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Unit Operations of Chemical Engineering Ali H. Abbar

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Unit operations

of

chemical engineering

Fourth year

Chemical Engineering Department

College of Engineering

AL-Qadesyia University

Lecturer: Ali H. Abbar


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Syllabus

1) Boundary layer theory

2) Transfer of heat, mass and momentum(the concept of universal velocity profile)

3) Transfer of heat, mass and momentum (Reynolds analogy)

4) Non-Newtonian fluids

5) Mixing

6) Fluid flow through packed column

7) Flow of particles though fluids

8) Fluidization

9) Filtration

10) Sedimentation

11) Centrifuge

12) Selective topics of unit operation application in renewable energy

References

1) Unit Operations of Chemical Engineering ,Seventh Edition, Prepared by Julian C. Smith ,Peter Harriott ,2009

2) Principles Of Unit Operation, second edition, By Alan S. Foust,1980

3) Chemical Engineering, Volume 1, Sixth edition, ByJ, M. Coulson and J. F. Richardson,1999

4) Chemical Engineering, Volume 2, fifth edition, ByJ, M. Coulson and J. F. Richardson,2002.


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CHAPTER (ONE)

The Boundary Layer

When a fluid flows over a surface that part of the stream which is

close to the surface suffers a significant retardation, and a velocity

profile develops in the fluid. The velocity gradients are steepest close to

the surface and become progressively smaller with distance from the

surface. It is convenient to divide the flow into two parts for practical

purposes:

1) A boundary layer close to the surface in which the velocity increases

from zero at the surface itself to a near constant stream velocity at its

outer boundary.

2) A region outside the boundary layer in which the velocity gradient in

a direction perpendicular to the surface is negligibly small and in which

the velocity is everywhere equal to the stream velocity.

The thickness of the boundary layer may be arbitrarily defined as

the distance from the surface at which the velocity reaches some

proportion (such as 0.9, 0.99, 0.999 ) of the undisturbed stream velocity.

The flow conditions in the boundary layer are of considerable

interest to chemical engineers because these influence, not only the drag

effect of the fluid on the surface, but also the heat or mass transfer rates

where a temperature or a concentration gradient exists.

1-1) BOUNDARY LAYER THEORY APPLIED TO A THIN PLATE

It is convenient first to consider the flow over a thin plate inserted

parallel to the flow of a fluid with a constant stream velocity us. It will

be assumed that the plate is sufficiently wide for conditions to be

constant across any finite width(w) of the plate which is being

considered. Furthermore, the extent of the fluid in a direction

perpendicular to the surface is considered as sufficiently large for the

velocity of the fluid remote from the surface to be unaffected and to

remain constant at the stream velocity us. The development of the

boundary layer over flat surface is illustrated in Figure (1-1 )


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Figure (1-1) Development of the boundary layer

On the assumption that there is no slip at the surface, the fluid

velocity at all points on the surface, where y = 0, will be zero. At some

position a distance x from the leading edge, the velocity will increase

from zero at the surface to approach the stream velocity us

asymptotically.

Near the leading edge of the surface where the boundary layer

thickness is small, the flow will be streamline, or laminar, and the shear

stresses will arise solely from viscous shear effects. When the boundary

layer thickness exceeds a critical value, the streamline flow ceases to be

stable and turbulence sets in.The important flow parameter is the

Reynolds number Reδ(= usδρ/µ). Because δ can be expressed as a

function of x(the distance from the leading edge of the surface) the usual

criterion is taken as the value of the Reynolds number Rex(= usxρ/µ). If

the location of the transition point is at a distance xc from the leading

edge, then [Rexc = 105].

When the flow in the boundary layer is turbulent, streamline flow

persists in a thin region close to the surface called the laminar sub-layer.

This region is of particular importance because, in heat or mass transfer,

it is where the greater part of the resistance to transfer lies. High heat

and mass transfer rates therefore depend on the laminar sub-layer being

thin. Separating the laminar sub-layer from the turbulent part of the

boundary layer is the buffer layer, in which the contributions of the

viscous effects and of the turbulent eddies are of comparable

magnitudes.


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1-1-A)The momentum equation

It will be assumed that a fluid of density (ρ) and viscosity (µ) flows over

a plane surface and the velocity of flow outside the boundary layer is us.

A boundary layer of thickness δ forms near the surface, and at a distance

y from the surface the velocity of the fluid is reduced to a value ux.as

shown in figure(1-2)

Figure (1-2) Element of boundary layer

The equilibrium is considered of an element of fluid bounded by

the planes 1-2 and 3-4 at distances x and x + dx respectively from the

leading edge; the element is of length (l) in the direction of flow and is

of depth w in the direction perpendicular to the plane 1-2-3-4. The

distance (l) is greater than the boundary layer thickness δ, and conditions

are constant over the width (w). The velocities and forces in the X-

direction are now considered.

At plane 1-2, mass rate of flow through a strip of thickness dy at

distance y from the surface = ρuxw dy

The total flow through plane 1 -2 ---------(1-1)

The rate of transfer of momentum through the elementary strip

=ρuxwdyux = wρux2dy

The total rate of transfer of momentum through plane 1 -2:


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---------------- (1-2)

In passing from plane 1-2 to plane 3-4, the mass flow changes by:

----------------- (1-3)

and the momentum flux changes by:

-------------- (1-4)

where Mii is the momentum flux across the plane 3-4.

A mass flow of fluid equal to the difference between the flows at

planes 3-4 and 1-2 must therefore occur through plane 2-4.

Since plane 2-4 lies outside the boundary layer, the fluid crossing

this plane must have a velocity us in the X-direction. Thus the rate of

transfer of momentum through plane 2-4 out of the element is:

------------- (1-5)

It will be noted that the derivative is negative, which indicates a

positive outflow of momentum from the element.

Thus, the net momentum flux out of the element Mex is given by:

Then, since us is assumed not to vary with x:

---------- (1-6)


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The net force acting F is just the retarding force attributable to the shear

stress at the surface only and Thus: F=R0wdx

Equating the net momentum flux out of the element to the net retarding

force and simplifying gives:

-------------- (1-7)

This expression, known as the momentum equation, may be

integrated provided that the relation between ux and y is known.

If the velocity of the main stream remains constant at us and the

density (ρ) may be taken as constant, the momentum equation then

becomes:

----------- (1-8)

Where Ro=-µ(∂ux/∂y)y=0

Ro is the shear stress in the fluid at the surface

1-1-B) The streamline portion of the boundary layer

In the streamline boundary layer the only forces acting within the

fluid are pure viscous forces and no transfer of momentum takes place

by eddy motion. Assuming that the relation between ux and y can be

expressed approximately by:

----------------- (1-9)

The coefficients a, b, c and uo may be evaluated because the boundary

conditions which the relation must satisfy are known as shown in figure

(1-3).


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Figure (1-3) Velocity distribution in streamline boundary layer

It is assumed here that the fluid in contact with the surface is at rest and

therefore uo must be zero. Furthermore, all the fluid close to the surface

is moving at very low velocity and therefore any changes in its

momentum as it flows parallel to the surface must be extremely small.

Thus the shear stress Ro in the fluid near the surface must approach a

constant value. Since Ro = -µ(dux/dy)y=0, dux/dy must also be constant at

small values of y and:

At the distant edge of the boundary layer it is assumed that the velocity

just equals the main stream velocity and that there is no discontinuity in

the velocity profile.

Thus, when y=δ:

Now with uo = 0, equation (1-9) becomes:

And

At y=0 thus b=0


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At y=δ

And thus a=-3cδ2

Hence

The equation for the velocity profile is therefore:

------------- (1-10)

Equation (1-10) corresponds closely to experimentally determined

velocity profiles in a laminar boundary layer. This relation applies over

the range 0 < y <δ. When y > δ, then: ux = us

The integral in the momentum equation (1-8) can now be evaluated

for the streamline boundary layer by considering the ranges

0 < y < δ and δ < y < l separately.

Thus:

-------------------------------- (1-11)

In addition ----------- (1-12)

Substitution from equations (1-11) and (1-12) in equation (1-8), gives:


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(Since δ =0 when x=0)

--------------- (1-13)

Thus

δ

And:

----------- (1-14)

1-1-C)Shear stress at the surface

The shear stress in the fluid at the surface is given by:

The shear stress R acting on the surface itself is equal and opposite to

the shear stress in the fluid at the surface; that is, R = -Ro. thus:

---------------- (1-15)


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Equation (1-15) gives the point values of R and (R/ρu2 )at x = x. In order

to calculate the total frictional force acting at the surface, it is necessary

to multiply the average value of R between x = 0 and x = x by the area

of the surface. The average value of (R/ρu2) denoted by the symbol

(R/ρu2) m is then given by;

-------- (1-16)

HOMEWORK: If the velocity profile in the laminar region may be represented

approximately by a sine function, calculate the boundary-layer thickness in terms

of distance from the leading edge of the surface.

1-1-D)The turbulent boundary layer

I) The turbulent portion

Equation (1-10) does not fit velocity profiles measured in a turbulent

boundary layer and an alternative approach must be used. In the

simplified treatment of the flow conditions within the turbulent

boundary layer the existence of the buffer layer is neglected and it is

assumed that the boundary layer consists of a laminar sub-layer, in

which momentum transfer is by molecular motion alone, outside which

there is a turbulent region in which transfer is effected entirely by eddy

motion (Figure1-4). The approach is based on the assumption that the

shear stress on the surface can be calculated from the simple power law

developed by Blasius.


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Figure (1-4) Turbulent boundary layer

Blasius has given the following approximate expression for the shear

stress at a plane smooth surface over which a fluid is flowing with a

velocity us, for conditions where Rex < 107:

---------------- (1-17)

Thus, the shear stress is expressed as a function of the boundary layer

thickness δ and it is therefore implicitly assumed that a certain velocity

profile exists in the fluid. As a first assumption, it may be assumed that a

simple power relation exists between the velocity and the distance from

the surface in the boundary layer, or:

-------------------- (1-18)

Hence

- (1-19)


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If the velocity profile is the same for all stream velocities, the shear

stress must be defined by specifying the velocity ux at any distance y

from the surface. The boundary layer thickness, determined by the

velocity profile, is then no longer an independent variable so that the

index of δ in equation (1-19) must be zero or:

Thus

----------------- (1-20)

Equation (1-20) is sometimes known as the Prandtl seventh power law.

Differentiating equation (1-20) with respect to y gives:

--------- (1-21)

This relation is not completely satisfactory in that it gives an infinite

velocity gradient at the surface, where the laminar sub-layer exists, and a

finite velocity gradient at the outer edge of the boundary layer. This is in

contradiction to the conditions which must exist in the stream. However,

little error is introduced by using this relation for the whole of the

boundary layer in the momentum equation because, firstly both the

velocities and hence the momentum fluxes near the surface are very low,

and secondly it gives the correct value of the velocity at the edge of the

boundary layer. Accepting equation for the limits 0 < y < δ, the integral

in equation (1-8) becomes:

-------------------------------- (1-22)


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From the Blasius equation:

Substituting from equations (1-17) and (1-22) into (1-8)

Putting the constant equal to zero, implies that δ= 0 when x = 0, that is

that the turbulent boundary layer extends to the leading edge of the

surface. An error is introduced by this assumption, but it is found to be

small except where the surface is only slightly longer than the critical

distance xc for the laminar-turbulent transition

------------------------ (1-23)

II) The laminar sub-layer

If at a distance x from the leading edge the laminar sub-layer is of

thickness δb and the total thickness of the boundary layer is δ, the

properties of the laminar sub-layer can be found by equating the shear

stress at the surface as given by the Blasius equation (1-17) to that

obtained from the velocity gradient near the surface.


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It has been noted that the shear stress and hence the velocity gradient are

almost constant near the surface. Since the laminar sub-layer is very

thin, the velocity gradient within it may therefore be taken as constant.

Thus the shear stress in the fluid at the surface,

Equating this to the value obtained from equation 1-17 gives

If the velocity at the edge of the laminar sub-layer is ub, that is, if ux =ub,

when y = δb

-------------- (1-24)

The velocity at the inner edge of the turbulent region must also be given

by the equation for the velocity distribution in the turbulent region.

-------------------------- (1-25)


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From equation (1-23)

------------------ (1-26)

The thickness of the laminar sub-layer is given by:

---------------------- (1-27)

Thus δb≈x0.1

that is, δb it increases very slowly as x increases. Further,

δb αus-0.9

and therefore decreases rapidly as the velocity is increased, and

heat and mass transfer coefficients are therefore considerably influenced

by the velocity.

III) Shear stress at the surface

The shear stress at the surface, at a distance x from the leading edge, is

given by:


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--------------------------- (1-28)

The mean value is:

---------- (1-29)

The total shear force acting on the surface is found by adding the forces

acting in the streamline (x < xc) and turbulent (x > xc) regions. This can

be done provided the critical value ReXc, is known.

A more accurate value for the mean value of (R/ρu

2s) m over the whole

surface can be obtained by using the expression for streamline

conditions over the range from x = 0 to x = xc (where xc is the critical

distance from the leading edge) and the expression for turbulent

conditions in the range x = xc to x= x,

-- (1-30)

---- (1-31)


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Example (1-1)

Calculate the thickness of the boundary layer at a distance of 150 mm

from the leading edge of a surface over which oil, of viscosity 0.05 N

s/m2 and density 1000 kg/m

3 flows with a velocity of 0.3 m/s.

Solution

Example(1-2)

Air is flowing at a velocity of 5 m/s over a plane surface. Derive an

expression for the thickness of the laminar sub-layer and calculate its

value at a distance of 1 m from the leading edge of the surface.

Assume that within the boundary layer outside the laminar sub-layer, the

velocity of flow is proportional to the one-seventh power of the distance

from the surface and that the shear stress R at the surface is given by:

(

) (

)

Where ρ is the density of the fluid (1.3 kg/m3 for air), µ is the viscosity

of the fluid (17 x 10-6

N s/m2 for air), us is the stream velocity (m/s), and

x is the distance from the leading edge (m).

Solution

The shear stress in the fluid at the surface: R =µux/y

From the equation given:

(

) (

)


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If the velocity at the edge of the laminar sub-layer is ub, ux= ub at y=δb

The velocity distribution is given by:


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1-2)BOUNDARY LAYER THEORY APPLIED TO PIPE FLOW

1-2-A)Entry conditions

When a fluid flowing with a uniform velocity enters a pipe, a

boundary layer forms at the walls and gradually thickens with distance

from the entry point. Since the fluid in the boundary layer is retarded

and the total flow remains constant, the fluid in the central stream is

accelerated. At a certain distance from the inlet, the boundary layers,

which have formed in contact with the walls, join at the axis of the pipe,

and, from that point onwards, occupy the whole cross-section and

consequently remain of a constant thickness. Fully developed flow then

exists. If the boundary layers are still streamline when fully developed

flow commences, the flow in the pipe remains streamline. On the other

hand, if the boundary layers are already turbulent, turbulent flow will

persist as shown in Figure 1-5.

Figure (1- 5) Conditions at entry to pipe

An approximate experimental expression for the inlet length Le for

laminar flow is:

------------------- (1-32)

where d is the diameter of the pipe and Re is the Reynolds group with

respect to pipe diameter, and based on the mean velocity of flow in the

pipe. This expression is only approximate, and is inaccurate for

Reynolds numbers in the region of 2500 because the boundary layer

thickness increases very rapidly in this region. An average value of Le

at a Reynolds number of 2500 is about 100d.


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1-2-B)Application of the boundary-layer theory

The velocity distribution and frictional resistance have been

calculated from purely theoretical considerations for the streamline flow

of a fluid in a pipe to be:

The boundary layer theory can now be applied in order to

calculate, approximately, the conditions when the fluid is turbulent. For

this purpose it is assumed that the boundary layer expressions may be

applied to flow over a cylindrical surface and that the flow conditions in

the region of fully developed flow are the same as those when the

boundary layers first join. The thickness of the boundary layer is thus

taken to be equal to the radius of the pipe and the velocity at the outer

edge of the boundary layer is assumed to be the velocity at the axis.

The velocity of the fluid may be assumed to obey the Prandtl one-

seventh power law, given by equation 1-20. If the boundary layer

thickness δ is replaced by the pipe radius r, this is then given by:

-------------------------- (1-33)

The shear stress at the walls is given by the Blasius equation (1-17) as:

Writing u = 0.817us, and d = 2r:

--- (1-34)

This equation is more usually written:

--------------------------- (1-35)

The discrepancy between the coefficients in equations 1-34 and 1-35 is

attributable to the fact that the effect of the curvature of the pipe wall has

not been taken into account in applying the equation for flow over a


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plane surface to flow through a pipe. In addition, it takes no account of

the existence of the laminar sub-layer at the walls.

Equation 1-35 is applicable for Reynolds numbers up to 105.

The velocity at the edge of the laminar sub-layer is given by:

------------------------- (1-36)

The thickness of the laminar sub-layer is given by:

-------------------- (1-37)

Example (1-3)

Calculate the thickness of the laminar sub-layer when benzene flows

through a pipe 50 mm in diameter at 2 1/s. What is the velocity of the

benzene at the edge of the laminar sub-layer? Assume that fully

developed flow exists within the pipe and that for benzene, ρ=870 kg/m3

andµ = 0.7 mN s/m2.


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Solution

The mass flow rate of benzene = (2 x 10-3

x 870) = 1.74 kg/s

1-3) THE BOUNDARY LAYER FOR HEAT TRANSFER

Where a fluid flows over a surface which is at a different

temperature, heat transfer occurs and a temperature profile is established

in the vicinity of the surface. A number of possible conditions may be

considered. At the outset, the heat transfer rate may be sufficient to

change the temperature of the fluid stream significantly or it may remain

at a substantially constant temperature. Furthermore, a variety of

conditions may apply at the surface. Thus the surface may be maintained


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at a constant temperature, particularly if it is in good thermal conduct

with a heat source or sink of high thermal capacity.

Alternatively, the heat flux at the surface may be maintained

constant, or conditions may be intermediate between the constant

temperature and the constant heat flux conditions

1-3-A)The heat balance

The procedure here is similar to that adopted previously for

momentum. A heat balance, as opposed to a momentum balance, is

taken over an element which extends beyond the limits of both the

velocity and thermal boundary layers. In this way, any fluid entering or

leaving the element through the face distant from the surface is at the

stream velocity us and stream temperature θs. A heat balance is made

therefore on the element shown in Figure (1-6) in which the length l is

greater than the velocity boundary layer thickness δ and the thermal

boundary layer thickness δt.

Figure (1-6) The thermal boundary layer

The rate of heat transfer through an element of width w of the plane 1 -2,

of thickness dy at a distance y from the surface is:

------------------ (1-38)

The total rate of transfer of heat through the plane, 1 -2 is then:

-------------- (1-39)

In the distance dx this heat flow changes by an amount given by:


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--------------- (1-40)

Since the plane, 2-4, lies outside the boundary layers, the heat leaving

the element through the plane as a result of this flow is:

------------------- (1-41)

Where θs is the temperature outside the thermal boundary layer.

The heat transferred by thermal conduction into the element through

plane, 1-3 is:

----------------------- (1-42)

If the temperature θs of the main stream is unchanged, a heat balance on

the element gives:

------(1-43)

------- (1-44)

where DH (= k/Cpρ) is the thermal diffusivity of the fluid.

I)Heat transfer for streamline flow over a plane surface

constant surface temperature

The flow of fluid over a plane surface, heated at distances greater than x0

from the leading edge, is now considered. As shown in Figure1- 7 the

velocity boundary layer starts at the leading edge and the thermal

boundary layer at a distance x0 from it. If the temperature of the heated

portion of the plate remains constant, this may be taken as the datum

temperature. It is assumed that the temperature at a distance y from the

surface may be represented by a polynomial of the form:


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--------------------- (1-45)

Figure (1- 7) Thermal boundary later- streamline flow

the heat transferred per unit area and unit time qo is given by:

the heat transfer rate into and out of the element must be the same, or:

At the outer edge of the thermal boundary layer, the temperature is θs

and the temperature gradient (dθ/dy) = 0 if there is to be no discontinuity

in the temperature profile.

if the thickness of the thermal boundary layer is δt, the temperature

distribution is given by:

-------------------- (1-46)

---------------- (1-47)

It is assumed that the velocity boundary layer is everywhere thicker than

the thermal boundary layer, so that δ >δt . Thus the velocity distribution

everywhere within the thermal boundary layer is given by equation1- 10.


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The integral in equation 1-44 clearly has a finite value within the

thermal boundary layer, although it is zero outside it. When the

expression for the temperature distribution in the boundary layer is

inserted, the upper limit of integration must be altered from l to δt

--------------- (1-48)

where σ=δt/δ

Since δt<δ, the second term is small compared with the first, and:

----------------- (1-49)

Substituting from equations1-47 and 1-49 in equation 1-44 gives:

------------------ (1-50)

It has already been shown that:


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where the Prandtl number Pr=cpµ/k

------ (1-51)

If the whole length of the plate is heated, x0=0 and:

------------------------ (1-52)

In this derivation, it has been assumed that σ < 1 .

For all liquids other than molten metals, Pr > 1 and hence, from equation

1-52, σ < 1,For gases, Pr < 0.6, so that σ > 1.18.Thus only a small error

is introduced when this expression is applied to gases.


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If h is the heat transfer coefficient, then:

--------------------------- (1-53)

-------------------------- (1-54)

If the surface is heated over its entire length, so that x0=0, then:

--------------- (1-55)

The mean value of (h) between x = 0 and x = x is given by:

Hence hm=2h

The mean value of the heat transfer coefficient between x = 0 and x = x

is equal to twice the point value at x = x. The mean value of the Nusselt

group is given by:

-------------------- (1-56)


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II)Heat transfer for streamline flow over a plane surface—constant

surface heat flux

Another important case is where the heat flux, as opposed to the

temperature at the surface, is constant; this may occur where the surface

is electrically heated. Then, the temperature difference │θs-θ0│will

increase in the direction of flow (x-direction) as the value of the heat

transfer coefficient decreases due to the thickening of the thermal

boundary layer.

The equation for the temperature profile in the boundary layer becomes:

------------------ (1-57)

------------- (1-58)

The value of the integral in the energy balance (equation 1-44) is again

given by equation 1-49 [substituting (θs-θ0) for θs]. The heat flux q0 at

the surface is now constant, and the right-hand side of equation 1-44

may be expressed as (-qo/ρCp).

Thus, for constant surface heat flux, equation 1-44 becomes:

------- (1-59)

Equation 1-59 cannot be integrated directly, however, because the

temperature driving force (θs-θ0) is not known as a function of location x

on the plate. The solution of equation 1-59 involves a quite complex

procedure and takes the following form:

-------------------- (1-60)

The average value of the Nusselt group (Nux)m is obtained by

integrating over the range x = 0 to x = x, giving:

------------- (1-61)


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1-4)THE BOUNDARY LAYER FOR MASS TRANSFER

If a concentration gradient exists within a fluid flowing over a

surface, mass transfer will take place, and the whole of the resistance to

transfer can be regarded as lying within a diffusion boundary layer in the

vicinity of the surface. For low concentrations of the diffusing

component, the effects of bulk flow will be small and the mass balance

equation for component A is:

--------------------- (1-62)

where CA and CAS are the molar concentrations of A at a distance y from

the surface and outside the boundary layer respectively, and l is a

distance at right angles to the surface which is greater than the thickness

of any of the boundary layers. Equation 1-62 is obtained in exactly the

same manner as equation 1-44 for heat transfer.

Again, the form of the concentration profile in the diffusion

boundary layer depends on the conditions which are assumed to exist at

the surface and in the fluid stream. For the conditions corresponding to

those used in consideration of the thermal boundary layer, that is

constant concentrations both in the stream outside the boundary layer

and at the surface, the concentration profile is of similar form to that

given by equation 1-57:

----------------- (1-63)

Where δD is the thickness of the concentration boundary layer,

CA is the concentration of A at y = y,

CAO is the concentration of A at the surface (y = 0), and

CAS is the concentration of A outside the boundary layer


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Substituting from equation 1-63 to evaluate the integral in equation 1-62,

assuming that mass transfer takes place over the whole length of the

surface (x0 = 0), by analogy with equation 1-52 gives:

------------ (1-64)

where Sc =µ/ρD is the Schmidt number. Equation 1-64 is applicable

provided that Sc > 1.

The point values of the Sherwood number Shx and mass transfer

coefficient hD are then given by:

--------------------- (1-65)

The mean value of the coefficient between x = 0 and x = x is then given

by:

----------------------- (1-66)

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