# Solved Problems: Gas Mixtures and Psychrometry

**1. ****Atmospheric air at 1.0132bar has a DBT of 30°C and a WBT of 25°C. Compute**

**i. ****The partial pressure of water vapour,**

**ii. ****The specific humidity,**

**iii. ****The dew point temperature,**

**iv. ****The relative humidity,**

**v. ****The degree of saturation,**

**vi. ****The density of air in the mixture,**

**vii. ****The density of vapour in the mixture and**

viii. **The enthalpy of the mixture. Use thermodynamic table.**

**2. Air at 20°C, 40% relative humidity is missed adiabatically with air at 40°C, 40% RH in the ratio of 1kg of former with 2kg of latter (on dry basis). Find the final condition (humidity and enthalpy) of air.**

**3.An air conditioning system is to take in outdoor air at 283K and 30percent relative humidity at a steady rate of 45 /min and to condition it to 298K and 60% relative humidity. The outdoor air is first heated to 295K in the heating section and then humidified by the injection of hot steam in the humidifying section. Assuming the entire process takes place at a pressure of 100kPa, determine (i) the rate of heat supply in the heating section and (ii) the mass flow rate of the steam required in the humidifying section.**

**Given data:**

= 10°C

= 30%

= 25°C

= 60%

= 22°C

= 45 /min = 100kPa

**Solution:**

Step 1:

The dry of air i.e 10°C of dry bulb temperature and 30% relative humidity is marked on the psychrometric chart at point

1. The horizontal line is drawn up to 22°C to obtain point

2. The dry of air i.e. 25°C dry bulb temperature and 60% relative humidity is marked on the psychrometric chart at point

Draw an inclined line from point 1 to 2. Read enthalpies and specific humidity

values at point 1,2 and 3 from psychrometric chart.

At point 1, enthalpy = 16.5kJ/kg

At point 2, enthalpy = 28kJ/kg

Mass of air,=

= 0.923kg/s

Heated added,

= 0.923 (28 –16.5)

= 10.615kJ

Specific humidity,

= 0.003kg/kg of dry air

= 0.012kg/kg of dry air

Moisture added,

= 0.012 –0.003 = 0.009kg/kg of dry air

We know that,

Specific humidity,

Mass of steam = 0.0083kg/s

**4. ****(i) What is the lowest temperature that air can attain in an evaporative cooler, if it enters at 1atm, 302K and 40% relative humidity? [Nov/Dec 2008]**

**Given data: **p = 1bar T = 302K = 40%

**Solution:**

From steam table, corresponding to dry bulb temperature 29°C, saturation pressure is 4.004kPa

Relative humidity,

= 1.601kPa

Lowest temperature that air attain in an evaporative cooler = 113.6°C which is corresponding to = 1.601kPa.

**(ii) Consider a room that contains air at 1atm, 308K and 40% relative humidity. Using the psychrometric chart, determine: the specific humidity, the enthalpy, the wet bulb temperature, the dew point temperature and the specific volume of air.**

**Given data:**

Pressure, p = 1atm

Relative humidity, = 40% Temperature, T = 308K = 35°C

**Solution:**

· Specific humidity,

From the point 1, draw a horizontal line with respect to temperature and relative humidity. At this point specific humidity is 0.0138kJ/kg. i.e. = 0.0138kJ/kg

· Enthalpy, h and Wet bulb temperature,

From the point 1, draw a inclined line along the constant wet bulb temperature line till it cuts enthalpy line. At this point, enthalpy is 52.5kJ/kg and wet bulb temperature is 23.9°C. i.e. h = 52.5kJ/kg and

= 23.9°C

Mark point 1 on the psychrometric chart by given dry bulb temperature (35°C) and relative humidity 40%.

From point 1, draw a horizontal line to the left till it cuts saturation curve.

At that point, temperature is 20°C and specific volume is 0.89 . i.e.

= 20°C and v = 0.89 .

**5. ****30 /min of moist air at 15°C DBT and 13°C WBT are mixed with 12 /min of moist air at 25°C DBT and 18°C WBT. Determine DBT and WBT of the mixture assuming barometric pressure is one atmosphere.**

Given data:

First steam of air,

Dry bulb temperature, = 15°C

Wet bulb temperature, = 13°C

Flow rate, = 30 /min

Second steam of air,

Dry bulb temperature, = 25°C

Wet bulb temperature, = 18°C

Flow rate, = 12 /min

**Solution:**

Step 1:

The first steam of air 15°C DBT and 13°C WBT is marked on the psychrometric chart at point 1.

Step 2:

The second stream of air 25°C DBT and 18°C WBT is marked on the psychrometric chart at point 2.

Join the points 1 and 2 from psychrometric chart. Step 3:

We know that,

Specific humidity of first steam of air, = 0.007kg/kg of dry air

Specific humidity of second steam of air,

= 0.00943kg/kg of dry air Specific humidity after mixing,

= 0.00943kg/kg of dry air

Step 5:

Draw a horizontal line from = 0.00943 till it cuts 1 –2 line.

Name the point 3.

From psychrometric chart, at point 3

Dry bulb temperature, = 24.02°C

Wet bulb temperature, = 18.2°C

**GAS MIXTURES AND PSYCHROMETRY**

**1.** **What is difference between air conditioning and refrigeration?**

Refrigeration is the process of providing and maintaining the temperature in space below atmospheric temperature.

Air conditioning is the process of supplying sufficient volume of clean air conditioning a specific amount of water vapor and maintaining the predetermined atmospheric condition with in a selected enclosure.

**2.Define dry bulb temperature (DBT)?**

The temperature which is measured by an ordinary thermometer is known as dry bulb temperature. It is generally denoted by .

**3.Define wet bulb temperature (WBT)?**

It is the temperature of air measured by a thermometer when its bulb is covered with wet cloth and exposed to a current rapidly moving air. It is denoted by .

**4.Define dew point temperature (DPT)?**

The temperature at which the water vapor present in air begins to condense when the air is cooled is known as dew point temperature. It is denoted by .

**5.Define Relative Humidity (RH) and Specify humidity?**

Relative Humidity is defined as the ratio of the mass of water vapor (m(v)) in a certain volume of moist air at a given temperature to the mass of water vapor m(vs) in the same volume of saturated air at the same temperature.

Specific Humidity is defined as the ratio of mass of water vapor (m(v)) to the mass of dry air in the given volume of mixture.

**6. Differentiate between absolute and relative humidity?**

Absolute humidity is defined as the ratio of the mass of water vapor (m(v)) in a certain volume of moist sir at a given temperature to the mass of water vapor at atmospheric conditions.

Relative Humidity is defined as the ratio of the mass of water vapor m(vs) in a certain volume of moist air at a given temperature to the mass of water vapor in the same volume of saturated air at the same temperature.

**7.Define degree of saturation?**

Degree of saturation is the ratio of specific humidity of moist air to the specific humidity of saturated air at temperature.

**8.What is dew point temperature? How is it related to dry bulb and wet bulb temperature at the saturation condition?**

It is the temperature at which the water vapor present in air begins to condense when the air is cooled.

For saturated air, the dry bulb, wet bulb and dew point temperature are all same.

**9.What is sensible heat factor?**

The ratio of sensible heat load to the total heat loaded is known as sensible heat factor or sensible heat ratio.

**10. What is humidification and dehumidification?**

Humidification is defined as the process of adding moisture at constant dry bulb temperature. So, but . So, the humidity ratio increases from .

Dehumidification in this case, the humidity ratio decreases by removing moisture from air. But this process is carried out at constant dry bulb temperature , but. The heat involved in dehumidification is also called as latent heat load.

**11. Explain the throttling process.**

When a gas or vapor expands and flows through an aperture of small size, the process is called as throttling process.

**12. What is psychrometry?**

Psychrometry is a study of properties of moist air. It deals with the state of atmosphere with respect to moisture content. On the other hand psychometrics deals with the thermal properties of air and the control and measurement s of the moisture content in air in addition to study the effects of atmospheric moisture on commodities and the human comforts.

**13. What is effective temperature?**

Effective temperature is defined as that temperature of saturated air at which the subject would experience the same feeling of comfort as experienced in the actual unsaturated environment.

**14. Describe the heating and humidification process.**

In case of winter air conditioning heating and humidification are very common. The system consists of a coil for sensible heating of air from state1 to state 3 then along the wet bulb temperature line through state2. Thereafter the humidification occurs along the wet bulb line until the required moisture is added to reach state2. The amount of moisture addition can again be calculated by m w = m (w2-w1). if the ignores the internal functioning during heating and humidification, one can reach state 1 to 2 by adding appropriate amount of moisture and heat as shown by process 1-2, dotted line

**15.Describe the Sensible Cooling and Heating process.**

The energy transfer in these processes occurs with change in temperature only, the specific humidity remaining constant. The heat transfer for this process can be obtained by the energy balance for the control volume for the known enthalpies, h1 and h2 and for the steady mass flow rate

The magnitude of is less than during cooling. Hence there is heat transfer from the air flowing over the coil to the cooling fluid. On the other hand during heating the energy is transferred to the air flowing over the coil.

**16. Define sensible heat and latent heat.**

Sensible heat is the heat that changes the temperature of the substance when added to it or when abstracted from it.

Latent heat is the heat that does not affect the temperature but change of state occurred by adding the heat or by abstracting the heat.

**17. What is meant by adiabatic mixing?**

The process of mixing two or more stream of air without any heat transfer to the surrounding is known as adiabatic mixing. It is happened in air conditioning system.

**PRE REQUEST DISCUSSION**

This page broadly deal with units and dimensions, properties of fluids and applications of control volume of continuity equation, energy equation, and momentum equation.

Man’s desire for knowledge of fluid phenomena began with his problems of watersupply, irrigation, navigation, and waterpower.

Matter exists in two states; the solid and the fluid, the fluid state being commonly divided into the liquid and gaseous states. Solids differ from liquids and liquids from gases in the spacing and latitude of motion of their molecules, these variables being large in a gas, smaller in a liquid, and extremely small in a solid. Thus it follows that intermolecular cohesive forces are large in a solid, smaller in a liquid, and extremely small in a gas.

**DIFFERENCES BETWEEN SOLIDS AND FLUIDS**

The differences between the behaviors of solids and fluids under an applied force are as follows:

i. For a solid, the strain is a function of the applied stress, providing that the elastic limit is not exceeded. For a fluid, the rate of strain is proportional to the applied stress.

ii. The strain in a solid is independent of the time over which the force is applied and, if the elastic limit is not exceeded, the deformation disappears when the force is removed. A fluid continues to flow as long as the force is applied and will not recover its original form when the force is removed.

**FLUID MECHANICS**

Fluid mechanics is that branch of science which deals with the behavior of fluids (liquids or gases) at rest as well as in motion. Thus this branch of science deals with the static, kinematics and dynamic aspects of fluids. The study of fluids at rest is called fluid statics. The study of fluids in motion, where pressure forces are not considered, is called fluid kinematics and if the pressure forces are also considered for the fluids in motion, that branch of science is called fluid dynamics.

**UNITS AND DIMENSIONS**

The word dimensions are used to describe basic concepts like mass, length, time, temperature and force.Units are the means of expressing the value of the dimension quantitatively or numerically.

Example – Kilogram, Metre, Second, Kelvin, Celcius.

The four examples are the fundamental units; other derived units are

Density = mass per unit volume = kg/m3

Force = mass x acceleration = kg.m/s2 = Newton or N

Pressure = force per unit area = N/m2 =Pascal or Pa

Other unit is‘ bar’ ,

where 1 bar =1 X105 Pa =100 Kpa = 0.1 Mpa

Work = force x distance = Newton x metre = N.m==J or Joule

Power = work done per unit time= J/s = Watt or W

Term Dimension Unit

Area L*L m2

Volume L*L*L m3

Velocity L* T-1 m/s

Acceleration L*T-2 m/s2

Force M*L*T-2 N

Pressure M*L-1*T-2 N/m2 = Pa

Work M*L2*T-2 Nm = J

Power M*L2*T-3 J/s = W

Density M*L-3 kg/m3

Viscosity M*L-1*T-1 kg/ms = N s/m2

Surface Tension M*T-1 N/m

** Quantity Representative symbol Dimensions **

Angular velocity w t-1

Area A L2

Density r M/L3

Force F ML/t2

Kinematic viscosity n L2/t

Linear velocity V L/t

Linear acceleration A L/t2

Mass flow rate m. M/t

Power P ML2/t3

Pressure P M/Lt2

Sonic velocity C L/t

Shear stress t M/Lt2

Surface tension s M/t2

Viscosity m M/Lt

Volume V L3

**Dimensions:**

Dimensions of the primary quantities:

**Fundamental dimension : Symbol**

Length L

Mass M

Time T

Temperature T

Dimensions of derived quantities can be expressed in terms of the fundamental dimensions.

**1.SYSTEM OF UNITS**

1. CGS Units

2. FPS Units

3. MKS Units

4. SI Units

**FLUID PROPERTIES**

**1 Density or Mass density:**

Density or mass density of a fluid is defined as the ratio of the mass of a fluid to its volume. Thus mass per unit volume of a is called density.

**2. Specific weight or weight density:**

Specific weight or weight density of a fluid is the ratio between the weight of a fluid to its volume. The weight per unit volume of a fluid is called weight density.

**3. Specific Volume:**

Specific volume of a fluid is defined as the volume of a fluid occupied by a unit mass orvolume per unit mass of a fluid

**4.Specific Gravity:**

Specific gravity is defined as the ratio of the weight density of a fluid to the weight density of a standard fluid.

**VISCOSITY**

Viscosity is defined as the property of a fluid which offers resistance to the

movement of one layer of fluid over adjacent layer of the fluid. When two layers of a fluid, a distance ‘dy’ apart, move one over the other at different velocities, say u

and u+du as shown in figure. The viscosity together with relative velocity causes a shear stress acting between the fluid layers

The top layer causes a shear stress on the adjacent lower layer while the lower layer causes a shear stress on the adjacent top layer. This shear stress is proportional to the rate of change of velocity with respect to y.

**VAPOUR PRESSURE**

The pressure at which a liquid will boil is called its vapor pressure. This pressure is a function 3 of temperature (vapor pressure increases with temperature). In this context we usually think about the temperature at which boiling occurs. For example, water boils at 100oC at sea-level atmospheric pressure (1 atm abs). However, in terms of vapor pressure, we can say that by increasing the temperature of water at sea level to 100 oC, we increase the vapor pressure to the point at which it is equal to the atmospheric pressure (1 atm abs), so that boiling occurs. It is easy to visualize that boiling can also occur in water at temperatures much below 100oC if the pressure in the water is reduced to its vapor pressure. For example, the vapor pressure of water at 10oC is 0.01 atm.

**1.CAVITATION**

Cavitation(flashing of the liquid into vapour) takes place when very low pressures are produced at certain locations of a flowing liquid. Cavitation results in the formation of vapour pockets or cavities which are carried away from the point of origin and collapse at the high pressure zone.

**COMPRESSIBILITY**

Compressibility is the reciprocal of the bulk modulus of elasticity, K which is defined as the ratio of compressive stress to volumetric strain.

Compressibility is given by = 1/K

**SURFACE TENSION**

Surface tension is defined as the tensile force acting on the surface of a liquid in contact with a gas or on the surface between two two immiscible liquids such that the contact surface behaves like a membrane under tension.

**1.A soap bubble 50 mm in diameter contains a pressure (in excess of atmospheric) of 2 bar. Find the surface tension in the soap film.**

**Data:**

Radius of soap bubble (r) = 25 mm = 0.025 m Dp = 2 Bar = 2 x 105 N/m2

**Formula:**

Pressure inside a soap bubble and surface tension (s) are related by, Dp = 4s/r

**Calculations:**

s = Dpr/4 = 2 x 105 x 0.025/4 = **1250 N/m**

**CAPILLARITY**

Capillarity is defined as a phenomenon of rise or fall of a liquid surface in a small tube relative to the adjacent general level of liquid when the tube is held vertically in the liquid. The rise of liquid surface is known as capillary rise while the fall of the liquid surface is known as capillary depression.

It is expressed in terms of cm or mm of liquid. Its value depends upon the specific weight of the liquid, diameter of the tube and surface tension of the liquid.

**1.Water has a surface tension of 0.4 N/m. In a 3 mm diameter vertical tube if the liquid rises 6 mm above the liquid outside the tube, calculate the contact angle.**

**Data:**

Surface tension (s) = 0.4 N/m

Dia of tube (d) = 3 mm = 0.003 m

Capillary rise (h) = 6 mm = 0.006 m

**Formula:**

Capillary rise due to surface tension is given by

h = 4 cos(q)/(rgd), where q is the contact angle.

**Calculations:**

cos(q) = hrgd/(4s) = 0.006 x 1000 x 9.812 x 0.003 / (4 x 0.4) = 0.11

Therfore, contact angle q = **83.7o**

**CONCEPT OF CONTROL VOLUME**

A specified large number of fluid and thermal devices have mass flow in and out of a system called as control volume.

**1.CONTINUITY EQUATION**

**Concepts**

The continuity equation is governed from the principle of conservation of mass.It states that the mass of fluid flowing through the pipe at the cross-section remains constants,if there is no fluid is added or removed from the pipe.

Let us make the mass balance for a fluid element as shown below: (an open-faced cube)

This is the continuity equation for every point in a fluid flow whether steady or unsteady , compressible or incompressible.

For steady, incompressible flow, the density r is constant and the equation simplifies to

For two dimensional incompressible flow this will simplify still further to

**2 EULER’S EQUATION OF MOTION**

This is known as Euler’s equation, giving, in differential form the relationship between p, *v*, rand elevation z, along a streamline for steady flow.

**3 BERNOULLI EQUATION**

**Concepts**

Bernoulli’s Equation relates velocity, pressure and elevation changes of a fluid in motion. Itmay be stated as follows “ In an ideal incompressible fluid when the flow is steady and

continuous the sum of pressure energy, kinetic energy and potential energy is constant along streamline”

This is the basic from of *Bernoulli equation* for steady incompressible inviscid flows. It may be written for any two points 1 and 2 on the same streamline as

The contstant of Bernoulli equation, can be named as *total head (ho)* has different values on different streamlines.

The total head may be regarded as the sum of the *piezometric head* h* = p/rg + z and the *kinetic head *v2/2g.

**Bernoullie equation is arrived from the following assumptions:**

1. Steady flow – common assumption applicable to many flows.

2. Incompressible flow – acceptable if the flow Mach number is less than 0.3.

3. Frictionless flow – very restrictive; solid walls introduce friction effects.

4. Valid for flow along a single streamline; i.e., different streamlines may have different ho.

5. No shaft work – no pump or turbines on the streamline.

6. No transfer of heat – either added or removed.

**Range of validity of the Bernoulli Equation:**

Bernoulli equation is valid along any streamline in any steady, inviscid, incompressible flow. There are no restrictions on the shape of the streamline or on the geometry of the overall flow. The equation is valid for flow in one, two or three dimensions.

**Modifications on Bernoulli equation:**

Bernoulli equation can be corrected and used in the following form for real cases.

**APPLICATIONS**

1.Venturimeter.

2.Orificemeter

3.Pitot Tube

**MOMENTUM EQUATION**

Net force acting on fluid in the direction of x=Rate of change of momentum in x direction

=Mass per sec×Change in velocity

p1A1-p2A2×cos θ–Fx=ρQ(v2cosθ-v1)

Fx=ρQ(v1–v2cosθ)–p2A2cosθ+p1A1

Similarlt,the momentum in y-direction is –p2A2sinθ+Fy=ρQ(v2sinθ-0)

Fy=ρQv2sinθ+p2A2 sinθ

Resultant force acting on the bend,

Fr=√Fx²+Fy²

**GLOSSARY**

Quantity : Unit

Mass in Kilogram : Kg

Length in Meter : M

Time in Second : s or as sec

Temperature in Kelvin : K

Mole : gmol or simply as mol

**Derived quantities:**

**Quantity : Unit**

Force in Newton (1 N = 1 kg.m/s2) : N

Pressure in Pascal (1 Pa = 1 N/m2) : N/m2

Work, energy in Joule ( 1 J = 1 N.m) : J

Power in Watt (1 W = 1 J/s) : W

The word dimensions are used to describe basic concepts like mass, length, time, temperature and force.Units are the means of expressing the value of the dimension quantitatively or numerically.

Example – Kilogram, Metre, Second, Kelvin, Celcius.

The four examples are the fundamental units; other derived units are

Density = mass per unit volume = kg/m3

Force = mass x acceleration = kg.m/s2 = Newton or N

Pressure = force per unit area = N/m2 =Pascal or Pa

Other unit is‘ bar’ ,

where 1 bar =1 X105 Pa =100 Kpa = 0.1 Mpa

Work = force x distance = Newton x metre = N.m==J or Joule

Power = work done per unit time= J/s = Watt or W

Term Dimension Unit

Area L*L m2

Volume L*L*L m3

Velocity L* T-1 m/s

Acceleration L*T-2 m/s2

Force M*L*T-2 N

Pressure M*L-1*T-2 N/m2 = Pa

Work M*L2*T-2 Nm = J

Power M*L2*T-3 J/s = W

Density M*L-3 kg/m3

Viscosity M*L-1*T-1 kg/ms = N s/m2

Surface Tension M*T-1 N/m

** Quantity Representative symbol Dimensions **

Angular velocity w t-1

Area A L2

Density r M/L3

Force F ML/t2

Kinematic viscosity n L2/t

Linear velocity V L/t

Linear acceleration A L/t2

Mass flow rate m. M/t

Power P ML2/t3

Pressure P M/Lt2

Sonic velocity C L/t

Shear stress t M/Lt2

Surface tension s M/t2

Viscosity m M/Lt

Volume V L3

**Dimensions:**

Dimensions of the primary quantities:

**Fundamental dimension : Symbol**

Length L

Mass M

Time T

Temperature T

Dimensions of derived quantities can be expressed in terms of the fundamental dimensions.

**1.SYSTEM OF UNITS**

1. CGS Units

2. FPS Units

3. MKS Units

4. SI Units

**1 Density or Mass density:**

Density or mass density of a fluid is defined as the ratio of the mass of a fluid to its volume. Thus mass per unit volume of a is called density.

**2. Specific weight or weight density:**

Specific weight or weight density of a fluid is the ratio between the weight of a fluid to its volume. The weight per unit volume of a fluid is called weight density.

**3. Specific Volume:**

Specific volume of a fluid is defined as the volume of a fluid occupied by a unit mass or volume per unit mass of a fluid

**4.Specific Gravity:**

Specific gravity is defined as the ratio of the weight density of a fluid to the weight density of a standard fluid.

**VISCOSITY**

Viscosity is defined as the property of a fluid which offers resistance to the

movement of one layer of fluid over adjacent layer of the fluid. When two layers of a fluid, a distance ‘dy’ apart, move one over the other at different velocities, say u

and u+du as shown in figure. The viscosity together with relative velocity causes a shear stress acting between the fluid layers

The top layer causes a shear stress on the adjacent lower layer while the lower layer causes a shear stress on the adjacent top layer. This shear stress is proportional to the rate of change of velocity with respect to y.

**VAPOUR PRESSURE**

The pressure at which a liquid will boil is called its vapor pressure. This pressure is a function 3 of temperature (vapor pressure increases with temperature). In this context we usually think about the temperature at which boiling occurs. For example, water boils at 100oC at sea-level atmospheric pressure (1 atm abs). However, in terms of vapor pressure, we can say that by increasing the temperature of water at sea level to 100 oC, we increase the vapor pressure to the point at which it is equal to the atmospheric pressure (1 atm abs), so that boiling occurs. It is easy to visualize that boiling can also occur in water at temperatures much below 100oC if the pressure in the water is reduced to its vapor pressure. For example, the vapor pressure of water at 10oC is 0.01 atm.

**1.CAVITATION**

Cavitation(flashing of the liquid into vapour) takes place when very low pressures are produced at certain locations of a flowing liquid. Cavitation results in the formation of vapour pockets or cavities which are carried away from the point of origin and collapse at the high pressure zone.

Surface tension is defined as the tensile force acting on the surface of a liquid in contact with a gas or on the surface between two two immiscible liquids such that the contact surface behaves like a membrane under tension.

**1.A soap bubble 50 mm in diameter contains a pressure (in excess of atmospheric) of 2 bar. Find the surface tension in the soap film.**

**Data:**

Radius of soap bubble (r) = 25 mm = 0.025 m Dp = 2 Bar = 2 x 105 N/m2

**Formula:**

Pressure inside a soap bubble and surface tension (s) are related by, Dp = 4s/r

**Calculations:**

s = Dpr/4 = 2 x 105 x 0.025/4 = **1250 N/m**

**CAPILLARITY**

Capillarity is defined as a phenomenon of rise or fall of a liquid surface in a small tube relative to the adjacent general level of liquid when the tube is held vertically in the liquid. The rise of liquid surface is known as capillary rise while the fall of the liquid surface is known as capillary depression.

It is expressed in terms of cm or mm of liquid. Its value depends upon the specific weight of the liquid, diameter of the tube and surface tension of the liquid.

**1.Water has a surface tension of 0.4 N/m. In a 3 mm diameter vertical tube if the liquid rises 6 mm above the liquid outside the tube, calculate the contact angle.**

**Data:**

Surface tension (s) = 0.4 N/m

Dia of tube (d) = 3 mm = 0.003 m

Capillary rise (h) = 6 mm = 0.006 m

**Formula:**

Capillary rise due to surface tension is given by

h = 4 cos(q)/(rgd), where q is the contact angle.

**Calculations:**

cos(q) = hrgd/(4s) = 0.006 x 1000 x 9.812 x 0.003 / (4 x 0.4) = 0.11

Therfore, contact angle q = **83.7o**

A specified large number of fluid and thermal devices have mass flow in and out of a system called as control volume.

**1.CONTINUITY EQUATION**

**Concepts**

The continuity equation is governed from the principle of conservation of mass.It states that the mass of fluid flowing through the pipe at the cross-section remains constants,if there is no fluid is added or removed from the pipe.

Let us make the mass balance for a fluid element as shown below: (an open-faced cube)

This is the continuity equation for every point in a fluid flow whether steady or unsteady , compressible or incompressible.

For steady, incompressible flow, the density r is constant and the equation simplifies to

For two dimensional incompressible flow this will simplify still further to

**2 EULER’S EQUATION OF MOTION**

This is known as Euler’s equation, giving, in differential form the relationship between p, *v*, r and elevation z, along a streamline for steady flow.

**3 BERNOULLI EQUATION**

**Concepts**

Bernoulli’s Equation relates velocity, pressure and elevation changes of a fluid in motion. It may be stated as follows “ In an ideal incompressible fluid when the flow is steady and

continuous the sum of pressure energy, kinetic energy and potential energy is constant along streamline”

This is the basic from of *Bernoulli equation* for steady incompressible inviscid flows. It may be written for any two points 1 and 2 on the same streamline as

The contstant of Bernoulli equation, can be named as *total head (ho)* has different values on different streamlines.

The total head may be regarded as the sum of the *piezometric head* h* = p/rg + z and the *kinetic head *v2/2g.

**Bernoullie equation is arrived from the following assumptions:**

1. Steady flow – common assumption applicable to many flows.

2. Incompressible flow – acceptable if the flow Mach number is less than 0.3.

3. Frictionless flow – very restrictive; solid walls introduce friction effects.

4. Valid for flow along a single streamline; i.e., different streamlines may have different ho.

5. No shaft work – no pump or turbines on the streamline.

6. No transfer of heat – either added or removed.

**Range of validity of the Bernoulli Equation:**

Bernoulli equation is valid along any streamline in any steady, inviscid, incompressible flow. There are no restrictions on the shape of the streamline or on the geometry of the overall flow. The equation is valid for flow in one, two or three dimensions.

**Modifications on Bernoulli equation:**

Bernoulli equation can be corrected and used in the following form for real cases.

**APPLICATIONS**

1.Venturimeter.

2.Orificemeter

3.Pitot Tube

**DIFFERENCES BETWEEN SOLIDS AND FLUIDS**

The differences between the behaviors of solids and fluids under an applied force are as follows:

i. For a solid, the strain is a function of the applied stress, providing that the elastic limit is not exceeded. For a fluid, the rate of strain is proportional to the applied stress.

ii. The strain in a solid is independent of the time over which the force is applied and, if the elastic limit is not exceeded, the deformation disappears when the force is removed. A fluid continues to flow as long as the force is applied and will not recover its original form when the force is removed.

**FLUID PROPERTIES AND FLOW CHARACTERISTICS**

**PART ****–**** A**

**1. Define fluids.**

Fluid may be defined as a substance which is capable of flowing. It has no definite shape of its own, but confirms to the shape of the containing vessel.

**2. What are the properties of ideal fluid?**

Ideal fluids have following properties i)It is incompressible

ii) It has zero viscosity

iii) Shear force is zero

**3. ****What are the properties of real fluid?**

Real fluids have following properties i)It is compressible

ii) They are viscous in nature

iii) Shear force exists always in such fluids.

**4. Explain the Density**

Density or mass density is defined as the ratio of the mass of the fluid to its volume. Thus mass per unit volume of a fluid is called density. It is denoted by the symbol (ρ).

Density = Mass of the fluid (kg) / Volume of the fluid (m3)

**5. Explain the Specific weight or weight density**

Specific weight or weight density of a fluid is the ratio between the weight of a fluid to its volume. Thus weight per uint volume of a fluid is called weight density and is denoted by the symbol (W).

(W) = Weight of the fluid / Volume of fluid

= Mass x Acceleration due to gravity /__ __Volume of fluid

W = pg

**6. Explain the Specific volume**

Specific volume of a fluid is defined as the volume of the fluid occupied by a unit Mass or volume per unit mass of a fluid is called specific volume.

Specific volume = Volume / Mass = m3 /kg = l/p

**7. Explain the Specific gravity**

Specific gravity is defined as the ratio of weight density of a fluid to the weight density of a standard fluid. For liquid, standard fluid is water and for gases, it is air.

Specific gravity = Weight density of any liquid or gas Weight / density of standard liquid or gas

**8.Define Viscosity.**

It is defined as the property of a liquid due to which it offers resistance to the movement of one layer of liquid over another adjacent layer.

**9. Define kinematic viscosity.**

It is defined as the ratio of dynamic viscosity to mass density. (m²/sec)

**10. Define Relative or Specific viscosity.**

It is the ratio of dynamic viscosity of fluid to dynamic viscosity of water at 20°C.

**11. State Newton’s law of viscosity and give examples.**

Newton’s law states that the shear stress ( ) on a fluid element layer is directly proportional to the rate of shear strain. The constant of proportionality is called co-

efficient of viscosity.

r = μ du / dy

**12. ****Give the importance of viscosity on fluid motion and its effect on temperature.**

Viscosity is the property of a fluid which offers resistance to the movement of one layer of fluid over another adjacent layer of the fluid. The viscosity is an important property which offers the fluid motion.

The viscosity of liquid decreases with increase in temperature and for gas it Increases with increase in temperature.

**13. ****Explain the Newtonian fluid**

The fluid which obeys the Newton’s law of viscosity i.e., the shear stress is directly

proportional to the rate of shear strain, is called Newtonian

fluid. r = μ du / dy

**14. Explain the Non-Newtonian fluid**

The fluids which does not obey the Newton’s law of viscosity i.e., the shear stress is not directly proportional to the ratio of shear strain, is called non-Newtonian fluid.

**15. Define compressibility.**

Compressibility is the reciprocal of bulk modulus of elasticity, k which is defined as the ratio of compressive stress to volume strain.

k = Increase of pressure / Volume strain

Compressibility = 1 / k = Volume of strain / Increase of pressure

**16. Define surface tension.**

Surface tension is defined as the tensile force acting on the surface of a liquid in Contact with a gas or on the surface between two immiscible liquids such that contact surface behaves like a membrane under tension.

**17. Define Capillarity.**

Capillary is a phenomenon of rise or fall of liquid surface relative to the adjacent general level of liquid.

**18. What is cohesion and adhesion in fluids?**

Cohesion is due to the force of attraction between the molecules of the same liquid. Adhesion is due to the force of attraction between the molecules of two different Liquids or between the molecules of the liquid and molecules of the solid boundary surface.

**19. State momentum of momentum equation?**

It states that the resulting torque acting on a rotating fluid is equal to the rate of change of moment of momentum.

**20. What is momentum equation**

It is based on the law of conservation of momentum or on the momentum principle It states that,the net force acting on a fluid mass is equal to the change in momentum of flow per unit time in that direction.

**21. What is Euler’s equation of motion**

This is the equation of motion in which forces due to gravity and pressure are taken into consideration. This is derived by considering the motion of a fluid element along a stream line.

**22. What is venturi meter?**

Venturi meter is a device for measuring the rate of fluid flow of a flowing fluid through a pipe. It consisits of three parts.

a. A short converging part b. Throat c.Diverging part. It is based on the principle of Bernoalli’s equation.

**23. What is an orifice meter?**

Orifice meter is the device used for measuring the rate of flow of a fluid through a pipe. it is a cheaper device as compared to venturi meter. it also works on the priniciple as that of venturi meter. It consists of a flat circular plate which has a circular sharp edged hole called orifice.

**24. What is a pitot tube?**

Pitot tube is a device for measuring the velocity of a flow at any point in a pipe or a channel. It is based on the principle that if the velocity of flow at a point becomes zero, the pressure there is increased due to the conversion of kinetic energy into pressure energy.

. What are the types of fluid flow?

Steady & unsteady fluid flow Uniform & Non-uniform flow

One dimensional, two-dimensional & three-dimensional flows Rotational & Irrotational flow

**25. State the application of Bernouillie****’****s equation ?**

It has the application on the following measuring devices.

1.Orifice meter.

2.Venturimeter.

3.Pitot tube.

**PRE REQUEST DISCUSSION**

This page has an in dept dealing of laminar flow through pipes, boundary layer concept, hydraulic and energy gradient, friction factor, minor losses, and flow through pipes in series and parallel.

Boundary layer is the region near a solid where the fluid motion is affected by the solid boundary. In the bulk of the fluid the flow is usually governed by the theory of ideal fluids. By contrast, viscosity is important in the boundary layer. The division of the problem of flow past an solid object into these two parts, as suggested by Prandtl in 1904 has proved to be of fundamental importance in fluid mechanics.

This concept of hydraulic gradient line and total energy line is very useful in the study of flow This concept of hydraulic gradient line and total energy line is very useful in the study of flow of fluids through pipes. f fluids through pipes.

**HYDRAULIC GRADIENT AND TOTAL ENERGY LINE**

__1.Hydraulic Gradient Line__It is defined as the line which gives the sum of pressure head (p/w) and datum head (z) of a flowing fluid in a pipe with respect to some reference line or it is the line which is obtained by joining the top of all vertical ordinates, showing the pressure head (p/w) of a flowing fluid in a pipe from the centre of the pipe. It is briefly written as H.G.L (Hydraulic Gradient Line).

__2.Total Energy Line__It is defined as the line which gives the sum of pressure head, datum head and kinetic head of a flowing fluid in a pipe with respect to some reference line. It is also defined as the line which is obtained by joining the tops of all vertical ordinates showing the sum of pressure head and kinetic head from the centre of the pipe. It is briefly written as T.E.L (Total Energy Line).

**BOUNDARY LAYER**

**Concepts**

The variation of velocity takes place in a narrow region in the vicinity of solid boundary. The fluid layer in the vicinity of the solid boundary where the effects of fluid friction i.e., variation of velocity are predominant is known as the *boundary layer.*

**1 FLOW OF VISCOUS FLUID THROUGH CIRCULAR PIPE**

For the flow of viscous fluid through circular pipe, the velocity distribution across a section, the ratio of maximum velocity to average velocity, the shear stress distribution and

drop of pressure fora given length is to be determined. The flow through circular pipe will be viscous or laminar, if the Reynold’s number is less than 2000.

**.2 DEVELOPMENT OF LAMINAR AND TURBULENT FLOWS IN CIRCULAR PIPES**

**1.Laminar Boundary Layer**

At the initial stage i.e, near the surface of the leading edge of the plate, the thickness of boundary layer is the small and the flow in the boundary layer is laminar though the main stream flows turbulent. So, the layer of the fluid is said to be *laminar boundary layer.*

**2.Turbulent Boundary Layer**

The thickness boundary layer increases with distance from the leading edge in the down-stream direction. Due to increases in thickness of boundary layer, the laminar boundary layer becomes unstable and the motion of the fluid is disturbed. It leads to a transition from laminar to turbulent boundary layer.

**3 BOUNDARY LAYER GROWTH OVER A FLAT PLATE**

Consider a continuous flow of fluid along the surface of a thin flat plate with its sharp leading edge set parallel to the flow direction as shown in figure 2.7.The fluid approaches the plate with uniform velocity U known as *free stream velocity* at the leading edge. As soon as the fluid comes in contact the leading edge of the plate,its velocity is reduced to zero as the fluid particles adhere to the plate boundary thereby satisfying no-slip condition.

**FLOW THROUGH CIRCULAR PIPES-****HAGEN POISEUILLE’S EQUATION**

Due to viscosity of the flowing fluid in a laminar flow,some losses of head take place.The equation which gives us the value of loss of head due to viscosity in a laminar flow is known as Hagen-Poiseuille’s law.

p1-p2=32μUL/D²

=128μQL/πD4

This equation is called as Hargen-Poiseuille equation for laminar flow in the circular pipes.

**DARCY’S EQUATION FOR LOSS OF HEAD DUE TO FRICTION IN PIPE**

A pipe is a closed conduit through which the fluid flows under pressure.When the fluid flows through the piping system,some of the potential energy is lost due to friction.

hƒ=4ƒLv²/2gD

**MOODY’S DIAGRAM**

Moody’s diagram is plotted between various values of *friction factor(*ƒ),*Reynolds*

*number(*Re) and* relative roughness*(R/K) as shown in figure 2.6.For any turbulent flowproblem,the values of friction factor(ƒ) can therefore be determined from Moody’s diagram,if the numerical values of R/K for the pipe and Rе of flow are known.

The Moody’s diagram has plotted from the equation

1/√ ƒ-2.0 log10(R/K)=1.74-2.0 log10[1+18.7/(R/K/Re/ ƒ)]

Where,R/K=relative roughness

ƒ=friction factor and Re=Reynolds number.

**CLASSIFICATION OF BOUNDARY LAYER THICKNESS**

1. Displacements thickness(δ*)

2. Momentum thickness(θ)

3. Energy thickness(δe)

**BOUNDARY LAYER SEPARATION**

The boundary layer leaves the surface and gets separated from it. This phenomenon is known as *boundary layer separation.*

**LOSSESS IN PIPES**

When a fluid flowing through a pipe, certain resistance is offered to the flowing fluid, it results in causing a loss of energy.

The loss is classified as:

1. Major losses

2. Minor losses

**Major Losses in Pipe Flow**

The major loss of energy is caused by friction in pipe. It may be computed by Darcy-weisbach equation.

**Minor Losses in Pipe Flow**

The loss of energy caused on account of the change in velocity of flowing fluid is called minor loss of energy.

**FLOW THOUGH PIPES IN SERIES AND PARALLEL**

**Pipes in Series**

The pipes of different diamers and lengths which are connected with one another to form a single pipeline.

**Pipes in Parallel**

When a main pipeline divides into two or more parallel pipes which again join together to form a single pipe and continuous as a main line

**GLOSSARY**

HGL –Hydraulic gradient line

TEL – Total energy line.

**Applications**

1. To find out friction factor in the flow through pipe.

2. To find out the losses in losses in the pipes.

**FLOW THROUGH CIRCULAR CONDUITS**

**1. Define viscosity (u).**

Viscosity is defined as the property of a fluid which offers resistance to the movement of one layer of fluid over another adjacent layer of the fluid.Viscosity is also defined as the shear stress required to produce unit rate of shear strain.

**2. Define kinematic viscosity.**

Kniematic viscosity is defined as the ratio between the dynamic viscosity and density of fluid. It is denoted by μ.

**3. What is minor energy loss in pipes?**

The loss of head or energy due to friction in a pipe is known as major loss while loss of energy due to change of velocity of fluid in magnitude or direction is called minor loss of energy. These include,

a. Loss of head due to sudden enlargement.

b. Loss of head due to sudden contraction.

c. Loss of head at entrance to a pipe.

d. Loss of head at exit of a pipe.

e. Loss of head due to an obstruction in a pipe.

f. Loss of head due to bend in a pipe.

g. Loss of head in various pipe fittings.

**4. What is total energy line?**

Total energy line is defined as the line which gives the sum of pressure head, datum head and kinetic head of a flowing fluid in a pipe with respect to some reference line. It is also defined as the line which is obtained by joining the tops of all vertical ordinates showing sum of the pressure head and kinetic head from the centre of the pipe.

**5. What is hydraulic gradient line?**

Hydraulic gradient line gives the sum of (p/w+z) with reference to datum line. Hence hydraulic gradient line is obtained by subtracting v2 / 2g from total energy line.

**6. What is meant by pipes in series?**

When pipes of different lengths and different diameters are connected end to end, pipes are called in series or compound pipe. The rate of flow through each pipe connected in series is same.

**7. What is meant by pipes in parallel?**

When the pipes are connected in parallel, the loss of head in each pipe is same. The rate of flow in main pipe is equal to the sum of rate of flow in each pipe, connected in parallel.

**8. What is boundary layer and boundary layer theory?**

When a solid body immersed in the flowing fluid, the variation of velocity from zero to free stream velocity in the direction normal to boundary takes place in a narrow region in the vicinity of solid boundary. This narrow region of fluid is called boundary layer. The theory dealing with boundary layer flow is called boundary layer theory.

**9. What is turbulent boundary layer?**

If the length of the plate is more then the distance x, the thickness of boundary layer will go on increasing in the downstream direction. Then laminar boundary becomes unstable and motion of fluid within it, is disturbed and irregular which leads to a transition from laminar to turbulent boundary layer.

**10. What is boundary layer thickness?**

Boundary layer thickness (S) is defined as the distance from boundary of the solid body measured in y-direction to the point where the velocity of fluid is approximately equal to 0.99 times the free steam (v) velocity of fluid.

**11. Define displacement thickness**

Displacement thickness (S*) is defined as the distances, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction inflow rate on account of boundary layer formation.

**12. What is momentum thickness?**

Momentum thickness (0) is defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in momentum of flowing fluid on account of boundary layer formation.

**13.Mention the general characteristics of laminar flow.**

• There is a shear stress between fluid layers

• ‘No slip’ at the boundary

• The flow is rotational

• There is a continuous dissipation of energy due to viscous shear

**14. ****What is Hagen poiseuill****e’****s formula ?**

P1-P2 / pg = h f = 32 µ UL / _gD^{2}

The expression is known as Hagen poiseuille formula .

Where P1-P2 / _g = Loss of pressure head U = Average velocity

µ = Coefficient of viscosity D = Diameter of pipe

L = Length of pipe

**15.What are the factors influencing the frictional loss in pipe flow ?**

Frictional resistance for the turbulent flow is

i. Proportional to vn where v varies from 1.5 to 2.0 . ii. Proportional to the density of fluid .

iii. Proportional to the area of surface in contact . iv. Independent of pressure .

v. Depend on the nature of the surface in contact .

**16. ****What is the expression for head loss due to friction in Darcy formula ?**

hf = 4fLV^{2} / 2gD

Where f = Coefficient of friction in pipe L = Length of the pipe

D = Diameter of pipe V = velocity of the fluid

**17. ****What do you understand by the terms**

a) major energy losses , b) minor energy losses Major energy losses : –

This loss due to friction and it is calculated by Darcy weis bach formula and chezy’s formula .

Minor energy losses :- This is due to

h. Sudden expansion in pipe .ii. Sudden contraction in pipe . iii. Bend in pipe .iv. Due to obstruction in pipe .

**18. Give an expression for loss of head due to sudden enlargement of the pipe :**

he = (V1-V2)^{2} /2g

Wherehe = Loss of head due to sudden enlargement of pipe . V1 = Velocity of flow at section 1-1

V2 = Velocity of flow at section 2-2

**19.Give an expression for loss of head due to sudden contraction : **hc =0.5 V^{2}/2g

Where hc = Loss of head due to sudden contraction . V = Velocity at outlet of pipe.

**20. Give an expression for loss of head at the entrance of the pipe **hi =0.5V2/2g

where hi = Loss of head at entrance of pipe .

V = Velocity of liquid at inlet and outlet of the pipe .

**21. What is sypon ? Where it is used: _**

Sypon is along bend pipe which is used to transfer liquid from a reservoir at a higher elevation to another reservoir at a lower level .

Uses of sypon : –

1. To carry water from one reservoir to another reservoir separated by a hill ridge .

2. To empty a channel not provided with any outlet sluice .

**PRE REQUEST DISCUSSION**

This Page deals with dimensional analysis,models and similitude,and application of dimensionless parameters.

Many important engineering problems cannot be solved completely by theoretical or mathematical methods. Problems of this type are especially common in fluid-flow, heat-flow, and diffusional operations. One method of attacking a problem for which no mathematical equation can be derived is that of empirical experimentations.

For example, the pressure loss from friction in a long, round, straight, smooth pipe depends on all these variables: the length and diameter of the pipe, the flow rate of the liquid, and the density and viscosity of the liquid. If any one of these variables is changed, the pressure drop also changes. The empirical method of obtaining an equation relating these factors to pressure drop requires that the effect of each separate variable be determined in turn by systematically varying that variable while keep all others constant. The procedure is laborious, and is difficult to organize or correlate the results so obtained into a useful relationship for calculations.

There exists a method intermediate between formal mathematical development and a completely empirical study. It is based on the fact that if a theoretical equation does exist among the variables affecting a physical process, that equation must be dimensionally homogeneous. Because of this requirement it is possible to group many factors into a smaller number of dimensionless groups of variables. The groups themselves rather than the separate factors appear in the final equation.

**Concepts**

Dimensional analysis drastically simplifies the task of fitting experimental data to design equations where a completely mathematical treatment is not possible; it is also useful in checking the consistency of the units in equations, in converting units, and in the scale-up of data obtained in physical models to predict the performance of full-scale model. The method is based on the concept of dimension and the use of *dimensional formulas*.

Dimensional analysis does not yield a numerical equation, and experiment is required to complete the solution of the problem. The result of a dimensional analysis is valuable in pointing a way to correlations of experimental data suitable for engineering use.

**METHODS OF DIMENSIONAL ANALYSIS**

If the number of variables involved in a physical phenomenon are known, then the relation among the variables can be determined by the following two methods.

1.Rayleigh’s method

2. Buckingham’s π theorem

**1****Rayleigh’s method**

This method is used for determining the expression for a variable which depends upon maximum three or four variables only. If the number of independent variables becomes more than four then it is very difficult to find the expression for the dependent variable.

**2 ****Buckingham’s**** ****π**** theorem.**

If there are n variables (independent and dependent variables) in a physical

phenomenon and if these variables contain m fundamental dimensions (M, L, T), then the variables are arranged into (n-m) dimensionless numbers. Each term is called Buckingham’s

π theorem.

**Applications**

v It is used to justify the dependency of one variable with the other.

v Usually this type of situation occurs in structures and hydraulic machines.

v To solve this problem efficiently, an excellent tool is identified called dimensional analysis.

**SMILITUDE ****–****TYPES OF SIMILARITIES**

Similitude is defined as the similarity between the model and its prototype in every respect, which means that the model and prototype are completely similar. Three types of similarities must exist between the model and prototype.

**Concepts**

Whenever it is necessary to perform tests on a model to obtain information that cannot be obtained by analytical means alone, the rules of similitude must be applied. *Similitude *is the theory and art of predicting prototype performance from model* *observations

**1. Geometric similarity **refers to linear dimensions. Two vessels of different sizes are** **geometrically similar if the ratios of the corresponding dimensions on the two scales are the same. If photographs of two vessels are completely super-impossible, they are geometrically similar.

**2.Kinematic similarity **refers to motion and requires geometric similarity and the same** **ratio of velocities for the corresponding positions in the vessels.

**3.Dynamic similarity **concerns forces and requires all force ratios for corresponding** **positions to be equal in kinematically similar vessels.

**SIGNIFICANCE**

The requirement for similitude of flow between model and prototype is that the significant dimensionless parameters must be equal for model and prototype

**DIMENSIONLESS PARAMETERS**

Since the inertia force is always present in a fluid flow, its ratio with each of the other forces provides a dimensionless number.

1. Reynold’s number

2. Froud’s number

3. Euler’s number

4. Weber’s number

5. Mach’s number

**Applications of dimensionless parameters**

1. Reynold’s model law

2. Froud’s model law

3. Euler’s model law

4. Weber’s model law

5. Mach’s model law

**Important Dimensionless Numbers in Fluid Mechanics:**

**MODEL ANALYSIS.**

**PRE REQUEST DISCUSSION**

Present engineering practice makes use of model tests more frequently than most people realize. For example, whenever a new airplane is designed, tests are made not only on the general scale model but also on various components of the plane. Numerous tests are made on individual wing sections as well as on the engine pods and tail sections

Models of automobiles and high-speed trains are also tested in wind tunnels to predict the drag and flow patterns for the prototype. Information derived from these model studies often indicates potential problems that can be corrected before prototype is built, thereby saving considerable time and expense in development of the prototype.

**Concepts**

Much time, mony and energy goes into the design construction and eradication of hydraulic structures and machines.

To minimize the chances of failure, it is always desired that the tests to be performed on small size models of the structures or machines. The model is the small scale replica of the actual structure or machine. The actual structure or machine is Called prototype.

**Applictions**

1. Civil engineering structures such as dams, canals etc.

2. Design of harbor, ships and submarines

3. Aero planes, rockets and machines.

4. Marine engineers make extensive tests on model shop hulls to predict the drag of the ships

**GLOSSARY**

**The three friction factor problems:**

The friction factor relates six parameters of the flow:

1. Pipe diameter

2. Average velocity

3. Fluid density

4. Fluid viscosity

5. Pipe roughness

6. The frictional losses per unit mass.

Therefore, given any five of these, we can use the friction-factor charts to find the sixth.

Most often, instead of being interested in the average velocity, we are interested in the volumetric flow rate Q = (p/4)D^{2}V

The three most common types of problems are the following:

Generally, type 1 can be solved directly, where as types 2 and 3 require simple trial and error.

Three fundamental problems which are commonly encountered in pipe-flow calculations: Constants: rho, mu, g, L

1. Given D, and v or Q, compute the pressure drop. (pressure-drop problem)

2. Given D, delP, compute velocity or flow rate (flow-rate problem)

3. Given Q, delP, compute the diameter D of the pipe (sizing problem)

**DIMENSIONAL ANALYSIS**

**1. What are the methods of dimensional analysis**

There are two methods of dimensional analysis. They are,

a. Rayleigh – Retz method

b. Buckingham’s theotem method.

Nowadays Buckingham’s theorem method is only used.

**2. Describe the Rayleigh’s method for dimensional analysis.**

Rayleigh’s method is used for determining the expression for a variable which depends upon maximum three or four variables only. If the number of independent variables becomes more than four, then it is very difficult to find the expression for dependent variable.

**3. What do you mean by dimensionless number**

Dimensionless numbers are those numbers which are obtained by dividing the inertia force by viscous force or gravity force or pressure force or surface tension or elastic force. As this is a ratio of one force to other force, it will be a dimensionless number.

**4. Name the different forces present in fluid flow**

Inertia force Viscous force Surface tension force Gravity force

**5. State Buckingham****’****s ****Π**** theorem**

It states that if there are ‘n’ variables in a dimensionally homogeneous equation and if these variables contain ‘m’ fundamental dimensions (M,L,T), then they are grouped into (n-m), dimensionless independent Π-terms.

**6. ****State the limitations of dimensional analysis.**

1. Dimensional analysis does not give any due regarding the selection of variables. 2.The complete information is not provided by dimensional analysis.

3.The values of coefficient and the nature of function can be obtained only by experiments or from mathematical analysis.

**7.****Define Similitude**

Similitude is defined as the complete similarity between the model and prototype.

**8. ****State Froude’s model law**

Only Gravitational force is more predominant force. The law states ‘The Froude’s number is same for both model and prototype’

**9.What are the similarities between model and prototype?**

(i) Geometric Similarity

(ii) Kinematicc Similarity

(iii) Dynamic Similarity

**10.Define Weber number.**

It is the ratio of the square root of the inertia force to the surface tension force.

**PRE REQUEST DISCUSSION**

Basic concepts of rot dynamic machines, velocity triangles for radial flow and axial flow machines, centrifugal pumps, turbines and Positive displacement pumps and rotary pumps its performance curves are discussed in this page.

The liquids used in the chemical industries differ considerably in physical and chemical properties. And it has been necessary to develop a wide variety of pumping equipment.

The two main forms are the *positive displacement type* and *centrifugal pumps*.

the former, the volume of liquid delivered is directly related to the displacement of the piston and therefore increases directly with speed and is not appreciably influenced by the pressure. In this group are the *reciprocating piston pump* and the *rotary gear pump*, both of which are commonly used for delivery against high pressures and where nearly constant delivery rates are required.

The centrifugal type depends on giving the liquid a high kinetic energy which is then converted as efficiently as possible into pressure energy.

**HEAD AND EFFICIANCES**

1. Gross head

2. Effective or Net head

3. Water and Bucket power

4. Hydraulic efficiency

5. Mechanical efficiency

6. Volume efficiency

7. Overall efficiency

**Concepts**

A pump is a device which converts the mechanical energy supplied into hydraulic energy by lifting water to higher levels.

**CENTRIFUGAL PUMP**

**Working principle**

If the mechanical energy is converted into pressure energy by means of centrifugal force acting

on the fluid, the hydraulic machine is called centrifugal pump. The centrifugal pump acts as a reversed of an inward radial flow reaction turbine

**Performance Characteristics of Pumps**

The fluid quantities involved in all hydraulic machines are the flow rate (Q) and the head (H), whereas the mechanical quantities associated with the machine itself are the power (P), speed (N), size (D) and efficiency (h ). Although they are of equal importance, the emphasis placed on certain of these quantities is different for different pumps. The output of a pump running at a given speed is the flow rate delivered by it and the head developed. Thus, a plot of head and flow rate at a given speed forms the fundamental performance characteristic of a pump. In order to achieve this performance, a power input is required which involves efficiency of energy transfer. Thus, it is useful to plot also the power P and the efficiency h against Q.

Over all efficiency of a pump (h ) = Fluid power output / Power input to the shaft = rgHQ / P

Type number or Specific speed of pump, n_{S} = NQ^{1/2} / (gH)^{3/4} (it is a dimensionless number)

**Centrifugal pump Performance**

In the volute of the centrifugal pump, the cross section of the liquid path is greater than in the impeller, and in an ideal frictionless pump the drop from the velocity V to the lower velocity is converted according to Bernoulli’s equation, to an increased pressure. This is the source of the discharge pressure of a centrifugal pump.

If the speed of the impeller is increased from N_{1} to N_{2} rpm, the flow rate will increase from Q_{1} to Q_{2} as per the given formula:

The head developed(H) will be proportional to the square of the quantity discharged, so that

The power consumed(W) will be the product of H and Q, and, therefore

These relationships, however, form only the roughest guide to the performance of centrifugal pumps.

**Characteristic curves**

Pump action and the performance of a pump are defined interms of their *characteristic curves*. These curves correlate the capacity of the pump in unit volume per* *unit time versus discharge or differential pressures. These curves usually supplied by pump manufacturers are for water only.

These curves usually shows the following relationships (for centrifugal pump).

· A plot of capacity versus differential head. The differential head is the difference in pressure between the suction and discharge.

· The pump efficiency as a percentage versus capacity.

· The break horsepower of the pump versus capacity.

The net poisitive head required by the pump versus capacity. The required NPSH for the pump is a characteristic determined by the manufacturer.

Centrifugal pumps are usually rated on the basis of head and capacity at the point of maximum efficiency.

**RECIPROCATING PUMPS**

**Working principle**

If the mechanical energy is converted into hydraulic energy (or pressure energy) by sucking the liquid into a cylinder in which a piston is reciprocating (moving backwards and forwards), which exerts the thrust on the liquid and increases its hydraulic energy (pressure energy), the pump is known as reciprocating pump

**Main ports of a reciprocating pump**

1.A cylinder with a piston, piston rod, connecting rod and a crank,

2. Suction pipe

3.Delivery pipe,

4. Suction valve and

5.Delivery valve.

**Slip of Reciprocating Pump**

Slip of a reciprocating pump is defined as the difference between the theoretical discharge and the actual discharge of the pump.

**Characteristic Curves Of Reciprocatring Pumps**

1.According to the water being on contact with one side or both sides of the piston

(i.) Single acting pump (ii.) Double-acting pump

2.According to the number of cylinders provided

(i.) Single acting pump (ii.) Double-acting pump (iii.) Triple-acting pump

**Reciprocating pumps Vs centrifugal pumps**

The advantages of reciprocating pumps in general over centrifugal pumps may be summarized as follows:

1. They can be designed for higher heads than centrifugal pumps.

2. They are not subject to air binding, and the suction may be under a pressure less than atmospheric without necessitating special devices for priming.

3. They are more flexible in operation than centrifugal pumps.

4. They operate at nearly constant efficiency over a wide range of flow rates.

The advantages of centrifugal pumps over reciprocating pumps are:

1. The simplest centrifugal pumps are cheaper than the simplest reciprocating pumps.

2. Centrifugal pumps deliver liquid at uniform pressure without shocks or pulsations.

3. They can be directly c onnected to motor derive without the use of gea rs or belts.

4. Valves in the discharg e line may be completely closed without injurin g them.

5. They can handle liquid s with large amounts of solids in suspension.

**Rotary Pumps**

The rotary pump is g ood for handling viscous liquids, nut because of the close tolerances needed, it can not be manufactured large enough to compete with centrifugal pumps for coping with very high flow rates.

Rotary pumps are available in a variety of configurations. • Double lobe pump

• Trible lobe pumps

• Gear pump

• Gear Pumps

• Spur Gear or Extern al-gear pump

External-gear pump (called as gear pump) consists essen tially of two intermeshing gears which are identical and which are surrounded by a closely fitting casing. One of the gears is driven directly by the prime mover while the other is allowed to rotate freel y. The fluid enters the spaces between the teeth and the casing and moves with the tee th along the outer periphery until it reaches the outlet where it is expelled from the pu mp.

External-gear p umps are used for flow rates up to about 400 m^{3}/hr working against pressures as high as 170 atm. The volumetric efficiency of gear pumps is in the order of 96 percen t at pressures of about 40 atm but decreases a s the pressure rises.

**Internal-gear Pump**

The above figure shows the operation of a internal gear pump. In the internal-gear pump a spur gear, or pinion, meshes with a ring gear with internal teeth. Both gears are inside the casing. The ring gear is coaxial with the inside of the casing, but the pinion, which is externally driven, is mounted eccentrically with respect to the center of the casing. A stationary metal crescent fills the space between the two gears. Liquid is carried from inlet to discharge by both gears, in the spaces between the gear teeth and the crescent.

**Lobe pumps**

In principle the lobe pump is similar to the external gear pump; liquid flows into the region created as the counter-rotating lobes unmesh. Displacement volumes are formed between the surfaces of each lobe and the casing, and the liquid is displaced by meshing of the lobes. Relatively large displacement volumes enable large solids (nonabrasive) to be handled. They also tend to keep liquid velocities and shear low, making the pump type suitable for high viscosity, shear-sensitive liquids.

Two lobe pump Three lobe pump

The choice of two or three lobe rotors depends upon solids size, liquid viscosity, and tolerance of flow pulsation. Two lobe handles larger solids and high viscosity but pulsates more. Larger lobe pumps cost 4-5 times a centrifugal pump of equal flow and head.

**Selection of Pumps**

The following factors influence the choice of pump for a particular operation:

1. *The quantity of liquid to be handled*: This primarily affects the size of the pump and* *determines whether it is desirable to use a number of pumps in parallel.

2. *The head against which the liquid is to be pumped*. This will be determined by the* *difference in pressure, the vertical height of the downstream and upstream reservoirs and by the frictional losses which occur in the delivery line. The suitability of a centrifugal pump and the number of stages required will largely be determined by this factor.

3. *The nature of the liquid to be pumped*. For a given throughput, the viscosity largely* *determines the frictional losses and hence the power required. The corrosive nature will determine the material of construction both for the pump and the packing. With suspensions, the clearance in the pump must be large compared with the size of the particles.

4. *The nature of power supply*. If the pump is to be driven by an electric motor or* *internal combustion engine, a high-speed centrifugal or rotary pump will be preferred as it can be coupled directly to the motor.

5. *If the pump is used only intermittently, *corrosion troubles are more likely than with* *continuous working.

**Applications**

The handling of liquids which are particularly corrosive or contain abrasive solids in suspension, compressed air is used as the motive force instead of a mechanical pump.

**Working principle**

If the mechanical energy is converted into pressure energy by means of centrifugal force acting

on the fluid, the hydraulic machine is called centrifugal pump. The centrifugal pump acts as a reversed of an inward radial flow reaction turbine

**Performance Characteristics of Pumps**

The fluid quantities involved in all hydraulic machines are the flow rate (Q) and the head (H), whereas the mechanical quantities associated with the machine itself are the power (P), speed (N), size (D) and efficiency (h ). Although they are of equal importance, the emphasis placed on certain of these quantities is different for different pumps. The output of a pump running at a given speed is the flow rate delivered by it and the head developed. Thus, a plot of head and flow rate at a given speed forms the fundamental performance characteristic of a pump. In order to achieve this performance, a power input is required which involves efficiency of energy transfer. Thus, it is useful to plot also the power P and the efficiency h against Q.

Over all efficiency of a pump (h ) = Fluid power output / Power input to the shaft = rgHQ / P

Type number or Specific speed of pump, nS = NQ1/2 / (gH)3/4 (it is a dimensionless number)

**Centrifugal pump Performance**

In the volute of the centrifugal pump, the cross section of the liquid path is greater than in the impeller, and in an ideal frictionless pump the drop from the velocity V to the lower velocity is converted according to Bernoulli’s equation, to an increased pressure. This is the source of the discharge pressure of a centrifugal pump.

If the speed of the impeller is increased from N1 to N2 rpm, the flow rate will increase from Q1 to Q2 as per the given formula:

The head developed(H) will be proportional to the square of the quantity discharged, so that

The power consumed(W) will be the product of H and Q, and, therefore

These relationships, however, form only the roughest guide to the performance of centrifugal pumps.

**Characteristic curves**

Pump action and the performance of a pump are defined interms of their *characteristic curves*. These curves correlate the capacity of the pump in unit volume per* *unit time versus discharge or differential pressures. These curves usually supplied by pump manufacturers are for water only.

These curves usually shows the following relationships (for centrifugal pump).

· A plot of capacity versus differential head. The differential head is the difference in pressure between the suction and discharge.

· The pump efficiency as a percentage versus capacity.

· The break horsepower of the pump versus capacity.

The net poisitive head required by the pump versus capacity. The required NPSH for the pump is a characteristic determined by the manufacturer.

Centrifugal pumps are usually rated on the basis of head and capacity at the point of maximum efficiency.

**RECIPROCATING PUMPS**

**Working principle**

If the mechanical energy is converted into hydraulic energy (or pressure energy) by sucking the liquid into a cylinder in which a piston is reciprocating (moving backwards and forwards), which exerts the thrust on the liquid and increases its hydraulic energy (pressure energy), the pump is known as reciprocating pump

**Main ports of a reciprocating pump**

1.A cylinder with a piston, piston rod, connecting rod and a crank,

2. Suction pipe

3.Delivery pipe,

4. Suction valve and

5.Delivery valve.

**Slip of Reciprocating Pump**

Slip of a reciprocating pump is defined as the difference between the theoretical discharge and the actual discharge of the pump.

**Characteristic Curves Of Reciprocatring Pumps**

1.According to the water being on contact with one side or both sides of the piston

(i.) Single acting pump (ii.) Double-acting pump

2.According to the number of cylinders provided

(i.) Single acting pump (ii.) Double-acting pump (iii.) Triple-acting pump

**Reciprocating pumps Vs centrifugal pumps**

The advantages of reciprocating pumps in general over centrifugal pumps may be summarized as follows:

1. They can be designed for higher heads than centrifugal pumps.

2. They are not subject to air binding, and the suction may be under a pressure less than atmospheric without necessitating special devices for priming.

3. They are more flexible in operation than centrifugal pumps.

4. They operate at nearly constant efficiency over a wide range of flow rates.

The advantages of centrifugal pumps over reciprocating pumps are:

1. The simplest centrifugal pumps are cheaper than the simplest reciprocating pumps.

2. Centrifugal pumps deliver liquid at uniform pressure without shocks or pulsations.

3. They can be directly c onnected to motor derive without the use of gea rs or belts.

4. Valves in the discharg e line may be completely closed without injurin g them.

5. They can handle liquid s with large amounts of solids in suspension.

The rotary pump is g ood for handling viscous liquids, nut because of the close tolerances needed, it can not be manufactured large enough to compete with centrifugal pumps for coping with very high flow rates.

Rotary pumps are available in a variety of configurations.

• Double lobe pump

• Trible lobe pumps

• Gear pump

• Gear Pumps

• Spur Gear or Extern al-gear pump

External-gear pump (called as gear pump) consists essen tially of two intermeshing gears which are identical and which are surrounded by a closely fitting casing. One of the gears is driven directly by the prime mover while the other is allowed to rotate freel y. The fluid enters the spaces between the teeth and the casing and moves with the tee th along the outer periphery until it reaches the outlet where it is expelled from the pu mp.

External-gear p umps are used for flow rates up to about 400 m3/hr working against pressures as high as 170 atm. The volumetric efficiency of gear pumps is in the order of 96 percen t at pressures of about 40 atm but decreases a s the pressure rises.

**Internal-gear Pump**

The above figure shows the operation of a internal gear pump. In the internal-gear pump a spur gear, or pinion, meshes with a ring gear with internal teeth. Both gears are inside the casing. The ring gear is coaxial with the inside of the casing, but the pinion, which is externally driven, is mounted eccentrically with respect to the center of the casing. A stationary metal crescent fills the space between the two gears. Liquid is carried from inlet to discharge by both gears, in the spaces between the gear teeth and the crescent.

In principle the lobe pump is similar to the external gear pump; liquid flows into the region created as the counter-rotating lobes unmesh. Displacement volumes are formed between the surfaces of each lobe and the casing, and the liquid is displaced by meshing of the lobes. Relatively large displacement volumes enable large solids (nonabrasive) to be handled. They also tend to keep liquid velocities and shear low, making the pump type suitable for high viscosity, shear-sensitive liquids.

Two lobe pump Three lobe pump

The choice of two or three lobe rotors depends upon solids size, liquid viscosity, and tolerance of flow pulsation. Two lobe handles larger solids and high viscosity but pulsates more. Larger lobe pumps cost 4-5 times a centrifugal pump of equal flow and head.

**Internal-gear Pump**

The above figure shows the operation of a internal gear pump. In the internal-gear pump a spur gear, or pinion, meshes with a ring gear with internal teeth. Both gears are inside the casing. The ring gear is coaxial with the inside of the casing, but the pinion, which is externally driven, is mounted eccentrically with respect to the center of the casing. A stationary metal crescent fills the space between the two gears. Liquid is carried from inlet to discharge by both gears, in the spaces between the gear teeth and the crescent.

The following factors influence the choice of pump for a particular operation:

1. *The quantity of liquid to be handled*: This primarily affects the size of the pump and* *determines whether it is desirable to use a number of pumps in parallel.

2. *The head against which the liquid is to be pumped*. This will be determined by the* *difference in pressure, the vertical height of the downstream and upstream reservoirs and by the frictional losses which occur in the delivery line. The suitability of a centrifugal pump and the number of stages required will largely be determined by this factor.

3. *The nature of the liquid to be pumped*. For a given throughput, the viscosity largely* *determines the frictional losses and hence the power required. The corrosive nature will determine the material of construction both for the pump and the packing. With suspensions, the clearance in the pump must be large compared with the size of the particles.

4. *The nature of power supply*. If the pump is to be driven by an electric motor or* *internal combustion engine, a high-speed centrifugal or rotary pump will be preferred as it can be coupled directly to the motor.

5. *If the pump is used only intermittently, *corrosion troubles are more likely than with* *continuous working.

**Applications**

The handling of liquids which are particularly corrosive or contain abrasive solids in suspension, compressed air is used as the motive force instead of a mechanical pump.

**1. What is meant by Pump?**

A pump is device which converts mechanical energy into hydraulic energy.

**2. Define a centrifugal pump**

If the mechanical energy is converted into pressure energy by means of centrifugal force cutting on the fluid, the hydraulic machine is called centrifugal pump.

**3. Define suction head (hs).**

Suction head is the vertical height of the centre lines of the centrifugal pump above the water surface in the tank or pump from which water is to be lifted. This height is also called suction lift and is denoted by hs.

**4. Define delivery head (hd).**

The vertical distance between the center line of the pump and the water surface in the tank to which water is delivered is known as delivery head. This is denoted by hd.

**5. Define static head (Hs).**

The sum of suction head and delivery head is known as static head. This is represented by ‘Hs’ and is written as,

Hs = hs+ hd

**6. ****Mention main components of Centrifugal pump.**

i) Impeller ii) Casing

iii) Suction pipe,strainer & Foot valve iv) Delivery pipe & Delivery valve

**7. What is meant by Priming?**

The delivery valve is closed and the suction pipe, casing and portion of the delivery pipe upto delivery valve are completely filled with the liquid so that no air pocket is left. This is called as priming.

**8. Define Manometric head.**

It is the head against which a centrifugal pump work.

**9. Describe multistage pump with**

**a. impellers in parallel b. impellers in series. In multi stage centrifugal pump,**

a. when the impellers are connected in series ( or on the same shaft) high head can be developed.

b. When the impellers are in parallel (or pumps) large quantity of liquid can be discharged.

**10.. Define specific speed of a centrifugal pump (Ns).**

The specific speed of a centrifugal pump is defined as the speed of a geometrically circular pump which would deliver one cubic meter of liquid per second against a head of one meter. It is denoted by ‘Ns’.

**11. What do you understand by characteristic curves of the pump?**

Characteristic curves of centrifugal pumps are defined those curves which are plotted from the results of a number of tests on the centrifugal pump.

**12. Why are centrifugal pumps used sometimes in series and sometimes in parallel?**

The centrifugal pumps used sometimes in series because for high heads and in parallel for high discharge

**13.Define Mechanical efficiency.**

I

t is defined as the ratio of the power actually delivered by the impeller to the power supplied to the shaft.

**14. Define overall efficiency.**

It is the ratio of power output of the pump to the power input to the pump.

**15. Define speed ratio, flow ratio.**

Speed ratio: It is the ratio of peripheral speed at outlet to the theoretical velocity of jet corresponding to manometric head.

Flow ratio: It is the ratio of the velocity of flow at exit to the theoretical velocity of jet corresponding to manometric head.

**16.. ****Mention main components of Reciprocating pump.**

# Piton or Plunger

# Suction and delivery pipe

# Crank and Connecting rod

**17.. Define Slip of reciprocating pump. When the negative slip does occur?**

The difference between the theoretical discharge and actual discharge is called slip of the pump.

But in sometimes actual discharge may be higher then theoretical discharge, in such a case coefficient of discharge is greater then unity and the slip will be negative called as negative slip.

**18. What is indicator diagram?**

Indicator diagram is nothing but a graph plotted between the pressure head in the cylinder and the distance traveled by the piston from inner dead center for one complete revolution of the crank

**19. What is meant by Cavitations?**

It is defined phenomenon of formation of vapor bubbles of a flowing liquid in a region where the pressure of the liquid falls below its vapor pressure and the sudden collapsing of theses vapor bubbles in a region of high pressure.

**20. What are rotary pumps?**

Rotary pumps resemble like a centrifugal pumps in appearance. But the working method differs. Uniform discharge and positive displacement can be obtained by using these rotary pumps, It has the combined advantages of both centrifugal and reciprocating pumps.

**PRE REQUEST DISCUSSION**

Hydraulic Machines are defined as those machines which convert either hydraulic energy (energy possessed by water) into mechanical energy (which is further converted into electrical energy) or mechanical energy into hydraulic energy.

The hydraulic machines, which convert the hydraulic energy into mechanical energy, are called turbines.

Turbines are defined as the hydraulic machines which convert hydraulic energy into mechanical energy. This mechanical energy is used in running an electric generator which is directly coupled to the shaft of the turbine. Thus the mechanical energy is converted into electrical energy. The electric power which is obtained from the hydraulic energy (energy of water) is known as Hydro- electro power.

In our subject point of view, the following turbines are important and will be discussed one by one.

1. Pelton wheel

2. Francis turbine

3. Kaplan turbine

**Concept**

Turbines are defined as the hydraulic machines which convert hydraulic energy into mechanical energy. This mechanical energy is used in running an electric generator which is directly coupled to the shaft of the turbine

**FLUID** **TYPES OF TURBINE**

Water Hydraulic Turbine

Steam Steam Turbine

Froen Vapour Turbine

Gas or air Gas Turbine

Wind Wind Mills

**CLASSIFICATION OF HYDRAULIC TURBINES**

1. According to the action of the water flowing

2. According to the main direction of flow of water

3. According to the head and quality of water required

4. According to the specific speed

**HEAD AND EFFICIANCES OF PELTON WHEEL**

1. Gross head

2. Effective or Net head

3. Water and Bucket power

4. Hydraulic efficiency

5. Mechanical efficiency

6. Volume efficiency

7. Overall efficiency

**IMPULSE TURBINE**

In an impulse turbine, all the energy available by water is converted into kinetic energy by passing a nozzle. The high velocity jet coming out of the nozzle then impinges on a series of buckets fixed around the rim of a wheel.

**Tangential Flow Turbine, Radial And Axial Turbines**

**1. Tangential flow turbine**

In a tangential flow turbine, water flows along the tangent to the path of runner. E.g. Pelton wheel

**2. Radial flow turbine**

In a radial flow turbine, water flows along the radial direction and mainly in the plane normal to the axis of rotation, as it passes through the runner. It may be either inward radial flow type or outward radial flow type.

**3. Axial flow turbine**

In axial flow turbines, water flows parallel to the axis of the turbine shaft. E.g. kaplan turbine

**4. Mixed flow turbine**

In a mixed flow turbine, the water enters the blades radiallsy and comes out axially and parallel to the turbine shaft .E.g. Modern Francis turbine.

In our subject point of view, the following turbines are important and will be discussed one by one

1. Pelton wheel

2. Francis turbine

3. Kaplan turbine

**PELTON WHEEL OR PELTON TURBINE**

The Pelton wheel is a tangential flow impulse turbine and now in common use. Leston A Pelton, an American engineer during 1880,develops this turbines. A pelton wheel consists of following main parts.

1. Penstock

2. Spear and nozzle

3. Runner with buckets

4. Brake nozzle

5. Outer casing

6. Governing mechanism

**1 VELOCITY TRIANGLES, WORKDONE, EFFICIENCY OF PELTON**

**WHEEL INLET AND OUTLET VECTOR DIAGRAMS**

Let V = Velocity of the jet

u = Velocity of the vane (cups) at the impact point u

= DN/ 60

where D = Diameter of the wheel corresponding to the impact point = Pitch circle diameter.

At inlet the shape of the vane is such that the direction of motion of the jet and the vane is the same.

i.e., Ȑ= 0, ș= 0

Relative velocity at inlet

Vr = V —u

**Hydraulic efficiency**

**This is the ratio of the workk done per second per head at inlet to the turbine.**

**Energy head at inlet = V ^{2}/2g**

**Condition for maximum hydraulic efficiency**

**For a given jet velocity for efficiency to be maximum, word done should be maximum**

**Work done per second per N of water**

**Hence for the condition of maximum hydraulic efficiency, the peripheral speed of the turbine should reach one half the jet speed.**

**SPECIFIC SPEED**

**[ The speed of any water turbine is represented by N rpm. A turbine has speed, known as specific speed and is represented by N**

**‘ ****Specific speed of a water turbine in the speed at which a geometrically similar**** ****turbine would run if producing unit power (1 kW) and working under a net head of 1 m. Such a turbine would be an imaginary one and is called specific turbine.**

**FRANCIS TURBINE**

Francis turbine is an i nward flow reaction turbine. It is developed b y the American engineer James B. Francis. In the earlier stages, Francis turbine had a purely ra dial floe runner. But the modern Francis turbine is a mixed flow reaction turbine in which the water enters the runner radially at its outer peripher y and leaves axially at its centre. This arrangement provides larger discharge area with prescribed diameter of the runner. The main parts such as

1. Penstock

2. Scroll or Spiral Casing

3. Speed ring or Stay ring

4. Guide vanes or Wickets gat es

5. Runner and runner blades

6. Draft tube

**KAPLAN TURBINE**

A Kaplan turbine is an axial flow reaction turbine which was developed by Austrian engineer V. Kaplan. It is suitable for relatively low heads. Hence, it requires a large quantity of water to develop large power. The main parts of Kaplan turbine, they are

1. Scroll casing

2. Stay ring

3. Guide vanes

4. Runner

5. Draft tube

**PERFORMANCE OF TURBINES**

Turbines are often required to work under varying conditions of head, speed, output and gate opening. In order to predict their behavior, it is essential to study the performance of the turbines under the varying conditions. The concept of unit quantities and specific quantities are required to

The behavior of a turbine is predicted working under different conditions.

Comparison is made between the performance of turbine of same type but of different sizes.

The performance of turbine is compared with different types.

**DRAFT TUBE**

The pressure at the exit of the runner of a reaction turbine is generally less than atmospheric pressure. Thus the water at the exit of the runner cannot be directly discharged to the tail race. A pipe o gradually increasing area is used for discharging water form the exit of the turbine to the tail race. This pipe of gradually increasing area is called a draft tube.

**SPECIFIC SPEED**

Homologus units are required in governing dimensionless groups to use scaled models in designing turbomachines, based geometric similitude.

Specific speed is the speed of a geometrically similar turbine, which will develop unit power when working under a unit head. The specific speed is used in comparing the different types of turbines as every type of turbine has different specific speed. In S.I. units, unit power is taken as one Kw and unit as one meter.

**GOVERNING OF TURBINES**

All the modern hydraulic turbines are directly coupled to the electric generators. The generators are always required to run at constant speed irrespective of the variations in the load. It is usually done by regulating the quantity of water flowing through the runner in accordance with the variations in the load. Such an operation of regulation of speed of turbine runner is known as governing of turbine and is usually done automatically by means of a governor.

**Applications**

1. To produce the power by water.

**GLOSSARY**

HP –Horse power

KW- Kilo watts

**1. Define hydraulic machines.**

Hydraulic machines which convert the energy of flowing water into mechanical energy.

**2. ****Give example for a low head, medium head and high head turbine.**

Low head turbine – Kaplan turbine

Medium head turbine – Modern Francis turbine High head turbine – Pelton wheel

**3. ****What is impulse turbine? Give example.**

In impulse turbine all the energy converted into kinetic energy. From these the turbine will develop high kinetic energy power. This turbine is called impulse turbine. Example: Pelton turbine

**4. What is reaction turbine? Give example.**

In a reaction turbine, the runner utilizes both potential and kinetic energies. Here portion of potential energy is converted into kinetic energy before entering into the turbine.

Example: Francis and Kaplan turbine.

**5. What is axial flow turbine?**

In axial flow turbine water flows parallel to the axis of the turbine shaft. Example: Kaplan turbine

**6. What is mixed flow turbine?**

In mixed flow water enters the blades radially and comes out axially, parallel to the turbine shaft. Example: Modern Francis turbine.

**7. What is the function of spear and nozzle?**

The nozzle is used to convert whole hydraulic energy into kinetic energy. Thus the nozzle delivers high speed jet. To regulate the water flow through the nozzle and to obtain a good jet of water spear or nozzle is arranged.

**8. Define gross head and net or effective head.**

Gross Head: The gross head is the difference between the water level at the reservoir and the level at the tailstock.

Effective Head: The head available at the inlet of the turbine.

**9. Define hydraulic efficiency.**

It is defined as the ratio of power developed by the runner to the power supplied by the water

jet.

**10. Define mechanical efficiency.**

It is defined as the ratio of power available at the turbine shaft to the power developed by the turbine runner.

**11. Define volumetric efficiency.**

It is defied as the volume of water actually striking the buckets to the total water supplied by the jet.

**12. Define over all efficiency.**

It is defined as the ratio of power available at the turbine shaft to the power available from the water jet.

**13. Define the terms**

(a) Hydraulic machines (b) Turbines (c) Pumps. a. Hydraulic machines:

Hydraulic machines are defined as those machines which convert either hydraulic energy into mechanical energy or mechanical energy into hydraulic energy.

b. Turbines;

The hydraulic machines which convert hydraulic energy into mechanical energy are called turbines.

c. Pumps:

The hydraulic Machines which convert mechanical energy into hydraulic energy are called

pumps.

**14. What do you mean by gross head?**

The difference between the head race level and tail race level when no water is flowing is known as gross head. It is denoted by Hg.

**15. What do you mean by net head?**

Net head is also known as effective head and is defined as the head available at the inlet of te turbine. It is denoted as H

**16. What is draft tube? why it is used in reaction turbine?**

The pressure at exit of runner of a reaction turbine is generally less than the atmospheric pressure. The water at exit cannot be directly discharged to tail race. A tube or pipe of gradually increasing area is used for discharging water from exit of turbine to tail race. This tube of increasing area is called draft tube.

**17. What is the significance of specific speed?**

Specific speed plays an important role for selecting the type of turbine. Also the performance of turbine can be predicted by knowing the specific speed of turbine.

**18.. What are unit quantities?**

Unit quantities are the quantities which are obtained when the head on the turbine is unity. They are unit speed, unit power unit discharge.

**19. Why unit quantities are important**

If a turbine is working under different heads, the behavior of turbine can be easily known from the values of unit quantities.

**20. What do you understand by characteristic curves of turbine?**

Characteristic curves of a hydraulic turbine are the curves, with the help of which the exact behavior and performance of turbine under different working conditions can be known.

**21. Define the term ‘governing of turbine’.**

Governing of turbine is defined as the operation by which the speed of the turbine is kept constant under all conditions of working. It is done by oil pressure governor.

**22. What are the types of draft tubes?**

The following are the important types of draft tubes which are commonly used.

a. Conical draft tubes

b. Simple elbow tubes

c. Moody spreading tubes and

d.Elbow draft tubes with circular inlet and rectangular outlet.

**The General Heat Conduction Equation in Cartesian coordinates and Polar coordinates**

Any physical phenomenon is generally accompanied by a change in space and time of its physical properties. The heat transfer by conduction in solids can only take place when there is a variation of temperature, in both space and time. Let us consider a small volume of a solid element as shown in Fig. 1.2 The dimensions are x-, Y-, and Z- coordinates.

Fig 1.1 Elemental volume in Cartesian coordinates

First we consider heat conduction the X-direction. Let T denote the temperature at the point P (x, y, z) located at the geometric centre of the element. The temperature gradient at the left hand face (x – ~x12) and at the right hand face (x + x/2) , using the Taylor’s series, can be written as:

¶T / x¶|_{L} =T/¶x ¶^{2}T /–x¶^{2}. x / 2¶+ higherD order terms.

¶T / x¶|_{R} =T/¶x ¶^{2}T /+x¶^{2}. x / 2¶ higherD order+ terms.

The net rate at which heat is conducted out of the element 10 X-direction assuming k as

constant and neglecting the higher order terms,

Similarly for Y- and Z-direction,

If there is heat generation within the element as Q, per unit volume and the internal energy of

the element changes with time, by making an energy balance, we write

is called the thermal diffusivity and is seen to be a physical property of the material of which the solid is composed.

The Eq. (2.la) is the general heat conduction equation for an isotropic solid with a constant

Under steady state or stationary condition, the temperature of a body does not vary with time, i.e. ¶T / t¶ 0=.And, with no internal generation, the equation (2.1) reduces to

Del^{2}T =0

It should be noted that Fourier law can always be used to compute the rate of heat transfer by conduction from the knowledge of temperature distribution even for unsteady condition and with internal heat generation.

**ONE DIMENSIONAL STEADY STATE EQUATION PLANE WALL ****:**

The term ‘one-dimensional’ is applied to heat conduction problem when:

(i) Only one space coordinate is required to describe the temperature distribution within a heat conducting body;

(ii) Edge effects are neglected;

(iii) The flow of heat energy takes place along the coordinate measured normal to the surface.

A plane wall is considered to be made out of a constant thermal conductivity material and extends to infinity in the Y- and Z-direction. The wall is assumed to be homogeneous and isotropic, heat flow is one-dimensional, under steady state conditions and losing negligible energy through the edges of the wall under the above mentioned assumptions the Eq. (2.2) reduces to

d^{2}T / dx^{2} = 0; the boundary conditions are: at x = 0, T = T_{1}

Integrating the above equation, x = L, T = T_{2}

T = C_{1}x + C_{2}, where C_{1} and C_{2} are two constants.

Substituting the boundary conditions, we get C_{2} = T_{1} and C_{1} = (T_{2} –T_{1})/L The temperature

distribution in the plane wall is given by

T = T1 –(T1 –T2) x/L (2.3)

which is linear and is independent of the material.

small for the same heat flow rate,”

**A Cylindrical Shell-Expression for Temperature Distribution**

In the cylindrical system, when the temperature is a function of radial distance only and is independent of azimuth angle or axial distance, the differential equation (2.2) would be, (Fig. 1.4)

d^{2}T /dr^{2} +(1/r) dT/dr = 0

with boundary conditions: at r = r_{l}, T = T_{1} and at r = r_{2}, T = T_{2}.

The differential equation can be written as:

upon integration, T = C_{1} ln (r) + C_{2}, where C_{1} and C_{2} are the arbitrary constants.

From Eq (2.5) It can be seen that the temperature varies 10gantJunically through the cylinder wall In contrast with the linear variation in the plane wall .

where A_{2} and A_{1} are the outside and inside surface areas respectively. The term A_{m} is called

‘Logarithmic Mean Area’ and the expression for heat low through a cylidercal wall has the same

form as that for a plane wall.

**ONE DIMENSIONAL STEADY STATE HEATCONDUCTION COMPOSITE SYSTEMS:**

**Composite Surfaces**

There are many practical situations where different materials are placed m layers to form composite surfaces, such as the wall of a building, cylindrical pipes or spherical shells having different layers of insulation. Composite surfaces may involve any number of series and parallel thermal circuits.

**Heat Transfer Rate through a Composite Wall**

Let us consider a general case of a composite wall as shown m Fig. 1.5 The different materials of thicknesses L_{1}, L_{2}, etc and having thermal conductivities k_{l}, k_{2}, etc. On one side of

the composite wall, there is a fluid A at temperature T_{A} and on the other side of the wall there is a fluid B at temperature T_{B}. The convective heat transfer coefficients on the two sides of the wall are h_{A} and h_{B} respectively. The system is analogous to a series of resistances as shown in the figure.

Fig 1.4 Heat transfer through a composite wall

**The Equivalent Thermal Conductivity**

The process of heat transfer through compos lie and plane walls can be more conveniently compared by introducing the concept of ‘equivalent thermal conductivity’, k_{eq}. It is defined as:

And, its value depends on the thermal and physical properties and the thickness of each constituent of the composite structure.

**An Expression for the Heat Transfer Rate through a Composite Cylindrical System**

Let us consider a composite cylindrical system consisting of two coaxial cylinders, radii r_{1}, r_{2} and r_{2} and r_{3}, thermal conductivities k_{l} and k_{2} the convective heat transfer coefficients at the inside andoutside surfaces h_{1} and h_{2} as shown in the figure. Assuming radial conduction under steady state

conditions we have: Fig 1.5

R_{1} =1/ h_{1}A_{1} =1/ 2 _{1} pLh_{1}

R_{2} =ln (r_{2} / r_{1} )2 pLk_{1}

R_{3} =ln (r_{3} / r_{2} )2 pLk_{2}

**CONDUCTION WITH INTERNAL HEAT GENERATION:**

Applications: current carrying conductor, chemically reacting systems, nuclear reactors. Energy generated per unit volume is given by *V Eq*.

__Plane wall with heat source:__ Assumptions: 1D, steady state, constant k, uniform

Consider one-dimensional, steady-state conduction in a plane wall of constant k, with uniform generation, and asymmetric surface conditions: Heat diffusion equation.

A medium through which heat is conducted may involve the conversion ofmechanical, electrical, nuclear, or chemical energy into heat (or thermal energy).In heat conduction analysis, such conversion processes are characterizedas **heat generation**.

For example, the temperature of a resistance wire rises rapidly when electriccurrent passes through it as a result of the electrical energy being convertedto heat at a rate of *I*2*R*, where *I* is the current and *R* is the electricalresistance of the wire The safe and effective removal of this heataway from the sites of heat generation (the electronic circuits) is the subjectof *electronics cooling,* which is one of the modern application areas of heat transfer.

Likewise, a large amount of heat is generated in the fuel elements of nuclearreactors as a result of nuclear fission that serves as the *heat source* for the nuclearpower plants. The natural disintegration of radioactive elements in nuclearwaste or other radioactive material also results in the generation of heat throughout the body. The heat generated in the sun as a result of the fusion ofhydrogen into helium makes the sun a large nuclear reactor that supplies heatto the earth.

Another source of heat generation in a medium is exothermic chemical reactionsthat may occur throughout the medium. The chemical reaction in thiscase serves as a *heat source* for the medium. In the case of endothermic reactions,however, heat is absorbed instead of being released during reaction, and thus the chemical reaction serves as a *heat sink.* The heat generation term becomes

a negative quantity in this case.

Often it is also convenient to model the absorption of radiation such as solarenergy or gamma rays as heat generation when these rays penetrate deepinto the body while being absorbed gradually. For example, the absorption ofsolar energy in large bodies of water can be treated as heat generation throughout the water at a rate equal to the rate of absorption, which varies withdepth. But the absorption of solar energy by an opaque bodyoccurs within a few microns of the surface, and the solar energy that penetratesinto the medium in this case can be treated as specified heat flux on the surface.

Note that heat generation is a *volumetric phenomenon.* That is, it occursthroughout the body of a medium. Therefore, the rate of heat generation in amedium is usually specified *per unit volume.*

The rate of heat generation in a medium may vary with time as well as positionwithin the medium. When the variation of heat generation with positionis known, the *total* rate of heat generation in a medium of volume V can be determinedfrom In the special case of *uniform* heat generation, as in the case of electricresistance heating throughout a homogeneous material, the relation in

reduces to *E ·*gen _ *e ·* genV*,* where Egen is the constant rate of heat generation per unit volume.

Convection: Heat transfer between a solid surface and a moving fluid is governed by the Newton’s cooling- law: q = hA(Ts -T . Therefore, to increase the convective heat transfer, one can Increase the temperature difference (Ts – T ) between the surace and the fluid.

Increase the convection coefficient h. This can be accomplished by increasing the fluid flow over the surface since h is a function of the flow velocity and the higher the velocity, the higher the h. Example: a cooling fan.

Increase the contact surface area A. Example: a heat sink with fin. Ac : the fin cross-sectional area.

P: the fin perimeter.

Many times, when the first option is not in our control and the second option (i.e. increasing *h*) is already stretched to its limit, we are left with the only alternative of increasing the effective surface area by using fins or extended surfaces. Fins are protrusions from the base surface into the cooling fluid, so that the extra surface of the protrusions is also in contact with the fluid. Most of you have encountered cooling fins on air-cooled engines (motorcycles, portable generators, etc.), electronic equipment (CPUs), automobile radiators, air conditioning equipment (condensers) and elsewhere.

The fin efficiency is defined as the ratio of the energy transferred through a real fin to that transferred through an ideal fin. An ideal fin is thought to be one made of a perfect or infinite conductor material. A perfect conductor has an infinite thermal conductivity so that the entire fin is at the base material temperature.

**Transient State Systems-Defined**

The process of heat transfer by conduction where the temperature varies with time and with space coordinates, is called ‘unsteady or transient’. All transient state systems may be broadly classified into two categories:

(a) Non-periodic Heat Flow System – the temperature at any point within the system changes as a non-linear function of time.

(b) Periodic Heat Flow System – the temperature within the system undergoes periodic changes which may be regular or irregular but definitely cyclic.

There are numerous problems where changes in conditions result in transient temperature distributions and they are quite significant. Such conditions are encountered in – manufacture of ceramics, bricks, glass and heat flow to boiler tubes, metal forming, heat treatment, etc.

** Biot and Fourier Modulus-Definition and Significance**

Let us consider an initially heated long cylinder (L >> R) placed in a moving stream of fluid at

T_{¥} <T_{s} , as shown In Fig. 3.1(a). The convective heat transfer coefficient at the surface is h, where,

Q = hA ( T_{s} –T_{¥})

This energy must be conducted to the surface, and therefore,

Q = -kA(dT / dr) r = R

or, h( Ts – Tm ) = -k(dT/dr)r=R = -k(Tc-Ts)/R

where Tc is the temperature at the axis of the cylinder

By rearranging,(Ts – Tc) / ( Ts – Tm ) h/Rk (3.1)

The term, hR/k, IS called the ‘BlOT MODULUS’. It is a dimensionless number and is the ratio of internal heat flow resistance to external heat flow resistance and plays a fundamental role in transient conduction problems involving surface convection effects. I t provides a measure 0 f the temperature drop in the solid relative to the temperature difference between the surface and the fluid.

For Bi << 1, it is reasonable to assume a uniform temperature distribution across a solid at any time during a transient process.

Founer Modulus – It is also a dimensionless number and is defind as

Fo= at/L^{2} (3.2)

where L is the characteristic length of the body, a is the thermal diffusivity, and t is the time

The Fourier modulus measures the magnitude of the rate of conduction relative to the change in temperature, i.e., the unsteady effect. If Fo << 1, the change in temperature will be experienced by a region very close to the surface.

Fig. 1.7 Effect of Biot Modulus on steady state temperature distribution in a plane wall with surface convection.

Fig. 1.8 (a) Nomenclature for Biot Modulus

Lumped Capacity System-Necessary Physical Assumptions

We know that a temperature gradient must exist in a material if heat energy is to be conducted into or out of the body. When Bi < 0.1, it is assumed that the internal thermal resistance of the body is very small in comparison with the external resistance and the transfer of heat energy is primarily controlled by the convective heat transfer at the surface. That is, the temperature within the body is approximately uniform. This idealised assumption is possible, if

(a) the physical size of the body is very small,

(b) the thermal conductivity of the material is very large, and

(c) the convective heat transfer coefficient at the surface is very small and there is a large

temperature difference across the fluid layer at the interface.

**An Expression for Evaluating the Temperature Variation in a Solid Using Lumped**

**Capacity Analysis**

Let us consider a small metallic object which has been suddenly immersed in a fluid during a heat treatment operation. By applying the first law of

Heat flowing out of the body = Decrease in the internal thermal energy of

during a time dt the body during that time dt

or, hA_{s}( T –T_{¥} )dt = – pCVdT

where A_{s} is the surface area of the body, V is the volume of the body and C is the specific heat capacity.

or, (hA/ rCV)dt = – dT /( T –T_{¥})

with the initial condition being: at t = 0, T = T_{s}

The solution is : ( T –T_{¥})/( T_{s} –T_{¥}) = exp(-hA / rCV)t

Fig. depicts the cooling of a body (temperature distribution time) using lumped thermal capacity system. The temperature history is seen to be an exponential decay.

We can express

Bi × Fo = (hL/k)×( at/L^{2}) = (hL/k)(k/ rC)(t/L^{2}) = (hA/ rCV)t,

where V / A is the characteristic length L.

And, the solution describing the temperature variation of the object with respect to time is given

( T –T_{¥})/( T_{s} –T_{¥}) = exp(-Bi· Fo)

A semi-infinite solid is an idealized body that has a *single plane surface* and extends to infinity in all directions, as shown in .This idealized body is used to indicate that the temperature change in the part of the body in which we are interested (the region close to the surface) is due to the thermal conditions on a single surface. The earth, for example, can be considered to be a semi-infinite medium in determining the variation of temperature near its surface. Also, a thick wall can be modeled as a semi-infinite medium if all we are interested in is the variation of temperature in the region near one of the surfaces, and the other surface is too far to have any impact on the region of interest during the time of observation. The temperature in the core region of the wall remains unchanged in this case.

For short periods of time, most bodies can be modeled as semi-infinite solids since heat does not have sufficient time to penetrate deep into the body,and the thickness of the body does not enter into the heat transfer analysis. A steel piece of any shape, for example, can be treated as a semi-infinite solid when it is quenched rapidly to harden its surface. A body whose surface is heated by a laser pulse can be treated the same way.

Consider a semi-infinite solid with constant thermo physical properties ,no internal heat generation,uniform thermal conditions on its exposed surface, and initially a uniform temperature of *Ti* throughout. Heat transfer in this case occurs only in the direction normal to the surface (the *x* direction), and thus it is one-dimensional. Differential equations are independent of the boundary or initial conditions, and thus for one-dimensional transient conduction in Cartesian coordinates applies. The

depth of the solid is large (*x* → _) compared to the depthphenomenathat h can be expressed mathematically as a boundary condition as *T*(*x* → *, t*) _ *Ti*.

Heat conduction in a semi-infinite solid is governed by the thermal condition simposed on the exposed surface, and thus the solution depends strongly on the boundary condition at *x* _ 0. Below we present a detailed analytical solution for the case of constant temperature *Ts* on the surface, and give the results for other more complicated boundary conditions. When the surface temperature is changed to *Ts* at *t* _ 0 and held constant at that value at all times, the formulation of the problem The separation of variables technique does not work in this case since the medium is infinite. But another clever approach that converts the partial differential equation into an ordinary differential equation by combining the two independent variables *x* and *t* into a single variable h, called the **similarity variable**, works well. For transient conduction in a semi-infinite medium, it is defined as Similarity variable.

**USE OF HEISLER CHARTS :**

There are three charts, associated with different geometries. For a plate/wall (Cartesian geometry) the Heisler chart

The first chart is to determine the temperature at the center 0 *T* at a given time.

By having the temperatureat the center 0 *T* at a given time, the second chart is to determine the temperature at other locations at the same time in terms of 0 *T* .

The third chart is to determine the total amount of heat transfer up to the time.

1. **A composite wall consists of three layers of thicknesses 300 mm, 200mm and100mm with** **thermal conductivities 1.5, 3.5 and is W/m K respectively. The inside surface is exposed to gases at 1200°C with convection heat transfer coefficient as 30W/m ^{2}K. The temperature of air on the other side of the wall is 30°C with convective heat transfer coefficient 10 Wm^{2}K. If the temperature at the outside surface of the wall is 180°C, calculate the temperature at other surface of the wall, the rate of heat transfer and the overall heat transfer coefficient.**

**Solution: **The composite wall and its equivalent thermal circuits is shown in the figure.

**2.Derivethe General Heat Conduction Equation for an Isotropic Solid with Constant**

**Thermal Conductivity in Cartesian coordinates. **

Any physical phenomenon is generally accompanied by a change in space and time of its physical properties. The heat transfer by conduction in solids can only take place when there is a variation of temperature, in both space and time. Let us consider a small volume of a solid

It should be noted that Fourier law can always be used to compute the rate of heat transfer by conduction from the knowledge of temperature distribution even for unsteady condition and with internal heat generation.

**A 20 cm thick slab of aluminums (k = 230 W/mK) is placed in contact with a 15 cm thick** **stainless steel plate (k = 15 W/mK). Due to roughness, 40 percent of the area is in direct contact and the gap (0.0002 m) is filled with air (k = 0.032 W/mK). The difference in temperature between the two outside surfaces of the plate is 200°C Estimate (i) the heat flow rate, (ii) the contact resistance, and (iii) the drop in temperature at the interface.**

**Solution: **Let us assume that out of 40% area m direct contact, half the surface area is** **occupied by steel and half is occupied by aluminums.

The physical system and its analogous electric circuits is shown in Fig. 1.3.

**4. A steel pipe. Inside diameter 100 mm, outside diameter 120 mm (k 50 W/m K) IS Insulated with a40mm thick hightemperature Insulation(k = 0.09 W/m K) and another Insulation 60 mm thick (k = 0.07 W/m K). The ambient temperature IS 25°C. The heattransfer coefficient for the inside and outside surfaces are 550 and 15 W/m ^{2}K respectively. The pipe carries steam at 300^{o}C. Calculate (1) the rate of heat loss by steam per unit length of the pipe (11) the temperature of the outside surface .**

**Solution: A **cross-section of the pipe with two layers of insulation is shown Fig. 1.4with its** **analogous electrical circuit.

Fig1.4A crosssection through an insulated cylinder, thermal resistances in series.

For L = 1.0 m.

**5. Steel balls 10 mm in diameter (k = 48 W/mK), (C = 600 J/kgK) are cooled in air at temperature 35°C from an initial temperature of 750°C. Calculate the time required for the temperature to drop to 150°C when h = 25 W/m2K and density p = 7800 kg/m3. (AU 2012).**

**Solution: **Characteristic length, L = VIA = 4/3** **pr^{3}/4** **pr^{2}** **= r/3 = 5 × 10^{-3}/3m** **Bi = hL/k = 25 × 5 × 10-^{3}/ (3 × 48) = 8.68 × 10^{-4}<< 0.1,

Since the internal resistance is negligible, we make use of lumped capacity analysis: Eq. (3.4),

( T –T_{¥}) / ( T_{s} –T_{¥})=exp(-Bi Fo) ; (150 35) / (750 35) = 0.16084

\Bi × Fo = 1827; Fo = 1.827/ (8. 68 × 10^{-4}) 2.1× 10^{3}

Or, at/ L^{2} = k/ ( rCL^{2})t = 2100 and t = 568 = 0.158 hour

We can also compute the change in the internal energy of the object as:

= -7800 × 600 × (4/3) p(5 × 10^{-3})^{3} (750-35) (0.16084 – 1)

= 1.47 × 10^{3} J = 1.47 kJ.

If we allow the time’t’ to go to infinity, we would have a situation that corresponds to steady state in the new environment. The change in internal energy will be U0 – U_{¥} = [ rCV( T_{s} –T_{¥}) exp(-

¥)- 1] = [ rCV( T_{s} –T_{¥}].

We can also compute the instantaneous heal transfer rate at any time.

Or. Q = – rVCdT/dt = – rVCd/dt[ T_{¥}+ ( T_{s} –T_{¥} )exp(-hAt/ rCV) ]

= hA( T_{s} –T_{¥})[exp(-hAt/ rCV)) and for t = 60s,

Q = 25 × 4 × 3.142 (5 × 10^{-3})^{2}(750 35) [exp( -25 × 3 × 60/5 × 10^{-3} × 7800 × 600)] = 4.63 W.

**6.Aluminums fins 1.5 cm wide and 10 mm thick are placed on a 2.5 cm diameter tube to dissipate the heat. The tube surface temperature is 170**°**C ambient temperatures is 20**°**C. calculate the heat loss per fin. Take h = 130 W/m ^{2} C and K = 200 W/m^{2} C for aluminums.**

**Given**

Wide of the fin b = 1.5 cm = 1.5 ´10^{-2} m

Thickness t = 10 mm = 10 ´10^{-3} m

Diameter of the tube d = 2.5 cm = 2.5 ´10^{-2} m

Surface temperature T_{b} = 170°C + 273 = 443 K

Ambient temperature T¥ = 20°C + 273 = 293 K

Heat transfer co-efficient h = 130 W/m^{2}°C

Thermal conductivity K = 200 W/m°C

**Solution**

Assume fin end is insulated, so this is short fin end insulated type problem.

Heat transfer [short fin, end insulated]

Q = (hPKA)^{1/2} (T_{b} – T¥) tan h(mL) ……..(1) [FromNoHMT.41] data book

Where

A –Area = Breadth ´thickness

1. **Define Heat Transfer.**

Heat transfer can be defined as transmission of energy from one region to another region due to temperature difference.

2**. What are the modes of heat transfer?**

Conduction,

Convection,Radiation.

3.**State Fourier law of conduction.**

The rate of heat conduction is proportional to the area measured normal to the direction of heat flow and to the temperature gradient in that direction.

Q = –kAdT / dx

**4. Define Thermal Conductivity.**

Thermal conductivity is defined as the ability of a substance to conduct heat.

**5. Write down the three dimensional heat conduction equations in cylindrical coordinates.**

The general three dimensional heat conduction equation in cylindrical coordinates

6. **List down the three types of boundary conditions.**

1. Prescribed temperature.

2. Prescribed heat flux.

3. Convection boundary conditions.

7. **State Newtons law of cooling.**

Heat transfer by convection is given by Newtons law of cooling

Q= h A ( T_{s} -T_{inf} )

Where A- Area exposed to heat transfer in

h- Heat transfer coefficient in W/ K

T- Temperature of the surface and fluid in K.

8**. What is meant by lumped heat analysis?**

In a Newtonian heating or cooling process the temperature throughout the solid is considered to be uniform at a given time. Such an analysis is called lumped heat capacity analysis.

9.**Define fin efficiency and fin effectiveness.**

The efficiency of a fin is defined as the ratio of actual heat transfer to the maximum possible heat transferred by the fin.

Ƞfin = Q fin / Q max.

Fin effectiveness is the ratio of heat transfer with fin to that without fin.

Fin effectiveness = Q with fin/ Q without fin.

10. **What is critical radius ofinsulation?( AU 2010)**

Addition of insulating material on a surface does not reduce the amount of heat transfer rate always .in fact under certain circumstances it actually increases the heat loss up to certain thickness of insulation. The radius of insulation for which the heat transfer is maximum is called critical radius of insulation.

11.**Write the** **Poisson’sequationfor heat conduction.**

When the temperature is not varying with respect to time, then the conduction is called as steady state condition.

**12.What is heat generation in solids? Give examples.**

The rate of energy generation per unit volume is known as internal heat generation in solids.

Examples: 1.Electric coils 2. Resistanceheater3. Nuclearreactor.

13. **A 3 mm wire of thermal conductivity 19 W / m K at a steady heat generation of 500**

Solution;

Critical / Centre temperature,

T _{c}= T_{inf} + qr^{2}/4K

= 298 + 500×X(0.015)^{ 2}

=298+14.8

Tc = 312.8 K.

**14. What are the factors affecting the thermal conductivity?**

Moisture,

Density of material,

Pressure,

Temperatures.

15. **What are Heislercharts ?**

In Heisler chart, the solutions for temperature distributions and heat flow in plane walls, long cylinders and spheres with finite internal and surface resistance are presented. Heisler chart nothing but a analytical solution in the form of graphs.