete to fall through a standard height.

The degree of compaction, called the compacting factor is measured by the density ratio i.e., the ratio of the density actually achieved in the test to density of same concrete fully compacted. The sample of concrete to be tested is placed in the upper hopper up to the brim. The trap-door is opened so that the concrete falls into the lower hopper. Then the trap-door of the lower hopper is opened and the concrete is allowed to fall into the cylinder. In the case of a dry-mix, it is likely that the concrete may not fall on opening the trap-door. In such a case, a slight poking by a rod may be required to set the concrete in motion. The excess concrete remaining above the top level of the cylinder is then cut off with the help of plane blades supplied with the apparatus. The outside of the cylinder is wiped clean. The concrete is filled up exactly up to the top level of the cylinder. It is weighed to the nearest 10 grams. This weight is known as ―Weight of partially compacted concrete.

The cylinder is emptied and then refilled with the concrete from the same sample in layers approximately 5 cm deep. The layers are heavily rammed or preferably vibrated so as to obtain full compaction. The top surface of the fully compacted concrete is then carefully struck off level with the top of the cylinder and weighed to the nearest 10 gm. This weight is known as Weight of fully compacted concrete. Usually, the range of compaction factor is from 0.78 to 0.95 and concrete with high fluidity has a higher compaction factor.

Theory

This test is adopted to determine workability of concrete where nominal size of aggregate does not exceed 40 mm and it is as per IS: 1199 – 1959. It is based on the definition, that workability is that property of concrete, which determines the amount of work required to produce full compaction. The test consists essentially of applying a standard amount of work to standard quantity of concrete and measuring the resulting compaction. The compaction factor is defined as the ratio of the weight of partially compacted concrete to the weight of fully compacted concrete. It shall be stated to the nearest second decimal place. The relationship between degree of workability and compaction factor are given below.

Table 1 Relationship between Degree of workability and Compaction Factor

Degree of workability	Compaction Factor
Very Low	0.75- 0.80
Low	0.80- 0.85
Medium	0.85- 0.92
High	> 0.92

Compaction factor test is more sensitive and precise than slump test and is particularly useful for concrete mixes of very low workability. Such concrete may show zero to very low slump value. Also, compaction factor (C.F.) test is able to indicate small variations in workability over a wide range. Compaction factor test proves the fact that with increase in the size of coarse aggregate, the workability will decrease. However, compaction factor test has certain limitations. When maximum size of aggregate is large as compare with mean particle size; the drop into bottom container will produce segregation and give unreliable comparison with other mixes of smaller maximum aggregate sizes. Moreover, the method of introducing concrete into mould bears no relationship to any of the more common methods of placing and compacting high concrete.

Compaction Factor Test Apparatus

Compaction factor test apparatus consists of two conical hoppers, A and B, mounted vertically above a cylindrical mould C. The upper hopper A has internal dimensions as: top diameter 250 mm; bottom diameter 125 mm and height 225 mm. The lower hopper B has internal dimensions as: top diameter 225 mm; bottom diameter 125 mm and height 225 mm. The cylinder has internal dimensions as: 150 mm diameter and 300 mm height. The distances between bottom of upper hopper and top of lower hopper, and bottom of lower hopper and top of cylinder are 200 mm in each case. The lower ends of the hoppers are fitted with quick release flap doors. The hoppers and cylinders are rigid in construction and rigidly mounted on a frame. These hoppers and cylinder are rigid and easily detachable from the frame. Other instruments used for this test consists of trowels, hand scoop (15.2 cm long), a rod of steel or other suitable material (1.6 cm diameter, 61 cm long rounded at one end) and a balance.

Fig.1 Compaction Factor Test Apparatus

Procedure

1. Prepare a concrete mix for testing workability. Consider a W/C ratio of 0.5 to 0.6 and design mix of proportion about 1:2:4 (it is presumed that a mix is designed already for the test). Weigh the quantity of cement, sand, aggregate and water correctly. Mix thoroughly. Use this freshly prepared concrete for the test.
2. Place the concrete into the upper hopper up to its brim.
3. Open the trapdoor of the upper hopper. The concrete will fall into the lower hopper.
4. Open the trapdoor of the lower hopper, so that concrete falls into the cylinder below.
5. Remove the excess concrete above the level of the top of the cylinder; clean the outside of the cylinder.
6. Weigh the concrete in the cylinder. This weight of concrete is the "weight of partially compacted concrete", (W1).
7. Empty the cylinder and refill with concrete in layers, compacting each layer well (or the same may be vibrated for full compaction). Top surface may be struck off level.
8. Find cut weight of the concrete in the fully compacted state. This weight is the “Weight of fully compacted concrete" (W2).

The degree of compaction, called the compacting factor is measured by the density ratio i.e., the ratio of the density actually achieved in the test to density of same concrete fully compacted.

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Equation of State: The Perfect Gas

The assumed properties of a perfect gas are closely matched by those of actual gases in many circumstances, although no actual gas is perfect. The molecules of a perfect gas would behave like tiny, perfectly elastic spheres in random motion and would influence one another only when they collided. Their total volume would be negligible in comparison with the space in which they moved. From these hypotheses the kinetic theory of gases indicates that, for equilibrium conditions, the absolute pressure p, density ‘ρ’, the volume V occupied by mass m and the absolute temperature T are related by the expression

p = ρRT

                                                                              or           pV = mRT                      (1)

in which ‘R’ is a constant called the gas constant, the value of which is constant for the gas concerned and ‘V’ is the volume occupied by the mass m of the gas. The absolute pressure is the pressure measured above absolute zero (or complete vacuum) and is given by

p_abs = p_gage + p_atm 

The absolute temperature is expressed in ‘kelvin’ i.e., K, when the temperature is measured in °C and it is given by

T°(abs) = T K = 273.15 + t°C

No actual gas is perfect. However, most gases (if at temperatures and pressures well away both from the liquid phase and from dissociation) obey this relation closely and hence their pressure, density and (absolute) temperature may to a good approximation, be related by Eq.1.

Similarly, air at normal temperature and pressure behaves closely in accordance with the equation of state. It may be noted that the gas constant R is defined by Eq. 1 as p/ρT and therefore, its dimensional expression is (FL/Mθ). Thus in SI units the gas constant R is expressed in Newton-metre per kilogram per kelvin i.e., (N.m/kg. K). Further, since 1 joule = 1 newton × 1 metre, the unit for R also becomes joule per kilogram per kelvin i.e., (J/kg. K). Again, since 1 N = 1 kg × 1 m/s², the unit for R becomes (m²/s² K).

In metric gravitational and absolute systems of units, the gas constant R is expressed in kilogram (f)-metre per metric slug per degree C absolute i.e., [kg(f)-m/msl deg. C abs.] and dyne-centimetre per gram (m) per degree C absolute i.e., [dyne-cm/gm(m) deg. C abs.] respectively. For air the value of R is 287 N-m/kg K, or 287 J/kg K, or 287 m²/s² K.

In metric gravitational system of units, the value of R for air is 287 kg(f)-m/msl deg. C abs. Further, since 1 msl = 9.81 kg (m), the value of R for air becomes (287/9.81) or 29.27 kg(f)-m/kg(m) deg. C abs.

Since specific volume may be defined as reciprocal of mass density, the equation of state may also be expressed in terms of specific volume of the gas as

pv = RT

in which v is specific volume.

The equation of state may also be expressed as

p = wRT

in which w is the specific weight of the gas. The unit for the gas constant R then becomes (m/K) or (m/deg. C abs). It may be shown that for air the value of R is 29.27 m/K. For a given temperature and pressure, Eq. 1 indicates that ρR = constant. By Avogadro’s hypothesis, all pure gases at the same temperature and pressure have the same number of molecules per unit volume. The density is proportional to the mass of an individual molecule and so the product of R and the ‘molecular weight’ M is constant for all perfect gases. This product MR is known as the universal gas constant. For real gases it is not strictly constant but for monatomic and diatomic gases its variation is slight. If M is the ratio of the mass of the molecule to the mass of a hydrogen atom, MR = 8310 J/kg K.

Any equation that relates p, ρ and T is known as an equation of state and equation of state is therefore termed the equation of state of a perfect gas. Most gases, if at temperatures and pressures well away both from the liquid phase and from dissociation, obey this relation closely and so their pressure, density and (absolute) temperature may, to a good approximation, be related by Eqn. 1. For example, air at normal temperatures and pressures behaves closely in accordance with the equation. But gases near to liquefaction – which are then usually termed vapours – depart markedly from the behaviour of a perfect gas. Equation 1 therefore does not apply to substances such as non-superheated steam and the vapours used in refrigerating plants. For such substances, corresponding values of pressure, temperature and density must be obtained from tables or charts.

It is usually assumed that the equation of state is valid not only when the fluid is in mechanical equilibrium and neither giving nor receiving heat, but also when it is not in mechanical or thermal equilibrium. This assumption seems justified because deductions based on it have been found to agree with experimental results.

Calorically Perfect Gas

A gas for which the specific heat capacity at constant volume, cv, is a constant is said to be calorically perfect. The term perfect gas, used without qualification, generally refers to a gas that is both thermally and calorically perfect.

Changes of State

A change of density may be achieved both by a change of pressure and by change of temperature. If the process is one in which the temperature is held constant, it is known as isothermal. On the other hand, the pressure may be held constant while the temperature is changed. In either of these two cases there must be a transfer of heat to or from the gas so as to maintain the prescribed conditions. If the density change occurs with no heat transfer to or from the gas, the process is said to be adiabatic.

If, in addition, no heat is generated within the gas (e.g. by friction) then the process is described as isentropic, and the absolute pressure and density of a perfect gas are related by the additional expression p/e^γ  = constant, where γ = cp/cv and cp and cv represent the specific heat capacities at constant pressure and constant volume respectively. For air and other diatomic gases in the usual ranges of temperature and pressure γ = 1.4.

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Pressure

When a fluid (either liquid or gas) is at rest, it exerts a force perpendicular to any surface in contact with it, such as a container wall or a body immersed in the fluid. While the fluid as a whole is at rest, the molecules that makes up the fluid are in motion, the force exerted by the fluid is due to molecules colliding with their surroundings. A fluid always has pressure as a result of innumerable molecular collisions. Pressure at any part of the fluid must experience forces exerted on it by adjoining fluid or by adjoining solid boundaries. If, therefore, part of the fluid is arbitrarily divided from the rest by an imaginary plane, there will be forces that may be considered as acting at that plane. As shown in Fig. 1, pressure occurs when a force is applied to an area. Fluid pressure is the force exerted by the fluid per unit area. Fluid pressure is transmitted with equal intensity in all directions and acts normal to any plane. In the same horizontal plane, the pressure intensities in a liquid are equal.

Fig.1 Force Producing a Pressure

The relationship between force, pressure and area is

Using the base units of Newton (N) for force and square metres (m²) for area, the unit of pressure is N/m², which for convenience is called a pascal (Pa). Because the pascal is an extremely small unit of pressure (car tyre pressure are around 200 000 Pa) the more commonly used unit is the kilopascal (kPa) or the megapascal (MPa). The pascal unit is used for the low pressures that occur in fans or in ventilation ducts. Kilopascals are used for normal gas and liquid pressures. The pressure in an oil hydraulic system would be measured in megapascals. The other multiple of the pascal that may b;e used on the weather forecast is the hectopascal (hPa), being used for the barometric pressure. A hectopascal is 10² pascals. A typical barometric pressure is 1013 hPa.

Pressures of large magnitude are often expressed in atmospheres (abbreviated to atm). For precise definition, one atmosphere is taken as 1.01325 × 10⁵ Pa. A pressure of 10⁵ Pa is called 1 bar. The thousand^th part of this unit, called a millibar (abbreviated to mbar), is commonly used by meteorologists. It should be noted that, although they are widely used, neither the atmosphere nor the bar are accepted for use with SI units. For pressures less than that of the atmosphere the units normally used are millimetres of mercury vacuum. These units refer to the difference between the height of a vertical column of mercury supported by the pressure considered and the height of one supported by the atmosphere. In the absence of shear forces, the direction of the plane over which the force due to the pressure acts has no effect on the magnitude of the pressure at a point. The fluid may even be accelerating in a particular direction provided that shear forces are absent – a condition that requires no relative motion between different particles of fluid. Many other pressure units are commonly encountered and their conversions are detailed below.

            1 bar =10⁵ N/m²

            1 atmosphere = 101325 N/m²

1 psi (1bf/in² - not SI unit) = 6895 N/m²

1 Torr = 133.3 N/m²

Pressure is determined from a calculation of the form (force divided by area), and so has the dimensions [F]/[L²] = [MLT⁻²]/[L²] = [ML⁻¹T⁻²]. Now although the force has direction, the pressure has not. The direction of the force also specifies the direction of the imaginary plane surface, since the latter is defined by the direction of a line perpendicular to, or normal to, the surface. Here, then, the force and the surface have the same direction and so in the equation Force = Pressure × Area of plane surface pressure must be a scalar quantity. Pressure is a property of the fluid at the point in question. Similarly, temperature and density are properties of the fluid and it is just as illogical to speak of ‘downward pressure’, for example, as of ‘downward temperature’ or ‘downward density’. To say that pressure acts in any direction, or even in all directions, is meaningless; pressure is a scalar quantity.

Terms commonly used in static pressure analysis include the following.

Pressure Head

The pressure intensity at the base of a column of homogenous fluid of a given height in metres.

Vacuum

A perfect vacuum is a completely empty space in which, therefore the pressure is zero.

Atmospheric Pressure

It is the pressure of earth's atmosphere. This changes with weather and elevation. The pressure at the surface of the earth due to the head of air above the surface is called atmospheric pressure. At sea level the atmospheric pressure is about 101.325 kN/m² (i.e. one atmosphere = 101.325 kN/m² is used as units of pressure).

Gauge Pressure

The pressure measured above or below atmospheric pressure is called Gauge pressure. Pressure cannot be measured directly; all instruments said to measure it in fact indicate a difference of pressure. This difference is frequently that between the pressure of the fluid under consideration and the pressure of the surrounding atmosphere. The pressure of the atmosphere is therefore commonly used as the reference or datum pressure that is the starting point of the scale of measurement. The difference in pressure recorded by the measuring instrument is then termed the gauge pressure.

Gauge pressure = Absolute pressure – Atmospheric pressure

Absolute Pressure

The pressure measured above absolute zero or vacuum is called Absolute pressure. The absolute pressure, that is the pressure considered relative to that of a perfect vacuum, is then given by

                        Absolute Pressure = Gauge Pressure + Atmospheric Pressure

pabs = pgauge +patm

The pressure of the atmosphere is not constant. For many engineering purposes the variation of atmospheric pressure (and therefore the variation of absolute pressure for a given gauge pressure, or vice versa) is of no consequence. In other cases, especially for the flow of gases – it is necessary to consider absolute pressures rather than gauge pressures and a knowledge of the pressure of the atmosphere is then required.

Positive and Negative Pressures

Because we are subjected to an atmospheric pressure, the pressure indicated on a gauge can be either positive (pressure) or negative (vacuum). Above atmospheric pressure is positive and called a gauge pressure for clarity. A typical pressure gauge would be calibrated in kPa. Below atmospheric pressure is negative and called a vacuum or a negative pressure. Fig.2 indicates the relationship between the pressure and vacuum ranges and introduces the concept of one pressure range starting from absolute zero pressure and called the absolute pressure range.

Fig. 2 Pressure/Vacuum Relationships

We normally express pressures in terms of gauge pressure and before these values may be used in calculations regarding the change of state of a gas, the gauge pressure must be changed into an absolute pressure. Changing to absolute values is done by adding the accepted value for atmospheric pressure, nominally 101.3 kPa.

Absolute Pressure (kPa) = Gauge Pressure (kPa) + 101.3 kPa

It is important, when specifying pressures or using pressures in a calculation, to determine if the values given are in terms of gauge pressure (sometimes written `kPa g’ or `kPa gauge’) or absolute pressure (sometimes written `kPa abs’).

Compressibility

A parameter describing the relationship between pressure and change in volume for a fluid. A compressible fluid is one which changes its volume appreciably under the application of pressure. Therefore, liquids are virtually incompressible whereas gases are easily compressed. The compressibility of a fluid is expressed by the bulk modulus of elasticity (K), which is the ratio of the change in unit pressure to the corresponding volume change per unit volume.

Vapour Pressure

When evaporation of a liquid having a free surface takes place within an enclosed space, the partial pressure created by the vapour molecules is called the vapour pressure. Vapour pressure of a liquid is the partial pressure of the vapour in contact with the saturated liquid at a given temperature. Vapour pressure increases with temperature.

All liquids possess a tendency to evaporate or vaporize i.e., to change from the liquid to the gaseous state. Such vaporization occurs because of continuous escaping of the molecules through the free liquid surface. When the liquid is confined in a closed vessel, the ejected vapour molecules get accumulated in the space between the free liquid surface and the top of the vessel. This accumulated vapour of the liquid exerts a partial pressure on the liquid surface which is known as vapour pressure of the liquid. As molecular activity increases with temperature, vapour pressure of the liquid also increases with temperature. If the external absolute pressure imposed on the liquid is reduced by some means to such an extent that it becomes equal to or less than the vapour pressure of the liquid, the boiling of the liquid starts, whatever be the temperature. Thus a liquid may boil even at ordinary temperature if the pressure above the liquid surface is reduced so as to be equal to or less than the vapour pressure of the liquid at that temperature.

If in any flow system the pressure at any point in the liquid approaches the vapour pressure, vaporization of liquid starts, resulting in the pockets of dissolved gases and vapours. The bubbles of vapour thus formed are carried by the flowing liquid into a region of high pressure where they collapse, giving rise to high impact pressure. The pressure developed by the collapsing bubbles is so high that the material from the adjoining boundaries gets eroded and cavities are formed on them. This phenomenon is known as cavitation. When the liquid pressure is dropped below the vapour pressure due to the flow phenomenon, we call the process cavitation. Mercury has a very low vapour pressure and hence it is an excellent fluid to be used in a barometer. On the contrary various volatile liquids like benzene etc., have high vapour pressure. Cavitation can cause serious problems, since the flow of liquid can sweep this cloud of bubbles on into an area of higher pressure where the bubbles will collapse suddenly. If this should occur in contact with a solid surface, very serious damage can result due to the very large force with which the liquid hits the surface. Cavitation can affect the performance of hydraulic machinery such as pumps, turbines and propellers, and the impact of collapsing bubbles can cause local erosion of metal surfaces.

Variation in Pressure with Depth

If the weight of the fluid can be neglected, the pressure in a fluid is the same throughout its volume. But often the fluid's weight is not negligible and under such condition pressure increases with increasing depth below the surface.

Let us now derive a general relation between the pressure ‘P’ at any point in a fluid at rest and the elevation ‘y’ of that point. We will assume that the density ′ρ′ and the acceleration due to gravity ‘g’ are same throughout the fluid. If the fluid is in equilibrium, every volume element is in equilibrium.

Consider a thin element of fluid with height ‘dy’. The bottom and top surfaces each have area ‘A’ and they are at elevations y and (y + dy) above some reference level where y = 0. The weight of the fluid element is

                                     dW = (volume) (density) (g)

                                            = (A dy) (ρ) (g)

             or                  dW = ρ g A dy

The pressure at the bottom surface P, the total y component of upward force is PA. The pressure at the top surface is P + dP and the total y-component of downward force on the top surface is (P + dP)A. The fluid element is in equilibrium, so the total y component of force including the weight and the forces at the bottom and top surfaces must be zero.

               Σ Fy = 0

                                                            PA – (P + dP)A – ρ g A dy = 0

This equation shows that when y increases, P decreases, i.e., as we move upward in the fluid pressure decreases.

If P1 and P2 be the pressures at elevations y1 and y2 and if ρ and g are constant, then integration of Equation (1), we get

                                   or                                     P2 – P1 = – ρ g (y2 – y1) (2)

It's often convenient to express equation (2) in terms of the depth below the surface of a fluid. Take point 1 at depth h below the surface of fluid and let P represents pressure at this point. Take point 2 at the surface of the fluid, where the pressure is P0 (for zero depth). The depth of point 1 below the surface is,

                                   h = y2 – y1

    and equation (2) becomes

                                           P0 – P = – ρ g (y2 – y1) = – ρgh

                                                      P = P0 + ρ gh                     (3)

Thus pressure increases linearly with depth, if ρ and g are uniform. A graph between P and h is shown below.

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Newtonian and Non-Newtonian Fluids

Fluids in which shear stress is directly proportional to the rate of deformation are “Newtonian fluids. Most common fluids such as water, air and gasoline are Newtonian under normal conditions. If the fluid is Newtonian, then

The constant of proportionality in Eq. (1) is the absolute (or dynamic) viscosity, μ. The Newton's law of viscosity is given for one-dimensional flow by

Note that, since the dimensions of ′τ′ are [F/L²] and the dimensions of dv/dy are [1/t], μ has dimensions [Ft/L²]. Since the dimensions of force, F, mass, m, length, L, and time, t, are related by Newton's second law of motion, the dimensions of μ, can also be expressed as [M/Lt]. In the British Gravitational system, the units of viscosity are lbf.s/ft² or slug/(ft .s). In the Absolute Metric system, the basic unit of viscosity is called a poise [1 poise = 1 g/(cm .s)]; in the SI system the units of viscosity are kg/(m. s) or Pa. s (1 Pa. s = 1 N. s/m²).

In fluid mechanics the ratio of absolute viscosity, μ, to density, 𝜌, often arises. This ratio is given the name kinematic viscosity and is represented by the symbol ′𝜗′. Since density has dimensions [M/L³], the dimensions of 𝜗 are [L²/t]. In the Absolute Metric system of units, the unit for 𝜗 is a stoke (1 stoke = 1 cm²/s). For gases, viscosity increases with temperature, whereas for liquids, viscosity decreases with increasing temperature. If one considers the deformation of two different Newtonian fluids, say Glycerine and water, one recognizes that they will deform at different rates under the action of same applied stress. Glycerine exhibits much more resistance to deformation than water. Thus we say it is more viscous.

Non-Newtonian Fluids

Fluids in which shear stress is not directly proportional to deformation rate are non- Newtonian. Many common fluids exhibit non-Newtonian behaviour. The familiar example is toothpaste. Toothpaste behaves as a "fluid" when squeezed from the tube. However, it does not run out by itself when the cap is removed. There is a threshold or yield stress below which toothpaste behaves as a solid. Non-Newtonian fluids commonly are classified as having time-independent or time-dependent behaviour.

Numerous empirical equations have been proposed to model the observed relations between τ and dv/dy for time-independent fluids. They may be adequately represented for many engineering applications by the power law model, which for one-dimensional flow becomes

where the exponent, n, is called the flow behaviour index and the coefficient, k, the consistency index. This equation reduces to Newton's law of viscosity for n = 1 with k = μ. To ensure that τ has the same sign as dv/dy, Eq. (2) is rewritten in the form

The idea behind Eq. (3) is that we end up with a viscosity 𝜂 that is used in a formula that is the same form as Eq. (2), in which the Newtonian viscosity μ is used. The big difference is that while μ is constant (except for temperature effects), 𝜂 depends on the shear rate. Most non-Newtonian fluids have apparent viscosities that are relatively high compared with the viscosity of water.

Fluids in which the apparent viscosity decreases with increasing deformation rate (n < 1) are called pseudo plastic (or shear thinning) fluids. Most non-Newtonian fluids fall into this group; examples include polymer solutions, colloidal suspensions, and paper pulp in water. If the apparent viscosity increases with increasing deformation rate (n > 1) the fluid is termed dilatant (or shear thickening). Suspensions of starch and of sand are examples of dilatant fluids.

A "fluid" that behaves as a solid until a minimum yield stress, τ, is exceeded and subsequently exhibits a linear relation between stress and rate of deformation is referred to as an ideal or Bingham plastic. Clay suspensions, drilling muds, and toothpaste are examples of substances exhibiting this behaviour. The study of non-Newtonian fluids is further complicated by the fact that the apparent viscosity may be time-dependent. Thixotropic fluids show a decrease in 𝜂 with time under a constant applied shear stress; many paints are thixotropic. Rheopectic fluids show an increase in 𝜂 with time. After deformation some fluids partially return to their original shape when the applied stress is released; such fluids are called viscoelastic.

Fig. 1 Variation of Shear Stress with Velocity Gradient

Non-Newtonian Liquids

For most fluids the dynamic viscosity is independent of the velocity gradient in straight and parallel flow, so Newton’s hypothesis is fulfilled. A graph of stress against rate of shear is a straight line through the origin with slope equal to μ. There is a fairly large category of liquids for which the viscosity is not independent of the rate of shear, and these liquids are referred to as non-Newtonian. Solutions (particularly of colloids) often have a reduced viscosity when the rate of shear is large, and such liquids are said to be pseudo-plastic. Gelatine, clay, milk, blood and liquid cement come in this category.

A few liquids exhibit the converse property of dilatancy; that is, their effective viscosity increases with increasing rate of shear. Concentrated solutions of sugar in water and aqueous suspensions of rice starch (in certain concentrations) are examples. Additional types of non-Newtonian behaviour may arise if the apparent viscosity changes with the time for which the shearing forces are applied. Liquids for which the apparent viscosity increases with the duration of the stress are termed rheopectic; those for which the apparent viscosity decreases with the duration are termed thixotropic.

A number of materials have the property of plasticity. Metals when strained beyond their elastic limit or when close to their melting points can deform continuously under the action of a constant force and thus in some degree behave like liquids of high viscosity. Their behaviour, is non-Newtonian, and most of the methods of mechanics of fluids are therefore inapplicable to them.

Viscoelastic materials possess both viscous and elastic properties; bitumen, nylon and flour dough are examples. In steady flow, that is, flow not changing with time, the   of shear is constant and may well be given by τ/μ where μ represents a constant dynamic viscosity as in a Newtonian fluid. Elasticity becomes evident when the shear stress is changed. A rapid increaseof stress from τ to τ +δτ causes the material to be sheared through an additional angle δτ /G where G represents an elastic modulus; the corresponding rate of shear is (1/G)∂τ/∂t so the total rate of shear in the material is (τ/μ) + (1/G)∂τ/∂t.

The fluids with which engineers most often have to deal are Newtonian, that is, their viscosity is not dependent on either the rate of shear or its duration, and the term mechanics of fluids is generally regarded as referring only to Newtonian fluids. The study of non-Newtonian liquids is termed rheology.

Inviscid Fluid

An important field of theoretical fluid mechanics involves the investigation of the motion of a hypothetical fluid having zero viscosity. Such a fluid is sometimes referred to as an ideal fluid. Although commonly adopted in the past, the use of this term is now discouraged as imprecise. A more meaningful term for a fluid of zero viscosity is inviscid fluid.

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Continuum Hypothesis and Transport Phenomena

Continuum Hypothesis 

A fluid, or any other substance for that matter, is composed of a large number of molecules in constant motion and undergoing collisions with each other. Matter is therefore discontinuous or discrete at microscopic scales. In principle, it is possible to study the mechanics of a fluid by studying the motion of the molecules themselves, as is done in kinetic theory or statistical mechanics. However, we are generally interested in the gross behaviour of the fluid, that is, in the average manifestation of the molecular motion. For example, forces are exerted on the boundaries of a container due to the constant bombardment of the molecules; the statistical average of this force per unit area is called pressure, a macroscopic property. So long as we are not interested in the mechanism of the origin of pressure, we can ignore the molecular motion and think of pressure as simply “force per unit area.” 

It is thus possible to ignore the discrete molecular structure of matter and replace it by a continuous distribution, called a continuum. For the continuum or macroscopic approach to be valid, the size of the flow system (characterized, for example, by the size of the body around which flow is taking place) must be much larger than the mean free path or the molecules. For ordinary cases, this is not a great restriction, since the mean free path is usually very small. For example, the mean free path for standard atmospheric air is ≈ 5 x 10^-8 m. In special situations, the mean free path of the molecules can be quite large and the continuum approach breaks down. In the upper altitudes of the atmosphere, for example, the mean free path of the molecules may be of the order of a matter, a kinetic theory approach is necessary for studying the dynamics of these rarefied gases. 

Transport Phenomena 

Consider a surface area AB within a mixture of two gases, say nitrogen and oxygen (Fig. 1), and assume that the concentration C of nitrogen (kilograms of nitrogen per cubic metre of mixture) varies across AB. 

Fig. 1 Mass flux qm due to concentration variation C(y) across AB

Random migration of molecules across AB in both directions will result in a net flux or nitrogen across AB, from the region  of higher C towards the region of lower C. Experiments show that, to a good approximation, the flux of one constituent in a mixture is proportional to its concentration gradient and it is given by 

                                       qm = - km Δ C                                  (1) 

Here the vector qm is the mass flux (kg m^-2 s^-1) of the constituent, Δ C is the concentration gradient of that constituent, and km is a constant of proportionality that depends on the particular pair of constituents in the mixture and the thermodynamic state. For example, km, for diffusion of nitrogen in a mixture with oxygen is different than km, for diffusion of nitrogen in a mixture with carbon dioxide. The linear relation (1) for mass diffusion is generally known as Fick's law. 

Relations like these are based on empirical evidence and are called phenomenological laws. Statistical mechanics can sometimes be used to derive such laws, but only for simple situations. The analogous relation for heat transport due to temperature gradient is Fourier's law and it is given by 

                                         q = - k ΔT                                         (2) 

where q is the heat flux (J m^-2 s^-1), ΔT is the temperature gradient, and k is the thermal conductivity of the material.

Fig. 2 Shear stress ‘τ’ on surface AB. Diffusion tends to decrease velocity gradients, so that the continuous line tends toward the dashed line

Next, consider the effect of velocity gradient du/dy (Fig. 2). It is clear that the macroscopic fluid velocity ‘u’ will tend to become uniform due to the random motion of the molecules, because of intermolecular collisions and the consequent exchange of molecular momentum. Imagine two railroad trains traveling on parallel . tracks at different speeds and workers shovelling coal from one train to the other. On the average, the impact of particles of coal going horn the slower to the faster train will tend to slow down the faster train, and similarly the coal going from the faster to the slower train will tend to speed up the latter. The net effect is a tendency to equalize the speeds of the two trains. An analogous process takes place in the fluid flow problem of Fig. 2. The velocity distribution here tends toward the dashed line, which can be described by saying that the x-momentum (determined by its “concentration” u) is being transferred downward. Such a momentum flux is equivalent to the existence of a shear stress in the fluid, just as the drag experienced by the two trains results from the momentum exchange through the transfer or coal particles. The fluid above AB tends to push the fluid underneath forward, whereas the fluid below AB tends to drag the upper fluid backward. Experiments show that the magnitude of the shear stress ‘τ’ along a surface such as AB is, to a good approximation, related to the velocity gradient by the linear relation 

which is called Newton’s law of friction. Here, the constant of proportionality ′µ′ (whose unit is kg m^-1 s^-l) is known as the dynamic viscosity, which is a strong function of temperature T. For ideal gases the random thermal speed is roughly proportional to √𝑇, the momentum transport, and consequently µ, also vary approximately as √𝑇. For liquids, on the other hand, the shear stress is caused more by the intermolecular cohesive forces than by the thermal motion of the molecules. These cohesive forces, and consequently µ for a liquid, decrease with temperature.

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Capillarity

If molecules of certain liquid possess, relatively, greater affinity for solid molecules, or in other words the liquid has greater adhesion than cohesion, then it will wet a solid surface with which it is in contact and will tend to rise at the point of contact, with the result that the liquid surface is concave upward and the angle of contact θ is less than 90° as shown in Fig. 1. Capillary action is the result of adhesion and surface tension. Adhesion of water to the walls of a vessel will cause an upward force on the liquid at the edges and result in a meniscus which turns upward. The surface tension acts to hold the surface intact, so instead of just the edges moving upward, the whole liquid surface is dragged upward.

For example, if a glass tube of small diameter is partially immersed in water, the water will wet the surface of the tube and it will rise in the tube to some height, above the normal water surface, with the angle of contact θ, being zero. The wetting of solid boundary by liquid results in creating decrease of pressure within the liquid and hence the rise in the liquid surface takes place, so that the pressure within the column at the elevation of the surrounding liquid surface is the same as the pressure at this elevation outside the column.

Fig. 1 Capillarity in Circular Glass Tubes

On the other hand, if for any liquid there is less attraction for solid molecule or in other words the cohesion predominates, then the liquid will not wet the solid surface and the liquid surface will be depressed at the point of contact, with the result that the liquid surface is concave downward and the angle of contact θ is greater than 90° as shown in Fig. 1. For instance, if the same glass tube is now inserted in mercury, since mercury does not wet the solid boundary in contact with it, the level of mercury inside the tube will be lower than the adjacent mercury level, with the angle of contact θ equal to about 130°. The tendency of the liquids which do not adhere to the solid surface, results in creating an increase of pressure across the liquid surface, (as in the case of a drop of liquid). It is because of the increased internal pressure, the elevation of the meniscus (curved liquid surface) in the tube is lowered to the level where the pressure is the same as that in the surrounding liquid.

Such a phenomenon of rise or fall of liquid surface relative to the adjacent general level of liquid is known as capillarity. Accordingly, the rise of liquid surface is designated as capillary rise and the lowering of liquid surface as capillary depression and it is expressed in terms of m or mm of liquid in SI units, in terms of cm or mm of liquid in the metric system of units and in terms of inch or ft of liquid in the English system of units.

The capillary rise (or depression) can be determined by considering the conditions of equilibrium in a circular tube of small diameter inserted in a liquid. It is supposed that the level of liquid has risen (or fallen) by h above (or below) the general liquid surface when a tube of radius r is inserted in the liquid. For the equilibrium of vertical forces acting on the mass of liquid lying above (or below) the general liquid level, the weight of liquid column h (or the total internal pressure in the case of capillary depression) must be balanced by the force, at surface of the liquid, due to surface tension ‘σ’. Thus equating these two forces we have

where ‘w’ is the specific weight of water, ‘s’ is specific gravity of liquid, and ‘θ’ is the contact angle between the liquid and the tube. The expression for ‘h’ the capillary rise (or depression) then becomes

As stated earlier, the contact angle ‘θ’ for water and glass is equal to zero. Thus the value of cos θ is equal to unity and hence h is given by the expression

This equation for capillary rise (or depression) indicates that the smaller the radius r the greater is the capillary rise (or depression). The above obtained expression for the capillary rise (or depression) is based on the assumption that the meniscus or the curved liquid surface is a section of a sphere. This is, true only in case of the tubes of small diameters (r < 2.5 mm) and as the size of the tube becomes larger, the meniscus becomes less spherical and also gravitational forces become more appreciable. Hence such simplified solution for computing the capillary rise (or depression) is possible only for the tubes of small diameters.

However, with increasing diameter of tube, the capillary rise (or depression) becomes much less. It has been observed that for tubes of diameters 6 mm or more the capillary rise (or depression) is negligible. Hence in order to avoid a correction for the effects of capillarity in manometers, used for measuring pressures, a tube of diameter 6 mm or more should be used. Another assumption made in deriving this equation is that the liquids and tube surfaces are extremely clean. In practice, such cleanliness is virtually never encountered and h will be found to be considerably smaller than that given by the above equation.

If a tube of radius r is inserted in mercury (specific gravity s1) above which a liquid of specific gravity s2 lies, then by considering the conditions of equilibrium it can be shown that the capillary depression h is given by

in which ‘σ’ is the surface tension of mercury in contact with the liquid.

Further if two vertical parallel plates ‘t’ distance apart and each of width ‘l’ are held partially immersed in a liquid of surface tension σ and specific gravity ‘s’, then the capillary rise (or depression) ‘h’ may be determined by equating the weight of the liquid column h (or the total internal pressure in the case of capillary depression) (swhlt) to the force due to surface tension (2σl cos θ). Thus we have

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