1B Chapter1 Sec.1.3 Color

8/13/2019 1B Chapter1 Sec.1.3 Color

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Relationship to Thermodynamics

Chapter One

Section 1.3


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Alternative Formulations

Alternative Formulations

Time Basis:

At an instant

or

Over a time interval

Type of System:

Control volume

Control surface

An important tool in heat transfer analysis, often

providing thebasis for determiningthe temperature

of a system.

CONSERVATION OF ENERGY

(FIRST LAW OF THERMODYNAMICS)


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At an Instant of Time:

Note representation of system by a

control surface (dashed line)at the boundaries.

Surface Phenomena

,

in out

: rate

energy transfer across the controlof thermal and/or mechanical

due to heat transfer, fluid flow and/or work interactions.surface

E E

Volumetric Phenomena

: rate of due to conversion from another energy form

(e.g., electrical, nuclear, or chemical); energy conversion proc

thermal ener

ess occurs w

gy generatio

ithin the sy e

n

m.

st

gE

stenergy: storate of rage inchang the se o ysf m.teE

Conserv

ation of Energy

in out stst

gdE

dtE E E E (1.12c)

Each term has units of J/s or W.

APPLICATION TO A CONTROL VOLUME

Over a Time Interval

Each term has units of J.

in out stgE E E E (1.12b)

CV at an Instant and over a Time Interval


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At an instant

tdUq Wdt

Special Cases (Linkages to Thermodynamics)

(i) TransientProcess for a Closed Systemof Mass (M) Assuming Heat Transfer

to the System (Inflow) and Work Done by the System (Outflow).

Over a time intervaltot

stQ EW (1.12a)

Closed System

For negligible changes in potential or kinetic energy

tQ W U

Internalthermal energy


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Example 1.4: Application to thermal response of a conductor with Ohmic

heating (generation):

Involves change in thermal energyand for an incompressible substance.

tdU dT

Mcdt dt

Heat transfer is from the conductor (negative )q

Generation may be viewed as electrical workdone on the system (negative )W

Example 1.4


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Example 1.6: Application to isothermal solid-liquid phase change in a container:

Latent Heat

of Fusion

latt sfU U Mh

Example 1.6


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(ii) Steady Statefor Flow through an Open Systemwithout Phase Change or

Generation:

flow ow rkpv enthalp ytu pv i

in out in out

ideal gas constant specific heFor an w :atith

pi i c T T

in out in out

in out

For an :incompressible liqu

0

id

u u c T T

pv pv

2 2

in

in

out0

2 2

0out

For systems with significant heat transfer:

V V

gz gz

At an Instant of Time:2

out

02t

m u pv V gz W

2

in2t

m u pv V gz q

(1.12d)

Open System


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Surface Energy Balance

A special case for which no volume or mass is encompassed by the control surface.

Conservation of Energy(Instant in Time):

outin 0E E (1.13)

Applies for steady-state and transient conditions.

Consider surface of wall with heat transfer by conduction, convection and radiation.

cond conv rad 0q q q

4 41 2 2 2 2 sur 0T T

k h T T T T L

With no mass and volume, energy storage and generation are not pertinent to the energy

balance, even if they occur in the medium bounded by the surface.

THE SURFACE ENERGY BALANCE


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Methodology

On a schematicof the system, represent the control surfaceby

dashed line(s).

Choose the appropriate time basis.

Identify relevant energytransport, generation and/or storage terms

by labeled arrowson the schematic.

Write the governing form of the Conservation of Energyrequirement.

Substitute appropriate expressions for terms of the energy equation.

Solve for the unknown quantity.

METHODOLOGY OF FIRST LAW ANALYSIS


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Problem 1.57: Thermal processing of silicon wafers in a two-zone furnace.

Determine (a) the initial rate of change of the wafer

temperature and (b) the steady-state temperature.

Problem: Silicon Wafer

KNOWN: Silicon wafer positioned in furnace with top and bottom surfaces exposed to hotand cool zones, respectively.

FIND: (a) Initial rate of change of the wafer temperature from a value of 300K,w,iT and (b)

steady-state temperature. Is convection significant? Sketch the variation of wafer temperature

with vertical distance.

SCHEMATIC:


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Problem: Silicon Wafer (cont.)

ASSUMPTIONS: (1) Wafer temperature is uniform, (2) Hot and cool zones have uniformtemperatures, (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 includes convection to the upper (u) and lower

(l) surfaces from the ambient gas, radiation exchange with the hot- and cool-zones and an energystorage term for the transient condition. Hence, from Eq. (1.12c),

in out stE E E

or, per unit surface area

rad, rad, cv, cv,w

h c u l d T

q q q q cd

dt

4 4 4 4sur,sur, ww c w u w l whd T

T T T T h T T h T T cd dt

(a) For the initial condition, the time rate of change of the wafer temperature is determined

using the foregoing energy balance with , 300K,w w iT T

8 2 4 4 4 8 2 4 4 4 440.65 5.67 10 W / m K 1500 300 K 0.65 5.67 10 W / m K 330 300 K

2 28W / m K 300 700 K 4W / m K 300 700 K

32700 kg / m 875J / kg K 0.00078 /w im dT dt

104 K/sw idT / dt

1B Chapter1 Sec.1.3 Color

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