Section 6. HEAT TRANSFER Dr. Congxiao Shang

6.1 Definitions Mechanisms of Heat (Thermal Energy) Transfer: Conduction: transmission of heat across matter, due to direct physical contact, e.g. in solids, liquids and gases. Radiation: heat transfer due to electromagnetic radiation across a space, even in a vacuum. Convection: heat transfer by “currents” in a gas or liquid, due to temperature differences or forced flow, an important mechanism of energy transfer between a solid surface and a liquid or a gas. All mechanisms may be involved in practical heat transfer, but the dominant mechanism differs in different cases. (Source of illustrations: http://sol.sci.uop.edu/~jfalward/ heattransfer/heattransfer.html)

Kelvin = Degree Celsius (oC) + 273 6.1 Definitions Concepts & Terminologies: THERMAL CAPACITY (of a system): Quantity of ENERGY required to heat a whole system by 1 K (Kelvin). Unit: J·K-1 SPECIFIC HEAT (or SPECIFIC HEAT CAPACITY): Quantity of ENERGY required to heat a unit mass of a system by 1 K. Unit: J·kg-1·K-1 Kelvin = Degree Celsius (oC) + 273 J= joule

(error in handout “m-3” ) 6.1 Definitions T1 T2 Q A d Temperature profile THERMAL CONDUCTIVITY, k : a measurement of heat flow through a body. It is the heat transmitted in unit time, in a direction normal to a surface of unit area, through a distance, d, across a unit temperature difference over the distance. Q, heat flow per unit time (Js-1) × d, distance (m) A, area (m2) × (T1-T2) temperature difference (K) k = (Eq. 6a) (Eq. 6a) Unit: Wm-1K-1 (or Wm-1 °C-1) (error in handout “m-3” ) (analogous to electrical conductivity or hydraulic permeability) ( note: 1W = 1 J s-1)

(T1-T2) temperature difference (K) 6.1 Definitions T1 T2 Q A d Temperature profile Another way of understanding the THERMAL CONDUCTIVITY, k, is to re-arrange the equation as : Q, heat flow per unit time (Js-1) A, area (m2) (T1-T2) temperature difference (K) d, distance (m) = k (Eq. 6b) The temperature difference per unit distance is called temperature gradient Heat flow per unit time per unit area is proportional to the temperature gradient; this proportionality is called thermal conductivity, k. The higher the thermal conductivity, the faster the heat flows

6.1 Definitions Why are diamonds so “cool”? Brick Styrofoam Thermal conductivity of other common materials: 6.1 Definitions Why are diamonds so “cool”? Material Thermal conductivity (298 K), W·m-1·K-1 Diamond 895-2300 ( the highest k) Carbon Nanotubes 1400 Silver 429 Copper 386 Gold 317 Aluminium 237 Iron 80.2 Brick Wood Wool Styrofoam (for building insulation) 0.15 – 0.6 0.12 -0.04 0.04 (very low k) 0.01 Why dose wool feel so “warm”?

where R is the resistance 6.1 Definitions THERMAL RESISTIVITY, ρ : reciprocal of conductivity, 1/k Unit: m K W-1 THERMAL Resistance of a system, R: where R is the resistance d is the thickness ρ is the resistivity R = d = d/k (error in handout, not divided by “A”) Note that the R-value above is a UNIT AREA THERMAL RESISTANCE (or thermal insulance), because the resistivity, ρ, is related to the conductivity, k, which is measured per unit area. Unit for R: m K W-1 m = Km2 W-1 (The reason for defining the thermal resistance, R, is that the R values are “additive” in multi-layer insulations and this makes calculations simpler. This will be explained later)

6.2 Conduction d Q. How much heat is conducted through a system ? Q A Temperature profile Q. How much heat is conducted through a system ? We know: -the larger the A, the larger the heat flow; -the larger the d, the smaller the heat flow. Therefore ; or Direct analogy with electricity:- Current (I) is equivalent to Heat Flow per unit area (Q/A) ; & Potential Difference (V1 - V2) or voltage is equivalent to temperature difference (T1-T2).

6.2 Conduction R = R1 + R2 + R3 + ........ i.e. resistances in series In most situations we have composite materials to deal with - e.g. a wall consisting of an outer skin (brick), a cavity, an inner skin and then plaster. Since the thermal resistance of each component has considered the thickness, the Total UNIT AREA THERMAL RESISTANCE is simply given by brick cavity plaster R = R1 + R2 + R3 + ........ i.e. resistances in series Electrical Analogue

6.2 Conduction “Resistance in Parallel”: e.g. conduction through a wall with a window, which is more complicated, as total (average) heat transfer depends on the thermal resistances and the relative areas of both components. Proportion of wall or window area to the total area The U - value is defined as 1/R, where R is resistance per unit area

R U clear ? 6.2 Conduction Thermal transmittance, the U value: The U value is simply defined as 1/R; Unit: W·K-1·m-2 (Remember: R is resistance per unit area, so U is transmittance per unit area as well) Both the R-value and the U-value are used to grade the insulation properties of a material or a system (e.g. a double-glazed assembly). R U clear ? A higher R-Value means the materials are more resistant to heat loss. A lower U-Value means the system will transmit less heat. Both mean the same thing & are desirable, from the insulation point of view.

6.2 Conduction Simple Example: Tbp Brick Brick wall 105 mm thick plaster Brick 20°C 0 °C Tbp 105mm 15mm Brick wall 105 mm thick plaster 15 mm thick on inside Internal temperature = 20°C External Temperature= 0 °C kbrick = 0.84 W m-1 K -1 kplaster = 0.50 W m-1 K -1 What is the U - value of the construction and also the temperature at the interface between the brick and the plaster?

6.2 Conduction Brick 20°C 0 °C Tbp 105mm 15mm Simple Example: plaster Brick 20°C 0 °C Tbp 105mm 15mm Simple Example: Total resistance, R = 0.125 + 0.030 = 0.155 m2 ºC W-1 and the U-value = 1/R = 1/0.155 = 6.45 W m-2 ºC-1 ============== Now heat flow in plaster = heat flow in brick=heat flow through the whole body So if Tbp is temperature at interface Hence Tbp = 129 * 0.125 = 16 . 1ºC =========