Module 1: Learning objectives

Transcription

1 Heat and Ma Tranfer Module 1: Learning objective Overview: Although much of the material of thi module will be dicued in greater detail, the objective of thi module i to give you a reaonable overview of heat tranfer Heat tranfer mode: You hould be aware of the everal mode of tranfer mode of tranfer and their phyical origin Phyical inight: Given a phyical ituation, you hould be able to perceive the relevant tranport phenomena The importance of developing thi inight mut not be underetimated You will be inveting a lot of time to acuire the tool needed to calculate heat tranfer phenomena However, before you can begin to ue thee tool to olve practical problem, you mut have the intuition to determine what i happening phyically In hort, you mut be able to look at a problem and identify the pertinent tranport phenomenon The example and problem at the end of thi module hould help you to begin developing thi intuition Rate euation and conervation law: You hould alo appreciate the ignificance of the rate euation and feel comfortable in uing them to compute tranport rate You mut alo recognize the importance of the conervation law and the need to carefully identify control volume With the rate euation, the conervation law may be ued to olve numerou heat tranfer problem Indian Intitute of Science Bangalore

2 Heat and Ma Tranfer MODULE I BASICS OF HEAT TRANSFER 11 Difference between heat and temperature In decribing heat tranfer problem, we often make the mitake of interchangeably uing the term heat and temperature Actually, there i a ditinct difference between the two Temperature i a meaure of the amount of energy poeed by the molecule of a ubtance It i a relative meaure of how hot or cold a ubtance i and can be ued to predict the direction of heat tranfer The uual ymbol for temperature i T The cale for meauring temperature in SI unit are the Celiu and Kelvin temperature cale On the other hand, heat i energy in tranit The tranfer of energy a heat occur at the molecular level a a reult of a temperature difference The uual ymbol for heat i Q Common unit for meauring heat are the Joule and calorie in the SI ytem What i Heat Tranfer? Energy in tranit due to temperature difference 1 Difference between thermodynamic and heat tranfer Thermodynamic tell u: how much heat i tranferred (δq) how much work i done (δw) final tate of the ytem Heat tranfer tell u: how (with what mode) δq i tranferred at what rate δq i tranferred temperature ditribution inide the body Heat tranfer complementary Thermodynamic 13 Mode of Heat Tranfer Conduction: An energy tranfer acro a ytem boundary due to a temperature difference by the mechanim of inter-molecular interaction Conduction need matter and doe not reuire any bulk motion of matter T 1 x A T k Indian Intitute of Science Bangalore

3 Heat and Ma Tranfer Conduction rate euation i decribed by the Fourier Law: = ka T where: = heat flow vector, (W) k = thermal conductivity, a thermodynamic property of the material (W/m K) A = Cro ectional area in direction of heat flow (m ) T = Gradient of temperature (K/m) = T/ x i + T/ y j + T/ z k Note: Since thi i a vector euation, it i often convenient to work with one component of the vector For example, in the x direction: x = - k A x dt/dx In circular coordinate it may convenient to work in the radial direction: r = - k A r dt/dr Convection: An energy tranfer acro a ytem boundary due to a temperature difference by the combined mechanim of intermolecular interaction and bulk tranport Convection need fluid matter moving fluid T T >T T Newton Law of Cooling: = h A ΔT where: = heat flow from urface, a calar, (W) h = heat tranfer coefficient (which i not a thermodynamic property of the material, but may depend on geometry of urface, flow characteritic, thermodynamic propertie of the fluid, etc (W/m K) A = Surface area from which convection i occurring (m ) ΔT = T S T = Temperature Difference between urface and coolant (K) Convection Free or natural convection (induced by buoyancy force) Forced convection (induced by external mean) May occur with phae change (boiling, condenation) Indian Intitute of Science Bangalore

4 Heat and Ma Tranfer Table 1 Typical value of h (W/m K) Free convection gae: - 5 liuid: Forced convection gae: 5-50 liuid: 50-0,000 Boiling/Condenation ,000 Radiation: Radiation heat tranfer involve the tranfer of heat by electromagnetic radiation that arie due to the temperature of the body Radiation doe not need matter Emiive power of a urface: E=σεT (W/ m ) where: ε = emiivity, which i a urface property (ε = 1 i black body) σ = Steffan Boltzman contant = 567 x 10-8 W/m K T = Abolute temperature of the urface (K) The above euation i derived from Stefan Boltzman law, which decribe a gro heat emiion rather than heat tranfer The expreion for the actual radiation heat tranfer rate between urface having arbitrary orientation can be uite complex, and will be dealt with in Module 9 However, the rate of radiation heat exchange between a mall urface and a large urrounding i given by the following expreion: r a d T u r co n v T Area = A = ε σ A (T T ur ) where: ε = Surface Emiivity A= Surface Area T = Abolute temperature of urface (K) T ur = Abolute temperature of urrounding(k) Indian Intitute of Science Bangalore

5 Heat and Ma Tranfer 1 Thermal Conductivity, k A noted previouly, thermal conductivity i a thermodynamic property of a material From the State Potulate given in thermodynamic, it may be recalled that thermodynamic propertie of pure ubtance are function of two independent thermodynamic intenive propertie, ay temperature and preure Thermal conductivity of real gae i largely independent of preure and may be conidered a function of temperature alone For olid and liuid, propertie are largely independent of preure and depend on temperature alone k = k (T) Table give the value of thermal conductivity for a variety of material Table Thermal Conductivitie of Selected Material at Room Temperature Material Thermal Conductivity, W/m K Copper 01 Silver 9 Gold 317 Aluminum 37 Steel 605 Limetone 15 Bakelite 1 Water 0613 Air 0063 It i important that the tudent gain a baic perpective of the magnitude of thermal conductivity for variou material The background for thi come from the introductory Chemitry coure Molecule of variou material gain energy through variou mechanim Gae exhibit energy through the kinetic energy of the molecule Energy i gained or lot through colluion of gaeou molecule a they travel through the medium Kinetic energy tranfer between gaeou molecule Lattice vibration may be tranferred between molecule a nuclei attract/repel each other Indian Intitute of Science Bangalore

6 Heat and Ma Tranfer Solid, being are much more tationary, cannot effectively tranfer energy through thee ame mechanim Intead, olid may exhibit energy through vibration or rotation of the nucleu Another important mechanim in which material maintain energy i by hifting electron into higher orbital ring In the cae of electrical conductor the electron are weakly bonded to the molecule and can drift from one molecule to another tranporting their energy with them Thi i an epecially effective tranport mechanim, o that material which are excellent electrical conductor are excellent thermal conductor Indian Intitute of Science Bangalore

7 Heat and Ma Tranfer Module 1: Worked out problem Problem 1: A freezer compartment conit of a cubical cavity that i m on a ide Aume the bottom to be perfectly inulated What i the minimum thickne of Styrofoam inulation (k=0030w/mk) which mut be applied to the top and ide wall to enure a heat load le than 500 W, when the inner and outer urface are -10 ºC and 35 0 C? Solution: Known: Dimenion of freezer component, inner and outer urface temperature Find:Thickne of Styrofoam inulation needed to maintain heat load below precribed value T =35 0 C T 1 =10 0 C L m=w Styrofoam k=0003w/ mk =500W Schematic: Aumption: (1) perfectly inulted bottom, () one-dimenional conduction through five wall of area A=m, (3) teady-tate condition Analyi: Uing Fourier law, the heat rate i given by '' ΔT = A = k A L Solving for L and recognizing that A total =5*W total 5kΔTW L = 5* 003W / mk * 5 C * m L = 500W L = 005m = 5mm 0 Indian Intitute of Science Bangalore

8 Heat and Ma Tranfer Comment: The corner will caue local departure from one dimenional conduction and, for a precribed value of L, a lightly larger heat lo Indian Intitute of Science Bangalore

9 Heat and Ma Tranfer Problem : A uare ilicon chip (k=150w/mk) i of width W=5mm on a ide and of thickne t=1mm the chip i mounted in a ubtrate uch that it ide and back urface are inulated, while the front urface i expoed to a coolant If W are being diipated in circuit mounted to the back urface of the chip, what i the teady-tate temperature difference between back and front urface? Known: Dimenion and thermal conductivity of a chip Power diipated on one urface Find: temperature drop acro the chip Schematic: W=5mm Subtrate Chip, k=150w/mk t=1mm P=W Aumption: (1) teady-tate condition, () contant propertie, (3) uniform diipation, () negligible heat lo from back and ide, (5) one-dimenional conduction in chip Analyi: All of the electrical power diipated at the back urface of the chip i tranferred by conduction through the chip Hence, Fourier law, ΔT P = = ka t tp 0001m * W ΔT = = kw 150W / mk(0005m ΔT = 11 0 C Comment: for fixed P, the temperature drop acro the chip decreae with increaing k and W, a well a with decreaing t ) Indian Intitute of Science Bangalore

10 Heat and Ma Tranfer Problem 3: Air at C flow over a plate of dimenion 050 m, by 05 m if the convection heat tranfer coefficient i 50 W/mK; determine the heat tranfer rate from the air to one ide of the plate when the plate i maintained at 0 0 C Known: air flow over a plate with precribed air and urface temperature and convection heat tranfer coefficient Find: heat tranfer rate from the air to the plate Schematic: Aumption: (1) temperature i uniform over plate area, () heat tranfer coefficient i uniform over plate area Analyi: the heat tranfer coefficient rate by convection from the airtream to the plate can be determined from Newton law of cooling written in the form, = A = ha(t T ) where A i the area of the plate Subtituting numerical value, = 50W / m = 815W '' K *(05*050)m (300 0) Comment: recognize that Newtown law of cooling implie a direction for the convection heat tranfer rate Written in the form above, the heat rate i from the air to plate 0 C Indian Intitute of Science Bangalore

11 Heat and Ma Tranfer Problem : A water cooled pherical object of diameter 10 mm and emiivity 09 i maintained at 000C What i the net tranfer rate from the oven wall to the object? Known: pherical object maintained at a precribed temperature within a oven Find: heat tranfer rate from the oven wall to the object Schematic: Sphere rad T =80 0 C ε=09, D=10mm Oven wall, T ur=00 0 C Aumption: (1) oven wall completely urround pherical object, () teady-tate condition, (3) uniform temperature for area of phere and oven wall, () oven encloure i evacuated and large compared to phere Analyi: heat tranfer rate will be only due to the radiation mode The rate euation i rad = εa σ(t ur T Where A=πD, the area of the phere, ubtituting numerical value, ) rad rad = 09* π(10 *10 = 30W 3 ) m *567 *10 8 W / m K[( ) ( ) ]K Comment: (1) thi rate euation i ueful for calculating the net heat exchange between a mall object and larger urface completely urround the maller one thi i an eential, retrictive condition () Recognize that the direction of the net heat exchange depend upon the manner in which T ur and T are written (3) When performing radiant heat tranfer calculation, it i alway neceary to have temperature in Kelvin (K) unit Indian Intitute of Science Bangalore

12 Heat and Ma Tranfer Problem 5: A urface of area 05m, emiivity 08 and temperature C i placed in a large, evacuated chamber whoe wall are maintained at 5 0 C What i the rate at which radiation i emitted by the urface? What i the net rate at which radiation i exchanged between the urface and the chamber wall? Known: Area, emiivity and temperature of a urface placed in a large, evacuated chamber of precribed temperature Find: (a) rate of urface radiation emiion, (b) net rate of radiation exchange between the urface and chamber wall Schematic: Tur=5 0 C A=05m T=150 0 C Aumption: (1) area of the encloed urface i much le than that of chamber wall Analyi (a) the rate at which radiation i emitted by the urface i emitted emit = emit emit = 08(05m emit = 76W A = εaσt )567 *10 8 W / m K [( )K] (b) The net rate at which radiation i tranferred from the urface to the chamber wall i = εaσ(t Turr) = 08(05m = 57W )567 *10 8 W / m K [(3K) (98K) Comment: the foregoing reult give the net heat lo from the urface which occur at the intant the urface i placed in the chamber The urface would, of coure, cool due to thi heat lo and it temperature, a well a the heat lo, would decreae with increaing time Steady-tate condition would eventually be achieved when the temperature of the urface reached that of the urrounding Indian Intitute of Science Bangalore

13 Heat and Ma Tranfer Problem 6: A olid aluminium phere of emiivityε i initially at an elevated temperature and i cooled by placing it in chamber The wall of the chamber are maintained at a lower temperature and a cold ga i circulated through the chamber Obtain an euation that could be ued to predict the variation of the aluminium temperature with time during the cooling proce Do not attempt to olve Known: Initial temperature, diameter and urface emiivity of a olid aluminium phere placed in a chamber whoe wall are maintained at lower temperature Temperature and convection coefficient aociated with ga flow over the phere Find: euation which could be ued to determine the aluminium temperature a a function of time during the cooling proce Schematic: Aluminium Sphere, D,T,,x Chamber wall,tur ε Ga h,t Eout Aumption: (1) at any time t, the temperature T of the phere i uniform, () contant propertie; (3) chamber wall are large relative to phere Analyi: applying an energy balance at an intant of time to a control volume about the phere, it follow that t E = E Identifying the heat rate out of the CV due to convection and radiation, the energy balance ha the form d ( ρvct) = ( conv + rad ) dt dt A = [h(t T ) + εσ(t Turr )] dt ρvc dt dt = 6 [h(t T ρcd out ) + εσ(t Where A=πD, V=πD 3 /6 and A/V=6/D for the phere T urr )] Comment: (1) knowing T=Ti at t =0, the foregoing euation could be olved by numerical integration to obtain T (t) () The validity of auming a uniform phere temperature depend upon h, D and the thermal conductivity of the olid (k) The validity of the aumption improve with increaing k and decreaing h and D Indian Intitute of Science Bangalore

14 Heat and Ma Tranfer Problem 7: In an orbiting pace tation, an electronic package i houed in a compartment having urface area A = 1m which i expoed to pace Under normal operating condition, the electronic diipate 1 kw, all of which mut be tranferred from the expoed urface to pace If the urface emiivity i 10 and the urface i not expoed to the un, what i it teady- tate temperature? If the urface i expoed to a olar flux of 750W/m and it aborptivity to olar radiation i 05, what i it teady tate temperature? Known: urface area of electronic package and power diipation by the electronic Surface emiivity and aborptivity to olar radiation Solar flux Find: urface temperature without and with incident olar radiation Schematic: Aumption: teady tate condition Analyi: applying conervation of energy to a control urface about the compartment, at any intant in E It follow that, with the olar input, α A '' = In the hade ( 0) T = 1m α T S S A E '' S '' S α = S out A *1*567 *10 g + E '' emit A εσt '' A S + P A εσ = 0 + P = 0 + P = W = 8 1 W / m K 36K In the un, T 1 = 05*1m * 750W / m W = 8 1m *1*567 *10 W / m K 380K Comment: in orbit, the pace tation would be continuouly cycling between hade, and a teady- tate condition would not exit Indian Intitute of Science Bangalore

15 Heat and Ma Tranfer Problem 8: The back ide of a metallic plate i perfectly inulated while the front ide aborb a olar radiant flux of 800 W/m The convection coefficient between the plate and the ambient air i 11 w /m K (a) Neglecting radiation exchange with the urrounding, calculate the temperature of the plate under teady-tate condition if the ambient air temperature i 0 0 C (b) For the ame ambient air temperature, calculate the temperature of the plate if it urface emiivity i 08 and the temperature of the urrounding i alo0 0 C Known: front urface of inulated plate aborb olar flux, and experience for cae (a) Convection proce with air at T and for cae (b): the ame convection proce and radiation exchange with urrounding at T ur Find: temperature of the plate, T, for the two cae Schematic: Aumption: (1) teady tate condition, () no heat lo out backide of plate, (3) urrounding large in comparion plate Analyi: (a) apply a urface energy balance, identifying the control urface a hown on the chematic For an intant of time the converation reuirement i Ein Eout =0 The relevant procee are convection between the plate and the air, conv, and aborbed olar flux, Conidering the plate to have an area A olve for T and ubtitute numerical value to find T T '' A = T ha + " (T / h T ) = W / m = 0 C + = 0 C C = 87 C 1W / m K (b) Conidering now the radiation exchange between the urface and it urrounding, the urface energy balance ha the form E E =0 in out Indian Intitute of Science Bangalore

16 Heat and Ma Tranfer '' "A W 800 m 1T A conv rad = 0 ha (T T W 1 (T m K + 536*10 8T ) εa (T W [0 + 73]K) 08*567 *10 8 m K = 6503 T ) = 0 (T [0 + 73]K ) = 0 By trial and error method, find that T =338K=65 0 C Comment: note that by conidering radiation exchange, T decreae a expected Note the manner in which conv i formulated uing conv i formulated uing Newton law of cooling: ince conv i hown leaving the control urface, the rate euation mut be h T T ) and not h ( T -T) ( Indian Intitute of Science Bangalore

17 Heat and Ma Tranfer Module 1: Short uetion 1 What i the driving force for (a) heat tranfer (b) electric current flow and (c) fluid flow? Which one of the following i not a property of the material? A thermal conductivity B heat tranfer coefficient C emiivity 3 What i the order of magnitude of thermal conductivity for (a) metal (b) olid inulating material (c) liuid (d) gae? What i the order of magnitude for the convection heat tranfer coefficient in free convection? Forced convection? Boiling? 5 When may one expect radiation heat tranfer to be important? 6 An ideal ga i heated from 50 ºC to 70 ºC (a) at contant volume and (b) at contant preure For which cae do you think the energy reuired will be greater? Why? 7 A peron claim that heat cannot be tranferred in a vacuum How do you repond to thi claim? 8 Dicu the mechanim of thermal conduction in gae, liuid and olid 9 Name ome good conductor of heat; ome poor conductor 10 Show that heat flow line mut be normal to iotherm in conduction heat tranfer Will it be true for convection heat tranfer? Indian Intitute of Science Bangalore