Chapter 3: Steady Heat Conduction
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1 3-1 Steady Heat Conduction in Plane Walls 3-2 Thermal Resistance 3-3 Steady Heat Conduction in Cylinders 3-4 Steady Heat Conduction in Spherical Shell 3-5 Steady Heat Conduction with Energy Generation 3-6 from Finned Surfaces 3-7 Bioheat Equation
2 3-1 Steady Heat Conduction in Plane Walls (1) 3-1
3 3-1 Steady Heat Conduction in Plane Walls (2) Assuming heat transfer is the only energy interaction and there is no heat generation, the energy balance can be expressed as Q in 0 dewall Qout 0 dt The rate of heat transfer through the wall must be constant ( ). Q, constant cond wall T T Q ka L 1 2 cond, wall (W) 3-2
4 3-2 Thermal Resistance (1) Q T T L Rwall ( C/W) ka 1 2 cond, wall (W) Rwall where R wall is the conduction resistance (or thermal resistance) Analogy to Electrical Current Flow: I V V 1 2 R e Electrical current flow Rate of heat transfer Electric current Thermal resistance Electrical resistance Temperature difference Voltage difference 3-3
5 3-2 Thermal Resistance (2) Thermal resistance can also be applied to convection processes. Newton s law of cooling for convection heat transfer rate ( ) can be rearranged as Q conv T s R T conv Qconv has Ts T (W) R conv is the convection resistance R conv 1 ha s ( C/W) 3-4
6 3-2 Thermal Resistance (2) Radiation Resistance: The rate of radiation heat transfer between a surface and the surrounding 4 4 T s Tsurr Qrad As Ts Tsurr hrad As ( Ts Tsurr ) (W) R R rad h 1 A rad s ( K/W) Q h T T T T (W/m K) rad rad s surr s surr As ( Ts Tsurr ) 3-5 rad
7 3-2 Thermal Resistance (3) Radiation and Convection Resistance A surface exposed to the surrounding might involves convection and radiation simultaneously. The convection and radiation resistances are parallel to each other. When T surr T, the radiation effect can properly be accounted for by replacing h in the convection resistance relation by h combined = h conv +h rad (W/m 2 K) 3-6
8 3-2 Thermal Resistance (4) Thermal Resistance Network Rate of heat convection into the wall Rate of heat conduction through the wall = = Rate of heat convection from the wall Q h A T T T 1,1 1 T L 1 2 ka h2 A T2 T,2 Q T T,1,2 R total (W) 1 L 1 Rtotal Rconv,1 Rwall Rconv,2 ( C/W) h A ka h A
9 3-2 Thermal Resistance (5) It is sometimes convenient to express heat transfer through a medium in an analogous manner to Newton s law of cooling as Q UAT (W) where U is the overall heat transfer coefficient. Note that 1 UA Rtotal ( C/K) 3-8
10 3-2 Thermal Resistance (6) Multilayer Plane Walls R R R R R total conv,1 wall,1 wall,2 conv,2 1 L L 1 h A k A k A h A
11 3-2 Thermal Resistance (7) Thermal Contact Resistance -In reality surfaces have some roughness. When two surfaces are pressed against each other, the peaks form good material contact but the valleys form voids filled with air. -As a result, an interface contains numerous air gaps of varying sizes that act as insulation because of the low thermal conductivity of air. -Thus, an interface offers some resistance to heat transfer, which is termed the thermal contact resistance, Rc. 3-10
12 3-2 Thermal Resistance (8) Generalized Thermal Resistance Network T T T T 1 1 Q Q Q T T R1 R2 R1 R2 R total RR R = Rtotal R R R R 1 2 total
13 3-2 Thermal Resistance (9) Combined Series-Parallel Arrangement Q T1 T R total R R R R RR R R 1 2 total 12 3 conv 3 conv R1 R2 L L L R R R R ; 2 ; 3 ; conv k1a1 k2a2 k3a3 ha
14 3-3 Steady Heat Conduction in Cylinders (1) Consider the long cylindrical layer Assumptions: the two surfaces of the cylindrical layer are maintained at constant temperatures T 1 and T 2, no heat generation, constant thermal conductivity, one-dimensional heat conduction. Fourier s law of heat conduction Q cond cyl dt, ka (W) dr 3-13
15 3-3 Steady Heat Conduction in Cylinders (2) dt Qcond, cyl ka (W) dr Separating the variables and integrating from r=r 1, where T(r 1 )=T 1, to r=r 2, where T(r 2 )=T 2 r2 T2 Q cond, cyl dr kdt A rr1 T T1 Substituting A =2prL and performing the integrations give T1 T2 Qcond, cyl 2 Lk ln r / r Since the heat transfer rate is constant T1 T2 Qcond, cyl R cyl
16 3-3 Steady Heat Conduction in Cylinders (3) Alternative method: 3-15
17 3-3 Steady Heat Conduction in Cylinders (3) Thermal Resistance with Convection Steady one-dimensional heat transfer through a cylindrical or spherical layer that is exposed to convection on both sides Q where T T,1,2 R total R R R R total conv,1 cyl conv, ln r / r 1 2 r L h 2 Lk 2 r L h
18 3-3 Steady Heat Conduction in Cylinders (4) Multilayered Cylinders R R R R R R total conv,1 cyl,1 cyl,3 cyl,3 conv,2 r r r r r r 1 ln / ln / ln / 1 2 r L h 2 Lk 2 Lk 2 Lk 2 r L h
19 3-3 Steady Heat Conduction in Cylinders (5) 3-18
20 3-3 Steady Heat Conduction in Cylinders (6) Critical Radius of Insulation Q 2 T1T T1T R ln r2/ r ins Rconv 1 1 2Lk h 2r L The value of r 2 at which reaches a maximum is determined by dq 0 dr k rcr, cylinder (m) h 2 2 d Q 2 dr 2 0 Thus, insulating the pipe may actually increase the rate of heat transfer instead of decreasing it. 3-19
21 3-3 Steady Heat Conduction in Cylinders (7) Critical Radius of Insulation -Adding more insulation to a wall or to the attic always decreases heat transfer. -Adding insulation to a cylindrical pipe or a spherical shell, however, is a different matter. -Adding insulation increases the conduction resistance of the insulation layer but decreases the convection resistance of the surface because of the increase in the outer surface area for convection. -The heat transfer from the pipe may increase or decrease, depending on which effect dominates. 3-20
22 3-4 Steady Heat Conduction in Spherical Shell Spherical Shell 3-21
23 3-5 Steady Heat Conduction with Energy Generation (1) 3-22
24 3-5 Steady Heat Conduction with Energy Generation (2) C 1 T s,2 T 2L s,1, C 2 q 2k L 2 T s,2 T 2L s,1 3-23
25 3-5 Steady Heat Conduction with Energy Generation (3) E out E g
26 3-5 Steady Heat Conduction with Energy Generation (4) Combined one-dimensional heat conduction equation n=0: plane wall (r >x) n=1: cylinder n=2: sphere 3-25
27 3-5 Steady Heat Conduction with Energy Generation (5) and E out E g qr 0 0 Ts T 3h 3-26
28 3-6 from Finned Surfaces (1) Newton s law of cooling Q conv has Ts T Two ways to increase the rate of heat transfer: increasing the heat transfer coefficient, increase the surface area fins Fins are the topic of this section. 3-27
29 3-6 from Finned Surfaces (2) Fin Equation Under steady conditions, the energy balance on this volume element can be expressed as Rate of heat conduction into the element at x or where Rate of heat conduction from the element at x+x = + Q Q Q cond, x cond, xx conv Q conv h px T T Substituting and dividing by x, we obtain Q Q Rate of heat convection from the element 3-28 x cond, xx cond, x hp T T 0
30 3-6 from Finned Surfaces (3) dq cond Q dx cond hp T T 0 dt kac dx 2 d 2 m 0 2 dx hp T T ; m ka c d dx dt kac hp T T dx 0 ( x) C e C e mx 1 2 mx 3-29
31 3-6 from Finned Surfaces (4) Boundary Conditions Several boundary conditions are typically employed: At the fin base Specified temperature boundary condition, expressed as: q(0)= q b = T b -T At the fin tip 1. Specified temperature 2. Infinitely Long Fin 3. Adiabatic tip 4. Convection (and combined convection and radiation). 3-30
32 3-6 from Finned Surfaces (5) Infinitely Long Fin For a sufficiently long fin the temperature at the fin tip approaches the ambient temperature Boundary condition: θ(l )=T(L)-T =0 When x so does e x=0: e mx =1 C 1 =0, C 2 = The temperature distribution: T x T T ( ) T mx x hp / kac b e e b dt heat transfer from the entire fin Q kac hpkac Tb T dx x0 3-31
33 3-6 from Finned Surfaces (6) Adiabatic Tip Boundary condition at fin tip: d 0 dx xl After some manipulations, the temperature distribution: T( x) T cosh m L x T T cosh ml heat transfer from the entire fin c c b dx x0 b dt Q ka hpka T T tanh ml 3-32
34 3-6 from Finned Surfaces (7) Convection (or Combined Convection and Radiation) from Fin Tip A practical way of accounting for the heat loss from the fin tip is to replace the fin length L in the relation for the insulated tip case by a corrected length defined as L c =L+A c /p For rectangular and cylindrical fins L c is L c,rectangular =L+t/2 L c,cylindrical =L+D/4 3-33
35 3-7 Bioheat Equation (1) Surroundings (convection, radiation) E, in E out Outer skin Inner skin Fat Muscle Bone (core) Conduction & No heat source heat generation (metabolic, perfusion, etc) E, g E st Ref: Fiala, D., Lomas, K., Stohrer, M., A Computer Model of Human Thermogulation for a Wide Range of Environmental Conditions: the Passive System, Journal of Physiology, Vol. 87, pp ,
36 3-7 Bioheat Equation (2) 1-D, Steady 2 d T qm q 2 dx k q c ( T p : metabolic heat generation rate p 0 : perfusion heat generation rate b b T ) a q m q p ω: perfusion rate, m 3 /s ρ b, c b : blood density and specific heat T a : arterial temperature 3-35
37 3-7 Bioheat Equation (3) 1-D, transient Governing Equation: 2 T k 2 r T r r q m W bl bl c bl ( Tbl, a T T) c t Initial condition: t=0 Boundary conditions: at surface 3-36
38 3-7 Bioheat Equation (4) Boundary Conditions Convection with ambient air, q c : Newton s cooling law Radiation with surrounding surfaces, q R Irradiation from high-temperature sources, q sr Evaporation of moisture from the skin, q e E in =E out q sk +q sr =q c +q R +q e T q c q R q sr q e T w Surroundings q sk q c q R q sr q e q sk Skin 3-37
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