Heat Transfer and Calorimetry IIT JEE Study Material

 

Heat is a form of energy that is transferred between a system and its surrounding as a result of temperature difference. The expansion due to increase in the temperature is known as thermal expansion. There are three types of thermal expansion, namely Linear, Superficial, and Volume expansion.

Types of Thermal Expansion:

1. Linear Expansion:

When there is any change in the length of a body due to heating then the expansion is called longitudinal or linear expansion.

Coefficient of Expansion:\(\mathbf{\Delta l}\) = Change in length and \(\mathbf{\Delta t}\) = Change in Temperature]

\(\mathbf{\alpha \;=\;\lim_{\Delta t\; \rightarrow \;0}\;\;\frac{1}{l_{0}}\;\;\frac{\Delta l}{\Delta t}\;\;and\;\;\Delta l\;=\;l_{0}\;\alpha \;\Delta t}\)

2. Superficial Expansion:

When there is any change in the area of a body due to heating then the expansion is called axial or superficial expansion.

Coefficient of Expansion: [\(\mathbf{\Delta A}\) = Change in Area and \(\mathbf{\Delta t}\) = Change in Temperature]

\(\mathbf{\beta \;=\;\lim_{\Delta t\; \rightarrow \;0}\;\;\frac{1}{A_{0}}\;\;\frac{\Delta A}{\Delta t}\;\;and\;\;\Delta A\;=\;A_{0}\;\beta \;\Delta t}\)

3. Volumetric Expansion:

When there is any change in the volume of a body due to heating then the expansion is called as volumetric or cubic expansion.

Coefficient of Expansion: [\(\mathbf{\Delta V}\) = Change in volume and \(\mathbf{\Delta t}\) = Change in Temperature]

\(\mathbf{\gamma \;=\;\lim_{\Delta t\; \rightarrow \;0}\;\;\frac{1}{V_{0}}\;\;\frac{\Delta V}{\Delta t}\;\;and\;\;\Delta V\;=\;V_{0}\;\gamma\;\Delta t}\)

If \(\mathbf{\alpha _{1},\;\alpha _{2},\;and\;\alpha _{3}}\) are coefficient of linear expansion in X, Y  and Z directions, then,

Case 1: For Isotropic Solids

\(\mathbf{\alpha \;=\;\alpha _{1}\;=\;\alpha _{2}\;=\;\alpha _{3}}\), \(\mathbf{\beta \;=\;2\;\alpha }\), and \(\mathbf{\gamma \;=\;3\;\alpha }\)

Case 2: For Anisotropic Solids

\(\mathbf{\beta\;=\;\alpha_{1}\;+\;\alpha _{2}\;\;and \;\;\gamma \;=\;\alpha_{1}\;+\;\alpha_{2}\;+\;\alpha_{3} }\)

Thermal Stress:

If temperature of a rod of length l0­ clamped between two fixed walls separated by same distance l0 is changed by amount \(\mathbf{\Delta t }\) then,

Case 1: \(\mathbf{\alpha}\) is constant

Stress = \(\mathbf{\frac{F}{A} }\) and Strain = \(\mathbf{\frac{\Delta l}{l_{0}} }\)

Therefore, Young’s Modulus =  \(\mathbf{\frac{\frac{F}{A}}{\frac{\Delta l}{l_{0}}}\;=\;\frac{F\;l_{0}}{A\;\Delta l}\;=\;\frac{F}{A\;\alpha \;\Delta t}}\)

i.e. \(\mathbf{F\;=\;Y\;A\;\alpha \;\Delta t}\)

Case 2: \(\mathbf{\alpha}\) is not constant

A. If \(\mathbf{\alpha}\) varies with distance [\(\mathbf{\alpha}\) = ax + b]

Total thermal expansion = \(\mathbf{\int_{0}^{1}\;\;\left ( ax\;+\;b \right )\;dx\;\Delta t}\)

B. If \(\mathbf{\alpha}\) varies with temperature [\(\mathbf{\alpha}\) = f (T)]

\(\mathbf{\Delta l\;=\;\int_{T_{1}}^{T_{2}}\;\alpha \;l_{0}\;dT}\)

Variation in Density: Density decreases with increase of temperature because of increase in volume and vice-versa i.e.

Density d = \(\mathbf{\frac{d_{0}}{1\;+\;\gamma \;\Delta t}}\)

Special Case: The Density water is maximum at 4 °C. Its density increases from 0 °C to 4 °C (g is positive). From 4 °C to higher temperatures g is positive.

Modes of Heat Transfer:

While conduction is the transfer of heat energy by direct contact, convection is the movement of heat by actual motion of matter; radiation is the transfer of energy with the help of electromagnetic waves.

Heat Transfer

Conduction

Convection

Radiation

Definition

Heat is transferred between objects by direct contact

Energy transition (heat transfer) occurs within the fluid

Heat is transmission is done without any physical contact between objects

Representation

How heat travels between objects in direct contact

How heat passes through fluids

How heat flows through empty spaces

Occurrence

Occurs in solids through molecular collisions.

Occurs in fluids by actual flow of matter.

Occurs at a distance and does not heat the intervening substance.

Causes

temperature difference

density difference

Occurs from all objects, at temperature greater than 0 Kelvin

Speed

Slow

Slow

Fast

Law of reflection and refraction

Does not follow

Does not follow

Follow

Transfer of heat

Uses heated solid substance

Uses intermediate substance

Uses electromagnetic waves

Differential form of Ohm’s law:

\(\mathbf{\frac{dQ}{dT}\;=\;K\;A\;\frac{dT}{dx}}\)

Where,

\(\mathbf{\frac{dT}{dx}}\) = Temperature Gradient

Stefan-Boltzmann law:

It states that the net radiant heat energy emitted from an object is proportional to the 4th power of its absolute temperature.

\(\mathbf{E\;=\;\sigma \;A\;T^{4}}\;\;J\; sec^{-1} \;m^{-2}\) 

Radiation Power:

\(\mathbf{\frac{dQ}{dT}\;=\;\sigma \;A\;T^{4}\;\;watt}\)

If Ts is the surrounding temperature:   [Black Body]

\(\mathbf{\frac{dQ}{dT}\;=\;\sigma \;A\;(T^{4}\;-\;T_{s}^{4})}\)

Emissive Power or Emissivity e = \(\mathbf{\frac{Heat\;from\;given \;body}{Heat\;from\;black\;body}}\)

Newton’s Law of cooling:

It states that the rate of change of the temperature (T) of an object is proportional to the difference between its own temperature and the temperature of its surroundings.

T(t) = \(\mathbf{T_s}\) + (\(\mathbf{T_0}\)\(\mathbf{T_s}\)) e – k t

Where,

\(\mathbf{T_s}\) = Surrounding Temperature

\(\mathbf{T_0}\) = Initial temperature of the body

T (t) = Temperature of body at time t

k = Cooling constant

Wien’s black body radiation:

At any temperature T greater than 0 Kelvin the body emits energy radiations of all wavelengths. According to the Wien’s displacement law, if the wavelength \(\mathbf{(\lambda) }\) corresponding to the maximum energy is \(\mathbf{\lambda_{max} }\) then,

\(\mathbf{\lambda_{max}\;T\;=\; b}\)

Where,

T = Temperature of body

b = Wien’s Constant

Calorimetry

It is the measurement of changes in the state variables of an object for the purpose of deriving the transfer of heat associated with changes in its state either due to phase transitions, physical changes, or chemical reactions, under specified constraints. A calorimeter is a device using which Calorimetry is performed.

The amount of heat required to raise the temperature of 1 gram of water from 14.5° C to 15.5°C at Standard Temperature and Pressure (STP) is 1 calorie.

\(\mathbf{Q\;=\;m\;\int_{T_{1}}^{T_{2}}\;\;C\;dt\;=\;m\;C\;\Delta T}\)

Heat transfer in phase change:

Q = mL

Where,

L = latent heat of substance in \(\mathbf{Cal \;gm ^{-1}\; °C^{-1}\; \;or\;in\;\; Kcal \;kg^{-1}\; °C^{-1}}\)

\(\mathbf{L_{steam}}\) = 540 cal/ gm

\(\mathbf{L_{ice}}\) = 80 cal/ gm for ice


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Unidirectional transmission of a nerve impulse through nerve fibre is due to the fact that