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Chapter 4 Fluid Flow, Heat Transfer, and Mass Transfer: Similarities and Coupling

4.1 Similarities among different types of transport4.1.1 Basic laws

[4.1-1] also [1.1-2]

The transfer of momentum, heat , and species A occurs in the direction of decreasing vz, T, and wA, as summarized in Fig. 4.1-1. according to Eqs. [1.1-2], [2.1-1], and [3.1-1]

[4.1-2] also [2.1-2]

[4.1-3] also[3.1-1]

zyz

d

dy

y

dTq k

dy

AAy A

dwj D

dy

Newton’s law of viscosity

Fourier’s law of conduction

Fick’s law of diffution

2

The three basic laws share the same form as follows:

Or

The three-dimensional forms of these basic laws are summarized in Table 4.1-1.

[4.1-4]

[4.1-5]

For constant physical properties, Eqs. [4.1-1] through [4.1-3] can be written as follows:

[4.1-6]

tan

Flux of gradient ofproportonal

transport transportcons t

property property

y

dj

dy

( )yz z

dv

dy

3

[4.1-7]

[4.1-8]

These equations share the same form listed as follows:

[4.1-9]

In other words, , , and DA are the diffusivities of momentum, heat, and mass, respectively, and vZ, CvT, and A are the concentration of z momentum, thermal energy, and species mass, respectively.

4.1.2 Coefficients of Transfer

Fig. 4.1-2 shows the transfer of z momentum, heat, and species A from an interface, where they are more abundant, to an adjacent fluid, and from an adjacent fluid, where they are more abundant, to an interface. The coefficients of transfer, according to Eqs. [1.1-35], [2.1-14], and [3.1-21], are defined as follows:

( )y v

dq C T

dy

( )Ay A A

dj D

dy

Flux of diffusivity gradient of

transport of transport transport property

property property concentration

4

[4.1-10]0 0'

0

( / )

0yz y z y

fz z

v yC

v v v

(momentum transfer coefficient)

5

[4.1-11]

[4.1-12]

[4.1-13]

[4.1-14]

[4.1-15]

As mentioned in Sec. 3.1.6, Eq. [4.1-12] is for low solubility of species A in the fluid. These coefficients share the same form listed as follows:

or

It is common to divide Cf by v/2 to make denominator appear in the form of the kinetic energy v2

∞/2. As shown in Eq. [1.1-36], the so-called friction coefficient is defined by

0 0

0 0

( / ) ( )

y y yq k T yh heat transfer coefficient

T T T T

0 0

0 0

( / ) (mass transfer coefficient)

Ay y A A y

mA A A A

j D w yk

w w w w

interface

Coefficient flux at the

of transfer difference in transport property

0 0

0 0

( / ) (mass transfer coefficient)

y yj yk

'0

21 1

2 2

yz yff

CC

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4.1.3 The Chilton-Colburn Analogy

The analogous behavior of momentum, heat, and mass transfer is apparent from Examples 1.4-6, 2.2-5, and 3.2-4, where laminar flow over a flat plate was considered.From Eqs. [1.4-67], [2.2-71], and [3.2-56], at a distance z from the leading edge of theplate,

(1 2)0.323Re2fz

z

C

1 3 1 20.323Pr Rezhz

k

1 3 1 20.323 Remz

A

k zSc

D

[4.1-16]

[4.1-17]

[4.1-18]

where

Rezz

(local Reynolds number)

Pr pCv

k

(Prandtl number)

A A

vSc

D D

(Schmidt number)

[4.1-19]

[4.1-20]

[4.1-21]

and υ∞ is the velocity of the fluid approaching the flat plate.

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Equations [4.1-16] through [4.1-18] can be rearranged as follows

(1 2)0.323Re2fz

z

C

2 3 (1 2)1Pr 0.323Re

Pr Re z

hz

k

2 3 1 210.323Re

Rem

zA z

k zSc

D Sc

[4.1-22]

[4.1-23]

[4.1-24]

Since these equations have the same RHS, we see

2 3 2 31 1Pr

2 Pr Re Refz m

A z

C k zhzSc

k D Sc

Substituting Eqs. [4.1-19] through [4.1-21] into Eq. [4.1-25], we obtain

2 3 2 3Pr2fz m

p

C khSc

C

[4.1-25]

[4.1-26]

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This equation, known as the Chilton-Colburn analogy,1 is ofen written as follows

2fz

H D

Cj j [4.1-27]

where the j factor for heat transfer

2/3PrHp

hj

v C

[4.1-28]

And the j factor for mass transfer

2/3mD

kj Sc

v [4.1-29]

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H Dj j [4.1-30]

The Chilton –Colburn analogy for momentum, heat and mass transfer has been derived here on the basis of laminar flow over a flat plate. However, it has been observed to be a reasonable approximation in laminar and turbulent flow in systems of other geometries provided no form drag is present . From drag, which has no counterpart in heat and mass transfer, makes Cf/2 greater than jH and jD, for example, in flow around (normal to) cylinders. However, when form drag is present, the Chilton –Colburn analogy between heat and mass transfer can still be valid, that is,

or

2/3 2 /3Pr m

p

khSc

v C v

[4.1-31]

These equations are considered valid for liquid and gases within the ranges

0.6 < Sc < 2500 and 0.6 < Pr < 100 . They have been observed to be a reasonable approximation for various geometries, such as flow over flat plates, flow around cylinders, and flow in pipes.

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The Chilton –Colburn analogy is useful in that it allows one unknown transfer coefficient to be evaluated from another transfer coefficient which is known or measured in the same geometry. For example, by use Eq. [4.1-26] the mass transfer coefficient km

(for low solubility of species A in the fluid) can be estimated from a heat transfer coefficient h already measured for the same geometry.

It is worth mentioning that for the limiting case of Pr=1, we see that from Eq.[4.1-26]

2fz

p

C h

C

[4.1-32]

Which is known as the Reynolds analogy , in honor of Reynolds’ first recognition of the analogous behavior of momentum and heat transfer in 1874.

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4.1.4 Integral-Balance Equations

The integral-balance equations governing momentum, heat , and species transfer, according to Eq. [1.4-3], [2.2-6], and [3.2-4], respectively, are as follows

( ) ( )bA Avd vv n dA ndA f p d

t

[4.1-33]

(momentum transfer)

( )v vA AC Td vC T ndA q ndA sd

t

( )A A A AA Aw vw ndA j ndA r d

t

(heat transfer)

(species transfer)

[4.1-34]

[4.1-35]

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In Eq. [4.1-33] the pressure term has been converted from a surface integral to a volume integral using a Gauss divergence type theorem (i.e., Eq. [A.4-2]). Furthermore, the body force fb and pressure gradient can be considered as the rate of momentum generation due to these force. In Eq. [4.1-34] the kinetic and potential energy, and the pressure, viscous, and shaft work are not included since they are either negligible or irrelevant in most materials processing problems. In Eq. [4.1-35] ρwA= ρA .

These integral balance equations share the same form as follows:

p

inf

inf

rate of net rate ofRate of rate of

low by otheraccumulation generation

convection net low

[4.1-36]

or

( )A A

v ndA j ndA s dt

[4.1-37]

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These equations are summarized in Table 4.1-2. The following integral mass-balance equation ,Eq.[1.2-4], is also included in the table:

( )A

d dAt

Ω v n [4.1-38]

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( ) ( ) ( )in out v p b

dm m

dt

Pv v F F F

TE( ) ( )v in v out

dmC T mC T Q S

dt

4.1.5 Overall Balance Equations

The overall balance equations for momentum, heat, and species transfer according to Eqs.[1.4-9], [2.2-8], and [3.2-7], respectively, are as follows

(momentum transfer) [4.1.-39]

( ) ( )AA in A out A A

dMmw mw J R

dt

(heat transfer) [4.1-40]

(species transfer) [4.1-41]

These overall balance equations share the same form as follows

Rate of rate of inflow rate of outflow

accumulation by convection by convection

rate of other net inflow rate of + +

from surroundings generation

[4.1-42]

or

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Where the total momentum, thermal energy, or species A in the control volume Ω is

= d

[4.1-44]

In Eq.[4.1-39] the viscous force Fv at the wall can be considered as the rate of momentum transfer through the wall by molecular diffusion. The pressure force Fp and the body force Fb , on the other hand, can be considered as the rate of momentum generation due to the action of these forces. In Eq.[4.1-40 ] Q is by conduction, which is similar to diffusion.

in out

d=(m ) -(m ) +J +S

dt [4.1-43]

The above equations are summarized in Table 4.1-3. The following overall mass balance equation (i.e. Eq [1.2-6]), is also included in the table

=( ) ( )in out

dMm m

dt [4.1-

45]

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b p( )= ( ) + ( )t

v vv f

( ) ( ) v vC T C T st

v q

4.1.6 Differential Balance Equations

The differential balance equations governing momentum, heat, and species transfer, according to Eqs. [1.5-6], [2.3-5] and [3.3-5], respectively, are as follows:

( ) ( )A A A Aw w rt

v j

(momentum transfer) [4.1-46]

(heat transfer) [4.1-47]

(species transfer) [4.1-48]

In Eq. [4.1-47] the viscous dissipation is neglected and in Eq. [4.1-48] ρwA =ρA

These differential balance equations share the same form as follows:

Rate of rate of net inflow rate of other rate of + +

accumulation by convection net inflow generation

[4.1-49]

or

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( ) ( ) st

v j

( ) ( )t

v

[4.1-50]These equations are summarized in Table 4.1-4. The following equation of

continuity, Eq. [1.3-4], is also included in the table:

[4.1-51]

Table 4.1-5 summarizes these equations for incompressible fluids.

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Example