20Transmission of Heat

📚
TRANSMISSION OF HEAT
▢ Modes of Heat Transfer:

Table 1: Three Modes of Heat Transmission

Mode
Medium required
Main occurrence
Speed
Conduction
Yes
Mostly solids
Slowest
Convection
Yes
Liquids and gases
Faster than conduction
Radiation
No
Vacuum / any transparent medium
Fastest; speed of light
▢ Conduction:
❖ Definition: Transfer of heat through a substance without actual transfer of mass
❖ Key Points:
  • •
    Usually takes place in solids
  • •
    Slowest process of heat transmission
  • •
    Path followed is irregular
  • •
    Only energy is transferred
  • •
    Neither mass nor momentum is transferred
  • •
    Possible in liquids and gases if heated from top
❖ Heat Flow Formula:

Table 1: Heat Flow in Conduction

Quantity
Formula / Meaning
Rate of heat flow
\(H=\frac{Q}{t}\)
Thermal resistance form
\(H=\frac{\Delta\theta}{R}\)
Using thermal resistivity
\(H=\frac{\theta_1-\theta_2}{\rho l/A}\)
Using thermal conductivity
\(H=\frac{KA(\theta_1-\theta_2)}{l}\)
Thermal resistance
\(R=\frac{l}{KA}=\frac{\rho l}{A}\)
Thermal resistivity
\(\rho=\frac{1}{K}\)
◉ Symbols:
  • •
    \(H\) = rate of heat flow
  • •
    \(Q\) = heat transferred
  • •
    \(t\) = time
  • •
    \(\theta_1\) = higher temperature
  • •
    \(\theta_2\) = lower temperature
  • •
    \(K\) = thermal conductivity
  • •
    \(\rho\) = thermal resistivity
  • •
    \(l\) = length
  • •
    \(A\) = cross-sectional area
❖ Thermal Conductivity:

Table 1: Thermal Conductivity

Point
Value / Description
Unit
\(W\ m^{-1}K^{-1}\)
Dimension
\([MLT^{-3}K^{-1}]\)
Good conductor
\(K\) high
Bad conductor
\(K\) nearly zero
Perfect heat conductor
\(K=\infty\)
Perfect heat insulator
\(K=0\)
Solids vs liquids vs gases
Solids > Liquids > Gases
◉ Examples:
    ■ **type: bullet
  1. Woolen clothes keep body warm because wool is bad conductor of heat
  2. Cloudy nights are warmer because clouds reduce heat loss
  3. Metal feels colder than non-metal in cold morning because metal has high thermal conductivity
  4. Metals are good conductors because they contain large number of free electrons
❖ Wiedemann-Franz Law:
◉ Statement: Ratio of thermal conductivity to electrical conductivity is directly proportional to absolute temperature
◉ Formula: \(\frac{K}{\sigma}\propto T\)
◉ Constant Form: \(\frac{K}{\sigma T}=constant\)
◉ At Constant Temperature: \(K\propto\sigma\)
◉ Conclusion:
    ■ **type: bullet
  1. Good conductor of heat is usually good conductor of electricity
  2. Bad conductor of heat is usually bad conductor of electricity
  3. Example of good conductor: Silver
  4. Example of bad conductor: Porcelain
❖ Joining of Rods:
◉ **table:
    ◉ Series Junction Temperature:
    ◈ General: \(T=\frac{\frac{K_1T_1}{l_1}+\frac{K_2T_2}{l_2}}{\frac{K_1}{l_1}+\frac{K_2}{l_2}}\)
    ◈ If Equal Lengths: \(T=\frac{K_1T_1+K_2T_2}{K_1+K_2}\)
    ❖ Steady State:
    ◉ Definition: State in which temperature of each cross-section of rod becomes steady/constant with time
    ◉ Variable State: Initially, when one end of metallic rod is heated, temperature of different points changes continuously
    ◉ Points:
    • •
      Different cross-sections may have different constant temperatures
    • •
      At steady state, temperature does not depend on thermal capacity
    • •
      Heat flow across body depends on thermal conductivity
    • •
      No heat is absorbed by cross-section at steady state
    • •
      Heat received by a cross-section is conducted to next section and partly radiated
    ❖ Searle's Method: In Searle's method, temperature gradient along the bar is same at all points along the bar
    ▢ Convection:
    ❖ Definition: Transfer of heat by actual motion of fluid particles
    ❖ Key Points:
    • •
      Takes place in liquids and gases
    • •
      Path followed is irregular
    • •
      Faster than conduction
    • •
      Mass transfer occurs

    Table 1: Natural vs Forced Convection

    Feature
    Natural convection
    Forced convection
    Rate of heat flow
    \(\frac{d\theta}{dt}=hA(\Delta T)^{5/4}\)
    \(\frac{d\theta}{dt}=hA\Delta T\)
    Depends on
    Density difference of medium
    External force / fan / pump
    Gravity-free space
    Does not occur
    Can occur
    Direction
    Bottom to top
    Any direction
    Examples
    Ventilation, land breeze, sea breeze, boiling water
    Fan, cooler, AC
    ❖ Convection Coefficient:
    ◉ Symbol: \(h\)
    ◉ Depends On:
    • •
      Nature of medium
    • •
      Specific heat
    • •
      Thermal properties
    • •
      Density
    • •
      Viscosity
    ▢ Radiation:
    ❖ Definition: Transfer of heat in the form of electromagnetic waves without material medium
    ❖ Key Points:
    • •
      No medium required
    • •
      Fastest mode of heat transfer
    • •
      Travels with speed of light
    • •
      Can be minimized but never completely stopped
    • •
      Occurs mainly by infra-red rays
    • •
      Does not change temperature of medium directly
    ❖ Detection:
    • •
      Thermocouple
    • •
      Thermopile
    • •
      Radiometer
    • •
      Pyrometer
    • •
      Bolometer
    ❖ Properties of Thermal Radiation:
    • •
      Invisible
    • •
      Travels in straight line
    • •
      Casts shadow
    • •
      Affects photographic plates
    • •
      Can be reflected by mirrors
    • •
      Can be refracted by lenses
    ▢ Reflection, Absorption and Transmission:
    ❖ Statement: When radiation falls on a body, part is reflected, part transmitted and part absorbed
    ❖ Formula: \(Q=R+A+T\)
    ❖ Fraction Form: \(1=r+a+t\)

    Table 1: Radiation Fractions

    Symbol
    Meaning
    \(Q\)
    Total incident radiant energy
    \(R\)
    Reflected energy
    \(A\)
    Absorbed energy
    \(T\)
    Transmitted energy
    \(r\)
    Reflectance
    \(a\)
    Absorptance / absorptive power
    \(t\)
    Transmittance
    ❖ Surface Properties:

    Table 1: Absorber and Reflector

    Surface
    Absorption
    Reflection
    White clothes
    Bad absorber
    Good reflector
    Polished black surface
    Bad absorber
    Good reflector
    Rough black surface
    Good absorber
    Bad reflector
    Smooth shining surface
    Bad absorber
    Good reflector
    Rough surface
    Good absorber
    Bad reflector
    ◉ General Rule: Good absorber is bad reflector and good reflector is bad absorber
    ▢ Black Body:
    ❖ Definition: A body that absorbs all radiations incident on it

    Table 1: Perfect Black Body

    Property
    Value / Point
    Absorptance
    \(a=1\)
    Reflectance
    \(r=0\)
    Transmittance
    \(t=0\)
    Emission
    Depends on surface temperature, not nature of material
    Spectrum
    Continuous spectrum
    When hot
    Emits all wavelengths and may appear white
    When cold
    Absorbs all radiations and appears black
    ❖ Fery's Black Body:
    • •
      Hollow double-walled metallic sphere
    • •
      Fine opening on one side
    • •
      Inner wall painted black
    • •
      Radiation entering it is absorbed completely
    • •
      Absorptive power = 1
    ▢ Emissive Power and Emissivity:

    Table 1: Emission Terms

    Term
    Meaning / Value
    Emissive power \(E\)
    Radiant energy emitted per unit area per second at a given temperature
    Depends on
    Nature of surface and temperature
    Maximum emissive power
    Perfect black body
    Minimum emissive power
    Smooth shining body
    Unit of emissive power
    \(W/m^2\)
    Dimension of emissive power
    \([MT^{-3}]\)
    Emissivity \(e\)
    Ratio of emissive power of body to emissive power of black body at same temperature
    Emissivity of black body
    1
    Emissivity
    Unitless and dimensionless
    ▢ Stefan's Law:
    ❖ Statement: Total energy radiated per second per unit area is directly proportional to fourth power of absolute temperature
    ❖ Validity: Best valid when body temperature is very large compared with surrounding temperature

    Table 1: Stefan-Boltzmann Law

    Quantity
    Formula
    Black body emissive power
    \(E=\sigma T^4\)
    Real body emissive power
    \(E=e\sigma T^4\)
    Power per unit area
    \(\frac{P}{A}=e\sigma T^4\)
    Total power radiated
    \(P=Ae\sigma T^4\)
    Stefan constant
    \(\sigma=5.67\times10^{-8}\ Wm^{-2}K^{-4}\)
    Net rate of cooling
    \(\frac{dT}{dt}=\frac{eA\sigma}{mc}(T^4-T_0^4)\)
    ❖ Rate of Cooling Depends On:

    Table 1: Cooling Dependence

    Factor
    Relation / Point
    Emissivity
    \(\frac{dT}{dt}\propto e\)
    Area
    \(\frac{dT}{dt}\propto A\)
    Mass
    \(\frac{dT}{dt}\propto\frac{1}{m}\)
    Specific heat
    \(\frac{dT}{dt}\propto\frac{1}{c}\)
    Body temperature increases
    Rate of cooling increases
    Surrounding temperature increases
    Rate of cooling decreases
    Perfect black body
    Maximum rate of cooling
    Rough sphere vs smooth sphere
    Rough sphere cools faster
    Hollow sphere vs solid sphere of same radius
    Hollow sphere cools faster because mass is less
    ❖ Temperature of Sun:
    ◉ Formula: \(T=\left[\frac{S}{\sigma}\left(\frac{r}{R}\right)^2\right]^{1/4}\)
    ◉ Symbols:
    • •
      \(r\) = average distance between Sun and Earth
    • •
      \(R\) = radius of Sun
    • •
      \(S\) = solar constant \(=1.4\ kW/m^2\)
    • •
      \(\sigma\) = Stefan constant
    ◉ Value: Surface temperature of Sun ≈ 5800 K
    ▢ Newton's Law of Cooling:
    ❖ Statement: Rate of heat loss of a hot body is directly proportional to temperature difference between body and surroundings when temperature difference is small
    ❖ Condition: \(T-T_0=\Delta T\) is small and \(T\approx T_0\)
    ❖ Derived From: Stefan's law

    Table 1: Newton Cooling Formulae

    Quantity
    Formula / Point
    Rate of heat loss
    \(\frac{dQ}{dt}\propto-\Delta T\)
    Cooling between \(T_1\) and \(T_2\) in time \(t\)
    \(\frac{T_1-T_2}{t}=K\left[\frac{T_1+T_2}{2}-T_0\right]\)
    If \(T=T_0\)
    \(\frac{dT}{dt}=0\)
    Conclusion
    A body cannot be cooled below surrounding temperature by radiation alone
    ❖ Use: Used to determine specific heat of liquids
    ❖ Also Applicable To: Forced convection losses
    ▢ Wien's Displacement Law:
    ❖ Statement: Wavelength corresponding to maximum energy radiation is inversely proportional to absolute temperature
    ❖ Formula: \(\lambda_m\propto\frac{1}{T}\)
    ❖ Constant Form: \(\lambda_m=\frac{b}{T}\)
    ❖ Wien Constant: \(b=2.89\times10^{-3}\ mK\)
    ❖ Approximation: \(\lambda_m\approx\frac{3\times10^{-3}}{T}\)
    ❖ Applications:
    • •
      Colour of a star determines its temperature
    • •
      Temperature of a star is determined by Wien's law
    • •
      On heating iron ball, colour changes from red to white
    ▢ Kirchhoff's Law:
    ❖ Statement: At a given temperature, ratio of emissive power to absorptive power is same for all surfaces and equals emissive power of black body
    ❖ Formula: \(\frac{E}{a}=E_b\)
    ❖ For Black Body: \(a=1\)
    ❖ Conclusion: Good absorber of a particular wavelength is also good emitter of that wavelength
    ❖ Applications:
    • •
      Deserts are hot during day and cold at night
    • •
      Sodium vapour absorbs yellow radiation which it emits when heated
    • •
      Explains Fraunhofer lines in Sun's spectrum
    • •
      White paper with black letters: on burning, letters appear whiter and paper appears black
    • •
      Red glass appears red because it reflects red and absorbs other colours; when heated, it emits complementary radiation
    ▢ Read and Digest:

    Table 1: Important Points

    Fact
    Point
    Black body spectrum
    Continuous spectrum
    Cooking pot material
    Low specific heat and high thermal conductivity
    Cracks in glass after heating/cooling
    Due to low thermal conductivity
    Radiation effect on medium
    Temperature of medium does not change
    Wire heated gradually
    First appears red because \(\lambda\propto1/T\)
    Rate of cooling for same mass, material and initial temperature
    Circular plate > Cube > Sphere
    Hollow sphere vs solid sphere cooling
    Hollow sphere cools faster
    Bolometer
    Detects heat radiation; resistance changes with heat radiation
    Good absorber
    Bad reflector
    Rough surface
    Good absorber and bad reflector
    Smooth surface
    Good reflector and bad absorber
    Black body
    Absorbs all incident radiation
    Metals as heat conductors
    Due to free electrons
    Vacuum
    Bad conductor of heat
    No atmosphere on Earth
    Earth would be very cold
    Star colour
    Determines temperature
    Temperature of Sun/planet
    Determined by Stefan's law
    Temperature of star
    Determined by Wien's law
    ▢ High-Yield Recall:
    ❖ **table:
      Q1.
      According to Wien's displacement law, maximum wavelength of emission is related to temperature as:
      📅BP 2011
      Q2.
      Body cools from 50°C to 49.9°C in 5s (surrounding=30°C). Time to cool from 40°C to 39.9°C?
      📅BP 2010
      Q3.
      Which can be considered a black body?
      📅BP 2010
      Q4.
      Two cylinders (same material) with diameter ratio 1:2 and length ratio 2:1. Heat conduction ratio?
      📅IOM 2012
      Q5.
      Heat energy by radiation is proportional to:
      📅IOM 2012
      Q6.
      Four colored articles (blue, red, black, white) heated then cooled. Which cools fastest?
      📅IOM 2011
      Q7.
      Object cools 80°C→70°C in 1min (room=30°C). Time for 50°C→40°C?
      📅MOE 2012
      Q8.
      Sun's λmax at T1,T2,T3 are 650nm,580nm,350nm. Correct relation?
      📅IOM 2010
      Q9.
      When ΔT=20°C, heat flow=273J/s. If ΔT becomes 20K, new heat flow?
      📅MOE 2009
      Q10.
      Ratio of energy emitted at 27°C and 600K?
      📅MOE 2010
      Q11.
      Stars A and B have λmax at 3600Å and 4800Å. Temperature ratio TA/TB?
      📅MOE 2012
      Q12.
      Stars radiate max at 3200Å and 4000Å. Temperature ratio?
      📅MOE 2013
      Q13.
      Liquid cools from 60°C to 50°C in 5min (surrounding=18°C). Temperature after next 5min?
      📅MOE 2014
      Q14.
      Black body radiates at rate E at T. When T doubles, new radiation rate?
      📅KU 2013
      Q15.
      Rate of heat loss is:
      📅KU 2011
      Q16.
      Kirchhoff's law states:
      📅KU 2010-2014
      Q17.
      Kirchhoff's law implies:
      📅KU 2013
      Q18.
      Two rods (K1/K2=5/3) joined end-to-end with ends at 100°C and 20°C. Junction temp?
      📅KU 2013
      Q19.
      Black spot on red-hot metal plate in dark room appears:
      📅KU 2014
      Q20.
      Refrigerator theory is based on:
      📅TE 2013
      Q21.
      If T halved, radiating power becomes:
      📅MOE 2009
      Q22.
      Correct emissive power relation:
      📅KU 2010
      Q23.
      Best cooking pot material has:
      📅BP 2016
      Q24.
      Stars with λmax=320nm and 400nm. Temperature ratio?
      📅IOM 2003
      Q25.
      Instrument measuring temperature by radiation:
      📅IOM 2000
      Q26.
      Rate of heat loss depends on:
      📅IOM 1998
      Q27.
      Two black bodies at T and T' emit λm and λm'. Correct ratio?
      📅MOE Curriculum
      Q28.
      Sun/Moon λmax ratio=1:400. Temperature ratio?
      📅MOE 2008
      Q29.
      Unit of thermal conductivity:
      📅MOE 2005
      Q30.
      If T increases 50%, radiation increase %:
      📅MOE 2006
      Q31.
      Water's thermal expansion coefficient at 0°C:
      📅Bangladesh Embassy
      Q32.
      If T doubles, radiated energy increases by factor:
      📅Bangladesh 2009
      Q33.
      Black body at 27°C vs 127°C radiation ratio:
      📅TE-2004
      Q34.
      Polished metal pot minimizes heat loss by:
      📅IE-2005
      Q35.
      Hot sand sensation due to:
      📅TE-2005
      Q36.
      Heat loss rate at 288K (σ=5.67×10-8 W/m2K4):
      📅
      Q37.
      Pond heating occurs mainly by:
      📅
      Q38.
      Which rod conducts most heat? (r=radius, l=length)
      📅
      Q39.
      To radiate 16× power, temperature change:
      📅
      Q40.
      Two spheres (r=1m@4000K vs r=4m@2000K):
      📅
      Q41.
      Two rods (K=2 and 3) in series. Equivalent K?
      📅
      Q42.
      Rod AB (150cm) with TA=100°C, TB=25°C. Temp at 50cm from B?
      📅
      Q43.
      Two-layer slab (K1, K2) of equal thickness. Equivalent K?
      📅
      Q44.
      Two walls (d1,K1 and d2,K2) in contact. Interface temp?
      📅
      Q45.
      Metallic rod radiates 10W at 77°C. Radiation at 227°C?
      📅
      Q46.
      Wall with layers A (3K) and B (K) of equal thickness. ΔT across A?
      📅
      Q47.
      Composite rod of materials K1 and K2. Equivalent K?
      📅
      Q48.
      Wall with layer A (2K) and B (K). Total ΔT=36°C. ΔT across A?
      📅
      Q49.
      Sphere cooling rate depends on:
      📅
      Q50.
      Two spheres (R1, R2) of same material cool under identical conditions. Rate ratio?
      📅
      Q51.
      Two identical rods: series vs parallel Keq?
      📅
      Q52.
      Equal ΔT across two rods. Equal heat transfer when:
      📅
      Q53.
      Sphere, cube, and plate (same material/mass) heated to 200°C. Which cools slowest?
      📅
      Q54.
      Two cylinders (diameters d1, d2) conduct equal heat when lengths relate as:
      📅
      Q55.
      Two spheres (big: 2r, t/4 vs small: r, t). Ice melts in 25min vs 16min. Kbig/Ksmall?
      📅
      Q56.
      Body cools 50°C→40°C in 10min (surrounding=20°C). Next 10min temp?
      📅
      Q57.
      Tea cools 80°C→60°C in 1min (ambient=30°C). Time for 60°C→50°C?
      📅IOM 2015
      Q58.
      Body cools 60°C→50°C in 10min (room=25°C). Next 10min temp?
      📅
      Q59.
      Cooling times t1(100→80°C), t2(80→60°C), t3(60→40°C) with T∞=27°C:
      📅
      Q60.
      Liquid loses 60cal/s at 80°C (room=20°C). Heat loss at 40°C?
      📅
      Q61.
      1cm Cu cube cools 100→99°C in 100s. 2cm cube cooling time?
      📅
      Q62.
      Black body at 2880K. U1(499-500nm), U2(999-1000nm), U3(1499-1500nm):
      📅
      Q63.
      Ice forms 1cm in 7h at -10°C. Time for 1→2cm?
      📅
      Q64.
      Body cools 65→60°C in 5min. Time for 60→55°C?
      📅IOM 2010
      Q65.
      If ΔT doubles, thermal conductivity:
      📅KU 2015
      Q66.
      Heat transfer by particle movement:
      📅IOM 2015
      Q67.
      Newton's cooling applies to:
      📅IOM 2017
      Q68.
      Greenhouse effect caused by:
      📅KU 2016