In a parallel thermal conductor arrangement:

1. The total heat current (i) equals the sum of individual heat currents: $i = i_1 + i_2$

2. Each individual heat current follows: $i_1 = \frac{T_1 - T_2}{R_1}$ and $i_2 = \frac{T_1 - T_2}{R_2}$

3. The total heat current can be expressed as: $i = (T_1 - T_2)(\frac{1}{R_1} + \frac{1}{R_2})$

This principle explains why heat transfer is more efficient through parallel pathways - the total thermal conductance increases with each additional parallel path.

A cavity-based blackbody works through multiple internal reflections that trap incoming radiation:

• When radiation enters through a small hole in an enclosed cavity, it undergoes multiple reflections inside • With each reflection, a portion of radiation is absorbed by the internal walls • The probability of radiation escaping back through the small entrance hole becomes extremely small • This results in nearly 100% absorption of incident radiation, regardless of the actual material properties of the cavity walls

The cavity design effectively creates an absorption coefficient approaching 1.0, making it the closest practical implementation of a theoretical perfect blackbody.

Note: This principle is used in laboratory blackbody standards for calibrating radiation measurement instruments.

Emissive power (E) is defined as the energy radiated per unit area per unit time per unit solid angle along the normal to a radiating surface.

Mathematically expressed as: $E = \frac{U}{(ΔA)(Δω)(Δt)}$

Where:

• U = energy radiated
• ΔA = small area of the radiating surface
• Δω = small solid angle about the normal
• Δt = time interval

Units: W·m⁻²·sr⁻¹ (watts per square meter per steradian)

Note: Unlike absorptive power (which is dimensionless), emissive power has physical dimensions.

The solid angle (Δω) is critical in radiative heat transfer calculations because:

• Radiation is emitted in three-dimensional space with varying intensity in different directions • The solid angle measures a conical section of this 3D space (measured in steradians) • For most real surfaces, radiation intensity varies with direction (non-Lambertian surfaces)

Importance of directional analysis:

1. Radiation is not uniform in all directions for real surfaces
2. The intensity often follows a directional distribution (e.g., cosine law for diffuse emitters)
3. When calculating total emitted energy, integration over all solid angles is necessary: $U_{total} = \int_{A}\int_{Ω}E(\theta,\phi)dωdA$

Example application: When designing thermal shields for spacecraft, engineers must account for the directional distribution of thermal radiation to properly protect sensitive components from heat sources that emit radiation preferentially in certain directions.

Starting with heat transfer equation: Q = KA(θ₁-θ₂)t/x

For ice melting experiment:

1. Heat transferred equals latent heat absorbed: Q = mL (where m = mass of ice melted, L = latent heat of fusion)

2. Substituting: mL = KA(θ₁-θ₂)t/x

3. Solving for K: K = (mL·x)/(A·(θ₁-θ₂)·t)

Where:

• m = mass of ice melted (kg)
• L = latent heat of fusion (J/kg)
• x = thickness of conducting slab (m)
• A = surface area (m²)
• (θ₁-θ₂) = temperature difference (°C)
• t = time (s)

This derivation shows how to experimentally determine thermal conductivity by measuring ice melted over time.

Step 1: Calculate total surface area A = 2(60×60 + 60×30 + 60×30) cm² = 1.44 m²

Step 2: Calculate rate of heat flow ΔQ/Δt = KA(θ₁-θ₂)/x = (0.04 W/m·°C)(1.44 m²)(40°C)/(0.015 m) = 154 W

Step 3: Calculate melting rate Rate = (Heat flow)/(Latent heat) = 154 W/(3.36×10⁵ J/kg) = 0.46 g/s

This represents the steady-state rate at which ice melts due to heat conduction through the container walls.

A thermopile maximizes sensitivity through:

1. Series Connection: Multiple thermocouples (typically antimony-bismuth pairs) connected in series to amplify the thermoelectric voltage

2. Strategic Junction Placement:

   • All hot junctions on one plane face
   • All cold junctions on the opposite face
   • Maximizes temperature gradient between junction sets

3. Surface Optimization:

   • Blackened hot junction face to increase radiation absorption
   • Metallic cover protecting cold junctions from direct radiation

4. Radiation Concentration:

   • Metallic cone to gather and focus incident radiation onto the hot junctions
   • Increases effective collection area

5. Junction Material Selection:

   • Specific metal pairs (antimony-bismuth) chosen for high Seebeck coefficient
   • Maximizes voltage per degree of temperature difference

A thermopile is a radiation detection device based on the Seebeck effect, consisting of:

• Multiple thermocouples (typically antimony and bismuth) connected in series
• Arranged with all hot junctions on one plane face (blackened to absorb radiation)
• All cold junctions on the opposite plane face (protected by a metallic cover)
• Connected to a galvanometer to measure the generated electrical current

The series connection significantly increases sensitivity compared to a single thermocouple. When radiation strikes the blackened hot junctions, a temperature difference develops between the hot and cold junctions, generating a voltage proportional to the radiation intensity.

Example application: Thermopiles are used in infrared thermometers, satellite thermal control systems, and scientific instruments measuring radiant heat flux.

Kirchhoff's Law of Thermal Radiation

Mathematical Formulation

Kirchhoff's Law states that the ratio of emissive power (E) to absorptive power (a) is constant for all bodies at a given temperature and equals the emissive power of a blackbody:

$\frac{E(\text{body})}{a(\text{body})} = E(\text{blackbody})$

Physical Interpretation

• Fundamental principle: Good absorbers are good emitters; poor absorbers are poor emitters
• Thermodynamic basis: Derived from principles of thermodynamic equilibrium
• Blackbody reference: A blackbody (a=1) represents the theoretical maximum emitter

Material Implications

1. High emissivity materials (dark, matte surfaces):

   • High absorptive power (~0.9)
   • Correspondingly high emissive power
   • Example: Black-painted surfaces, carbon black

2. Low emissivity materials (reflective surfaces):

   • Low absorptive power (~0.1)
   • Correspondingly low emissive power
   • Example: Polished metals, silver surfaces

3. Theoretical limits:

   • Perfect reflector (a=0): Cannot emit radiation (E=0)
   • Perfect blackbody (a=1): Maximum possible emission at any temperature

Practical Applications

• Thermal insulation design (reflective barriers)
• Radiator and heat sink engineering
• Solar collector optimization
• Infrared temperature measurement calibration
• Spacecraft thermal control systems

Core Mechanisms

• Bolometer: Operates on temperature-dependent electrical resistance change
• Thermopile: Functions via the Seebeck effect (thermoelectric principle) converting temperature gradients directly to voltage

Construction & Components

Bolometer:

• Wheatstone bridge configuration with four resistors
• Two resistors (R₁ and R₄) exposed to radiation
• Typically housed in evacuated glass bulb to minimize heat loss
• Requires external power source for bridge circuit operation

Thermopile:

• Multiple thermocouples (usually bismuth-antimony pairs) connected in series
• Distinct hot and cold junctions arranged in planes
• Blackened surface on hot junctions to maximize radiation absorption
• Self-powered due to thermoelectric properties

Signal Generation & Measurement

Bolometer:

• Radiation heats exposed resistors → resistance changes → bridge imbalance
• Signal detected as galvanometer deflection from bridge output
• Sensitivity enhanced by vacuum enclosure reducing convection losses
• Output requires bridge configuration to detect small resistance changes

Thermopile:

• Radiation creates temperature difference between hot and cold junctions
• Multiple junction pairs produce additive voltage outputs
• Signal directly proportional to radiation intensity
• Sensitivity enhanced by increasing number of thermocouple junctions

Key Operational Differences

• Energy transduction: Resistance change (bolometer) vs. Direct voltage generation (thermopile)
• Power requirements: External power needed (bolometer) vs. Self-generating (thermopile)
• Signal amplification: Electronic amplification required (bolometer) vs. Internal amplification via series design (thermopile)
• Sensing geometry: All elements on same plane (bolometer) vs. Separated hot/cold junction planes (thermopile)