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.
      Rate of heat flow through metal bar of area 1 m² with temperature gradient 1°C/m is called
      📅CEE MBBS 2025
      Q2.
      A cooking pot should have:
      📅CEE Nursing 2024
      Q3.
      Green house effect is caused by:
      📅CEE Nursing 2024
      Q4.
      Transmission of heat through the actual movement of particles of medium is:
      📅CEE Nursing 2024
      Q5.
      The radiated energy at 300 K is proportional to:
      📅CEE MBBS 2023
      Q6.
      At absolute zero temperature, a black body:
      📅CEE MBBS 2021
      Q7.
      A liquid cool down from 70°C to 60°C in 5 minutes. The time taken to cool it from 60°C to 50°C will be:
      📅CEE BDS 2024
      Q8.
      A black body at a temperature of 100 K radiates energy at the rate of 1 Wm⁻². At what temperature the energy radiates will be 256 Wm⁻²?
      📅CEE BDS 2024
      Q9.
      Greenhouse effect is caused by:
      📅CEE BDS 2021
      Q10.
      According to Wien’s displacement law, how does the wavelength of maximum blackbody emission \(\lambda_{\max}\) depend on absolute temperature \(T\)?
      📅BP 2011
      Q11.
      A body cools from \(50^\circ\mathrm C\) to \(49.9^\circ\mathrm C\) in \(5\,\mathrm s\) in surroundings at \(30^\circ\mathrm C\). Approximately how long will it take to cool from \(40^\circ\mathrm C\) to \(39.9^\circ\mathrm C\)?
      📅BP 2010
      Q12.
      Which is the best approximation to an ideal blackbody?
      📅BP 2010
      Q13.
      Two cylinders of the same material have diameters in the ratio 1:2 and lengths in the ratio 2:1. Under the same end-to-end temperature difference, find their heat-conduction-rate ratio.
      📅IOM 2012
      Q14.
      For a blackbody of fixed area, total emitted radiation power is proportional to which power of its absolute temperature \(T\)?
      📅IOM 2012
      Q15.
      Four otherwise identical opaque objects with blue, red, black and white surfaces are heated to the same temperature and then allowed to cool under the same conditions. Which is expected to lose heat by radiation fastest?
      📅IOM 2011
      Q16.
      A body cools from \(80^\circ\mathrm C\) to \(70^\circ\mathrm C\) in \(1\,\mathrm{min}\) at room temperature \(30^\circ\mathrm C\). Approximately how long does it take to cool from \(50^\circ\mathrm C\) to \(40^\circ\mathrm C\)?
      📅MOE 2012
      Q17.
      Three blackbodies have peak-emission wavelengths \(650\,\mathrm{nm}\), \(580\,\mathrm{nm}\), and \(350\,\mathrm{nm}\) at temperatures \(T_1,T_2,T_3\), respectively. Which ordering is correct?
      📅IOM 2010
      Q18.
      The heat flow through a rod is \(273\,\mathrm{J/s}\) for a temperature difference of \(20^\circ\mathrm C\). What is the rate for the same difference expressed as \(20\,\mathrm K\)?
      📅MOE 2009
      Q19.
      For identical blackbodies, find the ratio of total emitted power at \(27^\circ\mathrm C\) and at \(600\,\mathrm K\).
      📅MOE 2010
      Q20.
      Stars A and B have peak wavelengths \(3600\,\mathrm{\mathring A}\) and \(4800\,\mathrm{\mathring A}\). Find their absolute-temperature ratio \(T_A:T_B\).
      📅MOE 2012
      Q21.
      Two stars have peak wavelengths \(3200\,\mathrm{\mathring A}\) and \(4000\,\mathrm{\mathring A}\). Find the ratio of their absolute temperatures in the same order.
      📅MOE 2013
      Q22.
      A liquid cools from \(60^\circ\mathrm C\) to \(50^\circ\mathrm C\) in \(5\,\mathrm{min}\), while surroundings are at \(18^\circ\mathrm C\). What is its temperature after another \(5\,\mathrm{min}\), assuming Newton’s cooling law?
      📅MOE 2014
      Q23.
      A blackbody emits power \(E\) at absolute temperature \(T\). What is the emitted power at \(2T\), with its area unchanged?
      📅KU 2013
      Q24.
      For fixed emissivity and temperature difference, the rate of radiative heat loss is directly proportional to:
      📅KU 2011
      Q25.
      Kirchhoff’s law of thermal radiation, at a given wavelength and temperature, states that:
      📅KU 2010-2014
      Q26.
      For an opaque body in thermal equilibrium, which property generally accompanies high absorptivity at a given wavelength?
      📅KU 2013
      Q27.
      Two rods of equal length and cross-sectional area are joined in series. The hot end of rod 1 is at \(100^\circ\mathrm C\), and the cold end of rod 2 is at \(20^\circ\mathrm C\). If \(K_1/K_2=5/3\), find the steady-state junction temperature.
      📅KU 2013
      Q28.
      A blackened patch on a red-hot polished metal plate is viewed in a dark room. Relative to the glowing plate, how does the patch appear?
      📅KU 2014
      Q29.
      Cooling during throttling of a high-pressure gas in certain refrigeration processes is associated with the:
      📅TE 2013
      Q30.
      If the absolute temperature of a blackbody is halved while its area remains fixed, what percentage of the initial power does it emit?
      📅MOE 2009
      Q31.
      Which relation gives the total emissive power of an ideal blackbody in terms of absolute temperature \(T\)?
      📅KU 2010
      Q32.
      A desirable cooking-pot material combines which specific heat capacity and thermal conductivity characteristics?
      📅BP 2016
      Q33.
      Two stars have peak-emission wavelengths \(320\,\mathrm{nm}\) and \(400\,\mathrm{nm}\), respectively. What is their absolute-temperature ratio?
      📅IOM 2003
      Q34.
      Which instrument measures high temperatures from emitted thermal radiation?
      📅IOM 2000
      Q35.
      Under Newton’s law of cooling, with other conditions fixed, the rate of cooling is proportional to:
      📅IOM 1998
      Q36.
      Two blackbodies at absolute temperatures \(T\) and \(T^{\prime}\) have peak wavelengths \(\lambda_m\) and \(\lambda_m^{\prime}\). Which equation is correct?
      📅MOE Curriculum
      Q37.
      The ratio of peak-emission wavelengths of the Sun and Moon is \(1:400\). Treating both as blackbodies, find their absolute-temperature ratio.
      📅MOE 2008
      Q38.
      What is the SI unit of thermal conductivity?
      📅MOE 2005
      Q39.
      What is the percentage increase in blackbody radiated power when its absolute temperature increases by 50%?
      📅MOE 2006
      Q40.
      What is the sign of the volume-expansion coefficient of water near \(0^\circ\mathrm C\)?
      📅Bangladesh Embassy
      Q41.
      By what factor does the total power radiated by a blackbody increase when its absolute temperature doubles?
      📅Bangladesh 2009
      Q42.
      Find the ratio of total emitted powers of identical blackbodies at \(27^\circ\mathrm C\) and \(127^\circ\mathrm C\).
      📅TE-2004
      Q43.
      A polished metallic surface reduces heat loss primarily by reducing:
      📅IE-2005
      Q44.
      Why does dry sand often heat up rapidly in strong sunlight compared with water?
      📅TE-2005
      Q45.
      Find the emitted power per unit area of an ideal blackbody at \(288\,\mathrm K\). Take \(\sigma=5.67\times10^{-8}\,\mathrm{W/(m^2\,K^4)}\).
      Q46.
      Once sunlight has warmed the upper layers of a pond, heat is redistributed within the water mainly by:
      Q47.
      Four rods of identical material connect the same two temperatures. Which conducts heat fastest?
      Q48.
      For an ideal blackbody of constant surface area, what must happen to absolute temperature to multiply its radiation power by \(16\)?
      Q49.
      Two ideal blackbody spheres have radii and temperatures \((r,4000\,\mathrm K)\) and \((4r,2000\,\mathrm K)\). Compare their total emitted powers.
      Q50.
      Two rods with thermal conductivities \(K_1=2\) and \(K_2=3\) are joined end-to-end. Their lengths are unspecified. What is their effective thermal conductivity?
      Q51.
      A uniform rod \(AB\), length \(150\,\mathrm{cm}\), has steady-state end temperatures \(T_A=100^\circ\mathrm C\) and \(T_B=25^\circ\mathrm C\). Find the temperature at a point \(50\,\mathrm{cm}\) from B.
      Q52.
      Two layers of equal thickness, equal area, and conductivities \(K_1,K_2\) are arranged in series. Find their effective conductivity.
      Q53.
      Two planar slabs with thicknesses \(d_1,d_2\) and conductivities \(K_1,K_2\) are placed in series. Their outer faces are at \(T_1,T_2\). For equal areas and steady conduction, find the interface temperature.
      Q54.
      An idealised radiating body emits \(10\,\mathrm W\) at \(77^\circ\mathrm C\). Find the emitted power at \(227^\circ\mathrm C\), assuming constant emissivity and area and neglecting ambient back-radiation.
      Q55.
      A composite wall has two layers A and B of equal thickness and area, with conductivities \(K_A=3K\) and \(K_B=K\). If the total steady temperature difference is \(36^\circ\mathrm C\), find the drop across A.
      Q56.
      Two materials with conductivities \(K_1,K_2\) form equal-area, equal-length rods connected in parallel between the same temperatures. Find the effective conductivity of their combined cross section.
      Q57.
      Two equal-thickness wall layers with conductivities \(K_A=2K\), \(K_B=K\) have a total steady temperature difference \(36^\circ\mathrm C\). What is the drop across A?
      Q58.
      For geometrically similar spheres cooled under identical conditions, how does the initial *temperature-cooling rate* \(|dT/dt|\) depend on radius \(r\), density \(\rho\), and specific heat \(c\), with other factors fixed?
      Q59.
      Two spheres of the same material and surface character, radii \(R_1\) and \(R_2\), cool from the same temperature in identical surroundings. Find the ratio of their initial temperature-cooling rates \(|dT_1/dt|:|dT_2/dt|\).
      Q60.
      Two rods of the same homogeneous material have identical conductivity \(K\). They are combined in series or in parallel. How do the effective conductivities of the assemblies compare when their total length and cross-sectional area are used consistently?
      Q61.
      Two rods have the same length and the same end-to-end temperature difference. Under what condition are their steady heat-conduction rates equal? \(K_i\) denotes conductivity and \(A_i\) area.
      Q62.
      A sphere, a cube, and a thin plate made of the same material have equal masses and initial temperatures. Under similar external cooling conditions, which cools most slowly?
      Q63.
      Two cylinders of the same material at the same temperature difference conduct heat at equal rates. If their diameters are \(d_1,d_2\), what must be their length ratio \(L_1/L_2\)?
      Q64.
      Two spherical ice bodies of radii \(2r\) and \(r\) are covered with thin insulating layers of thicknesses \(d/4\) and \(d\), respectively. Under the same temperature difference, they melt completely in \(25\) and \(16\,\mathrm{min}\). Neglect curvature effects in heat conduction and find \(K_{\mathrm{big}}:K_{\mathrm{small}}\).
      Q65.
      A body cools from \(50^\circ\mathrm C\) to \(40^\circ\mathrm C\) in \(10\,\mathrm{min}\) at ambient temperature \(20^\circ\mathrm C\). What is its temperature after another \(10\,\mathrm{min}\)?
      Q66.
      Tea cools from \(80^\circ\mathrm C\) to \(60^\circ\mathrm C\) in \(1\,\mathrm{min}\) at ambient temperature \(30^\circ\mathrm C\). Approximately how long does it take to cool from \(60^\circ\mathrm C\) to \(50^\circ\mathrm C\)?
      📅IOM 2015
      Q67.
      A body cools from \(60^\circ\mathrm C\) to \(50^\circ\mathrm C\) in \(10\,\mathrm{min}\), with surroundings at \(25^\circ\mathrm C\). Find its temperature after another \(10\,\mathrm{min}\).
      Q68.
      A body cools successively from \(100\) to \(80^\circ\mathrm C\), from \(80\) to \(60^\circ\mathrm C\), and from \(60\) to \(40^\circ\mathrm C\) in surroundings at \(27^\circ\mathrm C\). If the times are \(t_1,t_2,t_3\), which ordering holds?
      Q69.
      An object loses heat at \(60\,\mathrm{cal/s}\) when at \(80^\circ\mathrm C\), with surroundings at \(20^\circ\mathrm C\). Under Newton’s law of cooling, what is its heat-loss rate at \(40^\circ\mathrm C\)?
      Q70.
      A copper cube of side \(1\,\mathrm{cm}\) cools from \(100^\circ\mathrm C\) to \(99^\circ\mathrm C\) in \(100\,\mathrm s\). Under identical surface and surrounding conditions, about how long will a cube of side \(2\,\mathrm{cm}\) take for the same small temperature drop?
      Q71.
      An ideal blackbody is at \(2880\,\mathrm K\). Let \(U_1,U_2,U_3\) denote energy emitted in equal-width wavelength intervals of \(499\)–\(500\,\mathrm{nm}\), \(999\)–\(1000\,\mathrm{nm}\), and \(1499\)–\(1500\,\mathrm{nm}\). Which statement is correct?
      Q72.
      Ice of thickness \(1\,\mathrm{cm}\) forms on a pond in \(7\,\mathrm h\) with air at \(-10^\circ\mathrm C\). Under the same idealised freezing conditions, how long does it take to increase its thickness from \(1\) to \(2\,\mathrm{cm}\)?
      Q73.
      A body cools from \(65^\circ\mathrm C\) to \(60^\circ\mathrm C\) in \(5\,\mathrm{min}\). If the ambient temperature is constant and below \(55^\circ\mathrm C\), how long will cooling from \(60\) to \(55^\circ\mathrm C\) take?
      📅IOM 2010
      Q74.
      If the temperature difference across a conductor doubles but its material and mean temperature are unchanged, what happens to its thermal conductivity?
      📅KU 2015
      Q75.
      Which mode of heat transfer takes place mainly through the bulk motion of fluid particles?
      📅IOM 2015
      Q76.
      Newton’s law of cooling can be used in experiments for determining which of the following?
      📅IOM 2017
      Q77.
      Which part of the electromagnetic spectrum is mainly absorbed and re-emitted by greenhouse gases, warming the lower atmosphere?
      📅KU 2016