Heat Transfer

HEAT TRANSFER

convection, and radiation.

BASIC HEAT TRANSFER RATE EQUATIONS Conduction

Fourier’s Law of Conduction , where

Qo = -kAdT_dx Qo = rate of heat transfer (W)

k = the thermal conductivity [W/(m·K)]

A = the surface area perpendicular to direction of heat

transfer (m2₎ Convection

Newton’s Law of Cooling , where

Qo =hA T_ w-T3i

h = the convection heat transfer coefficient of the fluid

[W/(m2·K)]

A = the convection surface area (m2₎ T_w = the wall surface temperature (K)

T_∞ = the bulk fluid temperature (K)

Radiation

The radiation emitted by a body is given by ,

Q_{o = fv}AT where4 ε = the emissivity of the body σ = the Stefan-Boltzmann constant = 5.67 × 10-8 _W/(m2·K4₎ A = the body surface area (m2₎ T = the absolute temperature (K) CONDUCTION

Conduction Through a Plane Wall T1 Q T2 k L , where Qo =-kA T

^

h

A = wall surface area normal to heat flow (m2₎ L = wall thickness (m)

T₁ = temperature of one surface of the wall (K)

T₂ = temperature of the other surface of the wall (K)

Conduction Through a Cylindrical Wall

T1 Q T2 r2 r1 k Cylinder (Length = L) ln Q r r kL T T 2 1 2 1 2 = r -o d

^

h

rcr= kinsulation_h 3 Thermal Resistance (R) Q _RT total = D o

Resistances in series are added: Rtotal= RR where, Plane Wall Conduction Resistance (K/W): R= _kAL, where

L = wall thickness

Cylindrical Wall Conduction Resistance (K/W): R ln _kLr ,

r 2 1 2 = _r d n where L = cylinder length Convection Resistance (K/W) : R= _hA1 Composite Plane Wall

L_B T1 T 2 h2 Q T2 T3 L_A k_B k_A Fluid 2 ∞ T 1 h1 Fluid 1 ∞ L k A k AL A A _AA _BB T 2 h1 h2 Q T3 T2 T1 ∞ T ∞1 1 1 h∞ rinsulation kinsulation h∞ rinsulation kinsulation

To evaluate Surface or Intermediate Temperatures: Q T_R T T _R T A B 1 2 2 3 = - = -o

Steady Conduction with Internal Energy Generation

The equation for one-dimensional steady conduction is , dx d T k Q 0 where gen 2 2 + = o

Qogen = the heat generation rate per unit volume (W/m3)

For a Plane Wall

Q" Ts1 Ts2 T(x) 1 k x gen Q Q"₂ L 0 L − T x Q L_k L x T T Lx T T 2 1 2 2 gen 2 s s s s 2 2 2 1 1 2 = o - + - + -^ h d n c mb l c m , Qo1"+Qo"2=2Q Logen where

Q_o" _{= the rate of heat transfer per area (heat flux) (W/m}2₎ Q" k dT_dx and Q" k dT_dx

L L

1= 2=

-o _{b l} o _{b l}

For a Long Circular Cylinder

Q' Ts r0 gen Q r drd rdTdr k Q 1 gen ₀ + = o b l T r Q r_k r r _T 4 1 gen s 02 02 2 = - + o

]

g

Qlo _{= the heat transfer rate from the cylinder per unit length of}

Transient Conduction Using the Lumped Capacitance Method

The lumped capacitance method is valid if ,

kA

hV ₁

Biot number, Bi where

s %

=

h = the convection heat transfer coefficient of the fluid

[W/(m2·K)]

V = the volume of the body (m3₎

k = thermal conductivity of the body [W/(m·K)]

A_s = the surface area of the body (m2₎

As

Body ρ, V, c , T P

T h,Fluid_∞

Constant Fluid Temperature

If the temperature may be considered uniform within the body at any time, the heat transfer rate at the body surface is given by

,

Qo =hA Ts^ -T3h= - tV c

^

h

T_∞ = the fluid temperature (K) ρ = the density of the body (kg/m3₎

c_P = the heat capacity of the body [J/(kg·K)] t = time (s)

The temperature variation of the body with time is , T-T3=_Ti- T e3i - bt where Vc hA P s = b _t and s 1 where time constant = = b _x x ^ h

The total heat transferred (Q_total) up to time t is ,

Qtotal= tVc TP_ i-T wherei T_i = initial body temperature (K)

Variable Fluid Temperature

If the ambient fluid temperature varies periodically according to the equation

cos

T3=T3,mean+ 1₂_T3,max-T3,mini ^~th

The temperature of the body, after initial transients have died away, is cos tan T 2 T T t T 1 , , , max min mean 2 2 1 = + -- + ~ b b ~ b ~ 3 3 3 -_ c i m ; = E G Fins

For a straight fin with uniform cross section (assuming negligible heat transfer from tip),

, tanh

Qo = hPkA Tc_ b-T3i _mL whereci

h = the convection heat transfer coefficient of the fluid

[W/(m2·K)]

P = perimeter of exposed fin cross section (m) k = fin thermal conductivity [W/(m·K)] A_c = fin cross-sectional area (m2₎ T_b = temperature at base of fin (K)

T_∞ = fluid temperature (K)

m _kAhP

c

=

L_c = L₊ A_Pc_{, corrected length of fin (m)}

Rectangular Fin L w w = 2w + 2t P = t Tb Ac T , h∞ t Pin Fin L D 2 = π P D 4 πD = Tb Ac T , h∞ CONVECTION Terms D = diameter (m)

h = average convection heat transfer coefficient of the fluid

[W/(m2·K)]

L = length (m)

Nu = average Nusselt number Pr = Prandtl number = cP_kn u_m = mean velocity of fluid (m/s)

u_∞ = free stream velocity of fluid (m/s) µ = dynamic viscosity of fluid [kg/(s·m)]

ρ = density of fluid (kg/m3₎ External Flow

In all cases, evaluate fluid properties at average temperature between that of the body and that of the flowing fluid. Flat Plate of Length L in Parallel Flow

. < . > Re Re Pr Re Re Pr Re u L Nu hL_k Nu hL_k 0 6640 10 0 0366 . 10 L L L L L L L 1 2 1 3 5 0 8 1 3 5 = = = = = n t 3 _ _ i i Cylinder of Diameter D in Cross Flow

, Re Re Pr u D Nu hD_k C where D D nD 1 3 = = = n t 3 Re_D C n 1 – 4 0.989 0.330 4 – 40 0.911 0.385 40 – 4,000 0.683 0.466 4,000 – 40,000 0.193 0.618 40,000 – 250,000 0.0266 0.805

Flow Over a Sphere of Diameter, D

. . , < < , ; . < < Re Pr Re Pr Nu hD_k 2 0 0 60 1 70 000 0 6 400 D D D 1 2 1 3 = = +

^

h

Laminar Flow in Circular Tubes

For laminar flow (Re_D < 2300), fully developed conditions

Nu_D = 4.36 (uniform heat flux)

For laminar flow (Re_D < 2300), combined entry length with constant surface temperature

. Re Pr , Nu DL 1 86 where . D D s b 1 3 0 14 =

_f

_p

µ_b = dynamic viscosity of fluid [kg/(s·m)] at

bulk temperature of fluid, T_b

µ_s = dynamic viscosity of fluid [kg/(s·m)] at

inside surface temperature of the tube, T_s

Turbulent Flow in Circular Tubes

For turbulent flow (Re_D > 104_{, Pr > 0.7) for either uniform}

surface temperature or uniform heat flux condition, Sieder-Tate equation offers good approximation:

. Re Pr Nu 0 027 . . D 0 8D 1 3 b_s 0 14 = d nn_n Non-Circular Ducts

In place of the diameter, D, use the equivalent (hydraulic) diameter (D_H) defined as

D

wetted perimeter 4 cross sectional area

H= #

-Circular Annulus (D_o > D_i)

In place of the diameter, D, use the equivalent (hydraulic) diameter (D_H) defined as

DH= Do-Di

Liquid Metals (0.003 < Pr < 0.05)

. . Re Pr

Nu 6 3 0 0167 . .

D= + 0 85D 0 93 (uniform heat flux)

. . Re Pr

NuD= 7 0+ 0 025 0 8D. 0 8. (constant wall temperature) Condensation of a Pure Vapor

On a Vertical Surface . , Nu hL_k k T T gh L 0 943 where . L l l sat s l fg 2 3 0 25 = = -n t _ i

>

H

ρ_l = density of liquid phase of fluid (kg/m3₎ g = gravitational acceleration (9.81 m/s2₎ h_fg = latent heat of vaporization [J/kg]

L = length of surface [m]

µ_l = dynamic viscosity of liquid phase of fluid [kg/(s·m)] k_l = thermal conductivity of liquid phase of fluid [W/(m·K)] T_sat = saturation temperature of fluid [K]

T_s = temperature of vertical surface [K]

Note: Evaluate all liquid properties at the average temperature between the saturated temperature, T_sat, and the surface temperature, T.

Outside Horizontal Tubes

. , Nu hD_k k T T gh D 0 729 where . D l l sat s l fg 2 3 0 25 = = -n t _ i

>

H

D = tube outside diameter (m)

Note: Evaluate all liquid properties at the average temperature between the saturated temperature, T_sat, and the surface temperature, T_s.

Natural (Free) Convection

Vertical Flat Plate in Large Body of Stationary Fluid

Equation also can apply to vertical cylinder of sufficiently large diameter in large body of stationary fluid.

,

h_{r =C Lb l}k Ra where_Ln

L = the length of the plate (cylinder) in the vertical

direction Ra_L = Rayleigh Number = Pr v g Ts T L 2 3 -b_ 3i T_s = surface temperature (K) T_∞ = fluid temperature (K)

β = coefficient of thermal expansion (1/K) (For an ideal gas: _T 2_T

s

= ₊

b

3 with T in absolute temperature) υ = kinematic viscosity (m2_/s)

Range of Ra_L C n

104 _{– 10}9 _{0.59 1/4}

109 _{– 10}13 _{0.10 1/3}

Long Horizontal Cylinder in Large Body of Stationary Fluid , h=_{C D}b lk Ra wherenD Pr v g T T D RaD s 2 3 = b_ - 3i Ra_D C n 10–3 _{– 10}2 _{1.02 0.148} 102 _{– 10}4 _{0.850 0.188} 104 _{– 10}7 _{0.480 0.250} 107 _{– 10}12 _{0.125 0.333} Heat Exchangers

The rate of heat transfer in a heat exchanger is ,

Qo =UAF T whereD lm

A = any convenient reference area (m2₎

F = heat exchanger configuration correction factor (F = 1 if temperature change of one fluid is negligible)

U = overall heat transfer coefficient based on area A and the log mean temperature difference [W/(m2·K)]

Heat Exchangers (cont.)

Overall Heat Transfer Coefficient for Concentric Tube and Shell-and-Tube Heat Exchangers

, ln UA hA A R kL D D A R h A 1 1 2 1 where i i i fi i o o fo o o = + + _r + + d n

A_i = inside area of tubes (m2₎ A_o = outside area of tubes (m2₎ D_i = inside diameter of tubes (m)

D_o = outside diameter of tubes (m)

h_i = convection heat transfer coefficient for inside of tubes [W/(m2·K)]

h_o = convection heat transfer coefficient for outside of tubes [W/(m2·K)]

k = thermal conductivity of tube material [W/(m·K)]

R_fi = fouling factor for inside of tube [(m2·K)/W]

R_fo = fouling factor for outside of tube [(m2·K)/W] Log Mean Temperature Difference (LMTD) For counterflow in tubular heat exchangers

ln T T T T T T T T T lm Hi Co Ho Ci Ho Ci Hi Co = -- - -D d _ _ n i i

For parallel flow in tubular heat exchangers , ln T T T T T T T T T where lm Hi Ci Ho Co Ho Co Hi Ci = -- - -D d _ _ n i i

∆T_lm = log mean temperature difference (K)

T_Hi = inlet temperature of the hot fluid (K) T_Ho = outlet temperature of the hot fluid (K)

T_Ci = inlet temperature of the cold fluid (K)

T_Co = outlet temperature of the cold fluid (K) Heat Exchanger Effectiveness, ε

Q Q

maximum possible heat transfer rateactual heat transfer rate max = = f _o o C T T C T T C T T C T T or min Hi Ci min H Hi Ho Hi Ci C Co Ci = -= -f f _ _ _ _ i i i i where

C = mco P =heat capacity rate (W/K)

C_min = smaller of C_C or C_H Number of Transfer Units (NTU)

NTU _CUA

min =

Effectiveness-NTU Relations

C _CC heat capacity ratio max

min

r= =

For parallel flow concentric tube heat exchanger exp ln C NTU C NTU _C C 1 1 1 1 1 1 r r r r = + - - + = -+ - + f f

^

^

h

h

For counterflow concentric tube heat exchanger

< < exp exp ln C NTU C NTU C C NTU NTU _C NTU _{C C} C NTU C 1 1 1 1 1 1 1 1 1 1 1 ₁ 1 1 r r r r r r r r r = - - -- - -= ₊ = = - -= _- = f f f f f f

^

^

_^

^

^

^

h

h

_h

h

h

h

For any body, α + ρ + τ = 1 , where

α = absorptivity (ratio of energy absorbed to incident energy) ρ = reflectivity (ratio of energy reflected to incident energy) τ = transmissivity (ratio of energy transmitted to incident energy)

Opaque Body

For an opaque body: α + ρ = 1 Gray Body

A gray body is one for which

α = ε, (0 < α < 1; 0 < ε < 1), where ε = the emissivity of the body

For a gray body: ε + ρ = 1

Real bodies are frequently approximated as gray bodies.

Black body

A black body is defined as one which absorbs all energy incident upon it. It also emits radiation at the maximum rate for a body of a particular size at a particular temperature. For such a body

Shape Factor (View Factor, Configuration Factor) Relations

Reciprocity Relations

A_iF_ij = A_jF_ji, where

A_i = surface area (m2_{) of surface i}

F_ij = shape factor (view factor, configuration factor); fraction of the radiation leaving surface i that is intercepted by surface j; 0 ≤ Fij ≤ 1

Summation Rule for N Surfaces

Fij 1 j N 1 = = !

Net Energy Exchange by Radiation between Two Bodies

Body Small Compared to its Surroundings ,

Qo12= fvA T` 14-T where24j

Qo12 = the net heat transfer rate from the body (W) ε = the emissivity of the body

σ = the Stefan-Boltzmann constant [σ = 5.67 × 10-8 _W/(m2·K4_)] A = the body surface area (m2₎

T₁ = the absolute temperature [K] of the body surface

T₂ = the absolute temperature [K] of the surroundings Net Energy Exchange by Radiation between Two Black Bodies

The net energy exchange by radiation between two black bodies that see each other is given by

Qo12= A F1 12v`T14-T24j

Net Energy Exchange by Radiation between Two Diffuse-Gray Surfaces that Form an Enclosure

Generalized Cases A2 T2 ε2 12 , , A2, ,T2 ε2 A1, ,T1 ε1 A1, ,T1 ε1 Q Q₁₂ Q A A F A T T 1 1 1 12 1 1 1 1 12 2 2 2 14 24 = -+ + -f f f f v o ` j

One-Dimensional Geometry with Thin Low-Emissivity Shield Inserted between Two Parallel Plates

A2 ε2 T2 , , A3,T3 ε3, 1 ε3, 2 A1 ε1 T1 , , 12 Q Radiation Shield Q A A F A A A F A T T 1 1 1 1 1 1 , , , , 12 1 1 1 1 13 3 1 3 3 1 3 2 3 3 2 3 32 2 2 2 14 24 = -+ + - + - + + -f f f f f f f f v o ` j Reradiating Surface

Reradiating Surfaces are considered to be insulated or adiabatic Q_ oR=0i. AR, ,TR εR A2, ,T2 ε2 A1, ,T1 ε1 12 Q Q A A F _{A F A F} A T T 1 1 1 1 1 R R 12 1 1 1 1 12 1 1 2 2 1 _{2 2}2 14 24 = -+ + + + -f f f f v -o c c ` m m j = G