IV. Transient Heat Conduction (UNSTEADY) – Chap. 5 of text … · 2019-10-23 · Chapter 4 – Page 21 HEAT TRANSFER FUNDAMENTALS Example 4.2 Given: Consider a 304 stainless steel

MAE 310 course notes – Fall 2011 Copyrighted by R. D. Gould

Chapter 4 – Page 1

HEAT TRANSFER FUNDAMENTALS

IV. Transient Heat Conduction (UNSTEADY) – Chap. 5 of text

Thus far we have considered only S.S. problems and have found T(x) for all time. Reconsider the 1-D transient plane wall problem:

Heat in Heat out qx qx+dx A q

dx

We derived the 1D transient HCE

x

k Tx

q C T x ttp

,

In general T = T(time,location)

Consider the problem where the initial condition (i.e. T(x t=0)), is known and then either the temperature, heat flux, or convective B.C. is suddenly altered at the boundary. The temperature will change within the body in space and time. The 1D transient HCE governs this behavior for 1D problems.

MAE 310 Muller Lec. 16 - 1




In this chapter we will consider:

1. Lumped system analysis T = f(t).

2. Transient temperature solutions (Heisler) – 1 term approx. of T = f(t,x).

3. Analytical 1D transient solutions for semi-infinite media.

4. Product solutions - superposition of two or three transient 1-D solutions to give a transient 2-D or 3-D solution.

5. Finite Difference solutions to transient heat conduction.





IV-1. Lumped Capacitance Method

[Temp = function(time only)]

Assumes negligible temperature gradient within the material

(i.e. uniform temperature as a function of space)

When is this a good assumption?

Answer: If, or R Rinternal externalL

kA h A

1

Bi hk

kh

1internal resistanceexternal resistance

0 1.

When Bi < 0.1 the following approximation holds: T x t T t, is within 5%!

We may assume that T is not a function of position! characteristic

lengthvolume

surface area





Consider a hot piece of material dropped suddenly into a cooling fluid.

HotMaterial

q t hA T t Tout wetted

at t=0, T(t=0) = Ti

The 1st law givesq W dU

dtq qnet net in out

dUdt

qout 0

dtdTC

dtdTmC

dtTmCd

dtdU

vvv





But C C cv p for incompressible liquids and solids

C dTdt

A h T t Tp s 0

Initial condition: T Ti at t = 0

dTdt

A hC

T t Ts

p

Let t T t T ddt

dTdt

i iT T at t = 0

ddt

A hC

s

p

Separating variables and integrating gives

dtmdtChAd

p

s





wherem A h

Cs

p

d m dti t

t

0

ln

i i

mtmt e

or

T t TT T

e A hC

ti

mt s

p

exp

Steady state is reached when T t T

T Ti

0

i.e., T t T





For example take to be ~ steady state T t TT Ti

0 01.

tmto reach

steady

4 61.

Note: As m increases time to reach S.S. decreases T t TT Ti

time, t

q t A h T t Tsconv but, T t T T e Timt

p

sis C

thATThAtq

expconv [Watts]

and thus,





The total heat transfer which occurs over length of time is

mTTC

dtC

thATTAhdttqQ

ip

o p

sis

o

exp1

expconvconv[Joules]

As

bodyip UTTCQQ maxconv





Example 4.1Given: A steel sphere 5 cm in diameter initially at uniform temperature of 450°C is suddenly placed in controlled environment of T= 100°C where h = 10 W/m2-C.

Find: Time required for the sphere to reach 150°C.

Solution:





IV-2. 1-D Transient Heat Transfer with Spatial Effect (use when Bi > 0.1)

The transient heat conduction problem for several simple shapes (constant k, no internal heat generation) subject to boundary conditions of practical importance have been computed. Analytic (infinite series) and graphical solutions are presented.

Geometries we will consider:

1. a long plane wall

2. a long solid cylinder

3. a sphere

all initially at a uniform temperature at t = 0 and with convection to a medium with fixed temperature at the exposed surface.

Other solutions are available.





Plane Wall

T, h T, h fluid flow fluid flow

k dTdx

hT hTx L

k dTdx

hT hTx L

x=-L x=0 x=L

This problem is symmetric both geometrically and thermally.

T, h fluid flow

dTdx x

0

0

k dTdx

hT hTx L

x=0 x=L

Governing equation (1-D HCE)

tTCq

xTk

x p





Assume: , k = constant

Recall thermal diffusivitypC

k

Governing equation:

2

21T

xTt

0 < x < L, t > 0

Left B.C. Tx 0 at x = 0, t > 0

Right B.C. k Tx

hT hT

at x = L, t > 0

Initial Condition T Ti for t = 0 in 0 x L

Note: There are 8 independent variables

x, t, L, k, h, , , Ti T

0q





We can minimize the number of independent variables by defining non-dimensional parameters.

TT

TtxTtxi

,,* dimensionless temp.

x xL

* dimensionless space coor.

Bi hLk

Biot number:

resistanceconvresistancecond

.

.

Fo t tL

* 2 dimensionless time (Fourier number)

The non-dimensional equations are:

2

2

*

*

*

*x t 0 < x* < 1, t* > 0

*

*x 0 at x* = 0, t* > 0

*

**

xBi 0 at x* = 1, t*> 0

* = 1 at t* = 0, 0 x* 1

Note:There are only 3 independent variables in the non-dimensional formulation: x*, t*, Bi





Fourier number:

CW,Lvolume

instorageheatofrateCW,LvolumeinL

acrossconductionheatofrate

3

321

2*

3

tLC

L

p

Lk

Ltt

Fo =

Large Fourier # deeper heat penetration into a solid over a given time

tTxXtx ,*Assuming that , and using separation of variables gives:





Exact Solution for Plane wall

TT

TtxTxFoCtxin

nnn,cosexp,

1 position

*

infotime

2*

nn

nnC

2sin2sin4

Binn tan

2LtFo

Lxx *where and

and the eigenvalues are the positive roots of the transcendental equation

Table 5.1 (p. 301) gives the first root to this equation (App. B.3 gives the first 4 roots)

[Eq. 5.42a]

[Eq. 5.42b]

[Eq. 5.42c]





Approximate (or one term) Solution for Plane wall

TT

TtxTxFoCtxi

,cosexp, *1

211

*

11

11 2sin2

sin4

C

Bi11 tan

2LtFo

Lxx *where and

and the eigenvalue is the positive root of the transcendental equation

Table 5.1 (p. 301) gives the first root to this equation

If the Fo > 0.2 the infinite series converges such that one term is sufficient

[Eq. 5.43a]





Approximate (one term) Solution for Plane wall – continued

TTTT

TTTtTFoCt

i

o

io

,0exp,0 211

*

The total energy transferred up to any time t is given by:

The non-dimensional centerline temperature (x* =0) is given by:

*

1

1sin1 o

oQQ

[Eq. 5.44]

[Eq. 5.49]









Example 4.2Given: Consider a 304 stainless steel plane wall having the following

properties and given thermal conditions.

5 cm

T Ci 200 , 3mkg7900

T C 70 , s

m10178.42

6

CmW680 2

h , Ckg

kJ515.0

pC

CmW17

k

Find: The temperature at a distance 1.25 cm from faces 1 minute after the plate has been exposed to the convective environment. Also determine how much energy has been removed per unit area from the plate during this time? Rework this problem for an aluminum slab ( = 2700, k = 213, Cp = .9, = 8.765 10-5).





Long Cylinder Case

Bi hrk

o * ,

T r t TT Ti

2ortFo

r rro

*

TT

TtrTrJFoCtri

o,exp, *

12

11*

TTTT

TTTtTFoCt

i

o

io

,0exp,0 211

*

The non-dimensional centerline temperature (r* =0) is given by:

The total energy transferred up to any time t is given by:

111

*21 J

QQ o

o

[Eq. 5.52a]

[Eq. 5.52c]

[Eq. 5.54]


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IV. Transient Heat Conduction (UNSTEADY) – Chap. 5 of text … · 2019-10-23 · Chapter 4 – Page 21 HEAT TRANSFER FUNDAMENTALS Example 4.2 Given: Consider a 304 stainless steel