2-1 General Relation for Fourier s Law of Heat Conduction 2-2 Heat Conduction Equation 2-3 Boundary Conditions and Initial Conditions 2-4 Variable Thermal Conductivity 2-5 Steady Heat Conduction in Plane Walls 2-6 Thermal Resistance 2-7 Steady Heat Conduction in Cylinders 2-8 Steady Heat Conduction in Spherical Shell 2-9 Steady Heat Conduction with Energy Generation 2-10 Heat Transfer from Finned Surfaces 2-11 Bioheat Equation
2-1 General Relation for Fourier s Law of Heat Conduction (1) The rate of heat conduction through a medium in a specified direction (say, in the x-direction) is expressed by Fourier s law of heat conduction for one-dimensional heat conduction as: Q& cond dt = ka dx (W) Heat is conducted in the direction of decreasing temperature, and thus the temperature gradient is negative when heat is conducted in the positive x-direction. 2-1
2-1 General Relation for Fourier s Law of Heat Conduction (2) The heat flux vector at a point P on the surface of the figure must be perpendicular to the surface, and it must point in the direction of decreasing temperature If n is the normal of the isothermal surface at point P, the rate of heat conduction at that point can be expressed by Fourier s law as Q& n dt = ka dn (W) 2-2
2-1 General Relation for Fourier s Law of Heat Conduction (3) In rectangular coordinates, the heat conduction vector can be expressed in terms of its components as r r r r Q& = Q& i + Q& j + Qk & n x y z which can be determined from Fourier s law as & & & T Qx = kax x = T Qy kay y = T Qz kaz z 2-3
2-2 Heat Conduction Equation (1) Rate of heat conduction at x Rate of heat conduction at x+δx Rate of heat generation inside the element - + = Rate of change of the energy content of the element Q & x Q& Q& x x & Q x +Δ x + E & gen, element 2 Q& Q& + Δ = Q& x x x x + Δx + ( Δx) 2 x 2! x Q& x+ Δx Q& x x = 1 2 T = ( ka ) = x x ΔE Δt +... T ( ka ) x x element Neglect higher orders 2-4
2-2 Heat Conduction Equation (2) ( ) ρ ( ) Δ E = E E = mc T T = caδx T T E & gen, element = e& genvelement = e& gen AΔx element t+δ t t t+δ t t t+δt t Q& Q & +Δ x x x t t t & = caδx + e AΔx gen T T ρ +Δ Δt Dividing by AΔx, taking the limit as Δx 0 and Δt 0, and from Fourier s law: 1 T T ka + e& gen = ρc A x x t The area A is constant for a plane wall the one dimensional transient heat conduction equation in a plane wall is T T & ρ Variable conductivity: k + egen = c x x t 2-5
2-2 Heat Conduction Equation (3) Constant conductivity: & 2 T e gen 1 T k + = ; α = 2 x k α t ρc The one-dimensional conduction equation may be reduces to the following forms under special conditions 1) Steady-state: 2 dt + e gen = 0 2 dx 2) Transient, no heat generation: & k 2 T 2 x 1 T = α t 3) Steady-state, no heat generation: 2 dt 0 2 dx = 2-6
2-2 Heat Conduction Equation (4) General Heat Conduction Equation Rate of heat conduction at x, y, and z Rate of heat conduction at x+δx, y+δy, and z+δz Rate of heat generation inside the element - + = Rate of change of the energy content of the element Q& + Q& + Q& x y z ( k T ) + e gen Q& Q& Q&, x+δ x y+δ y z+δz T = ρc t 1) Steady-state: 2) Transient, no heat generation: 3) Steady-state, no heat generation: & x y z k 2 2 2 T T T e gen + + + = 0 2 2 2 ΔE Δt + E element gen element = 2 2 2 1 + + = 2 2 2 T T T T x y z α t T T T x y z 2 2 2 + + = 0 2 2 2 2-7
2-2 Heat Conduction Equation (5) Cylindrical Coordinates 1 T 1 rk T k T k T + e T 2 + + & gen = ρc r r r r φ φ z z t 2-8
2-2 Heat Conduction Equation (6) Spherical Coordinates 1 2 T 1 T 1 kr k k sin T θ e T 2 + 2 2 + 2 + & gen = ρc r r r r sin θ φ φ r sinθ θ θ t 2-9
2-3 Boundary and Initial Conditions (1) Specified Temperature Boundary Condition Specified Heat Flux Boundary Condition Convection Boundary Condition Radiation Boundary Condition Interface Boundary Conditions Generalized Boundary Conditions 2-10
2-3 Boundary and Initial Conditions (2) Specified Temperature Boundary Condition For one-dimensional heat transfer through a plane wall of thickness L, for example, the specified temperature boundary conditions can be expressed as T(0, t) = T 1 T(L, t) = T 2 The specified temperatures can be constant, which is the case for steady heat conduction, or may vary with time. 2-11
2-3 Boundary and Initial Conditions (3) Specified Heat Flux Boundary Condition The heat flux in the positive x-direction anywhere in the medium, including the boundaries, can be expressed by Fourier s law of heat conduction as q& dt = k = dx Heat flux in the positive x- direction The sign of the specified heat flux is determined by inspection: positive if the heat flux is in the positive direction of the coordinate axis, and negative if it is in the opposite direction. 2-12
2-3 Boundary and Initial Conditions (4) Two Special Cases Insulated boundary Thermal symmetry k T(0, t) T(0, t) = 0 or = 0 x x T ( L, t) 2 x = 0 2-13
2-3 Boundary and Initial Conditions (5) Convection Boundary Condition Heat conduction at the surface in a selected direction = Heat convection at the surface in the same direction and T(0, t) k = h1 T 1 T t x [ (0, )] TLt (, ) k = h T L t T x [ (, ) ] 2 2 2-14
2-3 Boundary and Initial Conditions (6) Radiation Boundary Condition Heat conduction at the surface in a selected direction = Radiation exchange at the surface in the same direction and T(0, t) k = εσ T T(0, t) x 4 4 1 surr,1 T( L, t) k = εσ T( L, t) Tsurr x 4 4 2,2 2-15
2-3 Boundary and Initial Conditions (7) Interface Boundary Conditions At the interface the requirements are: (1) two bodies in contact must have the same temperature at the area of contact, (2) an interface (which is a surface) cannot store any energy, and thus the heat flux on the two sides of an interface must be the same. T A (x 0, t) = T B (x 0, t) and T ( x 0, t A ) T B( x 0, t ) ka = kb x x 2-16
2-4 Variable Thermal Conductivity (1) The thermal conductivity of a material, in general, varies with temperature. An average value for the thermal conductivity is commonly used when the variation is mild. This is also common practice for other temperaturedependent properties such as the density and specific heat. 2-17
2-4 Variable Thermal Conductivity (2) Variable Thermal Conductivity for One-Dimensional Cases When the variation of thermal conductivity with temperature k(t) is known, the average value of the thermal conductivity in the temperature range between T 1 and T 2 can be determined from T 2 ktdt ( ) T1 kave = T2 T1 The variation in thermal conductivity of a material with can often be approximated as a linear function and expressed as kt ( ) = k(1 + βt) 0 β : the temperature coefficient of thermal conductivity. 2-18
2-4 Variable Thermal Conductivity (3) Variable Thermal Conductivity For a plane wall the temperature varies linearly during steady onedimensional heat conduction when the thermal conductivity is constant. This is no longer the case when the thermal conductivity changes with temperature (even linearly). 2-19
2-4 Variable Thermal Conductivity (4) 2-20
2-4 Variable Thermal Conductivity (5) 2-21
2-4 Variable Thermal Conductivity (6) 2-22
2-5 Steady Heat Conduction in Plane Walls (1) 2-23
2-5 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 Q& out = = 0 dt The rate of heat transfer through the wall must be constant ( ). T T Q& ka L Q & cond, wall = constant 1 2 cond, wall = (W) 2-24
2-6 Thermal Resistance (1) T T Q& L o 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 Heat Transfer Electrical current flow Rate of heat transfer Electric current Thermal resistance Electrical resistance Temperature difference Voltage difference 2-25
2-6 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 o ( C/W) 2-26
2-6 Thermal Resistance (2) Radiation Resistance: The rate of radiation heat transfer between a surface and the surrounding ( 4 4 T ) s Tsurr Q& rad = εσ As Ts Tsurr = hrad As ( Ts Tsurr ) = (W) R R rad = h 1 A rad s ( K/W) Q& h = = εσ T + T T + T (W/m K) ( )( ) rad 2 2 2 rad s surr s surr As ( Ts Tsurr ) rad 2-27
2-6 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) 2-28
2-6 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 = h2a T2 T,2 Q& = T T,1,2 R total (W) 1 L 1 Rtotal Rconv,1 Rwall Rconv,2 (C/W) o = + + = + + ha ka ha 1 2 2-29
2-6 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& = UAΔT (W) where U is the overall heat transfer coefficient. Note that 1 UA = Rtotal o ( C/K) 2-30
2-6 Thermal Resistance (6) Multilayer Plane Walls R = R + R + R + R total conv,1 wall,1 wall,2 conv,2 1 L L 1 ha ka k A ha 1 2 = + + + 1 1 2 2 2-31
2-6 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. 2-32
2-6 Thermal Resistance (8) Generalized Thermal Resistance Network T T T T 1 1 Q& = Q& + Q& = + = ( T T ) + 1 1 2 1 2 1 2 1 2 R1 R2 R1 R2 R total 1 1 1 RR = + R = Rtotal R R R R 1 2 total 1 2 1+ 2 2-33
2-6 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 1 2 3 1 = ; 2 = ; 3 = ; conv = ka 1 1 ka 2 2 ka 3 3 ha3 1 2-34
2-7 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 2-35
2-7 Steady Heat Conduction in Cylinders (2) dt Q& cond, 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 r= r1 T= T1 Substituting A =2prL and performing the integrations give T1 T2 Q& cond, cyl = 2π Lk ln r / r ( ) 2 1 Since the heat transfer rate is constant T1 T2 Q& cond, cyl = R cyl 2-36
2-7 Steady Heat Conduction in Cylinders (3) Alternative method: 2-37
2-7 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,2 ( r r ) 1 ln / 1 2 1 = + + ( 2πrL 1 ) h1 2πLk ( 2πrL 2 ) h2 2-38
2-7 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 2πrL h 2πLk 2πLk 2πLk 2πrL h 2 1 3 2 4 3 = + + + + ( ) 1 1 1 2 3 2 2 2-39
2-7 Steady Heat Conduction in Cylinders (5) 2-40
2-7 Steady Heat Conduction in Cylinders (6) Critical Radius of Insulation T1 T T1 T Q& = = R ln ( r2 / r ins + Rconv 1) 1 + 2πLk h 2 r L dq& dr = 2 0 r k, = (m) h cr cylinder ( π ) The value of r 2 at which reaches a maximum is determined by 2 2 d Q & 2 dr 2 < 0 Thus, insulating the pipe may actually increase the rate of heat transfer instead of decreasing it. 2-41
2-7 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. 2-42
2-8 Steady Heat Conduction in Spherical Shell Spherical Shell 2-43
2-9 Steady Heat Conduction with Energy Generation (1) 2-44
2-9 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 2-45
2-9 Steady Heat Conduction with Energy Generation (3) E & out + E& g = 0 2-46
2-9 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 2-47
2-9 Steady Heat Conduction with Energy Generation (5) and qr E & & 0 out + E& g = 0 Ts = T + 3h 2-48
2-10 Heat Transfer from Finned Surfaces (1) Newton s law of cooling & Qconv = 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. 2-49
2-10 Heat Transfer 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+δx = + Q& = Q& + Q& Q& conv = h pδx T T cond, x cond, x+δx conv Substituting and dividing by Δx, we obtain Q& cond, x+δx cond, x Δx Q& ( )( ) Rate of heat convection from the element + hp T T = ( ) 0 2-50
2-10 Heat Transfer from Finned Surfaces (3) & dq cond Q& dx cond = + hp T = ( T ) 0 dt kac dx 2 d θ 2 m θ = 0 2 dx hp θ = T T ; m= ka c d dx dt kac hp T T = dx ( ) 0 θ ( x) = Ce + C e mx 1 2 mx 2-51
2-10 Heat Transfer 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). 2-52
2-10 Heat Transfer 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 mx @ 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 θ b dt heat transfer from the entire fin Q& = kac = hpkac ( Tb T ) dx x= 0 2-53
2-10 Heat Transfer from Finned Surfaces (6) Adiabatic Tip Boundary condition at fin tip: dθ = 0 dx x= L After some manipulations, the temperature distribution: heat transfer from the entire fin ( x) T( x) T cosh m L = T T cosh ml c c b dx x= 0 b dt Q& = ka = hpka T T tanh ml ( ) 2-54
2-10 Heat Transfer 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 2-55
2-11 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. 1957-1972, 1999. 2-56
2-11 Bioheat Equation (2) 1-D, Steady 2 d T q& m + q& + 2 dx k q& = ωρ c ( T p b b 0 T ) p a = q& m q& p : metabolic heat generation rate : perfusion heat generation rate ω: perfusion rate, m 3 /s ρ b, c b : blood density and specific heat T a : arterial temperature 2-57
2-11 Bioheat Equation (3) 1-D, transient Governing Equation: 2 T k 2 r ω T + r r + q m + ρ W bl bl c bl ( Tbl, a T ) T = ρc t Initial condition: t=0 Boundary conditions: at surface 2-58
2-11 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 = q + q q + sk c R sr q e q sk Skin 2-59