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Section 6. HEAT TRANSFER Dr. Congxiao Shang

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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. (Source of illustrations: heattransfer/heattransfer.html) All mechanisms may be involved in practical heat transfer, but the dominant mechanism differs in different cases.

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6.1 Definitions THERMAL CAPACITY (of a system): Quantity of ENERGY required to heat a whole system by 1 K (Kelvin). Unit: J·K -1 Concepts & Terminologies: 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 ( o C) J= joule

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6.1 Definitions ( note: 1W = 1 J s -1 ) 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. Unit: Wm -1 K -1 (or Wm -1 °C -1 ) (error in handout m -3 ) (analogous to electrical conductivity or hydraulic permeability) T1T1 T2T2 Q A d Temperature profile (Eq. 6a) Q, heat flow per unit time (Js -1 ) × d, distance (m) A, area (m 2 ) × ( T 1 -T 2 ) temperature difference (K) k =

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6.1 Definitions = k (T 1 -T 2 ) temperature difference (K) d, distance (m) Another way of understanding the THERMAL CONDUCTIVITY, k, is to re-arrange the equation as : Heat flow per unit time per unit area is proportional to the temperature gradient; this proportionality is called thermal conductivity, k. T1T1 T2T2 Q A d Temperature profile (Eq. 6b) Q, heat flow per unit time (Js -1 ) A, area (m 2 ) The temperature difference per unit distance is called temperature gradient The higher the thermal conductivity, the faster the heat flows

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6.1 Definitions Thermal conductivity of other common materials: MaterialThermal conductivity (298 K), W·m -1 ·K -1 Diamond ( the highest k) Carbon Nanotubes 1400 Silver429 Copper386 Gold317 Aluminium237 Iron80.2 Brick Wood Wool Styrofoam (for building insulation) 0.15 – (very low k) 0.01 Why are diamonds so cool? Why dose wool feel so warm?

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6.1 Definitions THERMAL Resistance of a system, R: where R is the resistance d is the thickness ρ is the resistivity 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 = Km 2 W -1 THERMAL RESISTIVITY, ρ : reciprocal of conductivity, 1/k Unit: m K W -1 R = d = d/k (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) (error in handout, not divided by A)

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6.2 Conduction Q. How much heat is conducted through a system ? T1T1 T2T2 Q A d Temperature profile We know: -the larger the A, the larger the heat flow; -the larger the d, the smaller the heat flow. Direct analogy with electricity:- Current (I) is equivalent to Heat Flow per unit area (Q/A) ; & Potential Difference (V 1 - V 2 ) or voltage is equivalent to temperature difference (T 1 -T 2 ). ; or Therefore

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6.2 Conduction 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 R = R 1 + R 2 + R i.e. resistances in series Electrical Analogue brick cavity brick plaster

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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. The U - value is defined as 1/R, where R is resistance per unit area Proportion of wall or window area to the total area

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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) A higher R-Value means the materials are more resistant to heat loss. 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). A lower U -Value means the system will transmit less heat. Both mean the same thing & are desirable, from the insulation point of view. R U clear ?

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6.2 Conduction Simple Example: Brick wall 105 mm thick plaster 15 mm thick on inside Internal temperature = 20°C External Temperature= 0 °C k brick = 0.84 W m -1 K -1 k plaster = 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? plaster Brick 20°C 0 °C T bp 105mm 15mm

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6.2 Conduction Simple Example: Total resistance, R = = m 2 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 T bp is temperature at interface Hence T bp = 129 * = 16. 1ºC ========= plaster Brick 20°C 0 °C T bp 105mm 15mm

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