Simplified analysis of heat transfer through a finned tube bundle in air cooled condenser-second assessment

2.1 Transverse fluid crossing of a finned tubes bank

The tubes on a bank are usually arranged in a row or triangle in the direction of flow, as shown in Figure 1. The outside diameter of the tube is taken as the characteristic length. The arrangement of tube bank is characterized by the transverse passage S_T, the longitudinal passage S_L and the diagonal step S_D between the centers of the tubes (see figure 1).

The diagonal step is determined from the following equation:

${{S}_{D}}=\sqrt{S_{L}^{2}+{{\left( \frac{{{S}_{T}}}{2} \right)}^{2}}}$ (1)

In equation (1) (see figure 1 for used notations):

S_T is the transverse step, in m

S_L is the longitudinal passage, in m

S_D is the diagonal step, in m

1.png

Figure 1. Tubes arrangement in a bank

As the fluid enters the bank, the flow area decreases from  ${{A}_{T}}={{S}_{T}}L$ to ${{A}_{T}}=\left( {{S}_{T}}-{{d}_{r}} \right)L$ between the tubes, and as a consequence, the flow velocity increases. In the stepped arrangement the velocity can increase further in the diagonal region if the rows of tubes are very close together. In tube banks the characteristics of the flow are dominated by the maximum velocity V_m which is held within the bank rather than by the approximate velocity V. Therefore, the Reynolds number is defined on the basis of the maximum velocity as:

${{\operatorname{Re}}_{d}}=\frac{\rho {{V}_{m}}d}{\mu }=\frac{{{V}_{m}}d}{\nu }$ (2)

The maximum velocity is determined based on the mass conservation requirement for the incompressible stationary flow. For alignment arrangement, the maximum velocity is in the minimum flow area between the tubes and the mass conservation can be expressed as (see Figure 1):

$\rho {{V}_{0}}{{A}_{1}}=\rho {{V}_{m}}{{A}_{T}}={{V}_{m}}\left( {{S}_{T}}-{{d}_{r}} \right)$ （3）

In equation (3) (see figure 1 for used notations):

• V₀ is the entry velocity of the refrigerant to the bundle, in m/s
• d_r is the outer diameter of the bare tubes (without fins), in m

Then from equation (3) it follows that the maximum velocity will be as:

${{V}_{Max}}=\frac{{{S}_{T}}}{{{S}_{T}}-d}{{V}_{0}}$  （4）

In the staggered arrangement, fluid approaching through the area A₁ (see Figure 1) passes through the area A_T and before for area 2A_D, as it is wound around the tube in the next row. If it is true that 2A_D>A_T, the maximum velocity still occurs in A_T, between the tubes and consequently, the V_m, in equation (4) can be used for staggered bundles tube.

If 2A_D<A_T [for $2\left( {{S}_{D}}-d \right)<\left( {{S}_{T}}-d \right)$ ], the maximum velocity will be in the diagonal cross-sections. In this case, maximum velocity is given by:

${{V}_{m}}=\frac{{{S}_{T}}}{2\left( {{S}_{D}}-{{d}_{r}} \right)}{{V}_{0}}$    (5)

The conservation of mass can be expressed as (see Figure 1):

$\rho {{V}_{0}}{{A}_{1}}=\rho {{V}_{m}}\left( 2{{A}_{D}} \right)$    (6)

Or what is the same.

${{V}_{0}}{{S}_{T}}=2{{V}_{m}}\left( {{S}_{D}}-{{d}_{r}} \right)$  (7)

Heat transfer peculiarities in cross flow over a tube bank are very similar to those found in a solitary tube. The variation of thermal transfer around a tube in a bench is determined by the flow pattern, which depends to a large extent on the arrangement of the tubes. Thus in line layout banks, there are two collision points, which results in two thermal transfer peaks. However in banks with staggered arrangement, the process of heat transfer is to some extent similar to that of a solitary tube.

Figure 2 provides a comparison of the quotient arising between the local heat transfer coefficient and the average heat transfer coefficient for a solitary tube and for third row of in-line and staggered arrangements.

2.png

Figure 2. Local heat transfer coefficient variation for different arrangements of finned tubes

Figure 2 shows that in in-line arrays the maximum local heat transfer is located at the point ϕ=50º, while the staggered arrangement has two maximum points, one at the point of fluid stagnation ϕ=50º and another in the flow separation. It is expected that the presence of two maximum points will cause the area under the curve to be larger in the staggered arrangements and therefore, taking into account also the low corrosion and deposition properties of the air, make preferable the staggered arrangement for ACC systems, because with this a higher average heat transfer coefficient is achieved. An inspection to ACC manufacturers' catalogs confirms that the staggered arrangement is the most used in all marketed equipment.

2.2 The j and f factors too many configurations

The j factor and fanning f factor are defined rather consistently in the literature by the following equations:

$j=\frac{\alpha {{\left( \Pr  \right)}^{{2}/{3}\;}}}{G\cdot Cp}=\frac{\alpha }{G\cdot Cp}\cdot {{\left( \frac{\mu \cdot Cp}{\lambda } \right)}^{{2}/{3}\;}}$   (8)

$\Delta p=4f\frac{L}{{{D}_{h}}}\frac{V_{m}^{2}}{2g}$    (9)

In equation (9) D_h is the equivalent diameter in diagonal cross-sections, and values of V_m is determined by the use of the following relations:

$\begin{align}  & if\quad 2\left( {{S}_{D}}-{{d}_{r}} \right)>\left( {{S}_{T}}-{{d}_{r}} \right)\quad ;\quad Vm=\frac{{{S}_{T}}}{{{S}_{T}}-{{d}_{r}}}{{V}_{0}} \\ & if\quad 2\left( {{S}_{D}}-{{d}_{r}} \right)\le \left( {{S}_{T}}-{{d}_{r}} \right)\quad ;\quad Vm=\frac{{{S}_{T}}}{2\left( {{S}_{D}}-{{d}_{r}} \right)}{{V}_{0}} \\\end{align}$(10)

The correlation given by Giedt [9] states that the thermal performance of a bare tube bank is given:

$j=\frac{0.376}{{{\operatorname{Re}}^{0.4}}}$       (11)

In-line tube bank, for a 10-row-deep in-line tube bank, the correlation for j is given by [9]:

$j=\frac{0.333}{{{\operatorname{Re}}^{0.4}}}$      (12)

Expressions (11) and (12) are valid for tube diameter from 6 to 50 mm, and Reynolds number range 100 -80 000.

For circular tube-fin arrangement, the basic geometry is shown in figure 3. Experimental heat-transfer and fluid-friction data for three circular finned tube surfaces are given in London et al. [8-10].

3.png

Figure 3. Tube-fins details

Briggs and Young Correlations [10]. For low-fin tube banks, the correlation based on experimental heat transfer data is:

$Nu=0.1507{{\operatorname{Re}}^{0.667}}{{\Pr }^{{1}/{3}\;}}{{\left( \frac{S}{{{l}_{F}}} \right)}^{0.164}}{{\left( \frac{S}{{{t}_{F}}} \right)}^{0.075}}$         (13)

where $S=\left[\frac{\left(l-N_{T} t_{F}\right)}{t_{F}}\right]$, with a standar deviation of 3.1%. This is applicable for a circular finned tube with low fin height and high fin density, Re = 1000-20 000, $N_{t} \geq 6$ and equilateral triangular pitch.

For high-fin tube banks the correlation based on experimental heat-transfer data with a standard deviation of 5.1% is:

$Nu=0.1378{{\operatorname{Re}}^{0.718}}{{\Pr }^{{1}/{3}\;}}{{\left( \frac{S}{{{l}_{F}}} \right)}^{0.296}}$   (14)

For all tube banks, the correlation based on regression analysis with a standard deviation of 5.1% is:

$\begin{align}  & Nu=0.134{{\operatorname{Re}}^{0.681}}{{\Pr }^{{1}/{3}\;}}{{\left( \frac{S}{{{l}_{F}}} \right)}^{0.2}}{{\left( \frac{S}{{{t}_{F}}} \right)}^{0.1134}} \\ & j=0.134{{\operatorname{Re}}^{-0.319}}{{\Pr }^{{1}/{3}\;}}{{\left( \frac{S}{{{l}_{F}}} \right)}^{0.2}}{{\left( \frac{S}{{{t}_{F}}} \right)}^{0.1134}} \\\end{align}$.      (15)

Rabas, Eckels and Sabatino Correlations [11] for tube banks arranged on an equilateral triangular pitch with low-finned tubes is:

$j=0.292{{\operatorname{Re}}^{-m}}{{\Pr }^{{1}/{3}\;}}{{\left( \frac{S}{{{d}_{F}}} \right)}^{1.116}}{{\left( \frac{S}{{{l}_{F}}} \right)}^{0.257}}{{\left( \frac{S}{{{t}_{F}}} \right)}^{-0.66}}\times \times {{\left( \frac{{{d}_{F}}}{{{d}_{r}}} \right)}^{0.47}}{{\left( \frac{{{d}_{F}}}{{{t}_{F}}} \right)}^{0.77}}{{\alpha }_{H}}{{\alpha }_{N}}$    (16)

where $m=0.415-0.0346 \ln \left(\frac{d_{F}}{S}\right)$

In Equation (16) α_H and α_N are a temperature-dependent fluid properties correction factor and a row correction factor respectively. The row correction factor has been covered earlier. Rabas and Taborek [12] recommend the following correlations for α_H and α_N:

${{\alpha }_{H}}={{\left( \frac{{{T}_{b}}+273}{{{T}_{W}}+273} \right)}^{0.25}}\cong {{\left( \frac{\Pr \,\left( {{T}_{b}} \right)}{\Pr \,\left( {{T}_{W}} \right)} \right)}^{0.26}}$  (17)

$\begin{align}  & for\ {{N}_{T}}\le 8\quad \to \ \ \quad {{\alpha }_{N}}=0,66{{\left( {{N}_{T}} \right)}^{0.21}} \\ & for\ {{N}_{T}}>8\quad \to \ \ \quad {{\alpha }_{N}}=1 \\\end{align}$ (18)

The fanning friction factor is given by:

\[f=3.805{{\operatorname{Re}}^{-0.23}}{{\left( \frac{S}{{{d}_{F}}} \right)}^{0.25}}{{\left( \frac{{{l}_{F}}}{S} \right)}^{0.76}}{{\left( \frac{{{d}_{r}}}{{{d}_{F}}} \right)}^{0.73}}\times {{\left( \frac{{{d}_{F}}}{{{S}_{T}}} \right)}^{0.71}}{{\left( \frac{{{S}_{L}}}{{{S}_{T}}} \right)}^{0.38}}\]   (19)

Equation (16) and (19) are applicable for $l_{F} \leq 6.35 m m, \quad 1000<R e \leq 25000 \quad, \quad 4.76<d_{r} \leq$ $31.75 m m, 246 \leq N_{F} \leq 1181$ fins/m $, \quad 15.08<S_{T} \leq$ $111 \mathrm{mm}, 10.32<S_{L} \leq 96.11 \mathrm{mm}, S_{L} \leq S_{T}, d_{F} / S<40$ and $N_{T} \geq 6$

Engineering Sciences Data Unit (ESDU) Correlations [4] is applicable for $10000<\mathrm{Re} \leq 100000$ and $N_{T} \geq 10$

$j=\frac{0.183{{\alpha }_{H}}{{\alpha }_{N}}{{\Pr }^{0.027}}{{\left( \frac{S}{{{l}_{F}}} \right)}^{0.36}}{{\left( \frac{{{S}_{T}}}{{{d}_{F}}} \right)}^{0.06}}}{{{\operatorname{Re}}^{0.3}}{{\left( \frac{{{d}_{F}}}{{{l}_{F}}} \right)}^{-0.11}}}$  (20)

The fanning friction factor is given by:

$f=\frac{4.71{{\left( \frac{{{l}_{F}}}{S} \right)}^{0.51}}{{\left( \frac{{{S}_{T}}-{{d}_{F}}}{{{S}_{L}}-{{d}_{F}}} \right)}^{0.54}}}{{{\operatorname{Re}}^{0.29}}{{\left( \frac{{{S}_{T}}}{{{d}_{F}}}-1 \right)}^{0.36}}}$  (21)

Gray and Webb Correlations [13] is applicable for $500<\mathrm{Re} \leq 2.47 \cdot 10^{5}$. To four-row array, j factor is determined by:

${{j}_{4}}=0.14{{\left( \frac{S}{{{d}_{r}}} \right)}^{0.03}}{{\operatorname{Re}}^{-0.33}}{{\left( {{{S}_{T}}}/{{{S}_{L}}}\; \right)}^{-0.5}}$  (22)

Equation (22) correlated within $\pm 10 \%$ in the 89% of availables data. Up to three rows:

$\frac{{{j}_{N}}}{{{j}_{4}}}=0.991\cdot {{\left[ 2.24{{\operatorname{Re}}^{-0.092}}{{\left( \frac{{{N}_{T}}}{4} \right)}^{-0.031}} \right]}^{0.607\cdot \left( 4-{{N}_{T}} \right)}}$ (23)

Equation (22) and (23) are valid for $1 \leq N_{T}, 1.27 \leq S_{T} / d_{r} \leq 2.55,1.4 \leq S_{L} / d_{r} \leq 2.58$ and $0.08 \leq S / d_{r} \leq 0.64$.

2.3 Fluid crossing of a finned tubes bank and his influence on local heat transfer coefficient

A fluid crossing a solitary cylinders (or a package of these) results in flow separation, which is difficult to handle analytically. Therefore, flows of this type should be studied experimentally or numerically. In fact, flow through cylinders and cylinder packs has been studied experimentally by numerous researchers and several empirical correlations have been developed for the determination of heat transfer coefficient [14-16]. The complicated pattern of the flow through a single finned cylinder greatly influences the heat transfer.

Many investigations have been carried out in order to clarify this problem. In figure 4 are given the average values of experimental results reported by [17-18]. This figure provides the variation of the local Nusselt number Nu  depending on the angle $\phi$, on the periphery of a finned cylinder bathed by a cross and perpendicular airflow.

A detailed analysis of figure 4 and the available experimental quantities allows us to establish that in the transversal bathing of a cylinder finned by a turbulent airflow, two basic regions are formed depending on the Reynolds dimensionless number. The first, in interval $4 \cdot 10^{3} \leq R e \leq 10^{5}$, and the second in interval $10^{5}<R e \leq 4 \cdot 10^{5}$.

4.png

Figure 4. Local Nusselt number variation as a function of the angle in air cross flow over finned cylinder

In the first interval, the value of  Nu starts relatively high at the stagnation point $\left(\varphi=0^{\circ}\right)$, reaching at this point its maximum value Nu_MAX. For $0^{\circ}<\varphi \approx 60^{\circ}$, Nu resents a decreasing tendency very close to a linear function with negative slope $m \approx-1.15$. In interval $60^{0}<\varphi \approx 80^{\circ}$ the downward trend undergoes a sharp change in slope taking now $-5<m \approx-6$. In both cases this decrease is caused by the thickening of the laminar boundary layer.

In the point $\varphi \approx 80^{\circ}$ separation of the laminar flow occurs and therefore it is logical that the number of  Nu reach its minimum value, causing this to also coincide in $\varphi \approx 80^{\circ}$ the presence of a positive inflection point. At this point it is true that $N u \approx\left(N u_{M A X}\right)^{0.84}$. For $80^{0}<\varphi \leq 180^{0}$, Nu present a clear tendency to grow as a result of intense mixing in the region of the separate stream (the wake). This growth follows approximately a positive slope line $m \approx 1,15$ until $\varphi \approx 180^{0}$, where it is true that  $N u \approx 0.9 N u_{M A X}$.

However the second region $10^{5}<\mathrm{Re} \leq 4 \cdot 10^{5}$ its behavior differs from the first. In the first interval, value of Nu  starts relatively high at the point of stagnation $\left(\varphi=0^{\circ}\right)$, reaching at this point a high value. $\left(N u_{0}\right)$ but not maximum as happened in the first region. For $0^{0}<\varphi \approx 90^{\circ} \quad N u$ number present a decreasing tendency very close to a linear function with negative slope $m \approx-1.8$. The point $\varphi \approx 90^{\circ}$ is where the laminar flow separation occurs, and therefore it is logical that the Nusselt number reaches its minimum value and this point is at once an inflection point. At this point it is true that $N u \approx\left(N u_{0}\right)^{0.92}$.

For $90^{0}<\varphi \leq 120^{\circ}$. Nu number present a clear tendency to growth being asymptotically guided by a positive slope whose value varies in the intervals.  12$\approx m \approx 14$. This increase in the local number is due to the transition from laminar to turbulent.  In $\varphi=120^{\circ}$ is where a maximum $\left(N u_{M A X}\right)$  and a second point of inflection are located, what is fulfilled $N u_{M A X}=\left(N u_{0}\right)^{1,05}$.

For $120^{\circ}<\varphi \leq 140^{0}$, Nu number present a pronounced decreasing behavior, not ruled by a single slope, until reaching $\varphi \approx 140^{0}$ , at which point the second minimum of the local Nu number is presented, being fulfilled that $N u=\left(N u_{0}\right)^{0.98}$.

This point is precisely where the separation of the flow occurs in the turbulent flow. For $140^{0}<\varphi \leq 180^{0}$, a new growth of the local number as a result of the intense mixing in the turbulent region of the wake, this growth presents an asymptotic behavior to a linear function with a positive slope that oscillates in the interval 4.1$\approx m \approx 4.8$, is also fulfilled in that, $\varphi=180^{\circ}$ that $N u=(16 / 15) \cdot\left(N u_{0}\right)$.

The discussions given in the previous paragraphs about the local heat transfer coefficients provide insight into the behavior of this at any point in the finned cylinder and for any operant flow regime, however, have little value in heat transfer calculations, since in these, the average heat transfer coefficient over the entire surface is required. The main author provides a detailed discussion of this problem in the reference [4].

The row variation in the staggered arrangements does not produce significant changes in local heat transfer coefficient. This criterion is confirmed by the authors [8-11], who examined experimentally arrangements with different dimensions and for a fixed value of $\operatorname{Re}=7\cdot {{10}^{4}}$. The results obtained confirm that the local heat transfer coefficient follows approximately a common pattern for all the arrangements examined. In figure 5 are plotted the averages available values.

5.png

Figure 5. Local heat transfer coefficient variation for different dimensions in ACC with staggered arrangement

The row variation in aligned arrangements does not produce significant changes in local heat transfer coefficient. This criterion is confirmed by the authors [12-14], who examined experimentally arrangements with different dimensions and for a fixed value of $\mathrm{Re}=3 \cdot 10^{4}$. The results obtained confirm that the local heat transfer coefficient follows approximately a common pattern for all the arrangements examined. In figure 6 are plotted the averages available values.

In any arrangement of tube bank, heat transfer coefficient increases from the first row to the third, and takes a constant value from the latter. In figure 7 and 8 are plotted for staggered and in-line arrangements, respectively, the average experimental values reported by the authors [19-22] for the first three rows of a tube package of an ACC, for $\mathrm{Re}=1.4 \cdot 10^{4}$ and one dimension $(2 x 2) d_{r}$.

6.png

Figure 6. Local heat transfer coefficient variation for different dimensions in ACC with aligned arrangement

7.png

Figure 7. Variation of a local heat transfer coefficient in ACC with staggered arrangement

8.png

Figure 8. Variation of a local heat transfer coefficient in ACC with aligned arrangement

2.4 Developments of new model and his validation through experimental available data

An analysis of 783 sets of available experimental data reported by a total of 10 authors linked to centers of high prestige [12-15], allowed to develop a dimensionally non-homogeneous expression, which was generated by an integral analysis of residues by cross-methods of breaks at intervals, (Breshnetzov’s method) describing with a mean error of the order $\pm 6.9 \%$ in the 85.3% of available experimental samples. This equation is given by the following expression:

$\alpha =\frac{\left[ 1+Log{{\left( {{{T}_{TBS}}}/{{{V}_{W}}}\; \right)}^{0.015}} \right]\cdot {{\left( {{V}_{m}} \right)}^{0.06}}{{\left( {{t}_{F}}\cdot {{l}_{F}} \right)}^{0.01}}}{\left( 0.15\cdot {{\left( {{S}_{T}}-{{d}_{r}} \right)}^{0.4}} \right)\cdot \left\{ 0.17Ln\left( F \right) \right\}}$         (24)

In Equation (24):

${{T}_{TBS}}$is the dry bulb temperature, in ⁰C

${{t}_{F}}$is the fins thickness, in mm

${{l}_{F}}$is the fins height, in mm

$F$is the number of fins per linear meter of the finned tube.

${{V}_{W}}$is the incident wind velocity over ACC installation, in km/h

9.png

This work was supported by Doctoral Research Program of Universidad Central “Marta Abreu” de las Villas, Cuba.