Yanán Camaraza-Medina
© 2021 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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In the present work, its show a summary of functional relationships developed for the application of dry condensation systems to Biomass Power Plants that present difficulties with access to water for condensation. The bibliographic review reveals the limitations of the analyzed works, in terms of the development of mathematical models and empirical correlations that allow evaluating the simultaneous effects of the surrounding meteorological variables on the average coefficient of heat transfer and the effect on the environment of the use of dry condensation. The analytical study is based on the weak solutions and their correlation with experimental quantities available in research already established in the area of action, a procedure is developed for the calculation of the average coefficients of heat transfer that includes the influence of local climatologically variables, the effect of the spatial distribution of the tubes package on the refrigerant and the confined confinement in inclined components, which increase the reliability of the thermo-hydraulic analysis and suppresses the need for the use of excess areas required by current methods. The proposed models and correlations allow the preparation of a procedure, by means of which all the possible operative variants are evaluated.
flow condensation, heat transfer coefficient, mathematical deduction
At the end of 2019, about 32 percent of water withdrawals for industrial purposes were used in wet condensers. In areas with limited access to water, the most widespread solution is the use of dry condensers, which achieve savings rates of water consumption close to 95 percent [1, 2].
The Cuban state has planned an investment that will allow the installation of 1650 MW of power, (solar, wind and biomass), which represents 24 percent of the national energy matrix. Of these, 875 MW correspond to 25 Biomass Power Plant (BPP) projects, which require high volumes of water for their condensation system, however, these requirements violate Law 124/2017 on the use of terrestrial waters [3].
Given the downturn of the water deficit and the potentiality of the use of biomass as an energy source, the use of ACC can be an effective solution, however in its evaluation there are inconveniences [4].
ACC uses air as refrigerant, which is why, the installation evidences a low global heat transfer coefficient, besides than the variation of the environmental temperature and the velocity of the wind affect this heat transfer coefficient perceptibly. The vapor condenses in the inside of inclined tubes, however, at the present time, does not count on a procedure that it enables getting from reliable way, the values of the heat transfer coefficients for this type of configuration [5-7].
The methods currently used in the evaluation of the global coefficient K in an ACC are imprecise, because they do not include the effect of local climatic variables, the influence of the geometry of the tube package on the refrigerant and the condensation confined in inclined components, so the use of excess area is required in the operation of these facilities [8-13].
For this reason, the main objective of the present work is to define a procedure for the calculation of the average heat transfer coefficients that includes the influence of local climatic variables, the effect of the spatial distribution of the tube package on the refrigerant and the condensation confined in inclined components, which increases the reliability of thermo-hydraulic analysis and eliminates the need to use excess areas required by current methods.
2.1 External heat transfer coefficient αL
An analysis of 783 experimental data sets reported in eight specialized papers on the subject, allowed to develop a dimensionally non-homogeneous expression, which was generated by an integral analysis of residues by cross-jump methods at intervals (Breshnetzov method), which correlates with an average error of ±6.9% in 84.8% percent of the available experimental samples. This equation is given by the following expression [14]:
${{\alpha }_{L}}=\frac{\left( 1+{{\log }_{10}}{{\left( {{{T}_{TBS}}}/{{{V}_{V}}}\; \right)}^{0.015\quad}} \right)\cdot {{\left( {{V}_{m}} \right)}^{0.06\quad}}{{\left( {{e}_{a}}\cdot {{h}_{a}} \right)}^{0.01}}}{{{\left( 0.15\cdot {{\left( {{S}_{T}}-{{d}_{e}} \right)}^{0.4}} \right)\quad}^{{}}}\cdot \ln {{\left( F \right)}^{0.28}}}$ (1)
where: ST is the transverse step, in m; VV is the velocity of the incident wind on the ACC installation, in m/s; Vm is the velocity in the narrowest section of the tube package, in m/s; ea is the fins thickness, in mm; de is the outer diameter of the bare tubes (without fins), in m; ha is the height of the fins, in mm; F is the number of fins per linear meter of finned tube length; TTBS is dry bulb temperature, in ℃. Value of Vm is determined by using the following criteria [15]:
if $2\left(S_{D}-d_{E}\right)>\left(S_{T}-D\right) \rightarrow V m=\frac{S_{T}}{S_{T}-d_{E}} V_{0}$
if $2\left(S_{D}-d_{E}\right) \leq\left(S_{T}-D\right) \rightarrow V m=\frac{S_{T}}{2\left(S_{D}-d_{E}\right)} V_{0}$ (2)
where: V0 is the rate of entry of the cooling agent into the tube package, in m/s; ST is the transverse step, in m; SL is the longitudinal step, in m; D is the outer diameter of the tubes + fins, which make up the package, in m; SD is the diagonal step, in m,
Table 1 gives the validity parameters of Eq. (1), while all the experimental data used in its development and validation are given by Camaraza-Medina et al. [14].
Table 1. Validity range of the Eq. (1)
|
Parameter |
Range |
Parameter |
Range |
|
Tube inclination ACC |
45-60o |
ha (mm) |
2.5-7.9 |
|
de (mm) |
19-50 |
V0 (m/s) |
0.1-100 |
|
ST/SL |
0.4-2 |
TTBS (℃) |
15-43 |
|
Vv (m/s) |
0-45 |
ea (mm) |
1.3-3.5 |
The deviation is determined by the following expression:
$E=100 \cdot\left|\frac{\alpha_{L}-\alpha_{\exp\quad}}{\alpha_{L}}\right|$ (3)
In Eq. (3) αexp is the average experimental heat transfer coefficient, while the mean absolute error (MAE) is determined as:
$M A E=\frac{1}{N} \sum_{N}\left|\frac{\alpha_{L}-\alpha_{\exp\quad}}{\alpha_{L}}\right|$ (4)
In Eq. (4), N is the amount of experimental data available. In Tables 2 and 3 the validity range of the Eq. (1) is fragmented in six zones, the average and maximum deviation obtained in its correlation with the available experimental data being given in each case.
In Figures 1 and 2, the values of MAE and Emax obtained in the correlation developed between available experimental data and selected models are given in graphical form [15].
The study shows that in the first and second range, the fundamentals results used in the comparison concentrate on two fundamental elements, described early (MAE and Emax). In these it is confirmed that the model given by Camaraza et al. (Eq. (3)) [14] have the best MAE adjustment values, showing an average error of 7.6% and 7.2% in the correlation with available experimental data for zones 1 and 6, respectively.
In the specialized literature it is established that Zukauskas's model correlates with an average error of 25%; however, the results obtained in the present study show an average error of 29.6% and 32.3% in the correlation with available experimental data for zones 1 and 6, respectively, proving that the values obtained in the present study are slightly higher to the values commonly attributed in the literature.
The most unfavorable indicators are obtained using the models of Griminson and Engineering Sciences Data Unit (ESDU), which provide MAE values of 33.2% and 29.1% respectively for zone 1, while the MAE values of 28.2% and 27.4% respectively for zone 6. The models of Gray-Webb, Briggs-Young and Rabas-Eckels allows to obtain convenient results, with MAE values of 15.3% to 27.1%, which agrees well with those results given by Camaraza [15].
The specialized literature does not count with reports that suggest the possible maximum executed error with the use of a determined model. In the present study the value of $E_{\max }$ generated with the use of every model for the six studied zones was obtained, by means of the correlation made between the experimental available data and the models selected.
The model developed by Camaraza et al. [14] shows the best Emax index, with 11.5% and 12.9% respectively in Zone 1, and 6. On the contrary, the most unfavorable indicators are obtained with the correlations of Griminson and Zukauskas, which provide $E_{\max }$ values of 41.4% and 38.5% respectively for Zone 1, increasing to 48.2% and 43.1% in Zone 6. The other models analized in this study provide fairly acceptable adjustments of correlation. The early elements allows affirming that the proposed model (Eq. (3)) constitutes a scientific novelty.
Table 2. Correlation of the first range of values with Eq. (1)
|
Valid for 0.4≤ST⁄SL ≤1; 19≤de≤50 mm and 1.3≤ea≤3.5 mm |
|||||
|
Zone |
Validity range |
Deviation |
Zone |
Validity range |
Deviation |
|
1 |
2.5≤ha≤4.7 0≤Vv≤5.4 0≤V0≤10 15≤TTBS≤18 115≤F≤194 |
MAE<7.6% for 89.9% of the data |
2 |
2.5≤ha≤5.1 0≤Vv≤10.1 0≤V0≤28.2 15≤TTBS≤24 115≤F≤254 |
MAE<6.8% for 88.1% of the data |
|
Emax<11.5% N=79 |
Emax<9.3% N=135 |
||||
|
3 |
2.5≤ha≤5.1 0≤Vv≤16.2 0≤V0≤41.4 15≤TTBS≤30 115≤F≤315 |
MAE<5.9% for 87.3% of the data |
4 |
2.5≤ha≤6.4 0≤Vv≤24.1 0≤V0≤65.6 15≤TTBS≤35 115≤F≤354 |
MAE<6.1% for 88.4% of the data |
|
Emax<11.2% N=192 |
Emax<12.1% N=232 |
||||
|
5 |
2.5≤ha≤7.9 0≤Vv≤38.2 0≤V0≤88.4 15≤TTBS≤40 115≤F≤354 |
MAE<6.2% for 89.7% of the data |
6 |
2.5≤ha≤7.9 0≤Vv≤45 0≤V0≤100 15≤TTBS≤43 115≤F≤394 |
MAE<7.5% for 90.5% of the data |
|
Emax<12.6% N=282 |
Emax<12.9% N=368 |
||||
Figure 1. MAE and Emax values obtained in the correlation data with other models for (0.4≤ST⁄SL≤1)
Table 3. Correlation of the second range of values with Eq. (1)
|
Valid for 1<ST⁄SL ≤2; 19≤de≤50 mm and 1.3≤ea≤3.5 mm |
|||||
|
Zone |
Validity range |
Deviation |
Zone |
Validity range |
Deviation |
|
1 |
2.5≤ha≤4.7 0≤Vv≤5.4 0≤V0≤10 15≤TTBS≤18 115≤F≤194 |
MAE<6.5% for 89.4% of the data |
2 |
2.5≤ha≤5.1 0≤Vv≤10.1 0≤V0≤28.2 15≤TTBS≤24 115≤F≤254 |
MAE<7.1% for 88.3% of the data |
|
Emax<11.8% N=82 |
Emax<10.9% N=146 |
||||
|
3 |
2.5≤ha≤5.1 0≤Vv≤16.2 0≤V0≤41.4 15≤TTBS≤30 115≤F≤315 |
MAE<7.3% for 87.6% of the data |
4 |
2.5≤ha≤6.4 0≤Vv≤24.1 0≤V0≤65.6 15≤TTBS≤35 115≤F≤354 |
MAE<6.0% for 90.1% of the data |
|
Emax<9.5% N=207 |
Emax<10.9% N=302 |
||||
|
5 |
2.5≤ha≤7.9 0≤Vv≤38.2 0≤V0≤88.4 15≤TTBS≤40 115≤F≤354 |
MAE<6.2% for 89.9% of the data |
6 |
2.5≤ha≤7.9 0≤Vv≤45 0≤V0≤100 15≤TTBS≤43 115≤F≤394 |
MAE<6.4%for 90.6% of the data |
|
Emax<11.3% N=356 |
Emax<11.8% N=15 |
||||
Figure 2. MAE and Emax values obtained in the correlation data with other models for $\left(1<S_{T} / S_{L} \leq 2\right)$
3.1 Heat transfer coefficient αT
The combination of the differential equations of the velocity profile and temperature distribution, and their subsequent homogenization by means of gamma functions, allowed to obtain a theoretical solution for the determination of the αT coefficient inside vertical tubes. The developed expression is given by Medina et al. [16]:
$N u_{\text {Vert }}=0.943 \cdot \sqrt[4]{d_{i}^{3} \frac{\sin \varphi \cdot g \cdot\left(\rho_{L}-\rho_{V}\right) \cdot r_{L V}}{v_{L} \cdot \lambda_{L} \cdot\left(T_{\text {sat }}-T_{P}\right)}}$ (5)
where: $d_{i}$ is the equivalent inner tube diameter, in $\mathrm{m} ; g$ is the gravitational acceleration, in $\mathrm{m} / \mathrm{s}^{2} ; \rho_{L}$ is the liquid density, in $\mathrm{kg} / \mathrm{m}^{3} ; \rho_{V}$ is the steam density, in $\mathrm{kg} / \mathrm{m}^{3} ; r_{L V}$ is the latent heat of vaporization, in $\mathrm{kJ} / \mathrm{kg} ; v_{L}$ is the liquid kinematic viscosity, in $\mathrm{m}^{2} / \mathrm{s} ; \lambda_{L}$ is the fluid thermal conductivity, in $\mathrm{W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right) ; T_{\text {sat }}$ is the steam saturation temperature, in ${ }^{\circ} \mathrm{C}$ and $T_{P}$ is the mean wall temperature, in ${ }^{\circ} \mathrm{C}$.
Subsequently, [17], the simultaneous solution of the differential equations of energy, momentum, continuity and conductivity was demonstrated, using the function of Tijonov (infinite line) as a weak substitute variable and linear discretization of a finite element one-dimensional three nodes. The definitive solution is reduced to the use of two dimensionless groups, whose combination allows to determine the αT coefficient for any orientation of the tubes. This element constitutes a scientific novelty, as no background of a similar method is found in the literature.
The analytical solution developed uses the dimensionless velocity criteria and the Crosser number. Through these two criteria, the intervals of applicability of the proposed methodology are established based on the orientation of the ducts, which was subsequently adjusted and correlated with 1192 experimental data reported by 20 researchers. The proposed model yields 11.8 percent of average deviation for horizontal tubes and 13.0 percent for inclined and vertical tubes. The formulation obtained is given and discussed [17]:
Adimensional velocity $J_{g}=\frac{x G}{\sqrt{g d_{i} \rho_{V}\left(\rho_{L}-\rho_{V}\right)}}$ (6)
Crosser number $Z=\left(\frac{1-x}{x}\right)^{0.8} P r_{L}^{0.4}$ (7)
For vertical and inclined tubes
Interval $1 J_{g} \geq \frac{1}{2.37 Z+0.728}$ (8)
Interval $20.927 e^{\left(-0.0868 Z^{-1.165\quad}\right)}<J_{g}<\frac{1}{2.37 Z+0.728}$ (9)
Interval $3 J_{g} \leq 0.927 e^{\left(-0.0868 Z^{-1.165\quad}\right)}$ (10)
For horizontal tubes
Interval $1 J_{g} \leq 0.979(Z+0.262)^{-0.618}$ (11)
Interval $1 J_{q}>0.979(Z+0.262)^{-0.618}$ (12)
In Eq. (6) to (12), $P r_{L}$ is the Prandtl number for single-phase; x is the thermodynamic vapor quality; G is the mass flux, in $\mathrm{kg} /\left(\mathrm{m}^{2} \mathrm{~s}\right)$.
The proposed expression for the determination of the heat transfer coefficient by condensation inside horizontal tubes is given by Camaraza-Medina et al. [18]:
$N u_{T}=N u_{L} \cdot\left\{4.9 x^{0.9}\left[(1-x)^{2}+\frac{(1-x)^{0.1}}{P r^{0.37}}\right]\right\}^{0.8}$ (13)
In Eq. (13) $\operatorname{Pr}$ is the reduced pressure. The mathematical deduction of the Eq. (13) and the elements associated are provided by Camaraza-Medina et al. [17].
For the determination of NuL, in the literature the use of Dittus-Boelter expression is a generalized criterion, however, recently the author showed that the correlation index could be improved by using a model derived from the Prandtl analogy, which is described as [19]:
$N u_{L}=\frac{\left(R e_{L}-10^{M}\right) \cdot P r_{L}}{A \cdot I^{2}-J \cdot I \cdot\left(1-P r_{L}^{2 / 3}\right)} \cdot\left(1+\left(\frac{d_{i}}{l}\right)^{2 / 3}\right) \cdot\left(\frac{\mu_{F}}{\mu_{F}}\right)^{0.14}$ (14)
where: $R e_{L}$ is the liquid Reynolds number, $\mu_{F}$ is the Fluid dynamic viscosity at mean fluid temperature, in $a \cdot s ; \mu_{P}$ is the fluid dynamic viscosity at wall temperature, in $P a \cdot s ; l$ is the length of the tube, in $\mathrm{m}$. The mathematical deduction of the Eq. (14) is given by Camaraza-Medina et al. [20], while the constants used are summarized in Table 4.
Table 4. Values of the constants used in Eq. (14)
|
|
$2.3 \cdot 10^{3}<\operatorname{Re}<1 \cdot 10^{4}$ |
$1 \cdot 10^{4} \leq R e\leq 1 \cdot 10^{4}$ |
|
A |
75.4 |
91.4 |
|
I |
$\log \left(R e_{L}^{0.56} / 3.196\right)$ |
|
|
J |
104 |
116.7 |
|
M |
$-0.027[\log R e]^{2}+0.2 \log R e+2.63$ |
0 |
Additional tests carried out allowed us to conclude that the combination of Eqns. (5) and (13) depending on the work zones, offers a better adjustment to the experimental values available for vertical and inclined tubes. This combination was developed is given by Camaraza-Medina et al. [18]:
Interval $1 \mathrm{Nu}=\mathrm{Nu}_{\text {Vert }}$ (15)
Interval $2 \mathrm{Nu}=\sqrt{\left(N u_{T}\right)^{2}+\left(N u_{V e r t}\right)^{2}}$ (16)
Interval $N u=N u_{V e r t}$ (17)
In Eq. (16), if $\operatorname{Re}_{V}<3.5 \cdot 10^{4}$, then $N u_{V e r t}=0$. Table 5 provides a detailed summary of parameters range that shows a satisfactory fit with Eqns. (15) to (17).
Table 5. Summary of the validity range for Eq. (15) to (17)
|
Parameter |
Range |
|
Fluids |
Water, R-22, R-32, R-113, R-123, R-125, R-134a, R-142b, R-404a, R-410a, R-502, R-507, isobutene, propylene, propane, benzene, ethanol, methanol, toluene and dowtherm 209. |
|
di (mm) |
2 - 50 |
|
$G\left(\mathrm{~kg} / \mathrm{m}^{2} \mathrm{~s}\right)$ |
4 - 850 |
|
$P r_{L}$ |
1 - 20 |
|
Z |
0.005 - 20 |
|
$R e_{L}$ |
60 - 84830 |
|
$R e_{V}$ |
8210 - 523980 |
|
$x$ |
0.01 - 1 |
|
$p_{r}$ |
0.0008 - 0.91 |
|
Tube orientation |
Horizontal, vertical and inclined |
|
$J_{g}$ |
0.6 - 20 |
In Tables 6 and 7, the validity range of Eqns. (15) to (17) is fragmented into four intervals, while in Figures 3 and 4 the values of MAE and Emax obtained in the correlation developed between available experimental data and selected models are given in graphical form [15].
For vertical and inclined tubes only the Shah model is used for comparison, since the remaining ones are only valid for horizontal tubes. An analysis of Figures 3 and 4 shows that the Chato's formulation generates a low adjustment of average values, which are between 28 and 43 percent, with a maximum error rate close to 60 percent in the four zones.
For horizontal tubes, the Chato’s Equation provides an MAE values in the range of 17 to 24 percent, with the best fit in zone 3, in which the values of pr and di agree very well with the precepts under which the Chato model was developed.
In Tables 8 and 9, the validity range of the Eq. (14) is divided into six intervals. In Figures 5 and 6 the values of MAE and Emax obtained in the correlation developed between available experimental data and selected models are given in graphical form [15].
For transition zone $2.4 \times 10^{3} \leq R e<10^{4}$ (see Figure 5), the most unfavorable indicators are obtained using the models of Dittus-Boelter and Mijeev, which provide MAE values respectively of 38.4% and 36.1% for 64.5% and 68.1% of the experimental data, respectively. Hausen's model allows to obtain convenient results, with MAE values of 19.3% in 75.1% of the examined data, which agrees well with those results given by Su An and Kim [21].
Figure 3. MAE and Emax values obtained in the correlation data with other models for vertical and inclined tubes
Table 6. Correlation of experimental values with respect to Eqns. (15) to (17)
|
Vertical and inclined tubes |
|||||
|
Zone |
Validity range |
Deviation |
Zone |
Validity range |
Deviation |
|
1 |
ReL≤9.8∙103 Rev≤1.7∙105 2≤di≤8.0 3≤G≤264 0.01≤x≤0.99 pr≤0.1 |
MAE<10.6% for 89.3% of the data |
2 |
ReL≤59∙103 Rev≤3.33∙105 2≤di≤19.3 3≤G≤468 0.01≤x≤0.99 pr≤0.25 |
MAE<12.2%for 85.4% of the data |
|
Emax<16.6% N=148 |
Emax<21.1% N=479 |
||||
|
3 |
ReL≤59∙103 Rev≤3.33∙105 2≤di≤30 3≤G≤538 0.01≤x≤0.99 pr≤0.44 |
MAE<12.4%for 82.9% of the data |
4 |
ReL≤59∙103 Rev≤3.33∙105 2≤di≤47.5 3≤G≤598 0.01≤x≤0.99 pr≤0.66 |
MAE<13% for 84.1% of the data |
|
Emax<22.4% N=542 |
Emax<23.4% N=584 |
||||
Figure 4. MAE and Emax values obtained in the correlation data with other models for horizontal tubes
Table 7. Correlation of experimental values with respect to Eqns. (15) to (17)
|
Horizontal tubes |
|||||
|
Zone |
Validity range |
Deviation |
Zone |
Validity range |
Deviation |
|
1 |
$R e_{L} \leq 6.4 \cdot 10^{4}$ $R e_{v} \leq 2.1 \cdot 10^{5}$ $2 \leq \mathrm{d}_{i} \leq 8.0$ $38 \leq \mathrm{G} \leq 300$ $0.01 \leq x \leq 0.99$ $p_{r} \leq 0.1$ |
$M A E<8.9 \%$ for 83.3% of the data |
2 |
$\operatorname{Re}_{L} \leq 7.0 \cdot 10^{4}$ $\operatorname{Re}_{v} \leq 4.8 \cdot 10^{5}$ $2 \leq \mathrm{d}_{\mathrm{i}} \leq 8.0$ $38 \leq \mathrm{G} \leq 600$ $0.01 \leq x \leq 0.99$ $p_{r} \leq 0.3$ |
$M A E<$ $10 \%$ for $31.2 \%$ of the data |
|
$E_{\max }<21.3 \%$ $N=265$ |
$E_{\max }<22.2 \%$ $N=436$ |
||||
|
3 |
$R e_{L} \leq 8.5 \cdot 10^{4}$ $R e_{v} \leq 5.9 \cdot 10^{5}$ $2 \leq \mathrm{d}_{\mathrm{i}} \leq 18.9$ $38 \leq \mathrm{G} \leq 750$ $0.01 \leq x \leq 0.99$ $p_{r} \leq 0.5$ |
$M A E<11.5 \%$ for $81.9 \%$ of the data |
4 |
$R e_{L} \leq 8.5 \cdot 10^{4}$ $R e_{v} \leq 6 \cdot 10^{5}$ $2 \leq \mathrm{d}_{\mathrm{i}} \leq 49$ $38 \leq \mathrm{G} \leq 850$ $0.01 \leq x \leq 0.99$ $p_{r} \leq 0.91$ |
$M A E<11.8 \% \text { for }$ $84.3 \%$ of the data |
|
$E_{\max }<22.7 \%$ $N=538$ |
$E_{\max }<23.6 \%$ $N=608$ |
||||
Figure 5. MAE and Emax values obtained in the correlation data with other models for $2.4 \cdot 10^{3} \leq R e<10^{4}$
Table 8. Correlation of the first range of values with Eq. (14)
|
Valid to $2.4 \cdot 10^{3} \leq R e_{L}<10^{4}$ (transient flow) |
|||||
|
Zone |
Validity range |
Deviation |
Zone |
Validity range |
Deviation |
|
1 |
$\operatorname{Pr}_{\mathrm{L}} \leq 10^{2}$ $\mu_{F} / \mu_{P} \leq 12.4$ |
$E M A<6.1 \%$ for $91.3 \%$ of the data |
2 |
$\operatorname{Pr}_{\mathrm{L}} \leq 2 \cdot 10^{2}$ $\mu_{F} / \mu_{P} \leq 18.4$ |
$E M A<7 \%$ for $90.4 \%$ of the data |
|
$E_{\max }<11.2 \%$ $N=209$ |
$E_{\max }<13.1 \%$ $N=387$ |
||||
|
3 |
$\mathrm{Pr}_{\mathrm{L}} \leq 2 \cdot 10^{3}$ $\mu_{F} / \mu_{P} \leq 22.2$ |
$E M A<8.7 \%$ for $89.1 \%$ of the data |
4 |
$\mathrm{Pr}_{\mathrm{L}} \leq 8.1 \cdot 10^{3}$ $\mu_{F} / \mu_{P} \leq 34.2$ |
$E M A<10 \%$ for $88.2 \%$ of the data |
|
$E_{\max }<14.7 \%$ $N=506$ |
$E_{\max }<16.8 \%$ $N=617$ |
||||
|
5 |
$\operatorname{Pr}_{\mathrm{L}} \leq 1.2 \cdot 10^{4}$ $\mu_{F} / \mu_{P} \leq 62.2$ |
$E M A<10.7 \%$ for $86.4 \%$ of the data |
6 |
$\operatorname{Pr}_{\mathrm{L}} \leq 4.7 \cdot 10^{4}$ $\mu_{F} / \mu_{P} \leq 177$ |
$E M A<13.6 \%$ for $80.4 \%$ of the data |
|
$E_{\max }<18.9 \%$ $N=789$ |
$E_{\max }<21.8 \%$ $N=1003$ |
||||
Figure 6. MAE and Emax values obtained in the correlation data with other models for $10^{4}<R e \leq 8.2 \cdot 10^{6}$
Table 9. Correlation of the second range of values with Eq. (14)
|
Valid to $10^{4}<R e_{L} \leq 8.2 \cdot 10^{6}$ (Turbulent flow) |
|||||
|
Zone |
Validity range |
Deviation |
Zone |
Validity range |
Deviation |
|
1 |
$\operatorname{Pr}_{\mathrm{L}} \leq 10^{2}$ $\mu_{F} / \mu_{P} \leq 12.4$ |
$E M A<13.1 \%$ for $89.4 \%$ of the data |
2 |
$\mathrm{Pr}_{\mathrm{L}} \leq 2 \cdot 10^{2}$ $\mu_{F} / \mu_{P} \leq 18.4$ |
$E M A<$ $14.8 \%$ for $88.3 \%$ of the data |
|
$E_{\max }<12.8 \%$ $N=419$ |
$E_{\max }<14.1 \%$ $N=795$ |
||||
|
3 |
$\mathrm{Pr}_{\mathrm{L}} \leq 2 \cdot 10^{3}$ $\mu_{F} / \mu_{P} \leq 22.2$ |
$E M A<8.3 \%$ for $86.1 \%$ of the data |
4 |
$\operatorname{Pr}_{\mathrm{L}} \leq 8.1 \cdot 10^{3}$ $\mu_{F} / \mu_{P} \leq 34.2$ |
$E M A<$ $10.2 \%$ for $85.3 \%$ of the data |
|
$E_{\max }<16.9 \%$ $N=1127$ |
$E_{\max }<18.7 \%$ $N=1414$ |
||||
|
5 |
$\operatorname{Pr}_{\mathrm{L}} \leq 1.2 \cdot 10^{4}$ $\mu_{F} / \mu_{P} \leq 62.2$ |
$E M A<11.3 \%$ for $83.1 \%$ of the data |
6 |
$\operatorname{Pr}_{\mathrm{L}} \leq 4.7 \cdot 10^{4}$ $\mu_{F} / \mu_{P} \leq 177$ |
$E M A<14 \%$ for $80.8 \%$ of the data |
|
$E_{\max }<20.9 \%$ $N=1715$ |
$E_{\max }<24.1 \%$ $N=2093$ |
||||
The correlations of Petukhov and Sieder-Tate are not valid in this interval; however, they provide an adequate result, with MAE values of 24.6 and 29.4% for 72.4 and 70.5% of the experimental data, respectively; this indicates that they can be used for rapid estimations of the heat transfer coefficients in the transition zone, which confirms the recommendations given by Su An and Kim [21].
For turbulent zone $10^{4} \leq R e<8.2 \times 10^{6}$ (see Figure 6), the most unfavorable indicators are obtained with the Dittus-Boelter and Sieder-Tate models. The first provides MAE values of 21.6% and 39.3% for 74.2% and 66.7% of the data, for Zones 1 and 6, respectively. The second model shows MAE values of 14.3% and 36.5% for 80.9% and 72.4% of the experimental data, for Zones 1 and 6, respectively.
The models by Mijeev and Hausen allow obtaining convenient correlation results. Hausen’s model provides MAE values of 9.8% and 20.4% for 85.2% and 74.2% of the experimental data, for Zones 1 and 6, respectively. The model by Mijeev shows MAE values of 15.5% and 32.8% for 77.6% and 70.2% of the experimental data, for Zones 1 and 6, respectively.
The correlation and adjustment of available experimental data facilitated the development of new models that allow reducing the uncertainty in the determination of the internal and external film coefficients, being established a comparison between the proposal and other existing relationships in the literature, which results in a lower dispersion margin for the models described in this work.
The average correlation error in the determination of the αL and αT coefficients is reduced for external and internal flow up to 6.9 and 13 percent respectively. Regarding the elements of study presented, there is no evidence of similar expressions in the available and known literature, which is why they are considered a scientific novelty.
The author is very grateful for the help provided by Professor S. Thomson, from the Department of Mathematics, Massachusetts Institute of Technology, USA and Dr. Moustafa Zeki Yilmazouglu, from the Department of Mechanical Engineering, Gazi University, Turkey.
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