Heat-Mass Transfer Analogy

8/8/2019 Heat-Mass Transfer Analogy

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Chapter 3.

Heat/Mass Transfer Analogy- Laminar Boundary Layer

As noted in the previous chapter, the analogous behaviors of heat and mass transfer have

been long recognized. In the field of gas turbine heat transfer, several experimental

studies have been done with mass transfer because of its experimental advantages. In

such cases, it was required that the heat/mass transfer analogy function be well known.

This chapter reviews the conventional heat/mass transfer analogy function and

numerically generates the analogy functions to examine their behaviors under the

influences of streamwise pressure gradients and different thermal boundary conditions

(BCs). For practical reasons, the fluid assumed in the heat transfer is air (Pr = 0.707) and

that of the mass transfer is air with naphthalene being the substance to be transported (Sc

= 2.28). Furthermore, since a typical use of heat/mass transfer analogy is to convert mass

transfer data to heat transfer data, discussions are focused mainly on such a direction of

the conversion.

3.1 Analytical Solutions

3.1.1 Conventional relationship


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The conventional relationship for laminar boundary layer flow is given by the following

simple expression.

31

Sc

Pr

Sh

Nu

= (3.1.1)

The conditions under which this equation is applicable are (1) no streamwise pressure

gradient (plane wall, also called flat plate), (2) the same Reynolds number and (3)

uniform temperature BCs. Note that this analogy function does not depend on

streamwise position; in other words, a single value can be applied to all streamwise

positions, provided that the flow is laminar. For this reason, the analogy functions are

sometimes called the analogy factor in the present study, referring to the ratio of Nusselt

number to Sherwood number. This section discusses how this analogy function comes

about.

Heat and mass transfer processes are governed by the partial differential

equations: energy (or temperature) and mass transfer equations, respectively. For a

steady-state and flow, two-dimensional, laminar, incompressible boundary layer, they

reduce to the following expressions.

2

2

yT

yT

vxT

u=

+

(3.1.2)

2

2

ym

Dym

vxm

u=

+

(3.1.3)

Since the velocity parameters, u and v, are involved, the continuity equation and the

momentum equation of the boundary layer must be introduced.

0yv

xu =

+

(3.1.4)

2

2

y

u

y

uv

x

uu

=

+

(3.1.5)

The above momentum equation assumes no streamwise pressure gradient, incompressible

flow and constant properties. Now we have four equations and four unknowns, u, v, T

and m. Comparing Eqns. (3.1.2) and (3.1.3), when = D, the temperature and the mass


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transfer equations are identical. Upon nondimensionalizing Eqns. (3.1.2) and (3.1.3), the

thermal diffussivity, , and the diffusion coefficient, D, are replaced by the dimensionlesstransport properties: Prandtl number, Pr, and Schmidt number, Sc. They are defined by:

=Pr ; DSc

= (3.1.6 and 7)

Solution methods are discussed in Eckert and Drake (1987) and Kays and

Crawford (1993). The resulting Nusselt number, Nu, distribution for a laminar boundary

layer as a function of streamwise position is well approximated as follows.

3 / 12 / 1xx PrRe332.0Nu = (3.1.8)

Similarly, for the mass transfer equation, the Sherwood number, Sh, distribution yields:

3 / 12 / 1xx ScRe332.0Sh = (3.1.9)

Equation (3.1.1) can now be obtained by dividing Eqn. (3.1.8) by Eqn. (3.1.9),

assuming the same x-Reynolds number. As is obvious from Eqns. (3.1.2) and (3.1.3), Nu

is equal to Sh when Pr is equal to Sc. Furthermore, in the case of air (Pr = 0.707) and

naphthalene (Sc = 2.28), the analogy function becomes:

677.0ShNu = (3.1.10)

As noted earlier, this relation assumes the same Reynolds number, the uniform-level BCs(uniform wall temperature for the heat transfer and uniform mass concentration for the

mass transfer) and no streamwise pressure gradients. Effects of these assumptions are

evaluated in the next section.

3.1.2 Analytical solution to the cross-boundary-condition analogy

The conventional relationship was derived assuming both processes having the identicallevel boundary condition (BC). This relation, Eqn. (3.1.1), may be sufficient in gas

turbine heat transfer studies as shown by the numerical simulations in the next section.

When results from a mass transfer experiment are used to estimate heat transfer situations

in an actual gas turbine engine, the conversion is from uniform-level-BC mass-transfer


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data to uniform-level-BC heat-transfer data, for which Eqn. (3.1.1) is most suited. The

gas turbine heat transfer situation is best simulated with a constant wall temperature

boundary condition. This is because variations in temperature at the airfoil surface are

relatively small compared to the difference between the surface and freestream

temperatures (e.g. pp.268-70 in Boyce, 1982). The next few sections show that this

relation is quite accurate for a surface with non-zero streamwise pressure gradients.

Another analytical solution is easy to derive. This is for the cases in which

boundary conditions do not match between the two processes: a uniform level for one and

a uniform flux for the other. Here, the relation is first derived, and discussions are made

on its meaning and possible applications.

Assuming a flat plate and laminar boundary layer, the distribution of Nusselt

number for a uniform heat flux BC is well approximated by the following (Kays and

Crawford 1993).

3 / 12 / 1xx PrRe453.0Nu = (3.1.11)

Dividing this by Eqn. (3.1.9) yields the analogy function for the cross-BC processes.

3

1

ScPr

36.1ShNu

= (3.1.12)

Corresponding to Eqn. (3.1.10), the above expression, for air and naphthalene, results in:

924.0ShNu = (3.1.13)

Again, this relationship is not a function of Reynolds number or streamwise position.

Noteworthy is that the value for the cross-BC processes, Eqn. (3.1.13), differs 36% from

the value for the conventional relationship, Eqn. (3.1.10).

The first discussion focuses on a significance of these results. Because the

governing partial differential equations are essentially identical for mass and heat transfer,

Eqn. (3.1.12) can, for example, be written for two processes, both being heat transfer:

31

5

6

5

6

PrPr

36.1NuNu

= (3.1.14)


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Here, the subscripts 6 and 5 designate the two different fluids and BCs; fluid 6 is at a

uniform flux BC, and fluid 5 is at a uniform level BC (corresponding to designations

introduced later, see Table 3.1). When these are the same fluids, the ratio of Nusselt

numbers becomes 1.36. This means that the heat transfer coefficient at any location on a

flat plate with uniform heat flux BC is always 36% greater than that of the plate with the

same Reynolds number (same location if 5 = 6) with a uniform temperature BC. Thiscan be realized by comparing Eqns. (3.1.1) and (3.1.12) since the denominators have the

same processes in the both equations.

One possible application of this analogy function is to convert data from a heat

transfer experiment with a uniform flux BC to uniform-temperature BC data. This would

be useful especially since many heat transfer experiments (such as steady state measure-

ments with liquid crystals) are done with uniform heat flux BCs. However, as shown in

the next section, streamwise pressure gradients have a greater effect on Eqn. (3.1.14) than

on Eqn. (3.1.1); the cross-BC functions for an airfoil profile deviate from Eqn. (3.1.14)

more than the matching-BC functions for an airfoil profile deviate from Eqn. (3.1.1). For

this type of an application, it is necessary to study how pressure gradients affect the

analogy functions.

Another possible case in which the cross-BC analogy function, Eqn. (3.1.12), may

be useful is when a heat transfer experiment with a uniform heat flux BC is directly

compared with a mass transfer experiment with a uniform mass concentration BC in

order to determine effects of streamwise pressure gradients. Because of the identical

governing PDEs, the mass transfer experiment is representative of both mass and heat

transfer processes with a uniform level BC. The results of the comparison should be

expressed as magnitudes of deviations from Eqn. (3.1.12). It will be shown that

converting the mass transfer data into the heat transfer data with both at a uniform level

BC should not be a problem even for a case with streamwise pressure gradients by the useof the conventional relationship, Eqn. (3.1.1). Once the generalized effect of pressure

gradients is determined, this relationship may be applied when boundary conditions do

not match.


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3.2 Analogy Functions on the CF6 Profile

3.2.1 Setup of cases

This chapter uses the numerical boundary layer code, TEXSTAN, to computationally

determine Nusselt number distributions for various flow and boundary conditions (BCs).

Heat/mass transfer analogy functions are obtained from numerically evaluated Nu and Sh

distributions under the flow and BCs of interest.

In this section, the streamwise pressure (or velocity) gradient of the General

Electric CF6 engine first-stage vane is implemented into the simulations, and results are

compared to the corresponding zero-pressure-gradient case. Also varied are BCs in order

to examine the effects on the analogy function. The primary reason that the different BCs

are tested is that a mismatch in BCs is frequent when heat transfer rates are

experimentally determined from mass transfer experiments. Many heat transfer

experiments are done with the BC of a uniform heat flux. If data from such experiments

are used to calculate the analogy function, consistency with Eqn. (3.1.1) would be lost,

for this equation was derived assuming uniform level BCs in both the mass and heat

transfer experiments. Therefore, it is important that one knows the effects of different

combinations of BCs.

Figure 3.1 shows the velocity profiles of the CF6 and the GE90 profiles on the

suction surfaces. The GE90 profile is used in the next section. The suction side profiles

are shown here, and all the analyses focus on the suction side. This is because suction

sides usually are of more interest, for they are the ones subjected to influences such as

flow separation, transition and three-dimensional secondary flows.

When the input data files to TEXSTAN are prepared, all possible combinations of

the three factors are sought, except cases of mass transfer with flux boundary conditions.

This accumulates the number of cases considered in this section to six, and they are

summarized in Table 3.1.


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Figure 3.1 Velocity profiles of the CF6 and GE90 blades, normalized on exit velocity.The CF6 from Chen (1988) and the GE90 from Wang (1997).

Table 3.1 Summary of cases for the CF6 profile

Case CF6-1 CF6-2 CF6-3 CF6-4 CF6-5 CF6-6

Profile CF6 Us=constantplane wall

Sc or Pr 2.28 0.707 2.28 0.707

B.C. T w qw Tw qw

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

x/C

U s /

U e x

CF6 profile

GE90 profile


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The following is a description of each of the cases simulated in this section. This

list can be applied also to the cases simulated in the next section with the GE90 profile.

Case CF6-1 --- simulation under the same conditions as Chen (1988; and Chen

and Goldstein, 1992), an experiment with the CF6 profile, a uniform mass

concentration and a naphthalene/air mixture being the fluid

Case CF6-2 --- case CF6-1 converted by Eqn. (3.1.1) to heat transfer to air

Case CF6-3 --- simulation under the same conditions as Chung (1992; and Chung

and Simon, 1993), an experiment with the CF6 profile, a uniform wall heat

flux and air being the fluid

Case CF6-4 --- the CF6-1 case with the pressure profile replaced with a uniform

velocity, a flat plate caseCase CF6-5 --- case CF6-4 converted by Eqn. (3.1.1) to heat transfer to air

Case CF6-6 --- Case CF6-3 with the pressure profile replaced with a uniform

velocity of a flat plate case

Two other possible cases, processes with uniform mass flux BCs, were not included

because, when a mass transfer experiment is done with naphthalene, the BC is of a

uniform mass concentration (a uniform level, equivalent to uniform temperature), and

usually, the conversion is performed from mass transfer to heat transfer. Thus, the cases

with uniform mass flux are not important.

Since two of the above cases correspond to heat transfer processes and the others

mass transfer, there are 8 combinations for which to produce some kind of analogy

functions. However, only the practical cases are analyzed. They are described below. As

the original analogy function is expressed in terms of the ratio, Nu/Sh, the numerators

below correspond to heat transfer processes and the denominators to mass transfer.

CF6-5/CF6-4 --- reproducing Eqn. (3.1.1) with the same boundary conditions.

CF6-2/CF6-1 --- same BCs as CF6-5/CF6-4 but with the CF6 pressure profile


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CF6-3/CF6-1 --- the conditions used to compare the Chens and the Chungs

experiments as done in the next chapter

CF6-6/CF6-4 --- same as CF6-3/CF6-1 but with absence of pressure gradient

effects on both the mass and the heat transfer processes.

CF6-4/CF6-1 --- both mass transfer processes. This shows effects of the

streamwise pressure gradient.

CF6-2/CF6-3 --- both heat transfer processes. This shows effects of the shift in

BCs. If this were a simple function, one could easily convert the data from

an experiment with a uniform heat flux BC to those of a uniform

temperature BC, the latter being more representative of actual engine

conditions.

For each of the above six different cases, an input data file to TEXSTAN was

prepared. They are presented in Appendix A. To introduce the computational code used

in the present study, TEXSTAN was originally developed as the STAN5 program by

Crawford and Kays (1976). The reference here provides theoretical backgrounds and

instructions for STAN5, and basically the same operations apply to TEXSTAN. These

programs solve transport equations such as momentum, energy and mass for two-

dimensional, internal and external boundary layer flows. The user of these programs is to

prepare an input file that lists all necessary parameters such as boundary conditions and

physical properties and directions for the outputs. The user solves the momentum

equation and, if desired, heat and/or mass transport equations.

3.2.2 Results and discussions

Figure 3.2 presents the variations of heat/mass transfer analogy functions versus

streamwise positions. In a numerical simulation of a boundary layer, heat (or mass)

transfer rates are calculated based on temperature (or mass concentration) gradients at the

surface. The calculation cannot proceed after the flow separates, at which point the

velocity gradient becomes zero. This is what happens to cases 3/1 and 2/1 near x/C of


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0.95. Looking at the velocity profile on Fig. 3.1, one finds that the flow separates just

after the mainstream flow starts to decelerate at x/C of 0.8.

There are two main trends seen in Fig. 3.2. Both the functions 2/1 and 5/4 are

essentially the same, as Eqn. (3.1.1) implies, the F ref value of Fig. 3.2, while functions 3/1

and 6/4 differ.

First of all, looking at the functions that are consistent with Eqn. (3.1.1), one

realizes that each pair consists of the same pressure profile (either the CF6 or flat plate)

and the same level BC (temperature and mass concentration). Also, the pair 2/1 is only

very weakly dependent on the non-zero pressure gradient. At this point, it can be

concluded that using Eqn. (3.1.1) to convert mass transfer data would yield a perfect

estimate of flat-plate heat transfer with uniform temperature BC if the mass transfer were

done with a flat plate. Similarly, it would yield an almost perfect estimate of heat transfer

with the CF6 profile and uniform temperature BC if the mass transfer were also done

with the CF6 profile. In other words, a single value from Eqn. (3.1.1) based only on Pr

and Sc can be used to convert Sh to Nu at any and all streamwise locations, assuming that

the heat transfer process has the same pressure profile, the same Re and a uniform-

temperature BC.

Second of all, the other two lines presented in Fig. 3.2, 3/1 and 6/4, form another

group; the 3/1 line appears to vary in the vicinity of the 6/4 line. Referring to Table 3.1,

one realizes that each line consists of heat transfer with a uniform flux BC and mass

transfer with a uniform level BC cross-BC analogy functions. To define the cross-BC

analogy functions, I would like to refer to heat transfer with a uniform flux BC and mass

transfer with a uniform level. Since a mass transfer experiment with naphthalene

provides a uniform level BC, this is often the case in converting mass transfer data to get

heat transfer coefficients using the analogy functions. Contrary to the behavior when

numerator and denominator had level BCs, cross-BC conversions cannot beapproximated by Eqn. (3.1.1) or Eqn. (3.1.12). Moreover, line 3/1 (both on the CF6

profile) is not as close to the line 6/4 (both on flat plate) as line2/1 (both on the CF6

profile) is to the line 5/4 (both on flat plate). Therefore, even when the line 6/4 is


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analytically derived, it is not a representative of cross-BC analogy functions for cases of

non-zero pressure gradient. The importance of this lies in the fact that many heat transfer

experiments are done with a uniform flux (a uniform heat flux).

3.3 Analogy Functions on the GE90 Profile

3.3.1 Setup of cases

In this section, distributions of analogy functions on the GE90 profile and the

corresponding flat plate cases are calculated, using the TEXSTAN program, and

discussed. The velocity profile of the GE90 blade is provided in Fig. 3.1, and the cases of

the computations are summarized in Table 3.2. The GE90 is a newer engine than the CF6

from General Electric, and its profile has a long-lasting acceleration region compared to

the CF6 profile, almost one half of the entire surface length. Table 3.2 is essentially the

Figure 3.2 Analogy functions for the CF6 profile and the corresponding flat plate cases.Re c=171,000 and P/C=0.77 (x/C=1.25 at the trailing edge). F ref corresponds to the value

from Eqn. (3.1.1). Case designations are simplified such as 3/1 for CF6-3/CF6-1.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

x/C

N u / S h

Fref

5/4

6/43/1

2/1


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same as Table 3.1, except that, where non-zero velocity profiles are implemented, the

GE90 replaces the CF6 profile. Therefore, the GE90-1 case and the CF6-1 case, for

instance, are under the same conditions except for the velocity profiles.

The case descriptions in the previous section can still be used, and all the six pairs

of analogy functions that were generated in the CF6 case are regenerated for the GE90

case.

3.3.2 Results corresponding to the Wangs experiment

Figure 3.3 presents four of the six analogy functions generated in this section, and the

other two are discussed in the next section. The cases with the non-zero pressure gradient

again indicate flow separation near x/C of 1.24. By this point, the flow has remained

attached for a quite a long distance against adverse pressure gradients downstream of the

beginning of deceleration near x/C of 0.7. This is remarkably longer than in the CF6 case

in which flow separates just after the pressure gradient sign reverses. This may be

Table 3.2 Summary of cases for the GE90 profile

Case No. Nu or Sh (profile #, fluid, BC )

GE90-1 Sh ( GE90, Sc=2.28, T w )

GE90-2 Nu (GE90, Pr=0.707, T w )

GE90-3 Nu (GE90, Pr=0.707, q )

GE90-4 Sh (f.p. *, Sc=2.28, T w )

GE90-5 Nu ( f.p. *, Pr=0.707, T w )

GE90-6 Nu (f.p. *, Pr=0.707, q )

# free-stream velocity profile* flat plate profile (uniform free-stream velocity)


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because of differences in the degree of deceleration and thinner boundary layers in the

case of the GE90, due to the prolonged strong acceleration.

Now, the results in Fig. 3.3 look quite similar to those in Fig. 3.2: lines 2/1 and

5/4 forming one group and lines 3/1 and 6/4 forming the other. The fist group behaves in

the same way as the one in the CF6 profile cases. The flat plate pair, 5/4, again confirms

the validity of Eqn. (3.1.1), and, together with the line 5/4 on Fig. 3.2, it is shown that the

reference value, F ref = 0.66, is not a function of Reynolds number. The GE90 profile pair,

2/1, varies only a little from the reference value, F ref , compared to lines, 3/1 and 6/4.

The other group of lines, 6/4 and 3/1, has the cross-BC analogy functions. The

flat plate function, 6/4, records the value 0.92, the same as the one from the line 6/4 in

Fig. 3.2. This shows that the analogy function for cross-BC cases with the absence of the

pressure gradient may not depend on Reynolds number. A simple relation such as Eqn.

(3.1.12) may be possible for cross-BC cases.

The deviations of lines 3/1 and 2/1 from the lines 6/4 and 5/4, respectively, in Fig.

3.3 seem greater than the corresponding deviations in Fig. 3.2 although line 2/1 still

deviates from F ref only slightly. Figures 3.2 and 3.3 are not enough to conclude that the

GE90 profile produces larger errors from the corresponding flat plate cases than does the

CF6 profile. This is because the GE90 cases are simulated at a greater chord Reynolds

number. The next section discusses whether these deviations are actually due to the

greater Reynolds number or the difference in the pressure profiles.


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Figure 3.4 Analogy functions, Nu/Sh, for the GE90 profile and the corresponding flatplate cases. Re c=171,000 and P/C=0.77 (x/C=1.38 at the trailing edge). F ref correspondsto the value from Eqn. (3.1.1). Case designations are simplified such as 3/1 for GE90-

3/GE90-1

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4x/C

N u / S h

5/4

6/4

3/1

2/1


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3.4 Heat/Heat and Mass/Mass Analogy Functions

3.4.1 Origins of non-uniform analogy functions

In the previous sections, all analyses were based on analogy functions between a fluidwith Pr=0.707 (representing air) and a fluid with Sc=2.28 (representing naphthalene), and

it was demonstrated that some pairs of processes resulted in rather uniform analogy

functions and others do not. This section attempts to further examine different effects

that form a non-uniform distribution of the analogy function. One of the most significant

effects is the difference in the thermal boundary conditions (BCs) as discussed in the

previous section. The first step here is to examine the effect of pressure profiles in cases

with a transport property of 2.28. The second step is to examine the effect of thermalboundary conditions in cases with a transport property of 0.707.

Obviously, a uniform distribution of the analogy function is advantageous, so it is

important to know under what circumstances the analogy function is not uniform and how

it may behave in such cases. When the function is uniform over a surface such as lines

6/4 and 5/4 in Figs. 3.2, 3.3 and 3.4, a single factor can be applied to the entire surface in

order to convert Sherwood numbers to Nusselt numbers, for example. When the function

is not uniform, e.g. line 3/1 in Figs. 3.2, 3.3 and 3.4, the conversion factor is dependent on

locations on the surface. Even if the variations along the surface cannot possibly be

predicted by knowing the effects of different causes, one should be able to determine

when the analogy function may behave with large deviations.

Pressure profiles may be thought to heavily affect uniformity of the analogy

function distribution. However, this is not always true. For example, line 3/1 in Figs.

3.2, 3.3 and 3.4 is a case with non-zero pressure gradient profiles and deviates from line

6/4 which is the identical case, except for the zero pressure gradient profiles. This large

deviation is not observed between the pairs 2/1 and 5/4. Therefore, the deviation of line

3/1 has a combined effect of thermal BCs and pressure profiles. Pair 4/1 is chosen to

examine the effects of pressure profiles since it is an intermediate case between 3/1 and


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6/4. Pair 2/3 is chosen to examine the effects of thermal boundary conditions since it may

become useful in converting flux-BC experimental data to level-BC data.

3.4.2 The effects of pressure profiles in mass/mass transfer analogy: line 4/1

Since the transport property of 2.28 was assumed to represent Sc of naphthalene, pair 4/1

might be called a mass/mass analogy. Line 4/1 in Figs 3.5, 3.6 and 3.7 is discussed here

as an example of a pair with level-BCs and identical transport properties. Looking at

each figure, the shape looks exactly like line 3/1 in the previous figures, except near the

leading edge. The resulting distributions clearly show that the trends are similar to lines

3/1 and 2/1 in Figs. 3.2, 3.3 and 3.4, except for the magnitudes of deviation. In other

words, when line 3/1 has a positive slope, line 2/1 also has one. Therefore, it can beconcluded that the shapes of lines 3/1 and 2/1 are due to the pressure gradients, and the

BCs are mainly responsible for the magnitudes of the deviations from the corresponding

plane wall cases.

Figure 3.5 CF6 Chen cases,Re c=171,000 and P/C=0.77 (x/C=1.25 at the trailing edge)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4x/C

N u

/ S h

4/1

2/3


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For case 1, which assumes non-zero pressure gradients, values of the analogy

function in the vicinity of the stagnation point can be analytically calculated using the

wedge flow solution (Eckert and Drake 1987). This solution method may be used where

velocity in the freestream, U s, can be expressed as:m

s xcU = (3.4.1)

The exponent, m, is described as follows.

dxdU

Ux

m ss

= (3.4.2)

According to Eckert and Drake (1987), the resulting Nusselt number for a boundary

condition (BC) of uniform wall temperature is:

3 / 12 / 1

xxPrRe)m(f Nu

=(3.4.3)

We know, at this point, that the above equation is the same as:

3 / 12 / 1xx ScRe)m(f Sh = (3.4.4)

This equation can now be used for line 4/1. Here, the coefficient, f(m), is a constant with

x but which depends on the acceleration parameter, m, in Eqn. (3.4.2), and is tabulated in

Kays and Crawford (1993) as the form of f(m) multiplied by Sc 1/3 .

One can apply Eqn. (3.4.4) to the stagnation flow region (small x/C) of the line

4/1. The numerator is the flat plate case where m = 0, and the stagnation flow is assumedfor the denominator, m = 1. Therefore, f(0) = 0.41 and f(1) = 0.72 yield (pp.167, Kays

and Crawford 1993). The following results since Sc is the same for the both cases.

1,s

4,s

1,s

4,s

1,s

4,s

3 / 1

3 / 1

1

4

U

U57.0

U

U

72.041.0

U

U

Sc)1(f Sc)0(f

ShSh === (3.4.5)

The numerator of the above equation is for the flat plate case for which m is zero. The

denominator is for the case with the non-zero pressure gradients for which stagnation

flow with m being 1.0 is assumed near the leading edge since the velocity increase islinear with x. The factor, therefore, is dependent on the ratio of freestream velocities.

The ratio theoretically begins with infinity at the leading edge since the velocity at the

stagnation point, U s,1 , is zero. For this reason, long-lasting acceleration, e.g. the GE90

profile, causes a continuous decrease seen until x/C of about 0.3 in Figs. 3.6 and 3.7.


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For ease, the GE90 profile can be looked at first. According to Fig. 3.1, the GE90

profile has a point at which no acceleration or deceleration is present (x/C of 0.68). Atthis streamwise position, the Sh/Sh values in Figs. 3.6 and 3.7 are about 0.8, moving

toward 1.0. At x/C of 1.0, line 4/1 in Fig. 3.6 crosses the Sh/Sh value of 1.0. This delay

of reaching 1.0 seems to be the effect of the flow history, which has suppressed the

boundary layers from growing. The flow in the GE90 profile has experienced a long

acceleration period when it ends. The boundary layer of such flow is still different from

the one having developed on a flat plate. Following the acceleration period in the GE90

profile is the deceleration period. Deceleration seems to strongly change boundary layercharacteristics, thereby increasing the Sh/Sh values. Finally, it reaches the maximum

value just before the flow separates.

The CF6 pressure profile is much more flat than the GE90 profile, as seen in Fig.

3.1. Therefore, the Sh/Sh values in Fig. 3.5 vary but remain close to closer to 1.0. The

Figure 3.7 GE90 profile with Chens Reynolds number,Re c=171,000 and P/C=0.77 (x/C=1.38 at the trailing edge)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

x/C

N u /

S h

4/1

2/3


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profile has a point of zero gradient just after the stagnation flow region near the leading

edge. The Sh/Sh values quickly adjust to near 1.0. From this point on, the profile has a

few points at which the gradient becomes zero and the Sh/Sh values tend to converge to

1.0.

Regardless of the type of pressure gradient, line 4/1 strongly responds to adverse

pressure gradients. Just before the flow separates, it goes beyond 1.0 and even over 1.2

for the GE90 cases. One must note here, however, that the local freestream velocity is

not equal to that of the plane wall case, and the normalized local freestream velocity at

separation of each pressure profile is also different. Therefore, to conclude that the

difference in the pressure profiles caused the GE90 profile to have higher analogy factors

at separation may be too hasty.

3.4.3 The effects of thermal BC in heat/heat transfer analogy: line 2/3

Line 2/3 in Figs. 3.5, 3.6 and 3.7 corresponds to both cases having the same non-zero

pressure gradients, the same transport property of 0.707 (presumably air) but different

BCs. To be precise, the numerator, case 2, is with a uniform level BC and the

denominator, case 3, with a uniform flux BC. If this analysis of line 2/3 yielded a

sufficiently flat profile of the analogy function, the function could be used together withthe analogy 2/1 to form the analogy 3/1, which is the analogy between uniform level mass

transfer data and uniform flux heat transfer data. When this combined process, 2/3 and

2/1, can be simply expressed, an experimental relationship for the cross-BC analogy

functions can be found. This is done in the next chapter. Besides different Reynolds

numbers, deriving a simple analogy function requires understanding the effects of

mismatching BCs. This section addresses the cross-BC effect as augmented by the non-

zero pressure gradients.

Unlike the pair 4 and 1, cases 2 and 3 have the same geometry. Therefore, the

flow fields are identical. If, by any means, the uniform level BC generated a similar

temperature field as that of the uniform flux BC, the Nu/Nu ratio again would be near 1.0.


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However, this is clearly not the case. The Nu/Nu values are confined in the region

between 0.6 and 1.0, smaller variations than in the previous study (4/1).

In the three figures, some trends can be observed. The distributions always begin

with the ratio of 1.0 at the leading edge and quickly decrease, without sudden changes,

toward values in the vicinity of 0.7 and 0.8. At the point of separation, each analogy

factor reaches the minimum of almost 0.6. Remaining always less than 1.0 is expected

because of the earlier discussion regarding the development of Eqn. (3.1.12). In fact, the

inverse of the equation can be used to explain the major part of the behavior of line 2/3 in

Figs. 3.5, 3.6 and 3.7. Because line 2/3 assumes the same transport property, the ratio is

unity. Therefore, Eqn. (3.1.12) is left only with the coefficient 1.36, and its inverse is

0.735. Originally, the equation is developed for a plane wall case with the Nusselt

number assumed to result from uniform-flux BC and the Sherwood number from a

uniform-level BC. The only difference from line 2/3 is the pressure profile, besides the

inverted ratio. Thus, for a flat plate case, 6/5, we expect a value of 1/1.36 (= 0.735, see

also Eqn. 3.1.14).

A general trend then is, except near the leading edge, that the local analogy factor

is greater than 0.735 when the flow accelerates and less when decelerating. The flow

history seems not to matter much here. When the velocity profiles get flat, the analogy

factors seem to quickly adjust to 0.735. The magnitudes of the deviations from the

reference value of 0.735 seem to be a function of flatness of the velocity profiles; the

flatter CF6 profile deviates less than does the less-flat GE90 profile. Finally, whether this

analogy function is useful in converting flux-BC heat transfer data to level-BC heat

transfer data for laminar flow regions is dependent on the type of pressure profile.

3.5 Considerations on Flow Transition from Laminar to Turbulent

As discussed in the next Chapter, the conventional analogy factor for turbulent boundary

layers is different from that of laminar boundary layers (compare Eqns. (4.1.2) and


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(3.1.10) ). Therefore, it would be interesting to find how the factors vary in the transition

region.

One way to do this may rely on experimental data. If there were two identical

experiments, except that one was heat transfer experiment and the other mass transfer, the

two data sets could be compared to determine analogy factors in the transition region.

This is a motivation to the next chapter in which the factors in transition regions are

discussed.

Another possibility to study variations of analogy factors in transition regions is to

extend the TEXSTAN calculations performed in this Chapter. The program has a

capacity to simulate transitions and turbulent flows. However, the calculations performed

so far had these features disabled so that no transition or turbulence modeling would be

necessary. Therefore, the calculations stopped when flows reach separation or the trailing

edge, remaining laminar throughout. By implementing transition and turbulence models,

the above calculations can be extended to the factors in transition regions and into

turbulent boundary layer regions. To do this, however, one must know which transition

model is best suited for the flow that is simulated.

An option available in TEXSTAN is to provide the program with a momentum-

thickness Reynolds number as an indicator for the beginning of transition. The Reynolds

number, Re m, increases with the streamwise positions, and transition is expected to begin

where Re m reaches a transition value. Therefore, with this option, the user of the program

must have, at least, an estimate of the transitional Reynolds number.

Momentum-thickness Reynolds number is a simple indicator to predict the

beginning of transition, and it has been discussed quite extensively. McDonald (1973)

establishes a relation between displacement-thickness Reynolds number and freestream

turbulence (Figure 1). Almost twenty years later, Mayle (1991a and 1991b) developed,

from more experimental data, an equation for transitional momentum-thickness Reynoldsnumber as a function of freestream turbulence intensity. Mayle emphasizes a good

agreement between the McDonalds relation, also called Abu-Ghannam and Shaw

relation, and the Mayles equation, which reads:


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43

8 / 5tr,m TI400Re

= (3.5.1)

Both McDonalds and Mayles relations are plotted in Fig. 3.8, along with momentum

thickness Reynolds numbers calculated by TEXSTAN for the CF6 velocity profile.

At a first glance of Fig. 3.8, one recognizes that the Re m from TEXSTAN does notreach either of the transition Reynolds numbers, even at the point of separation. Even at

this point, the TEXSTAN Re m is much lower than the transition values. This then

suggests that the transition take place after flow separation, since we know, from Chungs

and Chens experiments, that the boundary layers near the trailing edge of the CF6 profile

are turbulent. This is consistent with Chens discussions in his study (section 5.3 in

Chen, 1988). One reason for the difference in momentum thickness Reynolds numbers

between the models and the TEXSTAN Rem

is that the models assume no streamwise

pressure gradients. The TEXSTAN calculation prescribes adverse pressure gradients

toward the trailing edge, which considerably reduce transitional Reynolds numbers. This

analysis raises yet another question regarding the analogy function: What is the analogy

function in the region of flow separation? This is left for further studies in the analogy

function.


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3.6

ConclusionsThe heat/mass transfer analogy function is studied for laminar boundary layers. Transport

of heat and sublimation of naphthalene from heated and naphthalene-coated surfaces,

respectively, are assumed. Analytical and numerical calculations have resulted in various

values and distributions of the analogy function, depending on the combinations of

pressure profiles and thermal boundary conditions. Furthermore, to examine the effects

of these two, a mass/mass analogy function and a heat/heat analogy function are

developed.First, two analogy factors are developed analytically for plane wall cases. The

analogy factor, Nu/Sh, of 0.677 is a conventional relationship which assumes zero

pressure gradients and uniform-level boundary conditions for both heat and mass transfer

processes. It is a motivation to the present study to examine how the analogy function

Figure 3.8: Comparison of critical momentum thickness Reynolds numbers for the CF6velocity profile that is shown in Fig. 3.1

0

100

200

300

400

500

600

700

0.0 0.2 0.4 0.6 0.8 1.0x/C

R e m

McDonald

Mayle

TEXSTAN


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45

deviates from this conventional value by changing the boundary conditions and pressure

profiles. The other analogy factor developed is 0.924 (1.36 multiplied by 0.677). This

factor assumes zero pressure gradients for both and flux-BC for heat transfer and level-

BC for mass transfer. The effect is a 36% increase in the analogy factor due to the flux

boundary condition assumed in the heat transfer process. Still, the single value of 0.924

may be applied to an entire surface.

Second, the TEXSTAN program is employed to numerically calculate the analogy

functions for non-zero pressure gradients, namely the CF6 and GE90 profiles. The above

two analytical solutions are verified, and the non-zero gradient cases exhibit deviations

from the corresponding plane wall cases. The magnitudes of the deviations are different,

depending on the thermal BC assumed in the heat transfer processes. The deviations are

larger with a heat flux BC than with a uniform temperature BC. These behaviors are

similarly observed in both the CF6 and the GE90 cases.

Third, two pairs of processes that are not heat/mass analogies are considered in order to

further investigate the effects of pressure gradients and thermal BCs. To see the effects of

pressure profiles, mass/mass analogy is developed. The resulting shape of line 4/1 in

Figs. 3.5, 3.6 and 3.7 suggests that the shapes of lines 3/1 and 2/1 in Figs. 3.2, 3.3 and 3.4

are due to the pressure profiles, but the pressure profiles may not be responsible for the

deviations from the corresponding plane wall cases. A heat/heat analogy is developed to

see the effects of thermal boundary conditions. With the only difference being the

thermal boundary conditions, this analogy function examines a possibility of a conversion

of flux-BC heat transfer data to level-BC heat transfer data for non-zero pressure

gradients. If this analogy function happened to be completely a flat distribution, the

conversion would be very simple. However, this has not been the case. The deviation

from the corresponding plane wall factor of 0.735 in this case is larger than the deviation

of line 2/1 in Figs. 3.2, 3.3 and 3.4 from its corresponding plane wall factor of 0.677.This implies, for laminar boundary layers, that a mass transfer experiment with

naphthalene sublimation is going to be more accurate than a flux-BC heat transfer


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experiment since the boundary condition in an actual gas turbine heat transfer is

considered to be a uniform level boundary condition.

Finally, a few thoughts are given to the analogy function in the regions of flow separation

and transition. It should be interesting and useful to uncover how the analogy function

behaves under such conditions. The present study suggests that both numerical and

experimental investigations could be done for this purpose, and the experimental

approach is pursued in the next chapter.

Documents

Heat-Mass Transfer Analogy