FLUID DYNAMICS

In the order to describe the motion of a fluid, in principle, one might apply Newton’s laws to a particle (a small volume element of fluid) and follow its progress in time. This is a difficult approach. Instead, we consider the properties of the fluid, such as velocity, pressure, at fixed points in space. In order to simplify the discussion we take several assumptions:

(i) The fluid is non viscous        (ii) the flow is steady

(iii) The flow is non rotational   (IV) The fluid is incompressible.

42.  Viscosity:

It is an opposing force which comes into play when a layer of liquid slide over another layer of liquid.

Some phenomenon associated with viscosity.

     During swimming we experience some resistance that is due to viscosity.

     Water moves faster as compared to honey through a hole because viscosity of water is smaller than viscosity of honey.

     We can walk faster in air as compared to water due to viscosity.

Coefficient of Viscosity:

Suppose two layers of a liquid  and  moving with velocities  and  and  are the heights of the layer from the ground.

Then according to Newton the viscous force between two layers depends upon area of layers and velocity gradient as  

Combining eq. (i) and (ii)     

Or  

Where  is called coefficient of viscosity. Here  sign indicate that viscous opposes the change in motion.

If   and      

Then  

Hence coefficient of viscosity may be defined as the viscous force acting on unit area of the layer having unit velocity gradient perpendicular to direction of flow of the liquid.

            In   

             In SI unit of Nsm^-2 Or   decapoise

            1decapoise = 

            (1decapoise =1 Pascal second ())

Question22: A metal plate 1 m² in area rests on a layer of castor oil ( = 1 decapoise) 0.1 cm thick. Calculate the horizontal force required to move the plate with a speed of   3 cm/s. 

Solution:  where  = 15.5 poise A = 100 cm²            

  30s^–1 

  F = 1  1  30 = 30    

so force required = 30 N 

Question 23: A plate of area 2 m² moves horizontally with a speed of 2 m/s by applying a horizontal tangential force over the free surface of a liquid, the depth of the liquid is 1 m and the liquid in contact with the bed is stationary. Coefficient of viscosity of liquid is 0.01 poise. Find the tangential force needed to move the plate. 

Sol: Apply the Newton’s formula for the frictional force between two layers of a liquid.                     

Velocity gradient 

AS, |

So, to keep the plate moving, a force of  must be applied. 

43.  Effect of Temperature and Pressure on the Viscosity:

• The viscosity of a liquid decrease with increase in temperature and increase with decrease in temperature i.e.   
• The Viscosity of gases increase with increase in temperature and decrease with decrease in temperature i.e.    
• Except water, viscosity of liquid increase with increase in pressure and decrease with decrease in pressure.
• Viscosity of gases is independent of pressure.

44. Some Applications of Viscosity:

• Since viscosity of a liquid depends on temperature, so we can use proper lubricant in different reason (hot or cold).
• Liquid of high viscosity are used as shock absorber and buffers of train.
• Viscosity is also helpful in determining the molecular weight and shapes of organic molecules.
• It finds an important role in blood circulation in arteries and veins of human body.

45. Similarities and Differences between Viscosity and Solid Friction:

Similarities:       (1)   Both oppose the relative motions.

                            (2)   Both come into play when one layer move over the second layer.

                            (3)    Both are due to intermolecular interactions.

Dissimilarities:

• Viscosity depends upon the area of the layers while friction does not depend upon the area contact.
• Viscosity is independent on normal reaction between two layers but friction is directly proportional to normal reaction between two layers.
• Viscosity depends between relative velocities of two layers while friction does not depend upon the relative velocities of layers.

46. Poiseuille’s Formula:

According to Poiseuille’s Formula, the volume of a liquid flowing per second through a horizontal capillary tube of length, radius  under a pressure  applied across its end is given by

As    

This relation is called Poiseuille’s Formula.

Derivation of Poiseuille’s Formula on the Basis of Dimensional Analysis:

The volume of liquid flowing through capillary tube per second depends upon 

(i) Coefficient of Viscosity of the Liquid  

 (ii) Radius of the Tube   

(iii) Pressure Gradient set up along the capillary tube.        

 i.e. 

Where dimension formula of various quantities are   

Comparing the power of M, L, T on both sides we get    

On solving we get     

⟹   

Where K is constant of proportionality and its value is found  

so     

Which is the required expression for Poiseuille’s Formula?

47. Stokes Law:

According to stokes law, the force acting on a small sphere of radius  moving with uniform velocity  through fluid of viscosity  is given by      

Derivation of Stokes Law on the basis of dimensional Analysis:

It is found that viscous force on a body moving in liquid depends upon 

1.                Coefficient of viscosity  of the liquid.
2.             Radius  of the spherical body.
3.           Velocity  of the body

So let   

The above eq. may be written dimension as      

Now comparing the coefficient of M,L,T on both sides, we get

 On solving   

 ⟹  

Where K is constant of proportionality and its value is found   

So      

Which is the required stokes law.

Validity of Stokes Law:

     The fluid should have infinite extension and stream line.

     The body should not slip on liquid and perfectly rigid smooth.

     The size of the body should be small.

Question 24:  An air bubble of diameter 2 mm rises steadily through a solution of density  at the rate of.Calculate the coefficient of viscosity of the solution. The density of air is negligible.  

Sol: As the air bubble rises with constant velocity, the net force on it is zero. The force of buoyancy B is equal to the weight of the displaced liquid.

Thus

This force is upward. The viscous force acting downward is

.

 For uniform velocity

Question25: An air bubble of diameter 2 cm rises through a long cylindrical column of a viscous liquid, and

Travels at 0.21 cms^1. If the density of the liquid is 189 kg m^–3, find its coefficient of viscosity. 

Solution:  Weight of the bubble is equal to the viscous force.

Or     …(i)

Given: r = 10^2 m          v = 0.21  10^2 m/s g = 10 m/s² 

Substituting these values in (i) we have,  

  20 kgm^1 s^1

48. Terminal Velocity:

When a body falls in liquid then against the motion of body a viscous force and a upward thrust acts on the body. At this situation the maximum constant velocity acquired by the body while falling into the viscous medium is called terminal velocity. 

Suppose a spherical body of mass m, density  and radius  falls into a viscous fluid of density  and coefficient of viscosity. Then various forces acting on the body are:

Upward force T and Viscous force F. Downward forces W. Now net down ward force acting on the body is 

Or 

Or 

Or 

Or 

This is the required expression for terminal velocity.

Clearly if body is denser then liquid then body will move into the liquid and if density of body is smaller than density of liquid then body will float. 

Question 26:  Two spherical raindrops of equal size are falling vertically through air with a terminal velocity of   1 m/s. What would be the terminal speed if these two drops were to coalesce to form a large spherical drop? 

Sol: Use the formula for terminal velocity for spherical body.

 .

 Let r be the radius of small rain drops and R the radius of large drop.

Equating the volume, we have

Or 

49.Streamline, Laminar and Turbulent Flow 

(1) Stream line flow:  

 Stream line flow of a liquid is that flow in which each element of the liquid passing through a point travels along the same path and with the same velocity as the preceding element passes through that point. 

 Path ABC is streamline as shown in the figure and ,  and  are the velocity of the liquid particle at A, B and C point respectively. 

(2)  Laminar flow: 

 If a liquid is flowing over a horizontal surface with a steady flow and moves in the form of layers of different velocities which do not mix with each other, then the flow of liquid is called laminar flow. In this flow, the velocity of liquid flow is always less than the critical velocity of the liquid. The laminar flow is generally used synonymously with streamlined flow.   

Turbulent flow:

When a liquid moves with a velocity greater than its critical velocity, the motion of the particles of liquid becomes disordered or irregular. Such a flow is called a turbulent flow. In a turbulent flow, the path and the velocity of the particles of the liquid change continuously. 

50 . Critical Velocity and Reynolds’s Number 

The critical velocity is that velocity of liquid flow up to which its flow is streamlined and above which its flow becomes turbulent. 

Reynolds’s number is a pure number which determines the nature of flow of liquid through a pipe.  

It is defined as the ratio of the inertial force per unit area to the viscous force per unit area for a flowing fluid.  As            

If the value of Reynolds number 

    Lies between 0 to 2000, the flow of liquid is streamline or laminar. 

    Lies between 2000 to 3000 the flow of liquid is unstable and changing from streamline to turbulent flow.  

    Above 3000, the flow of liquid is definitely turbulent

51 .Equation of Continuity

It states that, the volume of liquid flowing per unit time is constant through different cross-sections of the container of the liquid. Thus, if  and  are velocities of fluid at respective points A and B of areas of cross-sections  and  and and  be the radius respectively. 

Then the equation of continuity is given by

   … (i) 

If the same liquid is flowing, then  

then the equation (i) can be written                

    .. (ii)

  ⇒       

⇒

• Equation of continuity represents the law of conservation of mass of moving fluids.

  (General equation of continuity)

• This equation is applicable to actual liquids or to other fluids which are not incompressible. 

Question 27: Water is flowing through a horizontal tube of non-uniform cross-section. At a place, the radius of the tube is 1.0 cm and the velocity of water is 2 m/s. What will be the velocity of water, where the radius of the pipe is 2.0 cm?  

Sol: Using equation of continuity,  

⇒ 

Question 28:  Figure shows a liquid being pushed out of a tube by pressing a piston. The area of cross-section of the piston is 1.0 cm² and that of the tube at the outlet is 20 mm². If the piston is pushed at a speed of 2 cm-s^-1, what is the speed of the outgoing liquid? 

­Sol: From the equation of continuity =10cm/s 

52. DIFFERENT FORM OF ENERGY IN FLUID FLOW

1. Pressure Energy

 If P is the pressure on the area A of a fluid, and the liquid moves through a distance due to this pressure, 

Then     

The volume of the liquid is Al. Pressure energy per unit volume of liquid

2.  Kinetic Energy

If a liquid of mass m and volume V is flowing with velocity v,

 then the kinetic energy is  

Kinetic energy per unit volume of liquid

 = =  

Here, ρ is the density of liquid.

3.  Potential energy

 If a liquid of mass m is at a height h from the reference line (h = 0), then its potential energy is mgh. 

∴ Potential energy per unit volume of the liquid

 = 

53 . BERNOULLI’S THEOREM

When a non-viscous and an incompressible fluid flows in a streamlined motion from one place to another in a container, then the total energy (pressure + kinetic + potential) of the fluid per unit volume is constant at every point of its path. 

Total energy = pressure energy + Kinetic energy + Potential energy 

=  

Where P is pressure, V is volume, M is mass and h is height from a reference level.

The total energy per unit volume 

= 

Where ρ is density, Thus if a liquid of density ρ, pressure  at a height  which flows with velocity  to another point in streamline motion where the liquid has pressure , at height  which flows with velocity ,

 Then   

Derivations:- 

To prove it, consider a liquid flowing steadily through a tube of non-uniform area of cross-section as shown in fig. If andare the pressures at the two ends of the tube respectively, work done in pushing  the volume V of incompressible fluid from point A to B through the tube will be

This work is used by the fluid in two ways. 

(a) In changing the potential energy of mass m (in the volume V) from  to   

i.e.    …… (ii)

(b) In changing the kinetic energy from

  to  ,  

i.e.,  K =

    (iii) 

Now as the fluid is non-viscous, by conservation of mechanical energy

  = 

i.e.,     +

Or    +

Or    +

Or   

Or 

This equation is the so called Bernoulli’s equation and represents conservation of mechanical energy in case of moving fluids. 

54 .Applications of Bernoulli’s Theorem:- 

1. Attraction between two closely parallel moving boats (or buses):

When two boats or buses move in the same direction, the water (or air) in the region between them moves faster than that on the remote sides. The pressure between them is reduced and hence due to pressure difference they are pulled towards each other creating the so called attraction. 

2. Working of an aeroplane:-

The wings of the aeroplane are of specific shape when the aeroplane runs, air passes at higher speed over it as compared to its lower surface. This difference of air speeds above and below the wings creates a pressure difference, due to which an upward force called ‘dynamic lift’ acts on the plane. If this force becomes greater than the weight of the plane, the plane will rise up.

3. Action of atomizer :- 

The action of carburetor, paint-gun, scent-spray or insect-sprayer is based on Bernoulli’s principle. In all these, by means of motion of a piston P in a cylinder C, high speed air is passed over a tube T dipped in liquid L to be sprayed. High speed air creates low pressure over the tube due to which liquid (paint, scent, insecticide or petrol) rises in it and is then blown off in very small droplets with expelled air.

4. Blowing off roofs by wind storms:-   

During a tornado or hurricane, when a high speed wind blows over a straw or tin roof, it creates a low pressure (P) in accordance with Bernoulli’s principle. However, the pressure below the roof (i.e., inside the room) is still atmospheric. So due to this difference of pressure, the roof is lifted up and is then blown off by the wind.

5. Heart attack 

due to access colestrol veins becomes narrows . presser outside the veins become more and blockage of veins occurs due to which heart stop working.  

6. Venturimeter

A  Venturimeter used to measure flow speed in a pipe of non-uniform cross-section. We apply Bernoulli’s equation to the wide (point 1) and narrow (point 2) parts of the pipe, with                     

From Bernoulli theorem 

From the continuity equation

Substituting and rearranging, we get

The pressure difference is also equal to ρgh, where h is the difference in liquid level in the two tubes. 

Substituting in equation (i), we get

Question 29: Calculate the rate of flow of glycerin of density  through the conical section of a pipe, if the radii of its ends are 0.1 m and 0.04 m and the pressure drop across its length is 

Sol: Apply the equation of continuity. Where area of cross-section is larger, the velocity of fluid is lesser and vice-versa. From continuity equation,

From Bernoulli’s equation, 

.  .. (ii) 

Solving equations (i) and (ii), we get 

 Rate of volume flow through the tube

Question 30: Figure shows a liquid of density 1200 kg m^–3 flowing steadily in a tube of varying cross section. The cross section at a point A is 1.0 cm² and that at B is 20 mm², the points A and B are in the same horizontal plane. The speed of the liquid at A is 10 cm s^-1. Calculate the difference in pressure at A and B.  

Sol: From equation of continuity. The speed  at B is given by,

  Or      

Or

By Bernoulli equation, 

Here.   

Thus     

55 .TORRICELLI’S THEOREM or Velocity of Efflux 

  If a liquid is filled in a vessel up to height H and a hole is made at a depth h below the free surface of the liquid as shown in fig. then taking the level of hole as reference level 

(i.e., zero point of potential energy) and applying Bernoulli’s principle to the liquid just inside and outside the hole (assuming the liquid to be at rest inside)

 We get  

⇒

This is same speed as that an object would acquire in falling from rest through a distance h and called velocity of efflux or velocity of flow

This result was first given by Torricelli, so this is known as Torricelli’s theorem. 

Range up to which liquid will fall = 

Or  

Now,  

⇒ 

Or   

⇒

• Range is max. If    

So   

⇒      

I.e. Range would be the same when the hole is at a height h or at a height H – h from the ground or from the top of the beaker.

R is maximum at   and  

    The velocity of efflux is independent of the nature of liquid, quantity of liquid in the vessel and the area of orifice. 

    Greater is the distance of the hole from the free surface of liquid, greater will be the velocity of efflux      [i.e. 

15. CALORIMETRY AND THERMAL EXPANSION

57 .  DEFINITION OF HEAT

Heat is energy in transient. Heat energy flows from one body to another body due to their temperature difference. It is measured in units of calories. 

The SI unit is Joule.   1 calorie = 4.2J

Illustration 26: What is the difference between heat and temperature?   

Sol:

• Temperature is associated with kinetic energy of atoms/molecule while heat is energy in transit.  
•  Temperature is the measure of the average kinetic energy of the molecules or atoms in a substance. Heat is the flow of energy from one body to another as a result of a temperature difference. 
•  It is important to point out that matter does not contain heat; it contains molecular kinetic energy and not heat. Heat flows and it is the energy that is being transferred. Once heat has been transferred to an object, it ceases to be heat. It becomes internal energy. 

58 .PRINCIPLE OF CALORIMETRY 

When two bodies at different temperatures are mixed, heat will pass from the body at a higher temperature to the body at a lower temperature until the temperature of the mixture becomes constant.

 The principle of calorimetry implies that heat lost by the body at a higher temperature is equal to the heat gained by the other body at a lower temperature assuming that there is no loss of heat in the surroundings.

59. HEAT CAPACITY 

The heat capacity of a body is defined as the amount of heat required to raise its temperature by 1°C. It is also known as the thermal capacity of the body. 

Suppose a body has mass m and specific heat c. Heat capacity = Heat required to raise the temperature of the body by 1°C = mc × 1 =mc

∴    Heat capacity =mc

Hence heat capacity of a body (solid or liquid) is equal to the product of its mass and specific heat. Clearly, the SI unit of heat capacity is J/°C or J/K. 

    The greater the mass of a body, the greater is its heat capacity. 

60. SPECIFIC HEAT CAPACITY 

When we supply heat to a solid substance (or liquid) its temperature increases.  It is found that the amount of heat Q absorbed by the solid substance (or liquid), is

(i) Directly proportional to the mass (m) of the substance

 i.e.,  Q ∝ m 

(ii) Directly proportional to the rise in temperature (∆T) 

i.e.    Q∝ ∆T  

Combining the two factors, 

we have,      ……. (i) 

Or     Q = cm ∆T 

Where c is constant of proportionality and is called specific heat capacity or simply specific heat of the substance. From eq. (i), 

we have 

If m = 1 kg and ∆T = 1°C, then c = Q.

Hence the specific heat of a solid (or liquid) may be defined as the amount of heat required to raise the temperature of 1kg of solid (or liquid) through 1°C (or 1K). SI unit of specific heat is J kg ^-1 °C^-1 or J kg ^-1 K^-1

• Specific heat capacity is the property of material and heat capacity is property of a given body. 

61 .DEFINITION OF CALORIE

The amount of heat needed to increase the temperature of 1 g of water from 14.5°C to 15.5°C at a pressure of   1 atm is called 1 calorie.       

1 kilo calorie =103 calories; 1 calorie = 4.186 Joule

    Thermal capacity of a body is the quantity of heat required to raise its temperature through 1°C and is equal to the product of mass and specific heat of the body.

    (Be careful about unit of temperature, use units according to the given units of s)

Question 31: A geyser heats water flowing at the rate of 3.0 liters per minute from 27°C to 77°C. If the geyser operates on a gas burner, what is the rate of combustion of the fuel if its heat of combustion is  J g^-1? 

Sol: The total heat required to increase the temperature of the water is equal to the heat supplied by the combustion of gas per minute. Mass of 3 liters of water =3kg 

∴Mass of water flowing per minute,     

Rise of temperature, 

Heat absorbed by water per minute= mc ∆ θ =3000×1× 50cal 

= 3000× 1× 50× 4.2  = 630000  

∴ Heat supply by gas burner= 630000 and heat of combustion of fuel = 4.0× 104 

 ∴ Rate of combustion of fuel = 

62. MOLAR SPECIFIC HEAT CAPACITY (C) FOR SOLIDS OR LIQUIDS

The molar specific heat of a solid (or liquid) is defined as the amount of heat required to raise the temperature of 1 mole of the solid (or liquid) through 1°C (or 1K). It is denoted by the symbol C.        

Therefore, the amount of heat Q required to raise the temperature of n moles of a solid (or liquid) through a temperature change ∆T   is given by;

Q =nC ∆T

 It is clear that SI unit of C is J mol^-1 K^-1. 

For any material of mass m and molecular mass M,

 the number of moles  

Eq. (i) gives the relation between molar specific heat C and the ordinary specific heat.

63. MOLAR SPECIFIC HEAT CAPACITY FOR THE GASES 

The amount of heat required to increase the temperature of 1 mole of a gas through 1°C is called molar heat capacity. 

The number of moles, n, in mass m of the gas is given by

1 Molar Specific Heat at Constant Volume, 

:If  is the heat required to raise the temperature of mass m gm or n moles of gas of molecular weight M at constant volume through temperature 

  ∆ T, 

Where   molar specific is heat at constant volume and is equal to   .

2 Molar specific heat at Constant Pressure, Cp:

If  is the heat required to raise the temperature of mass m gm or n moles of gas of molecular weight M at constant pressure through temperature

∆ T,  

Where   molar specific is heat at constant volume and is equal to  .

For monatomic gases,

  and 

For diatomic gas                

Mayer’s relation gives,      

Illustration 32 : How much heat is required to raise the temperature of an ideal mono atomic gas by 10 K if the gas is maintained at constant pressure?  

 Sol: The process is at constant pressure here. Formula for heat capacity of gas at constant pressure is used. 

The heat required is given by                          

Here        n=1 & ∆T  =10 K;   

 So  .  

64. LATENT HEAT

The amount of heat required to change a unit mass of a substance completely from one state to another at constant temperature is called the latent heat of the substance. 

If a substance of mass m required heat Q to change completely from one state to another at constant temperature, 

Then the   latent heat

The SI unit of latent heat of a substance is J kg^-1

. There are two types of latent heats: 

1.      Latent heat of fusion:.

The amount of heat required to change the unit mass of solid mass into its liquid state at constant temperature is called the latent heat of fusion of the solid.

• For example, the latent heat of fusion of ice is 334 J/kg.

 It means to change 1 kg of ice at 0°C into liquid water at 0°C, we must supply 334 KJ of heat. 

2.      Latent heat of vaporization:.

The amount of heat required to change the unit mass of a liquid into its gaseous state at constant temperature is called latent heat of vaporization of the liquid.

Question 33: A piece of ice of mass 100 g and at temperature 0°C is put in 200 g of water at 25°C. How much ice will melt as the temperature of the water reaches 0°C? The specific heat capacity of water =4200 JK ^-1 and the specific latent heat of fusion of ice= 3.4× 10⁵ JK ^-1

Sol:  Total heat lost by the water equal to the total heat gained by the ice.

The heat released as the water cools down from 25°C to 0°C is

The amount of ice melted by this much heat is given by

65. WATER EQUIVALENT:- 

The water equivalent of a body is defined as the mass of water that will absorb or lose the same amount of heat as the body for the same rise or fall in temperature. 

The water equivalent of a body is measured in kg in SI unit and in g in C.G.S. units. 

Consider a body of mass m and specific heat required to raise the temperature of the body through ∆T is  

 Q= cm ∆T   ……………… (i) 

Suppose w is water equivalent of this body. Then, by definition, 

From eqn. (i) And (ii), 

we have,  

⇒   

Thus the water equivalent of a body is numerically equal to the product of the mass of the body and its specific heat

    Water equivalent and heat capacity of a body are numerically equal.

Question 34: The water equivalent of a body is 10 kg. What does it mean?   

Sol: It means that if a body is heated through say 5°C, it will absorb the same amount of heat as absorbed by 10 kg of water when heated through 5°C.

66. MECHANICAL EQUIVALENT OF HEAT:-

If mechanical work W produces the same temperature change as heat H, 

We write, W=JH. 

Here J is called mechanical equivalent of heat. It is clear that if W and H are both measured in the same unit then J=1.

 If W is measured in joule (work done by a force of 1 N in displacing an object by 1 m in its direction) and H in calorie (heat required to raise the temperature of 1 g of water by 1°C) then J is expressed in joule per calorie. 

The value of J gives how many joules of mechanical work is needed to raise the temperature of 1 g of water by 1°C.

67. TEMPERATURE SCALES 

1 Kelvin Temperature Scale:- 

Kelvin is a temperature scale designed such that zero K is defined as absolute zero and the size of one unit is the same as the size of one degree Celsius. 

Water freezes at 273.15K; water boils at 373. 15K. for calculation purposes, 

we take 

• Absolute  zero : 

At absolute zero, a hypothetical temperature, all molecular movement stops

        All actual temperatures are above absolute zero.

2 Celsius Temperature Scale:-

Temperature Scale according to which the temperature difference between the reference temperature of the freezing and boiling of water is divided into 100 degrees. 

The freezing point is taken as zero degrees Celsius and the boiling point as 100 degrees Celsius. 

• The Celsius scale is widely known as the centigrade scale because it is divided into 100 degrees.

3 Fahrenheit Scale:-

Fahrenheit temperature scale is a scale based on 32 for the freezing point of water and 212 for the boiling point of water, the interval between the two being divided into 180 parts. 

For easy conversion of temperature units, remember the following equation

Here C, F and K are respectively temperature in Celsius, Fahrenheit and Kelvin scale. .

Question 35:  Express a temperature of 60° F in degree Celsius and in Kelvin.   

Question 36: Calculate the temperature which has the same value on (i) the Celsius and Fahrenheit (ii) Fahrenheit and Kelvin scales

Sol: (i)   Let the required temperature be x°, 

Now 

Or  

∴

⇒    

             Let the required temperature be x° from relation

On solving by taking 

We get,  X = 574.6

68.Triple Point of Water

The triple point of water is that unique temperature and pressure at which water can coexist in equilibrium between the solid, liquid and gaseous states. 

The pressure at the triple point of water is 4.58 mm of Hg and the temperature is 273.16K (or 0.01°C). 

THERMAL EXPANSION

69. DEFINITION OF THERMAL EXPANSION

It is the expansion due to increase in temperature. Most substances expand when they are heated. Thermal expansion is a result of the change in average separation between the atoms of an object.

 Atoms of an object can be imagined to be connected to one another by springs as shown in figure… As the temperature of the solid increases, the atoms oscillate with greater amplitudes, as a result the average separation between them increases, and the object expands, 

Thermal expansion arises from the nature of the potential energy curve. 

70. THERMAL EXPANSION OF SOLIDS

1 Linear Expansion

When a solid substance is heated, most of them generally expand. If a solid has a length and has a very small area of cross-section, at a temperature, its length increases to  when its temperature is increased by ∆T.

 The increase in length,   , 

Also     

So     

Where α is the coefficient of linear expansion which is given by

The coefficient of linear expansion is equal to the increase in length per unit length per degree rise of temperature.

 The SI unit of α is /°C or /K. 

    Its value is different for different solid materials.

 2 Superficial Expansion or Surface Expansion:-

If a solid plate of area  and of very small thickness is heated through a temperature ∆T so that its area increases to, then the increase in area ∆A is given by

Or  

Here β is called the coefficient of superficial expansion.  2 β= α

Hence the coefficient of superficial expansion of a solid may be defined as the fractional change in surface area     per degree change in temperature. 

Its SI unit is also / ⁰C or /K.

3 Volume Expansion:-

If a solid of initial volume  at any temperature is heated so that its volume is increased to with increase of temperature ∆T, 

The increase in volume, ∆V, is given by  

Or

Here γ is called the coefficient of volume or cubical expansion. γ = 3 α. 

As the temperature of solid increases, the oscillation of atoms increases which results in an increase of average distance between atoms due to which the volume increases.

• If  is the density of a solid at 0°C and  is its density, then for a constant mass m of the solid,  

     and      

       Where  and  are its respective volume at 0°C & T°.

       So    

       ⟹

Hence coefficient of cubical expansion of a solid may be defined as the fractional change in volume   per degree change in temperature. 

Its SI unit is / °C or /K. 

71 .RELATION BETWEEN COEFFICIENTS OF EXPANSION

We shall now show that for solid, the approximate relations between α, β and γ are: β= 2α, γ= 3α; 

1.        Relation between β and α. 

Consider a square plate of side  at 0°C   and  at   t°C  

As length     

Area of plate at t°C, 

Also Area of plate at t°C, 

⇒

Or    

Since the value of α is small, the term  may be neglected

So   

Or   

. ∴      β= 2 α  

    The result is altogether general because any flat surface can be regarded as a collection of small squares. 

          Relation between γ and α.

 Consider a cube of side    At 0°C and  at t°C 

As length     

Volume of cube at 0⁰C,   

Volume of cube at t °C,    

  Also Volume of cube at t °C, 

⇒      

Or  

Since the value of α is small, we can neglect the higher power of α. 

As     

∴     3 α t = γ t 

Or   γ =3α

Hence    

    Again, result is general because any solid can be regarded as a collection of small cubes.

72 . VARIATION OF DENSITY WITH TEMPERATURE 

As we know     (for a given mass)

So we may write  

∴   

Or     

By Binomial theorem as γ is small. 

 So 

• Most substances expand when they are heated,

 i.e.. Volume of a given mass of a substance increases on heating, so the density should decrease

Question 37:  A glass flask of volume 200 cm³ is just filled with mercury at 20° C. How much mercury will over flow when the temperature of the system is raised to 100°C? The coefficient of volume expansion of glass is 1.2×10^-5/°C and that of mercury is 18× 10^-5/°C.  

Sol:  The increase in the volume of the flask is 

The increase in the volume of the mercury is

∴ The volume of the mercury that will overflow

Question 38: A sheet of brass is 40 cm long and 8 cm broad at 0 °C. If the surface area at 100°C is 320.1 cm2, find the coefficient of linear expansion of brass.  

Sol: Surface area of sheet at 0°C,

Surface area of sheet at 100°C,

Rise in temperature

Increase in surface is

Coefficient of surface expansion β is given by

Coefficient of linear expansion,

73, THERMAL STRESS

If the ends of rods of length  are rigidly fixed and it is heated, its length  tends to increase due to increase in temperature ∆T , but it is prevented from expansion. It results in setting up compressive or tensile stress in the rod which is called the thermal stress.

 As              

74. PRACTICAL APPLICATION OF THERMAL EXPANSION OF SOLIDS

There are a large number of important practical applications of thermal expansion of solids. However, we shall brief only a few of them by way of illustration.

• While laying the railway tracks, a small gap is left between the successive lengths of the rails.  This gap is provided to allow for the expansion of the rails during summer. If no gap is left, these expansions cause the rails to buckle (tight).
• When the iron tyre of a wheel to be put on the wheel, the tyre is made slightly smaller in diameter than that of wheel. The iron tyre is first heated uniformly till its diameter becomes more than that of the wheel and is then slipped over the wheel. On cooling the tyre contracts and makes a tight fit on the wheel.
•  In bridges, one end is rigidly fastened to its abutment while the other rests on rollers. This provision allows the expansion and contraction to take place during changes in temperature.
• The concrete roads and floor are always made in sections and enough space is provided between the sections. This provision allows expansion and contraction to take place due to change in temperature.
• Effect of temperature on the time period of a pendulum:  As the temperature is increased length of the pendulum and hence, time period gets increased or a pendulum clock becomes slow and it’s loses the time. Similarly, if the temperature is decreased the length and hence, the time period gets decreased. A pendulum clock on this case runs fast and it gains the time. 

  • If a solid object has a hole in it, what happens to size of the hole, when the temperature of the object increases? 

A common misconception is that if the object expands, the hole will shrink because material expands into the hole. But the, truth is that if the Object expands the hole will expand too, because every linear dimension of an object change in the same way when the temperature changes. 

75. THERMAL EXPANSION OF LIQUIDS 

As a liquid in a vessel acquires the shape of the vessel, its heating increases the volume of the vessel initially due to expansion of the vessel which decreases the level of the liquid initially. When the temperature of the liquid is increased further, it increased the volume of the liquid. 

                                           Thus the apparent expansion of the liquid is lesser than the real expansion of the liquid which gives a value of coefficient of real expansion more than that for the coefficient of apparent expansion. 

The coefficient of real expansion, of a liquid is defined as the real increase in volume per degree rise of temperature per unit original volume of the liquid. , 

  Of a liquid is defined as the ratio of the observed increase in volume of the liquid with respect to the original level before heating per degree rise of temperature to the original volume of the liquid. 

So  

It is clear that   and both are measured unit °C^-1

. It can be shown that:  

Where is the coefficient of cubical expansion of glass (or material of the container). 

HEAT TRANSFER

76. INTRODUCTION

Heat can be transformed from one place to another place by the three processes – conduction, convection and radiation.

• In conduction, the heat flows from a place of higher temperature to a place of lower temperature through a stationary medium. The molecules of the medium oscillate about their equilibrium positions more violently at a place of higher temperature and collide with the molecules of adjacent position,. Thus heat can be transmitted by collision of molecules. 
• In the cases of liquids and gases, the heat is transferred not only by collision but also by motion of heated molecules which carry the heat in such media. This process is called convection. 
• Radiation is mode of transfer of heat in which the heat travels directly from one place to another without the role of any intervening medium. The heat from the sun propagates mostly through vacuum to reach the earth by the process of radiation.

77. CONDUCTION

In conduction, the heat flows from a place of higher temperature to a place of lower temperature through a stationary medium. 

The quantity of heat conducted Q in time t across a slab of length L, area of cross-section A and steady state temperature  and  at respective hot and cold ends is given by

,  …….(1)

Here k is the coefficient of thermal conductivity 

Coefficient of thermal conductivity is equal to the quantity of heat flowing per unit time through unit area of cross-section of a material per unit length along the direction of flow of heat.

Units of k are kilocalorie/meter second degree centigrade or J.m^-1sec^-1 K^-1. 

In C.G.S. units of k is expressed in calcm^-1 (℃)^-1 sec^– 

∴ From(1)  

Or

The quantity  is called the temperature gradient. 

The minus sign indicates that  is negative along the direction of the heat flow, i.e., heat flows from a higher temperature to a lower one.

Here ∆T = temperature difference (TD) and  Thermal resistance of the rod. 

This relation is mathematically equivalent to Ohm’s Law and can be used very effectively in solving problems effectively by considering temperature analogous to potential and heat transferred per unit time as current. Heat flow through a conducting rod Current flow through a resistance 

78. NEWTON’S LAW OF COOLING

The rate of cooling of a body is directly proportional to the difference of temperature of the body over its surroundings. If a body at temperature  is placed in surroundings at lower temperature,

The rate of cooling is given by

Where  is the quantity of heat lost in time dt. 

Newton’s law of cooling gives           

Where k is constant

If a body of mass m and specific heat s loses a temperature  in time, 

Then   

    Newton’s law of cooling can also be thought in the context of Stefan-Boltzmann law by considering the temperature difference between the body and the surroundings very close to zero, i.e. it can be considered as a special case of the latter. 

79. GROWTH OF ICE ON PONDS

When temperature of the atmosphere falls below 0°C, the water in the pond starts freezing. Let at time t thickness of ice in the pond is y and atmospheric temperature is -T°C. The temperature of water in contact with the lower surface of ice will be 0°C. 

 Using 

Or    

Where L = Latent heat of fusion

And hence time taken by ice to grow a thickness

  y 

• Time does not depend on the area of pond. Time taken by ice to grow on ponds is independent of area of the pond and it is only dependent only the thickness of ice sheet. 

80. CONVECTION

In this process, actual motion of heated material results in transfer of heat from one place to another. 

 For example, 

• In a hot air blower, air is heated by a heating element and is blown by a fan. The air carries the heat wherever it goes. 
• When water is kept in a vessel and heated on a stove, the water at the bottom gets heated due to conduction through the vessel’s bottom. Its density decreases and consequently it rises. Thus, the heat is carried from bottom to the top by the actual movement of the parts of the water. 

    If the heated material is forced to move, say by a blower or by a pump, the process of heat transfer is called forced convection.

    If the material moves due to difference in density, it is called natural or free convection. 

81. RADIATION

All objects radiate energy continuously in the form of electromagnetic waves, the transfer of energy in form of radiation which does not require a medium. 

     The best known example of this process is the radiation from Sun. 

Stefan’s law

The rate at which an object radiates energy is proportional to the fourth power of its absolute temperature. This is known as the Stefan’s law .

As    

e lies between 0 and 1 and is called emissivity of the object and σ is universal constant called Stefan’s constant, which has the value, 

82. PERFECTLY BLACK BODY

A body that absorbs the entire radiation incident upon it and has as emissivity equal to 1 is called a perfectly black body. A black body is also an ideal radiator. It implies that if a black body and an identical another body is kept at the same temperature, then the black body will radiate maximum power as is obvious from equation

This is also because e=1 for a perfectly black body while for any other body, e<1. 

• Always remember that black body is a perfect absorber and emitter of light. At temperatures higher than the surrounding, it is the most shining thing and at lower temperatures it is the darkest thing. 
• There is no perfect black body. Materials like black velvet or lamp black come close to being ideal black bodies, but the best practical realization of an ideal black body is a small hole leading into a cavity, as this absorbs 98% of the radiation incident on them. 

83. ABSORPTIVE POWER ‘a’

 “It is defined as the ratio of the radiant energy absorbed by a body in a given time to the total radiant energy incident on it in the same interval of time.”  

As a perfectly black body absorbs all radiations incident on it, the absorptive power of perfectly black body is maximum and unity. 

84. SPECTRAL ABSORPTIVE ‘’

This absorptive power ‘a’ refers to radiations of all wavelengths (or the total energy) while the spectral absorptive power is the ratio of radiant energy absorbed by a surface to the radiant energy incident on it for a particular wavelength λ. It may have different values for different wavelengths for a given surface. 

85. EMISSIVE POWER ‘e’

 “For a given surface it is defined as the radiant energy emitted per second per unit area of the surface.”

 It has the units of   ,

 for a black body 

Note: Absorptive power is dimensionless quantity where emissive power is not.

    Don’t confuse it with the emissivity e which is different from it, although both have the same symbols e.

86.  SPECTRAL EMISSIVE POWER 

 Similar to the definition of the spectral absorptive power, it is emissive power for a particular wavelength λ. 

Thus,   

87 . KIRCHHOFF’S LAW 

The ratio of emissive power to absorptive power is the same for all bodies at a given temperature and is equal to the emissive power E of a blackbody at that temperature.

Thus

    Kirchhoff’s law tells that if a body has high emissive power, it should also have high absorptive power to have the ratio e/a same. Similarly, a body having low emissive power should have low absorptive power. 

88. STEFANS-BOLTZMANN LAW

The energy of thermal radiation emitted per unit time by a blackbody of surface area A is given by

Here is a universal constant known as Stefan Boltzmann constant. 

The measured value of σ is 5.67×10^-8 Wm^-2 K^-4. 

89. WIEN’S DISPLACEMENT LAW

At ordinary temperatures (below about 600℃), the thermal radiation emitted by bodies is invisible, most of them lie in wavelengths longer than visible light.. As the temperature of the black body increases, two different behaviors are observed. The first effect is that the peak of the distribution shifts to shorter wavelengths. This shift is found to satisfy the following relationship called Wien’s displacement law. 

 .

 Here b is a constant called Wien’s constant. 

The value of this constant in SI unit is  . 

Question39:  The light from the sun is found to have a maximum intensity near the wavelength of 470 nm. Assuming that the surface of the sun emits as a blackbody, calculate the temperature of the surface of the sun. 

Sol: For a blackbody   

Thus, 

90. SOLAR CONSTANT AND TEMPERATURE OF SUN

Solar constant is defined as the amount of radiation received from the sun at the earth per minute per  of a surface placed at right angle to the solar radiation at a mean distance of the earth from the sun.

 Assuming that the absorption of solar radiation by the atmosphere near the earth is negligible,

 The value of solar constant, S, is equal to