Review for Exam #2. Specific Heat, Thermal Conductivity, and Thermal Diffusivity. Conduction

Transcription

1 Review for Exam # Speifi Heat, Thermal Condutivity, and Thermal Diffusivity Speifi heat ( p ) A measure of how muh energy is required to raise the temperature of an objet Thermal ondutivity (k) A measure of how quikly heat gets onduted from one part of an objet to another Thermal diffusivity () It ombines the effets of speifi heat, thermal ondutivity, and density of a material Thus, this one quantity an be used to determine how temperature hanges at various points within an objet Condution 3 1

2 Fourier s Law of Heat Condution Rate of heat transfer by ondution is given by Fourier s law of heat ondution as follows: Q = - ka (T/x) The negative sign is used to denote/determine the diretion of heat transfer (Left to right or right to left) Q: Energy transferred per unit time (W) k: Thermal ondutivity (W/m K); it is a +ve quantity A: Area of heat transfer (m ) T: Temperature differene aross the ends of solid (K) x: Distane aross whih heat transfer is taking plae (m) Q/A: Heat flux (W/m ) 4 Condution Aross a Slab or Cylinder Slab: Q = ka (T/x) Heat flow T 1 x T Cylinder: Q = ka lm (T/r) k: Thermal ondutivity (W/m K) A: Area aross whih heat transfer is taking plae (m ) T = T 1 T : Temperature differene (K) A lm : Logarithmi mean area (m ) r T 1 T Heat flow ote: A lm omes into play when the area for heat transfer at the two ends aross whih heat transfer is taking plae, is not the same 5 Logarithmi Mean Area (A lm ) r T 1 T L Heat flow Slab: Area for heat transfer is same at both ends Cylinder Area at one end (outside) is A o (= r o L) Area at other end (inside) is A i (= r i L) Whih area should be used in determining Q? A lm = (A o A i ) / ln (A o /A i ) = L (r o r i ) / [ln (r o /r i )] ote: A o > A lm > A i r i r o Q = ka lm (T/r) T = T 1 T r = r o -r i 6

3 Logarithmi Mean Temp Diff (T lm ) T w(o) T p(i) Double Tube Heat Exhanger Hot water T 1 T Produt T w(i) T p(o) T is OT onstant aross the length of tube T 1 = T w(o) T p(i), T = T w(i) T p(o) T lm = ( 1 ) / [ln ( 1 / )] ote: T lm lies between T 1 and T Subsripts: w for water; p for produt, i for inlet, o for outlet ote: T lm omes into play when the temperature differene aross the two ends where heat transfer is taking plae, is not the same 7 Convetion 8 ewton s Law of Cooling for Convetion Rate of heat transfer by onvetion (for heating or ooling) is given by ewton s law of ooling as follows: Q = h A (T s -T ) Q: Energy transferred per unit time (W) h: Convetive heat transfer oeffiient -- CHTC (W/m K) A: Surfae area available for heat transfer (m ) T = T s T : Temperature differene (K) T s : Surfae temperature of solid objet (K) T : Free stream (or bulk fluid) temperature of fluid (K) CHTC (h): Measure of rate of heat transfer by onvetion; OT a property; depends on fluid veloity, surfae harateristis (shape, size, smoothness), fluid properties (, k,, p ) 9 3

4 Free Convetion Fluid omes into ontat with hot solid Fluid temperature near solid inreases Fluid density near solid dereases This results in a buoyany fore that auses flow Rate of heat transfer (Q & h) depends on Temperature differene between fluid and surfae of solid Properties (,, k, p ) of fluid Dimensions and surfae harateristis (smoothness) of solid u = hd /k f = f ( Gr, Pr ) 10 usselt umber ( u ) u h d k f h: Convetive heat transfer oeffiient (W/m K) d : Charateristi dimension (m) k f : Thermal ondutivity of fluid (W/m K) usselt number represents the ratio of heat transfer by onvetion & ondution 11 Grashof ( Gr ) umber 3 g T T d Gr f f : Coeffiient of volumetri thermal expansion (K -1 ) g: Aeleration due to gravity (= 981 m/s ) f : Density of fluid (kg/m 3 ) T s : Surfae temperature of solid objet (K) T : Free stream temperature of fluid (K) d : Charateristi dimension of solid objet (m) (Obtained from tables based on shape & orientation of solid objet) f : Visosity of surrounding fluid (Pa s) Grashof number represents the ratio of buoyany and visous fores f s f 1 4

5 Prandtl umber ( Pr ) Pr p(f ) k f f p(f) : Speifi heat of fluid (J/kg K) f : Visosity of fluid (Pa s) k f : Thermal ondutivity of fluid (W/m K) Prandtl number represents the ratio of momentum and thermal diffusivities 13 Free Convetion (Plate) u = hd /k f = f ( Gr, Pr ) u = a ( Gr Pr ) m ; Ra = Gr Pr For vertial plate (d = plate height) a = 059, m = 050 (for 10 4 < Ra < 10 9 ) a = 010, m = 0333 (for 10 9 < Ra < ) For inlined plate (for Ra < 10 9 ) Use same eqn as vertial plate & replae g by g os in Gr For horizontal plate (d = Area/Perimeter) Upper surfae hot a = 054, m = 050 (for 10 4 < Ra < 10 7 ) a = 015, m = 0333 (for 10 7 < Ra < ) Lower surfae hot a = 07, m = 050 (for 10 5 < Ra < ) 14 Free Convetion (Cylinder) For vertial ylinder (d = ylinder height) Similar to vertial plate if D 35L/( Gr ) 05 For horizontal ylinder (d = ylinder diameter) For 10-5 < Ra < / 6 Ra u 9/ Pr 8/ 7 ote: Ra = Gr Pr 15 5

6 Free Convetion (Sphere) u 1/ Ra 9 / Pr 4 / 9 for Ra Ra = Gr Pr & 07 Pr For sphere, d = D/ ote 1: For all free onvetion situations, determine properties at the film temperature {T film = (T s + T )/} unless otherwise speified ote : For all free onvetion senarios, as the T between the fluid and surfae of solid inreases, Gr inreases Thus, u and h inrease 16 Fored Convetion Fluid is fored to move by an external fore (pump/fan) Rate of heat transfer (Q & h) depends on Properties (,, k, p ) of fluid Dimensions and surfae harateristis (smoothness) of solid h does OT depend on Temperature differene between fluid and surfae of solid h strongly depends on Reynolds number When all system and produt parameters are kept onstant, it is flow rate (a proess parameter) that strongly affets h u = hd /k f = f ( Re, Pr ) 17 Fored Convetion in a Pipe u = hd /k f = f ( Re, Pr ) Three sub-ategories of fored onvetion exist 1 Laminar flow ( Re < 100) A Constant surfae temperature of pipe u = 366 (for fully developed onditions) B Constant surfae heat flux u = 436 (for fully developed onditions) C Other situations (for entry region & fully developed) u = 186 ( Re x Pr x d /L) 033 ( b / w ) 014 Transitional flow (100 < Re < 4000) Frition fator (f) For smooth pipes: (f / 8)( Re 1000)Pr 1/ / 117(f / 8) ( 1) u 3 Pr d : ID of pipe, L: Length of pipe 1 f 0790 ln ( Re 164) For non-smooth pipes, use Moody hart (graph of: f, Re, /D) 18 6

7 Moody Diagram Frition Fator (f) Relative Roughness (/D) Reynolds umber ( Re ) = 59 x 10-6 m for ast iron; 1535 x 10-6 m for drawn tubing; 15 x 10-6 m for galvanized iron; 457 x 10-6 m for steel or wrought iron 19 Fored Convetion in a Pipe (ontd) 3 Turbulent flow ( Re > 4000) of a ewtonian fluid in a pipe, u = 003 ( Re ) 08 ( Pr ) 033 ( b / w ) 014 b: Visosity of fluid based on bulk fluid temperature w : Visosity of fluid based on wall temperature The term ( b / w ) is alled the visosity orretion fator and an be approximated to 10 in the absene of information on wall temperature ote: For flow in an annulus, use same eqn with d = 4 (A s /W p ) = d io d oi d io : Inside diameter of outside pipe d oi : Outside diameter of inner pipe ote: For all fored onvetion situations, use bulk temperature of fluid to determine properties (unless otherwise speified) 0 h fp for Fored Convetion over a Sphere u = hd /k f = f ( Re, Pr ) similar to flow in a pipe u = + 06 ( Re ) 05 ( Pr ) 033 For 1 < Re < 70,000 and 06 < Pr < 400 ote 1: d is the outside diameter of the sphere ote : Determine all properties at the film temperature {T film = (T s + T )/} 1 7

8 Comparison of Free and Fored Convetion Free onvetion [Q = ha T; u = hd /k f = f ( Gr, Pr )] Does not involve any external ageny in ausing flow Temperature differene (T) auses density differene; this auses flow Q & h depend on Temperature differene between surfae of solid and surrounding fluid (T) Properties (,, k, p ) of fluid Dimensions and surfae harateristis (smoothness) of solid Fored onvetion [Q = ha T; u = hd /k f = f ( Re, Pr )] External ageny suh as fan/pump auses flow Q & h depend on Properties (,, k, p ) of fluid Dimensions and surfae harateristis (smoothness) of solid Only Q and OT h depends on temperature differene between surfae of solid and surrounding fluid (T) Thermal Resistanes to Heat Transfer 3 Thermal Resistanes Condution: Q = ka T/x Single slab: Q = T/[(x/kA)] Multiple slabs: Q = T/[(x 1 /k 1 A) + (x /k A) + ] Cylindrial shell: Q = T/[(r/kA lm )] Multiple ylindrial shells: Q = T/[(r 1 /k 1 A lm(1) ) + (r /k A lm() )] Convetion: Q = ha T Single onvetion: Q = T/[(/hA)] Multiple onvetions: Q = T/[(/h 1 A 1 ) + (/h A ) + ] Combination of ondution and onvetion Multiple slabs Q = T/[(x 1 /k 1 A) + (x /k A) + (/h 1 A 1 ) + (/h A ) + ] Multiple ylindrial shells Q = T/[(/h 1 A 1 ) + (r 1 /k 1 A lm(1) ) + (r /k A lm() ) + (/h A )] Units of thermal resistane to heat transfer: K/W 4 8

9 Overall Heat Transfer Coeffiient (OHTC) 5 OHTC (or U) in Different Senarios Three ondutive heat transfers 1/(UA) = x 1 /(k 1 A) + x /(k A) + x 3 /(k 3 A) Two onvetive heat transfers 1/(UA) = 1/(h 1 A 1 ) + 1/(h A ) One ondutive and one onvetive heat transfer 1/(UA) = x 1 /(k 1 A) + 1/(hA) ote 1: 1/UA > 1/hA; Thus, U < h ote : If there is no ondutive resistane, U = h U: Overall heat transfer oeffiient (W/m K) 1/UA : Overall thermal resistane (K/W) 6 k, h, U, Resistanes, and Temperatures As thermal ondutivity (k) inreases, thermal resistane due to ondution (x/ka) dereases Thus, temperature differene between enter and surfae of objet dereases As onvetive heat transfer oeffiient (h) inreases, thermal resistane due to onvetion (1/hA) dereases Thus, temperature differene between the fluid and surfae of the solid objet dereases As overall heat transfer oeffiient (U) inreases, overall thermal resistane (1/UA) dereases Thus, temperature differene between the two points aross whih heat transfer is taking plae, dereases Thermal ondutivity: W/m K Convetive heat transfer oeffiient: W/m K Overall heat transfer oeffiient: W/m K Thermal resistane to heat transfer: K/W 7 9

10 Q Effet of Resistane on Temperature 95 C k x T Q h 5 C A heater is used to maintain the left end of the slab at 95 C Ambient air on right side is at 5 C What fators determine the magnitude of temperature at right end of slab? T is affeted by 95 C at left AD 5 C at right Resistane to heat transfer from left (by ondution) is x/ka Resistane to heat transfer from right (by onvetion) is 1/hA If both resistanes are equal, T = (95 + 5)/ = 50 C If ondutive resistane is less (ours when k is high), T > 50 C If onvetive resistane is less (ours when h is high), T < 50 C ote: The same Q flows through the slab and outside Thus, Q = ka (95 T)/x = ha(t 5) 8 Heat Exhangers (HX) 9 Heat Transfer in a Double Tube HX (Hot water heating a produt) T ho Produt T i, m, p() Q = h o A o [T hot water -T wall (outside) ] h o h i U Hot water T hi, m h, p(h) Q = ka lm [T wall (outside) -T wall (inside) ]/r Q = h i A i [T wall (inside) -T produt ] r ii T o r oi L Subsripts for T: for old, h for hot, i for inlet, o for outlet 30 10

11 Resistanes to Heat Transfer from Hot Water to Produt Produt h o h i Hot water Convetive resistane (1/h o A o ) Condutive resistane (r/k A lm ) Convetive resistane (1/h i A i ) U 31 Overall Heat Transfer Coeffiient (U) Q = T / [(1/h o A o ) + (r/ka lm ) + (1/h i A i )] Thermal resistanes have been added Denominator: Total thermal resistane 1/UA lm = (1/h o A o ) + (r/ka lm ) + (1/h i A i ) Thus, Q = UA lm T lm U: W/m K U: Aounts for all modes of heat transfer from hot water to produt U is OT a property; it is OT fixed for a HX; it depends on material properties, system dimensions, and proess parameters 3 Determination of U: Theoretial Method 1/UA lm = (1/h o A o ) + (r/ka lm ) + (1/h i A i ) h i and h o are usually determined using empirial orrelations k is a material property of the tube of HX A i, A o, and A lm are determined based on dimensions (length & radii) of heat exhanger tubes One A i, A o, A lm, h i, h o, and k are known, U is alulated using the above equation 33 11

12 Determination of U: Experimental Method (Hot Water as Heating Medium) Q = m p() T = m h p(h) T h = UA lm T lm Assumption: Heat loss = zero One the mass flow rates and temperatures of the produt and hot water are experimentally determined, U an be alulated If there is heat loss, Q lost by hot water = Q gained by produt + Q lost to outside 34 Tubular Heat Exhanger (Co- and Counter-Current) 1/UA lm = (1/h o A o ) + (r/ka lm ) + (1/h i A i ) and Q = UA lm T lm Temperature Temp profile in a o-urrent HX h: Hot T hi : Cold i: Inlet o: Outlet T ho T 1 T T o Temperature Temp profile in a ounter-urrent HX T o T hi T 1 h: Hot : Cold i: Inlet o: Outlet T ho T i T T i Distane along the tube Distane along the tube Used only when rapid initial ooling is needed Most ommonly used Lower heat transfer effiieny Higher heat transfer effiieny T o T ho always T o an be greater than T ho T btwn hot & old fluid de along length T btwn hot & old fluid does not A i = Inside surfae area of inner pipe = r ii L hange signifiantly along length A o = Outside surfae area of inner pipe = r oi L A lm = Logarithmi mean area of inner pipe = (A o A i ) / [ln (A o /A i )] = L (r oi r ii )/[ln (r oi /r ii )] T lm = Logarithmi mean temperature differene = (T 1 T )/[ln(t 1 /T )] 35 Hot water Co-Current Arrangement T hi, m h, p(h) h o Q Q 3 T ho Produt T i, m, p() U h i Q 1 T o r ii r oi L r = r oi -r ii Q 1 : Energy transferred from heating medium to produt (= energy gained by produt) Q : Energy lost by heating medium (= energy gained by produt and surroundings) Q 3 : Energy transferred from heating medium to surroundings (= energy gained by surroundings) ote: Q = Q 1 + Q 3 Common approximation: Q 3 = 0 (valid if HX is insulated) In this ase, Subsripts for m, p, T, T: Q 1 = m p() (T) = Q = m h p(h) (T) h = UA lm T lm h for hot, for old with 1/UA lm = 1/h i A i + r/ka lm + 1/h o A o i for inlet, o for outlet ote 1: (T) = (T o T i ); (T) h = (T hi T ho ) ote : (T) 1 = (T hi T i ); (T) = (T ho T o ) Subsripts for h, A: i for inside, o for outside 36 1

13 T ho Counter-Current Arrangement Q Q 3 h o T hi, m h, p(h) Hot water Produt T i, m, p() Q 1 h i U T o r ii r oi L r = r oi -r ii Q 1 : Energy transferred from heating medium to produt (= energy gained by produt) Q : Energy lost by heating medium (= energy gained by produt and surroundings) Q 3 : Energy transferred from heating medium to surroundings (= energy gained by surroundings) ote: Q = Q 1 + Q 3 Common approximation: Q 3 = 0 (valid if HX is insulated) In this ase, Subsripts for m, p, T, T: Q 1 = m p() (T) = Q = m h p(h) (T) h = UA lm T lm h for hot, for old with 1/UA lm = 1/h i A i + r/ka lm + 1/h o A o i for inlet, o for outlet ote 1: (T) = (T o T i ); (T) h = (T hi T ho ) ote : (T) 1 = (T ho T i ); (T) = (T hi T o ) Subsripts for h, A: i for inside, o for outside 37 Dimensionless umbers in Heat Transfer 38 usselt # ( u ) and Biot # ( Bi ) h d h D u Bi k k f s Both are denoted by hd/k usselt # Used in STEADY state heat transfer (to determine h ) d : Charateristi dimension (= pipe diameter for flow in a pipe) k f : Thermal ondutivity of FLUID Biot# Used in USTEADY state heat transfer (to determine the relative importane of ondution versus onvetion heat transfer) D : Distane between hottest and oldest point in solid objet k s : Thermal ondutivity of SOLID 39 13

14 Unsteady State Heat Transfer (Heat Condution to the Center of a Solid Objet) 40 Categories of Unsteady State Heat Transfer Categories are based on the magnitude of Biot # ( Bi = hd /k s ) 1 egligible internal (ondutive) resistane Bi < 01 (also alled lumped apaitane/parameter method) Finite internal and external resistanes 01 < Bi < 40 3 egligible external (onvetive) resistane Bi > 40 D for unsteady state heat transfer: Distane between points of maximum temperature differene within solid objet D for sphere: Radius of sphere D for an infinitely long ylinder: Radius of ylinder D for infinite slab with heat transfer from top & bottom: Half thikness of slab D for an infinite slab with heat transfer from top: Thikness of slab 41 Modes of Heat Transfer from Air to the Center of a Sphere Consider hot air (at 100 C) being blown over a old sphere (at 0 C) h 0 C 100 C air 0 C The two modes of heat transfer are Convetion (external) Q = ha T = T/(1/hA) Condution (internal) Q = ka T/x = T/(x/kA) 4 14

15 Bi Signifiane of Magnitude of Bi h D D / ks D / ksa Condutive Resis tan e k 1/ h 1/ ha Convetive Resis tan e s Bi < 01 (Cat #1) => Condutive resistane is low Ours when k s is very high (metals) OR D is very small t = 0 min t = min 100 C air 100 C air 0 C 0 C 50 C 49 C Bi > 40 (Cat #3) => Convetive resistane is low Ours when h is very high ( Re is high) OR D is large t = 0 min t = min 100 C air 100 C air 0 C 0 C 95 C 55 C 43 Signifiane of Magnitude of Bi (ontd) Bi h D k s D / k 1/ h s D / ksa Condutive Resis tan e 1/ ha Convetive Resis tan e 01 < Bi < 40 (Cat #) => either ondutive nor onvetive resistane is negligible (both are of the same order of magnitude) Ours when neither h nor k is very high t = 0 min t = min 100 C air 100 C air 0 C 0 C 70 C 45 C 44 Category #1 Temperature Ratio (TR) T T T T e h A t Vp i The above equation an be used to determine temperature, T, at time, t OR Based on time-temperature data, the equation an be used to determine h T i : Initial temperature of solid objet (K) T : Temperature of surrounding fluid (K) h: Convetive heat transfer oeffiient (W/m K) A: Surfae area for heat transfer (m ) : Density of solid objet (kg/m 3 ) V: Volume of solid objet (m 3 ) p : Speifi heat of solid objet (J/kg K) ote: V = mass of objet (kg) Shape Area Volume Brik (LW+LH+WH) LWH Cylinder RL + R R L Sphere 4R (4/3)R 3 L: Length of brik or ylinder W, H: Width, height of brik resp R: Radius of ylinder or sphere 45 15

16 Sample Heisler Chart 1 01 Bi h D k s Bi 1/ Bi = ks h D / Bi = Fo t D ote: s = k/( p ) = Thermal diffusivity of solid objet in m /s s 46 Summary of Categories of Unsteady State Heat Transfer h D Bi ks Category 1 Category Category 3 Bi Bi < < Bi < 40 Bi > 40 This ategory is enountered when Resistane that is negligible T that is small Solution approah k is high OR D is small Condutive (Internal) Btwn enter and surfae of solid Lumped parameter eqn either k nor h are high and D is not too small or too large one one Heisler hart(s) h is high OR D is large Convetive (External) Btwn fluid and surfae of solid Heisler hart(s) (with 1/ Bi = 0) 47 Finite Objets Finite objets (suh as a ylinder or brik an be obtained as an intersetion of infinite objets) TR (TR) Finite ylinder Infinite ylinder x(tr) Infnite slab TR (TR) Finite brik Infinite slab,length X (TR) Infinite slab,width X (TR) Infinite slab, height Heisler harts have to be used twie or thrie respetively to determine temperatures for finite ylinder (food in a an) and finite brik (food in a tray) ote: (t) finite ylinder (t) infinite ylinder + (t) infinite slab (t) finite brik (t) infinite slab, width + (t) infinite slab, depth + (t) infinite slab, height 48 16

17 Calulations for Finite Cylinder (Heisler Chart) Charateristi dimension (D ) Biot number ( Bi ) Bi = h D /k s 1/ Bi Thermal diffusivity () s = k s /( s p(s) ) Fourier number ( Fo ) Fo = s t/d Temperature ratio (TR) from Heisler hart (based on values of Fo & 1/ Bi ) TR Infinite Cylinder Infinite Slab If both Bi < 01, the lumped parameter method (eqn) an be used Finite ylinder (T T ) /(Ti T ) (TR ) Infinite ylinder x (TR ) Infnite slab Solve for T from the above equation 49 Calulations for Finite Brik (Heisler Chart) Charateristi dimension (D ) Biot number ( Bi ) Bi = h D /k s 1/ Bi Thermal diffusivity ( s ) s = k s /( s p(s) ) Fourier number ( Fo ) Fo = s t/d Temperature ratio (TR) from Heisler hart (based on values of Fo & 1/ Bi ) TR Infinite Slab #1 Infinite Slab # Infinite Slab #3 If all 3 Bi < 01, the lumped parameter method (eqn) an be used Finite brik (T T ) /(Ti T ) (TR) Infinite slab#1 x(tr) Infinite slab# x(tr) Infnite slab#3 Solve for T from the above equation 50 17