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HEAT EQUATION

Cp

T

t+ (u )T

=k2T+ Sv = k

Cpthermal diffusivity

t

CpT dV =

CV

Sv dVCS

(q n) dA

CpT(v n) dA

Rate heat generation heat flow surface heat loss from flow

THERMAL RESISTANCES

Q= T

RR=

ln routrin2Lk

CYLINDER R= L

kACONDUCT.

R= 1

hACONVEC. R=

1

4k

1

r1 1

r2

SPHERE

SIMPLE

q = k dTdx

Conduction

q = hA(T T) Convection

BIOT NUMBER

Bi = hL

k

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DILUTE SOLUTION (k - solvent) Assumes 1 wk Tg rubbery Rrelax >> Rdiff not limited by bending rate!

FICKIAN DIFFUSION.

T < Tg glassy Rdiff >> Rrelax limited by bending

CASE II Diffusion. Sharp front advances at const. rate.

CONVECTION-DIFFUSION-REACTION EQUATION Derivedusing RTT

c

t+ (u )c= D2c + S

Boundary Conditions:

Ni n= 0 impermeable wall Ni|1 n= Ni|2 n condition at interfacePi = Hici Henrys Law Ni n= Ri production at boundary

ci|1 = ci|2 liquid-porous with partition coefficientNi

n=

K(ci

|2

ci

|1) permeable boundary

TRANSIENT DIFFUSION Laplace Method

c

t=D

2c

x2 sC= D

2C

x2

BCs: x = 0 c= c0 C= c0s

|| x c, C 0

which has general soln. C= Ae

sD + Be

sD

Final solution in s-domain must be inverted.Concent. boundary layer can be defined e.g.= 0.01 c c0

DIFFUSION with CONVECTION Peclet Number P e = ULD

dif-fusion/convection time.Situation: Membrane in middle, flow through membrane. One side is at a

set conc. c0, flow is u in the x-direction.BC at the membrane: Ni n= K(ci|2 ci|1)

N =J+ cU= Dc + cu(i) =Dc

x+ cu

Hence our actual boundary cond. is: (for x = 0 i.e. at membrane )

Dcx

+ cu= K(c0 c)

Governing eq.2c

x2 u

D

c

x= 0

Gen. soln:c= Ae

uxD

MASS DIMENSIONLESS NUMBERS

Sc =

DSchmidt Number; MOM/MASS diffusivity

Shx = Kmx

D Re 12 Sc 13

km = N n

c=

Jy

c0flux per unit conc drop

PROD/ABSORPTION IN BULK

D2c

x2 kc = 0

No flux condition at a boundary gives cx

At permeable boundaries member that flux needs to be continuous, therefore equate.

REACTIONS Separate equation for each species. Assume reactirate k.

mA + nB pC SA = km[A]m[B]n

Now put into CDR.Reaction at an electrode has associated flux NA n= REAC

KROGH MODEL CDR reduces to:

D

r

r

r

c

r

R0

Where R0 is the rate at which cells take up oxygen. (EQ for TISSUE)

uc

z= 2

rcJwall

Governing equation IN capillary.Flux across wall:

Jwall = Dct

r|rc =K0(cc ct|rc)

i.e. the flux across wall IS proportional to the conc. difference.

GENERAL

Lct

=sC(s) c(0)

F(s) = Lf(t) =

0estf(t) dt Laplace transform

DT

Dt=

T

t+ v T Material derivative

Ni n dA= cuA Concentration flux i.e. CV analysis

dU =Q W First Law Isometric (no work) W = 0

Adiabatic means no heat transfer.

E= T4 q = (T4s T4surround)