Chapter 20 THERMAL PROPERTIES AND PROCESSES. Thermal Expansion

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1 Chapter 20 THERMAL PROPERTIES AND PROCESSES Thermal Expansion When the temperature goes up, the lengh of an object increases, ΔL/L = αδt, where the proportionality coefficient α is the coefficient of linear expansion. It depends on pressure and temperature; a stricter definition is α= lim[ ΔL/L ΔT ] ΔT 0= The coefficient of volume expansion β= lim[ ΔV/V ΔT ] ΔT 0= For a given material β= 3α. Example 1. Suppose we have an object with a circular hole. Will the radius of the hole decrease or increase with rising temperature? Tounderstandtheanswer,letusassumethatwehaveacircularholeona 1

2 ruler with the radius 2 cm and centered on a 4 cm mark on the ruler. Then the edges of the hole are at 2cm and 6 cm marks. Since the distance between the marks on the ruler increase with rising temperature, the radius of the hole increases as well. Thus the size of the holes always increases with temperature. Water is an important exception: In a narrow interval of temperatures it actually contracts when heated! 2

3 Example 2.What is the change in length of a 500 mi steel pipeline after it temperature raises by F? Then ΔT = 1800 = 100. = = = 0.55 Example 3. How much water will be spilled after you heat a 100L Pyrex can with 100L of water and heat the water from 20 0 C to50 0 C? The volume change of water! = " #! = 0, =0.63L For the glass,! = " '()*+! = 3 '()*+! = = The amount of spilled water is 0.63L = 0.60L 3

4 Example 4. Young modulus of copper is Y= 110GN/m 2 and its breaking stress is 230MN/m 2. By how much can you cool a copper rod with fixedends before it breaks? The stress F/Ais related to stretching as - =. = /. Solving for, 8 = 9 <=>?@ : = A <.; BB>C@ A < D E FG F =122K The van der Waals Equation and Liquid Vapor Isotherms Often,theidealgasequation H! = IJ shouldbereplacedbyamoreaccurate van der Waals equation of state, H + LIM! M! NI = IJ, Where b is the volume of one mole of gas molecules and a describes the interaction between molecules. 4

5 The value of the constant bcan be used to estimate the size of a molecule. For example, since the volume of 1mol of nitrogen molecules occupies the volume 38.7cm 3, the volume of one molecule O = N 38.7R S /UV = P MS UV*RWV*X/UV = FMS R S /UV*RWV* and the size of a molecule Z = O /S = 0.4I. 5

6 Example 5. A 50Ltank contains 1000 molof helium at H = 500L[. What fraction of the pressure is the term LIM and what fraction of the volume! M is NI? gh < >.>=ijk<lma = Ano < < Temperature of the gas EEEpqr M (E) < = 13.8atm; 13.8/500 = 2.8% NI = ( )1000mol =23.8 l; 23.8/50 = 47.6% pqr = (H + LIM! M )(! IN) IJ = 513.8L[ UV 0.082L[ UV F F = 164 6

Phase Diagrams At the critical point density of the condensed vapor becomes equal to that of the liquid and the phase difference disappears (the end point).

7 Phase Diagrams At the critical point density of the condensed vapor becomes equal to that of the liquid and the phase difference disappears (the end point). The curve O-B (liquid solid equilibrium) does not have an end point The direct change from a solid to a vapor below the curve A-O is called sublimation P-T phase diagram of H 2 O Liquid water cannot exist below the temperature of the triple point 7

8 The Transfer of Heat Heat is the transfer of energy due to a temperature difference. There are three mechanisms of this energy transfer: conduction, convection, and radiation For all mechanisms of heat transfer the rate of cooling is approximately proportional to the temperature difference between the body and its surrounding (the Newton law of cooling) During conduction, the energy is transferred due to interaction between molecules while the molecules themselves are not transported. The thermal current I u = v w = xy z, where dt/dxis the temperature gradient, Ais the cross-section of the conductor, and k is the coefficient of thermal conductivity Temperature gradually changes along the system The equation for the thermal current can be rewritten as = u + xy = uj where Ris the thermal resistanceand is the temperature drop. 8

If the thermal resistors are connected in series, M = uj, M S = uj M, and = S = u J + J M = uj } as for electric currents through the resistors connected in series If the resistors are connected in

9 If the thermal resistors are connected in series, M = uj, M S = uj M, and = S = u J + J M = uj } as for electric currents through the resistors connected in series If the resistors are connected in parallel, u wqwgr = u + u M + = =, J J M J } = + +, B < as for electric currents through the resistors connected in parallel Example 6. Find thermal current in each bar in the figure, the total current, and the equivalent thermal resistance of the two-bar system. The bars of length 5 cm have a rectangular cross-section 2cm 3cm. 9

The resistance of a lead bar J ƒ = + ƒ = x ƒ y ƒ For a silver bar, J - = + - x - y - = The currents are u ƒ = J ƒ = 0.05 35.3 /( ) (0.02 0.03) = 2.36 0.05 429 /( ) (0.02 0.03) = 0.

10 The resistance of a lead bar J ƒ = + ƒ = x ƒ y ƒ For a silver bar, J - = + - x - y - = The currents are u ƒ = J ƒ = /( ) ( ) = /( ) ( ) = = 42.4, 2.36 F u - = 100 = = J F The equivalent thermal resistance and the total current are 1 = 1 + 1, J J } J ƒ J } = F, - u wqw = u ƒ + u - =

11 In the building industry the thermal resistance of a square foot of cross-sectional area of material is called its R factor, R f. The R-factor is related to the thermal resistance R of the system measured in 0 F/(Btu/h) as R f = AR where the cross-section Ais measured in square feet. The R-factor can be also expressed as J = Jy = + x where Δx is the thickness of the sheet of material. The R-factors of various building materials are in the Table. Pay attention to the units! 11

12 Example 7.Find the equivalent R-factor of a roof consisting of a triple layer of asphalt shingles, 4in of roof insulation, and 1.5in-thick pine board. The R-factor of triple layer of shingles is R fsh = 3 (0.44h ft 2 0 F/Btu) The R-factor of 4in of roof insulation is R fin = 4 (2.8h ft 2 0 F/Btu) The R-factor of 1.5in of pine should be obtained from the conductivity table, R fp = Δx /k p = 1.5in/(0.78Btu in/h ft 2 0 F) = 1.92 h ft 2 0 F/Btu Since all these layers are assembled in series, the equivalent R-factor of the roof R feq = R fsh + R fin + R fp = 14.4 h ft 2 0 F/Btu The thermal conductivity of air is very low. The effective way of using air for thermal insulation is to trap air in small pockets to prevent convection. 12

13 Convection Convection is the transfer of heat by the transport material itself. Convection takes place mostly in fluids and gases and in responsible for ocean currents, air circulation, wind. Convection is used to heat (cool) the building by circulating the hot(cold) air. Mathematics of convection is rather complicated. Radiation All objects can emit and absorb electromagnetic radiation. The rate of radiation of energy is given by the Stefan-Boltzmann law, H = *ˆy, Where istheradiatedpower, Aisthesurfaceareaoftheradiatingbody, 0 e 1isthe emissivityofthesurface,andthe Stefan s constant ˆ = F M 13

14 The rate at which an object absorbs radiation is given by H g = *ˆy Œ 4 where Tsis the temperature of the sourceof radiation. If the object both emits and absorbs the radiation, the net radiated power is H h w = *ˆy( Œ4 ) An object that absorbs all incoming radiation e= 1 is called a blackbody which is also an ideal radiator. The positions of the maxima on the curves are given by the Wien s law: λ L+ = Power radiated by a blackbody as a function of the wavelength at different temperatures 14

15 Radiation from the Sun The radiation emitted by the surface of the Sun has the wavelength about 500 nm. According to the Wien s law this means that the temperature of the surface of the Sun is approximately 5800 K. For a body at room temperature T = 300 K, λ L+ =9.66 μm. 15

16 Review of Chapter 20 Coefficient of linear thermal expansion = Coefficient of volume thermal expansion " = The van der Waals equation of state H + gh< < = 3! IN = IJ The triple point is the unique point at which liquid, solid, and gas phases are at equilibrium with each other, For water w Ž r = Vapor pressure is the pressure at which liquid and gas phases are in equilibrium with each other. The critical point is the point in which the density of liquid and gas phases become equal to each other. For water = Heat transfer occurs by conduction, convection, and radiation 16

17 Thermal current u = v = xy (kis the coefficient of thermal z conductivity, A is the cross-section) Thermal resistance J = = z - Thermal conductors in series J } = J + J M + J S + Thermal conductors in parallel 1/J } = 1/J + 1/J M + 1/J S + The R-factor is the thermal resistance for a square foot of a slab of material in units of in ft 2 0 F/(Btu/h), J = Jy = z The Stefan-Boltzmann law of radiation H = *ˆy where the Stefan s constant ˆ = F p < i and the emissivity e is between 0 and 1 Net power radiated by a body at Tto its environment at T 0 H h w = *ˆy( E4 ) 17

18 Wien s law for the wavelength for the maximal radiation λ pgz = M. pp A blackbody absorbs all radiation, is a perfect radiator, and has the emissivity e= 1 18

Chapter 18 Temperature, Heat, and the First Law of Thermodynamics. Thermodynamics and Statistical Physics

Chapter 18 Temperature, Heat, and the First Law of Thermodynamics Thermodynamics and Statistical Physics Key contents: Temperature scales Thermal expansion Temperature and heat, specific heat Heat and