Heat/Mass Transfer Analogy  Laminar Boundary Layer


 Jane Berry
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1 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 Conventional relationship 20
2 The conventional relationship for laminar boundary layer flow is given by the following simple expression. 1 3 Nu Pr = (3.1.1) Sh Sc 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 steadystate and flow, twodimensional, laminar, incompressible boundary layer, they reduce to the following expressions. T T u + v x y m m u + v x y 2 T = α 2 y 2 m = D 2 y (3.1.2) (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. u v + = 0 (3.1.4) x y u u u + v x y 2 u = ν 2 y (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 α = D, the temperature and the mass 21
3 transfer equations are identical. Upon nondimensionalizing Eqns. (3.1.2) and (3.1.3), the thermal diffussivity, α, and the diffusion coefficient, D, are replaced by the dimensionless transport properties: Prandtl number, Pr, and Schmidt number, Sc. They are defined by: ν ν Pr = ; Sc = (3.1.6 and 7) α D 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. 1/ 2 1/ 3 Nu x = Re x Pr (3.1.8) Similarly, for the mass transfer equation, the Sherwood number, Sh, distribution yields: 1/ 2 1/ 3 Sh x = Re x Sc (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 xreynolds 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: Nu = (3.1.10) Sh As noted earlier, this relation assumes the same Reynolds number, the uniformlevel 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 Analytical solution to the crossboundarycondition analogy The conventional relationship was derived assuming both processes having the identical level 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 uniformlevelbc masstransfer 22
4 data to uniformlevelbc heattransfer 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 in Boyce, 1982). The next few sections show that this relation is quite accurate for a surface with nonzero 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). 1/ 2 1/ 3 Nu x = 0.453Re x Pr (3.1.11) Dividing this by Eqn. (3.1.9) yields the analogy function for the crossbc processes. 1 3 Nu Pr = 1.36 (3.1.12) Sh Sc Corresponding to Eqn. (3.1.10), the above expression, for air and naphthalene, results in: Nu = (3.1.13) Sh Again, this relationship is not a function of Reynolds number or streamwise position. Noteworthy is that the value for the crossbc 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: Nu Nu Pr6 = Pr (3.1.14) 5 23
5 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 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 ν 5 = ν 6 ) with a uniform temperature BC. This can 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 uniformtemperature BC data. This would be useful especially since many heat transfer experiments (such as steady state measurements 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 crossbc functions for an airfoil profile deviate from Eqn. (3.1.14) more than the matchingbc 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 crossbc 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 use of 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. 24
6 3.2 Analogy Functions on the CF6 Profile 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 firststage vane is implemented into the simulations, and results are compared to the corresponding zeropressuregradient 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 threedimensional 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
7 Us/Uex CF6 profile GE90 profile x/c 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 CF61 CF62 CF63 CF64 CF65 CF66 Profile CF6 U s =constant plane wall Sc or Pr B.C. T w q w T w q w 26
8 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 CF 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 CF case CF61 converted by Eqn. (3.1.1) to heat transfer to air Case CF 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 CF the CF61 case with the pressure profile replaced with a uniform velocity, a flat plate case Case CF case CF64 converted by Eqn. (3.1.1) to heat transfer to air Case CF Case CF63 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. CF65/CF reproducing Eqn. (3.1.1) with the same boundary conditions. CF62/CF same BCs as CF65/CF64 but with the CF6 pressure profile 27
9 CF63/CF the conditions used to compare the Chen s and the Chung s experiments as done in the next chapter CF66/CF same as CF63/CF61 but with absence of pressure gradient effects on both the mass and the heat transfer processes. CF64/CF both mass transfer processes. This shows effects of the streamwise pressure gradient. CF62/CF 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 twodimensional, 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 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 28
10 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 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 nonzero 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 flatplate 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 uniformtemperature 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 crossbc analogy functions. To define the crossbc 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 BC s, crossbc conversions cannot be approximated 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 29
11 analytically derived, it is not a representative of crossbc analogy functions for cases of nonzero 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) /1 6/4 Nu/Sh /1 5/4 F ref x/c 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 CF63/CF Analogy Functions on the GE90 Profile 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 longlasting acceleration region compared to the CF6 profile, almost one half of the entire surface length. Table 3.2 is essentially the 30
12 same as Table 3.1, except that, where nonzero velocity profiles are implemented, the GE90 replaces the CF6 profile. Therefore, the GE901 case and the CF61 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. Table 3.2 Summary of cases for the GE90 profile Case No. Nu or Sh (profile #, fluid, BC ) GE901 Sh ( GE90, Sc=2.28, T w ) GE902 Nu (GE90, Pr=0.707, T w ) GE903 Nu (GE90, Pr=0.707, q ) GE904 Sh (f.p. *, Sc=2.28, T w ) GE905 Nu ( f.p. *, Pr=0.707, T w ) GE906 Nu (f.p. *, Pr=0.707, q ) # freestream velocity profile * flat plate profile (uniform freestream velocity) Results corresponding to the Wang s 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 nonzero pressure gradient again indicate flow separation near x/c of 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 31
13 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 crossbc 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 This shows that the analogy function for crossbc 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 crossbc 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. 32
14 /1 6/4 Nu/Sh /1 F ref 0.4 5/ x/c Figure 3.3 Analogy functions, Nu/Sh, for the GE90 profile and the corresponding flat plate cases. Re c =540,000 and P/C=0.75 (x/c=1.38 at the trailing edge). F ref corresponds to the value from Eqn. (3.1.1). Case designations are simplified, such as 3/1 for GE903/GE Results corresponding to the Chen s chord Reynolds number This section examines the cases with the GE90 profile to which Chen s chordreynolds number is applied, i.e. the Reynolds number used in the section 3.2. The resulting analogy functions are compared with those in the section 3.2 (the results shown in Fig. 3.3 could not be compared with Fig. 3.2 because of the difference in the Reynolds numbers). The results of this section are presented in Fig. 3.4, and comparing with Fig. 3.2, will reveal effects of different pressure profiles. Any difference in the distributions shown in Fig. 3.4 from those in Fig. 3.3 is hard to distinguish. In fact, the two figures look exactly the same. This is expected from the discussions in section 3.1. The effects of Reynolds numbers cancel each other when the numerator and the denominator have the same profile. 33
15 /1 2/1 6/4 Nu/Sh / x/c Figure 3.4 Analogy functions, Nu/Sh, for the GE90 profile and the corresponding flat plate cases. Re c =171,000 and P/C=0.77 (x/c=1.38 at the trailing edge). F ref corresponds to the value from Eqn. (3.1.1). Case designations are simplified such as 3/1 for GE903/GE
16 3.4 Heat/Heat and Mass/Mass Analogy Functions Origins of nonuniform analogy functions In the previous sections, all analyses were based on analogy functions between a fluid with 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 nonuniform 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 The second step is to examine the effect of thermal boundary conditions in cases with a transport property of 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 nonzero 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 35
17 6/4. Pair 2/3 is chosen to examine the effects of thermal boundary conditions since it may become useful in converting fluxbc experimental data to levelbc data 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 levelbcs 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 be concluded 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 /1 Nu/Sh / x/c Figure 3.5 CF6 Chen cases, Re c =171,000 and P/C=0.77 (x/c=1.25 at the trailing edge) 36
18 For case 1, which assumes nonzero 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: U s m = c x (3.4.1) The exponent, m, is described as follows. x du s m = (3.4.2) U dx s According to Eckert and Drake (1987), the resulting Nusselt number for a boundary condition (BC) of uniform wall temperature is: Nu 1/ 2 1/ 3 x = f (m) Re x Pr (3.4.3) We know, at this point, that the above equation is the same as: Sh 1/ 2 1/ 3 x = f (m) Re x Sc (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 assumed for 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. Sh Sh 4 1 1/ 3 f (0)Sc U s, U s,4 U s,4 = = = 0.57 (3.4.5) 1/ 3 f (1)Sc U 0.72 U U s,1 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 nonzero pressure gradients for which stagnation flow with m being 1.0 is assumed near the leading edge since the velocity increase is linear 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, longlasting acceleration, e.g. the GE90 profile, causes a continuous decrease seen until x/c of about 0.3 in Figs. 3.6 and s,1 s,1
19 1.4 Nu/Sh /1 2/ x/c Figure 3.6 GE90 Wang cases, Re c =540,000 and P/C=0.75 (x/c=1.38 at the trailing edge) Once passing the acceleration regions, line 4/1 behaves differently, depending on the pressure profiles. The GE90 profile, with longlasting acceleration, tends to have lower values in this region whereas the CF6 profile jumps from the minimum to a high value. The differences between the figures seem to be greater between different pressure profiles (Fig. 3.5 and Fig. 3.6) than between different chord Reynolds numbers (Fig. 3.6 and Fig. 3.7). The only difference between cases 4 and 1 is the pressure profile, and, in case 1, there is a point (or two) at which pressure gradient, at least locally, is zero. This can be used to check the behavior of line 4/1 in the figures. If both had no pressure gradient, meaning that they are identical cases, the line would remain at Sh/Sh of 1.0 throughout the airfoil. Though the local mass transfer process depends on the local conditions as well as the flow history, one can observe that, when local pressure gradient becomes zero, the Sh/Sh values tend to adjust toward
20 /1 Nu/Sh / x/c Figure 3.7 GE90 profile with Chen s Reynolds number, Re c =171,000 and P/C=0.77 (x/c=1.38 at the trailing edge) 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). At this 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 layer characteristics, 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 Therefore, the Sh/Sh values in Fig. 3.5 vary but remain close to closer to 1.0. The 39
21 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 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 nonzero pressure gradients, the same transport property of (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 with the 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 crossbc 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 crossbc effect as augmented by the nonzero 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
22 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 Originally, the equation is developed for a plane wall case with the Nusselt number assumed to result from uniformflux BC and the Sherwood number from a uniformlevel 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 ). A general trend then is, except near the leading edge, that the local analogy factor is greater than 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 The magnitudes of the deviations from the reference value of seem to be a function of flatness of the velocity profiles; the flatter CF6 profile deviates less than does the lessflat GE90 profile. Finally, whether this analogy function is useful in converting fluxbc heat transfer data to levelbc 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 41
23 (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 momentumthickness 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. Momentumthickness 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 displacementthickness Reynolds number and freestream turbulence (Figure 1). Almost twenty years later, Mayle (1991a and 1991b) developed, from more experimental data, an equation for transitional momentumthickness Reynolds number as a function of freestream turbulence intensity. Mayle emphasizes a good agreement between the McDonald s relation, also called AbuGhannam and Shaw relation, and the Mayle s equation, which reads: 42
24 Re m,tr 5/ 8 = 400TI (3.5.1) Both McDonald s and Mayle s 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 not reach 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 Chung s and Chen s experiments, that the boundary layers near the trailing edge of the CF6 profile are turbulent. This is consistent with Chen s 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 Re m 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. 43
25 Rem McDonald Mayle TEXSTAN x/c Figure 3.8: Comparison of critical momentum thickness Reynolds numbers for the CF6 velocity profile that is shown in Fig Conclusions The heat/mass transfer analogy function is studied for laminar boundary layers. Transport of heat and sublimation of naphthalene from heated and naphthalenecoated 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 is a conventional relationship which assumes zero pressure gradients and uniformlevel boundary conditions for both heat and mass transfer processes. It is a motivation to the present study to examine how the analogy function 44
26 deviates from this conventional value by changing the boundary conditions and pressure profiles. The other analogy factor developed is (1.36 multiplied by 0.677). This factor assumes zero pressure gradients for both and fluxbc 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 may be applied to an entire surface. Second, the TEXSTAN program is employed to numerically calculate the analogy functions for nonzero pressure gradients, namely the CF6 and GE90 profiles. The above two analytical solutions are verified, and the nonzero 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 fluxbc heat transfer data to levelbc heat transfer data for nonzero 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 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 This implies, for laminar boundary layers, that a mass transfer experiment with naphthalene sublimation is going to be more accurate than a fluxbc heat transfer 45
27 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. 46
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