Temperature and Thermal Expansion

• Temperature is the degree of hotness or coldness, measured in Kelvin (SI unit).

Temperature Scales

• Tk = ToC + 273 converts Celsius to Kelvin.
• Conversion between Celsius (°C) and Fahrenheit (°F): C/5 = (F-32)/9.
• The formula (Reading - LFP) / (UFP - LFP) = constant relates temperature scales, where LFP is the lower fixed point, and UFP is the upper fixed point.

Thermal Expansion

• Linear expansion: L = L₀ + ΔL and L = L₀(1 + αΔT), where α is the coefficient of linear expansion.
• Area expansion: A = A₀ + ΔA and A = A₀(1 + βΔT), where β is the coefficient of area expansion.
• Volume expansion: V = V₀ + ΔV and V = V₀(1 + γΔT), where γ is the coefficient of volume expansion.
• ΔL = L₀ α ΔT relates change in length to original length and temperature change.
• ΔA = A₀ β ΔT relates change in area to original area and temperature change.
• ΔV = V₀ γ ΔT relates change in volume to original volume and temperature change.
• For isotropic materials: β = 2α and γ = 3α.
• The relationship between linear, area, and volume expansion coefficients for solids is α:β:γ = 1:2:3.
• The unit for α, β, and γ is 1/°C or 1/K.

Application of Linear Expansion

• In a pendulum clock, when temperature increases, the clock runs slow, and when temperature decreases, the clock runs fast.
• The loss of time in a given time interval is Δt = (1/2) α Δθ t.
• The time lost by a clock in a day is Δt = (1/2) α Δθ (86400) = 43200 α Δθ.
• In bimetallic strips, the strip with a higher α value will be on the convex side when temperature increases.
• For a bimetallic strip with αcu > αFe, when temperature increases, Δl of Cu > Δl of Fe.
• When a body with a cavity expands because of heating, the area of the hole increases.
• Expansion of an area of body is independent of shape & size of hole
• In scale readings subject to expansion or contraction, True Value = Scale Reading (1 + αΔθ).
• Error in Scale reading due to expansion or contraction: at θ'>0 True value > Scale Reading and at θ'<0 True value < Scale Reading

Expansion of Liquids

• γApp = γliq - γvessel is the apparent coefficient of volume expansion of a liquid.
• γApp = γliq - 3α, where the volume expansion coefficient of the vessel is 3α.
• If γliq > 3α, the water level will rise; if γliq < 3α, the water level will fall; if γliq = 3α, the water level will stay the same.

Heat

• Heat (Q) is the flow of energy due to a temperature difference.
• Units of heat are Joule (SI) and Erg (CGS), where 1 Joule = 10⁷ Ergs.
• Practical unit of heat is Calorie, where 1 Calorie = 4.2 Joule.
• ΔQ = m S ΔT, which is the amount of heat (Q) required to change the temperature. ΔQ is heat, m is mass, S is specific heat and ΔT is the change in temperature.
• Variation of density with temperature: Density α 1/Volume
• V' = V(1 + γΔθ) and then ρ' = ρ(1 - γΔθ).

Thermal Stress in a Rigidly Fixed Rod

• Thermal Stress = YαΔθ. Where Y is The Young's Modulus and α is the coefficient of linear expansion.
• Thermal Force = YAαΔθ. Where A is the area of rod.

Specific Heat Capacity

• Specific heat capacity, S = ΔQ / (mΔT).
• Heat required to raise the temperature of unit mass of a substance by unit degree.
• Unit: J/(kg·K) or J/(kg·°C). Common unit: cal/(gm·°C).
• Specific heat of water: Swater = 1 cal/(gm·°C) = 4200 J/(Kg·K).
• Sice = Ssteam = 0.5 cal/(gm·K) = 2100 J/(Kg·K).

Latent Heat

• Latent heat (L) is the heat required to change the phase of a unit mass of substance.
• For water, Lf (latent heat of fusion) = 80 cal/gm and Lv (latent heat of vaporization) = 540 cal/gm.
• Heat transfer for phase change: ΔQ = mL.
• Graph of heat transfer with temperature shows different phases and their transitions.

Water Equivalent

• Water equivalent is the mass of water that will absorb or lose the same quantity of heat as a given substance for the same change in temperature.
• mwsw = mbsb, where w is water and b is body.
• Heat supplied at a constant rate is represented by a graph and related equations.

Calorimetry

• Calorimetry is a device for the measurement of heat involved in a process.
• Principle of calorimetry: Heat lost by hotter bodies = Heat gained by colder bodies.
• Final equilibrium temperature: Teq = (ΣmsT) / (Σms).
• Cold

Ice-Water System

• ΔQ1 = m1siceθ1 is the heat to convert -θ1°C ice to 0°C ice.
• ΔQ2 = m1Lf is the heat to convert 0°C ice to 0°C water.
• ΔQ3 = m2Swaterθ2 is the heat to convert θ2°C water to 0°C water.

Heat Transfer

• Conduction is a heat transfer that occurs through a material medium.
• In conduction, heat flows from hot end to cold end.
• Conduction requires a medium and is a slow process.
• The equation for heat flow is Q/t = -KA(dT/dx), where K is the thermal conductivity.
• The unit of K is Watt/metre·°C or Watt/metre·K.
• The rate of heat flow is Q/t.
• A is the area of cross-section.
• dT/dx is the temperature gradient.

Conduction

• H = dQ/dt = ΔT/Rth = KA ΔT/L.
• I = (V2 - V1)/R.
• H = (T2 - T1)/ Rth.
• Electrical analogies: ΔV = IR and Thermal analogies: ΔT = H Rth.
• Resistance R = l/σA, where σ is electrical conductivity.
• Thermal resistance Rth = l/KA, where K is thermal conductivity.

Combination of rods

• Equivalent thermal when in series: Leq = (l1 + l2 + l3 + ... + ln) / (K1/l1 + K2/l2 + K3/l3 + ... + Kn/ln)
• If length are equal: Keq = (l1 + l2 + l3 + ... + ln) / (K1/l1 + K2/l2 + K3/l3 + ... + Kn/ln)
• Equivalent thermal when in parallel: Keq = (K1A1 + K2A2 + K3A3 + ... + KnAn) / (A1 + A2 + A3 + ... + An)

Temperature of Intermediate Junction

• T junction = (T1/R1 + T2/R2 + T3/R3) / (1/R1 + 1/R2 + 1/R3)

Heat Transfer

• Convection requires a medium, involves actual movement of fluid, and occurs naturally or forced and natural convection takes place due to the effect of gravity
• Sea Breeze: Wind blows from sea to land during day time.
• Land Breeze: Wind blows from land to sea during the night.

Radiation

• Absorption power(a)= Energy Absorbed/ Energy incident
• Absorption is the energy absorbed from the incident radiation.
• Radiation is the energy transmitted by virtue of temperature difference.
• Reflective power(r)= reflected Energy / incident Energy
• Transmittive powe(t)=Energy transmitted/Energy incident
• Qa + Qr + Qt =Q
• a + r + t = 1

Prevost Theory of Heat Exchange

• Every body radiates some radiation at all possible temperature.
• Emissivity (e) = Energy radiated by a general body / Energy radiated by a black body.
• Value of e ranges from 0 to almost 1 i.e. 0

Kirchhoff's Law

• Ratio of emissive power to absorption power is same for all surfaces at the same temperature and is equal to the emissive power of a perfectly black body at that temperature.
• E1/a1 = E2/a2 = Eb

Stefan’s Law

• Emissive power of body α absolute temp of the body
• Emissive Power E = σAT4
• Where σ is Stefan’s constant (5.67 x 10-8 Wm-2K-4)
• For ordinary body E= eσ T⁴ (e is emissivity)
• For body present in surrounding E=eσ (T⁴ -Tsurr⁴)

Newton’s Law Of Cooling

• Rate of cooling α Excess temperature of the body over the surrounding -(dT/dt) α (T-To)
• Tf - Ti/t= K(Ti + Tf/2- To)

Wein’s Displacement Law

• λm ∞ 1/T
• λmT = b = 2.89 x 10-3,
• Wavelength of maximum emission in meters
• T is temperature of Object in Kelvin,
• As the temp of the body increases, the spectral intensity is at maximum
• λmT is constant

Solan Constant

• Solar energy recived per unit area per unit is solar constant (1340 Watt /m²)
• S = σT4R²/d²
• Distance between earth and sun ~5800K