Dimensioning Heat Exchangers for Existing Dry Cooling Towers

 

 

J. D. Buys^I; D. G. Kroger^II

^IUniversity of Stellenbosch, Senior Lecturer, Department of Mathematics Stellenbosch, South Africa
^IIProfessor, Department of Mechanical Engineering, University of Stellenbosch, South Africa

 

 

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ABSTRACT

A method is proposed whereby the optimum dimensions of finned tube bundles for installation in an existing cooling tower are determined, subject to certain practical constraints and for a given cost structure. The influence of variations in the independent parameters is quantified.

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Introduction

Although dry air-cooled condensers have been used in relatively small power plants for many years, an increasing number of larger units have been brought into operation more recently. At present plants having outputs of almost 4 000 M We are under construction.

In this study the possibility of replacing the finned tube bundles of air-cooled systems with optimally dimensioned units is investigated. Replacement may be required where the original heat exchangers are damaged or have become ineffective due to corrosion, erosion or fouling. In addition to the fact that new types of finned tubes may be developed, the relative costs of materials, labour and energy change continually with time and location, resulting in correspondingly different optimised designs.

For purposes of illustration the proposed design procedure is applied to a hyperbolic natural draught cooling tower.

 

Analysis

Consider the example of a large natural draught dry cooling tower as shown schematically in figure 1. External to the tower the pressure difference in the stagnant air between ground level and the top of the tower, assuming constant density, is

 

 

Based on the assumption of the standard atmosphere, Montak-hab [1] shows that this approximate equation deviates by less than 2% from the true pressure difference over a height of 213 m.

The air accelerates from stagnant ambient conditions at 1 to plane 4 after the heat exchanger bundles with a resultant change in pressure due to friction, form drag, inlet and outlet losses and flow acceleration. If H₃ « H₄ a total pressure balance yields:

where K_e3 is the static pressure loss coefficient across the heat exchanger and K_ct is the tower pressure loss coefficient, both based on the tower cross-sectional inlet area A₃.

From 4 to 5 the air flows isentropically and with approximately constant density such that

Add equations (2) and (3) and equate the sum to equation (1)

For d₃ = d₄ this equation may be written in dimensionless form by dividing throughout by 0,5 p_a, v_a3² and by satisfying the continuity relation

Equation (5) is known as the draught equation for the natural convection cooling tower.

In order to make more effective use of the available tower base area, heat exchanger bundles may be arranged in the form of V-arrays or deltas. The air stream leaving this heat exchanger has a highly turbulent distorted velocity distribution in which further losses in total pressure occur above the outlet of the heat exchanger. These and other losses through the inclined heat exchangers are included in the total pressure drop given by Mo-handes [2],

where the total pressure loss coefficient through the heat exchanger under normal flow conditions can in general be expressed as [3]

Robinson and Briggs [4] propose the following correlation for the Euler number for bundles of staggered radially finned tubes:

A static pressure drop coefficient for the V-array of bundles in the cooling tower, based on conditions at the inlet plane 3 is defined as

such that with equation (6) and the continuity relation

With K_e3 = K_e03 for the V-array layout, substitute equation (10) in equation (5) and find

The average flow incidence angle may be expressed by the following empirical relation:

where the apex semi-angle θ is given in degrees.

Furthermore K_{i0 = 90°} is the heat exchanger inlet contraction pressure loss coefficient under normal flow conditions (K_{i0 = 90°} = 0,05 for many industrial finned tube heat exchangers), while the downstream pressure loss coefficient can be expressed approximately by the following empirical relation

According to Hempel et al [5] the cooling tower pressure loss coefficient may be expressed as:

This equation includes a value of 0,01 (d₃/H₃)² which makes provision for the approximate resistance due to tower supports.

By introducing the gas law, equation (11) may be expressed approximately in terms of the ambient pressure, and temperatures before and after the heat exchanger as well as the air flow rate through the tower.

The ultimate relation between the air flow rate through the tower and the outlet air temperature at given ambient conditions will be determined by the heat transfer characteristics of the finned tube bundles. The heat transfer rate from the warm water to the air stream is

or

where

For a clean bi-metallic radially finned tube as shown in figure 2,

According to Briggs and Young [6] the air-side heat transfer coefficient h_a through bundles of staggered radially finned tubes is given by

The effectiveness of the finned surface is expressed in terms of the fin efficiency, i.e.

According to Schmidt [7] the fin efficiency for radial fins can be determined approximately from

The water-side heat transfer coefficient is according to Gnie-linski [8]

where

and for a smooth tube [9],

The wetted inside tube surface area is

If the water mass flow rate per unit cross-sectional inside tube area is given by

the pressure drop per unit length of finned tube is

The total water pumping power is thus

The logarithmic mean temperature difference is defined as

According to Roetzel [10], the LMTD correction for crossflow conditions can be expressed as

where

The values of a_ik are individual to each heat exchanger configuration.

In the particular tower to be analysed the finned tube bundles are to be arranged radially in the form of V-arrays as shown in figure 3 in the base of the tower.

 

 

The final "optimised" retrofit finned tube geometry and layout within certain prescribed constraints is a function of material and other costs. For purposes of illustration it will be assumed that the total effective finned tube cost can be expressed as

where C_wf is a weighting factor

The core tube cost per unit length is

where C_tm is the tube material cost per unit mass.

Similarly the cost of the fin material per unit tube length is

where C_fm is the fin material cost per unit mass. The fixed cost of the tube per unit length C_fix, covers other costs incurred by the manufacturer. The cost of the manifolds may also be included in C_fix and C_wf.

The object of the final analysis will thus be to achieve the desired cooling at a minimum total cost, taking into consideration certain prescribed constraints. The results of such an analysis are best illustrated by means of a numerical example.

 

Numerical example

The heat exchanger bundles in an existing natural draught hyperbolic cooling tower are to be replaced at minimum cost. The following tower dimensions are specified:

Tower height H₅ = 120 m

Inlet height H₃ = 13,67 m

Inlet diameter d₃ = 82,958 m

Outlet (throat) diameter d₅ = 58 m.

According to the original design, 142 heat exchanger bundles are arranged radially in the form of V-arrays in the base of the tower. The particulars of the bundles are as follows:

Effective length (finned tube) of bundle L_t = 15 m Effective width of bundle W_b = 2,262 m

Bundle semi-apex angle θ = 30,75°

Number of water passes n_wp = 2

Number of tube rows n_r = 4

To minimise cost it is recommended that the original bundle base support be employed and that no changes be made to the water pipe layout to and from the bundles or the water pump. The bundle support structure imposes the restriction that L_b = W_b sin θ = 1,157 m.

The existing pump provides a water flow rate of 4 390 kg/s at a pressure drop of 16302 N/m² through the finned tubes and deviations from these values are not permitted. In effect this also means that the pumping power P_w remains unchanged.

Due to structural considerations the tube wall and the fin root thickness are assumed constant at 1,655 mm and 1,08 mm respectively. The contact resistance R_c is assumed to be negligible.

The cooling tower is required to remove 365,08 MW at an inlet water temperature of 61,45°C, an ambient air pressure at ground level of 84 600 N/m² and a corresponding air temperature of 15,6°C.

In order to achieve an optimum design, the independent parameters t_f, d_f, 0, d_i, P_t, P_l, P_f and n_wp may be varied subject to the constraints specified. A fin pitch of 2,35 mm is arbitrarily chosen as the design reference value. Where fouling is of significance very small fin pitches are avoided. The values of C_wf, C_tm, C_fm and C_fix are taken as 2, 0,8$/kg, 4,2$/kg and 2$/m respectively.

The cost minimisation problem may be solved numerically with a general purpose algorithm for non-linear constrained optimisation. A Generalised Reduced Gradient algorithm [11] and a Constrained Variable Metric algorithm [12], were investigated and it was found that the Variable Metric algorithm was very efficient when applied to this problem. All results presented in this paper were obtained using the FORTRAN routine VMCWD [13]. Derivatives of the cost function and constraints were calculated numerically. Variables and constraints were carefully scaled, since they differ greatly in magnitude. All calculations were performed in double precision (approximately 16 decimal digits) in order to obtain reasonably accurate values for the derivatives. On average, routine VMCWD called the routines which calculate the cost function and constraints and their gradients, 6 to 10 times for each complete optimisation.

The results of the cost minimisation are presented in table 1. The values of the independent parameters corresponding to the minimum cost are taken as reference values (first line of table 1).

In order to investigate the dependence of the minimum cost on each parameter, one parameter is then perturbed and a minimisation performed with respect to the other parameters. The influence of various parameters on the total cost are also shown graphically in figure 4.

 

 

Conclusions

According to the results shown in table 1 the optimum fin thickness is only 0,0903 mm while the fin diameter is 61,95 mm.

In practice somewhat thicker fins are preferred resulting in a corresponding increase in system cost as shown in table 1.

According to the analysis the finned tubes are closely packed at optimum conditions if a triangular pattern is assumed. For larger fin thicknesses a more open arrangement is preferable. The influence of the longitudinal pitch is shown for both a triangular tube layout and for the case where P_e is not tied to the transversal pitch. As P_l increases the system cost is reduced. This reduction is however only applicable within certain limits in which the finned tube correlations apply.

Where fouling is not serious, a reduction in fin pitch will generally result in a lower cost system.

The influence of variations in the cost structure are also shown in table 1.

The possibility of installing a four-pass heat exchanger was also investigated. The resultant optimum design is only fractionally less costly than the two-pass system. For the thicker fin the converse is true. Furthermore it is noted that the inside diameter of the tubes is considerably larger to ensure a pressure drop on the water-side that does not exceed that specified for the pump.

The authors have extended the above optimisation method to include the designs of complete cooling systems for new plants.

 

Nomenclature

A area m²

C cost factor

c_p specific heat J/kg °C

d diameter m

E Euler number

F temperature correction factor

f friction factor

G mass velocity kg/s m²

g gravitational acceleration m/s²

H height m

K loss coefficient

k thermal conductivity W/m °C

L length m

m mass flow rate kg/s

n number

Ρ pitch m, or power W

ρ pressure N/m²

Δ ρ pressure differential N/m²

Pr Prandtl number

Q heat transfer rate W

R_c thermal contact resistance m² K/W

Re Reynolds number

T temperature °C

ΔT_lm logarithmic mean temperature difference °C

U overall heat transfer coefficient W/m² °C

W width m

ν velocity m/s

Greek letters

ε effectiveness

η efficiency

ρ density kg/m³

μ dynamic viscosity kg/ms

σ contraction ratio

θ angle °

Subscripts

a air

av average

b bundles

ct cooling tower

e heat exchanger or effective

f fin

fr frontal

i inlet or inside

l longitudinal

ο outlet or outside

ρ passes

r tube rows or root

t total, transversal or tube

w water

 

References

1. Montakhab, A., Waste Heat Disposal to Air with Mechanical and Natural Draft Some Analytical Design Considerations, ASME Paper No. 78-WA/HT-17, 1978 and Jnl. of Eng. for Power, Vol. 102, 1980.         [ Links ]

2. Mohandes, M. A., The flow through heat exchanger banks, Ph.D thesis, University of Oxford, 1979.         [ Links ]

3. Kroger, D. G., Performance characteristics of industrial finned tubes presented in dimensional form, Int. Jnl. of Heat and Mass Transfer, Vol. 29, No. 8, pp. 1119-1125, 1986.         [ Links ]

4. Robinson, K. K. and E. D. Briggs, Pressure drop of air flowing across triangular pitch banks of finned tubes, Chem. Eng. Prog. Symp. Ser., Vol. 62, No. 64, 1964.         [ Links ]

5. Hempel, D. C, Stephan, J. and G. Hesse, Strömungswiderstand, Geschwindigkeits-verteilung und Optimale Höhe für den Lufteintritt in Naturzug-Kühltürmen, Brennst.-Wärme-Kraft, No. 11, Vol. 35, 1983.

6. Briggs, D. E. and E. H. Young, Convection heat transfer and pressure drop of air flowing across triangular pitch banks of finned tubes, Chem. Eng. Prog. Symp. Ser., Vol. 59, No. 41, 1963.         [ Links ]

7. Schmidt, T. E., La Production Calorifique des Surfaces Munies Dailettes, Annexe Du Bulletin De L'Institut International Du Froid, Annex G-5,1945-1946.

8. Gnielinski, V., Forsch, Ing. Wes., Vol. 41, No. 1, 1957.

9. Filonenko, G. K., Teploenergetika No. 4, 1954.

10. Roetzel, W., Berechnung der Waermeuebertragern, VDI-Waermeatlas, VDI, Duesseldorf, 1984.

11. Lasdon, L. S., Waren, A. D., Jain, A. and M. Ratner, Design and Testing of a Generalized Reduced Gradient Code for Nonlinear Programming, ACM Transactions on Mathematical Software 4, pp 34-50, 1978.

12. Powell, M. J. D., A Fast Algorithm for Nonlinearly Constrained Optimization Calculations, In: Numerical Analysis: Proceedings Dundee 1977, Springer Lecture Notes in Mathematics 630, G. A. Watson (ed), pp. 144-157.

13. Powell, M. J. D., VMCWD: A Fortran Subroutine for Constrained Optimization, SIGMAP Bulletin, No. 32, 1983.