Heat Transfer: Physical Physic al Origin Originss and Rate Equations Chapter One
• What is heat is heat transfer ? Heat transfer is thermal energy in transit due to a temperature difference.
• What is thermal is thermal energy? energy? Thermal energy is associated with the translation, rotation, vibration and electronic states of the atoms and molecules that comprise matter matter.. It represents the cumulati cumulative ve effect of microscopic activities and is directly linked to the temperature of matter.
DO NOT confuse or interchange the meanings of Thermal Thermal Energy Energy,, Temperature DEFINITIONS and NOMENCLATURE
Heat Transfer and Heat and Quantity
Heat Transfer Heat
Energy associated with microscopic behavior of matter
U or u
A means means of indirectly indirectly assessing assessing the amount of thermal energy stored in matter
K or °C
J or J/kg
Thermal energy transport due to temperature gradients
Amount of thermal energy transferred over a time interval t 0
Thermal energy transfer per unit time
Thermal energy transfer per unit time and surface area
+ U Thermal energy of system u
U Un nits
Thermal energy per unit mass of system 3
Modes of Heat Transfer MODES OF HEAT TRANSFER
Conductio Condu ction: n: Heat tra transfer nsfer in a solid or a stationary stationary flu fluid id (gas or liquid) liquid) due to the random the random motion of motion of its constituent atoms, molecules and /or electrons. MOLECULAR DIFFUSION Convection: Heat transfer due to the combined influe influence nce of bulk bulk and random motion for motion for fluid flow over a surface. MOLECULAR DIFFUSION + ADVECTION Radiation Radi ation::
Ener Energy gy that is emitted is emitted by matter due due to changes in the electron configurations of its atoms or molecules and is transported as electromagnetic waves
• Conduction and convection require the presence of temperature variations in a material medium. • Although radiation originates originates from matter matter,, its transport does not require a material medium and occurs most efficiently in a vacuum. 4
Conduction through a stagnant gas by molecular diffusion
Heat transfe transferr b molecu molecular lar diffusi diffusion on
General (vector) form of Fourier’s Fourier’s Law:
q k T Heat flux
Thermal conductivi ctivity ty Thermal condu
Tempe emperature rature gradi gradient ent
°C/m or K/m
Application to one-dimensional, to one-dimensional, steady conduction steady conduction across a plane wall of constant thermal ther mal conductivity: con ductivity:
q x k dT dx
k T2 T1
In this specific case, the temperature profile is linear.
q x k
rate (W): (W): Heat rate
T1 T 2 L
qx A 7
HEAT TRANSFER THROUGH A PLANE WALL Calculate the heat transfer rate.
Heat transfer by conduction PLUS advection
Relation of convec Relation convection tion to flow over a surfac surfacee and develop development ment of velocity and velocity and thermal thermal boundary layers:
Newton’ss law of cooling: Newton’
q h T
h : Convection heat transfer 2 K) coefficient (W/m
Examples of Convective Heat Transfer
THERMAL RADIATION: heat transfer by electromagnetic waves
The electromagnetic spectrum:
Heat transfer at a gas/surface interface involves radiation emission from emission from the surface and may also involve the radiation incident from the surroundings absorption of radiation incident (irradiation, irradiation, G ), as we well ll as con conve vect ction ion if T s T . due to emission: to emission: Energy outflow due Energy outflow 4 E Eb T s E : Emissive power r W/m2 :Surf ace emissi vity 0 1
E b : Emissive power of a blackbody (the perfect emitter)
: Stefan Stefan-B -Bol oltzm tzmann ann cons consta nt 5. 5.67 67×1 ×10 0-8 W/m W/m2 K 4 tant
due to irradiation: to irradiation: Energy absorption due Energy absorption Gabs G Gabs :Absorbed incident radiation ( W /m 2 )
Special case of case of surface exposed to large to large surroundings of surroundings of uniform temperature, T
G G sur
the net radiation heat flux from the
surface surfa ce due to exchange with the surroundings is: qrad
Ts G T s4 Tsu4r
<<< a very useful approximation
hr : Radiation heat transfer coefficient W/ m 2 K hr
Ts Tsur Ts2 Tsu2r
For combined convection and radiation,
h T s
Problem: Electronic Cooling
Problem Proble m 1.40: Power diss dissipation ipation from chips operati operating ng at a surfac surfacee temperatur temperaturee of 85C and in an enclosure whose walls and air are at 25 C for (a) free convection and (b) forced convect convection. ion. Schematic:
conditions, (2) Radiation exchange between a small surface and a large a large enclosure, (3) enclosure, (3) Assumptions: (1) Steady-state (1) Steady-state conditions, Negligible heat transfer from from sides of chip or from back of chip by chip by conduction through the substrate. Analysis:
Pe le c
q co n v
q ra d 2
4 hA T s T A Ts4 Tsur
2.25×10-4 m 2
(a) If heat transfer is by natural by natural convection, 5/ 4
T s T
= 4.2W /m 2 K 5/4 2.25 1 0-4 m 2 60K
0.60 2 2..25 10 m -4
= 0.158W 4
use absolute temperature in radiation calculations
0.158W + 0.065W = 0.223W
(b) If heat transfer is by forced by forced convection,
hA T s
3.375W 5W 3.37
250W /m 2 K 4 2.25 1 10 0-4 m 2 60K 3.375W
+ 0.065W 0.065W 3.44W 3.44W
CONSERVATION OF ENERGY (First Law of Thermodynamics)
ROL L VO VOLU LUME ME APPLICA APPL ICATION TION TO A CONT CONTRO •
At an Instant of Time: Note representation repr esentation of system by a line) at at the boundaries. boundaries. control surface (dashed line)
in out : ,
rate of thermal and/or m mechanic echanical al energy transfer across the control surface due to heat transfer, fluid flow and/or work interac tions.
g : E
rate of thermal ener gy generation due to conversion from another energy form (e.g., electrical, nuclear, or chemical); energy conversion proc ess occurs within the system.
rate of chang e o f energy storage in the system.
Conservation of Energy in E out E g dE st E st dt
Each term has units of J/s or W. • Ov Over er a Time Interval
Storage = Inflow - Outflow + Generation
Ein Eout E g E st Each term has units of J. 23
Examplee 1.4: Appli Exampl Application cation to thermal respon response se of a conductor with Ohmic heating (generation):
energy and and for an incompressible substance. • Involves Involves change change in thermal in thermal energy
dU t dt
• Heat transfer transfer is from the conductor conductor done on the system • Genera Generation tion may may be viewed viewed as electrical as electrical work done
THE SURFACE ENERGY BALANCE A specia speciall case for which no volume or mass is encompasse encompassed d by the control surf surface. ace. Conservation of Energy
out Ein E 0
also, there may be heat generation directly at the surface
• Applies Applies for steady-state steady-state and transient conditions. conditions.
Storage = 0 always !
• With With no mass and volume, volume, energy storage and generation generation are not pertinent pertinent to the energy balance, even if they occur in the medium bounded by the surface. Considerr surface of wall with heat transfer Conside transfer by conduction, conduction, convection convection and radiation.
qcond qconv qrad 0 k
T1 T 2
h T2 T 2 T24 Ts4ur 0
METHODOLOGY OF FIRST LAW LAW ANAL ANALYSIS YSIS • On a schematic schematic of of the system, represen representt the the control control surface surface by by dashed line(s).
• Choose the app appropriat ropriatee time basis. basis.
storage terms • Identify relevant energy transport, energy transport, generation and/or storage terms by labeled by labeled arrows on arrows on the schematic.
• Wr Write ite the go governing verning fform orm of the the Conservation Conservation of Energy Energy requirement. requirement.
• Subst Substitute itute appropr appropriate iate expres expressions sions for terms of the energ energy y equation.
• Solve for th thee unknown qua quantity ntity.. 29
PROBLEM 1.85 KNOWN:
Solar collector designed to heat heat water operating under pres prescribed cribed solar irradiation and loss loss
(a) Useful heat collected per unit area of the collector, collector, q ′′ u , (b) Temperature rise of the water
flow, To − Ti , and (c) Collector efficiency. SCHEMATIC:
(1) Steady-state conditions conditions,, (2) No heat losses out sides o orr back of collector, (3) Collector area is small compared to sky surroundings.
PROPERTIES: Table A.6 ,
Water (300K): c p = 4179 J/kg⋅K.
(a) Defining the collector as the the control volume and writi writing ng the conservation of energy requirement on a per unit area basis, find that & E in
& −E out +
& E gen
& E st .
Identifying processes as per above right sketch,
q s′′ola r
−q ′′ ′′ rad − q c′′on v − q ′′ u =
where q s′′ola r = 0.9 q s′′; that is, 90% of the solar flux flux is absorbed in the collecto collectorr (Eq. 1.6). Using the appropriate rate equations, the useful heat rate per unit area is
4 − h ( Ts − T∞ ) 0.9 qs′′ − εσ Tc4p − Tsky W W W o − 0.94 × 5.67 ×10 q′u′ = 0.9 × 700 10−8 3034 − 2634 K 4 − 10 30 − 25 ) C ( m2 m2 ⋅ K 4 m2 ⋅ K
q ′′ ′′ u = 630 W / m
386 W / m .
u ⋅ A. Defining a control volume about the water tubing, the (b) The total useful heat collected is q ′′ useful heat causes an enthalpy enthalpy change of the flowing water. That is, & c (T q′u′ ⋅ A=m p i
( Ti − To ) = 386 W/m2 × 3m 2 / 00..01kg/s × 4179J/kg ⋅ K=27.7
(c) The efficiency is η = q u / qS = 386 W/m COMMENTS:
) / ( 700 W /m )
0.55 or 55%.
Note how the sky has been treated as large surroundings at a uniform uniform temperature
Problem: Silicon Wafer
Problem Proble m 1.57: Therma Thermall processi processing ng of silicon wafers in a two-zone furnace. furnace. Determine (a) the initial rate of change of the wafer temperature and (b) the steady-state temperature.
wafer positioned positioned in furnace with top and bottom surfaces exposed exposed to hot and cool zones, respectively. FIND: (a)
Initial rate of change of the wafer temperature from a value of T w,i 300K, and (b)
steady-state temperature. Is convection significant? steady-state significant? Sketch the variation of wafer temperature with vertical distance. distance.
Problem: Silicon Wafer (cont.) ASSUMPTIONS: (1)
Wafer temperatu temperature re is uniform, (2) Hot and cool zones have uniform uniform temperatures, (3) Radiation exchange is between small surface (wafer) and large enclosure (chamber, hot or cold zone), and (4) Negligible heat losses from wafer to pin holder. ANALYSIS: The
energy balance on the wafer include includess convection to the upper (u) and lower (l) surfaces from the ambient gas, radiation exchange exchange with the hot- and cool-zones and an energy storage term for the transient condition. Hence, from Eq. (1.12c ),
st E in E out E
Notice how the convection terms are handled.
or, per unit surface area
d T w qrad, h qrad,c qcv,u qcv,l cd dt
4 4 4 Tw4 Tsur, u Tw T hl Tw T c Tw h Tsur, h
d T w dt
(a) For the initial condition, the time rate of change of the wafer temperature temperature is deter determined mined using the foregoing energy balance with Tw T w,i 300K, 0.65 5.67 10