xq UA T≡ Δ

1

tot

UAR

=

"",

x

BAct q

TTR −=

( )2 1,

ln /2t cond

r rR

Lkπ=

( ) 12211 ..... −∑=== tRAUAU

,1 2

1 1 14t condR

k r rπ⎛ ⎞

= −⎜ ⎟⎝ ⎠

( )W 2eg RIE =&

( )2

2

1 1 0c s

c c

dA dAd T dT h T Tdx A dx dx A k dx ∞

⎛ ⎞ ⎛ ⎞+ − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

sA Px=

2

c

hPmkA

=

bbc

ff hA

qθ

ε,

=

bf

fff hA

qqq

θη =≡

max

cf mLMq tanh=

( )/ 2cL L t= +

( )/ 4cL L D= +

c

cf mL

mLtanh=η

( ) ( ) 0625.02/ o / ≤khDkht

2 2 /m h kt=

tot

bt hAq

Rη

θ 10, ==

( )bcnctff ARhAC ,,1 /1 η+=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

1)( 11

CANA f

t

fco

ηη

( )ft

fo A

NAηη −−= 11

tcot

bct hAq

R)(

)(0,1

ηθ

==

2

1 1p

T T T Tkr k k q cr r r r z z t

ρφ φ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞+ + + =⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠

&

22 2 2 2 2

1 1 1p

T T T Tkr k ksen q cr r r r sen r sen t

θ ρθ φ φ θ θ θ

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞+ + + =⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠&

 

 

 

 

 

 

 

 

 

Conducción bidimensional en estado estable

1 2i

Mlq k T

N 1 2q Sk T

MlS

N 2121 TTT

2

,,1,1

,

2

2

)(

2

x

TTT

x

T nmnmnm

nm

2

,1,1,

,

2

2

)(

2

y

TTT

y

T nmnmnm

nm

MÉTODO DE LA RESISTENCIA INTERNA DESPRECIABLE

EFECTOS ESPACIALES

Pared plana

Cilindro infinito (L/ r0≥10)

Esfera

ln i

s

Vct

hA

1

exp1)(

t

VcQ i

exp s

i i

hAT Tt

T T Vc

TT )exp( FoBi

TT

TT

ii

dt

dTVcAqqEAq rcsradconvghss ),(

""

,

" )( " 4 4

, ( , )( ) ( )s s h g sur s c r

dTq A E h T T T T A Vc

dt

TT

TT

ii

*

L

xx * Fo

L

tt

2

*

1

*2* cosexpn

nnn xFoC nn

nnC

2sin2

sin4

Binn tan

*

1

2

11

* cosexp xFoC *

1

*

0

* cos x FoC 2

11

*

0 exp

TTcVQ i0

*

0

1

1

0

sin1

Q

Q

1

*

0

2* expn

nnn rJFoC

nn

n

n

nJJ

JC

2

1

2

0

12

BiJ

J

n

nn

0

1

*

10

2

11

* exp rJFoC *

10

*

0

* rJ FoC 2

11

*

0 exp

1

*

*

2* sin1

expn

n

n

nn rr

FoC

nn

nnnnC

2sin2

cossin4

Binn cot1

11

1

*

0

0

21

J

Q

Q

tt

s

CRVchA

)(

11

SÓLIDO SEMIINFINITO

Caso 1. Temperatura superficial constante

Caso 2. Flujo de calor superficial constante

Caso 3. Convección superficial

EFECTOS MULTIDIMENSIONALES

*

1*

1

2

11

* sin1

exp rr

FoC

*

1*

1

*

0

* sin1

rr

FoC 2

11

*

0 exp

1113

1

*

0

0

cossin3

1

Q

Q

"

,

"

, BsAs qq

2121

,

21

,

21

BA

iBBiAA

sckck

TckTckT

solidosemiinfinito

,,

i

T x t TS x t

T T

paredplana

,,

i

T x t TP x t

T T

cilindroinfinito

,,

i

T r t TC r t

T T

sTtT ,0

t

x

TT

TtxT

si

s

2erf

,

t

TTktq is

s

"

""

os qq

t

x

k

xq

t

x

k

tqTtxT oo

i

2erfc

4exp

2,

"221"

k

th

t

x

k

th

k

hx

t

x

TT

TtxT

i

i

2erfcexp

2erfc

,2

2

tTThx

Tk

x

,00

2,1,1,

2

,,1,1,

1

, 221

y

TTT

x

TTT

t

TT p

nm

p

nm

p

nm

p

nm

p

nm

p

nm

p

nm

p

nm

21

,

1

1,

1

1,

2

1

,

1

,1

1

,1,

1

, 221

y

TTT

x

TTT

t

TT p

nm

p

nm

p

nm

p

nm

p

nm

p

nm

p

nm

p

nm